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Front Matter

Preface

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PREFACE

BACKGROUND AND PURPOSE

As in the previous three editions, the primary objective of the fourth edition of Basic Econometrics is to provide an elementary but comprehensive introduction to econometrics without resorting to matrix algebra, calculus, or statistics beyond the elementary level. In this edition I have attempted to incorporate some of the developments in the theory and practice of econometrics that have taken place since the publication of the third edition in 1995. With the availability of sophisticated and user-friendly statistical packages, such as Eviews, Limdep, Microﬁt, Minitab, PcGive, SAS, Shazam, and Stata, it is now possible to discuss several econometric techniques that could not be included in the previous editions of the book. I have taken full advantage of these statistical packages in illustrating several examples and exercises in this edition. I was pleasantly surprised to ﬁnd that my book is used not only by economics and business students but also by students and researchers in several other disciplines, such as politics, international relations, agriculture, and health sciences. Students in these disciplines will ﬁnd the expanded discussion of several topics very useful.

THE FOURTH EDITION

The major changes in this edition are as follows: 1. In the introductory chapter, after discussing the steps involved in traditional econometric methodology, I discuss the very important question of how one chooses among competing econometric models. 2. In Chapter 1, I discuss very brieﬂy the measurement scale of economic variables. It is important to know whether the variables are ratio xxv Gujarati: Basic Econometrics, Fourth Edition

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PREFACE

scale, interval scale, ordinal scale, or nominal scale, for that will determine the econometric technique that is appropriate in a given situation. 3. The appendices to Chapter 3 now include the large-sample properties of OLS estimators, particularly the property of consistency. 4. The appendix to Chapter 5 now brings into one place the properties and interrelationships among the four important probability distributions that are heavily used in this book, namely, the normal, t, chi square, and F. 5. Chapter 6, on functional forms of regression models, now includes a discussion of regression on standardized variables. 6. To make the book more accessible to the nonspecialist, I have moved the discussion of the matrix approach to linear regression from old Chapter 9 to Appendix C. Appendix C is slightly expanded to include some advanced material for the beneﬁt of the more mathematically inclined students. The new Chapter 9 now discusses dummy variable regression models. 7. Chapter 10, on multicollinearity, includes an extended discussion of the famous Longley data, which shed considerable light on the nature and scope of multicollinearity. 8. Chapter 11, on heteroscedasticity, now includes in the appendix an intuitive discussion of White’s robust standard errors. 9. Chapter 12, on autocorrelation, now includes a discussion of the Newey–West method of correcting the OLS standard errors to take into account likely autocorrelation in the error term. The corrected standard errors are known as HAC standard errors. This chapter also discusses brieﬂy the topic of forecasting with autocorrelated error terms. 10. Chapter 13, on econometric modeling, replaces old Chapters 13 and 14. This chapter has several new topics that the applied researcher will ﬁnd particularly useful. They include a compact discussion of model selection criteria, such as the Akaike information criterion, the Schwarz information criterion, Mallows’s Cp criterion, and forecast chi square. The chapter also discusses topics such as outliers, leverage, inﬂuence, recursive least squares, and Chow’s prediction failure test. This chapter concludes with some cautionary advice to the practitioner about econometric theory and econometric practice. 11. Chapter 14, on nonlinear regression models, is new. Because of the easy availability of statistical software, it is no longer difﬁcult to estimate regression models that are nonlinear in the parameters. Some econometric models are intrinsically nonlinear in the parameters and need to be estimated by iterative methods. This chapter discusses and illustrates some comparatively simple methods of estimating nonlinear-in-parameter regression models. 12. Chapter 15, on qualitative response regression models, which replaces old Chapter 16, on dummy dependent variable regression models, provides a fairly extensive discussion of regression models that involve a dependent variable that is qualitative in nature. The main focus is on logit

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and probit models and their variations. The chapter also discusses the Poisson regression model, which is used for modeling count data, such as the number of patents received by a ﬁrm in a year; the number of telephone calls received in a span of, say, 5 minutes; etc. This chapter has a brief discussion of multinomial logit and probit models and duration models. 13. Chapter 16, on panel data regression models, is new. A panel data combines features of both time series and cross-section data. Because of increasing availability of panel data in the social sciences, panel data regression models are being increasingly used by researchers in many ﬁelds. This chapter provides a nontechnical discussion of the ﬁxed effects and random effects models that are commonly used in estimating regression models based on panel data. 14. Chapter 17, on dynamic econometric models, has now a rather extended discussion of the Granger causality test, which is routinely used (and misused) in applied research. The Granger causality test is sensitive to the number of lagged terms used in the model. It also assumes that the underlying time series is stationary. 15. Except for new problems and minor extensions of the existing estimation techniques, Chapters 18, 19, and 20 on simultaneous equation models are basically unchanged. This reﬂects the fact that interest in such models has dwindled over the years for a variety of reasons, including their poor forecasting performance after the OPEC oil shocks of the 1970s. 16. Chapter 21 is a substantial revision of old Chapter 21. Several concepts of time series econometrics are developed and illustrated in this chapter. The main thrust of the chapter is on the nature and importance of stationary time series. The chapter discusses several methods of ﬁnding out if a given time series is stationary. Stationarity of a time series is crucial for the application of various econometric techniques discussed in this book. 17. Chapter 22 is also a substantial revision of old Chapter 22. It discusses the topic of economic forecasting based on the Box–Jenkins (ARIMA) and vector autoregression (VAR) methodologies. It also discusses the topic of measuring volatility in ﬁnancial time series by the techniques of autoregressive conditional heteroscedasticity (ARCH) and generalized autoregressive conditional heteroscedasticity (GARCH). 18. Appendix A, on statistical concepts, has been slightly expanded. Appendix C discusses the linear regression model using matrix algebra. This is for the beneﬁt of the more advanced students. As in the previous editions, all the econometric techniques discussed in this book are illustrated by examples, several of which are based on concrete data from various disciplines. The end-of-chapter questions and problems have several new examples and data sets. For the advanced reader, there are several technical appendices to the various chapters that give proofs of the various theorems and or formulas developed in the text.

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PREFACE

ORGANIZATION AND OPTIONS

Changes in this edition have considerably expanded the scope of the text. I hope this gives the instructor substantial ﬂexibility in choosing topics that are appropriate to the intended audience. Here are suggestions about how this book may be used. One-semester course for the nonspecialist: Appendix A, Chapters 1 through 9, an overview of Chapters 10, 11, 12 (omitting all the proofs). One-semester course for economics majors: Appendix A, Chapters 1 through 13. Two-semester course for economics majors: Appendices A, B, C, Chapters 1 to 22. Chapters 14 and 16 may be covered on an optional basis. Some of the technical appendices may be omitted. Graduate and postgraduate students and researchers: This book is a handy reference book on the major themes in econometrics.

SUPPLEMENTS Data CD

Every text is packaged with a CD that contains the data from the text in ASCII or text format and can be read by most software packages.

Student Solutions Manual

Free to instructors and salable to students is a Student Solutions Manual (ISBN 0072427922) that contains detailed solutions to the 475 questions and problems in the text.

EViews

With this fourth edition we are pleased to provide sion 3.1 on a CD along with all of the data from the available from the publisher packaged with the text Eviews Student Version is available separately http://www.eviews.com for further information.

Web Site

Eviews Student Vertext. This software is (ISBN: 0072565705). from QMS. Go to

A comprehensive web site provides additional material to support the study of econometrics. Go to www.mhhe.com/econometrics/gujarati4.

ACKNOWLEDGMENTS

Since the publication of the ﬁrst edition of this book in 1978, I have received valuable advice, comments, criticism, and suggestions from a variety of people. In particular, I would like to acknowledge the help I have received

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from Michael McAleer of the University of Western Australia, Peter Kennedy of Simon Frazer University in Canada, and Kenneth White, of the University of British Columbia, George K. Zestos of Christopher Newport University, Virginia, and Paul Offner, Georgetown University, Washington, D.C. I am also grateful to several people who have inﬂuenced me by their scholarship. I especially want to thank Arthur Goldberger of the University of Wisconsin, William Greene of New York University, and the late G. S. Maddala. For this fourth edition I am especially grateful to these reviewers who provided their invaluable insight, criticism, and suggestions: Michael A. Grove at the University of Oregon, Harumi Ito at Brown University, Han Kim at South Dakota University, Phanindra V. Wunnava at Middlebury College, and George K. Zestos of Christopher Newport University. Several authors have inﬂuenced my writing. In particular, I am grateful to these authors: Chandan Mukherjee, director of the Centre for Development Studies, Trivandrum, India; Howard White and Marc Wuyts, both at the Institute of Social Studies in the Netherlands; Badi H. Baltagi, Texas A&M University; B. Bhaskara Rao, University of New South Wales, Australia; R. Carter Hill, Louisiana University; William E. Grifﬁths, University of New England; George G. Judge, University of California at Berkeley; Marno Verbeek, Center for Economic Studies, KU Leuven; Jeffrey Wooldridge, Michigan State University; Kerry Patterson, University of Reading, U.K.; Francis X. Diebold, Wharton School, University of Pennsylvania; Wojciech W. Charemza and Derek F. Deadman, both of the University of Leicester, U.K.; Gary Koop, University of Glasgow. I am very grateful to several of my colleagues at West Point for their support and encouragement over the years. In particular, I am grateful to Brigadier General Daniel Kaufman, Colonel Howard Russ, Lieutenant Colonel Mike Meese, Lieutenant Colonel Casey Wardynski, Major David Trybulla, Major Kevin Foster, Dean Dudley, and Dennis Smallwood. I would like to thank students and teachers all over the world who have not only used my book but have communicated with me about various aspects of the book. For their behind the scenes help at McGraw-Hill, I am grateful to Lucille Sutton, Aric Bright, and Catherine R. Schultz. George F. Watson, the copyeditor, has done a marvellous job in editing a rather lengthy and demanding manuscript. For that, I am much obliged to him. Finally, but not least important, I would like to thank my wife, Pushpa, and my daughters, Joan and Diane, for their constant support and encouragement in the preparation of this and the previous editions. Damodar N. Gujarati

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INTRODUCTION

I.1

WHAT IS ECONOMETRICS?

Literally interpreted, econometrics means “economic measurement.” Although measurement is an important part of econometrics, the scope of econometrics is much broader, as can be seen from the following quotations:

Econometrics, the result of a certain outlook on the role of economics, consists of the application of mathematical statistics to economic data to lend empirical support to the models constructed by mathematical economics and to obtain numerical results.1 . . . econometrics may be deﬁned as the quantitative analysis of actual economic phenomena based on the concurrent development of theory and observation, related by appropriate methods of inference.2 Econometrics may be deﬁned as the social science in which the tools of economic theory, mathematics, and statistical inference are applied to the analysis of economic phenomena.3 Econometrics is concerned with the empirical determination of economic laws.4

1 Gerhard Tintner, Methodology of Mathematical Economics and Econometrics, The University of Chicago Press, Chicago, 1968, p. 74. 2 P. A. Samuelson, T. C. Koopmans, and J. R. N. Stone, “Report of the Evaluative Committee for Econometrica,” Econometrica, vol. 22, no. 2, April 1954, pp. 141–146. 3 Arthur S. Goldberger, Econometric Theory, John Wiley & Sons, New York, 1964, p. 1. 4 H. Theil, Principles of Econometrics, John Wiley & Sons, New York, 1971, p. 1.

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The art of the econometrician consists in ﬁnding the set of assumptions that are both sufﬁciently speciﬁc and sufﬁciently realistic to allow him to take the best possible advantage of the data available to him.5 Econometricians . . . are a positive help in trying to dispel the poor public image of economics (quantitative or otherwise) as a subject in which empty boxes are opened by assuming the existence of can-openers to reveal contents which any ten economists will interpret in 11 ways.6 The method of econometric research aims, essentially, at a conjunction of economic theory and actual measurements, using the theory and technique of statistical inference as a bridge pier.7

I.2

WHY A SEPARATE DISCIPLINE?

As the preceding deﬁnitions suggest, econometrics is an amalgam of economic theory, mathematical economics, economic statistics, and mathematical statistics. Yet the subject deserves to be studied in its own right for the following reasons. Economic theory makes statements or hypotheses that are mostly qualitative in nature. For example, microeconomic theory states that, other things remaining the same, a reduction in the price of a commodity is expected to increase the quantity demanded of that commodity. Thus, economic theory postulates a negative or inverse relationship between the price and quantity demanded of a commodity. But the theory itself does not provide any numerical measure of the relationship between the two; that is, it does not tell by how much the quantity will go up or down as a result of a certain change in the price of the commodity. It is the job of the econometrician to provide such numerical estimates. Stated differently, econometrics gives empirical content to most economic theory. The main concern of mathematical economics is to express economic theory in mathematical form (equations) without regard to measurability or empirical veriﬁcation of the theory. Econometrics, as noted previously, is mainly interested in the empirical veriﬁcation of economic theory. As we shall see, the econometrician often uses the mathematical equations proposed by the mathematical economist but puts these equations in such a form that they lend themselves to empirical testing. And this conversion of mathematical into econometric equations requires a great deal of ingenuity and practical skill. Economic statistics is mainly concerned with collecting, processing, and presenting economic data in the form of charts and tables. These are the

E. Malinvaud, Statistical Methods of Econometrics, Rand McNally, Chicago, 1966, p. 514. Adrian C. Darnell and J. Lynne Evans, The Limits of Econometrics, Edward Elgar Publishing, Hants, England, 1990, p. 54. 7 T. Haavelmo, “The Probability Approach in Econometrics,” Supplement to Econometrica, vol. 12, 1944, preface p. iii.

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jobs of the economic statistician. It is he or she who is primarily responsible for collecting data on gross national product (GNP), employment, unemployment, prices, etc. The data thus collected constitute the raw data for econometric work. But the economic statistician does not go any further, not being concerned with using the collected data to test economic theories. Of course, one who does that becomes an econometrician. Although mathematical statistics provides many tools used in the trade, the econometrician often needs special methods in view of the unique nature of most economic data, namely, that the data are not generated as the result of a controlled experiment. The econometrician, like the meteorologist, generally depends on data that cannot be controlled directly. As Spanos correctly observes:

In econometrics the modeler is often faced with observational as opposed to experimental data. This has two important implications for empirical modeling in econometrics. First, the modeler is required to master very different skills than those needed for analyzing experimental data. . . . Second, the separation of the data collector and the data analyst requires the modeler to familiarize himself/herself thoroughly with the nature and structure of data in question.8

I.3

METHODOLOGY OF ECONOMETRICS

How do econometricians proceed in their analysis of an economic problem? That is, what is their methodology? Although there are several schools of thought on econometric methodology, we present here the traditional or classical methodology, which still dominates empirical research in economics and other social and behavioral sciences.9 Broadly speaking, traditional econometric methodology proceeds along the following lines: 1. 2. 3. 4. 5. 6. 7. 8. Statement of theory or hypothesis. Speciﬁcation of the mathematical model of the theory Speciﬁcation of the statistical, or econometric, model Obtaining the data Estimation of the parameters of the econometric model Hypothesis testing Forecasting or prediction Using the model for control or policy purposes.

To illustrate the preceding steps, let us consider the well-known Keynesian theory of consumption.

8 Aris Spanos, Probability Theory and Statistical Inference: Econometric Modeling with Observational Data, Cambridge University Press, United Kingdom, 1999, p. 21. 9 For an enlightening, if advanced, discussion on econometric methodology, see David F. Hendry, Dynamic Econometrics, Oxford University Press, New York, 1995. See also Aris Spanos, op. cit.

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1. Statement of Theory or Hypothesis

Keynes stated:

The fundamental psychological law . . . is that men [women] are disposed, as a rule and on average, to increase their consumption as their income increases, but not as much as the increase in their income.10

In short, Keynes postulated that the marginal propensity to consume (MPC), the rate of change of consumption for a unit (say, a dollar) change in income, is greater than zero but less than 1.

2. Speciﬁcation of the Mathematical Model of Consumption

Although Keynes postulated a positive relationship between consumption and income, he did not specify the precise form of the functional relationship between the two. For simplicity, a mathematical economist might suggest the following form of the Keynesian consumption function: Y = β1 + β2 X 0 < β2 < 1 (I.3.1)

where Y = consumption expenditure and X = income, and where β1 and β2 , known as the parameters of the model, are, respectively, the intercept and slope coefﬁcients. The slope coefﬁcient β2 measures the MPC. Geometrically, Eq. (I.3.1) is as shown in Figure I.1. This equation, which states that consumption is linY

Consumption expenditure

β2 = MPC 1

β1

Income FIGURE I.1 Keynesian consumption function.

X

10 John Maynard Keynes, The General Theory of Employment, Interest and Money, Harcourt Brace Jovanovich, New York, 1936, p. 96.

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early related to income, is an example of a mathematical model of the relationship between consumption and income that is called the consumption function in economics. A model is simply a set of mathematical equations. If the model has only one equation, as in the preceding example, it is called a single-equation model, whereas if it has more than one equation, it is known as a multiple-equation model (the latter will be considered later in the book). In Eq. (I.3.1) the variable appearing on the left side of the equality sign is called the dependent variable and the variable(s) on the right side are called the independent, or explanatory, variable(s). Thus, in the Keynesian consumption function, Eq. (I.3.1), consumption (expenditure) is the dependent variable and income is the explanatory variable.

3. Speciﬁcation of the Econometric Model of Consumption

The purely mathematical model of the consumption function given in Eq. (I.3.1) is of limited interest to the econometrician, for it assumes that there is an exact or deterministic relationship between consumption and income. But relationships between economic variables are generally inexact. Thus, if we were to obtain data on consumption expenditure and disposable (i.e., aftertax) income of a sample of, say, 500 American families and plot these data on a graph paper with consumption expenditure on the vertical axis and disposable income on the horizontal axis, we would not expect all 500 observations to lie exactly on the straight line of Eq. (I.3.1) because, in addition to income, other variables affect consumption expenditure. For example, size of family, ages of the members in the family, family religion, etc., are likely to exert some inﬂuence on consumption. To allow for the inexact relationships between economic variables, the econometrician would modify the deterministic consumption function (I.3.1) as follows: Y = β1 + β2 X + u (I.3.2)

where u, known as the disturbance, or error, term, is a random (stochastic) variable that has well-deﬁned probabilistic properties. The disturbance term u may well represent all those factors that affect consumption but are not taken into account explicitly. Equation (I.3.2) is an example of an econometric model. More technically, it is an example of a linear regression model, which is the major concern of this book. The econometric consumption function hypothesizes that the dependent variable Y (consumption) is linearly related to the explanatory variable X (income) but that the relationship between the two is not exact; it is subject to individual variation. The econometric model of the consumption function can be depicted as shown in Figure I.2.

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Y

Consumption expenditure

u

Income FIGURE I.2 Econometric model of the Keynesian consumption function.

X

4. Obtaining Data

To estimate the econometric model given in (I.3.2), that is, to obtain the numerical values of β1 and β2 , we need data. Although we will have more to say about the crucial importance of data for economic analysis in the next chapter, for now let us look at the data given in Table I.1, which relate to

TABLE I.1 DATA ON Y (PERSONAL CONSUMPTION EXPENDITURE) AND X (GROSS DOMESTIC PRODUCT, 1982–1996), BOTH IN 1992 BILLIONS OF DOLLARS Year 1982 1983 1984 1985 1986 1987 1988 1989 1990 1991 1992 1993 1994 1995 1996 Y 3081.5 3240.6 3407.6 3566.5 3708.7 3822.3 3972.7 4064.6 4132.2 4105.8 4219.8 4343.6 4486.0 4595.3 4714.1 X 4620.3 4803.7 5140.1 5323.5 5487.7 5649.5 5865.2 6062.0 6136.3 6079.4 6244.4 6389.6 6610.7 6742.1 6928.4

Source: Economic Report of the President, 1998, Table B–2, p. 282.

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5000

4500

PCE (Y)

4000

3500

3000 4000

5000 GDP (X)

6000

7000

FIGURE I.3

Personal consumption expenditure (Y ) in relation to GDP (X ), 1982–1996, both in billions of 1992 dollars.

the U.S. economy for the period 1981–1996. The Y variable in this table is the aggregate (for the economy as a whole) personal consumption expenditure (PCE) and the X variable is gross domestic product (GDP), a measure of aggregate income, both measured in billions of 1992 dollars. Therefore, the data are in “real” terms; that is, they are measured in constant (1992) prices. The data are plotted in Figure I.3 (cf. Figure I.2). For the time being neglect the line drawn in the ﬁgure.

5. Estimation of the Econometric Model

Now that we have the data, our next task is to estimate the parameters of the consumption function. The numerical estimates of the parameters give empirical content to the consumption function. The actual mechanics of estimating the parameters will be discussed in Chapter 3. For now, note that the statistical technique of regression analysis is the main tool used to obtain the estimates. Using this technique and the data given in Table I.1, we obtain the following estimates of β1 and β2 , namely, −184.08 and 0.7064. Thus, the estimated consumption function is: ˆ Y = −184.08 + 0.7064Xi (I.3.3)

The hat on the Y indicates that it is an estimate.11 The estimated consumption function (i.e., regression line) is shown in Figure I.3.

11 As a matter of convention, a hat over a variable or parameter indicates that it is an estimated value.

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As Figure I.3 shows, the regression line ﬁts the data quite well in that the data points are very close to the regression line. From this ﬁgure we see that for the period 1982–1996 the slope coefﬁcient (i.e., the MPC) was about 0.70, suggesting that for the sample period an increase in real income of 1 dollar led, on average, to an increase of about 70 cents in real consumption expenditure.12 We say on average because the relationship between consumption and income is inexact; as is clear from Figure I.3; not all the data points lie exactly on the regression line. In simple terms we can say that, according to our data, the average, or mean, consumption expenditure went up by about 70 cents for a dollar’s increase in real income.

6. Hypothesis Testing

Assuming that the ﬁtted model is a reasonably good approximation of reality, we have to develop suitable criteria to ﬁnd out whether the estimates obtained in, say, Eq. (I.3.3) are in accord with the expectations of the theory that is being tested. According to “positive” economists like Milton Friedman, a theory or hypothesis that is not veriﬁable by appeal to empirical evidence may not be admissible as a part of scientiﬁc enquiry.13 As noted earlier, Keynes expected the MPC to be positive but less than 1. In our example we found the MPC to be about 0.70. But before we accept this ﬁnding as conﬁrmation of Keynesian consumption theory, we must enquire whether this estimate is sufﬁciently below unity to convince us that this is not a chance occurrence or peculiarity of the particular data we have used. In other words, is 0.70 statistically less than 1? If it is, it may support Keynes’ theory. Such conﬁrmation or refutation of economic theories on the basis of sample evidence is based on a branch of statistical theory known as statistical inference (hypothesis testing). Throughout this book we shall see how this inference process is actually conducted.

7. Forecasting or Prediction

If the chosen model does not refute the hypothesis or theory under consideration, we may use it to predict the future value(s) of the dependent, or forecast, variable Y on the basis of known or expected future value(s) of the explanatory, or predictor, variable X. To illustrate, suppose we want to predict the mean consumption expenditure for 1997. The GDP value for 1997 was 7269.8 billion dollars.14 Putting

12 Do not worry now about how these values were obtained. As we show in Chap. 3, the statistical method of least squares has produced these estimates. Also, for now do not worry about the negative value of the intercept. 13 See Milton Friedman, “The Methodology of Positive Economics,” Essays in Positive Economics, University of Chicago Press, Chicago, 1953. 14 Data on PCE and GDP were available for 1997 but we purposely left them out to illustrate the topic discussed in this section. As we will discuss in subsequent chapters, it is a good idea to save a portion of the data to ﬁnd out how well the ﬁtted model predicts the out-of-sample observations.

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this GDP ﬁgure on the right-hand side of (I.3.3), we obtain: ˆ Y1997 = −184.0779 + 0.7064 (7269.8) = 4951.3167 (I.3.4)

or about 4951 billion dollars. Thus, given the value of the GDP, the mean, or average, forecast consumption expenditure is about 4951 billion dollars. The actual value of the consumption expenditure reported in 1997 was 4913.5 billion dollars. The estimated model (I.3.3) thus overpredicted the actual consumption expenditure by about 37.82 billion dollars. We could say the forecast error is about 37.82 billion dollars, which is about 0.76 percent of the actual GDP value for 1997. When we fully discuss the linear regression model in subsequent chapters, we will try to ﬁnd out if such an error is “small” or “large.” But what is important for now is to note that such forecast errors are inevitable given the statistical nature of our analysis. There is another use of the estimated model (I.3.3). Suppose the President decides to propose a reduction in the income tax. What will be the effect of such a policy on income and thereby on consumption expenditure and ultimately on employment? Suppose that, as a result of the proposed policy change, investment expenditure increases. What will be the effect on the economy? As macroeconomic theory shows, the change in income following, say, a dollar’s worth of change in investment expenditure is given by the income multiplier M, which is deﬁned as M= 1 1 − MPC (I.3.5)

If we use the MPC of 0.70 obtained in (I.3.3), this multiplier becomes about M = 3.33. That is, an increase (decrease) of a dollar in investment will eventually lead to more than a threefold increase (decrease) in income; note that it takes time for the multiplier to work. The critical value in this computation is MPC, for the multiplier depends on it. And this estimate of the MPC can be obtained from regression models such as (I.3.3). Thus, a quantitative estimate of MPC provides valuable information for policy purposes. Knowing MPC, one can predict the future course of income, consumption expenditure, and employment following a change in the government’s ﬁscal policies.

8. Use of the Model for Control or Policy Purposes

Suppose we have the estimated consumption function given in (I.3.3). Suppose further the government believes that consumer expenditure of about 4900 (billions of 1992 dollars) will keep the unemployment rate at its

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Economic theory

Mathematical model of theory Econometric model of theory

Data Estimation of econometric model

Hypothesis testing

Forecasting or prediction

FIGURE I.4

Anatomy of econometric modeling.

Using the model for control or policy purposes

current level of about 4.2 percent (early 2000). What level of income will guarantee the target amount of consumption expenditure? If the regression results given in (I.3.3) seem reasonable, simple arithmetic will show that 4900 = −184.0779 + 0.7064X (I.3.6)

which gives X = 7197, approximately. That is, an income level of about 7197 (billion) dollars, given an MPC of about 0.70, will produce an expenditure of about 4900 billion dollars. As these calculations suggest, an estimated model may be used for control, or policy, purposes. By appropriate ﬁscal and monetary policy mix, the government can manipulate the control variable X to produce the desired level of the target variable Y. Figure I.4 summarizes the anatomy of classical econometric modeling.

Choosing among Competing Models

When a governmental agency (e.g., the U.S. Department of Commerce) collects economic data, such as that shown in Table I.1, it does not necessarily have any economic theory in mind. How then does one know that the data really support the Keynesian theory of consumption? Is it because the Keynesian consumption function (i.e., the regression line) shown in Figure I.3 is extremely close to the actual data points? Is it possible that an-

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other consumption model (theory) might equally ﬁt the data as well? For example, Milton Friedman has developed a model of consumption, called the permanent income hypothesis.15 Robert Hall has also developed a model of consumption, called the life-cycle permanent income hypothesis.16 Could one or both of these models also ﬁt the data in Table I.1? In short, the question facing a researcher in practice is how to choose among competing hypotheses or models of a given phenomenon, such as the consumption–income relationship. As Miller contends:

No encounter with data is step towards genuine conﬁrmation unless the hypothesis does a better job of coping with the data than some natural rival. . . . What strengthens a hypothesis, here, is a victory that is, at the same time, a defeat for a plausible rival.17

How then does one choose among competing models or hypotheses? Here the advice given by Clive Granger is worth keeping in mind:18

I would like to suggest that in the future, when you are presented with a new piece of theory or empirical model, you ask these questions: (i) What purpose does it have? What economic decisions does it help with? and; (ii) Is there any evidence being presented that allows me to evaluate its quality compared to alternative theories or models? I think attention to such questions will strengthen economic research and discussion.

As we progress through this book, we will come across several competing hypotheses trying to explain various economic phenomena. For example, students of economics are familiar with the concept of the production function, which is basically a relationship between output and inputs (say, capital and labor). In the literature, two of the best known are the Cobb–Douglas and the constant elasticity of substitution production functions. Given the data on output and inputs, we will have to ﬁnd out which of the two production functions, if any, ﬁts the data well. The eight-step classical econometric methodology discussed above is neutral in the sense that it can be used to test any of these rival hypotheses. Is it possible to develop a methodology that is comprehensive enough to include competing hypotheses? This is an involved and controversial topic.

Milton Friedman, A Theory of Consumption Function, Princeton University Press, Princeton, N.J., 1957. 16 R. Hall, “Stochastic Implications of the Life Cycle Permanent Income Hypothesis: Theory and Evidence,” Journal of Political Economy, 1978, vol. 86, pp. 971–987. 17 R. W. Miller, Fact and Method: Explanation, Conﬁrmation, and Reality in the Natural and Social Sciences, Princeton University Press, Princeton, N.J., 1978, p. 176. 18 Clive W. J. Granger, Empirical Modeling in Economics, Cambridge University Press, U.K., 1999, p. 58.

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Introduction

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12

BASIC ECONOMETRICS

Econometrics

Theoretical

Applied

Classical FIGURE I.5

Bayesian

Classical

Bayesian

Categories of econometrics.

We will discuss it in Chapter 13, after we have acquired the necessary econometric theory.

I.4 TYPES OF ECONOMETRICS

As the classiﬁcatory scheme in Figure I.5 suggests, econometrics may be divided into two broad categories: theoretical econometrics and applied econometrics. In each category, one can approach the subject in the classical or Bayesian tradition. In this book the emphasis is on the classical approach. For the Bayesian approach, the reader may consult the references given at the end of the chapter. Theoretical econometrics is concerned with the development of appropriate methods for measuring economic relationships speciﬁed by econometric models. In this aspect, econometrics leans heavily on mathematical statistics. For example, one of the methods used extensively in this book is least squares. Theoretical econometrics must spell out the assumptions of this method, its properties, and what happens to these properties when one or more of the assumptions of the method are not fulﬁlled. In applied econometrics we use the tools of theoretical econometrics to study some special ﬁeld(s) of economics and business, such as the production function, investment function, demand and supply functions, portfolio theory, etc. This book is concerned largely with the development of econometric methods, their assumptions, their uses, their limitations. These methods are illustrated with examples from various areas of economics and business. But this is not a book of applied econometrics in the sense that it delves deeply into any particular ﬁeld of economic application. That job is best left to books written speciﬁcally for this purpose. References to some of these books are provided at the end of this book.

I.5 MATHEMATICAL AND STATISTICAL PREREQUISITES

Although this book is written at an elementary level, the author assumes that the reader is familiar with the basic concepts of statistical estimation and hypothesis testing. However, a broad but nontechnical overview of the basic statistical concepts used in this book is provided in Appendix A for

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INTRODUCTION

13

the beneﬁt of those who want to refresh their knowledge. Insofar as mathematics is concerned, a nodding acquaintance with the notions of differential calculus is desirable, although not essential. Although most graduate level books in econometrics make heavy use of matrix algebra, I want to make it clear that it is not needed to study this book. It is my strong belief that the fundamental ideas of econometrics can be conveyed without the use of matrix algebra. However, for the beneﬁt of the mathematically inclined student, Appendix C gives the summary of basic regression theory in matrix notation. For these students, Appendix B provides a succinct summary of the main results from matrix algebra.

I.6

THE ROLE OF THE COMPUTER

Regression analysis, the bread-and-butter tool of econometrics, these days is unthinkable without the computer and some access to statistical software. (Believe me, I grew up in the generation of the slide rule!) Fortunately, several excellent regression packages are commercially available, both for the mainframe and the microcomputer, and the list is growing by the day. Regression software packages, such as ET, LIMDEP, SHAZAM, MICRO TSP, MINITAB, EVIEWS, SAS, SPSS, STATA, Microﬁt, PcGive, and BMD have most of the econometric techniques and tests discussed in this book. In this book, from time to time, the reader will be asked to conduct Monte Carlo experiments using one or more of the statistical packages. Monte Carlo experiments are “fun” exercises that will enable the reader to appreciate the properties of several statistical methods discussed in this book. The details of the Monte Carlo experiments will be discussed at appropriate places.

I.7

SUGGESTIONS FOR FURTHER READING

The topic of econometric methodology is vast and controversial. For those interested in this topic, I suggest the following books: Neil de Marchi and Christopher Gilbert, eds., History and Methodology of Econometrics, Oxford University Press, New York, 1989. This collection of readings discusses some early work on econometric methodology and has an extended discussion of the British approach to econometrics relating to time series data, that is, data collected over a period of time. Wojciech W. Charemza and Derek F. Deadman, New Directions in Econometric Practice: General to Speciﬁc Modelling, Cointegration and Vector Autogression, 2d ed., Edward Elgar Publishing Ltd., Hants, England, 1997. The authors of this book critique the traditional approach to econometrics and give a detailed exposition of new approaches to econometric methodology. Adrian C. Darnell and J. Lynne Evans, The Limits of Econometrics, Edward Elgar Publishers Ltd., Hants, England, 1990. The book provides a somewhat

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BASIC ECONOMETRICS

balanced discussion of the various methodological approaches to econometrics, with renewed allegiance to traditional econometric methodology. Mary S. Morgan, The History of Econometric Ideas, Cambridge University Press, New York, 1990. The author provides an excellent historical perspective on the theory and practice of econometrics, with an in-depth discussion of the early contributions of Haavelmo (1990 Nobel Laureate in Economics) to econometrics. In the same spirit, David F. Hendry and Mary S. Morgan, The Foundation of Econometric Analysis, Cambridge University Press, U.K., 1995, have collected seminal writings in econometrics to show the evolution of econometric ideas over time. David Colander and Reuven Brenner, eds., Educating Economists, University of Michigan Press, Ann Arbor, Michigan, 1992, present a critical, at times agnostic, view of economic teaching and practice. For Bayesian statistics and econometrics, the following books are very useful: John H. Dey, Data in Doubt, Basic Blackwell Ltd., Oxford University Press, England, 1985. Peter M. Lee, Bayesian Statistics: An Introduction, Oxford University Press, England, 1989. Dale J. Porier, Intermediate Statistics and Econometrics: A Comparative Approach, MIT Press, Cambridge, Massachusetts, 1995. Arnold Zeller, An Introduction to Bayesian Inference in Econometrics, John Wiley & Sons, New York, 1971, is an advanced reference book.

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PART

ONE

SINGLE-EQUATION REGRESSION MODELS

Part I of this text introduces single-equation regression models. In these models, one variable, called the dependent variable, is expressed as a linear function of one or more other variables, called the explanatory variables. In such models it is assumed implicitly that causal relationships, if any, between the dependent and explanatory variables ﬂow in one direction only, namely, from the explanatory variables to the dependent variable. In Chapter 1, we discuss the historical as well as the modern interpretation of the term regression and illustrate the difference between the two interpretations with several examples drawn from economics and other ﬁelds. In Chapter 2, we introduce some fundamental concepts of regression analysis with the aid of the two-variable linear regression model, a model in which the dependent variable is expressed as a linear function of only a single explanatory variable. In Chapter 3, we continue to deal with the two-variable model and introduce what is known as the classical linear regression model, a model that makes several simplifying assumptions. With these assumptions, we introduce the method of ordinary least squares (OLS) to estimate the parameters of the two-variable regression model. The method of OLS is simple to apply, yet it has some very desirable statistical properties. In Chapter 4, we introduce the (two-variable) classical normal linear regression model, a model that assumes that the random dependent variable follows the normal probability distribution. With this assumption, the OLS estimators obtained in Chapter 3 possess some stronger statistical properties than the nonnormal classical linear regression model—properties that enable us to engage in statistical inference, namely, hypothesis testing.

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Chapter 5 is devoted to the topic of hypothesis testing. In this chapter, we try to ﬁnd out whether the estimated regression coefﬁcients are compatible with the hypothesized values of such coefﬁcients, the hypothesized values being suggested by theory and/or prior empirical work. Chapter 6 considers some extensions of the two-variable regression model. In particular, it discusses topics such as (1) regression through the origin, (2) scaling and units of measurement, and (3) functional forms of regression models such as double-log, semilog, and reciprocal models. In Chapter 7, we consider the multiple regression model, a model in which there is more than one explanatory variable, and show how the method of OLS can be extended to estimate the parameters of such models. In Chapter 8, we extend the concepts introduced in Chapter 5 to the multiple regression model and point out some of the complications arising from the introduction of several explanatory variables. Chapter 9 on dummy, or qualitative, explanatory variables concludes Part I of the text. This chapter emphasizes that not all explanatory variables need to be quantitative (i.e., ratio scale). Variables, such as gender, race, religion, nationality, and region of residence, cannot be readily quantiﬁed, yet they play a valuable role in explaining many an economic phenomenon.

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THE NATURE OF REGRESSION ANALYSIS

As mentioned in the Introduction, regression is a main tool of econometrics, and in this chapter we consider very brieﬂy the nature of this tool.

1.1

HISTORICAL ORIGIN OF THE TERM REGRESSION

The term regression was introduced by Francis Galton. In a famous paper, Galton found that, although there was a tendency for tall parents to have tall children and for short parents to have short children, the average height of children born of parents of a given height tended to move or “regress” toward the average height in the population as a whole.1 In other words, the height of the children of unusually tall or unusually short parents tends to move toward the average height of the population. Galton’s law of universal regression was conﬁrmed by his friend Karl Pearson, who collected more than a thousand records of heights of members of family groups.2 He found that the average height of sons of a group of tall fathers was less than their fathers’ height and the average height of sons of a group of short fathers was greater than their fathers’ height, thus “regressing” tall and short sons alike toward the average height of all men. In the words of Galton, this was “regression to mediocrity.”

1 Francis Galton, “Family Likeness in Stature,” Proceedings of Royal Society, London, vol. 40, 1886, pp. 42–72. 2 K. Pearson and A. Lee, “On the Laws of Inheritance,’’ Biometrika, vol. 2, Nov. 1903, pp. 357–462.

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1.2

THE MODERN INTERPRETATION OF REGRESSION

The modern interpretation of regression is, however, quite different. Broadly speaking, we may say

Regression analysis is concerned with the study of the dependence of one variable, the dependent variable, on one or more other variables, the explanatory variables, with a view to estimating and/or predicting the (population) mean or average value of the former in terms of the known or ﬁxed (in repeated sampling) values of the latter.

The full import of this view of regression analysis will become clearer as we progress, but a few simple examples will make the basic concept quite clear.

Examples

1. Reconsider Galton’s law of universal regression. Galton was interested in ﬁnding out why there was a stability in the distribution of heights in a population. But in the modern view our concern is not with this explanation but rather with ﬁnding out how the average height of sons changes, given the fathers’ height. In other words, our concern is with predicting the average height of sons knowing the height of their fathers. To see how this can be done, consider Figure 1.1, which is a scatter diagram, or scatter-

75

× Mean value

70 Son's height, inches

× × × × × × × × × × × × ×

65

60

× × × × × × × × × × × × × × × × × ×

× × × × × × × × × × × × × × × × × × ×

× × × × × × × × × × × × × × × × × × ×

60

65 70 Father's height, inches

75

FIGURE 1.1

Hypothetical distribution of sons’ heights corresponding to given heights of fathers.

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gram. This ﬁgure shows the distribution of heights of sons in a hypothetical population corresponding to the given or ﬁxed values of the father’s height. Notice that corresponding to any given height of a father is a range or distribution of the heights of the sons. However, notice that despite the variability of the height of sons for a given value of father’s height, the average height of sons generally increases as the height of the father increases. To show this clearly, the circled crosses in the ﬁgure indicate the average height of sons corresponding to a given height of the father. Connecting these averages, we obtain the line shown in the ﬁgure. This line, as we shall see, is known as the regression line. It shows how the average height of sons increases with the father’s height.3 2. Consider the scattergram in Figure 1.2, which gives the distribution in a hypothetical population of heights of boys measured at ﬁxed ages. Corresponding to any given age, we have a range, or distribution, of heights. Obviously, not all boys of a given age are likely to have identical heights. But height on the average increases with age (of course, up to a certain age), which can be seen clearly if we draw a line (the regression line) through the

70

Mean value

60 Height, inches

50

40

10

11

12 Age, years

13

14

FIGURE 1.2

Hypothetical distribution of heights corresponding to selected ages.

3 At this stage of the development of the subject matter, we shall call this regression line simply the line connecting the mean, or average, value of the dependent variable (son’s height) corresponding to the given value of the explanatory variable (father’s height). Note that this line has a positive slope but the slope is less than 1, which is in conformity with Galton’s regression to mediocrity. (Why?)

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circled points that represent the average height at the given ages. Thus, knowing the age, we may be able to predict from the regression line the average height corresponding to that age. 3. Turning to economic examples, an economist may be interested in studying the dependence of personal consumption expenditure on aftertax or disposable real personal income. Such an analysis may be helpful in estimating the marginal propensity to consume (MPC), that is, average change in consumption expenditure for, say, a dollar’s worth of change in real income (see Figure I.3). 4. A monopolist who can ﬁx the price or output (but not both) may want to ﬁnd out the response of the demand for a product to changes in price. Such an experiment may enable the estimation of the price elasticity (i.e., price responsiveness) of the demand for the product and may help determine the most proﬁtable price. 5. A labor economist may want to study the rate of change of money wages in relation to the unemployment rate. The historical data are shown in the scattergram given in Figure 1.3. The curve in Figure 1.3 is an example of the celebrated Phillips curve relating changes in the money wages to the unemployment rate. Such a scattergram may enable the labor economist to predict the average change in money wages given a certain unemployment rate. Such knowledge may be helpful in stating something about the inﬂationary process in an economy, for increases in money wages are likely to be reﬂected in increased prices.

+

Rate of change of money wages

0

Unemployment rate, %

–

FIGURE 1.3 Hypothetical Phillips curve.

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k=

Money Income

0 Inflation rate FIGURE 1.4 Money holding in relation to the inﬂation rate π.

π

6. From monetary economics it is known that, other things remaining the same, the higher the rate of inﬂation π, the lower the proportion k of their income that people would want to hold in the form of money, as depicted in Figure 1.4. A quantitative analysis of this relationship will enable the monetary economist to predict the amount of money, as a proportion of their income, that people would want to hold at various rates of inﬂation. 7. The marketing director of a company may want to know how the demand for the company’s product is related to, say, advertising expenditure. Such a study will be of considerable help in ﬁnding out the elasticity of demand with respect to advertising expenditure, that is, the percent change in demand in response to, say, a 1 percent change in the advertising budget. This knowledge may be helpful in determining the “optimum” advertising budget. 8. Finally, an agronomist may be interested in studying the dependence of crop yield, say, of wheat, on temperature, rainfall, amount of sunshine, and fertilizer. Such a dependence analysis may enable the prediction or forecasting of the average crop yield, given information about the explanatory variables. The reader can supply scores of such examples of the dependence of one variable on one or more other variables. The techniques of regression analysis discussed in this text are specially designed to study such dependence among variables.

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1.3

STATISTICAL VERSUS DETERMINISTIC RELATIONSHIPS

From the examples cited in Section 1.2, the reader will notice that in regression analysis we are concerned with what is known as the statistical, not functional or deterministic, dependence among variables, such as those of classical physics. In statistical relationships among variables we essentially deal with random or stochastic4 variables, that is, variables that have probability distributions. In functional or deterministic dependency, on the other hand, we also deal with variables, but these variables are not random or stochastic. The dependence of crop yield on temperature, rainfall, sunshine, and fertilizer, for example, is statistical in nature in the sense that the explanatory variables, although certainly important, will not enable the agronomist to predict crop yield exactly because of errors involved in measuring these variables as well as a host of other factors (variables) that collectively affect the yield but may be difﬁcult to identify individually. Thus, there is bound to be some “intrinsic” or random variability in the dependent-variable crop yield that cannot be fully explained no matter how many explanatory variables we consider. In deterministic phenomena, on the other hand, we deal with relationships of the type, say, exhibited by Newton’s law of gravity, which states: Every particle in the universe attracts every other particle with a force directly proportional to the product of their masses and inversely proportional to the square of the distance between them. Symbolically, F = k(m1 m2 /r 2 ), where F = force, m1 and m2 are the masses of the two particles, r = distance, and k = constant of proportionality. Another example is Ohm’s law, which states: For metallic conductors over a limited range of temperature the current C is proportional to the voltage V; that is, C = ( 1 )V where 1 is the constant of k k proportionality. Other examples of such deterministic relationships are Boyle’s gas law, Kirchhoff’s law of electricity, and Newton’s law of motion. In this text we are not concerned with such deterministic relationships. Of course, if there are errors of measurement, say, in the k of Newton’s law of gravity, the otherwise deterministic relationship becomes a statistical relationship. In this situation, force can be predicted only approximately from the given value of k (and m1 , m2 , and r), which contains errors. The variable F in this case becomes a random variable.

1.4 REGRESSION VERSUS CAUSATION

Although regression analysis deals with the dependence of one variable on other variables, it does not necessarily imply causation. In the words of Kendall and Stuart, “A statistical relationship, however strong and however

4 The word stochastic comes from the Greek word stokhos meaning “a bull’s eye.” The outcome of throwing darts on a dart board is a stochastic process, that is, a process fraught with misses.

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suggestive, can never establish causal connection: our ideas of causation must come from outside statistics, ultimately from some theory or other.”5 In the crop-yield example cited previously, there is no statistical reason to assume that rainfall does not depend on crop yield. The fact that we treat crop yield as dependent on rainfall (among other things) is due to nonstatistical considerations: Common sense suggests that the relationship cannot be reversed, for we cannot control rainfall by varying crop yield. In all the examples cited in Section 1.2 the point to note is that a statistical relationship in itself cannot logically imply causation. To ascribe causality, one must appeal to a priori or theoretical considerations. Thus, in the third example cited, one can invoke economic theory in saying that consumption expenditure depends on real income.6

1.5 REGRESSION VERSUS CORRELATION

Closely related to but conceptually very much different from regression analysis is correlation analysis, where the primary objective is to measure the strength or degree of linear association between two variables. The correlation coefﬁcient, which we shall study in detail in Chapter 3, measures this strength of (linear) association. For example, we may be interested in ﬁnding the correlation (coefﬁcient) between smoking and lung cancer, between scores on statistics and mathematics examinations, between high school grades and college grades, and so on. In regression analysis, as already noted, we are not primarily interested in such a measure. Instead, we try to estimate or predict the average value of one variable on the basis of the ﬁxed values of other variables. Thus, we may want to know whether we can predict the average score on a statistics examination by knowing a student’s score on a mathematics examination. Regression and correlation have some fundamental differences that are worth mentioning. In regression analysis there is an asymmetry in the way the dependent and explanatory variables are treated. The dependent variable is assumed to be statistical, random, or stochastic, that is, to have a probability distribution. The explanatory variables, on the other hand, are assumed to have ﬁxed values (in repeated sampling),7 which was made explicit in the deﬁnition of regression given in Section 1.2. Thus, in Figure 1.2 we assumed that the variable age was ﬁxed at given levels and height measurements were obtained at these levels. In correlation analysis, on the

5 M. G. Kendall and A. Stuart, The Advanced Theory of Statistics, Charles Grifﬁn Publishers, New York, 1961, vol. 2, chap. 26, p. 279. 6 But as we shall see in Chap. 3, classical regression analysis is based on the assumption that the model used in the analysis is the correct model. Therefore, the direction of causality may be implicit in the model postulated. 7 It is crucial to note that the explanatory variables may be intrinsically stochastic, but for the purpose of regression analysis we assume that their values are ﬁxed in repeated sampling (that is, X assumes the same values in various samples), thus rendering them in effect nonrandom or nonstochastic. But more on this in Chap. 3, Sec. 3.2.

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other hand, we treat any (two) variables symmetrically; there is no distinction between the dependent and explanatory variables. After all, the correlation between scores on mathematics and statistics examinations is the same as that between scores on statistics and mathematics examinations. Moreover, both variables are assumed to be random. As we shall see, most of the correlation theory is based on the assumption of randomness of variables, whereas most of the regression theory to be expounded in this book is conditional upon the assumption that the dependent variable is stochastic but the explanatory variables are ﬁxed or nonstochastic.8

1.6 TERMINOLOGY AND NOTATION

Before we proceed to a formal analysis of regression theory, let us dwell brieﬂy on the matter of terminology and notation. In the literature the terms dependent variable and explanatory variable are described variously. A representative list is:

Dependent variable ⇔ Explanatory variable ⇔ Independent variable Predictor Regressor Stimulus Exogenous Covariate Control variable ⇔ ⇔ ⇔ ⇔ ⇔ ⇔

Explained variable Predictand Regressand Response Endogenous Outcome Controlled variable ⇔ ⇔ ⇔ ⇔ ⇔ ⇔

Although it is a matter of personal taste and tradition, in this text we will use the dependent variable/explanatory variable or the more neutral, regressand and regressor terminology. If we are studying the dependence of a variable on only a single explanatory variable, such as that of consumption expenditure on real income, such a study is known as simple, or two-variable, regression analysis. However, if we are studying the dependence of one variable on more than

8 In advanced treatment of econometrics, one can relax the assumption that the explanatory variables are nonstochastic (see introduction to Part II).

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one explanatory variable, as in the crop-yield, rainfall, temperature, sunshine, and fertilizer examples, it is known as multiple regression analysis. In other words, in two-variable regression there is only one explanatory variable, whereas in multiple regression there is more than one explanatory variable. The term random is a synonym for the term stochastic. As noted earlier, a random or stochastic variable is a variable that can take on any set of values, positive or negative, with a given probability.9 Unless stated otherwise, the letter Y will denote the dependent variable and the X’s (X1 , X2 , . . . , Xk) will denote the explanatory variables, Xk being the kth explanatory variable. The subscript i or t will denote the ith or the tth observation or value. Xki (or Xkt ) will denote the ith (or tth) observation on variable Xk . N (or T) will denote the total number of observations or values in the population, and n (or t) the total number of observations in a sample. As a matter of convention, the observation subscript i will be used for crosssectional data (i.e., data collected at one point in time) and the subscript t will be used for time series data (i.e., data collected over a period of time). The nature of cross-sectional and time series data, as well as the important topic of the nature and sources of data for empirical analysis, is discussed in the following section.

1.7 THE NATURE AND SOURCES OF DATA FOR ECONOMIC ANALYSIS10

The success of any econometric analysis ultimately depends on the availability of the appropriate data. It is therefore essential that we spend some time discussing the nature, sources, and limitations of the data that one may encounter in empirical analysis.

Types of Data

Three types of data may be available for empirical analysis: time series, cross-section, and pooled (i.e., combination of time series and crosssection) data. Time Series Data The data shown in Table I.1 of the Introduction are an example of time series data. A time series is a set of observations on the values that a variable takes at different times. Such data may be collected at regular time intervals, such as daily (e.g., stock prices, weather reports), weekly (e.g., money supply ﬁgures), monthly [e.g., the unemployment rate, the Consumer Price Index (CPI)], quarterly (e.g., GDP), annually (e.g.,

See App. A for formal deﬁnition and further details. For an informative account, see Michael D. Intriligator, Econometric Models, Techniques, and Applications, Prentice Hall, Englewood Cliffs, N.J., 1978, chap. 3.

10 9

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government budgets), quinquennially, that is, every 5 years (e.g., the census of manufactures), or decennially (e.g., the census of population). Sometime data are available both quarterly as well as annually, as in the case of the data on GDP and consumer expenditure. With the advent of high-speed computers, data can now be collected over an extremely short interval of time, such as the data on stock prices, which can be obtained literally continuously (the so-called real-time quote). Although time series data are used heavily in econometric studies, they present special problems for econometricians. As we will show in chapters on time series econometrics later on, most empirical work based on time series data assumes that the underlying time series is stationary. Although it is too early to introduce the precise technical meaning of stationarity at this juncture, loosely speaking a time series is stationary if its mean and variance do not vary systematically over time. To see what this means, consider Figure 1.5, which depicts the behavior of the M1 money supply in the United States from January 1, 1959, to July 31, 1999. (The actual data are given in exercise 1.4.) As you can see from this ﬁgure, the M1 money supply shows a steady upward trend as well as variability over the years, suggesting that the M1 time series is not stationary.11 We will explore this topic fully in Chapter 21.

1200

1000

800

600

400

200

0 FIGURE 1.5

55

60

65

70

75

80

85

90

95

M1 money supply: United States, 1951:01–1999:09.

11 To see this more clearly, we divided the data into four time periods: 1951:01 to 1962:12; 1963:01 to 1974:12; 1975:01 to 1986:12, and 1987:01 to 1999:09: For these subperiods the mean values of the money supply (with corresponding standard deviations in parentheses) were, respectively, 165.88 (23.27), 323.20 (72.66), 788.12 (195.43), and 1099 (27.84), all ﬁgures in billions of dollars. This is a rough indication of the fact that the money supply over the entire period was not stationary.

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Cross-Section Data Cross-section data are data on one or more variables collected at the same point in time, such as the census of population conducted by the Census Bureau every 10 years (the latest being in year 2000), the surveys of consumer expenditures conducted by the University of Michigan, and, of course, the opinion polls by Gallup and umpteen other organizations. A concrete example of cross-sectional data is given in Table 1.1 This table gives data on egg production and egg prices for the 50 states in the union for 1990 and 1991. For each year the data on the 50 states are cross-sectional data. Thus, in Table 1.1 we have two cross-sectional samples. Just as time series data create their own special problems (because of the stationarity issue), cross-sectional data too have their own problems, speciﬁcally the problem of heterogeneity. From the data given in Table 1.1 we see that we have some states that produce huge amounts of eggs (e.g., Pennsylvania) and some that produce very little (e.g., Alaska). When we

TABLE 1.1

U.S. EGG PRODUCTION State AL AK AZ AR CA CO CT DE FL GA HI ID IL IN IA KS KY LA ME MD MA MI MN MS MO Y1 2,206 0.7 73 3,620 7,472 788 1,029 168 2,586 4,302 227.5 187 793 5,445 2,151 404 412 273 1,069 885 235 1,406 2,499 1,434 1,580 Y2 2,186 0.7 74 3,737 7,444 873 948 164 2,537 4,301 224.5 203 809 5,290 2,247 389 483 254 1,070 898 237 1,396 2,697 1,468 1,622 X1 92.7 151.0 61.0 86.3 63.4 77.8 106.0 117.0 62.0 80.6 85.0 79.1 65.0 62.7 56.5 54.5 67.7 115.0 101.0 76.6 105.0 58.0 57.7 87.8 55.4 X2 91.4 149.0 56.0 91.8 58.4 73.0 104.0 113.0 57.2 80.8 85.5 72.9 70.5 60.1 53.0 47.8 73.5 115.0 97.0 75.4 102.0 53.8 54.0 86.7 51.5 State MT NE NV NH NJ NM NY NC ND OH OK OR PA RI SC SD TN TX UT VT VA WA WV WI WY Y1 172 1,202 2.2 43 442 283 975 3,033 51 4,667 869 652 4,976 53 1,422 435 277 3,317 456 31 943 1,287 136 910 1.7 Y2 164 1,400 1.8 49 491 302 987 3,045 45 4,637 830 686 5,130 50 1,420 602 279 3,356 486 30 988 1,313 174 873 1.7 X1 68.0 50.3 53.9 109.0 85.0 74.0 68.1 82.8 55.2 59.1 101.0 77.0 61.0 102.0 70.1 48.0 71.0 76.7 64.0 106.0 86.3 74.1 104.0 60.1 83.0 X2 66.0 48.9 52.7 104.0 83.0 70.0 64.0 78.7 48.0 54.7 100.0 74.6 52.0 99.0 65.9 45.8 80.7 72.6 59.0 102.0 81.2 71.5 109.0 54.0 83.0

Note: Y1 = eggs produced in 1990 (millions) Y2 = eggs produced in 1991 (millions) X1 = price per dozen (cents) in 1990 X2 = price per dozen (cents) in 1991 Source: World Almanac, 1993, p. 119. The data are from the Economic Research Service, U.S. Department of Agriculture.

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160 Price of eggs per dozen (in cents) 140 120 100 80 60 40 FIGURE 1.6 Relationship between eggs produced and prices, 1990.

0

2000 4000 6000 8000 Number of eggs produced (millions)

include such heterogeneous units in a statistical analysis, the size or scale effect must be taken into account so as not to mix apples with oranges. To see this clearly, we plot in Figure 1.6 the data on eggs produced and their prices in 50 states for the year 1990. This ﬁgure shows how widely scattered the observations are. In Chapter 11 we will see how the scale effect can be an important factor in assessing relationships among economic variables. Pooled Data In pooled, or combined, data are elements of both time series and cross-section data. The data in Table 1.1 are an example of pooled data. For each year we have 50 cross-sectional observations and for each state we have two time series observations on prices and output of eggs, a total of 100 pooled (or combined) observations. Likewise, the data given in exercise 1.1 are pooled data in that the Consumer Price Index (CPI) for each country for 1973–1997 is time series data, whereas the data on the CPI for the seven countries for a single year are cross-sectional data. In the pooled data we have 175 observations—25 annual observations for each of the seven countries. Panel, Longitudinal, or Micropanel Data This is a special type of pooled data in which the same cross-sectional unit (say, a family or a ﬁrm) is surveyed over time. For example, the U.S. Department of Commerce carries out a census of housing at periodic intervals. At each periodic survey the same household (or the people living at the same address) is interviewed to ﬁnd out if there has been any change in the housing and ﬁnancial conditions of that household since the last survey. By interviewing the same household periodically, the panel data provides very useful information on the dynamics of household behavior, as we shall see in Chapter 16.

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The Sources of Data12

The data used in empirical analysis may be collected by a governmental agency (e.g., the Department of Commerce), an international agency (e.g., the International Monetary Fund (IMF) or the World Bank), a private organization (e.g., the Standard & Poor’s Corporation), or an individual. Literally, there are thousands of such agencies collecting data for one purpose or another. The Internet The Internet has literally revolutionized data gathering. If you just “surf the net” with a keyword (e.g., exchange rates), you will be swamped with all kinds of data sources. In Appendix E we provide some of the frequently visited web sites that provide economic and ﬁnancial data of all sorts. Most of the data can be downloaded without much cost. You may want to bookmark the various web sites that might provide you with useful economic data. The data collected by various agencies may be experimental or nonexperimental. In experimental data, often collected in the natural sciences, the investigator may want to collect data while holding certain factors constant in order to assess the impact of some factors on a given phenomenon. For instance, in assessing the impact of obesity on blood pressure, the researcher would want to collect data while holding constant the eating, smoking, and drinking habits of the people in order to minimize the inﬂuence of these variables on blood pressure. In the social sciences, the data that one generally encounters are nonexperimental in nature, that is, not subject to the control of the researcher.13 For example, the data on GNP, unemployment, stock prices, etc., are not directly under the control of the investigator. As we shall see, this lack of control often creates special problems for the researcher in pinning down the exact cause or causes affecting a particular situation. For example, is it the money supply that determines the (nominal) GDP or is it the other way round?

The Accuracy of Data14

Although plenty of data are available for economic research, the quality of the data is often not that good. There are several reasons for that. First, as noted, most social science data are nonexperimental in nature. Therefore, there is the possibility of observational errors, either of omission or commission. Second, even in experimentally collected data errors of measurement arise from approximations and roundoffs. Third, in questionnaire-type surveys, the problem of nonresponse can be serious; a researcher is lucky to

12 For an illuminating account, see Albert T. Somers, The U.S. Economy Demystiﬁed: What the Major Economic Statistics Mean and their Signiﬁcance for Business, D.C. Heath, Lexington, Mass., 1985. 13 In the social sciences too sometimes one can have a controlled experiment. An example is given in exercise 1.6. 14 For a critical review, see O. Morgenstern, The Accuracy of Economic Observations, 2d ed., Princeton University Press, Princeton, N.J., 1963.

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get a 40 percent response to a questionnaire. Analysis based on such partial response may not truly reﬂect the behavior of the 60 percent who did not respond, thereby leading to what is known as (sample) selectivity bias. Then there is the further problem that those who respond to the questionnaire may not answer all the questions, especially questions of ﬁnancially sensitive nature, thus leading to additional selectivity bias. Fourth, the sampling methods used in obtaining the data may vary so widely that it is often difﬁcult to compare the results obtained from the various samples. Fifth, economic data are generally available at a highly aggregate level. For example, most macrodata (e.g., GNP, employment, inﬂation, unemployment) are available for the economy as a whole or at the most for some broad geographical regions. Such highly aggregated data may not tell us much about the individual or microunits that may be the ultimate object of study. Sixth, because of conﬁdentiality, certain data can be published only in highly aggregate form. The IRS, for example, is not allowed by law to disclose data on individual tax returns; it can only release some broad summary data. Therefore, if one wants to ﬁnd out how much individuals with a certain level of income spent on health care, one cannot do that analysis except at a very highly aggregate level. But such macroanalysis often fails to reveal the dynamics of the behavior of the microunits. Similarly, the Department of Commerce, which conducts the census of business every 5 years, is not allowed to disclose information on production, employment, energy consumption, research and development expenditure, etc., at the ﬁrm level. It is therefore difﬁcult to study the interﬁrm differences on these items. Because of all these and many other problems, the researcher should always keep in mind that the results of research are only as good as the quality of the data. Therefore, if in given situations researchers ﬁnd that the results of the research are “unsatisfactory,” the cause may be not that they used the wrong model but that the quality of the data was poor. Unfortunately, because of the nonexperimental nature of the data used in most social science studies, researchers very often have no choice but to depend on the available data. But they should always keep in mind that the data used may not be the best and should try not to be too dogmatic about the results obtained from a given study, especially when the quality of the data is suspect.

A Note on the Measurement Scales of Variables15

The variables that we will generally encounter fall into four broad categories: ratio scale, interval scale, ordinal scale, and nominal scale. It is important that we understand each. Ratio Scale For a variable X, taking two values, X1 and X2 , the ratio X1 /X2 and the distance (X2 − X1 ) are meaningful quantities. Also, there is a

15 The following discussion relies heavily on Aris Spanos, Probability Theory and Statistical Inference: Econometric Modeling with Observational Data, Cambridge University Press, New York, 1999, p. 24.

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natural ordering (ascending or descending) of the values along the scale. Therefore, comparisons such as X2 ≤ X1 or X2 ≥ X1 are meaningful. Most economic variables belong to this category. Thus, it is meaningful to ask how big is this year’s GDP compared with the previous year’s GDP. Interval Scale An interval scale variable satisﬁes the last two properties of the ratio scale variable but not the ﬁrst. Thus, the distance between two time periods, say (2000–1995) is meaningful, but not the ratio of two time periods (2000/1995). Ordinal Scale A variable belongs to this category only if it satisﬁes the third property of the ratio scale (i.e., natural ordering). Examples are grading systems (A, B, C grades) or income class (upper, middle, lower). For these variables the ordering exists but the distances between the categories cannot be quantiﬁed. Students of economics will recall the indifference curves between two goods, each higher indifference curve indicating higher level of utility, but one cannot quantify by how much one indifference curve is higher than the others. Nominal Scale Variables in this category have none of the features of the ratio scale variables. Variables such as gender (male, female) and marital status (married, unmarried, divorced, separated) simply denote categories. Question: What is the reason why such variables cannot be expressed on the ratio, interval, or ordinal scales? As we shall see, econometric techniques that may be suitable for ratio scale variables may not be suitable for nominal scale variables. Therefore, it is important to bear in mind the distinctions among the four types of measurement scales discussed above.

1.8 SUMMARY AND CONCLUSIONS

1. The key idea behind regression analysis is the statistical dependence of one variable, the dependent variable, on one or more other variables, the explanatory variables. 2. The objective of such analysis is to estimate and/or predict the mean or average value of the dependent variable on the basis of the known or ﬁxed values of the explanatory variables. 3. In practice the success of regression analysis depends on the availability of the appropriate data. This chapter discussed the nature, sources, and limitations of the data that are generally available for research, especially in the social sciences. 4. In any research, the researcher should clearly state the sources of the data used in the analysis, their deﬁnitions, their methods of collection, and any gaps or omissions in the data as well as any revisions in the data. Keep in mind that the macroeconomic data published by the government are often revised.

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5. Since the reader may not have the time, energy, or resources to track down the data, the reader has the right to presume that the data used by the researcher are properly gathered and that the computations and analysis are correct. EXERCISES

1.1. Table 1.2 gives data on the Consumer Price Index (CPI) for seven industrialized countries with 1982–1984 = 100 as the base of the index. a. From the given data, compute the inﬂation rate for each country.16 b. Plot the inﬂation rate for each country against time (i.e., use the horizontal axis for time and the vertical axis for the inﬂation rate.) c. What broad conclusions can you draw about the inﬂation experience in the seven countries? d. Which country’s inﬂation rate seems to be most variable? Can you offer any explanation?

TABLE 1.2 CPI IN SEVEN INDUSTRIAL COUNTRIES, 1973–1997 (1982−1984 = 100) Year 1973 1974 1975 1976 1977 1978 1979 1980 1981 1982 1983 1984 1985 1986 1987 1988 1989 1990 1991 1992 1993 1994 1995 1996 1997 Canada 40.80000 45.20000 50.10000 53.90000 58.10000 63.30000 69.20000 76.10000 85.60000 94.90000 100.4000 104.7000 109.0000 113.5000 118.4000 123.2000 129.3000 135.5000 143.1000 145.3000 147.9000 148.2000 151.4000 153.8000 156.3000 France 34.60000 39.30000 43.90000 48.10000 52.70000 57.50000 63.60000 72.30000 81.90000 91.70000 100.4000 108.1000 114.4000 117.3000 121.1000 124.4000 128.7000 133.0000 137.2000 140.5000 143.5000 145.8000 148.4000 151.4000 153.2000 Germany 62.80000 67.10000 71.10000 74.20000 76.90000 79.00000 82.20000 86.70000 92.20000 97.10000 100.3000 102.7000 104.8000 104.7000 104.9000 106.3000 109.2000 112.2000 116.3000 122.1000 127.6000 131.1000 133.5000 135.5000 137.8000 Italy 20.60000 24.60000 28.80000 33.60000 40.10000 45.10000 52.10000 63.20000 75.40000 87.70000 100.8000 111.5000 121.1000 128.5000 134.4000 141.1000 150.4000 159.6000 169.8000 178.8000 186.4000 193.7000 204.1000 212.0000 215.7000 Japan 47.90000 59.00000 65.90000 72.20000 78.10000 81.40000 84.40000 90.90000 95.30000 98.10000 99.80000 102.1000 104.1000 104.8000 104.8000 105.6000 108.1000 111.4000 115.0000 116.9000 118.4000 119.3000 119.1000 119.3000 121.3000 U.K. 27.90000 32.30000 40.20000 46.80000 54.20000 58.70000 66.60000 78.50000 87.90000 95.40000 99.80000 104.8000 111.1000 114.9000 119.7000 125.6000 135.3000 148.2000 156.9000 162.7000 165.3000 169.4000 175.1000 179.4000 185.0000 U.S. 44.40000 49.30000 53.80000 56.90000 60.60000 65.20000 72.60000 82.40000 90.90000 96.50000 99.60000 103.9000 107.6000 109.6000 113.6000 118.3000 124.0000 130.7000 136.2000 140.3000 144.5000 148.2000 152.4000 156.9000 160.5000

16 Subtract from the current year’s CPI the CPI from the previous year, divide the difference by the previous year’s CPI, and multiply the result by 100. Thus, the inﬂation rate for Canada for 1974 is [(45.2 − 40.8)/40.8] × 100 = 10.78% (approx.).

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1.2. a. Plot the inﬂation rate of Canada, France, Germany, Italy, Japan, and the United Kingdom against the United States inﬂation rate. b. Comment generally about the behavior of the inﬂation rate in the six countries vis-à-vis the U.S. inﬂation rate. c. If you ﬁnd that the six countries’ inﬂation rates move in the same direction as the U.S. inﬂation rate, would that suggest that U.S. inﬂation “causes” inﬂation in the other countries? Why or why not? 1.3. Table 1.3 gives the foreign exchange rates for seven industrialized countries for years 1977–1998. Except for the United Kingdom, the exchange rate is deﬁned as the units of foreign currency for one U.S. dollar; for the United Kingdom, it is deﬁned as the number of U.S. dollars for one U.K. pound. a. Plot these exchange rates against time and comment on the general behavior of the exchange rates over the given time period. b. The dollar is said to appreciate if it can buy more units of a foreign currency. Contrarily, it is said to depreciate if it buys fewer units of a foreign currency. Over the time period 1977–1998, what has been the general behavior of the U.S. dollar? Incidentally, look up any textbook on macroeconomics or international economics to ﬁnd out what factors determine the appreciation or depreciation of a currency. 1.4. The data behind the M1 money supply in Figure 1.5 are given in Table 1.4. Can you give reasons why the money supply has been increasing over the time period shown in the table?

TABLE 1.3

EXCHANGE RATES FOR SEVEN COUNTRIES: 1977–1998 Year 1977 1978 1979 1980 1981 1982 1983 1984 1985 1986 1987 1988 1989 1990 1991 1992 1993 1994 1995 1996 1997 1998 Canada 1.063300 1.140500 1.171300 1.169300 1.199000 1.234400 1.232500 1.295200 1.365900 1.389600 1.325900 1.230600 1.184200 1.166800 1.146000 1.208500 1.290200 1.366400 1.372500 1.363800 1.384900 1.483600 France 4.916100 4.509100 4.256700 4.225100 5.439700 6.579400 7.620400 8.735600 8.980000 6.925700 6.012200 5.959500 6.380200 5.446700 5.646800 5.293500 5.666900 5.545900 4.986400 5.115800 5.839300 5.899500 Germany 2.323600 2.009700 1.834300 1.817500 2.263200 2.428100 2.553900 2.845500 2.942000 2.170500 1.798100 1.757000 1.880800 1.616600 1.661000 1.561800 1.654500 1.621600 1.432100 1.504900 1.734800 1.759700 Japan 268.6200 210.3900 219.0200 226.6300 220.6300 249.0600 237.5500 237.4600 238.4700 168.3500 144.6000 128.1700 138.0700 145.0000 134.5900 126.7800 111.0800 102.1800 93.96000 108.7800 121.0600 130.9900 Sweden 4.480200 4.520700 4.289300 4.231000 5.066000 6.283900 7.671800 8.270800 8.603200 7.127300 6.346900 6.137000 6.455900 5.923100 6.052100 5.825800 7.795600 7.716100 7.140600 6.708200 7.644600 7.952200 Switzerland 2.406500 1.790700 1.664400 1.677200 1.967500 2.032700 2.100700 2.350000 2.455200 1.797900 1.491800 1.464300 1.636900 1.390100 1.435600 1.406400 1.478100 1.366700 1.181200 1.236100 1.451400 1.450600 U.K. 1.744900 1.918400 2.122400 2.324600 2.024300 1.748000 1.515900 1.336800 1.297400 1.467700 1.639800 1.781300 1.638200 1.784100 1.767400 1.766300 1.501600 1.531900 1.578500 1.560700 1.637600 1.657300

Source: Economic Report of the President, January 2000 and January 2001.

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TABLE 1.4

SEASONALLY ADJUSTED M1 SUPPLY: 1959:01–1999:09 (BILLIONS OF DOLLARS) 1959:01 1959:07 1960:01 1960:07 1961:01 1961:07 1962:01 1962:07 1963:01 1963:07 1964:01 1964:07 1965:01 1965:07 1966:01 1966:07 1967:01 1967:07 1968:01 1968:07 1969:01 1969:07 1970:01 1970:07 1971:01 1971:07 1972:01 1972:07 1973:01 1973:07 1974:01 1974:07 1975:01 1975:07 1976:01 1976:07 1977:01 1977:07 1978:01 1978:07 1979:01 1979:07 1980:01 1980:07 1981:01 1981:07 1982:01 1982:07 1983:01 1983:07 138.8900 141.7000 139.9800 140.1800 141.0600 142.9200 145.2400 146.4600 148.2600 151.3400 153.7400 156.8000 160.7100 163.0500 169.0800 170.3100 171.8600 178.1300 184.3300 190.4900 198.6900 201.6600 206.2200 207.9800 215.5400 224.8500 230.0900 238.7900 251.4700 257.5400 263.7600 269.2700 273.9000 283.6800 288.4200 297.2000 308.2600 320.1900 334.4000 347.6300 358.6000 377.2100 385.8500 394.9100 410.8300 427.9000 442.1300 449.0900 476.6800 508.9600 139.3900 141.9000 139.8700 141.3100 141.6000 143.4900 145.6600 146.5700 148.9000 151.7800 154.3100 157.8200 160.9400 163.6800 169.6200 170.8100 172.9900 179.7100 184.7100 191.8400 199.3500 201.7300 205.0000 209.9300 217.4200 225.5800 232.3200 240.9300 252.1500 257.7600 265.3100 270.1200 275.0000 284.1500 290.7600 299.0500 311.5400 322.2700 335.3000 349.6600 359.9100 378.8200 389.7000 400.0600 414.3800 427.8500 441.4900 452.4900 483.8500 511.6000 139.7400 141.0100 139.7500 141.1800 141.8700 143.7800 145.9600 146.3000 149.1700 151.9800 154.4800 158.7500 161.4700 164.8500 170.5100 171.9700 174.8100 180.6800 185.4700 192.7400 200.0200 202.1000 205.7500 211.8000 218.7700 226.4700 234.3000 243.1800 251.6700 257.8600 266.6800 271.0500 276.4200 285.6900 292.7000 299.6700 313.9400 324.4800 336.9600 352.2600 362.4500 379.2800 388.1300 405.3600 418.6900 427.4600 442.3700 457.5000 490.1800 513.4100 139.6900 140.4700 139.5600 140.9200 142.1300 144.1400 146.4000 146.7100 149.7000 152.5500 154.7700 159.2400 162.0300 165.9700 171.8100 171.1600 174.1700 181.6400 186.6000 194.0200 200.7100 202.9000 206.7200 212.8800 220.0000 227.1600 235.5800 245.0200 252.7400 259.0400 267.2000 272.3500 276.1700 285.3900 294.6600 302.0400 316.0200 326.4000 339.9200 353.3500 368.0500 380.8700 383.4400 409.0600 427.0600 428.4500 446.7800 464.5700 492.7700 517.2100 140.6800 140.3800 139.6100 140.8600 142.6600 144.7600 146.8400 147.2900 150.3900 153.6500 155.3300 159.9600 161.7000 166.7100 171.3300 171.3800 175.6800 182.3800 187.9900 196.0200 200.8100 203.5700 207.2200 213.6600 222.0200 227.7600 235.8900 246.4100 254.8900 260.9800 267.5600 273.7100 279.2000 286.8300 295.9300 303.5900 317.1900 328.6400 344.8600 355.4100 369.5900 380.8100 384.6000 410.3700 424.4300 430.8800 446.5300 471.1200 499.7800 518.5300 141.1700 139.9500 139.5800 140.6900 142.8800 145.2000 146.5800 147.8200 150.4300 153.2900 155.6200 160.3000 162.1900 167.8500 171.5700 172.0300 177.0200 183.2600 189.4200 197.4100 201.2700 203.8800 207.5400 214.4100 223.4500 228.3200 236.6200 249.2500 256.6900 262.8800 268.4400 274.2000 282.4300 287.0700 296.1600 306.2500 318.7100 330.8700 346.8000 357.2800 373.3400 381.7700 389.4600 408.0600 425.5000 436.1700 447.8900 474.3000 504.3500 520.7900 (Continued )

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TABLE 1.4

(Continued) 1984:01 1984:07 1985:01 1985:07 1986:01 1986:07 1987:01 1987:07 1988:01 1988:07 1989:01 1989:07 1990:01 1990:07 1991:01 1991:07 1992:01 1992:07 1993:01 1993:07 1994:01 1994:07 1995:01 1995:07 1996:01 1996:07 1997:01 1997:07 1998:01 1998:07 1999:01 1999:07 524.4000 542.1300 555.6600 590.8200 620.4000 672.2000 729.3400 744.9600 755.5500 783.4000 784.9200 779.7100 794.9300 811.8000 826.7300 862.9500 910.4900 964.6000 1030.900 1085.880 1132.200 1151.490 1150.640 1146.500 1122.580 1112.340 1080.520 1067.570 1073.810 1075.370 1091.000 1099.530 526.9900 542.3900 562.4800 598.0600 624.1400 680.7700 729.8400 746.9600 757.0700 785.0800 783.4000 781.1400 797.6500 817.8500 832.4000 868.6500 925.1300 975.7100 1033.150 1095.560 1136.130 1151.390 1146.740 1146.100 1117.530 1102.180 1076.200 1072.080 1076.020 1072.210 1092.650 1102.400 530.7800 543.8600 565.7400 604.4700 632.8100 688.5100 733.0100 748.6600 761.1800 784.8200 782.7400 782.2000 801.2500 821.8300 838.6200 871.5600 936.0000 988.8400 1037.990 1105.430 1139.910 1152.440 1146.520 1142.270 1122.590 1095.610 1072.420 1064.820 1080.650 1074.650 1102.010 1093.460 534.0300 543.8700 569.5500 607.9100 640.3500 695.2600 743.3900 756.5000 767.5700 783.6300 778.8200 787.0500 806.2400 820.3000 842.7300 878.4000 943.8900 1004.340 1047.470 1113.800 1141.420 1150.410 1149.480 1136.430 1124.520 1082.560 1067.450 1062.060 1082.090 1080.400 1108.400 536.5900 547.3200 575.0700 611.8300 652.0100 705.2400 746.0000 752.8300 771.6800 784.4600 774.7900 787.9500 804.3600 822.0600 848.9600 887.9500 950.7800 1016.040 1066.220 1123.900 1142.850 1150.440 1144.650 1133.550 1116.300 1080.490 1063.370 1067.530 1078.170 1088.960 1104.750 540.5400 551.1900 583.1700 619.3600 661.5200 724.2800 743.7200 749.6800 779.1000 786.2600 774.2200 792.5700 810.3300 824.5600 858.3300 896.7000 954.7100 1024.450 1075.610 1129.310 1145.650 1149.750 1144.240 1126.730 1115.470 1081.340 1065.990 1074.870 1077.780 1093.350 1101.110

Source: Board of Governors, Federal Reserve Bank, USA.

1.5. Suppose you were to develop an economic model of criminal activities, say, the hours spent in criminal activities (e.g., selling illegal drugs). What variables would you consider in developing such a model? See if your model matches the one developed by the Nobel laureate economist Gary Becker.17 1.6. Controlled experiments in economics: On April 7, 2000, President Clinton signed into law a bill passed by both Houses of the U.S. Congress that lifted earnings limitations on Social Security recipients. Until then, recipients between the ages of 65 and 69 who earned more than $17,000 a year would lose 1 dollar’s worth of Social Security beneﬁt for every 3 dollars of income earned in excess of $17,000. How would you devise a study to assess the impact of this change in the law? Note: There was no income limitation for recipients over the age of 70 under the old law.

17 G. S. Becker, “Crime and Punishment: An Economic Approach,” Journal of Political Economy, vol. 76, 1968, pp. 169–217.

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TABLE 1.5

IMPACT OF ADVERTISING EXPENDITURE Firm 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21. Miller Lite Pepsi Stroh’s Fed’l Express Burger King Coca Cola McDonald’s MCl Diet Cola Ford Levi’s Bud Lite ATT/Bell Calvin Klein Wendy’s Polaroid Shasta Meow Mix Oscar Meyer Crest Kibbles ‘N Bits Impressions, millions 32.1 99.6 11.7 21.9 60.8 78.6 92.4 50.7 21.4 40.1 40.8 10.4 88.9 12.0 29.2 38.0 10.0 12.3 23.4 71.1 4.4 Expenditure, millions of 1983 dollars 50.1 74.1 19.3 22.9 82.4 40.1 185.9 26.9 20.4 166.2 27.0 45.6 154.9 5.0 49.7 26.9 5.7 7.6 9.2 32.4 6.1

Source: http://lib.stat.cmu.edu/DASL/Dataﬁles/tvadsdat.html

1.7. The data presented in Table 1.5 was published in the March 1, 1984 issue of the Wall Street Journal. It relates to the advertising budget (in millions of dollars) of 21 ﬁrms for 1983 and millions of impressions retained per week by the viewers of the products of these ﬁrms. The data are based on a survey of 4000 adults in which users of the products were asked to cite a commercial they had seen for the product category in the past week. a. Plot impressions on the vertical axis and advertising expenditure on the horizontal axis. b. What can you say about the nature of the relationship between the two variables? c. Looking at your graph, do you think it pays to advertise? Think about all those commercials shown on Super Bowl Sunday or during the World Series. Note: We will explore further the data given in Table 1.5 in subsequent chapters.

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In Chapter 1 we discussed the concept of regression in broad terms. In this chapter we approach the subject somewhat formally. Speciﬁcally, this and the following two chapters introduce the reader to the theory underlying the simplest possible regression analysis, namely, the bivariate, or twovariable, regression in which the dependent variable (the regressand) is related to a single explanatory variable (the regressor). This case is considered ﬁrst, not because of its practical adequacy, but because it presents the fundamental ideas of regression analysis as simply as possible and some of these ideas can be illustrated with the aid of two-dimensional graphs. Moreover, as we shall see, the more general multiple regression analysis in which the regressand is related to one or more regressors is in many ways a logical extension of the two-variable case.

2.1

A HYPOTHETICAL EXAMPLE1

As noted in Section 1.2, regression analysis is largely concerned with estimating and/or predicting the (population) mean value of the dependent variable on the basis of the known or ﬁxed values of the explanatory variable(s).2 To understand this, consider the data given in Table 2.1. The data

1 The reader whose statistical knowledge has become somewhat rusty may want to freshen it up by reading the statistical appendix, App. A, before reading this chapter. 2 The expected value, or expectation, or population mean of a random variable Y is denoted by the symbol E(Y). On the other hand, the mean value computed from a sample of values from ¯ the Y population is denoted as Y, read as Y bar.

37

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TABLE 2.1

WEEKLY FAMILY INCOME X, $ Y X→ ↓ 80 55 60 65 70 75 – – 325 65 100 65 70 74 80 85 88 – 462 77 120 79 84 90 94 98 – – 445 89 140 80 93 95 103 108 113 115 707 101 160 102 107 110 116 118 125 – 678 113 180 110 115 120 130 135 140 – 750 125 200 120 136 140 144 145 – – 685 137 220 135 137 140 152 157 160 162 1043 149 240 137 145 155 165 175 189 – 966 161 260 150 152 175 178 180 185 191 1211 173

Weekly family consumption expenditure Y, $

Total Conditional means of Y, E(Y |X )

in the table refer to a total population of 60 families in a hypothetical community and their weekly income (X) and weekly consumption expenditure (Y), both in dollars. The 60 families are divided into 10 income groups (from $80 to $260) and the weekly expenditures of each family in the various groups are as shown in the table. Therefore, we have 10 ﬁxed values of X and the corresponding Y values against each of the X values; so to speak, there are 10 Y subpopulations. There is considerable variation in weekly consumption expenditure in each income group, which can be seen clearly from Figure 2.1. But the general picture that one gets is that, despite the variability of weekly consump-

Weekly consumption expenditure, $

200 E(Y |X)

150

100

50

80

100

120

140 160 180 200 Weekly income, $

220

240

260

FIGURE 2.1

Conditional distribution of expenditure for various levels of income (data of Table 2.1).

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39

tion expenditure within each income bracket, on the average, weekly consumption expenditure increases as income increases. To see this clearly, in Table 2.1 we have given the mean, or average, weekly consumption expenditure corresponding to each of the 10 levels of income. Thus, corresponding to the weekly income level of $80, the mean consumption expenditure is $65, while corresponding to the income level of $200, it is $137. In all we have 10 mean values for the 10 subpopulations of Y. We call these mean values conditional expected values, as they depend on the given values of the (conditioning) variable X. Symbolically, we denote them as E(Y | X), which is read as the expected value of Y given the value of X (see also Table 2.2). It is important to distinguish these conditional expected values from the unconditional expected value of weekly consumption expenditure, E(Y). If we add the weekly consumption expenditures for all the 60 families in the population and divide this number by 60, we get the number $121.20 ($7272/60), which is the unconditional mean, or expected, value of weekly consumption expenditure, E(Y); it is unconditional in the sense that in arriving at this number we have disregarded the income levels of the various families.3 Obviously, the various conditional expected values of Y given in Table 2.1 are different from the unconditional expected value of Y of $121.20. When we ask the question, “What is the expected value of weekly consumption expenditure of a family,” we get the answer $121.20 (the unconditional mean). But if we ask the question, “What is the expected value of weekly consumption expenditure of a family whose monthly income is,

TABLE 2.2

CONDITIONAL PROBABILITIES p(Y | Xi) FOR THE DATA OF TABLE 2.1 p(Y | Xi) ↓ X→ 80

1 5 1 5 1 5 1 5 1 5

100

1 6 1 6 1 6 1 6 1 6 1 6

120

1 5 1 5 1 5 1 5 1 5

140

1 7 1 7 1 7 1 7 1 7 1 7 1 7

160

1 6 1 6 1 6 1 6 1 6 1 6

180

1 6 1 6 1 6 1 6 1 6 1 6

200

1 5 1 5 1 5 1 5 1 5

220

1 7 1 7 1 7 1 7 1 7 1 7 1 7

240

1 6 1 6 1 6 1 6 1 6 1 6

260

1 7 1 7 1 7 1 7 1 7 1 7 1 7

Conditional probabilities p(Y | Xi )

– – Conditional means of Y 65

– – 89

– – 137

– 77

– 113

– 125

– 161

101

149

173

3

As shown in App. A, in general the conditional and unconditional mean values are different.

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say, $140,” we get the answer $101 (the conditional mean). To put it differently, if we ask the question, “What is the best (mean) prediction of weekly expenditure of families with a weekly income of $140,” the answer would be $101. Thus the knowledge of the income level may enable us to better predict the mean value of consumption expenditure than if we do not have that knowledge.4 This probably is the essence of regression analysis, as we shall discover throughout this text. The dark circled points in Figure 2.1 show the conditional mean values of Y against the various X values. If we join these conditional mean values, we obtain what is known as the population regression line (PRL), or more generally, the population regression curve.5 More simply, it is the regression of Y on X. The adjective “population” comes from the fact that we are dealing in this example with the entire population of 60 families. Of course, in reality a population may have many families. Geometrically, then, a population regression curve is simply the locus of the conditional means of the dependent variable for the ﬁxed values of the explanatory variable(s). More simply, it is the curve connecting the means of the subpopulations of Y corresponding to the given values of the regressor X. It can be depicted as in Figure 2.2.

Y Conditional mean Weekly consumption expenditure, $ E(Y | Xi)

149 101 65 Distribution of Y given X = $220

80

140

220

X

Weekly income, $ FIGURE 2.2 Population regression line (data of Table 2.1).

4 I am indebted to James Davidson on this perspective. See James Davidson, Econometric Theory, Blackwell Publishers, Oxford, U.K., 2000, p. 11. 5 In the present example the PRL is a straight line, but it could be a curve (see Figure 2.3).

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This ﬁgure shows that for each X (i.e., income level) there is a population of Y values (weekly consumption expenditures) that are spread around the (conditional) mean of those Y values. For simplicity, we are assuming that these Y values are distributed symmetrically around their respective (conditional) mean values. And the regression line (or curve) passes through these (conditional) mean values. With this background, the reader may ﬁnd it instructive to reread the deﬁnition of regression given in Section 1.2.

2.2 THE CONCEPT OF POPULATION REGRESSION FUNCTION (PRF)

From the preceding discussion and Figures. 2.1 and 2.2, it is clear that each conditional mean E(Y | Xi ) is a function of Xi, where Xi is a given value of X. Symbolically, E(Y | Xi ) = f (Xi ) (2.2.1)

where f (Xi ) denotes some function of the explanatory variable X. In our example, E(Y | Xi ) is a linear function of Xi. Equation (2.2.1) is known as the conditional expectation function (CEF) or population regression function (PRF) or population regression (PR) for short. It states merely that the expected value of the distribution of Y given Xi is functionally related to Xi. In simple terms, it tells how the mean or average response of Y varies with X. What form does the function f (Xi ) assume? This is an important question because in real situations we do not have the entire population available for examination. The functional form of the PRF is therefore an empirical question, although in speciﬁc cases theory may have something to say. For example, an economist might posit that consumption expenditure is linearly related to income. Therefore, as a ﬁrst approximation or a working hypothesis, we may assume that the PRF E(Y | Xi ) is a linear function of Xi, say, of the type E(Y | Xi ) = β1 + β2 Xi (2.2.2)

where β1 and β2 are unknown but ﬁxed parameters known as the regression coefﬁcients; β1 and β2 are also known as intercept and slope coefﬁcients, respectively. Equation (2.2.1) itself is known as the linear population regression function. Some alternative expressions used in the literature are linear population regression model or simply linear population regression. In the sequel, the terms regression, regression equation, and regression model will be used synonymously. In regression analysis our interest is in estimating the PRFs like (2.2.2), that is, estimating the values of the unknowns β1 and β2 on the basis of observations on Y and X. This topic will be studied in detail in Chapter 3.

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2.3

THE MEANING OF THE TERM LINEAR

Since this text is concerned primarily with linear models like (2.2.2), it is essential to know what the term linear really means, for it can be interpreted in two different ways.

Linearity in the Variables

The ﬁrst and perhaps more “natural” meaning of linearity is that the conditional expectation of Y is a linear function of Xi, such as, for example, (2.2.2).6 Geometrically, the regression curve in this case is a straight line. In this interpretation, a regression function such as E(Y | Xi ) = β1 + β2 Xi2 is not a linear function because the variable X appears with a power or index of 2.

Linearity in the Parameters

The second interpretation of linearity is that the conditional expectation of Y, E(Y | Xi ), is a linear function of the parameters, the β’s; it may or may not be linear in the variable X.7 In this interpretation E(Y | Xi ) = β1 + β2 Xi2 is a linear (in the parameter) regression model. To see this, let us suppose X takes the value 3. Therefore, E(Y | X = 3) = β1 + 9β2 , which is obviously linear in β1 and β2 . All the models shown in Figure 2.3 are thus linear regression models, that is, models linear in the parameters. 2 Now consider the model E(Y | Xi ) = β1 + β2 Xi . Now suppose X = 3; then 2 we obtain E(Y | Xi ) = β1 + 3β2 , which is nonlinear in the parameter β2 . The preceding model is an example of a nonlinear (in the parameter) regression model. We will discuss such models in Chapter 14. Of the two interpretations of linearity, linearity in the parameters is relevant for the development of the regression theory to be presented shortly. Therefore, from now on the term “linear” regression will always mean a regression that is linear in the parameters; the β’s (that is, the parameters are raised to the ﬁrst power only). It may or may not be linear in the explanatory variables, the X’s. Schematically, we have Table 2.3. Thus, E(Y | Xi ) = β1 + β2 Xi , which is linear both in the parameters and variable, is a LRM, and so is E(Y | Xi ) = β1 + β2 Xi2 , which is linear in the parameters but nonlinear in variable X.

6 A function Y = f (X) is said to be linear in X if X appears with a power or index of 1 only √ (that is, terms such as X2, X, and so on, are excluded) and is not multiplied or divided by any other variable (for example, X · Z or X/Z, where Z is another variable). If Y depends on X alone, another way to state that Y is linearly related to X is that the rate of change of Y with respect to X (i.e., the slope, or derivative, of Y with respect to X, dY/dX) is independent of the value of X. Thus, if Y = 4X, dY/dX = 4, which is independent of the value of X. But if Y = 4X 2 , dY/dX = 8X, which is not independent of the value taken by X. Hence this function is not linear in X. 7 A function is said to be linear in the parameter, say, β1, if β1 appears with a power of 1 only and is not multiplied or divided by any other parameter (for example, β1β2, β2 /β1 , and so on).

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Y

Y

Quadratic Y = β1 + β2 X + β 3 X2 X Y

Exponential Y = e β1+β 2 X X

Cubic Y = β1 + β2 X + β 3 X2 + β 4 X3 X FIGURE 2.3 Linear-in-parameter functions.

TABLE 2.3

LINEAR REGRESSION MODELS Model linear in parameters? Model linear in variables? Yes Yes No

Note:

No LRM NLRM

LRM NLRM

LRM = linear regression model NLRM = nonlinear regression model

2.4

STOCHASTIC SPECIFICATION OF PRF

It is clear from Figure 2.1 that, as family income increases, family consumption expenditure on the average increases, too. But what about the consumption expenditure of an individual family in relation to its (ﬁxed) level of income? It is obvious from Table 2.1 and Figure 2.1 that an individual family’s consumption expenditure does not necessarily increase as the income level increases. For example, from Table 2.1 we observe that corresponding to the income level of $100 there is one family whose consumption expenditure of $65 is less than the consumption expenditures of two families whose weekly income is only $80. But notice that the average consumption

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expenditure of families with a weekly income of $100 is greater than the average consumption expenditure of families with a weekly income of $80 ($77 versus $65). What, then, can we say about the relationship between an individual family’s consumption expenditure and a given level of income? We see from Figure 2.1 that, given the income level of Xi , an individual family’s consumption expenditure is clustered around the average consumption of all families at that Xi , that is, around its conditional expectation. Therefore, we can express the deviation of an individual Yi around its expected value as follows: ui = Yi − E(Y | Xi ) or Yi = E(Y | Xi ) + ui (2.4.1)

where the deviation ui is an unobservable random variable taking positive or negative values. Technically, ui is known as the stochastic disturbance or stochastic error term. How do we interpret (2.4.1)? We can say that the expenditure of an individual family, given its income level, can be expressed as the sum of two components: (1) E(Y | Xi ), which is simply the mean consumption expenditure of all the families with the same level of income. This component is known as the systematic, or deterministic, component, and (2) ui , which is the random, or nonsystematic, component. We shall examine shortly the nature of the stochastic disturbance term, but for the moment assume that it is a surrogate or proxy for all the omitted or neglected variables that may affect Y but are not (or cannot be) included in the regression model. If E(Y | Xi ) is assumed to be linear in Xi , as in (2.2.2), Eq. (2.4.1) may be written as Yi = E(Y | Xi ) + ui = β1 + β2 Xi + ui (2.4.2)

Equation (2.4.2) posits that the consumption expenditure of a family is linearly related to its income plus the disturbance term. Thus, the individual consumption expenditures, given X = $80 (see Table 2.1), can be expressed as Y1 = 55 = β1 + β2 (80) + u1 Y2 = 60 = β1 + β2 (80) + u2 Y3 = 65 = β1 + β2 (80) + u3 Y4 = 70 = β1 + β2 (80) + u4 Y5 = 75 = β1 + β2 (80) + u5 (2.4.3)

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Now if we take the expected value of (2.4.1) on both sides, we obtain E(Yi | Xi ) = E[E(Y | Xi )] + E(ui | Xi ) = E(Y | Xi ) + E(ui | Xi ) (2.4.4)

where use is made of the fact that the expected value of a constant is that constant itself.8 Notice carefully that in (2.4.4) we have taken the conditional expectation, conditional upon the given X’s. Since E(Yi | Xi ) is the same thing as E(Y | Xi ), Eq. (2.4.4) implies that E(ui | Xi ) = 0 (2.4.5)

Thus, the assumption that the regression line passes through the conditional means of Y (see Figure 2.2) implies that the conditional mean values of ui (conditional upon the given X’s) are zero. From the previous discussion, it is clear (2.2.2) and (2.4.2) are equivalent forms if E(ui | Xi ) = 0.9 But the stochastic speciﬁcation (2.4.2) has the advantage that it clearly shows that there are other variables besides income that affect consumption expenditure and that an individual family’s consumption expenditure cannot be fully explained only by the variable(s) included in the regression model.

2.5 THE SIGNIFICANCE OF THE STOCHASTIC DISTURBANCE TERM

As noted in Section 2.4, the disturbance term ui is a surrogate for all those variables that are omitted from the model but that collectively affect Y. The obvious question is: Why not introduce these variables into the model explicitly? Stated otherwise, why not develop a multiple regression model with as many variables as possible? The reasons are many. 1. Vagueness of theory: The theory, if any, determining the behavior of Y may be, and often is, incomplete. We might know for certain that weekly income X inﬂuences weekly consumption expenditure Y, but we might be ignorant or unsure about the other variables affecting Y. Therefore, ui may be used as a substitute for all the excluded or omitted variables from the model. 2. Unavailability of data: Even if we know what some of the excluded variables are and therefore consider a multiple regression rather than a simple regression, we may not have quantitative information about these

8 See App. A for a brief discussion of the properties of the expectation operator E. Note that E(Y | Xi ), once the value of Xi is ﬁxed, is a constant. 9 As a matter of fact, in the method of least squares to be developed in Chap. 3, it is assumed explicitly that E(ui | Xi ) = 0. See Sec. 3.2.

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variables. It is a common experience in empirical analysis that the data we would ideally like to have often are not available. For example, in principle we could introduce family wealth as an explanatory variable in addition to the income variable to explain family consumption expenditure. But unfortunately, information on family wealth generally is not available. Therefore, we may be forced to omit the wealth variable from our model despite its great theoretical relevance in explaining consumption expenditure. 3. Core variables versus peripheral variables: Assume in our consumptionincome example that besides income X1, the number of children per family X2, sex X3, religion X4, education X5, and geographical region X6 also affect consumption expenditure. But it is quite possible that the joint inﬂuence of all or some of these variables may be so small and at best nonsystematic or random that as a practical matter and for cost considerations it does not pay to introduce them into the model explicitly. One hopes that their combined effect can be treated as a random variable ui .10 4. Intrinsic randomness in human behavior: Even if we succeed in introducing all the relevant variables into the model, there is bound to be some “intrinsic” randomness in individual Y’s that cannot be explained no matter how hard we try. The disturbances, the u’s, may very well reﬂect this intrinsic randomness. 5. Poor proxy variables: Although the classical regression model (to be developed in Chapter 3) assumes that the variables Y and X are measured accurately, in practice the data may be plagued by errors of measurement. Consider, for example, Milton Friedman’s well-known theory of the consumption function.11 He regards permanent consumption (Y p ) as a function of permanent income (X p ). But since data on these variables are not directly observable, in practice we use proxy variables, such as current consumption (Y ) and current income (X), which can be observable. Since the observed Y and X may not equal Y p and X p , there is the problem of errors of measurement. The disturbance term u may in this case then also represent the errors of measurement. As we will see in a later chapter, if there are such errors of measurement, they can have serious implications for estimating the regression coefﬁcients, the β’s. 6. Principle of parsimony: Following Occam’s razor,12 we would like to keep our regression model as simple as possible. If we can explain the behavior of Y “substantially” with two or three explanatory variables and if

10 A further difﬁculty is that variables such as sex, education, and religion are difﬁcult to quantify. 11 Milton Friedman, A Theory of the Consumption Function, Princeton University Press, Princeton, N.J., 1957. 12 “That descriptions be kept as simple as possible until proved inadequate,” The World of Mathematics, vol. 2, J. R. Newman (ed.), Simon & Schuster, New York, 1956, p. 1247, or, “Entities should not be multiplied beyond necessity,” Donald F. Morrison, Applied Linear Statistical Methods, Prentice Hall, Englewood Cliffs, N.J., 1983, p. 58.

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our theory is not strong enough to suggest what other variables might be included, why introduce more variables? Let ui represent all other variables. Of course, we should not exclude relevant and important variables just to keep the regression model simple. 7. Wrong functional form: Even if we have theoretically correct variables explaining a phenomenon and even if we can obtain data on these variables, very often we do not know the form of the functional relationship between the regressand and the regressors. Is consumption expenditure a linear (invariable) function of income or a nonlinear (invariable) function? If it is the former, Yi = β1 + B2 Xi + ui is the proper functional relationship between Y and X, but if it is the latter, Yi = β1 + β2 Xi + β3 Xi2 + ui may be the correct functional form. In two-variable models the functional form of the relationship can often be judged from the scattergram. But in a multiple regression model, it is not easy to determine the appropriate functional form, for graphically we cannot visualize scattergrams in multiple dimensions. For all these reasons, the stochastic disturbances ui assume an extremely critical role in regression analysis, which we will see as we progress.

2.6

THE SAMPLE REGRESSION FUNCTION (SRF)

By conﬁning our discussion so far to the population of Y values corresponding to the ﬁxed X’s, we have deliberately avoided sampling considerations (note that the data of Table 2.1 represent the population, not a sample). But it is about time to face up to the sampling problems, for in most practical situations what we have is but a sample of Y values corresponding to some ﬁxed X’s. Therefore, our task now is to estimate the PRF on the basis of the sample information. As an illustration, pretend that the population of Table 2.1 was not known to us and the only information we had was a randomly selected sample of Y values for the ﬁxed X’s as given in Table 2.4. Unlike Table 2.1, we now have only one Y value corresponding to the given X’s; each Y (given Xi) in Table 2.4 is chosen randomly from similar Y’s corresponding to the same Xi from the population of Table 2.1. The question is: From the sample of Table 2.4 can we predict the average weekly consumption expenditure Y in the population as a whole corresponding to the chosen X’s? In other words, can we estimate the PRF from the sample data? As the reader surely suspects, we may not be able to estimate the PRF “accurately” because of sampling ﬂuctuations. To see this, suppose we draw another random sample from the population of Table 2.1, as presented in Table 2.5. Plotting the data of Tables 2.4 and 2.5, we obtain the scattergram given in Figure 2.4. In the scattergram two sample regression lines are drawn so as

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TABLE 2.4 A RANDOM SAMPLE FROM THE POPULATION OF TABLE 2.1 Y 70 65 90 95 110 115 120 140 155 150 X 80 100 120 140 160 180 200 220 240 260

TABLE 2.5 ANOTHER RANDOM SAMPLE FROM THE POPULATION OF TABLE 2.1 Y 55 88 90 80 118 120 145 135 145 175 X 80 100 120 140 160 180 200 220 240 260

200 × First sample (Table 2.4) Weekly consumption expenditure, $ Second sample (Table 2.5) 150 × × Regression based on the second sample

SRF2 × SRF1

× 100 × × 50 × ×

Regression based on the first sample

80

100

120

140

160

180

200

220

240

260

Weekly income, $ FIGURE 2.4 Regression lines based on two different samples.

to “ﬁt” the scatters reasonably well: SRF1 is based on the ﬁrst sample, and SRF2 is based on the second sample. Which of the two regression lines represents the “true” population regression line? If we avoid the temptation of looking at Figure 2.1, which purportedly represents the PR, there is no way we can be absolutely sure that either of the regression lines shown in Figure 2.4 represents the true population regression line (or curve). The regression lines in Figure 2.4 are known as the sample regression lines. Sup-

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posedly they represent the population regression line, but because of sampling ﬂuctuations they are at best an approximation of the true PR. In general, we would get N different SRFs for N different samples, and these SRFs are not likely to be the same. Now, analogously to the PRF that underlies the population regression line, we can develop the concept of the sample regression function (SRF) to represent the sample regression line. The sample counterpart of (2.2.2) may be written as ˆ ˆ ˆ Yi = β1 + β2 Xi ˆ where Y is read as “Y-hat’’ or “Y-cap’’ ˆi = estimator of E(Y | Xi ) Y ˆ β1 = estimator of β1 ˆ β2 = estimator of β2 Note that an estimator, also known as a (sample) statistic, is simply a rule or formula or method that tells how to estimate the population parameter from the information provided by the sample at hand. A particular numerical value obtained by the estimator in an application is known as an estimate.13 Now just as we expressed the PRF in two equivalent forms, (2.2.2) and (2.4.2), we can express the SRF (2.6.1) in its stochastic form as follows: ˆ ˆ Yi = β1 + β2 Xi + ui ˆ (2.6.2) (2.6.1)

ˆ where, in addition to the symbols already deﬁned, ui denotes the (sample) ui is analogous to ui and can be regarded as ˆ residual term. Conceptually an estimate of ui . It is introduced in the SRF for the same reasons as ui was introduced in the PRF. To sum up, then, we ﬁnd our primary objective in regression analysis is to estimate the PRF Yi = β1 + β2 Xi + ui on the basis of the SRF ˆ ˆ Yi = β1 + βxi = ui ˆ (2.4.2)

(2.6.2)

because more often than not our analysis is based upon a single sample from some population. But because of sampling ﬂuctuations our estimate of

13 As noted in the Introduction, a hat above a variable will signify an estimator of the relevant population value.

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Y Yi ui Yi ui Yi E(Y | Xi) E(Y | Xi)

SRF: Yi = β1 + β2 Xi

Yi Weekly consumption expenditure, $

PRF: E(Y | Xi) = β1 + β2 Xi

A

Xi Weekly income, $ FIGURE 2.5 Sample and population regression lines.

X

the PRF based on the SRF is at best an approximate one. This approximation is shown diagrammatically in Figure 2.5. For X = Xi , we have one (sample) observation Y = Yi . In terms of the SRF, the observed Yi can be expressed as ˆ Yi = Yi + ui ˆ and in terms of the PRF, it can be expressed as Yi = E(Y | Xi ) + ui (2.6.4) (2.6.3)

ˆ Now obviously in Figure 2.5 Yi overestimates the true E(Y | Xi ) for the Xi shown therein. By the same token, for any Xi to the left of the point A, the SRF will underestimate the true PRF. But the reader can readily see that such over- and underestimation is inevitable because of sampling ﬂuctuations. The critical question now is: Granted that the SRF is but an approximation of the PRF, can we devise a rule or a method that will make this approximation as “close” as possible? In other words, how should the SRF be ˆ ˆ constructed so that β1 is as “close” as possible to the true β1 and β2 is as “close” as possible to the true β2 even though we will never know the true β1 and β2 ?

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The answer to this question will occupy much of our attention in Chapter 3. We note here that we can develop procedures that tell us how to construct the SRF to mirror the PRF as faithfully as possible. It is fascinating to consider that this can be done even though we never actually determine the PRF itself.

2.7

AN ILLUSTRATIVE EXAMPLE

We conclude this chapter with an example. Table 2.6 gives data on the level of education (measured by the number of years of schooling), the mean hourly wages earned by people at each level of education, and the number of people at the stated level of education. Ernst Berndt originally obtained the data presented in the table, and he derived these data from the current population survey conducted in May 1985.14 We will explore these data (with additional explanatory variables) in Chapter 3 (Example 3.3, p. 91). Plotting the (conditional) mean wage against education, we obtain the picture in Figure 2.6. The regression curve in the ﬁgure shows how mean wages vary with the level of education; they generally increase with the level of education, a ﬁnding one should not ﬁnd surprising. We will study in a later chapter how variables besides education can also affect the mean wage.

TABLE 2.6 MEAN HOURLY WAGE BY EDUCATION Mean wage Years of schooling 6 7 8 9 10 11 12 13 14 15 16 17 18 Mean wage, $ 4.4567 5.7700 5.9787 7.3317 7.3182 6.5844 7.8182 7.8351 11.0223 10.6738 10.8361 13.6150 13.5310 Total Number of people 3 5 15 12 17 27 218 37 56 13 70 24 31 528 14

Mean value

12 10 8 6 4 6 8 10 12 14 Education 16 18

FIGURE 2.6 Relationship between mean wages and education.

Source: Arthur S. Goldberger, Introductory Econometrics, Harvard University Press, Cambridge, Mass., 1998, Table 1.1, p. 5 (adapted).

14 Ernst R. Berndt, The Practice of Econometrics: Classic and Contemporary, Addison Wesley, Reading, Mass., 1991. Incidentally, this is an excellent book that the reader may want to read to ﬁnd out how econometricians go about doing research.

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2.8

SUMMARY AND CONCLUSIONS

1. The key concept underlying regression analysis is the concept of the conditional expectation function (CEF), or population regression function (PRF). Our objective in regression analysis is to ﬁnd out how the average value of the dependent variable (or regressand) varies with the given value of the explanatory variable (or regressor). 2. This book largely deals with linear PRFs, that is, regressions that are linear in the parameters. They may or may not be linear in the regressand or the regressors. 3. For empirical purposes, it is the stochastic PRF that matters. The stochastic disturbance term ui plays a critical role in estimating the PRF. 4. The PRF is an idealized concept, since in practice one rarely has access to the entire population of interest. Usually, one has a sample of observations from the population. Therefore, one uses the stochastic sample regression function (SRF) to estimate the PRF. How this is actually accomplished is discussed in Chapter 3.

EXERCISES

Questions 2.1. What is the conditional expectation function or the population regression function? 2.2. What is the difference between the population and sample regression functions? Is this a distinction without difference? 2.3. What is the role of the stochastic error term ui in regression analysis? What ˆ is the difference between the stochastic error term and the residual, ui ? 2.4. Why do we need regression analysis? Why not simply use the mean value of the regressand as its best value? 2.5. What do we mean by a linear regression model? 2.6. Determine whether the following models are linear in the parameters, or the variables, or both. Which of these models are linear regression models? Model a. b. c. d. e.

1 + ui Yi = β1 + β2 Xi Yi = β1 + β2 ln X i + ui ln Yi = β1 + β2 X i + ui ln Yi = ln β1 + β2 ln X i + ui 1 + ui ln Yi = β1 − β2 Xi

Descriptive title Reciprocal Semilogarithmic Inverse semilogarithmic Logarithmic or double logarithmic Logarithmic reciprocal

Note: ln = natural log (i.e., log to the base e); ui is the stochastic disturbance term. We will study these models in Chapter 6. 2.7. Are the following models linear regression models? Why or why not? a. Yi = e β1 +β2 X i +ui b. Yi =

1 1 + e β1 +β2 X i +ui

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2.8. What is meant by an intrinsically linear regression model? If β2 in exercise 2.7d were 0.8, would it be a linear or nonlinear regression model? *2.9. Consider the following nonstochastic models (i.e., models without the stochastic error term). Are they linear regression models? If not, is it possible, by suitable algebraic manipulations, to convert them into linear models?

1 β1 + β2 X i Xi b. Yi = β1 + β2 X i

1 + ui Xi d. Yi = β1 + (0.75 − β1 )e −β2 ( X i −2) + ui 3 e. Yi = β1 + β2 X i + ui

c. ln Yi = β1 + β2

a. Yi =

c. Yi =

1 1 + exp(−β1 − β2 X i )

2.10. You are given the scattergram in Figure 2.7 along with the regression line. What general conclusion do you draw from this diagram? Is the regression line sketched in the diagram a population regression line or the sample regression line?

12 Average growth in real manufacturing wages, % per year 10 8 6 4 2 0 –2 –4 –6 –0.08 –0.06 –0.04 –0.02 0.00 0.02 0.04 0.06 0.08

Average annual change in export-GNP ratio

East Asia and the Pacific Latin America and the Caribbean Middle East and North Africa South Asia Sub-Saharan Africa

FIGURE 2.7

Growth rates of real manufacturing wages and exports. Data are for 50 developing countries during 1970–90.

Source: The World Bank, World Development Report 1995, p. 55. The original source is UNIDO data, World Bank data.

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More manufactures in exports More raw materials in exports

4 3 2 1 0 –1 –2 –3 –4 –5 –6 –7 0 1 2 3 4 5 6 7 8 9 10 11 12

Abundant land; less skilled workers

Regional averages East Asia and the Pacific Industrial market economies South Asia

Scarce land; more skilled workers

Latin-America and the Caribbean Sub-Saharan Africa

FIGURE 2.8

Skill intensity of exports and human capital endowment. Data are for 126 industrial and developing countries in 1985. Values along the horizontal axis are logarithms of the ratio of the country’s average educational attainment to its land area: vertical axis values are logarithms of the ratio of manufactured to primary-products exports.

Source: World Bank, World Development Report 1995, p. 59. Original sources: Export data from United Nations Statistical Ofﬁce COMTRADE data base; education data from UNDP 1990; land data from the World Bank.

2.11. From the scattergram given in Figure 2.8, what general conclusions do you draw? What is the economic theory that underlies this scattergram? (Hint: Look up any international economics textbook and read up on the Heckscher–Ohlin model of trade.) 2.12. What does the scattergram in Figure 2.9 reveal? On the basis of this diagram, would you argue that minimum wage laws are good for economic well-being? 2.13. Is the regression line shown in Figure I.3 of the Introduction the PRF or the SRF? Why? How would you interpret the scatterpoints around the regression line? Besides GDP, what other factors, or variables, might determine personal consumption expenditure? Problems 2.14. You are given the data in Table 2.7 for the United States for years 1980–1996.

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Ratio of one year’s salary at minimum wage to GNP per capita 1.8 1.6 1.4 1.2 1.0 0.8 0.6 0.4 0.2 0 1 5 6 7 8 2 3 4 GNP per capita (thousands of dollars) 9 10

FIGURE 2.9

The minimum wage and GNP per capita. The sample consists of 17 developing countries. Years vary by country from 1988 to 1992. Data are in international prices.

Source: World Bank, World Development Report 1995, p. 75.

TABLE 2.7

LABOR FORCE PARTICIPATION DATA Year 1980 1981 1982 1983 1984 1985 1986 1987 1988 1989 1990 1991 1992 1993 1994 1995 1996 CLFPRM1 77.4 77.0 76.6 76.4 76.4 76.3 76.3 76.2 76.2 76.4 76.4 75.8 75.8 75.4 75.1 75.0 74.9 CLFPRF2 51.5 52.1 52.6 53.9 53.6 54.5 55.3 56.0 56.6 57.4 57.5 57.4 57.8 57.9 58.8 58.9 59.3 UNRM3 6.9 7.4 9.9 9.9 7.4 7.0 6.9 6.2 5.5 5.2 5.7 7.2 7.9 7.2 6.2 5.6 5.4 UNRF4 7.4 7.9 9.4 9.2 7.6 7.4 7.1 6.2 5.6 5.4 5.5 6.4 7.0 6.6 6.0 5.6 5.4 AHE825 7.78 7.69 7.68 7.79 7.80 7.77 7.81 7.73 7.69 7.64 7.52 7.45 7.41 7.39 7.40 7.40 7.43 AHE6 6.66 7.25 7.68 8.02 8.32 8.57 8.76 8.98 9.28 9.66 10.01 10.32 10.57 10.83 11.12 11.44 11.82

Source: Economic Report of the President, 1997. Table citations below refer to the source document. 1 CLFPRM, Civilian labor force participation rate, male (%), Table B-37, p. 343. 2 CLFPRF, Civilian labor force participation rate, female (%), Table B-37, p. 343. 3 UNRM, Civilian unemployment rate, male (%) Table B-40, p. 346. 4 UNRF, Civilian unemployment rate, female (%) Table B-40, p. 346. 5 AHE82, Average hourly earnings (1982 dollars), Table B-45, p. 352. 6 AHE, Average hourly earnings (current dollars), Table B-45, p. 352.

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a. Plot the male civilian labor force participation rate against male civilian unemployment rate. Eyeball a regression line through the scatter points. A priori, what is the expected relationship between the two and what is the underlying economic theory? Does the scattergram support the theory? b. Repeat part a for females. c. Now plot both the male and female labor participation rates against average hourly earnings (in 1982 dollars). (You may use separate diagrams.) Now what do you ﬁnd? And how would you rationalize your ﬁnding? d. Can you plot the labor force participation rate against the unemployment rate and the average hourly earnings simultaneously? If not, how would you verbalize the relationship among the three variables? 2.15. Table 2.8 gives data on expenditure on food and total expenditure, measured in rupees, for a sample of 55 rural households from India. (In early 2000, a U.S. dollar was about 40 Indian rupees.)

TABLE 2.8

FOOD AND TOTAL EXPENDITURE (RUPEES) Observation 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 Food expenditure 217.0000 196.0000 303.0000 270.0000 325.0000 260.0000 300.0000 325.0000 336.0000 345.0000 325.0000 362.0000 315.0000 355.0000 325.0000 370.0000 390.0000 420.0000 410.0000 383.0000 315.0000 267.0000 420.0000 300.0000 410.0000 220.0000 403.0000 350.0000 Total expenditure 382.0000 388.0000 391.0000 415.0000 456.0000 460.0000 472.0000 478.0000 494.0000 516.0000 525.0000 554.0000 575.0000 579.0000 585.0000 586.0000 590.0000 608.0000 610.0000 616.0000 618.0000 623.0000 627.0000 630.0000 635.0000 640.0000 648.0000 650.0000 Observation 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 Food expenditure 390.0000 385.0000 470.0000 322.0000 540.0000 433.0000 295.0000 340.0000 500.0000 450.0000 415.0000 540.0000 360.0000 450.0000 395.0000 430.0000 332.0000 397.0000 446.0000 480.0000 352.0000 410.0000 380.0000 610.0000 530.0000 360.0000 305.0000 Total expenditure 655.0000 662.0000 663.0000 677.0000 680.0000 690.0000 695.0000 695.0000 695.0000 720.0000 721.0000 730.0000 731.0000 733.0000 745.0000 751.0000 752.0000 752.0000 769.0000 773.0000 773.0000 775.0000 785.0000 788.0000 790.0000 795.0000 801.0000

Source: Chandan Mukherjee, Howard White, and Marc Wuyts, Econometrics and Data Analysis for Developing Countries, Routledge, New York, 1998, p. 457.

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a. Plot the data, using the vertical axis for expenditure on food and the horizontal axis for total expenditure, and sketch a regression line through the scatterpoints. b. What broad conclusions can you draw from this example? c. A priori, would you expect expenditure on food to increase linearly as total expenditure increases regardless of the level of total expenditure? Why or why not? You can use total expenditure as a proxy for total income. 2.16. Table 2.9 gives data on mean Scholastic Aptitude Test (SAT) scores for college-bound seniors for 1967–1990. a. Use the horizontal axis for years and the vertical axis for SAT scores to plot the verbal and math scores for males and females separately. b. What general conclusions can you draw? c. Knowing the verbal scores of males and females, how would you go about predicting their math scores? d. Plot the female verbal SAT score against the male verbal SAT score. Sketch a regression line through the scatterpoints. What do you observe?

TABLE 2.9 MEAN SCHOLASTIC APTITUDE TEST SCORES FOR COLLEGE-BOUND SENIORS, 1967–1990* Verbal Year 1967 1968 1969 1970 1971 1972 1973 1974 1975 1976 1977 1978 1979 1980 1981 1982 1983 1984 1985 1986 1987 1988 1989 1990 Males 463 464 459 459 454 454 446 447 437 433 431 433 431 428 430 431 430 433 437 437 435 435 434 429 Females 468 466 466 461 457 452 443 442 431 430 427 425 423 420 418 421 420 420 425 426 425 422 421 419 Total 466 466 463 460 455 453 445 444 434 431 429 429 427 424 424 426 425 426 431 431 430 428 427 424 Males 514 512 513 509 507 505 502 501 495 497 497 494 493 491 492 493 493 495 499 501 500 498 500 499 Math Females 467 470 470 465 466 461 460 459 449 446 445 444 443 443 443 443 445 449 452 451 453 455 454 455 Total 492 492 493 488 488 484 481 480 472 472 470 468 467 466 466 467 468 471 475 475 476 476 476 476

*Data for 1967–1971 are estimates. Source: The College Board. The New York Times, Aug. 28, 1990, p. B-5.

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TWO-VARIABLE REGRESSION MODEL: THE PROBLEM OF ESTIMATION

As noted in Chapter 2, our ﬁrst task is to estimate the population regression function (PRF) on the basis of the sample regression function (SRF) as accurately as possible. In Appendix A we have discussed two generally used methods of estimation: (1) ordinary least squares (OLS) and (2) maximum likelihood (ML). By and large, it is the method of OLS that is used extensively in regression analysis primarily because it is intuitively appealing and mathematically much simpler than the method of maximum likelihood. Besides, as we will show later, in the linear regression context the two methods generally give similar results.

3.1

THE METHOD OF ORDINARY LEAST SQUARES

The method of ordinary least squares is attributed to Carl Friedrich Gauss, a German mathematician. Under certain assumptions (discussed in Section 3.2), the method of least squares has some very attractive statistical properties that have made it one of the most powerful and popular methods of regression analysis. To understand this method, we ﬁrst explain the leastsquares principle. Recall the two-variable PRF: Yi = β1 + β2 Xi + ui (2.4.2)

However, as we noted in Chapter 2, the PRF is not directly observable. We

58

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estimate it from the SRF: ˆ ˆ Yi = β1 + β2 Xi + ui ˆ ˆ = Yi + ui ˆ (2.6.2) (2.6.3)

ˆ where Yi is the estimated (conditional mean) value of Yi . But how is the SRF itself determined? To see this, let us proceed as follows. First, express (2.6.3) as ˆ ui = Yi − Yi ˆ ˆ ˆ = Yi − β1 − β2 Xi (3.1.1)

ˆ which shows that the ui (the residuals) are simply the differences between the actual and estimated Y values. Now given n pairs of observations on Y and X, we would like to determine the SRF in such a manner that it is as close as possible to the actual Y. To this end, we may adopt the following criterion: Choose the SRF in such a ˆ ˆ way that the sum of the residuals ui = (Yi − Yi ) is as small as possible. Although intuitively appealing, this is not a very good criterion, as can be seen in the hypothetical scattergram shown in Figure 3.1. ui , Figure 3.1 shows that the ˆ If we adopt the criterion of minimizing ˆ ˆ ˆ ˆ residuals u2 and u3 as well as the residuals u1 and u4 receive the same weight ˆ ˆ ˆ ˆ in the sum (u1 + u2 + u3 + u4 ), although the ﬁrst two residuals are much closer to the SRF than the latter two. In other words, all the residuals receive

Y SRF Yi u1 u3 u4 u2 Yi = β1 + β2 Xi

X1 FIGURE 3.1

X2

X3

X4

X

Least-squares criterion.

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equal importance no matter how close or how widely scattered the individual observations are from the SRF. A consequence of this is that it is quite ˆ possible that the algebraic sum of the ui is small (even zero) although the ui are widely scattered about the SRF. To see this, let u1 , u2 , u3 , and u4 in ˆ ˆ ˆ ˆ ˆ Figure 3.1 assume the values of 10, −2, +2, and −10, respectively. The algeˆ ˆ braic sum of these residuals is zero although u1 and u4 are scattered more ˆ ˆ widely around the SRF than u2 and u3 . We can avoid this problem if we adopt the least-squares criterion, which states that the SRF can be ﬁxed in such a way that ui = ˆ2 = ˆ (Yi − Yi )2 ˆ ˆ (Yi − β1 − β2 Xi )2 (3.1.2)

ˆ2 ˆ is as small as possible, where ui are the squared residuals. By squaring ui , ˆ ˆ this method gives more weight to residuals such as u1 and u4 in Figure 3.1 ˆ ˆ ˆ than the residuals u2 and u3 . As noted previously, under the minimum ui ˆ criterion, the sum can be small even though the ui are widely spread about the SRF. But this is not possible under the least-squares procedure, for the ˆ u2 . A further justiﬁcation ˆi larger the ui (in absolute value), the larger the for the least-squares method lies in the fact that the estimators obtained by it have some very desirable statistical properties, as we shall see shortly. It is obvious from (3.1.2) that ˆ ˆ u2 = f (β1 , β2 ) ˆi (3.1.3)

that is, the sum of the squared residuals is some function of the estimaˆ ˆ ˆ tors β1 and β2 . For any given set of data, choosing different values for β1 and ˆ β2 will give different u’s and hence different values of ˆ u2 . To see this ˆi clearly, consider the hypothetical data on Y and X given in the ﬁrst two columns of Table 3.1. Let us now conduct two experiments. In experiment 1,

TABLE 3.1

EXPERIMENTAL DETERMINATION OF THE SRF Yi (1) 4 5 7 12 Sum: 28 Xt (2) 1 4 5 6 16 ˆ Y1i (3) 2.929 7.000 8.357 9.714 û1i (4) 1.071 −2.000 −1.357 2.286 0.0

2 û1i (5)

ˆ Y2i (6) 4 7 8 9

û2i (7) 0 −2 −1 3 0

2 û2i (8)

1.147 4.000 1.841 5.226 12.214

0 4 1 9 14

ˆ ˆ ˆ Notes: Y 1i = 1.572 + 1.357Xi (i.e., β1 = 1.572 and β2 = 1.357) ˆ ˆ ˆ Y 2i = 3.0 + 1.0Xi (i.e., β1 = 3 and β2 = 1.0) ˆ û 1i = (Yi − Y1i) ˆ û 2i = (Yi − Y2i)

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ˆ ˆ let β1 = 1.572 and β2 = 1.357 (let us not worry right now about how we got ˆ these values; say, it is just a guess).1 Using these β values and the X values given in column (2) of Table 3.1, we can easily compute the estimated Yi ˆ given in column (3) of the table as Y1i (the subscript 1 is to denote the ﬁrst experiment). Now let us conduct another experiment, but this time using ˆ ˆ the values of β1 = 3 and β2 = 1. The estimated values of Yi from this experiˆ ˆ ment are given as Y2i in column (6) of Table 3.1. Since the β values in the two experiments are different, we get different values for the estimated ˆ residuals, as shown in the table; u1i are the residuals from the ﬁrst experiˆ ment and u2i from the second experiment. The squares of these residuals are given in columns (5) and (8). Obviously, as expected from (3.1.3), these residual sums of squares are different since they are based on different sets ˆ of β values. ˆ ˆ Now which sets of β values should we choose? Since the β values of the u2 (= 12.214) than that obtained from ˆi ﬁrst experiment give us a lower ˆ ˆ the β values of the second experiment (= 14), we might say that the β’s of the ﬁrst experiment are the “best” values. But how do we know? For, if we had inﬁnite time and inﬁnite patience, we could have conducted many more ˆ such experiments, choosing different sets of β’s each time and comparing the ˆ ˆi resulting u2 and then choosing that set of β values that gives us the least u2 assuming of course that we have considered all the ˆi possible value of conceivable values of β1 and β2 . But since time, and certainly patience, are generally in short supply, we need to consider some shortcuts to this trialand-error process. Fortunately, the method of least squares provides us such ˆ ˆ a shortcut. The principle or the method of least squares chooses β1 and β2 ˆi in such a manner that, for a given sample or set of data, u2 is as small as possible. In other words, for a given sample, the method of least squares provides us with unique estimates of β1 and β2 that give the smallest possiˆi ble value of u2 . How is this accomplished? This is a straight-forward exercise in differential calculus. As shown in Appendix 3A, Section 3A.1, the process of differentiation yields the following equations for estimating β1 and β2 : ˆ ˆ Yi = nβ1 + β2 Xi (3.1.4)

ˆ Yi Xi = β1

ˆ Xi + β2

Xi2

(3.1.5)

where n is the sample size. These simultaneous equations are known as the normal equations.

1 For the curious, these values are obtained by the method of least squares, discussed shortly. See Eqs. (3.1.6) and (3.1.7).

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Solving the normal equations simultaneously, we obtain n ˆ β2 = = = Xi Yi − n Xi2 − Xi Xi

2

Yi

¯ ¯ (Xi − X)(Yi − Y) ¯ (Xi − X)2 xi yi xi2

(3.1.6)

¯ ¯ where X and Y are the sample means of X and Y and where we deﬁne xi = ¯ ¯ (Xi − X) and yi = (Yi − Y). Henceforth we adopt the convention of letting the lowercase letters denote deviations from mean values. ˆ β1 = Xi2 n Yi − Xi2 − Xi Xi Xi Yi

2

(3.1.7)

¯ ˆ ¯ = Y − β2 X The last step in (3.1.7) can be obtained directly from (3.1.4) by simple algebraic manipulations. Incidentally, note that, by making use of simple algebraic identities, formula (3.1.6) for estimating β2 can be alternatively expressed as ˆ β2 = = = xi yi 2 xi Xi2 Xi2 xi Yi ¯ − nX 2 X i yi ¯ − nX 2 (3.1.8)2

The estimators obtained previously are known as the least-squares estimators, for they are derived from the least-squares principle. Note the following numerical properties of estimators obtained by the method of OLS: “Numerical properties are those that hold as a consequence of the use

2 2 ¯ ¯ ¯ ¯ ¯ ¯ X2 = Xi − 2 Xi X + Xi − 2 X Xi + X 2 , since X Note 1: xi2 = (Xi − X)2 = ¯ and ¯ 2 = nX 2 since X is a constant, we ﬁnally ¯ ¯ Xi = nX X is a constant. Further noting that 2 ¯ xi2 = Xi − nX 2 . get ¯ ¯ ¯ ¯ ¯ xi yi = xi (Yi − Y) = xi Yi − Y xi = xi Yi − Y (Xi − X) = xi Yi , since Y Note 2: ¯ is a constant and since the sum of deviations of a variable from its mean value [e.g., (Xi − X)] ¯ yi = (Yi − Y) = 0. is always zero. Likewise,

2

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of ordinary least squares, regardless of how the data were generated.”3 Shortly, we will also consider the statistical properties of OLS estimators, that is, properties “that hold only under certain assumptions about the way the data were generated.”4 (See the classical linear regression model in Section 3.2.) I. The OLS estimators are expressed solely in terms of the observable (i.e., sample) quantities (i.e., X and Y). Therefore, they can be easily computed. II. They are point estimators; that is, given the sample, each estimator will provide only a single (point) value of the relevant population parameter. (In Chapter 5 we will consider the so-called interval estimators, which provide a range of possible values for the unknown population parameters.) III. Once the OLS estimates are obtained from the sample data, the sample regression line (Figure 3.1) can be easily obtained. The regression line thus obtained has the following properties: 1. It passes through the sample means of Y and X. This fact is obvious ¯ ˆ ¯ ˆ from (3.1.7), for the latter can be written as Y = β1 + β2 X, which is shown diagrammatically in Figure 3.2.

Y Yi = β1 + β2 Xi SRF

Y

X X FIGURE 3.2 Diagram showing that the sample regression line passes through the sample mean values of Y and X.

3 Russell Davidson and James G. MacKinnon, Estimation and Inference in Econometrics, Oxford University Press, New York, 1993, p. 3. 4 Ibid.

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ˆ 2. The mean value of the estimated Y = Yi is equal to the mean value of the actual Y for ˆ ˆ ˆ Yi = β1 + β2 Xi ¯ ˆ ¯ ˆ = (Y − β2 X) + β2 Xi ¯ ¯ ˆ = Y + β2 (Xi − X) Summing both sides of this last equality over the sample values and dividing through by the sample size n gives ¯ ˆ ¯ Y=Y (3.1.10)5 (3.1.9)

¯ where use is made of the fact that (Xi − X) = 0. (Why?) ˆ 3. The mean value of the residuals ui is zero. From Appendix 3A, Section 3A.1, the ﬁrst equation is −2 ˆ ˆ (Yi − β1 − β2 Xi ) = 0

ˆ ˆ ˆ But since ui = Yi − β1 − β2 Xi , the preceding equation reduces to ¯ = 0.6 −2 ui = 0, whence u ˆ ˆ As a result of the preceding property, the sample regression ˆ ˆ Yi = β1 + β2 Xi + ui ˆ (2.6.2)

can be expressed in an alternative form where both Y and X are expressed as deviations from their mean values. To see this, sum (2.6.2) on both sides to give ˆ ˆ Yi = nβ1 + β2 ˆ ˆ = nβ1 + β2 Xi + Xi ui ˆ since ui = 0 ˆ (3.1.11)

Dividing Eq. (3.1.11) through by n, we obtain ¯ ˆ ˆ ¯ Y = β1 + β2 X (3.1.12)

which is the same as (3.1.7). Subtracting Eq. (3.1.12) from (2.6.2), we obtain ¯ ¯ ˆ Yi − Y = β2 (Xi − X) + ui ˆ

5 Note that this result is true only when the regression model has the intercept term β1 in it. As App. 6A, Sec. 6A.1 shows, this result need not hold when β1 is absent from the model. 6 This result also requires that the intercept term β1 be present in the model (see App. 6A, Sec. 6A.1).

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or ˆ yi = β2 xi + ui ˆ (3.1.13)

where yi and xi , following our convention, are deviations from their respective (sample) mean values. Equation (3.1.13) is known as the deviation form. Notice that the ˆ intercept term β1 is no longer present in it. But the intercept term can always be estimated by (3.1.7), that is, from the fact that the sample regression line passes through the sample means of Y and X. An advantage of the deviation form is that it often simpliﬁes computing formulas. In passing, note that in the deviation form, the SRF can be written as ˆ yi = β2 xi ˆ (3.1.14)

ˆ ˆ ˆ whereas in the original units of measurement it was Yi = β1 + β2 Xi , as shown in (2.6.1). ˆ 4. The residuals ui are uncorrelated with the predicted Yi . This statement can be veriﬁed as follows: using the deviation form, we can write ˆ yi ui = β2 ˆ ˆ ˆ = β2 ˆ = β2 ˆ2 = β2 =0 ˆ xi yi / xi2 . where use is made of the fact that β2 = ˆ ui Xi = 0. This ˆ 5. The residuals ui are uncorrelated with Xi ; that is, fact follows from Eq. (2) in Appendix 3A, Section 3A.1.

3.2 THE CLASSICAL LINEAR REGRESSION MODEL: THE ASSUMPTIONS UNDERLYING THE METHOD OF LEAST SQUARES

xi ui ˆ ˆ xi (yi − β2 xi ) ˆ2 xi yi − β2 ˆ2 xi2 − β2 xi2 xi2 (3.1.15)

If our objective is to estimate β1 and β2 only, the method of OLS discussed in the preceding section will sufﬁce. But recall from Chapter 2 that in regresˆ ˆ sion analysis our objective is not only to obtain β1 and β2 but also to draw inferences about the true β1 and β2 . For example, we would like to know how ˆ ˆ ˆ close β1 and β2 are to their counterparts in the population or how close Yi is to the true E(Y | Xi ). To that end, we must not only specify the functional form of the model, as in (2.4.2), but also make certain assumptions about

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the manner in which Yi are generated. To see why this requirement is needed, look at the PRF: Yi = β1 + β2 Xi + ui . It shows that Yi depends on both Xi and ui . Therefore, unless we are speciﬁc about how Xi and ui are created or generated, there is no way we can make any statistical inference about the Yi and also, as we shall see, about β1 and β2 . Thus, the assumptions made about the Xi variable(s) and the error term are extremely critical to the valid interpretation of the regression estimates. The Gaussian, standard, or classical linear regression model (CLRM), which is the cornerstone of most econometric theory, makes 10 assumptions.7 We ﬁrst discuss these assumptions in the context of the two-variable regression model; and in Chapter 7 we extend them to multiple regression models, that is, models in which there is more than one regressor.

Assumption 1: Linear regression model. The regression model is linear in the parameters, as shown in (2.4.2) Yi = β1 + β2Xi + ui (2.4.2)

We already discussed model (2.4.2) in Chapter 2. Since linear-in-parameter regression models are the starting point of the CLRM, we will maintain this assumption throughout this book. Keep in mind that the regressand Y and the regressor X themselves may be nonlinear, as discussed in Chapter 2.8

Assumption 2: X values are ﬁxed in repeated sampling. Values taken by the regressor X are considered ﬁxed in repeated samples. More technically, X is assumed to be nonstochastic.

This assumption is implicit in our discussion of the PRF in Chapter 2. But it is very important to understand the concept of “ﬁxed values in repeated sampling,” which can be explained in terms of our example given in Table 2.1. Consider the various Y populations corresponding to the levels of income shown in that table. Keeping the value of income X ﬁxed, say, at level $80, we draw at random a family and observe its weekly family consumption expenditure Y as, say, $60. Still keeping X at $80, we draw at random another family and observe its Y value as $75. In each of these drawings (i.e., repeated sampling), the value of X is ﬁxed at $80. We can repeat this process for all the X values shown in Table 2.1. As a matter of fact, the sample data shown in Tables 2.4 and 2.5 were drawn in this fashion. What all this means is that our regression analysis is conditional regression analysis, that is, conditional on the given values of the regressor(s) X.

7 It is classical in the sense that it was developed ﬁrst by Gauss in 1821 and since then has served as a norm or a standard against which may be compared the regression models that do not satisfy the Gaussian assumptions. 8 However, a brief discussion of nonlinear-in-the-parameter regression models is given in Chap. 14.

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Assumption 3: Zero mean value of disturbance ui. Given the value of X, the mean, or expected, value of the random disturbance term ui is zero. Technically, the conditional mean value of ui is zero. Symbolically, we have E(ui |Xi) = 0 (3.2.1)

Assumption 3 states that the mean value of ui , conditional upon the given Xi , is zero. Geometrically, this assumption can be pictured as in Figure 3.3, which shows a few values of the variable X and the Y populations associated with each of them. As shown, each Y population corresponding to a given X is distributed around its mean value (shown by the circled points on the PRF) with some Y values above the mean and some below it. The distances above and below the mean values are nothing but the ui , and what (3.2.1) requires is that the average or mean value of these deviations corresponding to any given X should be zero.9 This assumption should not be difﬁcult to comprehend in view of the discussion in Section 2.4 [see Eq. (2.4.5)]. All that this assumption says is that the factors not explicitly included in the model, and therefore subsumed in ui , do not systematically affect the mean value of Y; so to speak, the positive ui

Y Mean

PRF: Yi = β1 + β2 Xi

+ui –ui

X1 FIGURE 3.3

X2

X3

X4

X

Conditional distribution of the disturbances ui.

9 For illustration, we are assuming merely that the u’s are distributed symmetrically as shown in Figure 3.3. But shortly we will assume that the u’s are distributed normally.

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values cancel out the negative ui values so that their average or mean effect on Y is zero.10 In passing, note that the assumption E(ui | Xi ) = 0 implies that E(Yi | Xi ) = βi + β2 Xi . (Why?) Therefore, the two assumptions are equivalent.

Assumption 4: Homoscedasticity or equal variance of ui. Given the value of X, the variance of ui is the same for all observations. That is, the conditional variances of ui are identical. Symbolically, we have var (ui | Xi) = E [ui − E(ui | Xi)]2 = E (ui2 | Xi ) because of Assumption 3 = σ2 where var stands for variance. (3.2.2)

Eq. (3.2.2) states that the variance of ui for each Xi (i.e., the conditional variance of ui ) is some positive constant number equal to σ 2 . Technically, (3.2.2) represents the assumption of homoscedasticity, or equal (homo) spread (scedasticity) or equal variance. The word comes from the Greek verb skedanime, which means to disperse or scatter. Stated differently, (3.2.2) means that the Y populations corresponding to various X values have the same variance. Put simply, the variation around the regression line (which is the line of average relationship between Y and X) is the same across the X values; it neither increases or decreases as X varies. Diagrammatically, the situation is as depicted in Figure 3.4. f (u) Probability density of ui

Y

X1

X2

Xi

PRF: Yi = β 1 + β 2 Xi X

FIGURE 3.4

Homoscedasticity.

10 For a more technical reason why Assumption 3 is necessary see E. Malinvaud, Statistical Methods of Econometrics, Rand McNally, Chicago, 1966, p. 75. See also exercise 3.3.

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f(u) Probability density of ui

Y

X1 X2 Xi X FIGURE 3.5 Heteroscedasticity.

β 1 + β 2 Xi b b

In contrast, consider Figure 3.5, where the conditional variance of the Y population varies with X. This situation is known appropriately as heteroscedasticity, or unequal spread, or variance. Symbolically, in this situation (3.2.2) can be written as var (ui | Xi ) = σi2 (3.2.3)

Notice the subscript on σ 2 in Eq. (3.2.3), which indicates that the variance of the Y population is no longer constant. To make the difference between the two situations clear, let Y represent weekly consumption expenditure and X weekly income. Figures 3.4 and 3.5 show that as income increases the average consumption expenditure also increases. But in Figure 3.4 the variance of consumption expenditure remains the same at all levels of income, whereas in Figure 3.5 it increases with increase in income. In other words, richer families on the average consume more than poorer families, but there is also more variability in the consumption expenditure of the former. To understand the rationale behind this assumption, refer to Figure 3.5. As this ﬁgure shows, var (u| X1 ) < var (u| X2 ), . . . , < var (u| Xi ). Therefore, the likelihood is that the Y observations coming from the population with X = X1 would be closer to the PRF than those coming from populations corresponding to X = X2 , X = X3 , and so on. In short, not all Y values corresponding to the various X’s will be equally reliable, reliability being judged by how closely or distantly the Y values are distributed around their means, that is, the points on the PRF. If this is in fact the case, would we not prefer to sample from those Y populations that are closer to their mean than those that are widely spread? But doing so might restrict the variation we obtain across X values.

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By invoking Assumption 4, we are saying that at this stage all Y values corresponding to the various X’s are equally important. In Chapter 11 we shall see what happens if this is not the case, that is, where there is heteroscedasticity. In passing, note that Assumption 4 implies that the conditional variances of Yi are also homoscedastic. That is, var (Yi | Xi ) = σ 2 (3.2.4)

2 Of course, the unconditional variance of Y is σY . Later we will see the importance of distinguishing between conditional and unconditional variances of Y (see Appendix A for details of conditional and unconditional variances).

Assumption 5: No autocorrelation between the disturbances. Given any two X values, Xi and Xj (i = j), the correlation between any two ui and uj (i = j) is zero. Symbolically, cov (ui, uj | Xi, Xj ) = E {[ui − E (ui)] | Xi }{[uj − E(uj)] | Xj } = E(ui | Xi)(uj | Xj) =0 where i and j are two different observations and where cov means covariance. (why?) (3.2.5)

In words, (3.2.5) postulates that the disturbances ui and uj are uncorrelated. Technically, this is the assumption of no serial correlation, or no autocorrelation. This means that, given Xi , the deviations of any two Y values from their mean value do not exhibit patterns such as those shown in Figure 3.6a and b. In Figure 3.6a, we see that the u’s are positively correlated, a positive u followed by a positive u or a negative u followed by a negative u. In Figure 3.6b, the u’s are negatively correlated, a positive u followed by a negative u and vice versa. If the disturbances (deviations) follow systematic patterns, such as those shown in Figure 3.6a and b, there is auto- or serial correlation, and what Assumption 5 requires is that such correlations be absent. Figure 3.6c shows that there is no systematic pattern to the u’s, thus indicating zero correlation. The full import of this assumption will be explained thoroughly in Chapter 12. But intuitively one can explain this assumption as follows. Suppose in our PRF (Yt = β1 + β2 Xt + ut ) that ut and ut−1 are positively correlated. Then Yt depends not only on Xt but also on ut−1 for ut−1 to some extent determines ut . At this stage of the development of the subject matter, by invoking Assumption 5, we are saying that we will consider the systematic effect, if any, of Xt on Yt and not worry about the other inﬂuences that might act on Y as a result of the possible intercorrelations among the u’s. But, as noted in Chapter 12, we will see how intercorrelations among the disturbances can be brought into the analysis and with what consequences.

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+ui

+ui

–ui

+ui

–ui

+ui

–ui (a) +ui

–ui (b)

–ui

+ui

–ui (c) FIGURE 3.6 Patterns of correlation among the disturbances. (a) positive serial correlation; (b) negative serial correlation; (c) zero correlation.

Assumption 6: Zero covariance between ui and Xi , or E(uiXi) = 0. Formally, cov (ui, Xi) = E [ui − E(ui)][Xi − E(Xi)] = E [ui (Xi − E(Xi))] = E (ui Xi) − E(Xi)E(ui) = E(ui Xi) =0 since E(ui) = 0 since E(Xi) is nonstochastic (3.2.6)

since E(ui) = 0

by assumption

Assumption 6 states that the disturbance u and explanatory variable X are uncorrelated. The rationale for this assumption is as follows: When we expressed the PRF as in (2.4.2), we assumed that X and u (which may represent the inﬂuence of all the omitted variables) have separate (and additive) inﬂuence on Y. But if X and u are correlated, it is not possible to assess their individual effects on Y. Thus, if X and u are positively correlated, X increases

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when u increases and it decreases when u decreases. Similarly, if X and u are negatively correlated, X increases when u decreases and it decreases when u increases. In either case, it is difﬁcult to isolate the inﬂuence of X and u on Y. Assumption 6 is automatically fulﬁlled if X variable is nonrandom or nonstochastic and Assumption 3 holds, for in that case, cov (u i , Xi ) = [Xi − E(Xi )]E[ui − E(ui )] = 0. (Why?) But since we have assumed that our X variable not only is nonstochastic but also assumes ﬁxed values in repeated samples,11 Assumption 6 is not very critical for us; it is stated here merely to point out that the regression theory presented in the sequel holds true even if the X ’s are stochastic or random, provided they are independent or at least uncorrelated with the disturbances ui .12 (We shall examine the consequences of relaxing Assumption 6 in Part II.)

Assumption 7: The number of observations n must be greater than the number of parameters to be estimated. Alternatively, the number of observations n must be greater than the number of explanatory variables.

This assumption is not so innocuous as it seems. In the hypothetical example of Table 3.1, imagine that we had only the ﬁrst pair of observations on Y and X (4 and 1). From this single observation there is no way to estimate the two unknowns, β1 and β2 . We need at least two pairs of observations to estimate the two unknowns. In a later chapter we will see the critical importance of this assumption.

Assumption 8: Variability in X values. The X values in a given sample must not all be the same. Technically, var (X ) must be a ﬁnite positive number.13

This assumption too is not so innocuous as it looks. Look at Eq. (3.1.6). ¯ If all the X values are identical, then Xi = X (Why?) and the denominator of that equation will be zero, making it impossible to estimate β2 and therefore β1 . Intuitively, we readily see why this assumption is important. Looking at

11 Recall that in obtaining the samples shown in Tables 2.4 and 2.5, we kept the same X values. 12 As we will discuss in Part II, if the X ’s are stochastic but distributed independently of ui , the properties of least estimators discussed shortly continue to hold, but if the stochastic X’s are merely uncorrelated with ui , the properties of OLS estimators hold true only if the sample size is very large. At this stage, however, there is no need to get bogged down with this theoretical point. 13 The sample variance of X is

var (X) = where n is sample size.

¯ (Xi − X)2 n− 1

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our family consumption expenditure example in Chapter 2, if there is very little variation in family income, we will not be able to explain much of the variation in the consumption expenditure. The reader should keep in mind that variation in both Y and X is essential to use regression analysis as a research tool. In short, the variables must vary!

Assumption 9: The regression model is correctly speciﬁed. Alternatively, there is no speciﬁcation bias or error in the model used in empirical analysis.

As we discussed in the Introduction, the classical econometric methodology assumes implicitly, if not explicitly, that the model used to test an economic theory is “correctly speciﬁed.” This assumption can be explained informally as follows. An econometric investigation begins with the speciﬁcation of the econometric model underlying the phenomenon of interest. Some important questions that arise in the speciﬁcation of the model include the following: (1) What variables should be included in the model? (2) What is the functional form of the model? Is it linear in the parameters, the variables, or both? (3) What are the probabilistic assumptions made about the Yi , the Xi , and the ui entering the model? These are extremely important questions, for, as we will show in Chapter 13, by omitting important variables from the model, or by choosing the wrong functional form, or by making wrong stochastic assumptions about the variables of the model, the validity of interpreting the estimated regression will be highly questionable. To get an intuitive feeling about this, refer to the Phillips curve shown in Figure 1.3. Suppose we choose the following two models to depict the underlying relationship between the rate of change of money wages and the unemployment rate: Yi = α1 + α2 Xi + ui Yi = β1 + β2 1 Xi + ui (3.2.7) (3.2.8)

where Yi = the rate of change of money wages, and Xi = the unemployment rate. The regression model (3.2.7) is linear both in the parameters and the variables, whereas (3.2.8) is linear in the parameters (hence a linear regression model by our deﬁnition) but nonlinear in the variable X. Now consider Figure 3.7. If model (3.2.8) is the “correct” or the “true” model, ﬁtting the model (3.2.7) to the scatterpoints shown in Figure 3.7 will give us wrong predictions: Between points A and B, for any given Xi the model (3.2.7) is going to overestimate the true mean value of Y, whereas to the left of A (or to the right of B) it is going to underestimate (or overestimate, in absolute terms) the true mean value of Y.

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Rate of change of money wages

Yi = b1 + b2 1 β β Xi

( )

A

Yi = α 1 + α 2 Xi b b

0 B

Unemployment rate, %

FIGURE 3.7

Linear and nonlinear Phillips curves.

The preceding example is an instance of what is called a speciﬁcation bias or a speciﬁcation error; here the bias consists in choosing the wrong functional form. We will see other types of speciﬁcation errors in Chapter 13. Unfortunately, in practice one rarely knows the correct variables to include in the model or the correct functional form of the model or the correct probabilistic assumptions about the variables entering the model for the theory underlying the particular investigation (e.g., the Phillips-type money wage change–unemployment rate tradeoff) may not be strong or robust enough to answer all these questions. Therefore, in practice, the econometrician has to use some judgment in choosing the number of variables entering the model and the functional form of the model and has to make some assumptions about the stochastic nature of the variables included in the model. To some extent, there is some trial and error involved in choosing the “right” model for empirical analysis.14 If judgment is required in selecting a model, what is the need for Assumption 9? Without going into details here (see Chapter 13), this assumption is there to remind us that our regression analysis and therefore the results based on that analysis are conditional upon the chosen model and to warn us that we should give very careful thought in formulating econometric

14 But one should avoid what is known as “data mining,” that is, trying every possible model with the hope that at least one will ﬁt the data well. That is why it is essential that there be some economic reasoning underlying the chosen model and that any modiﬁcations in the model should have some economic justiﬁcation. A purely ad hoc model may be difﬁcult to justify on theoretical or a priori grounds. In short, theory should be the basis of estimation. But we will have more to say about data mining in Chap. 13, for there are some who argue that in some situations data mining can serve a useful purpose.

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models, especially when there may be several competing theories trying to explain an economic phenomenon, such as the inﬂation rate, or the demand for money, or the determination of the appropriate or equilibrium value of a stock or a bond. Thus, econometric model-building, as we shall discover, is more often an art rather than a science. Our discussion of the assumptions underlying the classical linear regression model is now completed. It is important to note that all these assumptions pertain to the PRF only and not the SRF. But it is interesting to observe that the method of least squares discussed previously has some properties that are similar to the assumptions we have made about the PRF. For ¯ ui = 0, and, therefore, u = 0, is akin to the asˆ ˆ example, the ﬁnding that ˆ sumption that E(ui | Xi ) = 0. Likewise, the ﬁnding that ui Xi = 0 is similar to the assumption that cov (ui , Xi ) = 0. It is comforting to note that the method of least squares thus tries to “duplicate” some of the assumptions we have imposed on the PRF. Of course, the SRF does not duplicate all the assumptions of the CLRM. As we will show later, although cov (ui , uj ) = 0 (i = j) by assumption, it is ˆ ˆ not true that the sample cov (ui , uj ) = 0 (i = j). As a matter of fact, we will show later that the residuals not only are autocorrelated but also are heteroscedastic (see Chapter 12). When we go beyond the two-variable model and consider multiple regression models, that is, models containing several regressors, we add the following assumption.

Assumption 10: There is no perfect multicollinearity. That is, there are no perfect linear relationships among the explanatory variables.

We will discuss this assumption in Chapter 7, where we discuss multiple regression models.

A Word about These Assumptions

The million-dollar question is: How realistic are all these assumptions? The “reality of assumptions” is an age-old question in the philosophy of science. Some argue that it does not matter whether the assumptions are realistic. What matters are the predictions based on those assumptions. Notable among the “irrelevance-of-assumptions thesis” is Milton Friedman. To him, unreality of assumptions is a positive advantage: “to be important . . . a hypothesis must be descriptively false in its assumptions.”15 One may not subscribe to this viewpoint fully, but recall that in any scientiﬁc study we make certain assumptions because they facilitate the

15 Milton Friedman, Essays in Positive Economics, University of Chicago Press, Chicago, 1953, p. 14.

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development of the subject matter in gradual steps, not because they are necessarily realistic in the sense that they replicate reality exactly. As one author notes, “. . . if simplicity is a desirable criterion of good theory, all good theories idealize and oversimplify outrageously.”16 What we plan to do is ﬁrst study the properties of the CLRM thoroughly, and then in later chapters examine in depth what happens if one or more of the assumptions of CLRM are not fulﬁlled. At the end of this chapter, we provide in Table 3.4 a guide to where one can ﬁnd out what happens to the CLRM if a particular assumption is not satisﬁed. As a colleague pointed out to me, when we review research done by others, we need to consider whether the assumptions made by the researcher are appropriate to the data and problem. All too often, published research is based on implicit assumptions about problem and data that are likely not correct and that produce estimates based on these assumptions. Clearly, the knowledgeable reader should, realizing these problems, adopt a skeptical attitude toward the research. The assumptions listed in Table 3.4 therefore provide a checklist for guiding our research and for evaluating the research of others. With this backdrop, we are now ready to study the CLRM. In particular, we want to ﬁnd out the statistical properties of OLS compared with the purely numerical properties discussed earlier. The statistical properties of OLS are based on the assumptions of CLRM already discussed and are enshrined in the famous Gauss–Markov theorem. But before we turn to this theorem, which provides the theoretical justiﬁcation for the popularity of OLS, we ﬁrst need to consider the precision or standard errors of the least-squares estimates.

3.3 PRECISION OR STANDARD ERRORS OF LEAST-SQUARES ESTIMATES

From Eqs. (3.1.6) and (3.1.7), it is evident that least-squares estimates are a function of the sample data. But since the data are likely to change from sample to sample, the estimates will change ipso facto. Therefore, what is ˆ needed is some measure of “reliability” or precision of the estimators β1 ˆ and β2 . In statistics the precision of an estimate is measured by its standard error (se).17 Given the Gaussian assumptions, it is shown in Appendix 3A, Section 3A.3 that the standard errors of the OLS estimates can be obtained

16 Mark Blaug, The Methodology of Economics: Or How Economists Explain, 2d ed., Cambridge University Press, New York, 1992, p. 92. 17 The standard error is nothing but the standard deviation of the sampling distribution of the estimator, and the sampling distribution of an estimator is simply a probability or frequency distribution of the estimator, that is, a distribution of the set of values of the estimator obtained from all possible samples of the same size from a given population. Sampling distributions are used to draw inferences about the values of the population parameters on the basis of the values of the estimators calculated from one or more samples. (For details, see App. A.)

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as follows: σ2 ˆ var (β2 ) = xi2 σ ˆ se (β2 ) = xi2 ˆ var (β1 ) = ˆ se (β1 ) = Xi2 2 σ n xi2 Xi2 σ n xi2 (3.3.1) (3.3.2)

(3.3.3)

(3.3.4)

where var = variance and se = standard error and where σ 2 is the constant or homoscedastic variance of ui of Assumption 4. All the quantities entering into the preceding equations except σ 2 can be estimated from the data. As shown in Appendix 3A, Section 3A.5, σ 2 itself is estimated by the following formula: σ2 = ˆ u2 ˆi n− 2 (3.3.5)

ˆ where σ 2 is the OLS estimator of the true but unknown σ 2 and where the ˆi expression n − 2 is known as the number of degrees of freedom (df), u2 being the sum of the residuals squared or the residual sum of squares (RSS).18 ˆi ˆ ˆi Once u2 is known, σ 2 can be easily computed. u2 itself can be computed either from (3.1.2) or from the following expression (see Section 3.5 for the proof): u2 = ˆi ˆ2 yi2 − β2 xi2 (3.3.6)

Compared with Eq. (3.1.2), Eq. (3.3.6) is easy to use, for it does not require ˆ computing ui for each observation although such a computation will be useful in its own right (as we shall see in Chapters 11 and 12). Since xi yi ˆ β2 = xi2

18 The term number of degrees of freedom means the total number of observations in the sample (= n) less the number of independent (linear) constraints or restrictions put on them. In other words, it is the number of independent observations out of a total of n observations. ˆ ˆ For example, before the RSS (3.1.2) can be computed, β1 and β2 must ﬁrst be obtained. These two estimates therefore put two restrictions on the RSS. Therefore, there are n − 2, not n, independent observations to compute the RSS. Following this logic, in the three-variable regression RSS will have n − 3 df, and for the k-variable model it will have n − k df. The general rule is this: df = (n− number of parameters estimated).

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an alternative expression for computing

u2 is ˆi xi yi xi2

2

u2 = ˆi

yi2 −

(3.3.7)

ˆ In passing, note that the positive square root of σ 2 u2 ˆi n− 2

σ = ˆ

(3.3.8)

is known as the standard error of estimate or the standard error of the regression (se). It is simply the standard deviation of the Y values about the estimated regression line and is often used as a summary measure of the “goodness of ﬁt” of the estimated regression line, a topic discussed in Section 3.5. Earlier we noted that, given Xi , σ 2 represents the (conditional) variance of both ui and Yi . Therefore, the standard error of the estimate can also be called the (conditional) standard deviation of ui and Yi . Of course, as usual, 2 σY and σY represent, respectively, the unconditional variance and unconditional standard deviation of Y. Note the following features of the variances (and therefore the standard ˆ ˆ errors) of β1 and β2 . ˆ 1. The variance of β2 is directly proportional to σ 2 but inversely propor2 tional to xi . That is, given σ 2 , the larger the variation in the X values, the ˆ smaller the variance of β2 and hence the greater the precision with which β2 can be estimated. In short, given σ 2 , if there is substantial variation in the X values (recall Assumption 8), β2 can be measured more accurately than xi2 , the larger the variwhen the Xi do not vary substantially. Also, given 2 ance of σ , the larger the variance of β2 . Note that as the sample size n xi2 , will increase. As n inincreases, the number of terms in the sum, creases, the precision with which β2 can be estimated also increases. (Why?) ˆ Xi2 but in2. The variance of β1 is directly proportional to σ 2 and 2 xi and the sample size n. versely proportional to ˆ ˆ 3. Since β1 and β2 are estimators, they will not only vary from sample to sample but in a given sample they are likely to be dependent on each other, this dependence being measured by the covariance between them. It is shown in Appendix 3A, Section 3A.4 that ¯ ˆ ˆ ˆ cov (β1 , β2 ) = − X var (β2 ) ¯ = −X σ2 xi2 (3.3.9)

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ˆ Since var (β2 ) is always positive, as is the variance of any variable, the nature ¯ ¯ ˆ ˆ of the covariance between β1 and β2 depends on the sign of X. If X is positive, then as the formula shows, the covariance will be negative. Thus, if the slope coefﬁcient β2 is overestimated (i.e., the slope is too steep), the intercept coefﬁcient β1 will be underestimated (i.e., the intercept will be too small). Later on (especially in the chapter on multicollinearity, Chapter 10), we will see the utility of studying the covariances between the estimated regression coefﬁcients. How do the variances and standard errors of the estimated regression coefﬁcients enable one to judge the reliability of these estimates? This is a problem in statistical inference, and it will be pursued in Chapters 4 and 5.

3.4 PROPERTIES OF LEAST-SQUARES ESTIMATORS: THE GAUSS–MARKOV THEOREM19

As noted earlier, given the assumptions of the classical linear regression model, the least-squares estimates possess some ideal or optimum properties. These properties are contained in the well-known Gauss–Markov theorem. To understand this theorem, we need to consider the best linear unbiasedness property of an estimator.20 As explained in Appendix A, an ˆ estimator, say the OLS estimator β2 , is said to be a best linear unbiased estimator (BLUE) of β2 if the following hold: 1. It is linear, that is, a linear function of a random variable, such as the dependent variable Y in the regression model. ˆ 2. It is unbiased, that is, its average or expected value, E(β2 ), is equal to the true value, β2 . 3. It has minimum variance in the class of all such linear unbiased estimators; an unbiased estimator with the least variance is known as an efﬁcient estimator. In the regression context it can be proved that the OLS estimators are BLUE. This is the gist of the famous Gauss–Markov theorem, which can be stated as follows:

Gauss–Markov Theorem: Given the assumptions of the classical linear regression model, the least-squares estimators, in the class of unbiased linear estimators, have minimum variance, that is, they are BLUE.

The proof of this theorem is sketched in Appendix 3A, Section 3A.6. The full import of the Gauss–Markov theorem will become clearer as we move

19 Although known as the Gauss–Markov theorem, the least-squares approach of Gauss antedates (1821) the minimum-variance approach of Markov (1900). 20 The reader should refer to App. A for the importance of linear estimators as well as for a general discussion of the desirable properties of statistical estimators.

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E(β2 ) = β 2 β (a) Sampling distribution of β 2

β2

E(β*) = β 2 β2 (b) Sampling distribution of β* 2

β* 2

β2

β* 2 β2 (c) Sampling distributions of β 2 and β* b2 b2 β 2, β* 2

FIGURE 3.8

Sampling distribution of OLS estimator ∗ ˆ β 2 and alternative estimator β 2.

along. It is sufﬁcient to note here that the theorem has theoretical as well as practical importance.21 What all this means can be explained with the aid of Figure 3.8. In Figure 3.8(a) we have shown the sampling distribution of the OLS ˆ ˆ estimator β2 , that is, the distribution of the values taken by β2 in repeated sampling experiments (recall Table 3.1). For convenience we have assumed ˆ β2 to be distributed symmetrically (but more on this in Chapter 4). As the ˆ ˆ ﬁgure shows, the mean of the β2 values, E(β2 ), is equal to the true β2 . In this ˆ2 is an unbiased estimator of β2 . In Figure 3.8(b) we situation we say that β ∗ have shown the sampling distribution of β2 , an alternative estimator of β2

21 For example, it can be proved that any linear combination of the β’s, such as (β1 − 2β2 ), ˆ ˆ can be estimated by (β1 − 2β2 ), and this estimator is BLUE. For details, see Henri Theil, Introduction to Econometrics, Prentice-Hall, Englewood Cliffs, N.J., 1978, pp. 401–402. Note a technical point about the Gauss–Markov theorem: It provides only the sufﬁcient (but not necessary) condition for OLS to be efﬁcient. I am indebted to Michael McAleer of the University of Western Australia for bringing this point to my attention.

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obtained by using another (i.e., other than OLS) method. For convenience, * ˆ assume that β2 , like β2 , is unbiased, that is, its average or expected value is * ˆ equal to β2 . Assume further that both β2 and β2 are linear estimators, that is, * ˆ they are linear functions of Y. Which estimator, β2 or β2 , would you choose? To answer this question, superimpose the two ﬁgures, as in Figure 3.8(c). * * ˆ It is obvious that although both β2 and β2 are unbiased the distribution of β2 is more diffused or widespread around the mean value than the distribution * ˆ ˆ of β2 . In other words, the variance of β2 is larger than the variance of β2 . Now given two estimators that are both linear and unbiased, one would choose the estimator with the smaller variance because it is more likely to be close to β2 than the alternative estimator. In short, one would choose the BLUE estimator. The Gauss–Markov theorem is remarkable in that it makes no assumptions about the probability distribution of the random variable ui , and therefore of Yi (in the next chapter we will take this up). As long as the assumptions of CLRM are satisﬁed, the theorem holds. As a result, we need not look for another linear unbiased estimator, for we will not ﬁnd such an estimator whose variance is smaller than the OLS estimator. Of course, if one or more of these assumptions do not hold, the theorem is invalid. For example, if we consider nonlinear-in-the-parameter regression models (which are discussed in Chapter 14), we may be able to obtain estimators that may perform better than the OLS estimators. Also, as we will show in the chapter on heteroscedasticity, if the assumption of homoscedastic variance is not fulﬁlled, the OLS estimators, although unbiased and consistent, are no longer minimum variance estimators even in the class of linear estimators. The statistical properties that we have just discussed are known as ﬁnite sample properties: These properties hold regardless of the sample size on which the estimators are based. Later we will have occasions to consider the asymptotic properties, that is, properties that hold only if the sample size is very large (technically, inﬁnite). A general discussion of ﬁnite-sample and large-sample properties of estimators is given in Appendix A.

3.5 THE COEFFICIENT OF DETERMINATION r 2: A MEASURE OF “GOODNESS OF FIT”

Thus far we were concerned with the problem of estimating regression coefﬁcients, their standard errors, and some of their properties. We now consider the goodness of ﬁt of the ﬁtted regression line to a set of data; that is, we shall ﬁnd out how “well” the sample regression line ﬁts the data. From Figure 3.1 it is clear that if all the observations were to lie on the regression line, we would obtain a “perfect” ﬁt, but this is rarely the case. Generally, there will ˆ ˆ be some positive ui and some negative ui . What we hope for is that these residuals around the regression line are as small as possible. The coefﬁcient of determination r 2 (two-variable case) or R2 (multiple regression) is a summary measure that tells how well the sample regression line ﬁts the data.

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Y

X

Y

X

Y

X

(a)

(b)

(c)

Y

X Y=X Y (d) (e) X (f)

FIGURE 3.9

The Ballentine view of r 2: (a) r 2 = 0; (f ) r 2 = 1.

Before we show how r 2 is computed, let us consider a heuristic explanation of r 2 in terms of a graphical device, known as the Venn diagram, or the Ballentine, as shown in Figure 3.9.22 In this ﬁgure the circle Y represents variation in the dependent variable Y and the circle X represents variation in the explanatory variable X.23 The overlap of the two circles (the shaded area) indicates the extent to which the variation in Y is explained by the variation in X (say, via an OLS regression). The greater the extent of the overlap, the greater the variation in Y is explained by X. The r 2 is simply a numerical measure of this overlap. In the ﬁgure, as we move from left to right, the area of the overlap increases, that is, successively a greater proportion of the variation in Y is explained by X. In short, r 2 increases. When there is no overlap, r 2 is obviously zero, but when the overlap is complete, r 2 is 1, since 100 percent of the variation in Y is explained by X. As we shall show shortly, r 2 lies between 0 and 1. To compute this r 2 , we proceed as follows: Recall that ˆ Yi = Yi + ui ˆ or in the deviation form yi = yi + ui ˆ ˆ (3.5.1) (2.6.3)

where use is made of (3.1.13) and (3.1.14). Squaring (3.5.1) on both sides

22 See Peter Kennedy, “Ballentine: A Graphical Aid for Econometrics,” Australian Economics Papers, vol. 20, 1981, pp. 414–416. The name Ballentine is derived from the emblem of the wellknown Ballantine beer with its circles. 23 The term variation and variance are different. Variation means the sum of squares of the deviations of a variable from its mean value. Variance is this sum of squares divided by the appropriate degrees of freedom. In short, variance = variation/df.

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and summing over the sample, we obtain yi2 = = ˆ2 = β2 yi2 + ˆ yi2 + ˆ xi2 + ui + 2 ˆ2 ui ˆ2 u2 ˆi yi ui ˆ ˆ (3.5.2)

ˆ yi ui = 0 (why?) and yi = β2 xi . ˆ ˆ ˆ since The various sums of squares appearing in (3.5.2) can be described as ¯ yi2 = (Yi − Y)2 = total variation of the actual Y values about follows: their sample mean, which may be called the total sum of squares (TSS). ¯ ˆ ˆ ˆ ¯ ˆ 2 xi2 = variation of the estimated Y yi2 = (Yi − Y)2 = (Yi − Y)2 = β2 ˆ ¯ = Y), ˆ ¯ which appropriately may be called the values about their mean ( Y sum of squares due to regression [i.e., due to the explanatory variable(s)], or explained by regression, or simply the explained sum of squares (ESS). u2 = residual or unexplained variation of the Y values about the regresˆi sion line, or simply the residual sum of squares (RSS). Thus, (3.5.2) is TSS = ESS + RSS (3.5.3)

and shows that the total variation in the observed Y values about their mean value can be partitioned into two parts, one attributable to the regression line and the other to random forces because not all actual Y observations lie on the ﬁtted line. Geometrically, we have Figure 3.10.

Y

ui = due to residual Yi SRF B β 1 + B2 X i β (Yi –Y ) = total Yi (Yi –Y ) = due to regression

Y

FIGURE 3.10

Breakdown of the variation of Yi into two components.

0

Xi

X

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Now dividing (3.5.3) by TSS on both sides, we obtain 1= ESS RSS + TSS TSS ˆi − Y)2 ¯ (Y = + ¯ (Yi − Y)2

u2 ˆi ¯ (Yi − Y)2

(3.5.4)

We now deﬁne r 2 as r2 = or, alternatively, as r2 = 1 − u2 ˆi ¯ (Yi − Y)2 ˆ ¯ ESS (Yi − Y)2 = ¯ 2 TSS (Yi − Y) (3.5.5)

RSS =1− TSS

(3.5.5a)

The quantity r 2 thus deﬁned is known as the (sample) coefﬁcient of determination and is the most commonly used measure of the goodness of ﬁt of a regression line. Verbally, r 2 measures the proportion or percentage of the total variation in Y explained by the regression model. Two properties of r 2 may be noted: 1. It is a nonnegative quantity. (Why?) ˆ 2. Its limits are 0 ≤ r 2 ≤ 1. An r 2 of 1 means a perfect ﬁt, that is, Yi = Yi for each i. On the other hand, an r 2 of zero means that there is no relationˆ ship between the regressand and the regressor whatsoever (i.e., β2 = 0). In ˆ ¯ ˆ this case, as (3.1.9) shows, Yi = β1 = Y, that is, the best prediction of any Y value is simply its mean value. In this situation therefore the regression line will be horizontal to the X axis. Although r 2 can be computed directly from its deﬁnition given in (3.5.5), it can be obtained more quickly from the following formula: r2 = ESS TSS yi2 ˆ = yi2 ˆ β 2 x2 = 2 2i yi ˆ2 = β2 xi2 yi2

(3.5.6)

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If we divide the numerator and the denominator of (3.5.6) by the sample size n (or n − 1 if the sample size is small), we obtain

2 ˆ2 S r 2 = β2 x 2 Sy 2 2 where Sy and Sx are the sample variances of Y and X, respectively. ˆ xi2 , Eq. (3.5.6) can also be expressed as xi yi Since β2 =

(3.5.7)

r =

2

xi yi xi2

2

yi2

(3.5.8)

an expression that may be computationally easy to obtain. Given the deﬁnition of r 2 , we can express ESS and RSS discussed earlier as follows: ESS = r 2 · TSS (3.5.9) yi2 = r2 RSS = TSS − ESS = TSS(1 − ESS/TSS) = Therefore, we can write TSS = ESS + RSS yi2 = r 2 yi2 + (1 − r 2 ) yi2 (3.5.11) yi2 · (1 − r )

2

(3.5.10)

an expression that we will ﬁnd very useful later. A quantity closely related to but conceptually very much different from r 2 is the coefﬁcient of correlation, which, as noted in Chapter 1, is a measure of the degree of association between two variables. It can be computed either from √ r = ± r2 (3.5.12) or from its deﬁnition r= = n xi yi xi2 n Xi2 − yi2 Xi Yi − ( Xi

2

Xi )( n

Yi ) Yi

2

(3.5.13)

Yi2 −

which is known as the sample correlation coefﬁcient.24

24

The population correlation coefﬁcient, denoted by ρ, is deﬁned in App. A.

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Y r = +1

Y r = –1

Y r close to +1

X (a) Y r close to –1 Y (b) r positive but close to zero

X (c) Y r negative but close to zero

X

X (d) Y r=0 (e) Y

X (f )

X

Y = X2 but r = 0

X ( g) FIGURE 3.11 (h)

X

Correlation patterns (adapted from Henri Theil, Introduction to Econometrics, Prentice-Hall, Englewood Cliffs, N.J., 1978, p. 86).

Some of the properties of r are as follows (see Figure 3.11): 1. It can be positive or negative, the sign depending on the sign of the term in the numerator of (3.5.13), which measures the sample covariation of two variables. 2. It lies between the limits of −1 and +1; that is, −1 ≤ r ≤ 1. 3. It is symmetrical in nature; that is, the coefﬁcient of correlation between X and Y(r XY ) is the same as that between Y and X(rY X ). 4. It is independent of the origin and scale; that is, if we deﬁne Xi* = aXi + C and Yi* = bYi + d, where a > 0, b > 0, and c and d are constants,

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then r between X * and Y * is the same as that between the original variables X and Y. 5. If X and Y are statistically independent (see Appendix A for the deﬁnition), the correlation coefﬁcient between them is zero; but if r = 0, it does not mean that two variables are independent. In other words, zero correlation does not necessarily imply independence. [See Figure 3.11(h).] 6. It is a measure of linear association or linear dependence only; it has no meaning for describing nonlinear relations. Thus in Figure 3.11(h), Y = X 2 is an exact relationship yet r is zero. (Why?) 7. Although it is a measure of linear association between two variables, it does not necessarily imply any cause-and-effect relationship, as noted in Chapter 1. In the regression context, r 2 is a more meaningful measure than r, for the former tells us the proportion of variation in the dependent variable explained by the explanatory variable(s) and therefore provides an overall measure of the extent to which the variation in one variable determines the variation in the other. The latter does not have such value.25 Moreover, as we shall see, the interpretation of r (= R) in a multiple regression model is of dubious value. However, we will have more to say about r 2 in Chapter 7. In passing, note that the r 2 deﬁned previously can also be computed as the squared coefﬁcient of correlation between actual Yi and the estimated Yi , ˆ namely, Yi . That is, using (3.5.13), we can write r2 = That is, r2 = yi yi ˆ yi2

2

¯ ˆ ¯ 2 (Yi − Y)(Yi − Y) ¯ ˆ ¯ (Yi − Y)2 (Yi − Y)2

yi2 ˆ

(3.5.14)

¯ ¯ ˆ ˆ where Yi = actual Y, Yi = estimated Y, and Y = Y = the mean of Y. For proof, see exercise 3.15. Expression (3.5.14) justiﬁes the description of r 2 as a measure of goodness of ﬁt, for it tells how close the estimated Y values are to their actual values.

3.6 A NUMERICAL EXAMPLE

We illustrate the econometric theory developed so far by considering the Keynesian consumption function discussed in the Introduction. Recall that Keynes stated that “The fundamental psychological law . . . is that men

25 In regression modeling the underlying theory will indicate the direction of causality between Y and X, which, in the context of single-equation models, is generally from X to Y.

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TABLE 3.2

HYPOTHETICAL DATA ON WEEKLY FAMILY CONSUMPTION EXPENDITURE Y AND WEEKLY FAMILY INCOME X Y, $ 70 65 90 95 110 115 120 140 155 150 X, $ 80 100 120 140 160 180 200 220 240 260

[women] are disposed, as a rule and on average, to increase their consumption as their income increases, but not by as much as the increase in their income,” that is, the marginal propensity to consume (MPC) is greater than zero but less than one. Although Keynes did not specify the exact functional form of the relationship between consumption and income, for simplicity assume that the relationship is linear as in (2.4.2). As a test of the Keynesian consumption function, we use the sample data of Table 2.4, which for convenience is reproduced as Table 3.2. The raw data required to obtain the estimates of the regression coefﬁcients, their standard errors, etc., are given in Table 3.3. From these raw data, the following calculations are obtained, and the reader is advised to check them. ˆ β1 = 24.4545 ˆ β2 = 0.5091 ˆ var (β1 ) = 41.1370 ˆ var (β2 ) = 0.0013 ˆ ˆ cov (β1 , β2 ) = −0.2172 r 2 = 0.9621 and and

2

ˆ se (β1 ) = 6.4138 ˆ se (β2 ) = 0.0357 df = 8 (3.6.1)

σ = 42.1591 ˆ

r = 0.9809

The estimated regression line therefore is ˆ Yi = 24.4545 + 0.5091Xi (3.6.2)

which is shown geometrically as Figure 3.12. Following Chapter 2, the SRF [Eq. (3.6.2)] and the associated regression line are interpreted as follows: Each point on the regression line gives an estimate of the expected or mean value of Y corresponding to the chosen X ˆ ˆ value; that is, Yi is an estimate of E(Y | Xi ). The value of β2 = 0.5091, which measures the slope of the line, shows that, within the sample range of X between $80 and $260 per week, as X increases, say, by $1, the estimated increase in the mean or average weekly consumption expenditure amounts ˆ to about 51 cents. The value of β1 = 24.4545, which is the intercept of the

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TABLE 3.3 Yi (1) 70 65 90 95 110 115 120 140 155 150 Sum 1110 Mean 111 ˆ β2 = Xi (2) 80 100 120 140 160 180 200 220 240 260 1700 170 xiyi

RAW DATA BASED ON TABLE 3.2 YiXi (3) 5600 6500 10800 13300 17600 20700 24000 30800 37200 39000 205500 nc Xi2 (4) 6400 10000 14400 19600 25600 32400 40000 48400 57600 67600 322000 nc xi = ¯ Xi − X (5) −90 −70 −50 −30 −10 10 30 50 70 90 0 0 yi = ¯ Yi − Y (6) −41 −46 −21 −16 −1 4 9 29 44 39 0 0 xi2 (7) 8100 4900 2500 900 100 100 900 2500 4900 8100 33000 nc xiyi (8) 3690 3220 1050 480 10 40 270 1450 3080 3510 16800 nc ˆ Yi (9) 65.1818 75.3636 85.5454 95.7272 105.9090 116.0909 125.2727 136.4545 145.6363 156.8181 1109.9995 ≈ 1110.0 110 ûi = ˆ Yi −Yi (10) 4.8181 −10.3636 4.4545 −0.7272 4.0909 −1.0909 −6.2727 3.5454 8.3636 −6.8181 0 0 ˆ Yiûi (11) 314.0524 −781.0382 381.0620 −69.6128 433.2631 −126.6434 −792.0708 483.7858 1226.4073 −1069.2014 0.0040 ≈ 0.0 0

xi2 = 16,800/33,000 = 0.5091

¯ ˆ ˆ ¯ β 1 = Y − β 2X = 111 − 0.5091(170) = 24.4545

Notes: ≈ symbolizes “approximately equal to”; nc means “not computed.”

Y

Yi = 24.4545 + 0.5091 Xi

111 (Y) Y

β2 = 0.5091

1

24.4545 170 (X) FIGURE 3.12 Sample regression line based on the data of Table 3.2. X

89

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line, indicates the average level of weekly consumption expenditure when weekly income is zero. However, this is a mechanical interpretation of the intercept term. In regression analysis such literal interpretation of the intercept term may not be always meaningful, although in the present example it can be argued that a family without any income (because of unemployment, layoff, etc.) might maintain some minimum level of consumption expenditure either by borrowing or dissaving. But in general one has to use common sense in interpreting the intercept term, for very often the sample range of X values may not include zero as one of the observed values. Perhaps it is best to interpret the intercept term as the mean or average effect on Y of all the variables omitted from the regression model. The value of r 2 of 0.9621 means that about 96 percent of the variation in the weekly consumption expenditure is explained by income. Since r 2 can at most be 1, the observed r 2 suggests that the sample regression line ﬁts the data very well.26 The coefﬁcient of correlation of 0.9809 shows that the two variables, consumption expenditure and income, are highly positively correlated. The estimated standard errors of the regression coefﬁcients will be interpreted in Chapter 5.

3.7 ILLUSTRATIVE EXAMPLES

(PCE) goes up by about 71 cents. From Keynesian theory, the MPC is less than 1. The intercept value of about −184 tells us that if income were zero, the PCE would be about −184 billion dollars. Of course, such a mechanical interpretation of the intercept term does not make economic sense in the present instance because the zero income value is out of the range of values we are working with and does not represent a likely outcome (see Table I.1). As we will see on many an occasion, very often the intercept term may not make much economic sense. Therefore, in practice the intercept term may not be very meaningful, although on occasions it can be very meaningful, as we will see in some illustrative examples. The more meaningful value is the slope coefﬁcient, MPC in the present case. The r 2 value of 0.9984 means approximately 99 percent of the variation in the PCE is explained by variation in the GDP. Since r 2 at most can be 1, we can say that the regression line in (3.7.1), which is shown in Figure I.3, ﬁts our data extremely well; as you can see from that ﬁgure the actual data points are very tightly clustered around the estimated regression line. As we will see throughout this book, in regressions involving time series data one generally obtains high r 2 values. In the chapter on autocorrelation, we will see the reasons behind this phenomenon.

EXAMPLE 3.1 CONSUMPTION–INCOME RELATIONSHIP IN THE UNITED STATES, 1982–1996 Let us return to the consumption income data given in Table I.1 of the Introduction. We have already shown the data in Figure I.3 along with the estimated regression line (I.3.3). Now we provide the underlying OLS regression results. (The results were obtained from the statistical package Eviews 3.) Note: Y = personal consumption expenditure (PCE) and X = gross domestic product (GDP), all measured in 1992 billions of dollars. In this example, our data are time series data. ˆ Yi = −184.0780 + 0.7064Xi ˆ var (β1) = 2140.1707 ˆ var (β2) = 0.000061 r = 0.998406

2

(3.7.1)

ˆ se (β1) = 46.2619 ˆ se (β2) = 0.007827 σ 2 = 411.4913 ˆ

Equation (3.7.1) is the aggregate (i.e., for the economy as a whole) Keynesian consumption function. As this equation shows, the marginal propensity to consume (MPC) is about 0.71, suggesting that if income goes up by a dollar, the average personal consumption expenditure

26

A formal test of the signiﬁcance of r 2 will be presented in Chap. 8.

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EXAMPLE 3.2 FOOD EXPENDITURE IN INDIA Refer to the data given in Table 2.8 of exercise 2.15. The data relate to a sample of 55 rural households in India. The regressand in this example is expenditure on food and the regressor is total expenditure, a proxy for income, both ﬁgures in rupees. The data in this example are thus cross-sectional data. On the basis of the given data, we obtained the following regression: FoodExpi = 94.2087 + 0.4368 TotalExpi (3.7.2) ˆ var (β1) = 2560.9401 ˆ var (β2) = 0.0061 r 2 = 0.3698 ˆ se (β1) = 50.8563 ˆ se (β2) = 0.0783 σ 2 = 4469.6913 ˆ

From (3.7.2) we see that if total expenditure increases by 1 rupee, on average, expenditure on food goes up by about 44 paise (1 rupee = 100 paise). If total expenditure were zero, the average expenditure on food would be about 94 rupees. Again, such a mechanical interpretation of the intercept may not be meaningful. However, in this example one could argue that even if total expenditure is zero (e.g., because of loss of a job), people may still maintain some minimum level of food expenditure by borrowing money or by dissaving. The r 2 value of about 0.37 means that only 37 percent of the variation in food expenditure is explained by the total expenditure. This might seem a rather low value, but as we will see throughout this text, in crosssectional data, typically one obtains low r 2 values, possibly because of the diversity of the units in the sample. We will discuss this topic further in the chapter on heteroscedasticity (see Chapter 11).

EXAMPLE 3.3 THE RELATIONSHIP BETWEEN EARNINGS AND EDUCATION In Table 2.6 we looked at the data relating average hourly earnings and education, as measured by years of schooling. Using that data, if we regress27 average hourly earnings (Y ) on education (X ), we obtain the following results. ˆ Yi = −0.0144 + 0.7241 Xi ˆ ˆ var (β1) = 0.7649 se (β1) = 0.8746 ˆ ˆ var (β2) = 0.00483 se (β2) = 0.0695 r 2 = 0.9077 σ 2 = 0.8816 ˆ (3.7.3)

As the regression results show, there is a positive association between education and earnings, an unsurprising ﬁnding. For every additional year of schooling, the average hourly earnings go up by about 72 cents an hour. The intercept term is positive but it may have no economic meaning. The r 2 value suggests that about 89 percent of the variation in average hourly earnings is explained by education. For cross-sectional data, such a high r 2 is rather unusual.

3.8 A NOTE ON MONTE CARLO EXPERIMENTS

In this chapter we showed that under the assumptions of CLRM the leastsquares estimators have certain desirable statistical features summarized in the BLUE property. In the appendix to this chapter we prove this property

27 Every ﬁeld of study has its jargon. The expression “regress Y on X” simply means treat Y as the regressand and X as the regressor.

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more formally. But in practice how does one know that the BLUE property holds? For example, how does one ﬁnd out if the OLS estimators are unbiased? The answer is provided by the so-called Monte Carlo experiments, which are essentially computer simulation, or sampling, experiments. To introduce the basic ideas, consider our two-variable PRF: Yi = β1 + β2 Xi + ui A Monte Carlo experiment proceeds as follows: 1. Suppose the true values of the parameters are as follows: β1 = 20 and β2 = 0.6. 2. You choose the sample size, say n = 25. 3. You ﬁx the values of X for each observation. In all you will have 25 X values. 4. Suppose you go to a random number table, choose 25 values, and call them ui (these days most statistical packages have built-in random number generators).28 5. Since you know β1 , β2 , Xi , and ui , using (3.8.1) you obtain 25 Yi values. 6. Now using the 25 Yi values thus generated, you regress these on the 25 ˆ ˆ X values chosen in step 3, obtaining β1 and β2 , the least-squares estimators. 7. Suppose you repeat this experiment 99 times, each time using the same β1 , β2 , and X values. Of course, the ui values will vary from experiment to experiment. Therefore, in all you have 100 experiments, thus generating 100 values each of β1 and β2 . (In practice, many such experiments are conducted, sometimes 1000 to 2000.) ¯ ¯ ˆ ˆ 8. You take the averages of these 100 estimates and call them β 1 and β 2 . 9. If these average values are about the same as the true values of β1 and β2 assumed in step 1, this Monte Carlo experiment “establishes” that the least-squares estimators are indeed unbiased. Recall that under CLRM ˆ ˆ E(β1 ) = β1 and E(β2 ) = β2 . These steps characterize the general nature of the Monte Carlo experiments. Such experiments are often used to study the statistical properties of various methods of estimating population parameters. They are particularly useful to study the behavior of estimators in small, or ﬁnite, samples. These experiments are also an excellent means of driving home the concept of repeated sampling that is the basis of most of classical statistical inference, as we shall see in Chapter 5. We shall provide several examples of Monte Carlo experiments by way of exercises for classroom assignment. (See exercise 3.27.)

28 In practice it is assumed that ui follows a certain probability distribution, say, normal, with certain parameters (e.g., the mean and variance). Once the values of the parameters are speciﬁed, one can easily generate the ui using statistical packages.

(3.8.1)

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3.9

SUMMARY AND CONCLUSIONS

The important topics and concepts developed in this chapter can be summarized as follows. 1. The basic framework of regression analysis is the CLRM. 2. The CLRM is based on a set of assumptions. 3. Based on these assumptions, the least-squares estimators take on certain properties summarized in the Gauss–Markov theorem, which states that in the class of linear unbiased estimators, the least-squares estimators have minimum variance. In short, they are BLUE. 4. The precision of OLS estimators is measured by their standard errors. In Chapters 4 and 5 we shall see how the standard errors enable one to draw inferences on the population parameters, the β coefﬁcients. 5. The overall goodness of ﬁt of the regression model is measured by the coefﬁcient of determination, r 2 . It tells what proportion of the variation in the dependent variable, or regressand, is explained by the explanatory variable, or regressor. This r 2 lies between 0 and 1; the closer it is to 1, the better is the ﬁt. 6. A concept related to the coefﬁcient of determination is the coefﬁcient of correlation, r. It is a measure of linear association between two variables and it lies between −1 and +1. 7. The CLRM is a theoretical construct or abstraction because it is based on a set of assumptions that may be stringent or “unrealistic.” But such abstraction is often necessary in the initial stages of studying any ﬁeld of knowledge. Once the CLRM is mastered, one can ﬁnd out what happens if one or more of its assumptions are not satisﬁed. The ﬁrst part of this book is devoted to studying the CLRM. The other parts of the book consider the reﬁnements of the CLRM. Table 3.4 gives the road map ahead.

TABLE 3.4 WHAT HAPPENS IF THE ASSUMPTIONS OF CLRM ARE VIOLATED? Assumption number 1 2 3 4 5 6 7 8 9 10 11* Type of violation Nonlinearity in parameters Stochastic regressor(s) Nonzero mean of ui Heteroscedasticity Autocorrelated disturbances Nonzero covariance between disturbances and regressor Sample observations less than the number of regressors Insufﬁcient variability in regressors Speciﬁcation bias Multicollinearity Nonnormality of disturbances Where to study? Chapter 14 Introduction to Part II Introduction to Part II Chapter 11 Chapter 12 Introduction to Part II and Part IV Chapter 10 Chapter 10 Chapters 13, 14 Chapter 10 Introduction to Part II

*Note: The assumption that the disturbances ui are normally distributed is not a part of the CLRM. But more on this in Chapter 4.

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EXERCISES

Questions 3.1. Given the assumptions in column 1 of the table, show that the assumptions in column 2 are equivalent to them.

ASSUMPTIONS OF THE CLASSICAL MODEL (1) E(ui |Xi) = 0 cov (ui, uj) = 0 i = j var (ui |Xi) = σ 2 (2) E(Yi | Xi) = β2 + β2X cov (Yi, Yj) = 0 i = j var (Yi | Xi) = σ 2

ˆ ˆ 3.2. Show that the estimates β1 = 1.572 and β2 = 1.357 used in the ﬁrst experiment of Table 3.1 are in fact the OLS estimators. 3.3. According to Malinvaud (see footnote 10), the assumption that E(ui | X i ) = 0 is quite important. To see this, consider the PRF: Y = β1 + β2 X i + ui . Now consider two situations: (i) β1 = 0, β2 = 1, and E(ui ) = 0; and (ii) β1 = 1, β2 = 0, and E(ui ) = ( X i − 1). Now take the expectation of the PRF conditional upon X in the two preceding cases and see if you agree with Malinvaud about the signiﬁcance of the assumption E(ui | X i ) = 0. 3.4. Consider the sample regression ˆ ˆ Yi = β1 + β2 X i + ui ˆ ˆ ˆ Imposing the restrictions (i) ui = 0 and (ii) ui X i = 0, obtain the estiˆ ˆ mators β1 and β2 and show that they are identical with the least-squares estimators given in (3.1.6) and (3.1.7). This method of obtaining estimators is called the analogy principle. Give an intuitive justiﬁcation for imposing restrictions (i) and (ii). (Hint: Recall the CLRM assumptions about ui .) In passing, note that the analogy principle of estimating unknown parameters is also known as the method of moments in which sample moments (e.g., sample mean) are used to estimate population moments (e.g., the population mean). As noted in Appendix A, a moment is a summary statistic of a probability distribution, such as the expected value and variance. 3.5. Show that r 2 deﬁned in (3.5.5) ranges between 0 and 1. You may use the Cauchy–Schwarz inequality, which states that for any random variables X and Y the following relationship holds true: [E( XY)]2 ≤ E( X 2 )E(Y 2 ) ˆ ˆ 3.6. Let βY X and β X Y represent the slopes in the regression of Y on X and X on Y, respectively. Show that ˆ ˆ βY X β X Y = r 2

where r is the coefﬁcient of correlation between X and Y. ˆ ˆ 3.7. Suppose in exercise 3.6 that βY X β X Y = 1. Does it matter then if we regress Y on X or X on Y? Explain carefully.

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3.8. Spearman’s rank correlation coefﬁcient r s is deﬁned as follows: rs = 1 − 6 d2 n(n2 − 1)

where d = difference in the ranks assigned to the same individual or phenomenon and n = number of individuals or phenomena ranked. Derive r s from r deﬁned in (3.5.13). Hint: Rank the X and Y values from 1 to n. Note that the sum of X and Y ranks is n(n + 1)/2 each and therefore their means are (n + 1)/2. 3.9. Consider the following formulations of the two-variable PRF: Model I:

Yi = β1 + β2 X i + ui ¯ Model II: Yi = α1 + α2 ( X i − X) + ui

a. Find the estimators of β1 and α1 . Are they identical? Are their variances identical? b. Find the estimators of β2 and α2 . Are they identical? Are their variances identical? c. What is the advantage, if any, of model II over model I? 3.10. Suppose you run the following regression:

ˆ ˆ yi = β1 + β2 xi + ui ˆ

3.11.

3.12.

3.13. 3.14.

where, as usual, yi and xi are deviations from their respective mean values. ˆ ˆ What will be the value of β1 ? Why? Will β2 be the same as that obtained from Eq. (3.1.6)? Why? Let r 1 = coefﬁcient of correlation between n pairs of values (Yi , X i ) and r 2 = coefﬁcient of correlation between n pairs of values (a X i + b, cYi + d), where a, b, c, and d are constants. Show that r 1 = r 2 and hence establish the principle that the coefﬁcient of correlation is invariant with respect to the change of scale and the change of origin. Hint: Apply the deﬁnition of r given in (3.5.13). Note: The operations a X i , X i + b, and a X i + b are known, respectively, as the change of scale, change of origin, and change of both scale and origin. If r, the coefﬁcient of correlation between n pairs of values ( X i , Yi ), is positive, then determine whether each of the following statements is true or false: a. r between (− X i , −Yi ) is also positive. b. r between (−Xi , Yi ) and that between ( X i , −Yi ) can be either positive or negative. c. Both the slope coefﬁcients β yx and β xy are positive, where β yx = slope coefﬁcient in the regression of Y on X and β xy = slope coefﬁcient in the regression of X on Y. If X 1 , X 2 , and X 3 are uncorrelated variables each having the same standard deviation, show that the coefﬁcient of correlation between X 1 + X 2 and X 2 + X 3 is equal to 1 . Why is the correlation coefﬁcient not zero? 2 In the regression Yi = β1 + β2 X i + ui suppose we multiply each X value by a constant, say, 2. Will it change the residuals and ﬁtted values of Y? Explain. What if we add a constant value, say, 2, to each X value?

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3.15. Show that (3.5.14) in fact measures the coefﬁcient of determination. yi yi = ˆ Hint: Apply the deﬁnition of r given in (3.5.13) and recall that ( yi + ui ) yi = ˆ ˆ ˆ yi2 , and remember (3.5.6). ˆ 3.16. Explain with reason whether the following statements are true, false, or uncertain: a. Since the correlation between two variables, Y and X, can range from −1 to +1, this also means that cov (Y, X) also lies between these limits. b. If the correlation between two variables is zero, it means that there is no relationship between the two variables whatsoever. ˆ c. If you regress Yi on Yi (i.e., actual Y on estimated Y), the intercept and slope values will be 0 and 1, respectively. 3.17. Regression without any regressor. Suppose you are given the model: Yi = β1 + ui . Use OLS to ﬁnd the estimator of β1 . What is its variance and the RSS? Does the estimated β1 make intuitive sense? Now consider the two-variable model Yi = β1 + β2 X i + ui . Is it worth adding X i to the model? If not, why bother with regression analysis?

Problems

3.18. In Table 3.5, you are given the ranks of 10 students in midterm and ﬁnal examinations in statistics. Compute Spearman’s coefﬁcient of rank correlation and interpret it. 3.19. The relationship between nominal exchange rate and relative prices. From the annual observations from 1980 to 1994, the following regression results were obtained, where Y = exchange rate of the German mark to the U.S. dollar (GM/$) and X = ratio of the U.S. consumer price index to the German consumer price index; that is, X represents the relative prices in the two countries:

ˆ Yt = 6.682 − 4.318 X t se = (1.22)(1.333) r 2 = 0.528

a. Interpret this regression. How would you interpret r 2 ? b. Does the negative value of X t make economic sense? What is the underlying economic theory? c. Suppose we were to redeﬁne X as the ratio of German CPI to the U.S. CPI. Would that change the sign of X? And why? 3.20. Table 3.6 gives data on indexes of output per hour (X) and real compensation per hour (Y) for the business and nonfarm business sectors of the U.S. economy for 1959–1997. The base year of the indexes is 1982 = 100 and the indexes are seasonally adjusted.

TABLE 3.5 Rank Midterm Final A 1 3 B 3 2 C 7 8 D 10 7

Student E 9 9 F 5 6 G 4 5 H 8 10 I 2 1 J 6 4

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TABLE 3.6

PRODUCTIVITY AND RELATED DATA, BUSINESS SECTOR, 1959–98 [Index numbers, 1992 = 100; quarterly data seasonally adjusted] Output per hour of all persons1 Year or quarter 1959 ……… 1960 ……… 1961 ……… 1962 ……… 1963 ……… 1964 ……… 1965 ……… 1966 ……… 1967 ……… 1968 ……… 1969 ……… 1970 ……… 1971 ……… 1972 ……… 1973 ……… 1974 ……… 1975 ……… 1976 ……… 1977 ……… 1978 ……… 1979 ……… 1980 ……… 1981 ……… 1982 ……… 1983 ……… 1984 ……… 1985 ……… 1986 ……… 1987 ……… 1988 ……… 1989 ……… 1990 ……… 1991 ……… 1992 ……… 1993 ……… 1994 ……… 1995 ……… 1996 ……… 1997 ………

1 2

Compensation per hour2 Business sector 13.1 13.7 14.2 14.8 15.4 16.2 16.8 17.9 18.9 20.5 21.9 23.6 25.1 26.7 29.0 31.8 35.1 38.2 41.2 44.9 49.2 54.5 59.6 64.1 66.8 69.7 73.1 76.8 79.8 83.6 85.9 90.8 95.1 100.0 102.5 104.4 106.8 110.7 114.9 Nonfarm business sector 13.7 14.3 14.8 15.4 15.9 16.7 17.2 18.2 19.3 20.8 22.2 23.8 25.4 27.0 29.2 32.1 35.3 38.4 41.5 45.2 49.5 54.8 60.2 64.6 67.3 70.2 73.4 77.2 80.1 83.7 86.0 90.7 95.1 100.0 102.2 104.2 106.7 110.4 114.5

Business sector 50.5 51.4 53.2 55.7 57.9 60.6 62.7 65.2 66.6 68.9 69.2 70.6 73.6 76.0 78.4 77.1 79.8 82.5 84.0 84.9 84.5 84.2 85.8 85.3 88.0 90.2 91.7 94.1 94.0 94.7 95.5 96.1 96.7 100.0 100.1 100.7 101.0 103.7 105.4

Nonfarm business sector 54.2 54.8 56.6 59.2 61.2 63.8 65.8 68.0 69.2 71.6 71.7 72.7 75.7 78.3 80.7 79.4 81.6 84.5 85.8 87.0 86.3 86.0 87.0 88.3 89.9 91.4 92.3 94.7 94.5 95.3 95.8 96.3 97.0 100.0 100.1 100.6 101.2 103.7 105.1

Output refers to real gross domestic product in the sector. Wages and salaries of employees plus employers’ contributions for social insurance and private beneﬁt plans. Also includes an estimate of wages, salaries, and supplemental payments for the self-employed. Source: Economic Report of the President, 1999, Table B-49, p. 384.

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a. Plot Y against X for the two sectors separately. b. What is the economic theory behind the relationship between the two variables? Does the scattergram support the theory? c. Estimate the OLS regression of Y on X. Save the results for a further look after we study Chapter 5. 3.21. From a sample of 10 observations, the following results were obtained:

Yi = 1110 X i = 1700 X i Yi = 205,500 Yi2 = 132,100

X i2 = 322,000

with coefﬁcient of correlation r = 0.9758 . But on rechecking these calculations it was found that two pairs of observations were recorded:

Y 90 140 X 120 220 instead of Y 80 150 X 110 210

What will be the effect of this error on r? Obtain the correct r. 3.22. Table 3.7 gives data on gold prices, the Consumer Price Index (CPI), and the New York Stock Exchange (NYSE) Index for the United States for the period 1977–1991. The NYSE Index includes most of the stocks listed on the NYSE, some 1500 plus.

TABLE 3.7 Price of gold at New York, $ per troy ounce 147.98 193.44 307.62 612.51 459.61 376.01 423.83 360.29 317.30 367.87 446.50 436.93 381.28 384.08 362.04 Consumer Price Index (CPI), 1982–84 = 100 60.6 65.2 72.6 82.4 90.9 96.5 99.6 103.9 107.6 109.6 113.6 118.3 124.0 130.7 136.2

Year 1977 1978 1979 1980 1981 1982 1983 1984 1985 1986 1987 1988 1989 1990 1991

New York Stock Exchange (NYSE) Index, Dec. 31, 1965 = 100 53.69 53.70 58.32 68.10 74.02 68.93 92.63 92.46 108.90 136.00 161.70 149.91 180.02 183.46 206.33

Source: Data on CPI and NYSE Index are from the Economic Report of the President, January 1993, Tables B-59 and B-91, respectively. Data on gold prices are from U.S. Department of Commerce, Bureau of Economic Analysis, Business Statistics, 1963–1991, p. 68.

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a. Plot in the same scattergram gold prices, CPI, and the NYSE Index. b. An investment is supposed to be a hedge against inﬂation if its price and/or rate of return at least keeps pace with inﬂation. To test this hypothesis, suppose you decide to ﬁt the following model, assuming the scatterplot in a suggests that this is appropriate:

Gold price t = β1 + β2 CPI t + ut NYSE index t = β1 + β2 CPI t + ut

3.23. Table 3.8 gives data on gross domestic product (GDP) for the United States for the years 1959–1997. a. Plot the GDP data in current and constant (i.e., 1992) dollars against time. b. Letting Y denote GDP and X time (measured chronologically starting with 1 for 1959, 2 for 1960, through 39 for 1997), see if the following model ﬁts the GDP data:

Yt = β1 + β2 X t + ut

Estimate this model for both current and constant-dollar GDP. c. How would you interpret β2 ? d. If there is a difference between β2 estimated for current-dollar GDP and that estimated for constant-dollar GDP, what explains the difference?

TABLE 3.8 NOMINAL AND REAL GDP, UNITED STATES, 1959–1997 Year 1959 1960 1961 1962 1963 1964 1965 1966 1967 1968 1969 1970 1971 1972 1973 1974 1975 1976 1977 1978 NGDP 507.2000 526.6000 544.8000 585.2000 617.4000 663.0000 719.1000 787.7000 833.6000 910.6000 982.2000 1035.600 1125.400 1237.300 1382.600 1496.900 1630.600 1819.000 2026.900 2291.400 RGDP 2210.200 2262.900 2314.300 2454.800 2559.400 2708.400 2881.100 3069.200 3147.200 3293.900 3393.600 3397.600 3510.000 3702.300 3916.300 3891.200 3873.900 4082.900 4273.600 4503.000 Year 1979 1980 1981 1982 1983 1984 1985 1986 1987 1988 1989 1990 1991 1992 1993 1994 1995 1996 1997 NGDP 2557.500 2784.200 3115.900 3242.100 3514.500 3902.400 4180.700 4422.200 4692.300 5049.600 5438.700 5743.800 5916.700 6244.400 6558.100 6947.000 7269.600 7661.600 8110.900 RGDP 4630.600 4615.000 4720.700 4620.300 4803.700 5140.100 5323.500 5487.700 5649.500 5865.200 6062.000 6136.300 6079.400 6244.400 6389.600 6610.700 6761.700 6994.800 7269.800

Note: NGDP = nominal GDP (current dollars in billions). RGDP = real GDP (1992 billions of dollars). Source: Economic Report of the President, 1999, Tables B-1 and B-2, pp. 326–328.

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3.24. 3.25.

3.26. 3.27.

e. From your results what can you say about the nature of inﬂation in the United States over the sample period? Using the data given in Table I.1 of the Introduction, verify Eq. (3.7.1). For the S.A.T. example given in exercise 2.16 do the following: a. Plot the female verbal score against the male verbal score. b. If the scatterplot suggests that a linear relationship between the two seems appropriate, obtain the regression of female verbal score on male verbal score. c. If there is a relationship between the two verbal scores, is the relationship causal? Repeat exercise 3.24, replacing math scores for verbal scores. Monte Carlo study classroom assignment: Refer to the 10 X values given in Table 3.2. Let β1 = 25 and β2 = 0.5. Assume ui ≈ N(0, 9), that is, ui are normally distributed with mean 0 and variance 9. Generate 100 samples using these values, obtaining 100 estimates of β1 and β2 . Graph these estimates. What conclusions can you draw from the Monte Carlo study? Note: Most statistical packages now can generate random variables from most well-known probability distributions. Ask your instructor for help, in case you have difﬁculty generating such variables.

APPENDIX 3A

3A.1 DERIVATION OF LEAST-SQUARES ESTIMATES

ˆ ˆ Differentiating (3.1.2) partially with respect to β1 and β2 , we obtain ∂ u2 ˆi ˆ ∂ β2 u2 ˆi ˆ ∂ β1 ∂ = −2 ˆ ˆ (Yi − β1 − β2 Xi ) = −2 ˆ ˆ (Yi − β1 − β2 Xi )Xi = −2 ui ˆ ui Xi ˆ (1)

= −2

(2)

Setting these equations to zero, after algebraic simpliﬁcation and manipulation, gives the estimators given in Eqs. (3.1.6) and (3.1.7).

3A.2 LINEARITY AND UNBIASEDNESS PROPERTIES OF LEAST-SQUARES ESTIMATORS

From (3.1.8) we have ˆ β2 = where ki = xi xi2 xi Yi = xi2 ki Yi (3)

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ˆ which shows that β2 is a linear estimator because it is a linear function of Y; actually it is a weighted average of Yi with ki serving as the weights. It can ˆ similarly be shown that β1 too is a linear estimator. Incidentally, note these properties of the weights ki : 1. Since the Xi are assumed to be nonstochastic, the ki are nonstochastic too. ki = 0. 2. k2 = 1 xi2 . 3. i ki xi = ki Xi = 1. These properties can be directly veriﬁed from 4. the deﬁnition of ki . For example, ki = xi xi2 = = 0, 1 xi2 xi , since for a given sample xi2 is known

since xi , the sum of deviations from the mean value, is always zero

Now substitute the PRF Yi = β1 + β2 Xi + ui into (3) to obtain ˆ β2 = = β1 = β2 + ki (β1 + β2 Xi + ui ) ki + β2 ki ui ki Xi + ki ui (4)

where use is made of the properties of ki noted earlier. Now taking expectation of (4) on both sides and noting that ki , being nonstochastic, can be treated as constants, we obtain ˆ E(β2 ) = β2 + = β2 ki E(ui ) (5)

ˆ since E(ui ) = 0 by assumption. Therefore, β2 is an unbiased estimator of β2 . ˆ1 is also an unbiased estimator of β1 . Likewise, it can be proved that β

3A.3 VARIANCES AND STANDARD ERRORS OF LEAST-SQUARES ESTIMATORS

Now by the deﬁnition of variance, we can write ˆ ˆ ˆ var (β2 ) = E[β2 − E(β2 )]2 ˆ = E(β2 − β2 )2 =E ki ui

2

ˆ since E(β2 ) = β2 using Eq. (4) above

(6)

2 2 2 = E k1 u2 + k2 u2 + · · · + kn u2 + 2k1 k2 u1 u2 + · · · + 2kn−1 knun−1 un 1 2 n

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Since by assumption, E(u2 ) = σ 2 for each i and E(ui uj ) = 0, i = j, it follows i that ˆ var (β2 ) = σ 2 = σ2 xi2 k2 i (using the deﬁnition of k2 ) i (7)

= Eq. (3.3.1) ˆ The variance of β1 can be obtained following the same line of reasoning ˆ ˆ already given. Once the variances of β1 and β2 are obtained, their positive square roots give the corresponding standard errors.

3A.4

ˆ ˆ COVARIANCE BETWEEN β1 AND β2 By deﬁnition, ˆ ˆ ˆ ˆ ˆ ˆ cov (β1 , β2 ) = E{[β1 − E(β1 )][β2 − E(β2 )]} ˆ ˆ = E(β1 − β1 )(β2 − β2 ) ¯ ˆ = − X E(β2 − β2 ) ¯ ˆ = − X var (β2 ) = Eq. (3.3.9) ¯ ¯ ¯ ˆ ˆ ¯ ˆ where use is made of the fact that β1 = Y − β2 X and E(β1 ) = Y − β2 X, giving ¯ ˆ ˆ ˆ ˆ β1 − E(β1 ) = − X(β2 − β2 ). Note: var (β2 ) is given in (3.3.1).

2

(Why?) (8)

3A.5

THE LEAST-SQUARES ESTIMATOR OF σ2

Recall that Yi = β1 + β2 Xi + ui Therefore, ¯ ¯ Y = β1 + β2 X + u ¯ Subtracting (10) from (9) gives yi = β2 xi + (ui − u) ¯ Also recall that ˆ ui = yi − β2 xi ˆ Therefore, substituting (11) into (12) yields ˆ ¯ ui = β2 xi + (ui − u) − β2 xi ˆ (13) (12) (11) (10) (9)

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Collecting terms, squaring, and summing on both sides, we obtain ˆ u2 = (β2 − β2 )2 ˆi xi2 + ˆ (ui − u)2 − 2(β2 − β2 ) ¯ xi (ui − u) (14) ¯

Taking expectations on both sides gives E u2 = ˆi ˆ xi2 E(β2 − β2 )2 + E = ˆ (ui − u)2 − 2E (β2 − β2 ) ¯ ki ui (xi ui ) (15) xi (ui − u) ¯

ˆ xi2 var (β2 ) + (n − 1) var (ui ) − 2E ki xi u2 i

= σ 2 + (n − 1) σ 2 − 2E = σ 2 + (n − 1) σ 2 − 2σ 2 = (n − 2)σ 2

where, in the last but one step, use is made of the deﬁnition of ki given in Eq. (3) and the relation given in Eq. (4). Also note that E (ui − u)2 = E ¯ =E =E = nσ 2 − u2 − nu2 ¯ i u2 − n i u2 − i 1 n ui n u2 i

2

n 2 σ = (n − 1)σ 2 n

where use is made of the fact that the ui are uncorrelated and the variance of each ui is σ 2 . Thus, we obtain E Therefore, if we deﬁne σ2 = ˆ its expected value is E(σ 2 ) = ˆ 1 E n− 2 u2 = σ 2 ˆi using (16) (18) u2 ˆi n− 2 (17) u2 = (n − 2)σ 2 ˆi (16)

ˆ which shows that σ 2 is an unbiased estimator of true σ 2 .

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3A.6 MINIMUM-VARIANCE PROPERTY OF LEAST-SQUARES ESTIMATORS

It was shown in Appendix 3A, Section 3A.2, that the least-squares estimator ˆ ˆ β2 is linear as well as unbiased (this holds true of β1 too). To show that these estimators are also minimum variance in the class of all linear unbiased ˆ estimators, consider the least-squares estimator β2 : ˆ β2 = where ki = ¯ Xi − X = ¯ (Xi − X)2 xi xi2 (see Appendix 3A.2) (19) ki Yi

ˆ which shows that β2 is a weighted average of the Y ’s, with ki serving as the weights. Let us deﬁne an alternative linear estimator of β2 as follows:

∗ β2 =

wi Yi

(20)

where wi are also weights, not necessarily equal to ki . Now

∗ E(β2 ) =

wi E(Yi ) wi (β1 + β2 Xi ) wi + β2 wi Xi (21)

= = β1

∗ Therefore, for β2 to be unbiased, we must have

wi = 0 and wi Xi = 1 Also, we may write

∗ var (β2 ) = var

(22) (23)

wi Yi

2 wi var Yi 2 wi

= = σ2 = σ2 = σ2 = σ2

[Note: var Yi = var ui = σ 2 ] [Note: cov (Yi , Yj ) = 0 (i = j)] xi + xi2 xi xi2 xi xi2

2

wi − wi − wi −

xi xi2 + σ2

2

(Note the mathematical trick) xi2

2 xi2

+ 2σ 2

wi −

xi xi2

xi xi2 (24)

2

+ σ2

1 xi2

because the last term in the next to the last step drops out. (Why?)

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* Since the last term in (24) is constant, the variance of (β2 ) can be minimized only by manipulating the ﬁrst term. If we let xi wi = xi2 Eq. (24) reduces to * var (β2 ) =

σ2 xi2

(25)

ˆ = var (β2 ) In words, with weights wi = ki , which are the least-squares weights, the * variance of the linear estimator β2 is equal to the variance of the leastˆ ˆ2 ; otherwise var (β * ) > var (β2 ). To put it differently, if squares estimator β 2 there is a minimum-variance linear unbiased estimator of β2 , it must be the ˆ least-squares estimator. Similarly it can be shown that β1 is a minimumvariance linear unbiased estimator of β1 .

3A.7 CONSISTENCY OF LEAST-SQUARES ESTIMATORS

We have shown that, in the framework of the classical linear regression model, the least-squares estimators are unbiased (and efﬁcient) in any sample size, small or large. But sometimes, as discussed in Appendix A, an estimator may not satisfy one or more desirable statistical properties in small samples. But as the sample size increases indeﬁnitely, the estimators possess several desirable statistical properties. These properties are known as the large sample, or asymptotic, properties. In this appendix, we will discuss one large sample property, namely, the property of consistency, which is discussed more fully in Appendix A. For the two-variable model ˆ we have already shown that the OLS estimator β2 is an unbiased estimator ˆ2 is also a consistent estimator of β2 . As of the true β2 . Now we show that β ˆ shown in Appendix A, a sufﬁcient condition for consistency is that β2 is unbiased and that its variance tends to zero as the sample size n tends to inﬁnity. Since we have already proved the unbiasedness property, we need only ˆ show that the variance of β2 tends to zero as n increases indeﬁnitely. We know that σ2 σ 2 /n ˆ var (β2 ) = = (26) xi2 xi2 /n By dividing the numerator and denominator by n, we do not change the equality. Now ˆ lim var (β2 ) = lim n→ ∞

σ 2 /n xi2 /n n→ ∞

=0

(27)

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where use is made of the facts that (1) the limit of a ratio quantity is the limit of the quantity in the numerator to the limit of the quantity in the denominator (refer to any calculus book); (2) as n tends to inﬁnity, σ 2 /n tends to zero because σ 2 is a ﬁnite number; and [( xi2 )/n] = 0 because the variance of X has a ﬁnite limit because of Assumption 8 of CLRM. ˆ The upshot of the preceding discussion is that the OLS estimator β2 is a ˆ consistent estimator of true β2 . In like fashion, we can establish that β1 is also a consistent estimator. Thus, in repeated (small) samples, the OLS estimators are unbiased and as the sample size increases indeﬁnitely the OLS estimators are consistent. As we shall see later, even if some of the assumptions of CLRM are not satisﬁed, we may be able to obtain consistent estimators of the regression coefﬁcients in several situations.

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4

CLASSICAL NORMAL LINEAR REGRESSION MODEL (CNLRM)

What is known as the classical theory of statistical inference consists of two branches, namely, estimation and hypothesis testing. We have thus far covered the topic of estimation of the parameters of the (twovariable) linear regression model. Using the method of OLS we were able to estimate the parameters β1, β2, and σ2. Under the assumptions of the classical linear regression model (CLRM), we were able to show that the ˆ ˆ ˆ estimators of these parameters, β1 , β2 , and σ 2 , satisfy several desirable statistical properties, such as unbiasedness, minimum variance, etc. (Recall the BLUE property.) Note that, since these are estimators, their values will change from sample to sample. Therefore, these estimators are random variables. But estimation is half the battle. Hypothesis testing is the other half. Recall that in regression analysis our objective is not only to estimate the sample regression function (SRF), but also to use it to draw inferences about the population regression function (PRF), as emphasized in Chapter ˆ 2. Thus, we would like to ﬁnd out how close β1 is to the true β1 or how close σ 2 is to the true σ 2 . For instance, in Example 3.2, we estimated the SRF ˆ as shown in Eq. (3.7.2). But since this regression is based on a sample of 55 families, how do we know that the estimated MPC of 0.4368 represents the (true) MPC in the population as a whole? ˆ ˆ ˆ Therefore, since β1 , β2 , and σ 2 are random variables, we need to ﬁnd out their probability distributions, for without that knowledge we will not be able to relate them to their true values.

107

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4.1

THE PROBABILITY DISTRIBUTION OF DISTURBANCES ui

To ﬁnd out the probability distributions of the OLS estimators, we proceed ˆ as follows. Speciﬁcally, consider β2 . As we showed in Appendix 3A.2, ˆ β2 = ki Yi (4.1.1)

where ki = xi / xi2 . But since the X’s are assumed ﬁxed, or nonstochastic, because ours is conditional regression analysis, conditional on the ﬁxed valˆ ues of Xi, Eq. (4.1.1) shows that β2 is a linear function of Yi, which is random by assumption. But since Yi = β1 + β2 Xi + ui , we can write (4.1.1) as ˆ β2 = ki (β1 + β2 Xi + ui ) (4.1.2)

ˆ Because ki, the betas, and Xi are all ﬁxed, β2 is ultimately a linear function of the random variable ui, which is random by assumption. Therefore, the ˆ ˆ probability distribution of β2 (and also of β1 ) will depend on the assumption made about the probability distribution of ui . And since knowledge of the probability distributions of OLS estimators is necessary to draw inferences about their population values, the nature of the probability distribution of ui assumes an extremely important role in hypothesis testing. Since the method of OLS does not make any assumption about the probabilistic nature of ui , it is of little help for the purpose of drawing inferences about the PRF from the SRF, the Gauss–Markov theorem notwithstanding. This void can be ﬁlled if we are willing to assume that the u’s follow some probability distribution. For reasons to be explained shortly, in the regression context it is usually assumed that the u’s follow the normal distribution. Adding the normality assumption for ui to the assumptions of the classical linear regression model (CLRM) discussed in Chapter 3, we obtain what is known as the classical normal linear regression model (CNLRM).

4.2

THE NORMALITY ASSUMPTION FOR ui

The classical normal linear regression model assumes that each ui is distributed normally with Mean: Variance: cov (ui, uj): E(ui ) = 0 E[ui − E(ui )]2 = E u2 = σ 2 i E{[(ui − E(ui )][uj − E(uj )]} = E(ui uj ) = 0 i = j (4.2.1) (4.2.2) (4.2.3)

The assumptions given above can be more compactly stated as ui ∼ N(0, σ 2 ) (4.2.4)

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where the symbol ∼ means distributed as and N stands for the normal distribution, the terms in the parentheses representing the two parameters of the normal distribution, namely, the mean and the variance. As noted in Appendix A, for two normally distributed variables, zero covariance or correlation means independence of the two variables. Therefore, with the normality assumption, (4.2.4) means that ui and uj are not only uncorrelated but are also independently distributed. Therefore, we can write (4.2.4) as ui ∼ NID(0, σ 2 ) where NID stands for normally and independently distributed.

Why the Normality Assumption?

(4.2.5)

Why do we employ the normality assumption? There are several reasons: 1. As pointed out in Section 2.5, ui represent the combined inﬂuence (on the dependent variable) of a large number of independent variables that are not explicitly introduced in the regression model. As noted, we hope that the inﬂuence of these omitted or neglected variables is small and at best random. Now by the celebrated central limit theorem (CLT) of statistics (see Appendix A for details), it can be shown that if there are a large number of independent and identically distributed random variables, then, with a few exceptions, the distribution of their sum tends to a normal distribution as the number of such variables increase indeﬁnitely.1 It is the CLT that provides a theoretical justiﬁcation for the assumption of normality of ui . 2. A variant of the CLT states that, even if the number of variables is not very large or if these variables are not strictly independent, their sum may still be normally distributed.2 3. With the normality assumption, the probability distributions of OLS estimators can be easily derived because, as noted in Appendix A, one property of the normal distribution is that any linear function of normally distributed variables is itself normally distributed. As we discussed earlier, ˆ ˆ OLS estimators β1 and β2 are linear functions of ui . Therefore, if ui are norˆ ˆ mally distributed, so are β1 and β2 , which makes our task of hypothesis testing very straightforward. 4. The normal distribution is a comparatively simple distribution involving only two parameters (mean and variance); it is very well known and

1 For a relatively simple and straightforward discussion of this theorem, see Sheldon M. Ross, Introduction to Probability and Statistics for Engineers and Scientists, 2d ed., Harcourt Academic Press, New York, 2000, pp. 193–194. One exception to the theorem is the Cauchy distribution, which has no mean or higher moments. See M. G. Kendall and A. Stuart, The Advanced Theory of Statistics, Charles Grifﬁn & Co., London, 1960, vol. 1, pp. 248–249. 2 For the various forms of the CLT, see Harald Cramer, Mathematical Methods of Statistics, Princeton University Press, Princeton, N.J., 1946, Chap. 17.

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its theoretical properties have been extensively studied in mathematical statistics. Besides, many phenomena seem to follow the normal distribution. 5. Finally, if we are dealing with a small, or ﬁnite, sample size, say data of less than 100 observations, the normality assumption assumes a critical role. It not only helps us to derive the exact probability distributions of OLS estimators but also enables us to use the t, F, and χ 2 statistical tests for regression models. The statistical properties of t, F, and χ 2 probability distributions are discussed in Appendix A. As we will show subsequently, if the sample size is reasonably large, we may be able to relax the normality assumption. A cautionary note: Since we are “imposing” the normality assumption, it behooves us to ﬁnd out in practical applications involving small sample size data whether the normality assumption is appropriate. Later, we will develop some tests to do just that. Also, later we will come across situations where the normality assumption may be inappropriate. But until then we will continue with the normality assumption for the reasons discussed previously.

4.3 PROPERTIES OF OLS ESTIMATORS UNDER THE NORMALITY ASSUMPTION

With the assumption that ui follow the normal distribution as in (4.2.5), the OLS estimators have the following properties; Appendix A provides a general discussion of the desirable statistical properties of estimators. 1. They are unbiased. 2. They have minimum variance. Combined with 1, this means that they are minimum-variance unbiased, or efﬁcient estimators. 3. They have consistency; that is, as the sample size increases indeﬁnitely, the estimators converge to their true population values. ˆ 4. β1 (being a linear function of ui) is normally distributed with Mean: ˆ var (β1 ): Or more compactly,

2 ˆ β1 ∼ N β1 , σβ ˆ

1

ˆ E(β1 ) = β1

2 σβ = ˆ

1

(4.3.1) = (3.3.3) (4.3.2)

Xi2 2 σ n xi2

Then by the properties of the normal distribution the variable Z, which is deﬁned as Z= ˆ β1 − β1 σβ1 ˆ (4.3.3)

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follows the standard normal distribution, that is, a normal distribution with zero mean and unit ( = 1) variance, or Z ∼ N(0, 1) ˆ 5. β2 (being a linear function of ui) is normally distributed with Mean: ˆ var (β2 ) : Or, more compactly,

2 ˆ β2 ∼ N β2 , σβ ˆ

2

ˆ E(β2 ) = β2

2 σβ = ˆ

2

(4.3.4) = (3.3.1) (4.3.5)

σ2 xi2

Then, as in (4.3.3), Z= ˆ β2 − β2 σβ2 ˆ (4.3.6)

also follows the standard normal distribution. ˆ ˆ Geometrically, the probability distributions of β1 and β2 are shown in Figure 4.1. f(β1) β f (B2)

Density Density E(β1) = β1 f(Z)

β1

E(b 2) = b 2 β β f(Z) Density

β2

Density

0 FIGURE 4.1

Z=

β1 – β1 σβ

1

0

β 2 – β2 Z= σ β2

ˆ ˆ Probability distributions of β1 and β2 .

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ˆ 6. (n − 2)(σ 2 /σ 2 ) is distributed as the χ 2 (chi-square) distribution with (n − 2)df.3 This knowledge will help us to draw inferences about the true σ 2 from the estimated σ 2, as we will show in Chapter 5. (The chi-square distribution and its properties are discussed in Appendix A.) ˆ ˆ ˆ 7. (β1 , β2 ) are distributed independently of σ 2 . The importance of this will be explained in the next chapter. ˆ ˆ 8. β1 and β2 have minimum variance in the entire class of unbiased estimators, whether linear or not. This result, due to Rao, is very powerful because, unlike the Gauss–Markov theorem, it is not restricted to the class of linear estimators only.4 Therefore, we can say that the least-squares estimators are best unbiased estimators (BUE); that is, they have minimum variance in the entire class of unbiased estimators. To sum up: The important point to note is that the normality assumption ˆ ˆ enables us to derive the probability, or sampling, distributions of β1 and β2 2 ˆ (both normal) and σ (related to the chi square). As we will see in the next chapter, this simpliﬁes the task of establishing conﬁdence intervals and testing (statistical) hypotheses. In passing, note that, with the assumption that ui ∼ N(0, σ 2 ), Yi , being a linear function of ui, is itself normally distributed with the mean and variance given by E(Yi ) = β1 + β2 Xi var (Yi ) = σ More neatly, we can write Yi ∼ N(β1 + β2 Xi , σ 2 )

4.4 THE METHOD OF MAXIMUM LIKELIHOOD (ML)

2

(4.3.7) (4.3.8)

(4.3.9)

A method of point estimation with some stronger theoretical properties than the method of OLS is the method of maximum likelihood (ML). Since this method is slightly involved, it is discussed in the appendix to this chapter. For the general reader, it will sufﬁce to note that if ui are assumed to be normally distributed, as we have done for reasons already discussed, the ML and OLS estimators of the regression coefﬁcients, the β’s, are identical, and this is true of simple as well as multiple regressions. The ML estiˆ2 mator of σ 2 is ui /n. This estimator is biased, whereas the OLS estimator

3 The proof of this statement is slightly involved. An accessible source for the proof is Robert V. Hogg and Allen T. Craig, Introduction to Mathematical Statistics, 2d ed., Macmillan, New York, 1965, p. 144. 4 C. R. Rao, Linear Statistical Inference and Its Applications, John Wiley & Sons, New York, 1965, p. 258.

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ˆ2 of σ 2 = ui /(n − 2), as we have seen, is unbiased. But comparing these two estimators of σ 2 , we see that as the sample size n gets larger the two estimators of σ 2 tend to be equal. Thus, asymptotically (i.e., as n increases indeﬁnitely), the ML estimator of σ 2 is also unbiased. Since the method of least squares with the added assumption of normality of ui provides us with all the tools necessary for both estimation and hypothesis testing of the linear regression models, there is no loss for readers who may not want to pursue the maximum likelihood method because of its slight mathematical complexity.

4.5 SUMMARY AND CONCLUSIONS

1. This chapter discussed the classical normal linear regression model (CNLRM). 2. This model differs from the classical linear regression model (CLRM) in that it speciﬁcally assumes that the disturbance term ui entering the regression model is normally distributed. The CLRM does not require any assumption about the probability distribution of ui; it only requires that the mean value of ui is zero and its variance is a ﬁnite constant. 3. The theoretical justiﬁcation for the normality assumption is the central limit theorem. 4. Without the normality assumption, under the other assumptions discussed in Chapter 3, the Gauss–Markov theorem showed that the OLS estimators are BLUE. 5. With the additional assumption of normality, the OLS estimators are not only best unbiased estimators (BUE) but also follow well-known probability distributions. The OLS estimators of the intercept and slope are themselves normally distributed and the OLS estimator of the variance of ui ( = σ 2 ) is related to the chi-square distribution. ˆ 6. In Chapters 5 and 8 we show how this knowledge is useful in drawing inferences about the values of the population parameters. 7. An alternative to the least-squares method is the method of maximum likelihood (ML). To use this method, however, one must make an assumption about the probability distribution of the disturbance term ui. In the regression context, the assumption most popularly made is that ui follows the normal distribution. 8. Under the normality assumption, the ML and OLS estimators of the intercept and slope parameters of the regression model are identical. However, the OLS and ML estimators of the variance of ui are different. In large samples, however, these two estimators converge. 9. Thus the ML method is generally called a large-sample method. The ML method is of broader application in that it can also be applied to regression models that are nonlinear in the parameters. In the latter case, OLS is generally not used. For more on this, see Chapter 14.

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10. In this text, we will largely rely on the OLS method for practical reasons: (a) Compared to ML, the OLS is easy to apply; (b) the ML and OLS estimators of β1 and β2 are identical (which is true of multiple regressions too); and (c) even in moderately large samples the OLS and ML estimators of σ 2 do not differ vastly. However, for the beneﬁt of the mathematically inclined reader, a brief introduction to ML is given in the appendix to this chapter and also in Appendix A.

APPENDIX 4A

4A.1 MAXIMUM LIKELIHOOD ESTIMATION OF TWO-VARIABLE REGRESSION MODEL

Assume that in the two-variable model Yi = β1 + β2 Xi + ui the Yi are normally and independently distributed with mean = β1 + β2 Xi and variance = σ 2 . [See Eq. (4.3.9).] As a result, the joint probability density function of Y1 , Y2 , . . . , Yn, given the preceding mean and variance, can be written as f (Y1 , Y2 , . . . , Yn | β1 + β2 Xi , σ 2 ) But in view of the independence of the Y’s, this joint probability density function can be written as a product of n individual density functions as f (Y1 , Y2 , . . . , Yn | β1 + β2 Xi , σ 2 ) = f (Y1 | β1 + β2 Xi , σ 2 ) f (Y2 | β1 + β2 Xi , σ 2 ) · · · f (Yn | β1 + β2 Xi , σ 2 ) where f (Yi ) = 1 1 (Yi − β1 − β2 Xi )2 exp − √ 2 σ2 σ 2π (2) (1)

which is the density function of a normally distributed variable with the given mean and variance. (Note: exp means e to the power of the expression indicated by {}.) Substituting (2) for each Yi into (1) gives f (Yi , Y2 , . . . , Yn | β1 + β2 Xi , σ 2 ) = 1 √ n σ 2π n exp −

1 2

(Yi − β1 − β2 Xi )2 σ2 (3)

If Y1 , Y2 , . . . , Yn are known or given, but β1 , β2 , and σ 2 are not known, the function in (3) is called a likelihood function, denoted by LF(β1 , β2 , σ 2 ),

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and written as1 LF(β1 , β2 , σ 2 ) = 1 √ σ n 2π exp − 1 2 (Yi − β1 − β2 Xi )2 σ2 (4)

n

The method of maximum likelihood, as the name indicates, consists in estimating the unknown parameters in such a manner that the probability of observing the given Y’s is as high (or maximum) as possible. Therefore, we have to ﬁnd the maximum of the function (4). This is a straightforward exercise in differential calculus. For differentiation it is easier to express (4) in the log term as follows.2 (Note: ln = natural log.) ln LF = −nln σ − n 1 ln (2π) − 2 2 (Yi − β1 − β2 Xi )2 σ2 (Yi − β1 − β2 Xi )2 σ2 (5)

n n 1 = − ln σ 2 − ln (2π) − 2 2 2

Differentiating (5) partially with respect to β1 , β2 , and σ 2, we obtain 1 ∂ ln LF =− 2 ∂β1 σ ∂ ln LF 1 =− 2 ∂β2 σ (Yi − β1 − β2 Xi )(−1) (Yi − β1 − β2 Xi )(−Xi ) (Yi − β1 − β2 Xi )2 (6) (7) (8)

∂ ln LF n 1 =− 2 + ∂σ 2 2σ 2σ 4

Setting these equations equal to zero (the ﬁrst-order condition for opti˜ ˜ ˜ mization) and letting β1 , β2 , and σ 2 denote the ML estimators, we obtain3 1 σ2 ˜ 1 σ2 ˜ n 1 − 2+ 2σ ˜ 2σ 4 ˜ ˜ ˜ (Yi − β1 − β2 Xi ) = 0 ˜ ˜ (Yi − β1 − β2 Xi )Xi = 0 ˜ ˜ (Yi − β1 − β2 Xi )2 = 0 (9) (10) (11)

1 Of course, if β1 , β2 , and σ 2 are known but the Yi are not known, (4) represents the joint probability density function—the probability of jointly observing the Yi . 2 Since a log function is a monotonic function, ln LF will attain its maximum value at the same point as LF. 3 We use ˜ (tilde) for ML estimators and ˆ (cap or hat) for OLS estimators.

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After simplifying, Eqs. (9) and (10) yield ˜ ˜ Yi = nβ1 + β2 ˜ Yi Xi = β1 Xi Xi2 (12) (13)

˜ Xi + β2

which are precisely the normal equations of the least-squares theory ob˜ tained in (3.1.4) and (3.1.5). Therefore, the ML estimators, the β’s, are the ˆ given in (3.1.6) and (3.1.7). This equalsame as the OLS estimators, the β’s, ity is not accidental. Examining the likelihood (5), we see that the last term enters with a negative sign. Therefore, maximizing (5) amounts to minimizing this term, which is precisely the least-squares approach, as can be seen from (3.1.2). Substituting the ML ( = OLS) estimators into (11) and simplifying, we ˜ obtain the ML estimator of σ 2 as σ2 = ˜ = = 1 n 1 n 1 n ˜ ˜ (Yi − β1 − β2 Xi )2 ˆ ˆ (Yi − β1 − β2 Xi )2 ui ˆ2 (14)

˜ From (14) it is obvious that the ML estimator σ 2 differs from the OLS ˆ ˆ2 estimator σ 2 = [1/(n − 2)] ui , which was shown to be an unbiased estimator of σ 2 in Appendix 3A, Section 3A.5. Thus, the ML estimator of σ 2 is biased. The magnitude of this bias can be easily determined as follows. Taking the mathematical expectation of (14) on both sides, we obtain E(σ 2 ) = ˜ = 1 E n ui ˆ2 using Eq. (16) of Appendix 3A, Section 3A.5 (15)

n− 2 2 σ n 2 2 σ n

= σ2 −

˜ which shows that σ 2 is biased downward (i.e., it underestimates the true σ 2 ) in small samples. But notice that as n, the sample size, increases indeﬁnitely, the second term in (15), the bias factor, tends to be zero. There˜ fore, asymptotically (i.e., in a very large sample), σ 2 is unbiased too, that ˜ ˜ is, lim E(σ 2 ) = σ 2 as n → ∞. It can further be proved that σ 2 is also a

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˜ consistent estimator4; that is, as n increases indeﬁnitely σ 2 converges to its true value σ 2 .

4A.2 MAXIMUM LIKELIHOOD ESTIMATION OF FOOD EXPENDITURE IN INDIA

Return to Example 3.2 and regression (3.7.2), which gives the regression of food expenditure on total expenditure for 55 rural households in India. Since under the normality assumption the OLS and ML estimators of the regression coefﬁcients are the same, we obtain the ML estimators as ˆ ˜ ˆ ˜ β1 = β1 = 94.2087 and β2 = β2 = 0.4386. The OLS estimator of σ 2 is σ 2 = 4469.6913, but the ML estimator is σ 2 = 4407.1563, which is smaller ˆ ˜ than the OLS estimator. As noted, in small samples the ML estimator is downward biased; that is, on average it underestimates the true variance σ 2 . Of course, as you would expect, as the sample size gets bigger, the difference between the two estimators will narrow. Putting the values of the estimators in the log likelihood function, we obtain the value of −308.1625. If you want the maximum value of the LF, just take the antilog of −308.1625. No other values of the parameters will give you a higher probability of obtaining the sample that you have used in the analysis.

APPENDIX 4A EXERCISES 4.1. “If two random variables are statistically independent, the coefﬁcient of correlation between the two is zero. But the converse is not necessarily true; that is, zero correlation does not imply statistical independence. However, if two variables are normally distributed, zero correlation necessarily implies statistical independence.” Verify this statement for the following joint probability density function of two normally distributed variables Y1 and Y2 (this joint probability density function is known as the bivariate normal probability density function): f (Y1 , Y2 ) = 1 2π σ1 σ2 1 − × Y1 − µ 1 σ1 ρ2

2

exp −

1 2(1 − ρ 2 ) Y2 − µ 2 σ2

2

− 2ρ

(Y1 − µ 1 )(Y2 − µ 2 ) + σ1 σ2

4 See App. A for a general discussion of the properties of the maximum likelihood estimators as well as for the distinction between asymptotic unbiasedness and consistency. Roughly ˜2 speaking, in asymptotic unbiasedness we try to ﬁnd out the lim E(σn ) as n tends to inﬁnity, where n is the sample size on which the estimator is based, whereas in consistency we try to ˜2 ﬁnd out how σn behaves as n increases indeﬁnitely. Notice that the unbiasedness property is a repeated sampling property of an estimator based on a sample of given size, whereas in consistency we are concerned with the behavior of an estimator as the sample size increases indeﬁnitely.

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where µ 1 = mean of Y1 µ 2 = mean of Y2 σ1 = standard deviation of Y1 σ2 = standard deviation of Y2 ρ = coefﬁcient of correlation between Y1 and Y2 4.2. By applying the second-order conditions for optimization (i.e., secondderivative test), show that the ML estimators of β1 , β2 , and σ 2 obtained by solving Eqs. (9), (10), and (11) do in fact maximize the likelihood function (4). 4.3. A random variable X follows the exponential distribution if it has the following probability density function (PDF): f ( X) = (1/θ )e − X/θ =0 for X > 0

elsewhere

where θ > 0 is the parameter of the distribution. Using the ML method, ˆ show that the ML estimator of θ is θ = X i /n, where n is the sample size. ¯ That is, show that the ML estimator of θ is the sample mean X.

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TWO-VARIABLE REGRESSION: INTERVAL ESTIMATION AND HYPOTHESIS TESTING

Beware of testing too many hypotheses; the more you torture the data, the more likely they are to confess, but confession obtained under duress may not be admissible in the court of scientiﬁc opinion.1

As pointed out in Chapter 4, estimation and hypothesis testing constitute the two major branches of classical statistics. The theory of estimation consists of two parts: point estimation and interval estimation. We have discussed point estimation thoroughly in the previous two chapters where we introduced the OLS and ML methods of point estimation. In this chapter we ﬁrst consider interval estimation and then take up the topic of hypothesis testing, a topic intimately related to interval estimation.

5.1 STATISTICAL PREREQUISITES

Before we demonstrate the actual mechanics of establishing conﬁdence intervals and testing statistical hypotheses, it is assumed that the reader is familiar with the fundamental concepts of probability and statistics. Although not a substitute for a basic course in statistics, Appendix A provides the essentials of statistics with which the reader should be totally familiar. Key concepts such as probability, probability distributions, Type I and Type II errors, level of signiﬁcance, power of a statistical test, and conﬁdence interval are crucial for understanding the material covered in this and the following chapters.

1 Stephen M. Stigler, “Testing Hypothesis or Fitting Models? Another Look at Mass Extinctions,” in Matthew H. Nitecki and Antoni Hoffman, eds., Neutral Models in Biology, Oxford University Press, Oxford, 1987, p. 148.

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5.2

INTERVAL ESTIMATION: SOME BASIC IDEAS

To ﬁx the ideas, consider the hypothetical consumption-income example of Chapter 3. Equation (3.6.2) shows that the estimated marginal propensity to consume (MPC) β2 is 0.5091, which is a single (point) estimate of the unknown population MPC β2 . How reliable is this estimate? As noted in Chapter 3, because of sampling ﬂuctuations, a single estimate is likely to differ from the true value, although in repeated sampling its mean value is ˆ expected to be equal to the true value. [Note: E(β2 ) = β2 .] Now in statistics the reliability of a point estimator is measured by its standard error. Therefore, instead of relying on the point estimate alone, we may construct an interval around the point estimator, say within two or three standard errors on either side of the point estimator, such that this interval has, say, 95 percent probability of including the true parameter value. This is roughly the idea behind interval estimation. To be more speciﬁc, assume that we want to ﬁnd out how “close” is, say, ˆ β2 to β2 . For this purpose we try to ﬁnd out two positive numbers δ and α, the latter lying between 0 and 1, such that the probability that the random ˆ ˆ interval (β2 − δ, β2 + δ) contains the true β2 is 1 − α. Symbolically, ˆ ˆ Pr (β2 − δ ≤ β2 ≤ β2 + δ) = 1 − α (5.2.1)

Such an interval, if it exists, is known as a conﬁdence interval; 1 − α is known as the conﬁdence coefﬁcient; and α (0 < α < 1) is known as the level of signiﬁcance.2 The endpoints of the conﬁdence interval are known ˆ as the conﬁdence limits (also known as critical values), β2 − δ being the ˆ lower conﬁdence limit and β2 + δ the upper conﬁdence limit. In passing, note that in practice α and 1 − α are often expressed in percentage forms as 100α and 100(1 − α) percent. Equation (5.2.1) shows that an interval estimator, in contrast to a point estimator, is an interval constructed in such a manner that it has a speciﬁed probability 1 − α of including within its limits the true value of the parameter. For example, if α = 0.05, or 5 percent, (5.2.1) would read: The probability that the (random) interval shown there includes the true β2 is 0.95, or 95 percent. The interval estimator thus gives a range of values within which the true β2 may lie. It is very important to know the following aspects of interval estimation: 1. Equation (5.2.1) does not say that the probability of β2 lying between the given limits is 1 − α. Since β2 , although an unknown, is assumed to be some ﬁxed number, either it lies in the interval or it does not. What (5.2.1)

2 Also known as the probability of committing a Type I error. A Type I error consists in rejecting a true hypothesis, whereas a Type II error consists in accepting a false hypothesis. (This topic is discussed more fully in App. A.) The symbol α is also known as the size of the (statistical) test.

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states is that, for the method described in this chapter, the probability of constructing an interval that contains β2 is 1 − α. 2. The interval (5.2.1) is a random interval; that is, it will vary from one ˆ sample to the next because it is based on β2 , which is random. (Why?) 3. Since the conﬁdence interval is random, the probability statements attached to it should be understood in the long-run sense, that is, repeated sampling. More speciﬁcally, (5.2.1) means: If in repeated sampling conﬁdence intervals like it are constructed a great many times on the 1 − α probability basis, then, in the long run, on the average, such intervals will enclose in 1 − α of the cases the true value of the parameter. ˆ 4. As noted in 2, the interval (5.2.1) is random so long as β2 is not known. But once we have a speciﬁc sample and once we obtain a speciﬁc numerical ˆ value of β2 , the interval (5.2.1) is no longer random; it is ﬁxed. In this case, we cannot make the probabilistic statement (5.2.1); that is, we cannot say that the probability is 1 − α that a given ﬁxed interval includes the true β2 . In this situation β2 is either in the ﬁxed interval or outside it. Therefore, the probability is either 1 or 0. Thus, for our hypothetical consumption-income example, if the 95% conﬁdence interval were obtained as (0.4268 ≤ β2 ≤ 0.5914), as we do shortly in (5.3.9), we cannot say the probability is 95% that this interval includes the true β2 . That probability is either 1 or 0. How are the conﬁdence intervals constructed? From the preceding discussion one may expect that if the sampling or probability distributions of the estimators are known, one can make conﬁdence interval statements such as (5.2.1). In Chapter 4 we saw that under the assumption of normalˆ ˆ ity of the disturbances ui the OLS estimators β1 and β2 are themselves 2 ˆ normally distributed and that the OLS estimator σ is related to the χ 2 (chisquare) distribution. It would then seem that the task of constructing conﬁdence intervals is a simple one. And it is!

5.3 CONFIDENCE INTERVALS FOR REGRESSION COEFFICIENTS β1 AND β2 Conﬁdence Interval for β2

It was shown in Chapter 4, Section 4.3, that, with the normality assumpˆ ˆ tion for ui, the OLS estimators β1 and β2 are themselves normally distributed with means and variances given therein. Therefore, for example, the variable Z= ˆ β2 − β2 ˆ se (β2 ) ˆ (β2 − β2 ) σ xi2 (5.3.1)

=

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as noted in (4.3.6), is a standardized normal variable. It therefore seems that we can use the normal distribution to make probabilistic statements about β2 provided the true population variance σ 2 is known. If σ 2 is known, an important property of a normally distributed variable with mean µ and variance σ 2 is that the area under the normal curve between µ ± σ is about 68 percent, that between the limits µ ± 2σ is about 95 percent, and that between µ ± 3σ is about 99.7 percent. But σ 2 is rarely known, and in practice it is determined by the unbiased ˆ ˆ estimator σ 2 . If we replace σ by σ , (5.3.1) may be written as t= ˆ β2 − β2 estimator − parameter = ˆ estimated standard error of estimator se (β2 ) ˆ (β2 − β2 ) σ ˆ xi2

(5.3.2)

=

ˆ where the se (β2 ) now refers to the estimated standard error. It can be shown (see Appendix 5A, Section 5A.2) that the t variable thus deﬁned follows the t distribution with n − 2 df. [Note the difference between (5.3.1) and (5.3.2).] Therefore, instead of using the normal distribution, we can use the t distribution to establish a conﬁdence interval for β2 as follows: Pr (−tα/2 ≤ t ≤ tα/2 ) = 1 − α (5.3.3)

where the t value in the middle of this double inequality is the t value given by (5.3.2) and where tα/2 is the value of the t variable obtained from the t distribution for α/2 level of signiﬁcance and n − 2 df; it is often called the critical t value at α/2 level of signiﬁcance. Substitution of (5.3.2) into (5.3.3) yields Pr −tα/2 ≤ Rearranging (5.3.4), we obtain ˆ ˆ ˆ ˆ Pr [β2 − tα/2 se (β2 ) ≤ β2 ≤ β2 + tα/2 se (β2 )] = 1 − α

3

ˆ β2 − β2 ≤ tα/2 = 1 − α ˆ se (β2 )

(5.3.4)

(5.3.5)3

Some authors prefer to write (5.3.5) with the df explicitly indicated. Thus, they would write ˆ ˆ ˆ ˆ Pr [β2 − t(n−2),α/2 se (β2 ) ≤ β2 ≤ β2 + t(n−2)α/2 se (β2 )] = 1 − α

But for simplicity we will stick to our notation; the context clariﬁes the appropriate df involved.

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Equation (5.3.5) provides a 100(1 − α) percent conﬁdence interval for β2 , which can be written more compactly as 100(1 − α)% conﬁdence interval for β2 : ˆ ˆ β2 ± tα/2 se (β2 ) (5.3.6)

Arguing analogously, and using (4.3.1) and (4.3.2), we can then write: ˆ ˆ ˆ ˆ Pr [β1 − tα/2 se (β1 ) ≤ β1 ≤ β1 + tα/2 se (β1 )] = 1 − α or, more compactly, 100(1 − α)% conﬁdence interval for β1 : ˆ ˆ β1 ± tα/2 se (β1 ) (5.3.8) (5.3.7)

Notice an important feature of the conﬁdence intervals given in (5.3.6) and (5.3.8): In both cases the width of the conﬁdence interval is proportional to the standard error of the estimator. That is, the larger the standard error, the larger is the width of the conﬁdence interval. Put differently, the larger the standard error of the estimator, the greater is the uncertainty of estimating the true value of the unknown parameter. Thus, the standard error of an estimator is often described as a measure of the precision of the estimator, i.e., how precisely the estimator measures the true population value. Returning to our illustrative consumption–income example, in Chapter 3 ˆ ˆ (Section 3.6) we found that β2 = 0.5091, se (β2 ) = 0.0357, and df = 8. If we assume α = 5%, that is, 95% conﬁdence coefﬁcient, then the t table shows that for 8 df the critical tα/2 = t0.025 = 2.306. Substituting these values in (5.3.5), the reader should verify that the 95% conﬁdence interval for β2 is as follows: 0.4268 ≤ β2 ≤ 0.5914 Or, using (5.3.6), it is 0.5091 ± 2.306(0.0357) that is, 0.5091 ± 0.0823 (5.3.10) (5.3.9)

The interpretation of this conﬁdence interval is: Given the conﬁdence coefﬁcient of 95%, in the long run, in 95 out of 100 cases intervals like

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(0.4268, 0.5914) will contain the true β2 . But, as warned earlier, we cannot say that the probability is 95 percent that the speciﬁc interval (0.4268 to 0.5914) contains the true β2 because this interval is now ﬁxed and no longer random; therefore, β2 either lies in it or does not: The probability that the speciﬁed ﬁxed interval includes the true β2 is therefore 1 or 0.

Conﬁdence Interval for β1

Following (5.3.7), the reader can easily verify that the 95% conﬁdence interval for β1 of our consumption–income example is 9.6643 ≤ β1 ≤ 39.2448 Or, using (5.3.8), we ﬁnd it is 24.4545 ± 2.306(6.4138) that is, 24.4545 ± 14.7902 (5.3.12) (5.3.11)

Again you should be careful in interpreting this conﬁdence interval. In the long run, in 95 out of 100 cases intervals like (5.3.11) will contain the true β1 ; the probability that this particular ﬁxed interval includes the true β1 is either 1 or 0.

Conﬁdence Interval for β1 and β2 Simultaneously

There are occasions when one needs to construct a joint conﬁdence interval for β1 and β2 such that with a conﬁdence coefﬁcient (1 − α), say, 95%, that interval includes β1 and β2 simultaneously. Since this topic is involved, the interested reader may want to consult appropriate references.4 We will touch on this topic brieﬂy in Chapters 8 and 10.

5.4

CONFIDENCE INTERVAL FOR σ2

As pointed out in Chapter 4, Section 4.3, under the normality assumption, the variable χ 2 = (n − 2) σ2 ˆ σ2 (5.4.1)

4 For an accessible discussion, see John Neter, William Wasserman, and Michael H. Kutner, Applied Linear Regression Models, Richard D. Irwin, Homewood, Ill., 1983, Chap. 5.

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f(χ2)

Density 2.5%

95%

2.5%

2.1797 χ2

0.975

17.5346 χ2

0.025

χ2

FIGURE 5.1

The 95% conﬁdence interval for χ 2 (8 df).

follows the χ 2 distribution with n − 2 df.5 Therefore, we can use the χ 2 distribution to establish a conﬁdence interval for σ 2

2 2 Pr χ1−α/2 ≤ χ 2 ≤ χα/2 = 1 − α

(5.4.2)

where the χ 2 value in the middle of this double inequality is as given by (5.4.1) 2 2 and where χ1−α/2 and χα/2 are two values of χ 2 (the critical χ 2 values) obtained from the chi-square table for n − 2 df in such a manner that they cut off 100(α/2) percent tail areas of the χ 2 distribution, as shown in Figure 5.1. Substituting χ 2 from (5.4.1) into (5.4.2) and rearranging the terms, we obtain Pr (n − 2) σ2 ˆ σ2 ˆ ≤ σ 2 ≤ (n − 2) 2 2 χα/2 χ1−α/2 =1−α

(5.4.3)

which gives the 100(1 − α)% conﬁdence interval for σ 2 . To illustrate, consider this example. From Chapter 3, Section 3.6, we obˆ tain σ 2 = 42.1591 and df = 8. If α is chosen at 5 percent, the chi-square table 2 2 for 8 df gives the following critical values: χ0.025 = 17.5346, and χ0.975 = 2.1797. These values show that the probability of a chi-square value exceeding 17.5346 is 2.5 percent and that of 2.1797 is 97.5 percent. Therefore, the interval between these two values is the 95% conﬁdence interval for χ 2 , as shown diagrammatically in Figure 5.1. (Note the skewed characteristic of the chi-square distribution.)

5 For proof, see Robert V. Hogg and Allen T. Craig, Introduction to Mathematical Statistics, 2d ed., Macmillan, New York, 1965, p. 144.

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Substituting the data of our example into (5.4.3), the reader should verify that the 95% conﬁdence interval for σ 2 is as follows: 19.2347 ≤ σ 2 ≤ 154.7336 (5.4.4)

The interpretation of this interval is: If we establish 95% conﬁdence limits on σ 2 and if we maintain a priori that these limits will include true σ 2 , we shall be right in the long run 95 percent of the time.

5.5 HYPOTHESIS TESTING: GENERAL COMMENTS

Having discussed the problem of point and interval estimation, we shall now consider the topic of hypothesis testing. In this section we discuss brieﬂy some general aspects of this topic; Appendix A gives some additional details. The problem of statistical hypothesis testing may be stated simply as follows: Is a given observation or ﬁnding compatible with some stated hypothesis or not? The word “compatible,” as used here, means “sufﬁciently” close to the hypothesized value so that we do not reject the stated hypothesis. Thus, if some theory or prior experience leads us to believe that the true slope coefﬁcient β2 of the consumption–income example is unity, is the obˆ served β2 = 0.5091 obtained from the sample of Table 3.2 consistent with the stated hypothesis? If it is, we do not reject the hypothesis; otherwise, we may reject it. In the language of statistics, the stated hypothesis is known as the null hypothesis and is denoted by the symbol H0 . The null hypothesis is usually tested against an alternative hypothesis (also known as maintained hypothesis) denoted by H1 , which may state, for example, that true β2 is different from unity. The alternative hypothesis may be simple or composite.6 For example, H1 : β2 = 1.5 is a simple hypothesis, but H1 : β2 = 1.5 is a composite hypothesis. The theory of hypothesis testing is concerned with developing rules or procedures for deciding whether to reject or not reject the null hypothesis. There are two mutually complementary approaches for devising such rules, namely, conﬁdence interval and test of signiﬁcance. Both these approaches predicate that the variable (statistic or estimator) under consideration has some probability distribution and that hypothesis testing involves making statements or assertions about the value(s) of the parameter(s) of such distribution. For example, we know that with the normality assumpˆ tion β2 is normally distributed with mean equal to β2 and variance given by (4.3.5). If we hypothesize that β2 = 1, we are making an assertion about one

6 A statistical hypothesis is called a simple hypothesis if it speciﬁes the precise value(s) of the parameter(s) of a probability density function; otherwise, it is called a composite hy√ pothesis. For example, in the normal pdf (1/σ 2π) exp {− 1 [(X − µ)/σ ]2 }, if we assert that 2 H1 : µ = 15 and σ = 2, it is a simple hypothesis; but if H1 : µ = 15 and σ > 15, it is a composite hypothesis, because the standard deviation does not have a speciﬁc value.

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of the parameters of the normal distribution, namely, the mean. Most of the statistical hypotheses encountered in this text will be of this type—making assertions about one or more values of the parameters of some assumed probability distribution such as the normal, F, t, or χ 2 . How this is accomplished is discussed in the following two sections.

5.6 HYPOTHESIS TESTING: THE CONFIDENCE-INTERVAL APPROACH Two-Sided or Two-Tail Test

To illustrate the conﬁdence-interval approach, once again we revert to the consumption–income example. As we know, the estimated marginal propenˆ sity to consume (MPC), β2 , is 0.5091. Suppose we postulate that H0 : β2 = 0.3 H1 : β2 = 0.3 that is, the true MPC is 0.3 under the null hypothesis but it is less than or greater than 0.3 under the alternative hypothesis. The null hypothesis is a simple hypothesis, whereas the alternative hypothesis is composite; actually it is what is known as a two-sided hypothesis. Very often such a two-sided alternative hypothesis reﬂects the fact that we do not have a strong a priori or theoretical expectation about the direction in which the alternative hypothesis should move from the null hypothesis. ˆ Is the observed β2 compatible with H0 ? To answer this question, let us refer to the conﬁdence interval (5.3.9). We know that in the long run intervals like (0.4268, 0.5914) will contain the true β2 with 95 percent probability. Consequently, in the long run (i.e., repeated sampling) such intervals provide a range or limits within which the true β2 may lie with a conﬁdence coefﬁcient of, say, 95%. Thus, the conﬁdence interval provides a set of plausible null hypotheses. Therefore, if β2 under H0 falls within the 100(1 − α)% conﬁdence interval, we do not reject the null hypothesis; if it lies outside the interval, we may reject it.7 This range is illustrated schematically in Figure 5.2.

Decision Rule: Construct a 100(1 − α)% conﬁdence interval for β2. If the β2 under H 0 falls within this conﬁdence interval, do not reject H 0, but if it falls outside this interval, reject H 0.

Following this rule, for our hypothetical example, H0 : β2 = 0.3 clearly lies outside the 95% conﬁdence interval given in (5.3.9). Therefore, we can reject

7 Always bear in mind that there is a 100α percent chance that the conﬁdence interval does not contain β2 under H0 even though the hypothesis is correct. In short, there is a 100α percent chance of committing a Type I error. Thus, if α = 0.05, there is a 5 percent chance that we could reject the null hypothesis even though it is true.

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Values of β 2 lying in this interval are plausible under H0 with 100(1 – α )% confidence. Hence, do not reject H0 if β2 lies in this region.

β2 – tα se( β 2) α/2

FIGURE 5.2 A 100(1 − α)% conﬁdence interval for β2.

β 2 + tα se( β 2) α/2

the hypothesis that the true MPC is 0.3, with 95% conﬁdence. If the null hypothesis were true, the probability of our obtaining a value of MPC of as much as 0.5091 by sheer chance or ﬂuke is at the most about 5 percent, a small probability. In statistics, when we reject the null hypothesis, we say that our ﬁnding is statistically signiﬁcant. On the other hand, when we do not reject the null hypothesis, we say that our ﬁnding is not statistically signiﬁcant. Some authors use a phrase such as “highly statistically signiﬁcant.” By this they usually mean that when they reject the null hypothesis, the probability of committing a Type I error (i.e., α) is a small number, usually 1 percent. But as our discussion of the p value in Section 5.8 will show, it is better to leave it to the researcher to decide whether a statistical ﬁnding is “signiﬁcant,” “moderately signiﬁcant,” or “highly signiﬁcant.”

One-Sided or One-Tail Test

Sometimes we have a strong a priori or theoretical expectation (or expectations based on some previous empirical work) that the alternative hypothesis is one-sided or unidirectional rather than two-sided, as just discussed. Thus, for our consumption–income example, one could postulate that H0 : β2 ≤ 0.3 and H1 : β2 > 0.3

Perhaps economic theory or prior empirical work suggests that the marginal propensity to consume is greater than 0.3. Although the procedure to test this hypothesis can be easily derived from (5.3.5), the actual mechanics are better explained in terms of the test-of-signiﬁcance approach discussed next.8

8 If you want to use the conﬁdence interval approach, construct a (100 − α)% one-sided or one-tail conﬁdence interval for β2. Why?

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5.7 HYPOTHESIS TESTING: THE TEST-OF-SIGNIFICANCE APPROACH Testing the Signiﬁcance of Regression Coefﬁcients: The t Test

An alternative but complementary approach to the conﬁdence-interval method of testing statistical hypotheses is the test-of-signiﬁcance approach developed along independent lines by R. A. Fisher and jointly by Neyman and Pearson.9 Broadly speaking, a test of signiﬁcance is a procedure by which sample results are used to verify the truth or falsity of a null hypothesis. The key idea behind tests of signiﬁcance is that of a test statistic (estimator) and the sampling distribution of such a statistic under the null hypothesis. The decision to accept or reject H0 is made on the basis of the value of the test statistic obtained from the data at hand. As an illustration, recall that under the normality assumption the variable t= ˆ β2 − β2 ˆ se (β2 ) ˆ (β2 − β2 ) σ ˆ xi2

(5.3.2)

=

follows the t distribution with n − 2 df. If the value of true β2 is speciﬁed under the null hypothesis, the t value of (5.3.2) can readily be computed from the available sample, and therefore it can serve as a test statistic. And since this test statistic follows the t distribution, conﬁdence-interval statements such as the following can be made: Pr −tα/2 ≤

* ˆ β2 − β2 ≤ tα/2 = 1 − α ˆ se (β2 )

(5.7.1)

* where β2 is the value of β2 under H0 and where −tα/2 and tα/2 are the values of t (the critical t values) obtained from the t table for (α/2) level of signiﬁcance and n − 2 df [cf. (5.3.4)]. The t table is given in Appendix D. Rearranging (5.7.1), we obtain * * ˆ ˆ ˆ Pr [β2 − tα/2 se (β2 ) ≤ β2 ≤ β2 + tα/2 se (β2 )] = 1 − α

(5.7.2)

ˆ which gives the interval in which β2 will fall with 1 − α probability, given * β2 = β2 . In the language of hypothesis testing, the 100(1 − α)% conﬁdence interval established in (5.7.2) is known as the region of acceptance (of

9 Details may be found in E. L. Lehman, Testing Statistical Hypotheses, John Wiley & Sons, New York, 1959.

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the null hypothesis) and the region(s) outside the conﬁdence interval is (are) called the region(s) of rejection (of H0 ) or the critical region(s). As noted previously, the conﬁdence limits, the endpoints of the conﬁdence interval, are also called critical values. The intimate connection between the conﬁdence-interval and test-ofsigniﬁcance approaches to hypothesis testing can now be seen by comparing (5.3.5) with (5.7.2). In the conﬁdence-interval procedure we try to establish a range or an interval that has a certain probability of including the true but unknown β2 , whereas in the test-of-signiﬁcance approach we hypotheˆ size some value for β2 and try to see whether the computed β2 lies within reasonable (conﬁdence) limits around the hypothesized value. Once again let us revert to our consumption–income example. We know ˆ ˆ that β2 = 0.5091, se (β2 ) = 0.0357, and df = 8. If we assume α = 5 percent, * tα/2 = 2.306. If we let H0 : β2 = β2 = 0.3 and H1 : β2 = 0.3, (5.7.2) becomes ˆ Pr (0.2177 ≤ β2 ≤ 0.3823) = 0.95 (5.7.3)10

ˆ as shown diagrammatically in Figure 5.3. Since the observed β2 lies in the β2 = 0.3. critical region, we reject the null hypothesis that true In practice, there is no need to estimate (5.7.2) explicitly. One can compute the t value in the middle of the double inequality given by (5.7.1) and see whether it lies between the critical t values or outside them. For our example, t= f ( β2)

0.5091 − 0.3 = 5.86 0.0357

(5.7.4)

Density

Critical region 2.5% 0.2177 FIGURE 5.3 0.3 0.3823

β2 b b2 = 0.5091 lies in this critical region 2.5%

ˆ β2

ˆ The 95% conﬁdence interval for β 2 under the hypothesis that β2 = 0.3.

10 In Sec. 5.2, point 4, it was stated that we cannot say that the probability is 95 percent that the ﬁxed interval (0.4268, 0.5914) includes the true β2. But we can make the probabilistic stateˆ ment given in (5.7.3) because β2 , being an estimator, is a random variable.

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f(t)

Density

Critical region 2.5%

95% Region of acceptance

t = 5.86 lies in this critical region 2.5% t

– 2.306 FIGURE 5.4

0

+2.306

The 95% conﬁdence interval for t(8 df).

which clearly lies in the critical region of Figure 5.4. The conclusion remains the same; namely, we reject H0 . ˆ Notice that if the estimated β2 ( = β2 ) is equal to the hypothesized β2 , the t value in (5.7.4) will be zero. However, as the estimated β2 value departs from the hypothesized β2 value, |t| (that is, the absolute t value; note: t can be positive as well as negative) will be increasingly large. Therefore, a “large” |t| value will be evidence against the null hypothesis. Of course, we can always use the t table to determine whether a particular t value is large or small; the answer, as we know, depends on the degrees of freedom as well as on the probability of Type I error that we are willing to accept. If you take a look at the t table given in Appendix D, you will observe that for any given value of df the probability of obtaining an increasingly large |t| value becomes progressively smaller. Thus, for 20 df the probability of obtaining a |t| value of 1.725 or greater is 0.10 or 10 percent, but for the same df the probability of obtaining a |t| value of 3.552 or greater is only 0.002 or 0.2 percent. Since we use the t distribution, the preceding testing procedure is called appropriately the t test. In the language of signiﬁcance tests, a statistic is said to be statistically signiﬁcant if the value of the test statistic lies in the critical region. In this case the null hypothesis is rejected. By the same token, a test is said to be statistically insigniﬁcant if the value of the test statistic lies in the acceptance region. In this situation, the null hypothesis is not rejected. In our example, the t test is signiﬁcant and hence we reject the null hypothesis. Before concluding our discussion of hypothesis testing, note that the testing procedure just outlined is known as a two-sided, or two-tail, testof-signiﬁcance procedure in that we consider the two extreme tails of the relevant probability distribution, the rejection regions, and reject the null hypothesis if it lies in either tail. But this happens because our H1 was a

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two-sided composite hypothesis; β2 = 0.3 means β2 is either greater than or less than 0.3. But suppose prior experience suggests to us that the MPC is expected to be greater than 0.3. In this case we have: H0 : β2 ≤ 0.3 and H1 : β2 > 0.3. Although H1 is still a composite hypothesis, it is now one-sided. To test this hypothesis, we use the one-tail test (the right tail), as shown in Figure 5.5. (See also the discussion in Section 5.6.) The test procedure is the same as before except that the upper conﬁdence limit or critical value now corresponds to tα = t.05 , that is, the 5 percent level. As Figure 5.5 shows, we need not consider the lower tail of the t distribution in this case. Whether one uses a two- or one-tail test of signiﬁcance will depend upon how the alternative hypothesis is formulated, which, in turn, may depend upon some a priori considerations or prior empirical experience. (But more on this in Section 5.8.) We can summarize the t test of signiﬁcance approach to hypothesis testing as shown in Table 5.1.

f( β 2) b

Density

95% Region of acceptance

β2 b b2 = 0.5091 lies in this critical region 2.5% β2 b

0.3 f(t)

0.3664

[b2 + 1.860 se( β 2 )] b b β* b2

Density

95% Region of acceptance

0

1.860

t = 5.86 lies in this critical region 5% t

t 0.05 (8 df ) FIGURE 5.5 One-tail test of signiﬁcance.

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TABLE 5.1

THE t TEST OF SIGNIFICANCE: DECISION RULES Type of hypothesis Two-tail Right-tail Left-tail H0: the null hypothesis β2 = β* 2 β2 ≤ β* 2 β2 ≥ β* 2 H1: the alternative hypothesis β2 = β* 2 β2 > β* 2 β2 < β* 2 Decision rule: reject H0 if |t | > tα/2,df t > tα,df t < −tα,df

* Notes: β2 is the hypothesized numerical value of β2. |t | means the absolute value of t. tα or t α/2 means the critical t value at the α or α/2 level of signiﬁcance. df: degrees of freedom, (n − 2) for the two-variable model, (n − 3) for the threevariable model, and so on. The same procedure holds to test hypotheses about β1.

Testing the Signiﬁcance of σ2: The χ2 Test

As another illustration of the test-of-signiﬁcance methodology, consider the following variable: χ 2 = (n − 2) σ2 ˆ σ2 (5.4.1)

which, as noted previously, follows the χ 2 distribution with n − 2 df. For the ˆ hypothetical example, σ 2 = 42.1591 and df = 8. If we postulate that H0 : σ 2 = 85 vs. H1 : σ 2 = 85, Eq. (5.4.1) provides the test statistic for H0 . Substituting the appropriate values in (5.4.1), it can be found that under H0 , χ 2 = 3.97. If we assume α = 5%, the critical χ 2 values are 2.1797 and 17.5346. Since the computed χ 2 lies between these limits, the data support the null hypothesis and we do not reject it. (See Figure 5.1.) This test procedure is called the chi-square test of signiﬁcance. The χ 2 test of signiﬁcance approach to hypothesis testing is summarized in Table 5.2.

TABLE 5.2

A SUMMARY OF THE χ2 TEST H0: the null hypothesis

2 σ 2 = σ0 2 σ 2 = σ0 2 σ 2 = σ0

H1: the alternative hypothesis

2 σ 2 > σ0 2 σ 2 < σ0 2 σ 2 = σ0

Critical region: reject H0 if df(σ 2) ˆ 2 > χα,df 2 σ0 df(σ 2) ˆ 2 < χ(1−α),df 2 σ0 df(σ 2) ˆ 2 > χα/2,df 2 σ0 2 or < χ(1−α/2),df

Note: σ 0 is the value of σ 2 under the null hypothesis. The ﬁrst subscript on χ2 in the last column is the level of signiﬁcance, and the second subscript is the degrees of freedom. These are critical chi-square values. Note that df is (n − 2) for the two-variable regression model, (n − 3) for the three-variable regression model, and so on.

2

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5.8

HYPOTHESIS TESTING: SOME PRACTICAL ASPECTS

The Meaning of “Accepting” or “Rejecting” a Hypothesis

If on the basis of a test of signiﬁcance, say, the t test, we decide to “accept” the null hypothesis, all we are saying is that on the basis of the sample evidence we have no reason to reject it; we are not saying that the null hypothesis is true beyond any doubt. Why? To answer this, let us revert to our consumption–income example and assume that H0 : β2 (MPC) = 0.50. Now ˆ ˆ the estimated value of the MPC is β2 = 0.5091 with a se (β2 ) = 0.0357. Then on the basis of the t test we ﬁnd that t = (0.5091 − 0.50)/0.0357 = 0.25, which is insigniﬁcant, say, at α = 5%. Therefore, we say “accept” H0 . But now let us assume H0 : β2 = 0.48. Applying the t test, we obtain t = (0.5091 − 0.48)/0.0357 = 0.82, which too is statistically insigniﬁcant. So now we say “accept” this H0 . Which of these two null hypotheses is the “truth”? We do not know. Therefore, in “accepting” a null hypothesis we should always be aware that another null hypothesis may be equally compatible with the data. It is therefore preferable to say that we may accept the null hypothesis rather than we (do) accept it. Better still,

. . . just as a court pronounces a verdict as “not guilty” rather than “innocent,” so the conclusion of a statistical test is “do not reject” rather than “accept.”11

The “Zero” Null Hypothesis and the “2-t” Rule of Thumb

A null hypothesis that is commonly tested in empirical work is H0 : β2 = 0, that is, the slope coefﬁcient is zero. This “zero” null hypothesis is a kind of straw man, the objective being to ﬁnd out whether Y is related at all to X, the explanatory variable. If there is no relationship between Y and X to begin with, then testing a hypothesis such as β2 = 0.3 or any other value is meaningless. This null hypothesis can be easily tested by the conﬁdence interval or the t-test approach discussed in the preceding sections. But very often such formal testing can be shortcut by adopting the “2-t” rule of signiﬁcance, which may be stated as

“2-t” Rule of Thumb. If the number of degrees of freedom is 20 or more and if α, the level of signiﬁcance, is set at 0.05, then the null hypothesis β2 = 0 can be rejected if the t value ˆ ˆ [ = β 2 /se (β 2)] computed from (5.3.2) exceeds 2 in absolute value.

The rationale for this rule is not too difﬁcult to grasp. From (5.7.1) we know that we will reject H0 : β2 = 0 if ˆ ˆ t = β2 /se (β2 ) > tα/2

11

ˆ when β2 > 0

Jan Kmenta, Elements of Econometrics, Macmillan, New York, 1971, p. 114.

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or ˆ ˆ t = β2 /se (β2 ) < −tα/2 or when |t| = ˆ β2 > tα/2 ˆ se (β2 ) (5.8.1) ˆ when β2 < 0

for the appropriate degrees of freedom. Now if we examine the t table given in Appendix D, we see that for df of about 20 or more a computed t value in excess of 2 (in absolute terms), say, 2.1, is statistically signiﬁcant at the 5 percent level, implying rejection of the null hypothesis. Therefore, if we ﬁnd that for 20 or more df the computed t value is, say, 2.5 or 3, we do not even have to refer to the t table to assess the signiﬁcance of the estimated slope coefﬁcient. Of course, one can always refer to the t table to obtain the precise level of signiﬁcance, and one should always do so when the df are fewer than, say, 20. In passing, note that if we are testing the one-sided hypothesis β2 = 0 versus β2 > 0 or β2 < 0, then we should reject the null hypothesis if |t| = ˆ β2 > tα ˆ se (β2 ) (5.8.2)

If we ﬁx α at 0.05, then from the t table we observe that for 20 or more df a t value in excess of 1.73 is statistically signiﬁcant at the 5 percent level of signiﬁcance (one-tail). Hence, whenever a t value exceeds, say, 1.8 (in absolute terms) and the df are 20 or more, one need not consult the t table for the statistical signiﬁcance of the observed coefﬁcient. Of course, if we choose α at 0.01 or any other level, we will have to decide on the appropriate t value as the benchmark value. But by now the reader should be able to do that.

Forming the Null and Alternative Hypotheses12

Given the null and the alternative hypotheses, testing them for statistical signiﬁcance should no longer be a mystery. But how does one formulate these hypotheses? There are no hard-and-fast rules. Very often the phenomenon under study will suggest the nature of the null and alternative hypotheses. For example, consider the capital market line (CML) of portfolio theory, which postulates that Ei = β1 + β2 σi , where E = expected return on portfolio and σ = the standard deviation of return, a measure of risk. Since return and risk are expected to be positively related—the higher the risk, the

12 For an interesting discussion about formulating hypotheses, see J. Bradford De Long and Kevin Lang, “Are All Economic Hypotheses False?” Journal of Political Economy, vol. 100, no. 6, 1992, pp. 1257–1272.

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higher the return—the natural alternative hypothesis to the null hypothesis that β2 = 0 would be β2 > 0. That is, one would not choose to consider values of β2 less than zero. But consider the case of the demand for money. As we shall show later, one of the important determinants of the demand for money is income. Prior studies of the money demand functions have shown that the income elasticity of demand for money (the percent change in the demand for money for a 1 percent change in income) has typically ranged between 0.7 and 1.3. Therefore, in a new study of demand for money, if one postulates that the income-elasticity coefﬁcient β2 is 1, the alternative hypothesis could be that β2 = 1, a two-sided alternative hypothesis. Thus, theoretical expectations or prior empirical work or both can be relied upon to formulate hypotheses. But no matter how the hypotheses are formed, it is extremely important that the researcher establish these hypotheses before carrying out the empirical investigation. Otherwise, he or she will be guilty of circular reasoning or self-fulﬁlling prophesies. That is, if one were to formulate hypotheses after examining the empirical results, there may be the temptation to form hypotheses that justify one’s results. Such a practice should be avoided at all costs, at least for the sake of scientiﬁc objectivity. Keep in mind the Stigler quotation given at the beginning of this chapter!

Choosing α, the Level of Signiﬁcance

It should be clear from the discussion so far that whether we reject or do not reject the null hypothesis depends critically on α, the level of signiﬁcance or the probability of committing a Type I error—the probability of rejecting the true hypothesis. In Appendix A we discuss fully the nature of a Type I error, its relationship to a Type II error (the probability of accepting the false hypothesis) and why classical statistics generally concentrates on a Type I error. But even then, why is α commonly ﬁxed at the 1, 5, or at the most 10 percent levels? As a matter of fact, there is nothing sacrosanct about these values; any other values will do just as well. In an introductory book like this it is not possible to discuss in depth why one chooses the 1, 5, or 10 percent levels of signiﬁcance, for that will take us into the ﬁeld of statistical decision making, a discipline unto itself. A brief summary, however, can be offered. As we discuss in Appendix A, for a given sample size, if we try to reduce a Type I error, a Type II error increases, and vice versa. That is, given the sample size, if we try to reduce the probability of rejecting the true hypothesis, we at the same time increase the probability of accepting the false hypothesis. So there is a tradeoff involved between these two types of errors, given the sample size. Now the only way we can decide about the tradeoff is to ﬁnd out the relative costs of the two types of errors. Then,

If the error of rejecting the null hypothesis which is in fact true (Error Type I) is costly relative to the error of not rejecting the null hypothesis which is in fact

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false (Error Type II), it will be rational to set the probability of the ﬁrst kind of error low. If, on the other hand, the cost of making Error Type I is low relative to the cost of making Error Type II, it will pay to make the probability of the ﬁrst kind of error high (thus making the probability of the second type of error low).13

Of course, the rub is that we rarely know the costs of making the two types of errors. Thus, applied econometricians generally follow the practice of setting the value of α at a 1 or a 5 or at most a 10 percent level and choose a test statistic that would make the probability of committing a Type II error as small as possible. Since one minus the probability of committing a Type II error is known as the power of the test, this procedure amounts to maximizing the power of the test. (See Appendix A for a discussion of the power of a test.) But all this problem with choosing the appropriate value of α can be avoided if we use what is known as the p value of the test statistic, which is discussed next.

The Exact Level of Signiﬁcance: The p Value

As just noted, the Achilles heel of the classical approach to hypothesis testing is its arbitrariness in selecting α. Once a test statistic (e.g., the t statistic) is obtained in a given example, why not simply go to the appropriate statistical table and ﬁnd out the actual probability of obtaining a value of the test statistic as much as or greater than that obtained in the example? This probability is called the p value (i.e., probability value), also known as the observed or exact level of signiﬁcance or the exact probability of committing a Type I error. More technically, the p value is deﬁned as the lowest signiﬁcance level at which a null hypothesis can be rejected. To illustrate, let us return to our consumption–income example. Given the null hypothesis that the true MPC is 0.3, we obtained a t value of 5.86 in (5.7.4). What is the p value of obtaining a t value of as much as or greater than 5.86? Looking up the t table given in Appendix D, we observe that for 8 df the probability of obtaining such a t value must be much smaller than 0.001 (one-tail) or 0.002 (two-tail). By using the computer, it can be shown that the probability of obtaining a t value of 5.86 or greater (for 8 df) is about 0.000189.14 This is the p value of the observed t statistic. This observed, or exact, level of signiﬁcance of the t statistic is much smaller than the conventionally, and arbitrarily, ﬁxed level of signiﬁcance, such as 1, 5, or 10 percent. As a matter of fact, if we were to use the p value just computed,

Jan Kmenta, Elements of Econometrics, Macmillan, New York, 1971, pp. 126–127. One can obtain the p value using electronic statistical tables to several decimal places. Unfortunately, the conventional statistical tables, for lack of space, cannot be that reﬁned. Most statistical packages now routinely print out the p values.

14 13

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and reject the null hypothesis that the true MPC is 0.3, the probability of our committing a Type I error is only about 0.02 percent, that is, only about 2 in 10,000! As we noted earlier, if the data do not support the null hypothesis, |t| obtained under the null hypothesis will be “large” and therefore the p value of obtaining such a |t| value will be “small.” In other words, for a given sample size, as |t| increases, the p value decreases, and one can therefore reject the null hypothesis with increasing conﬁdence. What is the relationship of the p value to the level of signiﬁcance α? If we make the habit of ﬁxing α equal to the p value of a test statistic (e.g., the t statistic), then there is no conﬂict between the two values. To put it differently, it is better to give up ﬁxing α arbitrarily at some level and simply choose the p value of the test statistic. It is preferable to leave it to the reader to decide whether to reject the null hypothesis at the given p value. If in an application the p value of a test statistic happens to be, say, 0.145, or 14.5 percent, and if the reader wants to reject the null hypothesis at this (exact) level of signiﬁcance, so be it. Nothing is wrong with taking a chance of being wrong 14.5 percent of the time if you reject the true null hypothesis. Similarly, as in our consumption–income example, there is nothing wrong if the researcher wants to choose a p value of about 0.02 percent and not take a chance of being wrong more than 2 out of 10,000 times. After all, some investigators may be risk-lovers and some risk-averters! In the rest of this text, we will generally quote the p value of a given test statistic. Some readers may want to ﬁx α at some level and reject the null hypothesis if the p value is less than α. That is their choice.

Statistical Signiﬁcance versus Practical Signiﬁcance

Let us revert to our consumption–income example and now hypothesize that the true MPC is 0.61 (H0 : β2 = 0.61). On the basis of our sample result ˆ of β2 = 0.5091, we obtained the interval (0.4268, 0.5914) with 95 percent conﬁdence. Since this interval does not include 0.61, we can, with 95 percent conﬁdence, say that our estimate is statistically signiﬁcant, that is, signiﬁcantly different from 0.61. But what is the practical or substantive signiﬁcance of our ﬁnding? That is, what difference does it make if we take the MPC to be 0.61 rather than 0.5091? Is the 0.1009 difference between the two MPCs that important practically? The answer to this question depends on what we really do with these estimates. For example, from macroeconomics we know that the income multiplier is 1/(1 − MPC). Thus, if MPC is 0.5091, the multiplier is 2.04, but it is 2.56 if MPC is equal to 0.61. That is, if the government were to increase its expenditure by $1 to lift the economy out of a recession, income will eventually increase by $2.04 if the MPC is 0.5091 but by $2.56 if the MPC is 0.61. And that difference could very well be crucial to resuscitating the economy.

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The point of all this discussion is that one should not confuse statistical signiﬁcance with practical, or economic, signiﬁcance. As Goldberger notes:

When a null, say, β j = 1, is speciﬁed, the likely intent is that β j is close to 1, so close that for all practical purposes it may be treated as if it were 1. But whether 1.1 is “practically the same as” 1.0 is a matter of economics, not of statistics. One cannot resolve the matter by relying on a hypothesis test, because the test statisˆ tic [t = ] (bj − 1)/σbj measures the estimated coefﬁcient in standard error units, which are not meaningful units in which to measure the economic parameter β j − 1. It may be a good idea to reserve the term “signiﬁcance” for the statistical concept, adopting “substantial” for the economic concept.15

The point made by Goldberger is important. As sample size becomes very large, issues of statistical signiﬁcance become much less important but issues of economic signiﬁcance become critical. Indeed, since with very large samples almost any null hypothesis will be rejected, there may be studies in which the magnitude of the point estimates may be the only issue.

The Choice between Conﬁdence-Interval and Test-of-Signiﬁcance Approaches to Hypothesis Testing

In most applied economic analyses, the null hypothesis is set up as a straw man and the objective of the empirical work is to knock it down, that is, reject the null hypothesis. Thus, in our consumption–income example, the null hypothesis that the MPC β2 = 0 is patently absurd, but we often use it to dramatize the empirical results. Apparently editors of reputed journals do not ﬁnd it exciting to publish an empirical piece that does not reject the null hypothesis. Somehow the ﬁnding that the MPC is statistically different from zero is more newsworthy than the ﬁnding that it is equal to, say, 0.7! Thus, J. Bradford De Long and Kevin Lang argue that it is better for economists

. . . to concentrate on the magnitudes of coefﬁcients and to report conﬁdence levels and not signiﬁcance tests. If all or almost all null hypotheses are false, there is little point in concentrating on whether or not an estimate is indistinguishable from its predicted value under the null. Instead, we wish to cast light on what models are good approximations, which requires that we know ranges of parameter values that are excluded by empirical estimates.16

In short, these authors prefer the conﬁdence-interval approach to the test-ofsigniﬁcance approach. The reader may want to keep this advice in mind.17

15 Arthur S. Goldberger, A Course in Econometrics, Harvard University Press, Cambridge, ˆ Massachusetts, 1991, p. 240. Note bj is the OLS estimator of β j and σbj is its standard error. For a corroborating view, see D. N. McCloskey, “The Loss Function Has Been Mislaid: The Rhetoric of Signiﬁcance Tests,” American Economic Review, vol. 75, 1985, pp. 201–205. See also D. N. McCloskey and S. T. Ziliak, “The Standard Error of Regression,” Journal of Economic Literature, vol. 37, 1996, pp. 97–114. 16 See their article cited in footnote 12, p. 1271. 17 For a somewhat different perspective, see Carter Hill, William Grifﬁths, and George Judge, Undergraduate Econometrics, Wiley & Sons, New York, 2001, p. 108.

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5.9

REGRESSION ANALYSIS AND ANALYSIS OF VARIANCE

In this section we study regression analysis from the point of view of the analysis of variance and introduce the reader to an illuminating and complementary way of looking at the statistical inference problem. In Chapter 3, Section 3.5, we developed the following identity: yi2 = yi2 + ˆ ˆ2 ui = β2 ˆ2 xi2 + ui ˆ2 (3.5.2)

that is, TSS = ESS + RSS, which decomposed the total sum of squares (TSS) into two components: explained sum of squares (ESS) and residual sum of squares (RSS). A study of these components of TSS is known as the analysis of variance (ANOVA) from the regression viewpoint. Associated with any sum of squares is its df, the number of independent observations on which it is based. TSS has n − 1 df because we lose 1 df in ¯ computing the sample mean Y. RSS has n − 2 df. (Why?) (Note: This is true only for the two-variable regression model with the intercept β1 present.) ESS has 1 df (again true of the two-variable case only), which follows from ˆ 2 xi2 is a function of β2 only, since ˆ xi2 is known. the fact that ESS = β2 Let us arrange the various sums of squares and their associated df in Table 5.3, which is the standard form of the AOV table, sometimes called the ANOVA table. Given the entries of Table 5.3, we now consider the following variable: F= = = ˆ2 β2 σ2 ˆ MSS of ESS MSS of RSS ˆ 2 xi2 β2 ui (n − 2) ˆ2 xi2 (5.9.1)

If we assume that the disturbances ui are normally distributed, which we do under the CNLRM, and if the null hypothesis (H0 ) is that β2 = 0, then it can be shown that the F variable of (5.9.1) follows the F distribution with

TABLE 5.3 ANOVA TABLE FOR THE TWO-VARIABLE REGRESSION MODEL Source of variation Due to regression (ESS) Due to residuals (RSS) TSS SS* ˆ2 y i2 = β2 ˆ u i2 ˆ y i2 x i2 1 n−2 n−1 df ˆ2 β2 MSS† x i2

u i2 ˆ = σ2 ˆ n−2

*SS means sum of squares. † Mean sum of squares, which is obtained by dividing SS by their df.

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1 df in the numerator and (n − 2) df in the denominator. (See Appendix 5A, Section 5A.3, for the proof. The general properties of the F distribution are discussed in Appendix A.) What use can be made of the preceding F ratio? It can be shown18 that ˆ2 E β2 and E ui ˆ2 = E(σ 2 ) = σ 2 ˆ n− 2 (5.9.3)

2 xi2 = σ 2 + β2

xi2

(5.9.2)

(Note that β2 and σ 2 appearing on the right sides of these equations are the true parameters.) Therefore, if β2 is in fact zero, Eqs. (5.9.2) and (5.9.3) both provide us with identical estimates of true σ 2 . In this situation, the explanatory variable X has no linear inﬂuence on Y whatsoever and the entire variation in Y is explained by the random disturbances ui . If, on the other hand, β2 is not zero, (5.9.2) and (5.9.3) will be different and part of the variation in Y will be ascribable to X. Therefore, the F ratio of (5.9.1) provides a test of the null hypothesis H0 : β2 = 0. Since all the quantities entering into this equation can be obtained from the available sample, this F ratio provides a test statistic to test the null hypothesis that true β2 is zero. All that needs to be done is to compute the F ratio and compare it with the critical F value obtained from the F tables at the chosen level of signiﬁcance, or obtain the p value of the computed F statistic. To illustrate, let us continue with our consumption–income example. The ANOVA table for this example is as shown in Table 5.4. The computed F value is seen to be 202.87. The p value of this F statistic corresponding to 1 and 8 df cannot be obtained from the F table given in Appendix D, but by using electronic statistical tables it can be shown that the p value is 0.0000001, an extremely small probability indeed. If you decide to choose the level-of-signiﬁcance approach to hypothesis testing and ﬁx α at 0.01, or a 1 percent level, you can see that the computed F of 202.87 is obviously signiﬁcant at this level. Therefore, if we reject the null hypothesis that β2 = 0, the probability of committing a Type I error is very small. For all practical

TABLE 5.4 ANOVA TABLE FOR THE CONSUMPTION–INCOME EXAMPLE Source of variation Due to regression (ESS) Due to residuals (RSS) TSS SS 8552.73 337.27 8890.00 df 1 8 9 MSS 8552.73 42.159 F= 8552.73 42.159

= 202.87

18 For proof, see K. A. Brownlee, Statistical Theory and Methodology in Science and Engineering, John Wiley & Sons, New York, 1960, pp. 278–280.

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purposes, our sample could not have come from a population with zero β2 value and we can conclude with great conﬁdence that X, income, does affect Y, consumption expenditure. Refer to Theorem 5.7 of Appendix 5A.1, which states that the square of the t value with k df is an F value with 1 df in the numerator and k df in the denominator. For our consumption–income example, if we assume H0 : β2 = 0, then from (5.3.2) it can be easily veriﬁed that the estimated t value is 14.26. This t value has 8 df. Under the same null hypothesis, the F value was 202.87 with 1 and 8 df. Hence (14.24)2 = F value, except for the rounding errors. Thus, the t and the F tests provide us with two alternative but complementary ways of testing the null hypothesis that β2 = 0. If this is the case, why not just rely on the t test and not worry about the F test and the accompanying analysis of variance? For the two-variable model there really is no need to resort to the F test. But when we consider the topic of multiple regression we will see that the F test has several interesting applications that make it a very useful and powerful method of testing statistical hypotheses.

5.10 APPLICATION OF REGRESSION ANALYSIS: THE PROBLEM OF PREDICTION

On the basis of the sample data of Table 3.2 we obtained the following sample regression: ˆ Yi = 24.4545 + 0.5091Xi (3.6.2)

ˆ where Yt is the estimator of true E(Yi) corresponding to given X. What use can be made of this historical regression? One use is to “predict” or “forecast” the future consumption expenditure Y corresponding to some given level of income X. Now there are two kinds of predictions: (1) prediction of the conditional mean value of Y corresponding to a chosen X, say, X0 , that is the point on the population regression line itself (see Figure 2.2), and (2) prediction of an individual Y value corresponding to X0 . We shall call these two predictions the mean prediction and individual prediction.

Mean Prediction19

To ﬁx the ideas, assume that X0 = 100 and we want to predict E(Y | X0 = 100). Now it can be shown that the historical regression (3.6.2) provides the point estimate of this mean prediction as follows: ˆ ˆ ˆ Y0 = β1 + β2 X0 = 24.4545 + 0.5091(100) = 75.3645

19

(5.10.1)

For the proofs of the various statements made, see App. 5A, Sec. 5A.4.

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ˆ where Y0 = estimator of E(Y | X0 ). It can be proved that this point predictor is a best linear unbiased estimator (BLUE). ˆ Since Y0 is an estimator, it is likely to be different from its true value. The difference between the two values will give some idea about the prediction or forecast error. To assess this error, we need to ﬁnd out the sampling ˆ ˆ distribution of Y0 . It is shown in Appendix 5A, Section 5A.4, that Y0 in Eq. (5.10.1) is normally distributed with mean (β1 + β2 X0 ) and the variance is given by the following formula: ˆ var (Y0 ) = σ 2 ¯ 1 (X0 − X)2 + 2 n xi (5.10.2)

ˆ By replacing the unknown σ 2 by its unbiased estimator σ 2 , we see that the variable t= ˆ Y0 − (β1 + β2 X0 ) ˆ se (Y0 ) (5.10.3)

follows the t distribution with n − 2 df. The t distribution can therefore be used to derive conﬁdence intervals for the true E(Y0 | X0 ) and test hypotheses about it in the usual manner, namely, ˆ ˆ ˆ ˆ ˆ ˆ Pr [β1 + β2 X0 − tα/2 se (Y0 ) ≤ β1 + β2 X0 ≤ β1 + β2 X0 + tα/2 se (Y0 )] = 1 − α (5.10.4) ˆ where se (Y0 ) is obtained from (5.10.2). For our data (see Table 3.3), ˆ var (Y0 ) = 42.159 = 10.4759 and ˆ se (Y0 ) = 3.2366 Therefore, the 95% conﬁdence interval for true E(Y | X0 ) = β1 + β2 X0 is given by 75.3645 − 2.306(3.2366) ≤ E(Y0 | X = 100) ≤ 75.3645 + 2.306(3.2366) that is, 67.9010 ≤ E(Y | X = 100) ≤ 82.8381 (5.10.5) 1 (100 − 170)2 + 10 33,000

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Y

170 150 130 110 90 70 68 50 0 58 Confidence interval for mean Y

Yi = 24.4545 + 0.5091Xi

92 83

Confidence interval for individual Y

0

60

80

100

120

140

160 X

180

200

220

240

260

X

FIGURE 5.6

Conﬁdence intervals (bands) for mean Y and individual Y values.

Thus, given X0 = 100, in repeated sampling, 95 out of 100 intervals like (5.10.5) will include the true mean value; the single best estimate of the true mean value is of course the point estimate 75.3645. If we obtain 95% conﬁdence intervals like (5.10.5) for each of the X values given in Table 3.2, we obtain what is known as the conﬁdence interval, or conﬁdence band, for the population regression function, which is shown in Figure 5.6.

Individual Prediction

If our interest lies in predicting an individual Y value, Y0 , corresponding to a given X value, say, X0 , then, as shown in Appendix 5, Section 5A.3, a best linear unbiased estimator of Y0 is also given by (5.10.1), but its variance is as follows: ¯ 1 (X0 − X)2 ˆ ˆ var (Y0 − Y0 ) = E[Y0 − Y0 ]2 = σ 2 1 + + 2 n xi (5.10.6)

It can be shown further that Y0 also follows the normal distribution with ˆ mean and variance given by (5.10.1) and (5.10.6), respectively. Substituting σ 2

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for the unknown σ 2, it follows that t= ˆ Y0 − Y0 ˆ se (Y0 − Y0 )

also follows the t distribution. Therefore, the t distribution can be used to draw inferences about the true Y0 . Continuing with our consumption–income example, we see that the point prediction of Y0 is 75.3645, the same as that ˆ of Y0 , and its variance is 52.6349 (the reader should verify this calculation). Therefore, the 95% conﬁdence interval for Y0 corresponding to X0 = 100 is seen to be (58.6345 ≤ Y0 | X0 = 100 ≤ 92.0945) (5.10.7)

Comparing this interval with (5.10.5), we see that the conﬁdence interval for individual Y0 is wider than that for the mean value of Y0 . (Why?) Computing conﬁdence intervals like (5.10.7) conditional upon the X values given in Table 3.2, we obtain the 95% conﬁdence band for the individual Y values corresponding to these X values. This conﬁdence band along with the conﬁˆ dence band for Y0 associated with the same X’s is shown in Figure 5.6. Notice an important feature of the conﬁdence bands shown in Figure 5.6. ¯ The width of these bands is smallest when X0 = X. (Why?) However, the ¯ (Why?) This change would width widens sharply as X0 moves away from X. suggest that the predictive ability of the historical sample regression line ¯ falls markedly as X0 departs progressively from X. Therefore, one should exercise great caution in “extrapolating” the historical regression line to predict E(Y | X0) or Y0 associated with a given X0 that is far removed ¯ from the sample mean X.

5.11 REPORTING THE RESULTS OF REGRESSION ANALYSIS

There are various ways of reporting the results of regression analysis, but in this text we shall use the following format, employing the consumption– income example of Chapter 3 as an illustration: ˆ Yi = 24.4545 se = (6.4138) t = (3.8128) p = (0.002571) + 0.5091Xi (0.0357) (14.2605) (0.000000289) r 2 = 0.9621 df = 8 F1,8 = 202.87 (5.11.1)

In Eq. (5.11.1) the ﬁgures in the ﬁrst set of parentheses are the estimated standard errors of the regression coefﬁcients, the ﬁgures in the second set are estimated t values computed from (5.3.2) under the null hypothesis that

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the true population value of each regression coefﬁcient individually is zero (e.g., 3.8128 = 24.4545 ÷ 6.4138), and the ﬁgures in the third set are the estimated p values. Thus, for 8 df the probability of obtaining a t value of 3.8128 or greater is 0.0026 and the probability of obtaining a t value of 14.2605 or larger is about 0.0000003. By presenting the p values of the estimated t coefﬁcients, we can see at once the exact level of signiﬁcance of each estimated t value. Thus, under the null hypothesis that the true population intercept value is zero, the exact probability (i.e., the p value) of obtaining a t value of 3.8128 or greater is only about 0.0026. Therefore, if we reject this null hypothesis, the probability of our committing a Type I error is about 26 in 10,000, a very small probability indeed. For all practical purposes we can say that the true population intercept is different from zero. Likewise, the p value of the estimated slope coefﬁcient is zero for all practical purposes. If the true MPC were in fact zero, our chances of obtaining an MPC of 0.5091 would be practically zero. Hence we can reject the null hypothesis that the true MPC is zero. Earlier we showed the intimate connection between the F and t statistics, 2 namely, F1,k = tk . Under the null hypothesis that the true β2 = 0, (5.11.1) shows that the F value is 202.87 (for 1 numerator and 8 denominator df) and the t value is about 14.24 (8 df); as expected, the former value is the square of the latter value, except for the roundoff errors. The ANOVA table for this problem has already been discussed.

5.12 EVALUATING THE RESULTS OF REGRESSION ANALYSIS

In Figure I.4 of the Introduction we sketched the anatomy of econometric modeling. Now that we have presented the results of regression analysis of our consumption–income example in (5.11.1), we would like to question the adequacy of the ﬁtted model. How “good” is the ﬁtted model? We need some criteria with which to answer this question. First, are the signs of the estimated coefﬁcients in accordance with theoretical or prior expectations? A priori, β2 , the marginal propensity to consume (MPC) in the consumption function, should be positive. In the present example it is. Second, if theory says that the relationship should be not only positive but also statistically signiﬁcant, is this the case in the present application? As we discussed in Section 5.11, the MPC is not only positive but also statistically signiﬁcantly different from zero; the p value of the estimated t value is extremely small. The same comments apply about the intercept coefﬁcient. Third, how well does the regression model explain variation in the consumption expenditure? One can use r 2 to answer this question. In the present example r 2 is about 0.96, which is a very high value considering that r 2 can be at most 1. Thus, the model we have chosen for explaining consumption expenditure behavior seems quite good. But before we sign off, we would like to ﬁnd out

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whether our model satisﬁes the assumptions of CNLRM. We will not look at the various assumptions now because the model is patently so simple. But there is one assumption that we would like to check, namely, the normality of the disturbance term, ui . Recall that the t and F tests used before require that the error term follow the normal distribution. Otherwise, the testing procedure will not be valid in small, or ﬁnite, samples.

Normality Tests

Although several tests of normality are discussed in the literature, we will consider just three: (1) histogram of residuals; (2) normal probability plot (NPP), a graphical device; and (3) the Jarque–Bera test. Histogram of Residuals. A histogram of residuals is a simple graphic device that is used to learn something about the shape of the PDF of a random variable. On the horizontal axis, we divide the values of the variable of interest (e.g., OLS residuals) into suitable intervals, and in each class interval we erect rectangles equal in height to the number of observations (i.e., frequency) in that class interval. If you mentally superimpose the bellshaped normal distribution curve on the histogram, you will get some idea as to whether normal (PDF) approximation may be appropriate. A concrete example is given in Section 5.13 (see Figure 5.8). It is always a good practice to plot the histogram of the residuals as a rough and ready method of testing for the normality assumption. Normal Probability Plot. A comparatively simple graphical device to study the shape of the probability density function (PDF) of a random variable is the normal probability plot (NPP) which makes use of normal probability paper, a specially designed graph paper. On the horizontal, or ˆ x, axis, we plot values of the variable of interest (say, OLS residuals, ui ), and on the vertical, or y, axis, we show the expected value of this variable if it were normally distributed. Therefore, if the variable is in fact from the normal population, the NPP will be approximately a straight line. The NPP of the residuals from our consumption–income regression is shown in Figure 5.7, which is obtained from the MINITAB software package, version 13. As noted earlier, if the ﬁtted line in the NPP is approximately a straight line, one can conclude that the variable of interest is normally distributed. In Figure 5.7, we see that residuals from our illustrative example are approximately normally distributed, because a straight line seems to ﬁt the data reasonably well. MINITAB also produces the Anderson–Darling normality test, known as the A2 statistic. The underlying null hypothesis is that the variable under consideration is normally distributed. As Figure 5.7 shows, for our example, the computed A2 statistic is 0.394. The p value of obtaining such a value of A2 is 0.305, which is reasonably high. Therefore, we do not reject the

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.999 .99 .95 .80 Probability .50 .20 .05 .01 .001 –10 –5 0 RESI1 5 A2 P-value 0.394 0.305

Average 0.0000000 StDev 6.12166 N 10 FIGURE 5.7 Residuals from consumption–income regression.

hypothesis that the residuals from our consumption–income example are normally distributed. Incidentally, Figure 5.7 shows the parameters of the (normal) distribution, the mean is approximately 0 and the standard deviation is about 6.12. Jarque–Bera (JB) Test of Normality.20 The JB test of normality is an asymptotic, or large-sample, test. It is also based on the OLS residuals. This test ﬁrst computes the skewness and kurtosis (discussed in Appendix A) measures of the OLS residuals and uses the following test statistic: JB = n S2 (K − 3)2 + 6 24 (5.12.1)

where n = sample size, S = skewness coefﬁcient, and K = kurtosis coefﬁcient. For a normally distributed variable, S = 0 and K = 3. Therefore, the JB test of normality is a test of the joint hypothesis that S and K are 0 and 3, respectively. In that case the value of the JB statistic is expected to be 0. Under the null hypothesis that the residuals are normally distributed, Jarque and Bera showed that asymptotically (i.e., in large samples) the JB statistic given in (5.12.1) follows the chi-square distribution with 2 df. If the computed p value of the JB statistic in an application is sufﬁciently low, which will happen if the value of the statistic is very different from 0, one can reject the hypothesis that the residuals are normally distributed. But if

20 See C. M. Jarque and A. K. Bera, “A Test for Normality of Observations and Regression Residuals,” International Statistical Review, vol. 55, 1987, pp. 163–172.

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the p value is reasonably high, which will happen if the value of the statistic is close to zero, we do not reject the normality assumption. The sample size in our consumption–income example is rather small. Hence, strictly speaking one should not use the JB statistic. If we mechanically apply the JB formula to our example, the JB statistic turns out to be 0.7769. The p value of obtaining such a value from the chi-square distribution with 2 df is about 0.68, which is quite high. In other words, we may not reject the normality assumption for our example. Of course, bear in mind the warning about the sample size.

Other Tests of Model Adequacy

Remember that the CNLRM makes many more assumptions than the normality of the error term. As we examine econometric theory further, we will consider several tests of model adequacy (see Chapter 13). Until then, keep in mind that our regression modeling is based on several simplifying assumptions that may not hold in each and every case.

A CONCLUDING EXAMPLE Let us return to Example 3.2 about food expenditure in India. Using the data given in (3.7.2) and adopting the format of (5.11.1), we obtain the following expenditure equation: FoodExpi = 94.2087 + 0.4368 TotalExpi se = (50.8563) t = (1.8524) p = (0.0695) r2 = 0.3698; F1,53 = 31.1034 (0.0783) (5.5770) (0.0000)* df = 53 (p value = 0.0000)* (5.12.2)

were true, what is the probability of obtaining a value of 0.4368? Under the null hypothesis, we observe from (5.12.2) that the t value is 5.5770 and the p value of obtaining such a t value is practically zero. In other words, we can reject the null hypothesis resoundingly. But suppose the null hypothesis were that β2 = 0.5. Now what? Using the t test we obtain: t= 0.4368 − 0.5 = −0.8071 0.0783

where * denotes extremely small. First, let us interpret this regression. As expected, there is a positive relationship between expenditure on food and total expenditure. If total expenditure went up by a rupee, on average, expenditure on food increased by about 44 paise. If total expenditure were zero, the average expenditure on food would be about 94 rupees. Of course, this mechanical interpretation of the intercept may not make much economic sense. The r 2 value of about 0.37 means that 37 percent of the variation in food expenditure is explained by total expenditure, a proxy for income. Suppose we want to test the null hypothesis that there is no relationship between food expenditure and total expenditure, that is, the true slope coefﬁcient β2 = 0. The estimated value of β2 is 0.4368. If the null hypothesis

The probability of obtaining a |t | of 0.8071 is greater than 20 percent. Hence we do not reject the hypothesis that the true β2 is 0.5. Notice that, under the null hypothesis, the true slope coefﬁcient is zero, the F value is 31.1034, as shown in (5.12.2). Under the same null hypothesis, we obtained a t value of 5.5770. If we square this value, we obtain 31.1029, which is about the same as the F value, again showing the close relationship between the t and the F statistic. (Note: The numerator df for the F statistic must be 1, which is the case here.) Using the estimated residuals from the regression, what can we say about the probability distribution of the error term? The information is given in Figure 5.8. As the ﬁgure shows, the residuals from the food expenditure regression seem to be symmetrically distributed. Application of the Jarque–Bera test shows that the JB statistic is about 0.2576, and the probability of obtaining such a statistic under the normality assumption is about (Continued)

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A CONCLUDING EXAMPLE 14 12 Number of Observations 10 8 6 4 2 0

–150

(Continued)

Series: Residuals Sample 1 55 Observations 55 Mean Median Maximum Minimum Std. dev. Skewness Kurtosis Jarque–Bera Probability –100 –50 –1.19 10–14 7.747849 171.5859 –153.7664 66.23382 0.119816 3.234473 0.257585 0.879156

0 50 Residuals

100

150

FIGURE 5.8

Residuals from the food expenditure regression. We leave it to the reader to establish conﬁdence intervals for the two regression coefﬁcients as well as to obtain the normal probability plot and do mean and individual predictions.

88 percent. Therefore, we do not reject the hypothesis that the error terms are normally distributed. But keep in mind that the sample size of 55 observations may not be large enough.

5.13

SUMMARY AND CONCLUSIONS

1. Estimation and hypothesis testing constitute the two main branches of classical statistics. Having discussed the problem of estimation in Chapters 3 and 4, we have taken up the problem of hypothesis testing in this chapter. 2. Hypothesis testing answers this question: Is a given ﬁnding compatible with a stated hypothesis or not? 3. There are two mutually complementary approaches to answering the preceding question: conﬁdence interval and test of signiﬁcance. 4. Underlying the conﬁdence-interval approach is the concept of interval estimation. An interval estimator is an interval or range constructed in such a manner that it has a speciﬁed probability of including within its limits the true value of the unknown parameter. The interval thus constructed is known as a conﬁdence interval, which is often stated in percent form, such as 90 or 95%. The conﬁdence interval provides a set of plausible hypotheses about the value of the unknown parameter. If the null-hypothesized value lies in the conﬁdence interval, the hypothesis is not rejected, whereas if it lies outside this interval, the null hypothesis can be rejected. 5. In the signiﬁcance test procedure, one develops a test statistic and examines its sampling distribution under the null hypothesis. The test statistic usually follows a well-deﬁned probability distribution such as the normal, t, F, or chi-square. Once a test statistic (e.g., the t statistic) is computed

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from the data at hand, its p value can be easily obtained. The p value gives the exact probability of obtaining the estimated test statistic under the null hypothesis. If this p value is small, one can reject the null hypothesis, but if it is large one may not reject it. What constitutes a small or large p value is up to the investigator. In choosing the p value the investigator has to bear in mind the probabilities of committing Type I and Type II errors. 6. In practice, one should be careful in ﬁxing α, the probability of committing a Type I error, at arbitrary values such as 1, 5, or 10 percent. It is better to quote the p value of the test statistic. Also, the statistical signiﬁcance of an estimate should not be confused with its practical signiﬁcance. 7. Of course, hypothesis testing presumes that the model chosen for empirical analysis is adequate in the sense that it does not violate one or more assumptions underlying the classical normal linear regression model. Therefore, tests of model adequacy should precede tests of hypothesis. This chapter introduced one such test, the normality test, to ﬁnd out whether the error term follows the normal distribution. Since in small, or ﬁnite, samples, the t, F, and chi-square tests require the normality assumption, it is important that this assumption be checked formally. 8. If the model is deemed practically adequate, it may be used for forecasting purposes. But in forecasting the future values of the regressand, one should not go too far out of the sample range of the regressor values. Otherwise, forecasting errors can increase dramatically. EXERCISES

Questions 5.1. State with reason whether the following statements are true, false, or uncertain. Be precise. a. The t test of signiﬁcance discussed in this chapter requires that the ˆ ˆ sampling distributions of estimators β1 and β2 follow the normal distribution. b. Even though the disturbance term in the CLRM is not normally distributed, the OLS estimators are still unbiased. ˆ c. If there is no intercept in the regression model, the estimated ui ( = ui ) will not sum to zero. d. The p value and the size of a test statistic mean the same thing. e. In a regression model that contains the intercept, the sum of the residuals is always zero. f. If a null hypothesis is not rejected, it is true. ˆ g. The higher the value of σ 2 , the larger is the variance of β2 given in (3.3.1). h. The conditional and unconditional means of a random variable are the same things. i. In the two-variable PRF, if the slope coefﬁcient β2 is zero, the intercept ¯ β1 is estimated by the sample mean Y . j. The conditional variance, var (Yi | X i ) = σ 2, and the unconditional vari2 ance of Y, var (Y) = σY , will be the same if X had no inﬂuence on Y.

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5.2. Set up the ANOVA table in the manner of Table 5.4 for the regression model given in (3.7.2) and test the hypothesis that there is no relationship between food expenditure and total expenditure in India. 5.3. From the data given in Table 2.6 on earnings and education, we obtained the following regression [see Eq. (3.7.3)]: Meanwagei = 0.7437 + 0.6416 Educationi se = (0.8355) t=( ) ( ) r2 = 0.8944 n = 13 (9.6536)

a. Fill in the missing numbers. b. How do you interpret the coefﬁcient 0.6416? c. Would you reject the hypothesis that education has no effect whatsoever on wages? Which test do you use? And why? What is the p value of your test statistic? d. Set up the ANOVA table for this example and test the hypothesis that the slope coefﬁcient is zero. Which test do you use and why? e. Suppose in the regression given above the r2 value was not given to you. Could you have obtained it from the other information given in the regression? 5.4. Let ρ 2 represent the true population coefﬁcient of correlation. Suppose you want to test the hypothesis that ρ 2 = 0 . Verbally explain how you would test this hypothesis. Hint: Use Eq. (3.5.11). See also exercise 5.7. 5.5. What is known as the characteristic line of modern investment analysis is simply the regression line obtained from the following model: r it = αi + βi r mt + ut

where r it = the rate of return on the ith security in time t r mt = the rate of return on the market portfolio in time t ut = stochastic disturbance term In this model βi is known as the beta coefﬁcient of the ith security, a measure of market (or systematic) risk of a security.* On the basis of 240 monthly rates of return for the period 1956–1976, Fogler and Ganapathy obtained the following characteristic line for IBM stock in relation to the market portfolio index developed at the University of Chicago†: rit = 0.7264 + 1.0598rmt ˆ se = (0.3001 ) (0.0728 ) r 2 = 0.4710 df = 238 F1,238 = 211.896

a. A security whose beta coefﬁcient is greater than one is said to be a volatile or aggressive security. Was IBM a volatile security in the time period under study?

* See Haim Levy and Marshall Sarnat, Portfolio and Investment Selection: Theory and Practice, Prentice-Hall International, Englewood Cliffs, N.J., 1984, Chap. 12. † H. Russell Fogler and Sundaram Ganapathy, Financial Econometrics, Prentice Hall, Englewood Cliffs, N.J., 1982, p. 13.

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b. Is the intercept coefﬁcient signiﬁcantly different from zero? If it is, what is its practical meaning? 5.6. Equation (5.3.5) can also be written as

ˆ ˆ ˆ ˆ Pr [β2 − tα/2 se (β2 ) < β2 < β2 + tα/2 se (β2 )] = 1 − α

That is, the weak inequality (≤ ) can be replaced by the strong inequality (< ). Why? 5.7. R. A. Fisher has derived the sampling distribution of the correlation coefﬁcient deﬁned in (3.5.13). If it is assumed that the variables X and Y are jointly normally distributed, that is, if they come from a bivariate normal distribution (see Appendix 4A, exercise 4.1), then under the assumption that the population correlation coefﬁcient ρ is zero, it can be shown that √ √ t = r n − 2/ 1 − r 2 follows Student’s t distribution with n − 2 df.* Show that this t value is identical with the t value given in (5.3.2) under the null hypothesis that β2 = 0. Hence establish that under the same null hypothesis F = t 2 . (See Section 5.9.)

Problems 5.8. Consider the following regression output†:

ˆ Yi = 0.2033 + 0.6560X t

se = (0.0976) (0.1961) r 2 = 0.397 RSS = 0.0544 ESS = 0.0358 where Y = labor force participation rate (LFPR) of women in 1972 and X = LFPR of women in 1968. The regression results were obtained from a sample of 19 cities in the United States. a. How do you interpret this regression? b. Test the hypothesis: H0 : β2 = 1 against H1 : β2 > 1 . Which test do you use? And why? What are the underlying assumptions of the test(s) you use? c. Suppose that the LFPR in 1968 was 0.58 (or 58 percent). On the basis of the regression results given above, what is the mean LFPR in 1972? Establish a 95% conﬁdence interval for the mean prediction. d. How would you test the hypothesis that the error term in the population regression is normally distribute? Show the necessary calculations. 5.9. Table 5.5 gives data on average public teacher pay (annual salary in dollars) and spending on public schools per pupil (dollars) in 1985 for 50 states and the District of Columbia.

* If ρ is in fact zero, Fisher has shown that r follows the same t distribution provided either X or Y is normally distributed. But if ρ is not equal to zero, both variables must be normally distributed. See R. L. Anderson and T. A. Bancroft, Statistical Theory in Research, McGraw-Hill, New York, 1952, pp. 87–88. † Adapted from Samprit Chatterjee, Ali S. Hadi, and Bertram Price, Regression Analysis by Example, 3d ed., Wiley Interscience, New York, 2000, pp. 46–47.

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TABLE 5.5

AVERAGE SALARY AND PER PUPIL SPENDING (DOLLARS), 1985 Observation 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 Salary 19,583 20,263 20,325 26,800 29,470 26,610 30,678 27,170 25,853 24,500 24,274 27,170 30,168 26,525 27,360 21,690 21,974 20,816 18,095 20,939 22,644 24,624 27,186 33,990 23,382 20,627 Spending 3346 3114 3554 4642 4669 4888 5710 5536 4168 3547 3159 3621 3782 4247 3982 3568 3155 3059 2967 3285 3914 4517 4349 5020 3594 2821 Observation 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 Salary 22,795 21,570 22,080 22,250 20,940 21,800 22,934 18,443 19,538 20,460 21,419 25,160 22,482 20,969 27,224 25,892 22,644 24,640 22,341 25,610 26,015 25,788 29,132 41,480 25,845 Spending 3366 2920 2980 3731 2853 2533 2729 2305 2642 3124 2752 3429 3947 2509 5440 4042 3402 2829 2297 2932 3705 4123 3608 8349 3766

Source: National Education Association, as reported by Albuquerque Tribune, Nov. 7, 1986.

To ﬁnd out if there is any relationship between teacher’s pay and per pupil expenditure in public schools, the following model was suggested: Payi = β1 + β2 Spendi + ui, where Pay stands for teacher’s salary and Spend stands for per pupil expenditure. a. Plot the data and eyeball a regression line. b. Suppose on the basis of a you decide to estimate the above regression model. Obtain the estimates of the parameters, their standard errors, r2, RSS, and ESS. c. Interpret the regression. Does it make economic sense? d. Establish a 95% conﬁdence interval for β2 . Would you reject the hypothesis that the true slope coefﬁcient is 3.0? e. Obtain the mean and individual forecast value of Pay if per pupil spending is $5000. Also establish 95% conﬁdence intervals for the true mean and individual values of Pay for the given spending ﬁgure. f. How would you test the assumption of the normality of the error term? Show the test(s) you use. 5.10. Refer to exercise 3.20 and set up the ANOVA tables and test the hypothesis that there is no relationship between productivity and real wage

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5.11.

5.12.

5.13.

5.14.

5.15.

compensation. Do this for both the business and nonfarm business sectors. Refer to exercise 1.7. a. Plot the data with impressions on the vertical axis and advertising expenditure on the horizontal axis. What kind of relationship do you observe? b. Would it be appropriate to ﬁt a bivariate linear regression model to the data? Why or why not? If not, what type of regression model will you ﬁt the data to? Do we have the necessary tools to ﬁt such a model? c. Suppose you do not plot the data and simply ﬁt the bivariate regression model to the data. Obtain the usual regression output. Save the results for a later look at this problem. Refer to exercise 1.1. a. Plot the U.S. Consumer Price Index (CPI) against the Canadian CPI. What does the plot show? b. Suppose you want to predict the U.S. CPI on the basis of the Canadian CPI. Develop a suitable model. c. Test the hypothesis that there is no relationship between the two CPIs. Use α = 5% . If you reject the null hypothesis, does that mean the Canadian CPI “causes” the U.S. CPI? Why or why not? Refer to exercise 3.22. a. Estimate the two regressions given there, obtaining standard errors and the other usual output. b. Test the hypothesis that the disturbances in the two regression models are normally distributed. c. In the gold price regression, test the hypothesis that β2 = 1 , that is, there is a one-to-one relationship between gold prices and CPI (i.e., gold is a perfect hedge). What is the p value of the estimated test statistic? d. Repeat step c for the NYSE Index regression. Is investment in the stock market a perfect hedge against inﬂation? What is the null hypothesis you are testing? What is its p value? e. Between gold and stock, which investment would you choose? What is the basis of your decision? Table 5.6 gives data on GNP and four deﬁnitions of the money stock for the United States for 1970–1983. Regressing GNP on the various deﬁnitions of money, we obtain the results shown in Table 5.7. The monetarists or quantity theorists maintain that nominal income (i.e., nominal GNP) is largely determined by changes in the quantity or the stock of money, although there is no consensus as to the “right” deﬁnition of money. Given the results in the preceding table, consider these questions: a. Which deﬁnition of money seems to be closely related to nominal GNP? b. Since the r2 terms are uniformly high, does this fact mean that our choice for deﬁnition of money does not matter? c. If the Fed wants to control the money supply, which one of these money measures is a better target for that purpose? Can you tell from the regression results? Suppose the equation of an indifference curve between two goods is

X i Yi = β1 + β2 X i

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TABLE 5.6

GNP AND FOUR MEASURES OF MONEY STOCK Money stock measure, $ billion Year 1970 1971 1972 1973 1974 1975 1976 1977 1978 1979 1980 1981 1982 1983 GNP, $ billion 992.70 1,077.6 1,185.9 1,326.4 1,434.2 1,549.2 1,718.0 1,918.3 2,163.9 2,417.8 2,631.7 2,957.8 3,069.3 3,304.8 M1 216.6 230.8 252.0 265.9 277.6 291.2 310.4 335.4 363.1 389.1 414.9 441.9 480.5 525.4 M2 628.2 712.8 805.2 861.0 908.5 1,023.3 1,163.6 1,286.7 1,389.1 1,498.5 1,632.6 1,796.6 1,965.4 2,196.3 M3 677.5 776.2 886.0 985.0 1,070.5 1,174.2 1,311.9 1,472.9 1,647.1 1,804.8 1,990.0 2,238.2 2,462.5 2,710.4 L 816.3 903.1 1,023.0 1,141.7 1,249.3 1,367.9 1,516.6 1,704.7 1,910.6 2,117.1 2,326.2 2,599.8 2,870.8 3,183.1

Deﬁnitions: M1 = currency + demand deposits + travelers checks and other checkable deposits (OCDs) M2 = M1 + overnight RPs and Eurodollars + MMMF (money market mutual fund) balances + MMDAs (money market deposit accounts) + savings and small deposits M3 = M2 + large time deposits + term RPs + Institutional MMMF L = M3 + other liquid assets Source: Economic Report of the President, 1985, GNP data from Table B-1, p. 232; money stock data from Table B-61, p. 303.

TABLE 5.7

GNP–MONEY STOCK REGRESSIONS, 1970–1983 1) 2) 3) 4) GNPt = −787.4723 + 8.0863 M1t (77.9664) (0.2197) GNPt = −44.0626 + 1.5875 M2t (61.0134) (0.0448) GNPt = 159.1366 + 1.2034 M3t (42.9882) (0.0262) GNPt = 164.2071 + 1.0290 Lt (44.7658) (0.0234) r 2 = 0.9912 r 2 = 0.9905 r 2 = 0.9943 r 2 = 0.9938

Note: The ﬁgures in parentheses are the estimated standard errors.

TABLE 5.8

Consumption of good X: Consumption of good Y:

1 4

2 3.5

3 2.8

4 1.9

5 0.8

How would you estimate the parameters of this model? Apply the preceding model to the data in Table 5.8 and comment on your results. 5.16. Since 1986 the Economist has been publishing the Big Mac Index as a crude, and hilarious, measure of whether international currencies are at their “correct” exchange rate, as judged by the theory of purchasing power parity (PPP). The PPP holds that a unit of currency should be able

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to buy the same bundle of goods in all countries. The proponents of PPP argue that, in the long run, currencies tend to move toward their PPP. The Economist uses McDonald’s Big Mac as a representative bundle and gives the information in Table 5.9. Consider the following regression model:

Yi = β1 + β2 X i + ui

where Y = actual exchange rate and X = implied PPP of the dollar.

TABLE 5.9 THE HAMBURGER STANDARD Big Mac prices In local currency United States† Argentina Australia Brazil Britain Canada Chile China Czech Rep Denmark Euro area France Germany Italy Spain Hong Kong Hungary Indonesia Japan Malaysia Mexico New Zealand Philippines Poland Russia Singapore South Africa South Korea Sweden Switzerland Taiwan Thailand $2.54 Peso2.50 A$3.00 Real3.60 £1.99 C$3.33 Peso1260 Yuan9.90 Koruna56.00 DKr24.75 2.57 FFr18.5 DM5.10 Lire4300 Pta395 HK$10.70 Forint399 Rupiah14700 ¥294 M$4.52 Peso21.9 NZ$3.60 Peso59.00 Zloty5.90 Rouble35.00 S$3.30 Rand9.70 Won3000 SKr24.0 SFr6.30 NT$70.0 Baht55.0 In dollars 2.54 2.50 1.52 1.64 2.85 2.14 2.10 1.20 1.43 2.93 2.27 2.49 2.30 1.96 2.09 1.37 1.32 1.35 2.38 1.19 2.36 1.46 1.17 1.46 1.21 1.82 1.19 2.27 2.33 3.65 2.13 1.21 Implied PPP* of the dollar – 0.98 1.18 1.42 1.28‡ 1.31 496 3.90 22.0 9.74 0.99§ 7.28 2.01 1693 156 4.21 157 5787 116 1.78 8.62 1.42 23.2 2.32 13.8 1.30 3.82 1181 9.45 2.48 27.6 21.7 Actual $ exchange rate April 17, 2001 – 1.00 1.98 2.19 1.43‡ 1.56 601 8.28 39.0 8.46 0.88§ 7.44 2.22 2195 189 7.80 303 10855 124 3.80 9.29 2.47 50.3 4.03 28.9 1.81 8.13 1325 10.28 1.73 32.9 45.5 Under (−)/ over (+) valuation against the dollar, % – −2 −40 −35 12 −16 −17 −53 −44 15 −11 −2 −9 −23 −18 −46 −48 −47 −6 −53 −7 −43 −54 −42 −52 −28 −53 −11 −8 44 −16 −52

*Purchasing power parity: local price divided by price in the United States. † Average of New York, Chicago, San Francisco, and Atlanta. ‡ Dollars per pound. § Dollars per euro. Source: McDonald’s; The Economist, April 21, 2001.

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a. If the PPP holds, what values of β1 and β2 would you expect a priori? b. Do the regression results support your expectation? What formal test do you use to test your hypothesis? c. Should the Economist continue to publish the Big Mac Index? Why or why not? 5.17. Refer to the S.A.T. data given in exercise 2.16. Suppose you want to predict the male math (Y) scores on the basis of the female math scores (X) by running the following regression:

Yt = β1 + β2 X t + ut

a. Estimate the preceding model. b. From the estimated residuals, ﬁnd out if the normality assumption can be sustained. c. Now test the hypothesis that β2 = 1, that is, there is a one-to-one correspondence between male and female math scores. d. Set up the ANOVA table for this problem. 5.18. Repeat the exercise in the preceding problem but let Y and X denote the male and female verbal scores, respectively. 5.19. Table 5.10 gives annual data on the Consumer Price Index (CPI) and the Wholesale Price Index (WPI), also called Producer Price Index (PPI), for the U.S. economy for the period 1960–1999. a. Plot the CPI on the vertical axis and the WPI on the horizontal axis. A priori, what kind of relationship do you expect between the two indexes? Why?

TABLE 5.10 CPI AND WPI, UNITED STATES, 1960–1999 Year 1960 1961 1962 1963 1964 1965 1966 1967 1968 1969 1970 1971 1972 1973 1974 1975 1976 1977 1978 1979 CPI 29.8 30.0 30.4 30.9 31.2 31.8 32.9 33.9 35.5 37.7 39.8 41.1 42.5 46.2 51.9 55.5 58.2 62.1 67.7 76.7 WPI 31.7 31.6 31.6 31.6 31.7 32.8 33.3 33.7 34.6 36.3 37.1 38.6 41.1 47.4 57.3 59.7 62.5 66.2 72.7 83.4 Year 1980 1981 1982 1983 1984 1985 1986 1987 1988 1989 1990 1991 1992 1993 1994 1995 1996 1997 1998 1999 CPI 86.3 94.0 97.6 101.3 105.3 109.3 110.5 115.4 120.5 126.1 133.8 137.9 141.9 145.8 149.7 153.5 158.6 161.3 163.9 168.3 WPI 93.8 98.8 100.5 102.3 103.5 103.6 99.70 104.2 109.0 113.0 118.7 115.9 117.6 118.6 121.9 125.7 128.8 126.7 122.7 128.0

Source: Economic Report of the President, 2000, pp. 373 and 379.

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b. Suppose you want to predict one of these indexes on the basis of the other index. Which will you use as the regressand and which as the regressor? Why? c. Run the regression you have decided in b. Show the standard output. Test the hypothesis that there is a one-to-one relationship between the two indexes. d. From the residuals obtained from the regression in c, can you entertain the hypothesis that the true error term is normally distributed? Show the tests you use.

APPENDIX 5A

5A.1 PROBABILITY DISTRIBUTIONS RELATED TO THE NORMAL DISTRIBUTION

The t, chi-square (χ 2 ), and F probability distributions, whose salient features are discussed in Appendix A, are intimately related to the normal distribution. Since we will make heavy use of these probability distributions in the following chapters, we summarize their relationship with the normal distribution in the following theorem; the proofs, which are beyond the scope of this book, can be found in the references.1 Theorem 5.1. If Z1 , Z2 , . . . , Zn are normally and independently distributed random variables such that Zi ∼ N(µi , σi2 ), then the sum Z = ki Zi , where k i are constants not all zero, is also distributed normally with k i µi and variance k2 σi2 ; that is, Z ∼ N( ki µi , ki2 σi2 ). Note: mean i µ denotes the mean value. In short, linear combinations of normal variables are themselves normally distributed. For example, if Z1 and Z2 are normally and independently distributed as Z1 ∼ N(10, 2) and Z2 ∼ N(8, 1.5), then the linear combination Z = 0.8Z1 + 0.2Z2 is also normally distributed with mean = 0.8(10) + 0.2(8) = 9.6 and variance = 0.64(2) + 0.04(1.5) = 1.34, that is, Z ∼ (9.6, 1.34). Theorem 5.2. If Z1 , Z2 , . . . , Zn are normally distributed but are not independent, the sum Z = ki Zi , where ki are constants not all ki µi and variance zero, is also normally distributed with mean [ ki2 σi2 + 2 ki kj cov (Zi , Z j ), i = j]. Thus, if Z1 ∼ N(6, 2) and Z2 ∼ N(7, 3) and cov (Z1 , Z2 ) = 0.8, then the linear combination 0.6Z1 + 0.4Z2 is also normally distributed with mean = 0.6(6) + 0.4(7) = 6.4 and variance = [0.36(2) + 0.16(3) + 2(0.6)(0.4)(0.8)] = 1.584.

1 For proofs of the various theorems, see Alexander M. Mood, Franklin A. Graybill, and Duane C. Bose, Introduction to the Theory of Statistics, 3d ed., McGraw-Hill, New York, 1974, pp. 239–249.

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Theorem 5.3. If Z1 , Z2 , . . . , Zn are normally and independently distributed random variables such that each Zi ∼ N(0, 1), that is, a standardized 2 2 2 Zi2 = Z1 + Z2 + · · · + Zn follows the chi-square normal variable, then 2 2 Zi ∼ χn , where n denotes the distribution with n df. Symbolically, degrees of freedom, df. In short, “the sum of the squares of independent standard normal variables has a chi-square distribution with degrees of freedom equal to the number of terms in the sum.”2 Theorem 5.4. If Z1 , Z2 , . . . , Zn are independently distributed random variables each following chi-square distribution with ki df, then the Zi = Z1 + Z2 + · · · + Zn also follows a chi-square distribution with sum k = ki df. Thus, if Z1 and Z2 are independent χ 2 variables with df of k1 and k2 , respectively, then Z = Z1 + Z2 is also a χ 2 variable with (k1 + k2 ) degrees of freedom. This is called the reproductive property of the χ 2 distribution. Theorem 5.5. If Z1 is a standardized normal variable [Z1 ∼ N(0, 1)] and another variable Z2 follows the chi-square distribution with k df and is independent of Z1 , then the variable deﬁned as √ Z1 Z1 k standard normal variable t=√ = ∼ tk √ = √ Z2 Z2 / k independent chi-square variable/df follows Student’s t distribution with k df. Note: This distribution is discussed in Appendix A and is illustrated in Chapter 5. Incidentally, note that as k, the df, increases indeﬁnitely (i.e., as k → ∞), the Student’s t distribution approaches the standardized normal distribution.3 As a matter of convention, the notation tk means Student’s t distribution or variable with k df. Theorem 5.6. If Z1 and Z2 are independently distributed chi-square variables with k1 and k2 df, respectively, then the variable F= Z1 /k1 ∼ Fk1 ,k2 Z2 /k2

has the F distribution with k1 and k2 degrees of freedom, where k1 is known as the numerator degrees of freedom and k2 the denominator degrees of freedom.

Ibid., p. 243. For proof, see Henri Theil, Introduction to Econometrics, Prentice-Hall, Englewood Cliffs, N.J., 1978, pp. 237–245.

3 2

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Again as a matter of convention, the notation Fk1 ,k2 means an F variable with k1 and k2 degrees of freedom, the df in the numerator being quoted ﬁrst. In other words, Theorem 5.6 states that the F variable is simply the ratio of two independently distributed chi-square variables divided by their respective degrees of freedom. Theorem 5.7. The square of (Student’s) t variable with k df has an F distribution with k1 = 1 df in the numerator and k2 = k df in the denominator.4 That is,

2 F1,k = tk

Note that for this equality to hold, the numerator df of the F variable 2 2 must be 1. Thus, F1,4 = t4 or F1,23 = t23 and so on. As noted, we will see the practical utility of the preceding theorems as we progress. Theorem 5.8. For large denominator df, the numerator df times the F value is approximately equal to the chi-square value with the numerator df. Thus,

2 m Fm,n = χm

as n → ∞

Theorem 5.9. For sufﬁciently large df, the chi-square distribution can be approximated by the standard normal distribution as follows: √ Z = 2χ 2 − 2k − 1 ∼ N(0, 1) where k denotes df.

5A.2 DERIVATION OF EQUATION (5.3.2)

Let Z1 = and Z2 = (n − 2) σ2 ˆ σ2 (2) ˆ (β2 − β2 ) xi2 ˆ β2 − β2 = ˆ σ se (β2 ) (1)

Provided σ is known, Z1 follows the standardized normal distribution; that is, Z1 ∼ N(0, 1). (Why?) Z2 , follows the χ 2 distribution with (n − 2) df.5

For proof, see Eqs. (5.3.2) and (5.9.1). For proof, see Robert V. Hogg and Allen T. Craig, Introduction to Mathematical Statistics, 2d ed., Macmillan, New York, 1965, p. 144.

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Furthermore, it can be shown that Z2 is distributed independently of Z1 .6 Therefore, by virtue of Theorem 5.5, the variable √ Z1 n − 2 t= √ (3) Z2 follows the t distribution with n − 2 df. Substitution of (1) and (2) into (3) gives Eq. (5.3.2).

5A.3 DERIVATION OF EQUATION (5.9.1)

Equation (1) shows that Z1 ∼ N(0, 1). Therefore, by Theorem 5.3, the preceding quantity

2 Z1 =

ˆ (β2 − β2 )2 σ2

xi2

follows the χ 2 distribution with 1 df. As noted in Section 5A.1, Z2 = (n − 2) ui ˆ2 σ2 ˆ = 2 2 σ σ

also follows the χ 2 distribution with n − 2 df. Moreover, as noted in Section 4.3, Z2 is distributed independently of Z1 . Then from Theorem 5.6, it follows that F=

2 ˆ Z1 1 (β2 − β2 )2 ( xi2 ) = Z2 /(n − 2) ui (n − 2) ˆ2

follows the F distribution with 1 and n − 2 df, respectively. Under the null hypothesis H0 : β2 = 0, the preceding F ratio reduces to Eq. (5.9.1).

5.A.4 DERIVATIONS OF EQUATIONS (5.10.2) AND (5.10.6)

Variance of Mean Prediction

Given Xi = X0 , the true mean prediction E(Y0 | X0 ) is given by E(Y0 | X0 ) = β1 + β2 X0 We estimate (1) from ˆ ˆ ˆ Y0 = β1 + β2 X0 Taking the expectation of (2), given X0 , we get ˆ ˆ ˆ E(Y0 ) = E(β1 ) + E(β2 )X0 = β1 + β2 X0

6 For proof, see J. Johnston, Econometric Methods, McGraw-Hill, 3d ed., New York, 1984, pp. 181–182. (Knowledge of matrix algebra is required to follow the proof.)

(1) (2)

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ˆ ˆ because β1 and β2 are unbiased estimators. Therefore, ˆ E(Y0 ) = E(Y0 | X0 ) = β1 + β2 X0 (3) ˆ That is, Y0 is an unbiased predictor of E(Y0 | X0 ). Now using the property that var (a + b) = var (a) + var (b) + 2 cov (a, b), we obtain

2 ˆ ˆ ˆ ˆ ˆ var (Y0 ) = var (β1 ) + var (β2 )X0 + 2 cov (β1 β2 )X0

(4)

ˆ ˆ Using the formulas for variances and covariance of β1 and β2 given in (3.3.1), (3.3.3), and (3.3.9) and manipulating terms, we obtain ¯ 1 (X0 − X)2 ˆ + var (Y0 ) = σ 2 2 n xi

Variance of Individual Prediction

= (5.10.2)

We want to predict an individual Y corresponding to X = X0 ; that is, we want to obtain Y0 = β1 + β2 X0 + u0 We predict this as ˆ ˆ ˆ Y0 = β1 + β2 X0 ˆ The prediction error, Y0 − Y0 , is ˆ ˆ ˆ Y0 − Y0 = β1 + β2 X0 + u0 − (β1 + β2 X0 ) ˆ ˆ = (β1 − β1 ) + (β2 − β2 )X0 + u0 Therefore, ˆ ˆ ˆ E(Y0 − Y0 ) = E(β1 − β1 ) + E(β2 − β2 )X0 − E(u0 ) =0 ˆ ˆ because β1 , β2 are unbiased, X0 is a ﬁxed number, and E(u0 ) is zero by assumption. ˆ Squaring (7) on both sides and taking expectations, we get var (Y0 − Y0 ) = ˆ1 ) + X 2 var (β2 ) + 2X0 cov (β1 , β2 ) + var (uo). Using the variance and coˆ var (β 0 ˆ ˆ variance formulas for β1 and β2 given earlier, and noting that var (u0 ) = σ 2, we obtain ˆ var (Y0 − Y0 ) = σ 2 1 + ¯ 1 (X0 − X)2 + 2 n xi = (5.10.6) (7) (6) (5)

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EXTENSIONS OF THE TWO-VARIABLE LINEAR REGRESSION MODEL

Some aspects of linear regression analysis can be easily introduced within the framework of the two-variable linear regression model that we have been discussing so far. First we consider the case of regression through the origin, that is, a situation where the intercept term, β1, is absent from the model. Then we consider the question of the units of measurement, that is, how the Y and X variables are measured and whether a change in the units of measurement affects the regression results. Finally, we consider the question of the functional form of the linear regression model. So far we have considered models that are linear in the parameters as well as in the variables. But recall that the regression theory developed in the previous chapters requires only that the parameters be linear; the variables may or may not enter linearly in the model. By considering models that are linear in the parameters but not necessarily in the variables, we show in this chapter how the two-variable models can deal with some interesting practical problems. Once the ideas introduced in this chapter are grasped, their extension to multiple regression models is quite straightforward, as we shall show in Chapters 7 and 8.

6.1 REGRESSION THROUGH THE ORIGIN

There are occasions when the two-variable PRF assumes the following form: Yi = β2 Xi + ui (6.1.1)

In this model the intercept term is absent or zero, hence the name regression through the origin.

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As an illustration, consider the Capital Asset Pricing Model (CAPM) of modern portfolio theory, which, in its risk-premium form, may be expressed as1 (ER i − r f ) = βi (ERm − r f ) (6.1.2) where ERi = expected rate of return on security i ERm = expected rate of return on the market portfolio as represented by, say, the S&P 500 composite stock index rf = risk-free rate of return, say, the return on 90-day Treasury bills βi = the Beta coefficient, a measure of systematic risk, i.e., risk that cannot be eliminated through diversification. Also, a measure of the extent to which the ith security’s rate of return moves with the market. A βi > 1 implies a volatile or aggressive security, whereas a βi < 1 a defensive security. (Note: Do not confuse this βi with the slope coefficient of the twovariable regression, β2.) If capital markets work efficiently, then CAPM postulates that security i’s expected risk premium (= ERi − rf) is equal to that security’s β coefficient times the expected market risk premium (= ERm − rf). If the CAPM holds, we have the situation depicted in Figure 6.1. The line shown in the figure is known as the security market line (SML). For empirical purposes, (6.1.2) is often expressed as R i − rf = βi (R m − rf ) + u i or R i − rf = αi + βi (R m − rf ) + u i

ER i – rf

(6.1.3) (6.1.4)

Security market line

ER i – rf 1

0 FIGURE 6.1 Systematic risk.

βi

1 See Haim Levy and Marshall Sarnat, Portfolio and Investment Selection: Theory and Practice, Prentice-Hall International, Englewood Cliffs, N.J., 1984, Chap. 14.

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R i – rf Security risk premium 0 Systematic risk

βi

FIGURE 6.2

The Market Model of Portfolio Theory (assuming αi = 0).

The latter model is known as the Market Model.2 If CAPM holds, αi is expected to be zero. (See Figure 6.2.) In passing, note that in (6.1.4) the dependent variable, Y, is (Ri − rf) and the explanatory variable, X, is βi, the volatility coefficient, and not (Rm − rf). Therefore, to run regression (6.1.4), one must first estimate βi, which is usually derived from the characteristic line, as described in exercise 5.5. (For further details, see exercise 8.28.) As this example shows, sometimes the underlying theory dictates that the intercept term be absent from the model. Other instances where the zerointercept model may be appropriate are Milton Friedman’s permanent income hypothesis, which states that permanent consumption is proportional to permanent income; cost analysis theory, where it is postulated that the variable cost of production is proportional to output; and some versions of monetarist theory that state that the rate of change of prices (i.e., the rate of inflation) is proportional to the rate of change of the money supply. How do we estimate models like (6.1.1), and what special problems do they pose? To answer these questions, let us first write the SRF of (6.1.1), namely, ˆ Yi = β2 Xi + ui ˆ (6.1.5)

Now applying the OLS method to (6.1.5), we obtain the following formuˆ las for β2 and its variance (proofs are given in Appendix 6A, Section 6A.1): ˆ β2 = Xi Yi Xi2 (6.1.6)

2 See, for instance, Diana R. Harrington, Modern Portfolio Theory and the Capital Asset Pricing Model: A User’s Guide, Prentice Hall, Englewood Cliffs, N.J., 1983, p. 71.

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ˆ var (β2 ) = where σ 2 is estimated by σ2 = ˆ

σ2 Xi2

(6.1.7)

ui ˆ2 n− 1

(6.1.8)

It is interesting to compare these formulas with those obtained when the intercept term is included in the model: ˆ β2 = xi yi xi2 (3.1.6) (3.3.1) (3.3.5)

σ2 ˆ var (β2 ) = xi2 σ2 = ˆ ui ˆ2 n− 2

The differences between the two sets of formulas should be obvious: In the model with the intercept term absent, we use raw sums of squares and cross products but in the intercept-present model, we use adjusted (from ˆ mean) sums of squares and cross products. Second, the df for computing σ 2 is (n − 1) in the first case and (n − 2) in the second case. (Why?) Although the interceptless or zero intercept model may be appropriate on occasions, there are some features of this model that need to be noted. First, ui , which is always zero for the model with the intercept term (the ˆ conventional model), need not be zero when that term is absent. In short, ui need not be zero for the regression through the origin. Second, r2, the ˆ coefficient of determination introduced in Chapter 3, which is always nonnegative for the conventional model, can on occasions turn out to be negative for the interceptless model! This anomalous result arises because the r2 introduced in Chapter 3 explicitly assumes that the intercept is included in the model. Therefore, the conventionally computed r2 may not be appropriate for regression-through-the-origin models.3 r 2 for Regression-through-Origin Model

As just noted, and as further discussed in Appendix 6A, Section 6A.1, the conventional r2 given in Chapter 3 is not appropriate for regressions that do not contain the intercept. But one can compute what is known as the raw r2 for such models, which is defined as raw r 2 = Xi Yi Xi2

2

Yi2

(6.1.9)

3 For additional discussion, see Dennis J. Aigner, Basic Econometrics, Prentice Hall, Englewood Cliffs, N.J., 1971, pp. 85–88.

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Note: These are raw (i.e., not mean-corrected) sums of squares and cross products. Although this raw r2 satisfies the relation 0 < r2 < 1, it is not directly comparable to the conventional r2 value. For this reason some authors do not report the r2 value for zero intercept regression models. Because of these special features of this model, one needs to exercise great caution in using the zero intercept regression model. Unless there is very strong a priori expectation, one would be well advised to stick to the conventional, intercept-present model. This has a dual advantage. First, if the intercept term is included in the model but it turns out to be statistically insignificant (i.e., statistically equal to zero), for all practical purposes we have a regression through the origin.4 Second, and more important, if in fact there is an intercept in the model but we insist on fitting a regression through the origin, we would be committing a specification error, thus violating Assumption 9 of the classical linear regression model.

AN ILLUSTRATIVE EXAMPLE: THE CHARACTERISTIC LINE OF PORTFOLIO THEORY Table 6.1 gives data on the annual rates of return (%) on Afuture Fund, a mutual fund whose primary investment objective is maximum capital gain, and on the market portfolio, as measured by the Fisher Index, for the period 1971–1980. In exercise 5.5 we introduced the characteristic line of investment analysis, which can be written as Yi = αi + βi Xi + ui (6.1.10) TABLE 6.1 ANNUAL RATES OF RETURN ON AFUTURE FUND AND ON THE FISHER INDEX (MARKET PORTFOLIO), 1971–1980 Return on Afuture Fund, % Y 67.5 19.2 −35.2 −42.0 63.7 19.3 3.6 20.0 40.3 37.5 Return on Fisher Index, % X 19.5 8.5 −29.3 −26.5 61.9 45.5 9.5 14.0 35.3 31.0

Year 1971 1972 1973 1974 1975 1976 1977 1978 1979 1980

where Yi = annual rate of return (%) on Afuture Fund Xi = annual rate of return (%) on the market portfolio βi = slope coefficient, also known as the Beta coefficient in portfolio theory, and αi = the intercept In the literature there is no consensus about the prior value of αi . Some empirical results have shown it to be positive and statistically significant and some have shown it to be not statistically significantly different from zero; in the latter case we could write the model as Yi = βi Xi + ui that is, a regression through the origin. (6.1.11)

Source: Haim Levy and Marshall Sarnat, Portfolio and Investment Selection: Theory and Practice, Prentice-Hall International, Englewood Cliffs, N.J., 1984, pp. 730 and 738. These data were obtained by the authors from Weisenberg Investment Service, Investment Companies, 1981 edition.

(Continued)

4 Henri Theil points out that if the intercept is in fact absent, the slope coefficient may be estimated with far greater precision than with the intercept term left in. See his Introduction to Econometrics, Prentice Hall, Englewood Cliffs, N.J., 1978, p. 76. See also the numerical example given next.

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AN ILLUSTRATIVE EXAMPLE

(Continued) Note: The r 2 values of (6.1.12) and (6.1.13) are not directly comparable. From these results one cannot reject the hypothesis that the true intercept is equal to zero, thereby justifying the use of (6.1.1), that is, regression through the origin. In passing, note that there is not a great deal of difference in the results of (6.1.12) and (6.1.13), although ˆ the estimated standard error of β is slightly lower for the regression-through-the-origin model, thus supporting Theil’s argument given in footnote 4 that if αi is in fact zero, the slope coefficient may be measured with greater precision: using the data given in Table 6.1 and the regression results, the reader can easily verify that the 95% confidence interval for the slope coefficient of the regression-through-the-origin model is (0.6566, 1.5232) whereas for the model (6.1.13) it is (0.5195, 1.6186); that is, the former confidence interval is narrower that the latter.

If we decide to use model (6.1.11), we obtain the following regression results

ˆ Yi = 1.0899 Xi

(0.1916) t = (5.6884) which shows that βi is significantly greater than zero. The interpretation is that a 1 percent increase in the market rate of return leads on the average to about 1.09 percent increase in the rate of return on Afuture Fund. How can we be sure that model (6.1.11), not (6.1.10), is appropriate, especially in view of the fact that there is no strong a priori belief in the hypothesis that αi is in fact zero? This can be checked by running the regression (6.1.10). Using the data given in Table 6.1, we obtained the following results: raw r 2 = 0.7825 (6.1.12)

ˆ Yi = 1.2797 + 1.0691X i

(7.6886) t = (0.1664) (0.2383) (4.4860) r = 0.7155

2

(6.1.13)

6.2

SCALING AND UNITS OF MEASUREMENT

To grasp the ideas developed in this section, consider the data given in Table 6.2, which refers to U.S. gross private domestic investment (GPDI) and gross domestic product (GDP), in billions as well as millions of (chained) 1992 dollars.

TABLE 6.2

GROSS PRIVATE DOMESTIC INVESTMENT AND GDP, UNITED STATES, 1988–1997 Observation 1988 1989 1990 1991 1992 1993 1994 1995 1996 1997 GPDIBL 828.2000 863.5000 815.0000 738.1000 790.4000 863.6000 975.7000 996.1000 1084.1000 1206.4000 GPDIM 828200.0 863500.0 815000.0 738100.0 790400.0 863600.0 975700.0 996100.0 1084100.0 1206400.0 GDPB 5865.200 6062.000 6136.300 6079.400 6244.400 6389.600 6610.700 6761.600 6994.800 7269.800 GDPM 5865200 6062000 6136300 6079400 6244400 6389600 6610700 6761600 6994800 7269800

Note: GPDIBL = gross private domestic investment, billions of 1992 dollars. GPDIM = gross private domestic investments, millions of 1992 dollars. GDPB = gross domestic product, billions of 1992 dollars. GDPM = gross domestic product, millions of 1992 dollars. Source: Economic Report of the President, 1999, Table B-2, p. 328.

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Suppose in the regression of GPDI on GDP one researcher uses data in billions of dollars but another expresses data in millions of dollars. Will the regression results be the same in both cases? If not, which results should one use? In short, do the units in which the regressand and regressor(s) are measured make any difference in the regression results? If so, what is the sensible course to follow in choosing units of measurement for regression analysis? To answer these questions, let us proceed systematically. Let ˆ ˆ Yi = β1 + β2 Xi + ui ˆ where Y = GPDI and X = GDP. Define Yi* = w1 Yi Xi* = w2 Xi (6.2.2) (6.2.3) (6.2.1)

where w1 and w2 are constants, called the scale factors; w1 may equal w2 or be different. From (6.2.2) and (6.2.3) it is clear that Yi* and Xi* are rescaled Yi and Xi . Thus, if Yi and Xi are measured in billions of dollars and one wants to express them in millions of dollars, we will have Yi* = 1000 Yi and Xi* = 1000 Xi ; here w1 = w2 = 1000. Now consider the regression using Yi* and Xi* variables: ˆ* ˆ* Yi* = β1 + β2 Xi* + ui ˆ* (6.2.4)

ˆ* ˆ where Yi* = w1 Yi , Xi* = w2 Xi , and ui = w1 ui . (Why?) We want to find out the relationships between the following pairs: 1. 2. 3. 4. 5. 6. ˆ ˆ* β1 and β1 ˆ ˆ* β2 and β2 ˆ ˆ* var (β1 ) and var (β1 ) ˆ2 ) and var (β * ) ˆ var (β 2 2 *2 σ and σ ˆ ˆ 2 2 r xy and r x* y*

From least-squares theory we know (see Chapter 3) that ¯ ˆ ˆ ¯ β1 = Y − β2 X ˆ β2 = ˆ var (β1 ) = xi yi xi2 Xi2 · σ2 n xi2 (6.2.5) (6.2.6)

(6.2.7)

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σ2 ˆ var (β2 ) = xi2 σ2 = ˆ ui ˆ2 n− 2

(6.2.8)

(6.2.9)

Applying the OLS method to (6.2.4), we obtain similarly ¯ ˆ* ˆ* ¯ β1 = Y * − β2 X * ˆ* β2 = ˆ* var (β1 ) = xi* yi* xi*2 Xi*2 · σ *2 n xi*2 (6.2.10) (6.2.11) (6.2.12) (6.2.13)

σ *2 ˆ* var (β2 ) = xi*2 σ *2 = ˆ ui ˆ *2 (n − 2)

(6.2.14)

From these results it is easy to establish relationships between the two sets of parameter estimates. All that one has to do is recall these definitional ˆ* ˆ relationships: Yi* = w1 Yi (or yi* = w1 yi ); Xi* = w2 Xi (or xi* = w2 xi ); ui = w1 ui ; * * ¯ ¯ ¯ ¯ Y = w1 Y and X = w2 X. Making use of these definitions, the reader can easily verify that w1 ˆ β2 w2 ˆ* ˆ β1 = w1 β1 ˆ* β2 = σ *2 = w2 σ 2 ˆ 1ˆ ˆ* var (β1 ) = w2 1 ˆ var (β1 )

2

(6.2.15) (6.2.16) (6.2.17) (6.2.18) (6.2.19) (6.2.20)

ˆ* var (β2 ) =

w1 w2

ˆ var (β2 )

2 2 r xy = r x* y*

From the preceding results it should be clear that, given the regression results based on one scale of measurement, one can derive the results based on another scale of measurement once the scaling factors, the w’s, are known. In practice, though, one should choose the units of measurement sensibly; there is little point in carrying all those zeros in expressing numbers in millions or billions of dollars.

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From the results given in (6.2.15) through (6.2.20) one can easily derive some special cases. For instance, if w1 = w2 , that is, the scaling factors are identical, the slope coefficient and its standard error remain unaffected in going from the (Yi , Xi ) to the (Yi* , Xi* ) scale, which should be intuitively clear. However, the intercept and its standard error are both multiplied by w1 . But if the X scale is not changed (i.e., w2 = 1) and the Y scale is changed by the factor w1 , the slope as well as the intercept coefficients and their respective standard errors are all multiplied by the same w1 factor. Finally, if the Y scale remains unchanged (i.e., w1 = 1) but the X scale is changed by the factor w2 , the slope coefficient and its standard error are multiplied by the factor (1/w2 ) but the intercept coefficient and its standard error remain unaffected. It should, however, be noted that the transformation from the (Y, X) to the (Y * , X * ) scale does not affect the properties of the OLS estimators discussed in the preceding chapters.

A NUMERICAL EXAMPLE: THE RELATIONSHIP BETWEEN GPDI AND GDP, UNITED STATES, 1988–1997 To substantiate the preceding theoretical results, let us return to the data given in Table 6.2 and examine the following results (numbers in parentheses are the estimated standard errors). Both GPDI and GDP in billions of dollars: GPDIt = −1026.498 se = (257.5874) + 0.3016 GDPt (0.0399) r 2 = 0.8772 (6.2.21)

Both GPDI and GDP in millions of dollars: GPDIt = −1,026,498 se = (257,587.4) + 0.3016 GDPt (0.0399) r 2 = 0.8772 (6.2.22)

Notice that the intercept as well as its standard error is 1000 times the corresponding values in the regression (6.2.21) (note that w1 = 1000 in going from billions to millions of dollars), but the slope coefficient as well as its standard error is unchanged, in accordance with theory. GPDI in billions of dollars and GDP in millions of dollars: GPDIt = −1026.498 se = (257.5874) + 0.000301 GDPt (0.0000399) r 2 = 0.8772 (6.2.23)

As expected, the slope coefficient as well as its standard error is 1/1000 its value in (6.2.21), since only the X, or GDP, scale is changed. GPDI in millions of dollars and GDP in billions of dollars: GPDIt = −1,026,498 se = (257,587.4) + 301.5826 GDPt (39.89989) r 2 = 0.8772 (6.2.24)

Again notice that both the intercept and the slope coefficients as well as their respective standard errors are 1000 times their values in (6.2.21), in accordance with our theoretical results. Notice that in all the regressions presented above the r 2 value remains the same, which is not surprising because the r 2 value is invariant to changes in the unit of measurement, as it is a pure, or dimensionless, number.

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A Word about Interpretation

Since the slope coefficient β2 is simply the rate of change, it is measured in the units of the ratio Units of the dependent variable Units of the explanatory variable Thus in regression (6.2.21) the interpretation of the slope coefficient 0.3016 is that if GDP changes by a unit, which is 1 billion dollars, GPDI on the average changes by 0.3016 billion dollars. In regression (6.2.23) a unit change in GDP, which is 1 million dollars, leads on average to a 0.000302 billion dollar change in GPDI. The two results are of course identical in the effects of GDP on GPDI; they are simply expressed in different units of measurement.

6.3 REGRESSION ON STANDARDIZED VARIABLES

We saw in the previous section that the units in which the regressand and regressor(s) are expressed affect the interpretation of the regression coefficients. This can be avoided if we are willing to express the regressand and regressor(s) as standardized variables. A variable is said to be standardized if we subtract the mean value of the variable from its individual values and divide the difference by the standard deviation of that variable. Thus, in the regression of Y and X, if we redefine these variables as Yi* = Xi* = ¯ Yi − Y SY ¯ Xi − X SX (6.3.1) (6.3.2)

¯ ¯ where Y = sample mean of Y, SY = sample standard deviation of Y, X = sample mean of X, and SX is the sample standard deviation of X; the variables Yi* and Xi* are called standardized variables. An interesting property of a standardized variable is that its mean value is always zero and its standard deviation is always 1. (For proof, see Appendix 6A, Section 6A.2.) As a result, it does not matter in what unit the regressand and regressor(s) are measured. Therefore, instead of running the standard (bivariate) regression: Yi = β1 + β2 Xi + ui we could run regression on the standardized variables as

* * * Yi* = β1 + β2 Xi* + ui * * = β2 Xi* + ui

(6.3.3)

(6.3.4) (6.3.5)

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since it is easy to show that, in the regression involving standardized regressand and regressor(s), the intercept term is always zero.5 The regression * * coefficients of the standardized variables, denoted by β1 and β2 , are known 6 in the literature as the beta coefficients. Incidentally, notice that (6.3.5) is a regression through the origin. How do we interpret the beta coefficients? The interpretation is that if the (standardized) regressor increases by one standard deviation, on average, the * (standardized) regressand increases by β2 standard deviation units. Thus, unlike the traditional model (6.3.3), we measure the effect not in terms of the original units in which Y and X are expressed, but in standard deviation units. To show the difference between (6.3.3) and (6.3.5), let us return to the GPDI and GDP example discussed in the preceding section. The results of (6.2.21) discussed previously are reproduced here for convenience. GPDIt = −1026.498 se = (257.5874) + 0.3016 GDPt (0.0399) r 2 = 0.8872 (6.3.6)

where GPDI and GDP are measured in billions of dollars. The results corresponding to (6.3.5) are as follows, where the starred variables are standardized variables: GPDIt = 0.9387 GDP* t se = (0.1149)

*

(6.3.7)

We know how to interpret (6.3.6): If GDP goes up by a dollar, on average GPDI goes up by about 30 cents. How about (6.3.7)? Here the interpretation is that if the (standardized) GDP increases by one standard deviation, on average, the (standardized) GPDI increases by about 0.94 standard deviations. What is the advantage of the standardized regression model over the traditional model? The advantage becomes more apparent if there is more than one regressor, a topic we will take up in Chapter 7. By standardizing all regressors, we put them on equal basis and therefore can compare them directly. If the coefficient of a standardized regressor is larger than that of another standardized regressor appearing in that model, then the latter contributes more relatively to the explanation of the regressand than the latter. In other words, we can use the beta coefficients as a measure of relative strength of the various regressors. But more on this in the next two chapters. Before we leave this topic, two points may be noted. First, for the standardized regression (6.3.7) we have not given the r2 value because this is a regression through the origin for which the usual r2 is not applicable, as pointed out in Section 6.1. Second, there is an interesting relationship between the β coefficients of the conventional model and the beta coefficients.

5 Recall from Eq. (3.1.7) that intercept = mean value of the dependent variable − slope times the mean value of the regressor. But for the standardized variables the mean values of the dependent variable and the regressor are zero. Hence the intercept value is zero. 6 Do not confuse these beta coefficients with the beta coefficients of finance theory.

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For the bivariate case, the relationship is as follows: ˆ* ˆ Sx β2 = β2 Sy (6.3.8)

where Sx = the sample standard deviation of the X regressor and Sy = the sample standard deviation of the regressand. Therefore, one can crisscross between the β and beta coefficients if we know the (sample) standard deviation of the regressor and regressand. We will see in the next chapter that this relationship holds true in the multiple regression also. It is left as an exercise for the reader to verify (6.3.8) for our illustrative example.

6.4 FUNCTIONAL FORMS OF REGRESSION MODELS

As noted in Chapter 2, this text is concerned primarily with models that are linear in the parameters; they may or may not be linear in the variables. In the sections that follow we consider some commonly used regression models that may be nonlinear in the variables but are linear in the parameters or that can be made so by suitable transformations of the variables. In particular, we discuss the following regression models: 1. 2. 3. 4. The log-linear model Semilog models Reciprocal models The logarithmic reciprocal model

We discuss the special features of each model, when they are appropriate, and how they are estimated. Each model is illustrated with suitable examples.

6.5 HOW TO MEASURE ELASTICITY: THE LOG-LINEAR MODEL

Consider the following model, known as the exponential regression model: Yi = β1 Xi 2 eui which may be expressed alternatively as7 ln Yi = ln β1 + β2 ln Xi + ui where ln = natural log (i.e., log to the base e, and where e = 2.718).8 If we write (6.5.2) as ln Yi = α + β2 ln Xi + ui (6.5.3) (6.5.2) β (6.5.1)

7 Note these properties of the logarithms: (1) ln ( AB) = ln A + ln B, (2) ln ( A/B) = ln A − ln B, and (3) ln ( Ak ) = k ln A, assuming that A and B are positive, and where k is some constant. 8 In practice one may use common logarithms, that is, log to the base 10. The relationship between the natural log and common log is: lne X = 2.3026 log10 X. By convention, ln means natural logarithm, and log means logarithm to the base 10; hence there is no need to write the subscripts e and 10 explicitly.

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where α = ln β1 , this model is linear in the parameters α and β2, linear in the logarithms of the variables Y and X, and can be estimated by OLS regression. Because of this linearity, such models are called log-log, double-log, or loglinear models. If the assumptions of the classical linear regression model are fulfilled, the parameters of (6.5.3) can be estimated by the OLS method by letting Yi* = α + β2 Xi* + ui (6.5.4) ˆ ˆ where Yi* = ln Yi and Xi* = ln Xi . The OLS estimators α and β2 obtained will be best linear unbiased estimators of α and β2, respectively. One attractive feature of the log-log model, which has made it popular in applied work, is that the slope coefficient β2 measures the elasticity of Y with respect to X, that is, the percentage change in Y for a given (small) percentage change in X.9 Thus, if Y represents the quantity of a commodity demanded and X its unit price, β2 measures the price elasticity of demand, a parameter of considerable economic interest. If the relationship between quantity demanded and price is as shown in Figure 6.3a, the double-log

Y Quantity demanded ln Y Log of quantity demanded

Y= β1 X i–β 2

lnY = ln β1 – β 2 ln Xi

Price ( a) FIGURE 6.3 Constant-elasticity model.

X

Log of price (b)

ln X

9 The elasticity coefficient, in calculus notation, is defined as (dY/Y)/(dX/ X) = [(dY/dX)(X/Y)]. Readers familiar with differential calculus will readily see that β2 is in fact the elasticity coefficient. A technical note: The calculus-minded reader will note that d(ln X)/dX = 1/ X or d(ln X) = dX/ X, that is, for infinitesimally small changes (note the differential operator d) the change in ln X is equal to the relative or proportional change in X. In practice, though, if the . change in X is small, this relationship can be written as: change in ln X = relative change in X, . where = means approximately. Thus, for small changes, . (ln Xt − ln Xt−1 ) = (Xt − Xt−1 )/ Xt−1 = relative change in X

Incidentally, the reader should note these terms, which will occur frequently: (1) absolute change, (2) relative or proportional change, and (3) percentage change, or percent growth rate. Thus, (Xt − Xt−1 ) represents absolute change, (Xt − Xt−1 )/ Xt−1 = (Xt / Xt−1 − 1) is relative or proportional change and [(Xt − Xt−1 )/ Xt−1 ]100 is the percentage change, or the growth rate. Xt and Xt−1 are, respectively, the current and previous values of the variable X.

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transformation as shown in Figure 6.3b will then give the estimate of the price elasticity (−β2). Two special features of the log-linear model may be noted: The model assumes that the elasticity coefficient between Y and X, β2, remains constant throughout (why?), hence the alternative name constant elasticity model.10 In other words, as Figure 6.3b shows, the change in ln Y per unit change in ln X (i.e., the elasticity, β2) remains the same no matter at which ln X we measure the elasticity. Another feature of the model is that alˆ ˆ though α and β2 are unbiased estimates of α and β2, β1 (the parameter enterˆ ˆ ing the original model) when estimated as β1 = antilog (α) is itself a biased estimator. In most practical problems, however, the intercept term is of secondary importance, and one need not worry about obtaining its unbiased estimate.11 In the two-variable model, the simplest way to decide whether the loglinear model fits the data is to plot the scattergram of ln Yi against ln Xi and see if the scatter points lie approximately on a straight line, as in Figure 6.3b.

AN ILLUSTRATIVE EXAMPLE: EXPENDITURE ON DURABLE GOODS IN RELATION TO TOTAL PERSONAL CONSUMPTION EXPENDITURE Table 6.3 presents data on total personal consumption expenditure (PCEXP), expenditure on durable goods (EXPDUR), expenditure on nondurable goods (EXPNONDUR), and expenditure on services (EXPSERVICES), all measured in 1992 billions of dollars.12 Suppose we wish to find the elasticity of expenditure on durable goods with respect to total personal consumption expenditure. Plotting the log of expenditure on durable goods against the log of total personal consumption expenditure, you will see that the relationship between the two variables is linear. Hence, the doublelog model may be appropriate. The regression results

are as follows: ln EXDURt = −9.6971 + 1.9056 ln PCEXt (0.0514) (37.0962)* (6.5.5) r 2 = 0.9849 se = (0.4341) t = (−22.3370)*

where * indicates that the p value is extremely small. As these results show, the elasticity of EXPDUR with respect to PCEX is about 1.90, suggesting that if total personal expenditure goes up by 1 percent, on average, the expenditure on durable goods goes up by about 1.90 percent. Thus, expenditure on durable goods is very responsive to changes in personal consumption expenditure. This is one reason why producers of durable goods keep a keen eye on changes in personal income and personal consumption expenditure. In exercises 6.17 and 6.18, the reader is asked to carry out a similar exercise for nondurable goods expenditure and expenditure on services. (Continued)

10 A constant elasticity model will give a constant total revenue change for a given percentage change in price regardless of the absolute level of price. Readers should contrast this result with the elasticity conditions implied by a simple linear demand function, Yi = β1 + β2 Xi + ui . However, a simple linear function gives a constant quantity change per unit change in price. Contrast this with what the log-linear model implies for a given dollar change in price. 11 Concerning the nature of the bias and what can be done about it, see Arthur S. Goldberger, Topics in Regression Analysis, Macmillan, New York, 1978, p. 120. 12 Durable goods include motor vehicles and parts, furniture, and household equipment; nondurable goods include food, clothing, gasoline and oil, fuel oil and coal; and services include housing, electricity and gas, transportation, and medical care.

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AN ILLUSTRATIVE EXAMPLE: . . . (Continued) TABLE 6.3 TOTAL PERSONAL EXPENDITURE AND CATEGORIES Observation 1993-I 1993-II 1993-III 1993-IV 1994-I 1994-II 1994-III 1994-IV 1995-I 1995-II 1995-III 1995-IV 1996-I 1996-II 1996-III 1996-IV 1997-I 1997-II 1997-III 1997-IV 1998-I 1998-II 1998-III EXPSERVICES 2445.3 2455.9 2480.0 2494.4 2510.9 2531.4 2543.8 2555.9 2570.4 2594.8 2610.3 2622.9 2648.5 2668.4 2688.1 2701.7 2722.1 2743.6 2775.4 2804.8 2829.3 2866.8 2904.8 EXPDUR 504.0 519.3 529.9 542.1 550.7 558.8 561.7 576.6 575.2 583.5 595.3 602.4 611.0 629.5 626.5 637.5 656.3 653.8 679.6 648.8 710.3 729.4 733.7 EXPNONDUR 1337.5 1347.8 1356.8 1361.8 1378.4 1385.5 1393.2 1402.5 1410.4 1415.9 1418.5 1425.6 1433.5 1450.4 1454.7 1465.1 1477.9 1477.1 1495.7 1494.3 1521.2 1540.9 1549.1 PCEXP 4286.8 4322.8 4366.6 4398.0 4439.4 4472.2 4498.2 4534.1 4555.3 4593.6 4623.4 4650.0 4692.1 4746.6 4768.3 4802.6 4853.4 4872.7 4947.0 4981.0 5055.1 5130.2 5181.8

Note: EXPSERVICES = expenditure on services, billions of 1992 dollars. EXPDUR = expenditure on durable goods, billions of 1992 dollars. EXPNONDUR = expenditure on nondurable goods, billions of 1992 dollars. PCEXP = total personal consumption expenditure, billions of 1992 dollars. Source: Economic Report of the President, 1999, Table B-17, p. 347.

6.6

SEMILOG MODELS: LOG–LIN AND LIN–LOG MODELS

How to Measure the Growth Rate: The Log–Lin Model

Economists, businesspeople, and governments are often interested in finding out the rate of growth of certain economic variables, such as population, GNP, money supply, employment, productivity, and trade deficit. Suppose we want to find out the growth rate of personal consumption expenditure on services for the data given in Table 6.3. Let Yt denote real expenditure on services at time t and Y0 the initial value of the expenditure on services (i.e., the value at the end of 1992-IV). You may recall the following well-known compound interest formula from your introductory course in economics. Yt = Y0 (1 + r)t (6.6.1)

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where r is the compound (i.e., over time) rate of growth of Y. Taking the natural logarithm of (6.6.1), we can write ln Yt = ln Y0 + t ln (1 + r) Now letting β1 = ln Y0 β2 = ln (1 + r) we can write (6.6.2) as ln Yt = β1 + β2 t Adding the disturbance term to (6.6.5), we obtain13 ln Yt = β1 + β2 t + ut (6.6.6) (6.6.5) (6.6.3) (6.6.4) (6.6.2)

This model is like any other linear regression model in that the parameters β1 and β2 are linear. The only difference is that the regressand is the logarithm of Y and the regressor is “time,” which will take values of 1, 2, 3, etc. Models like (6.6.6) are called semilog models because only one variable (in this case the regressand) appears in the logarithmic form. For descriptive purposes a model in which the regressand is logarithmic will be called a log-lin model. Later we will consider a model in which the regressand is linear but the regressor(s) are logarithmic and call it a lin-log model. Before we present the regression results, let us examine the properties of model (6.6.5). In this model the slope coefficient measures the constant proportional or relative change in Y for a given absolute change in the value of the regressor (in this case the variable t), that is,14 β2 = relative change in regressand absolute change in regressor (6.6.7)

If we multiply the relative change in Y by 100, (6.6.7) will then give the percentage change, or the growth rate, in Y for an absolute change in X, the regressor. That is, 100 times β2 gives the growth rate in Y; 100 times β2 is

13 We add the error term because the compound interest formula will not hold exactly. Why we add the error after the logarithmic transformation is explained in Sec. 6.8. 14 Using differential calculus one can show that β2 = d(ln Y)/dX = (1/Y)(dY/dX) = (dY/Y)/dX, which is nothing but (6.6.7). For small changes in Y and X this relation may be approximated by

(Yt − Yt−1 )/Yt−1 (Xt − Xt−1 ) Note: Here X = t.

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known in the literature as the semielasticity of Y with respect to X. (Question: To get the elasticity, what will we have to do?)

AN ILLUSTRATIVE EXAMPLE: THE RATE OF GROWTH EXPENDITURE ON SERVICES To illustrate the growth model (6.6.6), consider the data on expenditure on services given in Table 6.3. The regression results are as follows: ln EXSt = se = 7.7890 + (0.0023) 0.00743t (0.00017)

2

the end of the fourth quarter of 1992). The regression line obtained in Eq. (6.6.8) is sketched in Figure 6.4.

Log of expenditure on services

8.00 7.96 7.92 7.88 7.84 7.80

(6.6.8)

t = (3387.619)*

(44.2826)* r = 0.9894

Note: EXS stands for expenditure on services and * denotes that the p value is extremely small. The interpretation of Eq. (6.6.8) is that over the quarterly period 1993:1 to 1998:3, expenditure on services increased at the (quarterly) rate of 0.743 percent. Roughly, this is equal to an annual growth rate of 2.97 percent. Since 7.7890 = log of EXS at the beginning of the study period, by taking its antilog we obtain 2413.90 (billion dollars) as the beginning value of EXS (i.e., the value at

0

4

8

12 16 Time

20

24

FIGURE 6.4

Instantaneous versus Compound Rate of Growth. The coefficient of the trend variable in the growth model (6.6.6), β2, gives the instantaneous (at a point in time) rate of growth and not the compound (over a period of time) rate of growth. But the latter can be easily found from (6.6.4) by taking the antilog of the estimated β2 and subtracting 1 from it and multiplying the difference by 100. Thus, for our illustrative example, the estimated slope coefficient is 0.00743. Therefore, [antilog(0.00743) − 1] = 0.00746 or 0.746 percent. Thus, in the illustrative example, the compound rate of growth on expenditure on services was about 0.746 percent per quarter, which is slightly higher than the instantaneous growth rate of 0.743 percent. This is of course due to the compounding effect. Linear Trend Model. Instead of estimating model (6.6.6), researchers sometimes estimate the following model: Yt = β1 + β2 t + ut (6.6.9)

That is, instead of regressing the log of Y on time, they regress Y on time, where Y is the regressand under consideration. Such a model is called a linear trend model and the time variable t is known as the trend variable. If the slope coefficient in (6.6.9) is positive, there is an upward trend in Y, whereas if it is negative, there is a downward trend in Y.

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For the expenditure on services data that we considered earlier, the results of fitting the linear trend model (6.6.9) are as follows: EXSt = 2405.848 t = (322.9855) + 19.6920t (36.2479) r 2 = 0.9843 (6.6.10)

In contrast to Eq. (6.6.8), the interpretation of Eq. (6.6.10) is as follows: Over the quarterly period 1993-I to 1998-III, on average, expenditure on services increased at the absolute (note: not relative) rate of about 20 billion dollars per quarter. That is, there was an upward trend in the expenditure on services. The choice between the growth rate model (6.6.8) and the linear trend model (6.6.10) will depend upon whether one is interested in the relative or absolute change in the expenditure on services, although for comparative purposes it is the relative change that is generally more relevant. In passing, observe that we cannot compare the r2 values of models (6.6.8) and (6.6.10) because the regressands in the two models are different. We will show in Chapter 7 how one compares the R2’s of models like (6.6.8) and (6.6.10).

The Lin–Log Model

Unlike the growth model just discussed, in which we were interested in finding the percent growth in Y for an absolute change in X, suppose we now want to find the absolute change in Y for a percent change in X. A model that can accomplish this purpose can be written as: Yi = β1 + β2 ln Xi + ui For descriptive purposes we call such a model a lin–log model. Let us interpret the slope coefficient β2.15 As usual, β2 = = change in Y change in ln X change in Y relative change in X (6.6.11)

The second step follows from the fact that a change in the log of a number is a relative change.

15

Again, using differential calculus, we have dY 1 = β2 dX X

Therefore, β2 = dY dX X

= (6.6.12)

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Symbolically, we have β2 = where, as usual, equivalently, as Y X/X (6.6.12)

denotes a small change. Equation (6.6.12) can be written, Y = β2 ( X/X) (6.6.13)

This equation states that the absolute change in Y ( = Y) is equal to slope times the relative change in X. If the latter is multiplied by 100, then (6.6.13) gives the absolute change in Y for a percentage change in X. Thus, if ( X/X) changes by 0.01 unit (or 1 percent), the absolute change in Y is 0.01(β2 ); if in an application one finds that β2 = 500, the absolute change in Y is (0.01)(500) = 5.0. Therefore, when regression (6.6.11) is estimated by OLS, do not forget to multiply the value of the estimated slope coefficient by 0.01, or, what amounts to the same thing, divide it by 100. If you do not keep this in mind, your interpretation in an application will be highly misleading. The practical question is: When is a lin–log model like (6.6.11) useful? An interesting application has been found in the so-called Engel expenditure models, named after the German statistician Ernst Engel, 1821–1896. (See exercise 6.10.) Engel postulated that “the total expenditure that is devoted to food tends to increase in arithmetic progression as total expenditure increases in geometric progression.”16

AN ILLUSTRATIVE EXAMPLE As an illustration of the lin–log model, let us revisit our example on food expenditure in India, Example 3.2. There we fitted a linear-in-variables model as a first approximation. But if we plot the data we obtain the plot in Figure 6.5. As this figure suggests, food expenditure increases more slowly as total expenditure increases, perhaps giving credence to Engel’s law. The results of fitting the lin–log model to the data are as follows: FoodExpi = −1283.912 t= (−4.3848)* + 257.2700 ln TotalExpi (5.6625)* r 2 = 0.3769 (6.6.14) Expenditure on food (Rs.)

700 600 500 400 300 200 100 300 400 500 600 700 800 900 Total expenditure (Rs.) FIGURE 6.5 (Continued)

Note: * denotes an extremely small p value.

16 See Chandan Mukherjee, Howard White, and Marc Wuyts, Econometrics and Data Analysis for Developing Countries, Routledge, London, 1998, p. 158. This quote is attributed to H. Working, “Statistical Laws of Family Expenditure,” Journal of the American Statistical Association, vol. 38, 1943, pp. 43–56.

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AN ILLUSTRATIVE EXAMPLE

(Continued) coefficient given before, namely, Elasticity = dY X dX Y

Interpreted in the manner described earlier, the slope coefficient of about 257 means that an increase in the total food expenditure of 1 percent, on average, leads to about 2.57 rupees increase in the expenditure on food of the 55 families included in the sample. (Note: We have divided the estimated slope coefficient by 100.) Before proceeding further, note that if you want to compute the elasticity coefficient for the log–lin or lin–log models, you can do so from the definition of the elasticity

As a matter of fact, once the functional form of a model is known, one can compute elasticities by applying the preceding definition. (Table 6.6, given later, summarizes the elasticity coefficients for the various models.)

6.7

RECIPROCAL MODELS

Models of the following type are known as reciprocal models. Yi = β1 + β2 1 Xi + ui

(6.7.1)

Although this model is nonlinear in the variable X because it enters inversely or reciprocally, the model is linear in β1 and β2 and is therefore a linear regression model.17 This model has these features: As X increases indefinitely, the term β2(l/X) approaches zero (note: β2 is a constant) and Y approaches the limiting or asymptotic value β1. Therefore, models like (6.7.1) have built in them an asymptote or limit value that the dependent variable will take when the value of the X variable increases indefinitely.18 Some likely shapes of the curve corresponding to (6.7.1) are shown in Figure 6.6. As an illustration of Figure 6.6a, consider the data given in Table 6.4. These are cross-sectional data for 64 countries on child mortality and a few other variables. For now, concentrate on the variables, child mortality (CM) and per capita GNP, which are plotted in Figure 6.7. As you can see, this figure resembles Figure 6.6a: As per capita GNP increases, one would expect child mortality to decrease because people can afford to spend more on health care, assuming all other factors remain constant. But the relationship is not a straight line one: As per capita GNP increases, initially there is dramatic drop in CM but the drop tapers off as per capita GNP continues to increase.

∗ 17 If we let Xi = (1/ Xi ), then (6.7.1) is linear in the parameters as well as the variables Yi ∗ and Xi . 18 The slope of (6.7.1) is: dY/dX = −β2 (1/ X 2 ), implying that if β2 is positive, the slope is negative throughout, and if β2 is negative, the slope is positive throughout. See Figures 6.6a and 6.6c, respectively.

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Y

Y

Y

β2 > 0 β1 > 0

β2 > 0 β1 < 0

β1

β2 < 0

β1 0 X 0 –β1 (a) FIGURE 6.6 The reciprocal model: Y = β1 + β2 1 . X

Child Mortality and PGNP

X

0

–β2 β1 (c)

X

(b)

400

300

CM

200

100

0 FIGURE 6.7 Relationship between child mortality and per capita GNP in 66 countries.

0

5000

10000 PGNP

15000

20000

If we try to fit the reciprocal model (6.7.1), we obtain the following regression results: CMi = 81.79436 + 27,273.17 se = (10.8321) t = (7.5511) (3759.999) (7.2535) r 2 = 0.4590 1 PGNPi

(6.7.2)

As per capita GNP increases indefinitely, child mortality approaches its asymptotic value of about 82 deaths per thousand. As explained in footnote 18, the positive value of the coefficient of (l/PGNPt) implies that the rate of change of CM with respect to PGNP is negative. One of the important applications of Figure 6.6b is the celebrated Phillips curve of macroeconomics. Using the data on percent rate of change of money wages (Y) and the unemployment rate (X) for the United Kingdom

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TABLE 6.4

FERTILITY AND OTHER DATA FOR 64 COUNTRIES Observation 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 CM 128 204 202 197 96 209 170 240 241 55 75 129 24 165 94 96 148 98 161 118 269 189 126 12 167 135 107 72 128 27 152 224 FLFP 37 22 16 65 76 26 45 29 11 55 87 55 93 31 77 80 30 69 43 47 17 35 58 81 29 65 87 63 49 63 84 23 PGNP 1870 130 310 570 2050 200 670 300 120 290 1180 900 1730 1150 1160 1270 580 660 420 1080 290 270 560 4240 240 430 3020 1420 420 19830 420 530 TFR 6.66 6.15 7.00 6.25 3.81 6.44 6.19 5.89 5.89 2.36 3.93 5.99 3.50 7.41 4.21 5.00 5.27 5.21 6.50 6.12 6.19 5.05 6.16 1.80 4.75 4.10 6.66 7.28 8.12 5.23 5.79 6.50 Observation 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 CM 142 104 287 41 312 77 142 262 215 246 191 182 37 103 67 143 83 223 240 312 12 52 79 61 168 28 121 115 186 47 178 142 FLFP 50 62 31 66 11 88 22 22 12 9 31 19 88 35 85 78 85 33 19 21 79 83 43 88 28 95 41 62 45 85 45 67 PGNP 8640 350 230 1620 190 2090 900 230 140 330 1010 300 1730 780 1300 930 690 200 450 280 4430 270 1340 670 410 4370 1310 1470 300 3630 220 560 TFR 7.17 6.60 7.00 3.91 6.70 4.20 5.43 6.50 6.25 7.10 7.10 7.00 3.46 5.66 4.82 5.00 4.74 8.49 6.50 6.50 1.69 3.25 7.17 3.52 6.09 2.86 4.88 3.89 6.90 4.10 6.09 7.20

Note: CM = Child mortality, the number of deaths of children under age 5 in a year per 1000 live births. FLFP = Female literacy rate, percent. PGNP = per capita GNP in 1980. TFR = total fertility rate, 1980–1985, the average number of children born to a woman, using agespecific fertility rates for a given year. Source: Chandan Mukherjee, Howard White, and Marc Whyte, Econometrics and Data Analysis for Developing Countries, Routledge, London, 1998, p. 456.

for the period 1861–1957, Phillips obtained a curve whose general shape resembles Figure 6.6b (Figure 6.8).19 As Figure 6.8 shows, there is an asymmetry in the response of wage changes to the level of the unemployment rate: Wages rise faster for a unit change in unemployment if the unemployment rate is below U n, which is

19 A. W. Phillips, “The Relationship between Unemployment and the Rate of Change of Money Wages in the United Kingdom, 1861–1957,” Economica, November 1958, vol. 15, pp. 283–299. Note that the original curve did not cross the unemployment rate axis, but Fig. 6.8 represents a later version of the curve.

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Rate of change of money wages, %

The natural rate of unemployment

0

UN

Unemployment rate, %

–β1

FIGURE 6.8

The Phillips curve.

called the natural rate of unemployment by economists [defined as the rate of unemployment required to keep (wage) inflation constant], and then they fall for an equivalent change when the unemployment rate is above the natural rate, β1, indicating the asymptotic floor for wage change. This particular feature of the Phillips curve may be due to institutional factors, such as union bargaining power, minimum wages, unemployment compensation, etc. Since the publication of Phillips’ article, there has been very extensive research on the Phillips curve at the theoretical as well as empirical levels. Space does not permit us to go into the details of the controversy surrounding the Phillips curve. The Phillips curve itself has gone through several incarnations. A comparatively recent formulation is provided by Olivier Blanchard.20 If we let πt denote the inflation rate at time t, which is defined as the percentage change in the price level as measured by a representative price index, such as the Consumer Price Index (CPI), and UNt denote the unemployment rate at time t, then a modern version of the Phillips curve can be expressed in the following format: πt − πte = β2 (UNt − U n) + ut where πt = actual inflation rate at time t πte = expected inflation rate at time t, the expectation being formed in year (t − 1) (6.7.3)

20 See Olivier Blanchard, Macroeconomics, Prentice Hall, Englewood Cliffs, N.J., 1997, Chap. 17.

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UNt = actual unemployment rate prevailing at time t U n = natural rate of unemployment at time t ut = stochastic error term21 Since πte is not directly observable, as a starting point one can make the simplifying assumption that πte = πt−1 ; that is, the inflation expected this year is the inflation rate that prevailed in the last year; of course, more complicated assumptions about expectations formation can be made, and we will discuss this topic in Chapter 17, on distributed lag models. Substituting this assumption into (6.7.3) and writing the regression model in the standard form, we obtain the following estimating equation: πt − πt−1 = β1 + β2 UNt + ut (6.7.4)

where β1 = −β2 U n. Equation (6.7.4) states that the change in the inflation rate between two time periods is linearly related to the current unemployment rate. A priori, β2 is expected to be negative (why?) and β1 is expected to be positive (this figures, since β2 is negative and U n is positive). Incidentally, the Phillips relationship given in (6.7.3) is known in the literature as the modified Phillips curve, or the expectations-augmented Phillips curve (to indicate that πt−1 stands for expected inflation), or the acceleratonist Phillips curve (to suggest that a low unemployment rate leads to an increase in the inflation rate and hence an acceleration of the price level). As an illustration of the modified Phillips curve, we present in Table 6.5 data on inflation as measured by year-to-year percentage in the Consumer Price Index (CPIflation) and the unemployment rate for the period 1960–1998. The unemployment rate represents the civilian unemployment rate. From these data we obtained the change in the inflation rate (πt − πt−1) and plotted it against the civilian unemployment rate; we are using the CPI as a measure of inflation. The resulting graph appears in Figure 6.9. As expected, the relation between the change in inflation rate and the unemployment rate is negative—a low unemployment rate leads to an increase in the inflation rate and therefore an acceleration of the price level, hence the name accelerationist Phillips curve. Looking at Figure 6.9, it is not obvious whether a linear (straight line) regression model or a reciprocal model fits the data; there may be a curvilinear relationship between the two variables. We present below regressions based on both the models. However, keep in mind that for the reciprocal model the intercept term is expected to be negative and the slope positive, as noted in footnote 18. Linear model: (πt − πt−1) = 4.1781 − t = (3.9521) 0.6895 UNt (−4.0692) r2 = 0.3150 (6.7.5)

21 Economists believe this error term represents some kind of supply shock, such as the OPEC oil embargoes of 1973 and 1979.

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TABLE 6.5

INFLATION RATE AND UNEMPLOYMENT RATE, UNITED STATES, 1960–1998 Observation 1960 1961 1962 1963 1964 1965 1966 1967 1968 1969 1970 1971 1972 1973 1974 1975 1976 1977 1978 1979 INFLRATE 1.7 1.0 1.0 1.3 1.3 1.6 2.9 3.1 4.2 5.5 5.7 4.4 3.2 6.2 11.0 9.1 5.8 6.5 7.6 11.3 UNRATE 5.5 6.7 5.5 5.7 5.2 4.5 3.8 3.8 3.6 3.5 4.9 5.9 5.6 4.9 5.6 8.5 7.7 7.1 6.1 5.8 Observation 1980 1981 1982 1983 1984 1985 1986 1987 1988 1989 1990 1991 1992 1993 1994 1995 1996 1997 1998 INFLRATE 13.5 10.3 6.2 3.2 4.3 3.6 1.9 3.6 4.1 4.8 5.4 4.2 3.0 3.0 2.6 2.8 3.0 2.3 1.6 UNRATE 7.1 7.6 9.7 9.6 7.5 7.2 7.0 6.2 5.5 5.3 5.6 6.8 7.5 6.9 6.1 5.6 5.4 4.9 4.5

Note: The inflation rate is the percent year-to-year change in CPI. The unemployment rate is the civilian unemployment rate. Source: Economic Report of the President, 1999, Table B-63, p. 399, for CPI changes and Table B-42, p. 376, for the unemployment rate.

6 Change in inflation rate 4 2 0 –2 –4 –6 FIGURE 6.9 The modified Phillips curve. 3 4 5 6 7 8 9 10

Unemployment rate (%)

Reciprocal model: (πt − πt−1 ) = −3.2514 + 18.5508 t = (−2.9715) (3.0625) 1 UN t r = 0.2067

2

(6.7.6)

All the estimated coefficients in both the models are individually statistically significant, all the p values being lower than the 0.005 level.

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Model (6.7.5) shows that if the unemployment rate goes down by 1 percentage point, on average, the change in the inflation rate goes up by about 0.7 percentage points, and vice versa. Model (6.7.6) shows that even if the unemployment rate increases indefinitely, the most the change in the inflation rate will go down will be about 3.25 percentage points. Incidentally, from Eq. (6.7.5), we can compute the underlying natural rate of unemployment as: ˆ β1 4.1781 Un = = = 6.0596 (6.7.7) ˆ 0.6895 −β2 That is, the natural rate of unemployment is about 6.06%. Economists put the natural rate between 5 to 6%, although in the recent past in the United States the actual rate has been much below this rate.

Log Hyperbola or Logarithmic Reciprocal Model

We conclude our discussion of reciprocal models by considering the logarithmic reciprocal model, which takes the following form: ln Yi = β1 − β2 1 Xi + ui (6.7.8)

Its shape is as depicted in Figure 6.10. As this figure shows, initially Y increases at an increasing rate (i.e., the curve is initially convex) and then it increases at a decreasing rate (i.e., the curve becomes concave).22 Such a

Y

X FIGURE 6.10 The log reciprocal model.

22

From calculus, it can be shown that d 1 (ln Y) = −β2 − 2 dX X

= β2

1 X2

But d 1 dY (ln Y) = dX Y dX Making this substitution, we obtain Y dY = β2 2 dX X which is the slope of Y with respect to X.

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model may therefore be appropriate to model a short-run production function. Recall from microeconomics that if labor and capital are the inputs in a production function and if we keep the capital input constant but increase the labor input, the short-run output–labor relationship will resemble Figure 6.10. (See Example 7.4, Chapter 7.)

6.8 CHOICE OF FUNCTIONAL FORM

In this chapter we discussed several functional forms an empirical model can assume, even within the confines of the linear-in-parameter regression models. The choice of a particular functional form may be comparatively easy in the two-variable case, because we can plot the variables and get some rough idea about the appropriate model. The choice becomes much harder when we consider the multiple regression model involving more than one regressor, as we will discover when we discuss this topic in the next two chapters. There is no denying that a great deal of skill and experience are required in choosing an appropriate model for empirical estimation. But some guidelines can be offered: 1. The underlying theory (e.g., the Phillips curve) may suggest a particular functional form. 2. It is good practice to find out the rate of change (i.e., the slope) of the regressand with respect to the regressor as well as to find out the elasticity of the regressand with respect to the regressor. For the various models considered in this chapter, we provide the necessary formulas for the slope and elasticity coefficients of the various models in Table 6.6. The knowledge of these formulas will help us to compare the various models.

TABLE 6.6 Model Equation Slope =

dY dX

Elasticity = X * Y

dY X dX Y

Linear Log–linear Log–lin Lin–log Reciprocal Log reciprocal

Y = β1 + β2X lnY = β1 + β2 ln X lnY = β1 + β2 X Y = β1 + β2 ln X Y = β1 + β2 lnY = β1 − β2 1 X 1 X

β2 β2 Y X 1 X 1 X2 Y X2

β2 β2

β2 (Y ) β2 −β2 β2

β2 (X )* β2 −β2 β2 1 * Y 1 * XY 1 * X

Note: * indicates that the elasticity is variable, depending on the value taken by X or Y or both. When no X and Y values are specified, in practice, very often these elasticities are measured at the mean values of these ¯ ¯ variables, namely, X and Y.

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3. The coefficients of the model chosen should satisfy certain a priori expectations. For example, if we are considering the demand for automobiles as a function of price and some other variables, we should expect a negative coefficient for the price variable. 4. Sometime more than one model may fit a given set of data reasonably well. In the modified Phillips curve, we fitted both a linear and a reciprocal model to the same data. In both cases the coefficients were in line with prior expectations and they were all statistically significant. One major difference was that the r 2 value of the linear model was larger than that of the reciprocal model. One may therefore give a slight edge to the linear model over the reciprocal model. But make sure that in comparing two r 2 values the dependent variable, or the regressand, of the two models is the same; the regressor(s) can take any form. We will explain the reason for this in the next chapter. 5. In general one should not overemphasize the r 2 measure in the sense that the higher the r 2 the better the model. As we will discuss in the next chapter, r 2 increases as we add more regressors to the model. What is of greater importance is the theoretical underpinning of the chosen model, the signs of the estimated coefficients and their statistical significance. If a model is good on these criteria, a model with a lower r 2 may be quite acceptable. We will revisit this important topic in greater depth in Chapter 13.

*6.9 A NOTE ON THE NATURE OF THE STOCHASTIC ERROR TERM: ADDITIVE VERSUS MULTIPLICATIVE STOCHASTIC ERROR TERM

Consider the following regression model, which is the same as (6.5.1) but without the error term: Yi = β1 X β2 (6.9.1)

For estimation purposes, we can express this model in three different forms: Yi = β1 Xi 2 ui Yi = β1 Xi 2 eui Yi = β β1 Xi 2 β β

(6.9.2) (6.9.3) (6.9.4)

+ ui

Taking the logarithms on both sides of these equations, we obtain ln Yi = α + β2 ln Xi + ln ui ln Yi = α + β2 ln Xi + ui ln Yi = ln β1 Xi 2 + ui where α = ln β1.

*Optional

β

(6.9.2a) (6.9.3a) (6.9.4a)

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Models like (6.9.2) are intrinsically linear (in-parameter) regression models in the sense that by suitable (log) transformation the models can be made linear in the parameters α and β2. (Note: These models are nonlinear in β1.) But model (6.9.4) is intrinsically nonlinear-in-parameter. There is no simple way to take the log of (6.9.4) because ln ( A + B) = ln A + ln B. Although (6.9.2) and (6.9.3) are linear regression models and can be estimated by OLS or ML, we have to be careful about the properties of the stochastic error term that enters these models. Remember that the BLUE property of OLS requires that ui has zero mean value, constant variance, and zero autocorrelation. For hypothesis testing, we further assume that ui follows the normal distribution with mean and variance values just discussed. In short, we have assumed that ui ∼ N(0, σ 2 ). Now consider model (6.9.2). Its statistical counterpart is given in (6.9.2a). To use the classical normal linear regression model (CNLRM), we have to assume that ln ui ∼ N(0, σ 2 ) (6.9.5)

Therefore, when we run the regression (6.9.2a), we will have to apply the normality tests discussed in Chapter 5 to the residuals obtained from this regression. Incidentally, note that if ln ui follows the normal distribution with zero mean and constant variance, then statistical theory shows that ui 2 in (6.9.2) must follow the log-normal distribution with mean eσ /2 and 2 2 variance eσ (eσ − 1). As the preceding analysis shows, one has to pay very careful attention to the error term in transforming a model for regression analysis. As for (6.9.4), this model is a nonlinear-in-parameter regression model and will have to be solved by some iterative computer routine. Model (6.9.3) should not pose any problems for estimation. To sum up, pay very careful attention to the disturbance term when you transform a model for regression analysis. Otherwise, a blind application of OLS to the transformed model will not produce a model with desirable statistical properties.

6.10

SUMMARY AND CONCLUSIONS

This chapter introduced several of the finer points of the classical linear regression model (CLRM). 1. Sometimes a regression model may not contain an explicit intercept term. Such models are known as regression through the origin. Although the algebra of estimating such models is simple, one should use such modui is nonzero; ˆ els with caution. In such models the sum of the residuals additionally, the conventionally computed r 2 may not be meaningful. Unless

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there is a strong theoretical reason, it is better to introduce the intercept in the model explicitly. 2. The units and scale in which the regressand and the regressor(s) are expressed are very important because the interpretation of regression coefficients critically depends on them. In empirical research the researcher should not only quote the sources of data but also state explicitly how the variables are measured. 3. Just as important is the functional form of the relationship between the regressand and the regressor(s). Some of the important functional forms discussed in this chapter are (a) the log-linear or constant elasticity model, (b) semilog regression models, and (c) reciprocal models. 4. In the log-linear model both the regressand and the regressor(s) are expressed in the logarithmic form. The regression coefficient attached to the log of a regressor is interpreted as the elasticity of the regressand with respect to the regressor. 5. In the semilog model either the regressand or the regressor(s) are in the log form. In the semilog model where the regressand is logarithmic and the regressor X is time, the estimated slope coefficient (multiplied by 100) measures the (instantaneous) rate of growth of the regressand. Such models are often used to measure the growth rate of many economic phenomena. In the semilog model if the regressor is logarithmic, its coefficient measures the absolute rate of change in the regressand for a given percent change in the value of the regressor. 6. In the reciprocal models, either the regressand or the regressor is expressed in reciprocal, or inverse, form to capture nonlinear relationships between economic variables, as in the celebrated Phillips curve. 7. In choosing the various functional forms, great attention should be paid to the stochastic disturbance term ui. As noted in Chapter 5, the CLRM explicitly assumes that the disturbance term has zero mean value and constant (homoscedastic) variance and that it is uncorrelated with the regressor(s). It is under these assumptions that the OLS estimators are BLUE. Further, under the CNLRM, the OLS estimators are also normally distributed. One should therefore find out if these assumptions hold in the functional form chosen for empirical analysis. After the regression is run, the researcher should apply diagnostic tests, such as the normality test, discussed in Chapter 5. This point cannot be overemphasized, for the classical tests of hypothesis, such as the t, F, and χ2, rest on the assumption that the disturbances are normally distributed. This is especially critical if the sample size is small. 8. Although the discussion so far has been confined to two-variable regression models, the subsequent chapters will show that in many cases the extension to multiple regression models simply involves more algebra without necessarily introducing more fundamental concepts. That is why it is so very important that the reader have a firm grasp of the two-variable regression model.

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EXERCISES

Questions 6.1. Consider the regression model yi = β1 + β2 xi + ui ¯ ¯ where yi = (Yi − Y) and xi = ( X i − X). In this case, the regression line must pass through the origin. True or false? Show your calculations. 6.2. The following regression results were based on monthly data over the period January 1978 to December 1987: ˆ Yt = 0.00681

+ 0.75815X t (0.27009) (2.80700) (0.0186) r 2 = 0.4406

se = (0.02596) t = (0.26229) p value = (0.7984)

ˆ Yt = 0.76214Xt

se = (0.265799) t = (2.95408) p value = (0.0131) r 2 = 0.43684

where Y = monthly rate of return on Texaco common stock, %, and X = monthly market rate of return,%.* a. What is the difference between the two regression models? b. Given the preceding results, would you retain the intercept term in the first model? Why or why not? c. How would you interpret the slope coefficients in the two models? d. What is the theory underlying the two models? e. Can you compare the r 2 terms of the two models? Why or why not? f. The Jarque–Bera normality statistic for the first model in this problem is 1.1167 and for the second model it is 1.1170. What conclusions can you draw from these statistics? g. The t value of the slope coefficient in the zero intercept model is about 2.95, whereas that with the intercept present is about 2.81. Can you rationalize this result? 6.3. Consider the following regression model:

1 = β1 + β2 Yi 1 Xi + ui

Note: Neither Y nor X assumes zero value. a. Is this a linear regression model? b. How would you estimate this model?

* The underlying data were obtained from the data diskette included in Ernst R. Berndt, The Practice of Econometrics: Classic and Contemporary, Addison-Wesley, Reading, Mass., 1991.

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c. What is the behavior of Y as X tends to infinity? d. Can you give an example where such a model may be appropriate? 6.4. Consider the log-linear model: ln Yi = β1 + β2 ln X i + ui

Plot Y on the vertical axis and X on the horizontal axis. Draw the curves showing the relationship between Y and X when β2 = 1, and when β2 > 1, and when β2 < 1. 6.5. Consider the following models: Model I: Model II:

Yi = β1 + β2 X i + ui Yi* = α1 + α2 X i* + ui

ˆ ˆ where Y* and X* are standardized variables. Show that α2 = β2 (Sx /Sy ) and hence establish that although the regression slope coefficients are independent of the change of origin they are not independent of the change of scale. 6.6. Consider the following models: ln Yi* = α1 + α2 ln X i* + u* i ln Yi = β1 + β2 ln X i + ui

6.7. 6.8. 6.9. 6.10.

where Yi* = w 1 Yi and X i* = w 2 X i , the w’s being constants. a. Establish the relationships between the two sets of regression coefficients and their standard errors. b. Is the r2 different between the two models? Between regressions (6.6.8) and (6.6.10), which model do you prefer? Why? For the regression (6.6.8), test the hypothesis that the slope coefficient is not significantly different from 0.005. From the Phillips curve given in (6.7.3), is it possible to estimate the natural rate of unemployment? How? The Engel expenditure curve relates a consumer’s expenditure on a commodity to his or her total income. Letting Y = consumption expenditure on a commodity and X = consumer income, consider the following models:

Yi = β1 + β2 X i + ui Yi = β1 + β2 (1/X i ) + ui ln Yi = ln β1 + β2 ln X i + ui ln Yi = ln β1 + β2 (1/X i ) + ui Yi = β1 + β2 ln X i + ui

Which of these model(s) would you choose for the Engel expenditure curve and why? (Hint: Interpret the various slope coefficients, find out the expressions for elasticity of expenditure with respect to income, etc.)

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6.11. Consider the following model:

Yi = e β1 +β2 X i 1 + e β1 +β2 X i

As it stands, is this a linear regression model? If not, what “trick,” if any, can you use to make it a linear regression model? How would you interpret the resulting model? Under what circumstances might such a model be appropriate? 6.12. Graph the following models (for ease of exposition, we have omitted the observation subscript, i): a. Y = β1 X β2, for β2 > 1, β2 = 1, 0 < β2 < 1, . . . . b. Y = β1 e β2 X, for β2 > 0 and β2 < 0. Discuss where such models might be appropriate. Problems 6.13. You are given the data in Table 6.7.* Fit the following model to these data and obtain the usual regression statistics and interpret the results:

1 100 = β1 + β2 100 − Yi Xi

TABLE 6.7 Yi Xi 86 3 79 7 76 12 69 17 65 25 62 35 52 45 51 55 51 70 48 120

6.14. To measure the elasticity of substitution between capital and labor inputs Arrow, Chenery, Minhas, and Solow, the authors of the now famous CES (constant elasticity of substitution) production function, used the following model†: log V L = log β1 + β2 log W + u

where (V/L) = value added per unit of labor L = labor input W = real wage rate The coefficient β2 measures the elasticity of substitution between labor and capital (i.e., proportionate change in factor proportions/proportionate change in relative factor prices). From the data given in Table 6.8, verify that the estimated elasticity is 1.3338 and that it is not statistically significantly different from 1. 6.15. Table 6.9 gives data on the GDP (gross domestic product) deflator for domestic goods and the GDP deflator for imports for Singapore for the period 1968–1982. The GDP deflator is often used as an indicator of inflation in place of the CPI. Singapore is a small, open economy, heavily dependent on foreign trade for its survival.

*Source: Adapted from J. Johnston, Econometric Methods, 3d ed., McGraw-Hill, New York, 1984, p. 87. Actually this is taken from an econometric examination of Oxford University in 1975. † “Capital-Labor Substitution and Economic Efficiency,” Review of Economics and Statistics, August 1961, vol. 43, no. 5, pp. 225–254.

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TABLE 6.8 Industry Wheat flour Sugar Paints and varnishes Cement Glass and glassware Ceramics Plywood Cotton textiles Woolen textiles Jute textiles Chemicals Aluminum Iron and steel Bicycles Sewing machines log(V/L) 3.6973 3.4795 4.0004 3.6609 3.2321 3.3418 3.4308 3.3158 3.5062 3.2352 3.8823 3.7309 3.7716 3.6601 3.7554 log W 2.9617 2.8532 3.1158 3.0371 2.8727 2.9745 2.8287 3.0888 3.0086 2.9680 3.0909 3.0881 3.2256 3.1025 3.1354

Source: Damodar Gujarati, “A Test of ACMS Production Function: Indian Industries, 1958,” Indian Journal of Industrial Relations, vol. 2, no. 1, July 1966, pp. 95–97.

TABLE 6.9 GDP deflator for domestic goods, Y 1000 1023 1040 1087 1146 1285 1485 1521 1543 1567 1592 1714 1841 1959 2033 GDP deflator for imports, X 1000 1042 1092 1105 1110 1257 1749 1770 1889 1974 2015 2260 2621 2777 2735

Year 1968 1969 1970 1971 1972 1973 1974 1975 1976 1977 1978 1979 1980 1981 1982

Source: Colin Simkin, “Does Money Matter in Singapore?” The Singapore Economic Review, vol. XXIX, no.1, April 1984, Table 6, p. 8.

To study the relationship between domestic and world prices, you are given the following models: 1. Yt = α1 + α2 X t + ut 2. Yt = β2 X t + ut where Y = GDP deflator for domestic goods and X = GDP deflator for imports.

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a. How would you choose between the two models a priori? b. Fit both models to the data and decide which gives a better fit. c. What other model(s) might be appropriate for the data? 6.16. Refer to the data given in exercise 6.15. The means of Y and X are 1456 and 1760, respectively, and the corresponding standard deviations are 346 and 641. Estimate the following regression:

Yt* = α1 + α2 X t* + ut

where the starred variables are standardized variables, and interpret the results. 6.17. Refer to Table 6.3. Find out the rate of growth of expenditure on durable goods. What is the estimated semielasticity? Interpret your results. Would it make sense to run a double-log regression with expenditure on durable goods as the regressand and time as the regressor? How would you interpret the slope coefficient in this case. 6.18. From the data given in Table 6.3, find out the growth rate of expenditure on nondurable goods and compare your results with those obtained from problem 6.17. 6.19. Revisit exercise 1.7. Now that you know several functional forms, which one might be appropriate to study the relationship between advertising impressions retained and the amount of money spent on advertising? Show the necessary calculations.

APPENDIX 6A

6A.1 DERIVATION OF LEAST-SQUARES ESTIMATORS FOR REGRESSION THROUGH THE ORIGIN

We want to minimize ui = ˆ2 ˆ (Yi − β2 Xi )2 (1)

ˆ with respect to β2 . ˆ Differentiating (1) with respect to β2 , we obtain d ˆ dβ2 ui ˆ2 =2 ˆ (Yi − β2 Xi )(−Xi ) (2)

Setting (2) equal to zero and simplifying, we get ˆ β2 = Xi Yi Xi2 (6.1.6) = (3)

Now substituting the PRF: Yi = β2 Xi + ui into this equation, we obtain ˆ β2 = Xi (β2 Xi + ui ) Xi2 Xi ui Xi2

(4)

= β2 +

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ˆ [Note: E(β2 ) = β2 .] Therefore, ˆ E(β2 − β2 )2 = E Xi ui Xi2

2

(5)

Expanding the right-hand side of (5) and noting that the Xi are nonstochastic and the ui are homoscedastic and uncorrelated, we obtain ˆ ˆ var (β2 ) = E(β2 − β2 )2 = σ2 Xi2 (6.1.7) = (6)

Incidentally, note that from (2) we get, after equating it to zero ui Xi = 0 ˆ (7)

From Appendix 3A, Section 3A.1 we see that when the intercept term is preˆ sent in the model, we get in addition to (7) the condition ui = 0. From the mathematics just given it should be clear why the regression through the ˆ origin model may not have the error sum, ui , equal to zero. ˆ Suppose we want to impose the condition that ui = 0. In that case we have ˆ Yi = β2 ˆ = β2 Xi + Xi , ui ˆ (8) since ui = 0 by construction ˆ

This expression then gives ˆ β2 = Yi Xi

¯ Y mean value of Y = = ¯ mean value of X X

(9)

ˆ But this estimator is not the same as (3) above or (6.1.6). And since the β2 ˆ2 of (9) cannot be unbiased. of (3) is unbiased (why?), the β The upshot is that, in regression through the origin, we cannot have both ui Xi and ui equal to zero, as in the conventional model. The only conˆ ˆ ˆ dition that is satisfied is that ui Xi is zero. Recall that ˆ Yi = Yi + u ˆ (2.6.3)

Summing this equation on both sides and dividing by N, the sample size, we obtain ¯ ¯ ¯ ˆ ˆ Y =Y+u (10) Since for the zero intercept model it then follows that ¯ ui and, therefore u, need not be zero, ˆ ˆ (11)

¯ ˆ ¯ Y=Y

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that is, the mean of actual Y values need not be equal to the mean of the estimated Y values; the two mean values are identical for the intercept-present model, as can be seen from (3.1.10). It was noted that, for the zero-intercept model, r2 can be negative, whereas for the conventional model it can never be negative. This condition can be shown as follows. Using (3.5.5a), we can write r2 = 1 − RSS =1− TSS ui ˆ2 yi2 (12)

Now for the conventional, or intercept-present, model, Eq. (3.3.6) shows that RSS = ˆ2 ui = ˆ2 yi2 − β2 xi2 ≤ yi2 (13)

ˆ unless β2 is zero (i.e., X has no influence on Y whatsoever). That is, for the conventional model, RSS ≤ TSS, or, r2 can never be negative. For the zero-intercept model it can be shown analogously that RSS = ui = ˆ2 ˆ2 Yi2 − β2 Xi2 (14)

(Note: The sums of squares of Y and X are not mean-adjusted.) Now there is ¯ yi2 = Yi2 − NY 2 (the no guarantee that this RSS will always be less than TSS), which suggests that RSS can be greater than TSS, implying that r 2, as conventionally defined, can be negative. Incidentally, notice that in this case ¯ ˆ 2 Xi2 < NY 2 . RSS will be greater than TSS if β2

6A.2 PROOF THAT A STANDARDIZED VARIABLE HAS ZERO MEAN AND UNIT VARIANCE

¯ Consider the random variable (r.v.) Y with the (sample) mean value of Y and (sample) standard deviation of Sy. Define Yi* = ¯ Yi − Y Sy (15)

Hence Yi* is a standardized variable. Notice that standardization involves a dual operation: (1) change of the origin, which is the numerator of (15), and (2) change of scale, which is the denominator. Thus, standardization involves both a change of the origin and change of scale. Now 1 ¯ Yi* = Sy ¯ (Yi − Y) =0 n (16)

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since the sum of deviation of a variable from its mean value is always zero. Hence the mean value of the standardized value is zero. (Note: We could pull out the Sy term from the summation sign because its value is known.) Now ¯ (Yi − Y)2 /(n − 1) 2 Sy* = 2 Sy = = Note that

2 Sy =

1 2 (n − 1)Sy

2 (n − 1)Sy 2 (n − 1)Sy

¯ (Yi − Y)2 =1

(17)

¯ (Yi − Y)2 n− 1

which is the sample variance of Y.

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MULTIPLE REGRESSION ANALYSIS: THE PROBLEM OF ESTIMATION

The two-variable model studied extensively in the previous chapters is often inadequate in practice. In our consumption–income example, for instance, it was assumed implicitly that only income X affects consumption Y. But economic theory is seldom so simple for, besides income, a number of other variables are also likely to affect consumption expenditure. An obvious example is wealth of the consumer. As another example, the demand for a commodity is likely to depend not only on its own price but also on the prices of other competing or complementary goods, income of the consumer, social status, etc. Therefore, we need to extend our simple two-variable regression model to cover models involving more than two variables. Adding more variables leads us to the discussion of multiple regression models, that is, models in which the dependent variable, or regressand, Y depends on two or more explanatory variables, or regressors. The simplest possible multiple regression model is three-variable regression, with one dependent variable and two explanatory variables. In this and the next chapter we shall study this model. Throughout, we are concerned with multiple linear regression models, that is, models linear in the parameters; they may or may not be linear in the variables.

7.1 THE THREE-VARIABLE MODEL: NOTATION AND ASSUMPTIONS

Generalizing the two-variable population regression function (PRF) (2.4.2), we may write the three-variable PRF as Yi = β1 + β2 X2i + β3 X3i + ui

202

(7.1.1)

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where Y is the dependent variable, X2 and X3 the explanatory variables (or regressors), u the stochastic disturbance term, and i the ith observation; in case the data are time series, the subscript t will denote the tth observation.1 In Eq. (7.1.1) β1 is the intercept term. As usual, it gives the mean or average effect on Y of all the variables excluded from the model, although its mechanical interpretation is the average value of Y when X2 and X3 are set equal to zero. The coefﬁcients β2 and β3 are called the partial regression coefﬁcients, and their meaning will be explained shortly. We continue to operate within the framework of the classical linear regression model (CLRM) ﬁrst introduced in Chapter 3. Speciﬁcally, we assume the following: Zero mean value of ui , or E (ui | X2i , X3i ) = 0 No serial correlation, or cov (ui , uj ) = 0 Homoscedasticity, or var (ui ) = σ 2 Zero covariance between ui and each X variable, or cov (ui , X2i ) = cov (ui , X3i ) = 0 No speciﬁcation bias, or The model is correctly speciﬁed No exact collinearity between the X variables, or No exact linear relationship between X2 and X3 (7.1.7) (7.1.6) (7.1.5)2 (7.1.4) i= j (7.1.3) for each i (7.1.2)

In addition, as in Chapter 3, we assume that the multiple regression model is linear in the parameters, that the values of the regressors are ﬁxed in repeated sampling, and that there is sufﬁcient variability in the values of the regressors. The rationale for assumptions (7.1.2) through (7.1.6) is the same as that discussed in Section 3.2. Assumption (7.1.7), that there be no exact linear relationship between X2 and X3 , technically known as the assumption of

1

For notational symmetry, Eq. (7.1.1) can also be written as Yi = β1 X1i + β2 X2i + β3 X3i + ui

with the provision that X1i = 1 for all i. 2 This assumption is automatically fulﬁlled if X2 and X3 are nonstochastic and (7.1.2) holds.

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no collinearity or no multicollinearity if more than one exact linear relationship is involved, is new and needs some explanation. Informally, no collinearity means none of the regressors can be written as exact linear combinations of the remaining regressors in the model. Formally, no collinearity means that there exists no set of numbers, λ2 and λ3 , not both zero such that λ2 X2i + λ3 X3i = 0 (7.1.8)

If such an exact linear relationship exists, then X2 and X3 are said to be collinear or linearly dependent. On the other hand, if (7.1.8) holds true only when λ2 = λ3 = 0, then X2 and X3 are said to be linearly independent. Thus, if X2i = −4X3i or X2i + 4X3i = 0 (7.1.9)

the two variables are linearly dependent, and if both are included in a regression model, we will have perfect collinearity or an exact linear relationship between the two regressors. Although we shall consider the problem of multicollinearity in depth in Chapter 10, intuitively the logic behind the assumption of no multicollinearity is not too difﬁcult to grasp. Suppose that in (7.1.1) Y, X2 , and X3 represent consumption expenditure, income, and wealth of the consumer, respectively. In postulating that consumption expenditure is linearly related to income and wealth, economic theory presumes that wealth and income may have some independent inﬂuence on consumption. If not, there is no sense in including both income and wealth variables in the model. In the extreme, if there is an exact linear relationship between income and wealth, we have only one independent variable, not two, and there is no way to assess the separate inﬂuence of income and wealth on consumption. To see this clearly, let X3i = 2X2i in the consumption–income–wealth regression. Then the regression (7.1.1) becomes Yi = β1 + β2 X2i + β3 (2X2i ) + ui = β1 + (β2 + 2β3 )X2i + ui = β1 + α X2i + ui where α = (β2 + 2β3 ). That is, we in fact have a two-variable and not a threevariable regression. Moreover, if we run the regression (7.1.10) and obtain α, there is no way to estimate the separate inﬂuence of X2 ( = β2 ) and X3 ( = β3 ) on Y, for α gives the combined inﬂuence of X2 and X3 on Y.3

3 Mathematically speaking, α = (β2 + 2β3 ) is one equation in two unknowns and there is no unique way of estimating β2 and β3 from the estimated α.

(7.1.10)

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In short the assumption of no multicollinearity requires that in the PRF we include only those variables that are not exact linear functions of one or more variables in the model. Although we will discuss this topic more fully in Chapter 10, a couple of points may be noted here. First, the assumption of no multicollinearity pertains to our theoretical (i.e., PRF) model. In practice, when we collect data for empirical analysis there is no guarantee that there will not be correlations among the regressors. As a matter of fact, in most applied work it is almost impossible to ﬁnd two or more (economic) variables that may not be correlated to some extent, as we will show in our illustrative examples later in the chapter. What we require is that there be no exact relationships among the regressors, as in Eq. (7.1.9). Second, keep in mind that we are talking only about perfect linear relationships between two or more variables. Multicollinearity does not rule out 2 nonlinear relationships between variables. Suppose X3i = X2i . This does not violate the assumption of no perfect collinearity, as the relationship between the variables here is nonlinear.

7.2 INTERPRETATION OF MULTIPLE REGRESSION EQUATION

Given the assumptions of the classical regression model, it follows that, on taking the conditional expectation of Y on both sides of (7.1.1), we obtain E(Yi | X2i , X3i ) = β1 + β2 X2i + β3i X3i (7.2.1)

In words, (7.2.1) gives the conditional mean or expected value of Y conditional upon the given or ﬁxed values of X2 and X3 . Therefore, as in the two-variable case, multiple regression analysis is regression analysis conditional upon the ﬁxed values of the regressors, and what we obtain is the average or mean value of Y or the mean response of Y for the given values of the regressors.

7.3 THE MEANING OF PARTIAL REGRESSION COEFFICIENTS

As mentioned earlier, the regression coefﬁcients β2 and β3 are known as partial regression or partial slope coefﬁcients. The meaning of partial regression coefﬁcient is as follows: β2 measures the change in the mean value of Y, E(Y), per unit change in X2 , holding the value of X3 constant. Put differently, it gives the “direct” or the “net” effect of a unit change in X2 on the mean value of Y, net of any effect that X3 may have on mean Y. Likewise, β3 measures the change in the mean value of Y per unit change in X3 , holding the value of X2 constant.4 That is, it gives the “direct” or “net” effect of a unit

4 The calculus-minded reader will notice at once that β2 and β3 are the partial derivatives of E(Y | X2 , X3 ) with respect to X2 and X3.

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change in X3 on the mean value of Y, net of any effect that X2 may have on mean Y.5 How do we actually go about holding the inﬂuence of a regressor constant? To explain this, let us revert to our child mortality example. Recall that in that example, Y = child mortality (CM), X2 = per capita GNP (PGNP), and X3 = female literacy rate (FLR). Let us suppose we want to hold the inﬂuence of FLR constant. Since FLR may have some effect on CM as well as PGNP in any given concrete data, what we can do is to remove the (linear) inﬂuence of FLR from both CM and PGNP by running the regression of CM on FLR and that of PGNP on FLR separately and then looking at the residuals obtained from these regressions. Using the data given in Table 6.4, we obtain the following regressions: CMi = 263.8635 − 2.3905 FLRi + u1i ˆ se = (12.2249) (0.2133) r 2 = 0.6695 (7.3.1)

ˆ where u1i represents the residual term of this regression. PGNPi = −39.3033 + 28.1427 FLRi + u2i ˆ se = (734.9526) (12.8211) r 2 = 0.0721 (7.3.2)

ˆ where u2i represents the residual term of this regression. Now u1i = (CMi − 263.8635 + 2.3905 FLRi ) ˆ (7.3.3)

represents that part of CM left after removing from it the (linear) inﬂuence of FLR. Likewise, u2i = (PGNPi + 39.3033 − 28.1427 FLRi ) ˆ (7.3.4)

represents that part of PGNP left after removing from it the (linear) inﬂuence of FLR. ˆ ˆ Therefore, if we now regress u1i on u2i , which are “puriﬁed” of the (linear) inﬂuence of FLR, wouldn’t we obtain the net effect of PGNP on CM? That is indeed the case (see Appendix 7A, Section 7A.2). The regression results are as follows: ˆ u1i = −0.0056u2i ˆ ˆ se = (0.0019) r 2 = 0.1152 (7.3.5)

Note: This regression has no intercept term because the mean value of the ˆ ˆ OLS residuals u1i and u2i is zero (why?)

5 Incidentally, the terms holding constant, controlling for, allowing or accounting for the inﬂuence of, correcting the inﬂuence of, and sweeping out the inﬂuence of are synonymous and will be used interchangeably in this text.

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The slope coefﬁcient of −0.0056 now gives the “true” or net effect of a unit change in PGNP on CM or the true slope of CM with respect to PGNP. That is, it gives the partial regression coefﬁcient of CM with respect to PGNP, β2 . Readers who want to get the partial regression coefﬁcient of CM with respect to FLR can replicate the above procedure by ﬁrst regressing CM on ˆ PGNP and getting the residuals from this regression (u1i ), then regressing ˆ FLR on PGNP and obtaining the residuals from this regression (u2i ), and ˆ ˆ then regressing u1i on u2i . I am sure readers get the idea. Do we have to go through this multistep procedure every time we want to ﬁnd out the true partial regression coefﬁcient? Fortunately, we do not have to do that, for the same job can be accomplished fairly quickly and routinely by the OLS procedure discussed in the next section. The multistep procedure just outlined is merely for pedagogic purposes to drive home the meaning of “partial” regression coefﬁcient.

7.4 OLS AND ML ESTIMATION OF THE PARTIAL REGRESSION COEFFICIENTS

To estimate the parameters of the three-variable regression model (7.1.1), we ﬁrst consider the method of ordinary least squares (OLS) introduced in Chapter 3 and then consider brieﬂy the method of maximum likelihood (ML) discussed in Chapter 4.

OLS Estimators

To ﬁnd the OLS estimators, let us ﬁrst write the sample regression function (SRF) corresponding to the PRF of (7.1.1) as follows: ˆ ˆ ˆ Yi = β1 + β2 X2i + β3 X3i + ui ˆ (7.4.1)

ˆ where ui is the residual term, the sample counterpart of the stochastic disturbance term ui . As noted in Chapter 3, the OLS procedure consists in so choosing the valˆ2 ues of the unknown parameters that the residual sum of squares (RSS) ui is as small as possible. Symbolically, min ui = ˆ2 ˆ ˆ ˆ (Yi − β1 − β2 X2i − β3 X3i )2 (7.4.2)

where the expression for the RSS is obtained by simple algebraic manipulations of (7.4.1).

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The most straightforward procedure to obtain the estimators that will minimize (7.4.2) is to differentiate it with respect to the unknowns, set the resulting expressions to zero, and solve them simultaneously. As shown in Appendix 7A, Section 7A.1, this procedure gives the following normal equations [cf. Eqs. (3.1.4) and (3.1.5)]: ¯ ˆ ¯ ˆ ¯ ˆ Y = β1 + β2 X2 + β3 X3 ˆ Yi X2i = β1 ˆ Yi X3i = β1 ˆ X2i + β2 ˆ X3i + β2

2 ˆ X2i + β3

(7.4.3) X2i X3i

2 X3i

(7.4.4) (7.4.5)

ˆ X2i X3i + β3

From Eq. (7.4.3) we see at once that ¯ ˆ ˆ ¯ ˆ ¯ β1 = Y − β2 X2 − β3 X3 (7.4.6)

which is the OLS estimator of the population intercept β1 . Following the convention of letting the lowercase letters denote deviations from sample mean values, one can derive the following formulas from the normal equations (7.4.3) to (7.4.5): ˆ β2 = ˆ β3 = yi x2i

2 x2i 2 x3i − 2 x3i 2 x2i − 2 x2i 2 x3i

yi x3i − − x2i x3i yi x2i x2i x3i

x2i x3i

2

(7.4.7)6

yi x3i

x2i x3i

2

(7.4.8)

which give the OLS estimators of the population partial regression coefﬁcients β2 and β3 , respectively. In passing, note the following: (1) Equations (7.4.7) and (7.4.8) are symmetrical in nature because one can be obtained from the other by interchanging the roles of X2 and X3 ; (2) the denominators of these two equations are identical; and (3) the three-variable case is a natural extension of the two-variable case.

Variances and Standard Errors of OLS Estimators

Having obtained the OLS estimators of the partial regression coefﬁcients, we can derive the variances and standard errors of these estimators in the manner indicated in Appendix 3A.3. As in the two-variable case, we need the standard errors for two main purposes: to establish conﬁdence intervals and

6

This estimator is equal to that of (7.3.5), as shown in App. 7A, Sec. 7A.2.

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to test statistical hypotheses. The relevant formulas are as follows:7 ˆ var (β1 ) = ¯ X2 1 + 2 n

2 ¯2 x3i + X3 2 x2i 2 ¯ ¯ x2i − 2 X2 X3 2 x3i

x2i x3i

2

−

x2i x3i

· σ 2 (7.4.9) (7.4.10)

ˆ ˆ se (β1 ) = + var (β1 ) ˆ var (β2 ) = or, equivalently, ˆ var (β2 ) = σ2 2 2 x2i 1 − r2 3

2 x3i 2 x2i 2 x3i −

x2i x3i

2

σ2

(7.4.11)

(7.4.12)

where r2 3 is the sample coefﬁcient of correlation between X2 and X3 as deﬁned in Chapter 3.8 ˆ ˆ se (β2 ) = + var (β2 ) ˆ var (β3 ) = or, equivalently, ˆ var (β3 ) = σ2 2 2 x3i 1 − r2 3 (7.4.15) (7.4.16)

2 2 x3i 2 x2i 2 x2i 2 x3i

(7.4.13)

2

−

x2i x3i

σ2

(7.4.14)

ˆ ˆ se (β3 ) = + var (β3 ) ˆ ˆ cov (β2 , β3 ) = −r2 3 σ

2 1 − r2 3 2 x2i

(7.4.17)

In all these formulas σ 2 is the (homoscedastic) variance of the population disturbances ui . Following the argument of Appendix 3A, Section 3A.5, the reader can verify that an unbiased estimator of σ 2 is given by σ2 = ˆ ui ˆ2 n− 3 (7.4.18)

7 The derivations of these formulas are easier using matrix notation. Advanced readers may refer to App. C. 8 Using the deﬁnition of r given in Chap. 3, we have

2 r2 3 =

x2i x3i

2 x2t

2

2 x3t

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Note the similarity between this estimator of σ 2 and its two-variable counˆ ˆ2 terpart [σ 2 = ( ui )/(n − 2)]. The degrees of freedom are now (n − 3) ˆ2 because in estimating ui we must ﬁrst estimate β1 , β2 , and β3 , which consume 3 df. (The argument is quite general. Thus, in the four-variable case the df will be n − 4.) ˆ The estimator σ 2 can be computed from (7.4.18) once the residuals are available, but it can also be obtained more readily by using the following relation (for proof, see Appendix 7A, Section 7A.3): ui = ˆ2 ˆ yi2 − β2 ˆ yi x2i − β3 yi x3i (7.4.19)

which is the three-variable counterpart of the relation given in (3.3.6).

Properties of OLS Estimators

The properties of OLS estimators of the multiple regression model parallel those of the two-variable model. Speciﬁcally: 1. The three-variable regression line (surface) passes through the means ¯ ¯ ¯ Y, X2 , and X3 , which is evident from (7.4.3) [cf. Eq. (3.1.7) of the twovariable model]. This property holds generally. Thus in the k-variable linear regression model [a regressand and (k − 1) regressors] Yi = β1 + β2 X2i + β3 X3i + · · · + βk Xki + ui we have ¯ ¯ ˆ ¯ ˆ β1 = Y − β2 X2 − β3 X3 − · · · − βk Xk (7.4.21) ˆ 2. The mean value of the estimated Yi ( = Yi ) is equal to the mean value Yi , which is easy to prove: of the actual ˆ ˆ ˆ ˆ Yi = β1 + β2 X2i + β3 X3i ¯ ˆ ¯ ˆ ¯ ˆ ˆ = (Y − β2 X2 − β3 X3 ) + β2 X2i + β3 X3i ¯ ¯ ¯ ˆ ˆ = Y + β2 (X2i − X2 ) + β3 (X3i − X3 ) ¯ ˆ ˆ = Y + β2 x2i + β3 x3i where as usual small letters indicate values of the variables as deviations from their respective means. Summing both sides of (7.4.22) over the sample values and dividing ¯ ˆ ¯ x2i = x3i = 0. Why?) through by the sample size n gives Y = Y. (Note: Notice that by virtue of (7.4.22) we can write (7.4.23) ˆi − Y). ¯ ˆ where yi = (Y Therefore, the SRF (7.4.1) can be expressed in the deviation form as ˆ ˆ yi = yi + ui = β2 x2i + β3 x3i + ui ˆ ˆ ˆ (7.4.24) ˆ ˆ yi = β2 x2i + β3 x3i ˆ (7.4.20)

(Why?) (7.4.22)

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¯ ui = u = 0, which can be veriﬁed from (7.4.24). [Hint: Sum both ˆ ˆ 3. sides of (7.4.24) over the sample values.] ˆ ˆ 4. The residuals ui are uncorrelated with X2i and X3i , that is, ui X2i = ui X3i = 0 (see Appendix 7A.1 for proof). ˆ ˆ ui Yi = 0. Why? ˆ ˆ ˆ 5. The residuals ui are uncorrelated with Yi ; that is, ˆ [Hint: Multiply (7.4.23) on both sides by ui and sum over the sample values.] 6. From (7.4.12) and (7.4.15) it is evident that as r2 3 , the correlation ˆ coefﬁcient between X2 and X3 , increases toward 1, the variances of β2 and 2 2 ˆ β3 increase for given values of σ 2 and x2i or x3i . In the limit, when r2 3 = 1 (i.e., perfect collinearity), these variances become inﬁnite. The implications of this will be explored fully in Chapter 10, but intuitively the reader can see that as r2 3 increases it is going to be increasingly difﬁcult to know what the true values of β2 and β3 are. [More on this in the next chapter, but refer to Eq. (7.1.10).] 7. It is also clear from (7.4.12) and (7.4.15) that for given values of r2 3 2 2 and x2i or x3i , the variances of the OLS estimators are directly propor2 tional to σ ; that is, they increase as σ 2 increases. Similarly, for given values 2 ˆ of σ 2 and r2 3 , the variance of β2 is inversely proportional to x2i ; that is, the greater the variation in the sample values of X2 , the smaller the variance of ˆ β2 and therefore β2 can be estimated more precisely. A similar statement can ˆ be made about the variance of β3 . 8. Given the assumptions of the classical linear regression model, which are spelled out in Section 7.1, one can prove that the OLS estimators of the partial regression coefﬁcients not only are linear and unbiased but also have minimum variance in the class of all linear unbiased estimators. In short, they are BLUE: Put differently, they satisfy the Gauss-Markov theorem. (The proof parallels the two-variable case proved in Appendix 3A, Section 3A.6 and will be presented more compactly using matrix notation in Appendix C.)

Maximum Likelihood Estimators

We noted in Chapter 4 that under the assumption that ui , the population disturbances, are normally distributed with zero mean and constant variance σ 2 , the maximum likelihood (ML) estimators and the OLS estimators of the regression coefﬁcients of the two-variable model are identical. This equality extends to models containing any number of variables. (For proof, see Appendix 7A, Section 7A.4.) However, this is not true of the estimator ui /n regardless of ˆ2 of σ 2 . It can be shown that the ML estimator of σ 2 is the number of variables in the model, whereas the OLS estimator of σ 2 is ui /(n − 2) in the two-variable case, ui /(n − 3) in the three-variable case, ˆ2 ˆ2 2 ui /(n − k) in the case of the k-variable model (7.4.20). In short, the ˆ and OLS estimator of σ 2 takes into account the number of degrees of freedom, whereas the ML estimator does not. Of course, if n is very large, the ML and OLS estimators of σ 2 will tend to be close to each other. (Why?)

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7.5 THE MULTIPLE COEFFICIENT OF DETERMINATION R 2 AND THE MULTIPLE COEFFICIENT OF CORRELATION R

In the two-variable case we saw that r 2 as deﬁned in (3.5.5) measures the goodness of ﬁt of the regression equation; that is, it gives the proportion or percentage of the total variation in the dependent variable Y explained by the (single) explanatory variable X. This notation of r 2 can be easily extended to regression models containing more than two variables. Thus, in the threevariable model we would like to know the proportion of the variation in Y explained by the variables X2 and X3 jointly. The quantity that gives this information is known as the multiple coefﬁcient of determination and is denoted by R2 ; conceptually it is akin to r 2 . To derive R2 , we may follow the derivation of r 2 given in Section 3.5. Recall that ˆ ˆ ˆ Yi = β1 + β2 X2i + β3 X3i + ui ˆ ˆ = Yi + ui ˆ

(7.5.1)

ˆ where Yi is the estimated value of Yi from the ﬁtted regression line and is an estimator of true E(Yi | X2i , X3i ). Upon shifting to lowercase letters to indicate deviations from the mean values, Eq. (7.5.1) may be written as ˆ ˆ yi = β2 x2i + β3 x3i + ui ˆ = yi + ui ˆ ˆ

(7.5.2)

Squaring (7.5.2) on both sides and summing over the sample values, we obtain yi2 = = yi2 + ˆ yi2 + ˆ ui + 2 ˆ2 ui ˆ2 yi ui ˆ ˆ (7.5.3) (Why?)

Verbally, Eq. (7.5.3) states that the total sum of squares (TSS) equals the explained sum of squares (ESS) + the residual sum of squares (RSS). Now ˆ2 substituting for ui from (7.4.19), we obtain yi2 = yi2 + ˆ ˆ yi2 − β2 ˆ yi x2i − β3 yi x3i

which, on rearranging, gives ESS = ˆ yi2 = β2 ˆ ˆ yi x2i + β3 yi x3i (7.5.4)

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Now, by deﬁnition R2 = ESS TSS ˆ ˆ β2 yi x2i + β3 = 2 yi

yi x3i

(7.5.5)9

[cf. (7.5.5) with (3.5.6)]. Since the quantities entering (7.5.5) are generally computed routinely, R2 can be computed easily. Note that R2 , like r 2 , lies between 0 and 1. If it is 1, the ﬁtted regression line explains 100 percent of the variation in Y. On the other hand, if it is 0, the model does not explain any of the variation in Y. Typically, however, R2 lies between these extreme values. The ﬁt of the model is said to be “better’’ the closer is R2 to 1. Recall that in the two-variable case we deﬁned the quantity r as the coefﬁcient of correlation and indicated that it measures the degree of (linear) association between two variables. The three-or-more-variable analogue of r is the coefﬁcient of multiple correlation, denoted by R, and it is a measure of the degree of association between Y and all the explanatory variables jointly. Although r can be positive or negative, R is always taken to be positive. In practice, however, R is of little importance. The more meaningful quantity is R2 . Before proceeding further, let us note the following relationship between R2 and the variance of a partial regression coefﬁcient in the k-variable multiple regression model given in (7.4.20): σ2 ˆ var (β j ) = xj2 1 1 − R2 j (7.5.6)

ˆ where β j is the partial regression coefﬁcient of regressor X j and R2 is the R2 j in the regression of X j on the remaining (k − 2) regressors. [Note: There are (k − 1) regressors in the k-variable regression model.] Although the utility of Eq. (7.5.6) will become apparent in Chapter 10 on multicollinearity, observe that this equation is simply an extension of the formula given in (7.4.12) or (7.4.15) for the three-variable regression model, one regressand and two regressors.

7.6 EXAMPLE 7.1: CHILD MORTALITY IN RELATION TO PER CAPITA GNP AND FEMALE LITERACY RATE

In Chapter 6 we considered the behavior of child mortality (CM) in relation to per capita GNP (PGNP). There we found that PGNP has a negative impact on CM, as one would expect. Now let us bring in female literacy as measured

9

Note that R2 can also be computed as follows: R2 = 1 − RSS =1− TSS ui ˆ2 yi2 =1− (n − 3)σ 2 ˆ 2 (n − 1)Sy

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by the female literacy rate (FLR). A priori, we expect that FLR too will have a negative impact on CM. Now when we introduce both the variables in our model, we need to net out the inﬂuence of each of the regressors. That is, we need to estimate the (partial) regression coefﬁcients of each regressor. Thus our model is: CMi = β1 + β2 PGNPi + β3 FLRi + ui (7.6.1)

The necessary data are given in Table 6.4. Keep in mind that CM is the number of deaths of children under ﬁve per 1000 live births, PGNP is per capita GNP in 1980, and FLR is measured in percent. Our sample consists of 64 countries. Using the Eviews3 statistical package, we obtained the following results: CMi = 263.6416 − 0.0056 PGNPi − 2.2316 FLR i se = (11.5932) (0.0019) (0.2099) R2 = 0.7077 (7.6.2) ¯ R2 = 0.6981* where ﬁgures in parentheses are the estimated standard errors. Before we interpret this regression, observe the partial slope coefﬁcient of PGNP, namely, −0.0056. Is it not precisely the same as that obtained from the three-step procedure discussed in the previous section [see Eq. (7.3.5)]? But should that surprise you? Not only that, but the two standard errors are precisely the same, which is again unsurprising. But we did so without the three-step cumbersome procedure. Let us now interpret these regression coefﬁcients: −0.0056 is the partial regression coefﬁcient of PGNP and tells us that with the inﬂuence of FLR held constant, as PGNP increases, say, by a dollar, on average, child mortality goes down by 0.0056 units. To make it more economically interpretable, if the per capita GNP goes up by a thousand dollars, on average, the number of deaths of children under age 5 goes down by about 5.6 per thousand live births. The coefﬁcient −2.2316 tells us that holding the inﬂuence of PGNP constant, on average, the number of deaths of children under 5 goes down by about 2.23 per thousand live births as the female literacy rate increases by one percentage point. The intercept value of about 263, mechanically interpreted, means that if the values of PGNP and FLR rate were ﬁxed at zero, the mean child mortality would be about 263 deaths per thousand live births. Of course, such an interpretation should be taken with a grain of salt. All one could infer is that if the two regressors were ﬁxed at zero, child mortality will be quite high, which makes practical sense. The R2 value of about 0.71 means that about 71 percent of the variation in child mortality is explained by PGNP and FLR, a fairly high value considering that the maximum value of R2 can at most be 1. All told, the regression results make sense.

*On this, see Sec. 7.8.

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What about the statistical signiﬁcance of the estimated coefﬁcients? We will take this topic up in Chapter 8. As we will see there, in many ways this chapter will be an extension of Chapter 5, which dealt with the two-variable model. As we will also show, there are some important differences in statistical inference (i.e., hypothesis testing) between the two-variable and multivariable regression models.

Regression on Standardized Variables

In the preceding chapter we introduced the topic of regression on standardized variables and stated that the analysis can be extended to multivariable regressions. Recall that a variable is said to be standardized or in standard deviation units if it is expressed in terms of deviation from its mean and divided by its standard deviation. For our child mortality example, the results are as follows:

* * CM = − 0.2026 PGNPi − 0.7639 FLRi *

se = (0.0713)

(0.0713)

r 2 = 0.7077

(7.6.3)

Note: The starred variables are standardized variables. Also note that there is no intercept in the model for reasons already discussed in the previous chapter. As you can see from this regression, with FLR held constant, a standard deviation increase in PGNP leads, on average, to a 0.2026 standard deviation decrease in CM. Similarly, holding PGNP constant, a standard deviation increase in FLR, on average, leads to a 0.7639 standard deviation decrease in CM. Relatively speaking, female literacy has more impact on child mortality than per capita GNP. Here you will see the advantage of using standardized variables, for standardization puts all variables on equal footing because all standardized variables have zero means and unit variances.

7.7 SIMPLE REGRESSION IN THE CONTEXT OF MULTIPLE REGRESSION: INTRODUCTION TO SPECIFICATION BIAS

Recall that assumption (7.1.6) of the classical linear regression model states that the regression model used in the analysis is “correctly” speciﬁed; that is, there is no speciﬁcation bias or speciﬁcation error (see Chapter 3 for some introductory remarks). Although the topic of speciﬁcation error will be discussed more fully in Chapter 13, the illustrative example given in the preceding section provides a splendid opportunity not only to drive home the importance of assumption (7.1.6) but also to shed additional light on the meaning of partial regression coefﬁcient and provide a somewhat informal introduction to the topic of speciﬁcation bias.

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Assume that (7.6.1) is the “true” model explaining the behavior of child mortality in relation to per capita GNP and female literacy rate (FLR). But suppose we disregard FLR and estimate the following simple regression: Yi = α1 + α2 X2i + u1i (7.7.1)

where Y = CM and X2 = PGNP. Since (7.6.1) is the true model, estimating (7.7.1) would constitute a speciﬁcation error; the error here consists in omitting the variable X3, the female literacy rate. Notice that we are using different parameter symbols (the alphas) in (7.7.1) to distinguish them from the true parameters (the betas) given in (7.6.1). Now will α2 provide an unbiased estimate of the true impact of PGNP, ˆ which is given by β2 in model (7.6.1)? In other words, will E(α2 ) = β2 , where α2 is the estimated value of α2 ? In other words, will the coefﬁcient of PGNP ˆ in (7.7.1) provide an unbiased estimate of the true impact of PGNP on CM, knowing that we have omitted the variable X3 (FLR) from the model? As you ˆ would suspect, in general α2 will not be an unbiased estimator of the true β2 . To give a glimpse of the bias, let us run the regression (7.7.1), which gave the following results. CMi = 157.4244 − 0.0114 PGNPi se = (9.8455) (0.0032) r2 = 0.1662 (7.7.2)

Observe several things about this regression compared to the “true” multiple regression (7.6.1): 1. In absolute terms (i.e., disregarding the sign), the PGNP coefﬁcient has increased from 0.0056 to 0.0114, almost a two-fold increase. 2. The standard errors are different. 3. The intercept values are different. 4. The r 2 values are dramatically different, although it is generally the case that, as the number of regressors in the model increases, the r 2 value increases. Now suppose that you regress child mortality on female literacy rate, disregarding the inﬂuence of PGNP. You will obtain the following results: CMi = 263.8635 − 2.3905 FLR i se = (21.2249) (0.2133) r 2 = 0.6696 (7.7.3)

Again if you compare the results of this (misspeciﬁed) regression with the “true” multiple regression, you will see that the results are different, although the difference here is not as noticeable as in the case of regression (7.7.2).

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The important point to note is that serious consequences can ensue if you misﬁt a model. We will look into this topic more thoroughly in Chapter 13, on speciﬁcation errors.

7.8 R 2 AND THE ADJUSTED R 2

An important property of R2 is that it is a nondecreasing function of the number of explanatory variables or regressors present in the model; as the number of regressors increases, R2 almost invariably increases and never decreases. Stated differently, an additional X variable will not decrease R2. Compare, for instance, regression (7.7.2) or (7.7.3) with (7.6.2). To see this, recall the deﬁnition of the coefﬁcient of determination: R2 = ESS TSS RSS TSS ui ˆ2 yi2 (7.8.1)

=1− =1−

yi2 is independent of the number of X variables in the model because Now ¯ ˆ2 it is simply (Yi − Y)2 . The RSS, ui , however, depends on the number of regressors present in the model. Intuitively, it is clear that as the number of ˆ2 X variables increases, ui is likely to decrease (at least it will not increase); 2 hence R as deﬁned in (7.8.1) will increase. In view of this, in comparing two regression models with the same dependent variable but differing number of X variables, one should be very wary of choosing the model with the highest R2. To compare two R2 terms, one must take into account the number of X variables present in the model. This can be done readily if we consider an alternative coefﬁcient of determination, which is as follows: ui (n − k) ˆ2 yi2 (n − 1)

¯ R2 = 1 −

(7.8.2)

where k = the number of parameters in the model including the intercept term. (In the three-variable regression, k = 3. Why?) The R2 thus deﬁned ¯ is known as the adjusted R2, denoted by R2 . The term adjusted means adjusted for the df associated with the sums of squares entering into ˆ2 (7.8.1): ui has n − k df in a model involving k parameters, which include

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yi2 has n − 1 df. (Why?) For the three-variable the intercept term, and 2 ˆ case, we know that ui has n − 3 df. Equation (7.8.2) can also be written as σ2 ˆ ¯ R2 = 1 − 2 SY (7.8.3)

2 ˆ where σ 2 is the residual variance, an unbiased estimator of true σ 2 , and SY is the sample variance of Y. ¯ It is easy to see that R2 and R2 are related because, substituting (7.8.1) into (7.8.2), we obtain

n− 1 ¯ R2 = 1 − (1 − R2 ) n− k

(7.8.4)

¯ It is immediately apparent from Eq. (7.8.4) that (1) for k > 1, R2 < R2 which implies that as the number of X variables increases, the adjusted R2 in¯ creases less than the unadjusted R2; and (2) R2 can be negative, although R2 10 ¯ 2 turns out to be negative in an appliis necessarily nonnegative. In case R cation, its value is taken as zero. Which R2 should one use in practice? As Theil notes:

¯ . . . it is good practice to use R2 rather than R2 because R2 tends to give an overly optimistic picture of the ﬁt of the regression, particularly when the number of explanatory variables is not very small compared with the number of observations.11

But Theil’s view is not uniformly shared, for he has offered no general theo¯ retical justiﬁcation for the “superiority’’ of R2 . For example, Goldberger ar2 gues that the following R , call it modiﬁed R2, will do just as well12: Modiﬁed R2 = (1 − k/n)R2 (7.8.5)

His advice is to report R2, n, and k and let the reader decide how to adjust R2 by allowing for n and k. Despite this advice, it is the adjusted R2, as given in (7.8.4), that is reported by most statistical packages along with the conventional R2. The ¯ reader is well advised to treat R2 as just another summary statistic.

10 ¯ ¯ Note, however, that if R2 = 1, R2 = R2 = 1. When R2 = 0, R2 = (1 − k)/(n − k), in which ¯ case R2 can be negative if k > 1. 11 Henri Theil, Introduction to Econometrics, Prentice Hall, Englewood Cliffs, N.J., 1978, p. 135. 12 Arthur S. Goldberger, A Course in Econometrics, Harvard University Press, Cambridge, Mass., 1991, p. 178. For a more critical view of R2, see S. Cameron, “Why is the R Squared Adjusted Reported?”, Journal of Quantitative Economics, vol. 9, no. 1, January 1993, pp. 183–186. He argues that “It [R2] is NOT a test statistic and there seems to be no clear intuitive justiﬁcation for its use as a descriptive statistic. Finally, we should be clear that it is not an effective tool for the prevention of data mining” (p. 186).

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Incidentally, for the child mortality regression (7.6.2), the reader should ¯ verify that R2 is 0.6981, keeping in mind that in this example (n − 1) = 63 ¯ and (n − k) = 60. As expected, R2 of 0.6981 is less than R2 of 0.7077. 2 2 Besides R and adjusted R as goodness of ﬁt measures, other criteria are often used to judge the adequacy of a regression model. Two of these are Akaike’s Information criterion and Amemiya’s Prediction criteria, which are used to select between competing models. We will discuss these criteria when we consider the problem of model selection in greater detail in a later chapter (see Chapter 13).

Comparing Two R 2 Values

It is crucial to note that in comparing two models on the basis of the coefﬁcient of determination, whether adjusted or not, the sample size n and the dependent variable must be the same; the explanatory variables may take any form. Thus for the models ln Yi = β1 + β2 X2i + β3 X3i + ui Yi = α1 + α2 X2i + α3 X3i + ui (7.8.6) (7.8.7)

the computed R2 terms cannot be compared. The reason is as follows: By deﬁnition, R2 measures the proportion of the variation in the dependent variable accounted for by the explanatory variable(s). Therefore, in (7.8.6) R2 measures the proportion of the variation in ln Y explained by X2 and X3, whereas in (7.8.7) it measures the proportion of the variation in Y, and the two are not the same thing: As noted in Chapter 6, a change in ln Y gives a relative or proportional change in Y, whereas a change in Y gives an abˆ solute change. Therefore, var Yi /var Yi is not equal to var (ln Y i )/var (ln Yi ); that is, the two coefﬁcients of determination are not the same.13 How then does one compare the R2’s of two models when the regressand is not in the same form? To answer this question, let us ﬁrst consider a numerical example.

13

From the deﬁnition of R2, we know that 1 − R2 = RSS = TSS ¯ (Yi − Y)2 ui ˆ2 (ln Yi − ln Y)2 ui ˆ2

for the linear model and 1 − R2 =

for the log model. Since the denominators on the right-hand sides of these expressions are different, we cannot compare the two R2 terms directly. As shown in Example 7.2, for the linear speciﬁcation, the RSS = 0.1491 (the residual sum of squares of coffee consumption), and for the log–linear speciﬁcation, the RSS = 0.0226 (the residual sum of squares of log of coffee consumption). These residuals are of different orders of magnitude and hence are not directly comparable.

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EXAMPLE 7.2 COFFEE CONSUMPTION IN THE UNITED STATES, 1970–1980 Consider the data in Table 7.1. The data pertain to consumption of cups of coffee per day (Y ) and real retail price of coffee (X ) in the United States for years 1970–1980. Applying OLS to the data, we obtain the following regression results: ˆ Yt = 2.6911 − 0.4795Xt se = (0.1216) (0.1140) RSS = 0.1491; r 2 = 0.6628 (7.8.8)

The results make economic sense: As the price of coffee increases, on average, coffee consumption goes down by about half a cup per day. The r 2 value of about 0.66 means that the price of coffee explains about 66 percent of the variation in coffee consumption. The reader can readily verify that the slope coefﬁcient is statistically signiﬁcant. From the same data, the following double log, or constant elasticity, model can be estimated: lnYt = 0.7774 − 0.2530 ln Xt se = (0.0152) (0.0494) RSS = 0.0226; r 2 = 0.7448 (7.8.9)

Since this is a double log model, the slope coefﬁcient gives a direct estimate of the price elasticity coefﬁcient. In the present instance, it tells us that if the price of coffee per pound goes up by 1 percent, on average, per day coffee consumption goes down by about 0.25 percent. Remember that in the linear model (7.8.8) the slope coefﬁcient only gives the rate of change of coffee consumption with respect to price. (How will you estimate the price elasticity for the TABLE 7.1 U.S. COFFEE CONSUMPTION (Y ) IN RELATION TO AVERAGE REAL RETAIL PRICE (X),* 1970–1980 Y, Cups per person per day 2.57 2.50 2.35 2.30 2.25 2.20 2.11 1.94 1.97 2.06 2.02 X, $ per lb 0.77 0.74 0.72 0.73 0.76 0.75 1.08 1.81 1.39 1.20 1.17

Year 1970 1971 1972 1973 1974 1975 1976 1977 1978 1979 1980

*Note: The nominal price was divided by the Consumer Price Index (CPI) for food and beverages, 1967 = 100. Source: The data for Y are from Summary of National Coffee Drinking Study, Data Group, Elkins Park, Penn., 1981; and the data on nominal X (i.e., X in current prices) are from Nielsen Food Index, A. C. Nielsen, New York, 1981. I am indebted to Scott E. Sandberg for collecting the data.

(Continued)

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EXAMPLE 7.2 (Continued) linear model?) The r 2 value of about 0.74 means that about 74 percent of the variation in the log of coffee demand is explained by the variation in the log of coffee price. Since the r 2 value of the linear model of 0.6628 is smaller than the r 2 value of 0.7448 of the log–linear model, you might be tempted to choose the latter model because of its high r 2 value. But for reasons already noted, we cannot do so. But if you do want to compare the two r 2 values, you may proceed as follows: 1. Obtain lnYt from (7.8.9) for each observation; that is, obtain the estimated log value of each observation from this model. Take the antilog of these values and then compute r 2 between these antilog values and actual Yt in the manner indicated by Eq. (3.5.14). This r 2 value is comparable to the r 2 value of the linear model (7.8.8). 2. Alternatively, assuming all Y values are positive, take logarithms of the Y values, ln Y. ˆ Obtain the estimated Y values, Yt , from the linear model (7.8.8), take the logarithms of these ˆ t ) and compute the r 2 between (ln Yt ) and (ln Yt ) in the manner ˆ estimated Y values (i.e., ln Y indicated in Eq. (3.5.14). This r 2 value is comparable to the r 2 value obtained from (7.8.9). For our coffee example, we present the necessary raw data to compute the comparable r 2’s in Table 7.2. To compare the r 2 value of the linear model (7.8.8) with that of (7.8.9), we ﬁrst ˆ obtain log of (Yt ) [given in column (6) of Table 7.2], then we obtain the log of actual Y values [given in column (5) of the table], and then compute r 2 between these two sets of values using Eq. (3.5.14). The result is an r 2 value of 0.7318, which is now comparable with the r 2 value of the log–linear model of 0.7448. Now the difference between the two r 2 values is very small. On the other hand, if we want to compare the r 2 value of the log–linear model with the linear model, we obtain lnYt for each observation from (7.8.9) [given in column (3) of the table], obtain their antilog values [given in column (4) of the table], and ﬁnally compute r 2 between these antilog values and the actual Y values, using formula (3.5.14). This will give an r 2 value of 0.7187, which is slightly higher than that obtained from the linear model (7.8.8), namely, 0.6628. Using either method, it seems that the log–linear model gives a slightly better ﬁt. TABLE 7.2 RAW DATA FOR COMPARING TWO R 2 VALUES ˆ Yt (2) 2.321887 2.336272 2.345863 2.341068 2.326682 2.331477 2.173233 1.823176 2.024579 2.115689 2.130075 Antilog of lnYt (4) 2.324616 2.348111 2.364447 2.356209 2.332318 2.340149 2.133882 1.872508 2.001884 2.077742 2.091096 ˆ ln (Y t) (6) 0.842380 0.848557 0.852653 0.850607 0.844443 0.846502 0.776216 0.600580 0.705362 0.749381 0.756157

Year 1970 1971 1972 1973 1974 1975 1976 1977 1978 1979 1980

Yt (1) 2.57 2.50 2.35 2.30 2.25 2.20 2.11 1.94 1.97 2.06 2.02

lnYt (3) 0.843555 0.853611 0.860544 0.857054 0.846863 0.850214 0.757943 0.627279 0.694089 0.731282 0.737688

ln Yt (5) 0.943906 0.916291 0.854415 0.832909 0.810930 0.788457 0.746688 0.662688 0.678034 0.722706 0.703098

Notes: Column (1): Actual Y values from Table 7.1 Column (2): Estimated Y values from the linear model (7.8.8) Column (3): Estimated log Y values from the double-log model (7.8.9) Column (4): Antilog of values in column (3) Column (5): Log values of Y in column (1) ˆ Column (6): Log values ofY t in column (2)

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Allocating R 2 among Regressors

Let us return to our child mortality example. We saw in (7.6.2) that the two regressors PGNP and FLR explain 0.7077 or 70.77 percent of the variation in child mortality. But now consider the regression (7.7.2) where we dropped the FLR variable and as a result the r 2 value dropped to 0.1662. Does that mean the difference in the r 2 value of 0.5415 (0.7077 − 0.1662) is attributable to the dropped variable FLR? On the other hand, if you consider regression (7.7.3), where we dropped the PGNP variable, the r 2 value drops to 0.6696. Does that mean the difference in the r 2 value of 0.0381 (0.7077 − 0.6696) is due to the omitted variable PGNP? The question then is: Can we allocate the multiple R2 of 0.7077 between the two regressors, PGNP and FLR, in this manner? Unfortunately, we cannot do so, for the allocation depends on the order in which the regressors are introduced, as we just illustrated. Part of the problem here is that the two regressors are correlated, the correlation coefﬁcient between the two being 0.2685 (verify it from the data given in Table 6.4). In most applied work with several regressors, correlation among them is a common problem. Of course, the problem will be very serious if there is perfect collinearity among the regressors. The best practical advice is that there is little point in trying to allocate the R2 value to its constituent regressors.

¯ The “Game’’ of Maximizing R 2

In concluding this section, a warning is in order: Sometimes researchers play ¯ the game of maximizing R2 , that is, choosing the model that gives the high¯ 2 . But this may be dangerous, for in regression analysis our objective is est R ¯ not to obtain a high R2 per se but rather to obtain dependable estimates of the true population regression coefﬁcients and draw statistical inferences ¯ about them. In empirical analysis it is not unusual to obtain a very high R2 but ﬁnd that some of the regression coefﬁcients either are statistically insigniﬁcant or have signs that are contrary to a priori expectations. Therefore, the researcher should be more concerned about the logical or theoretical relevance of the explanatory variables to the dependent variable and their sta¯ tistical signiﬁcance. If in this process we obtain a high R2 , well and good; on ¯ 2 is low, it does not mean the model is necessarily bad.14 the other hand, if R

14 Some authors would like to deemphasize the use of R2 as a measure of goodness of ﬁt as well as its use for comparing two or more R2 values. See Christopher H. Achen, Interpreting and Using Regression, Sage Publications, Beverly Hills, Calif., 1982, pp. 58–67, and C. Granger and P. Newbold, “R2 and the Transformation of Regression Variables,” Journal of Econometrics, vol. 4, 1976, pp. 205–210. Incidentally, the practice of choosing a model on the basis of highest R2, a kind of data mining, introduces what is known as pretest bias, which might destroy some of the properties of OLS estimators of the classical linear regression model. On this topic, the reader may want to consult George G. Judge, Carter R. Hill, William E. Grifﬁths, Helmut Lütkepohl, and Tsoung-Chao Lee, Introduction to the Theory and Practice of Econometrics, John Wiley, New York, 1982, Chap. 21.

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As a matter of fact, Goldberger is very critical about the role of R2 . He has said:

From our perspective, R2 has a very modest role in regression analysis, being a measure of the goodness of ﬁt of a sample LS [least-squares] linear regression in a body of data. Nothing in the CR [CLRM] model requires that R2 be high. Hence a high R2 is not evidence in favor of the model and a low R2 is not evidence against it. In fact the most important thing about R2 is that it is not important in the CR model. The CR model is concerned with parameters in a population, not with goodness of ﬁt in the sample. . . . If one insists on a measure of predictive success (or rather failure), then σ 2 might sufﬁce: after all, the parameter σ 2 is the expected squared forecast error that would result if the population CEF [PRF] were used as the predictor. Alternatively, the squared standard error of forecast . . . at relevant values of x [regressors] may be informative.15

7.9 EXAMPLE 7.3: THE COBB–DOUGLAS PRODUCTION FUNCTION: MORE ON FUNCTIONAL FORM

In Section 6.4 we showed how with appropriate transformations we can convert nonlinear relationships into linear ones so that we can work within the framework of the classical linear regression model. The various transformations discussed there in the context of the two-variable case can be easily extended to multiple regression models. We demonstrate transformations in this section by taking up the multivariable extension of the twovariable log–linear model; others can be found in the exercises and in the illustrative examples discussed throughout the rest of this book. The speciﬁc example we discuss is the celebrated Cobb–Douglas production function of production theory. The Cobb–Douglas production function, in its stochastic form, may be expressed as Yi = β1 X2i2 X3i3 eui where Y = output X2 = labor input X3 = capital input u = stochastic disturbance term e = base of natural logarithm From Eq. (7.9.1) it is clear that the relationship between output and the two inputs is nonlinear. However, if we log-transform this model, we obtain:

15

β

β

(7.9.1)

Arther S. Goldberger, op. cit., pp. 177–178.

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ln Yi = ln β1 + β2 ln X2i + β3 ln X3i + ui = β0 + β2 ln X2i + β3 ln X3i + ui

(7.9.2)

where β0 = ln β1 . Thus written, the model is linear in the parameters β0 , β2 , and β3 and is therefore a linear regression model. Notice, though, it is nonlinear in the variables Y and X but linear in the logs of these variables. In short, (7.9.2) is a log-log, double-log, or log-linear model, the multiple regression counterpart of the two-variable log-linear model (6.5.3). The properties of the Cobb–Douglas production function are quite well known: 1. β2 is the (partial) elasticity of output with respect to the labor input, that is, it measures the percentage change in output for, say, a 1 percent change in the labor input, holding the capital input constant (see exercise 7.9). 2. Likewise, β3 is the (partial) elasticity of output with respect to the capital input, holding the labor input constant. 3. The sum (β2 + β3 ) gives information about the returns to scale, that is, the response of output to a proportionate change in the inputs. If this sum is 1, then there are constant returns to scale, that is, doubling the inputs will double the output, tripling the inputs will triple the output, and so on. If the sum is less than 1, there are decreasing returns to scale—doubling the inputs will less than double the output. Finally, if the sum is greater than 1, there are increasing returns to scale—doubling the inputs will more than double the output. Before proceeding further, note that whenever you have a log–linear regression model involving any number of variables the coefﬁcient of each of the X variables measures the (partial) elasticity of the dependent variable Y with respect to that variable. Thus, if you have a k-variable log-linear model: ln Yi = β0 + β2 ln X2i + β3 ln X3i + · · · + βk ln Xki + ui (7.9.3)

each of the (partial) regression coefﬁcients, β2 through βk, is the (partial) elasticity of Y with respect to variables X2 through Xk.16 To illustrate the Cobb–Douglas production function, we obtained the data shown in Table 7.3; these data are for the agricultural sector of Taiwan for 1958–1972. Assuming that the model (7.9.2) satisﬁes the assumptions of the classical linear regression model,17 we obtained the following regression by the OLS

16 To see this, differentiate (7.9.3) partially with respect to the log of each X variable. Therefore, ∂ ln Y/∂ ln X2 = (∂Y/∂ X2 )(X2 /Y) = β2 , which, by deﬁnition, is the elasticity of Y with respect to X2 and ∂ ln Y/∂ ln X3 = (∂Y/∂ X3 )(X3 /Y) = β3 , which is the elasticity of Y with respect to X3, and so on. 17 Notice that in the Cobb–Douglas production function (7.9.1) we have introduced the stochastic error term in a special way so that in the resulting logarithmic transformation it enters in the usual linear form. On this, see Sec. 6.9.

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TABLE 7.3

REAL GROSS PRODUCT, LABOR DAYS, AND REAL CAPITAL INPUT IN THE AGRICULTURAL SECTOR OF TAIWAN, 1958–1972 Year 1958 1959 1960 1961 1962 1963 1964 1965 1966 1967 1968 1969 1970 1971 1972 Real gross product (millions of NT $)*, Y 16,607.7 17,511.3 20,171.2 20,932.9 20,406.0 20,831.6 24,806.3 26,465.8 27,403.0 28,628.7 29,904.5 27,508.2 29,035.5 29,281.5 31,535.8 Labor days (millions of days), X2 275.5 274.4 269.7 267.0 267.8 275.0 283.0 300.7 307.5 303.7 304.7 298.6 295.5 299.0 288.1 Real capital input (millions of NT $), X3 17,803.7 18,096.8 18,271.8 19,167.3 19,647.6 20,803.5 22,076.6 23,445.2 24,939.0 26,713.7 29,957.8 31,585.9 33,474.5 34,821.8 41,794.3

Source: Thomas Pei-Fan Chen, “Economic Growth and Structural Change in Taiwan—1952–1972, A Production Function Approach,” unpublished Ph.D. thesis, Dept. of Economics, Graduate Center, City University of New York, June 1976, Table II. *New Taiwan dollars.

method (see Appendix 7A, Section 7A.5 for the computer printout): ln Y i = −3.3384 + 1.4988 ln X2i + 0.4899 ln X3i (2.4495) t = (−1.3629) (0.5398) (2.7765) ¯2 (0.1020) (4.8005) df = 12 (7.9.4)

R2 = 0.8890 R = 0.8705

From Eq. (7.9.4) we see that in the Taiwanese agricultural sector for the period 1958–1972 the output elasticities of labor and capital were 1.4988 and 0.4899, respectively. In other words, over the period of study, holding the capital input constant, a 1 percent increase in the labor input led on the average to about a 1.5 percent increase in the output. Similarly, holding the labor input constant, a 1 percent increase in the capital input led on the average to about a 0.5 percent increase in the output. Adding the two output elasticities, we obtain 1.9887, which gives the value of the returns to scale parameter. As is evident, over the period of the study, the Taiwanese agricultural sector was characterized by increasing returns to scale.18

18 We abstain from the question of the appropriateness of the model from the theoretical viewpoint as well as the question of whether one can measure returns to scale from time series data.

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From a purely statistical viewpoint, the estimated regression line ﬁts the data quite well. The R2 value of 0.8890 means that about 89 percent of the variation in the (log of) output is explained by the (logs of) labor and capital. In Chapter 8, we shall see how the estimated standard errors can be used to test hypotheses about the “true” values of the parameters of the Cobb– Douglas production function for the Taiwanese economy.

7.10 POLYNOMIAL REGRESSION MODELS

We now consider a class of multiple regression models, the polynomial regression models, that have found extensive use in econometric research relating to cost and production functions. In introducing these models, we further extend the range of models to which the classical linear regression model can easily be applied. To ﬁx the ideas, consider Figure 7.1, which relates the short-run marginal cost (MC) of production (Y) of a commodity to the level of its output (X). The visually-drawn MC curve in the ﬁgure, the textbook U-shaped curve, shows that the relationship between MC and output is nonlinear. If we were to quantify this relationship from the given scatterpoints, how would we go about it? In other words, what type of econometric model would capture ﬁrst the declining and then the increasing nature of marginal cost? Geometrically, the MC curve depicted in Figure 7.1 represents a parabola. Mathematically, the parabola is represented by the following equation: Y = β0 + β1 X + β2 X 2 (7.10.1)

which is called a quadratic function, or more generally, a second-degree polynomial in the variable X—the highest power of X represents the degree of the polynomial (if X3 were added to the preceding function, it would be a third-degree polynomial, and so on).

Y MC

Marginal cost

FIGURE 7.1

The U-shaped marginal cost curve.

Output

X

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The stochastic version of (7.10.1) may be written as Yi = β0 + β1 Xi + β2 Xi2 + ui which is called a second-degree polynomial regression. The general kth degree polynomial regression may be written as Yi = β0 + β1 Xi + β2 Xi2 + · · · + βk Xik + ui (7.10.3) (7.10.2)

Notice that in these types of polynomial regressions there is only one explanatory variable on the right-hand side but it appears with various powers, thus making them multiple regression models. Incidentally, note that if Xi is assumed to be ﬁxed or nonstochastic, the powered terms of Xi also become ﬁxed or nonstochastic. Do these models present any special estimation problems? Since the second-degree polynomial (7.10.2) or the kth degree polynomial (7.10.13) is linear in the parameters, the β’s, they can be estimated by the usual OLS or ML methodology. But what about the collinearity problem? Aren’t the various X’s highly correlated since they are all powers of X? Yes, but remember that terms like X 2, X 3, X 4, etc., are all nonlinear functions of X and hence, strictly speaking, do not violate the no multicollinearity assumption. In short, polynomial regression models can be estimated by the techniques presented in this chapter and present no new estimation problems.

EXAMPLE 7.4 ESTIMATING THE TOTAL COST FUNCTION As an example of the polynomial regression, consider the data on output and total cost of production of a commodity in the short run given in Table 7.4. What type of regression model will ﬁt these data? For this purpose, let us ﬁrst draw the scattergram, which is shown in Figure 7.2. From this ﬁgure it is clear that the relationship between total cost and output resembles the elongated S curve; notice how the total cost curve ﬁrst increases gradually and then rapidly, as predicted by the celebrated law of diminishing returns. This S shape of the total cost curve can be captured by the following cubic or third-degree polynomial: Yi = β0 + β1 Xi + β2 Xi2 + β3 Xi3 + ui where Y = total cost and X = output. (7.10.4)

TABLE 7.4

TOTAL COST (Y ) AND OUTPUT (X ) Total cost, $ 193 226 240 244 257 260 274 297 350 420

Output 1 2 3 4 5 6 7 8 9 10

Given the data of Table 7.4, we can apply the OLS method to estimate the parameters of (7.10.4). But before we do that, let us ﬁnd out what economic theory has to say about the short-run cubic cost function (7.10.4). Elementary price theory shows that in the short run the (Continued)

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EXAMPLE 7.4 (Continued) 450 400 350 300 250 Output 200 Y 150 1 2 3 4 5 6 7 Output 8 9 10 X Cost AC MC X Cost Y Y TC

FIGURE 7.2

Total cost of production

The total cost curve.

marginal cost (MC) and average cost (AC) curves of production are typically U-shaped—initially, as output increases both MC and AC decline, but after a certain level of output they both turn upward, again the consequence of the law of diminishing return. This can be seen in Figure 7.3 (see also Figure 7.1). And since the MC and AC curves are derived from the total cost curve, the U-shaped nature of these curves puts some restrictions on the parameters of the total cost curve (7.10.4). As a matter of fact, it can be shown that the parameters of (7.10.4) must satisfy the following restrictions if one is to observe the typical U-shaped short-run marginal and average cost curves:19 1. β0 , β1 , and β3 > 0 2. β2 < 0

2 3. β2 < 3β1 β3

Output FIGURE 7.3 Short-run cost functions.

X

wrong model), we will have to modify our theory or look for a new theory and start the empirical enquiry all over again. But as noted in the Introduction, this is the nature of any empirical investigation. Empirical Results When the third-degree polynomial regression was ﬁtted to the data of Table 7.4, we obtained the following results: ˆ Yi = 141.7667 + 63.4776Xi − 12.9615X i2 + 0.9396X i3 (6.3753) (4.7786) (0.9857) (0.0591) R 2 = 0.9983 (7.10.6) (Continued)

(7.10.5)

All this theoretical discussion might seem a bit tedious. But this knowledge is extremely useful when we examine the empirical results, for if the empirical results do not agree with prior expectations, then, assuming we have not committed a speciﬁcation error (i.e., chosen the

19 See Alpha C. Chiang, Fundamental Methods of Mathematical Economics, 3d ed., McGrawHill, New York, 1984, pp. 250–252.

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EXAMPLE 7.4 (Continued) (Note: The ﬁgures in parentheses are the estimated standard errors.) Although we will examine the statistical signiﬁcance of these results in the next chapter, the reader can verify that they are in conformity with the theoretical expectations listed in (7.10.5). We leave it as an exercise for the reader to interpret the regression (7.10.6).

EXAMPLE 7.5 GDP GROWTH RATE, 1960–1985 AND RELATIVE PER CAPITA GDP, IN 119 DEVELOPING COUNTRIES As an additional economic example of the polynomial regression model, consider the following regression results20: GDPGi = 0.013 + 0.062 RGDP − 0.061 RGDP2 se = (0.004) (0.027)

2

(0.033) R = 0.053 adj R = 0.036

2

(7.10.7)

where GDPG = GDP growth rate, percent (average for 1960–1985), and RGDP = relative per capita GDP, 1960 (percentage of U.S. GDP per capita, 1960). The adjusted R 2 (adj R 2) tells us that, after taking into account the number of regressors, the model explains only about 3.6 percent of the variation in GDPG. Even the unadjusted R 2 of 0.053 seems low. This might sound a disappointing value but, as we shall show in the next chapter, such low R 2’s are frequently encountered in cross-sectional data with a large number of observations. Besides, even an apparently low R 2 value can be statistically signiﬁcant (i.e., different from zero), as we will show in the next chapter. As this regression shows, GDPG in developing countries increased as RGDP increased, but at a decreasing rate; that is, developing economies were not catching up with advanced economies.21 This example shows how relatively simple econometric models can be used to shed light on important economic phenomena.

*7.11 PARTIAL CORRELATION COEFFICIENTS Explanation of Simple and Partial Correlation Coefﬁcients

In Chapter 3 we introduced the coefﬁcient of correlation r as a measure of the degree of linear association between two variables. For the three-variable

20 Source: The East Asian Economic Miracle: Economic Growth and Public Policy, A World Bank Policy Research Report, Oxford University Press, U.K, 1993, p. 29. 21 If you take the derivative of (7.10.7), you will obtain

dGDPG = 0.062 − 0.122 RGDP dRGDP showing that the rate of change of GDPG with respect to RGDP is declining. If you set this derivative to zero, you will get RGDP ≈ 0.5082. Thus, if a country’s GDP reaches about 51 percent of the U.S. GDP, the rate of growth of GDPG will crawl to zero. * Optional.

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regression model we can compute three correlation coefﬁcients: r1 2 (correlation between Y and X2), r1 3 (correlation coefﬁcient between Y and X3), and r2 3 (correlation coefﬁcient between X2 and X3); notice that we are letting the subscript 1 represent Y for notational convenience. These correlation coefﬁcients are called gross or simple correlation coefﬁcients, or correlation coefﬁcients of zero order. These coefﬁcients can be computed by the deﬁnition of correlation coefﬁcient given in (3.5.13). But now consider this question: Does, say, r 1 2 in fact measure the “true” degree of (linear) association between Y and X2 when a third variable X3 may be associated with both of them? This question is analogous to the following question: Suppose the true regression model is (7.1.1) but we omit from the model the variable X3 and simply regress Y on X2, obtaining the slope coefﬁcient of, say, b1 2. Will this coefﬁcient be equal to the true coefﬁcient β2 if the model (7.1.1) were estimated to begin with? The answer should be apparent from our discussion in Section 7.7. In general, r1 2 is not likely to reﬂect the true degree of association between Y and X2 in the presence of X3. As a matter of fact, it is likely to give a false impression of the nature of association between Y and X2, as will be shown shortly. Therefore, what we need is a correlation coefﬁcient that is independent of the inﬂuence, if any, of X3 on X2 and Y. Such a correlation coefﬁcient can be obtained and is known appropriately as the partial correlation coefﬁcient. Conceptually, it is similar to the partial regression coefﬁcient. We deﬁne r1 2.3 = partial correlation coefﬁcient between Y and X2, holding X3 constant r1 3.2 = partial correlation coefﬁcient between Y and X3, holding X2 constant r2 3.1 = partial correlation coefﬁcient between X2 and X3, holding Y constant These partial correlations can be easily obtained from the simple or zeroorder, correlation coefﬁcients as follows (for proofs, see the exercises)22: r1 2.3 = r1 3.2 = r2 3.1 = r1 2 − r1 3 r2 3

2 1 − r1 3 2 1 − r2 3

(7.11.1)

r1 3 − r1 2 r2 3

2 1 − r1 2 2 1 − r2 3

(7.11.2)

r2 3 − r1 2 r1 3

2 1 − r1 2 2 1 − r1 3

(7.11.3)

The partial correlations given in Eqs. (7.11.1) to (7.11.3) are called ﬁrstorder correlation coefﬁcients. By order we mean the number of secondary

22 Most computer programs for multiple regression analysis routinely compute the simple correlation coefﬁcients; hence the partial correlation coefﬁcients can be readily computed.

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subscripts. Thus r1 2.3 4 would be the correlation coefﬁcient of order two, r1 2.3 4 5 would be the correlation coefﬁcient of order three, and so on. As noted previously, r1 2, r1 3, and so on are called simple or zero-order correlations. The interpretation of, say, r1 2.3 4 is that it gives the coefﬁcient of correlation between Y and X2, holding X3 and X4 constant.

Interpretation of Simple and Partial Correlation Coefﬁcients

In the two-variable case, the simple r had a straightforward meaning: It measured the degree of (linear) association (and not causation) between the dependent variable Y and the single explanatory variable X. But once we go beyond the two-variable case, we need to pay careful attention to the interpretation of the simple correlation coefﬁcient. From (7.11.1), for example, we observe the following: 1. Even if r1 2 = 0, r1 2.3 will not be zero unless r1 3 or r2 3 or both are zero. 2. If r1 2 = 0 and r1 3 and r2 3 are nonzero and are of the same sign, r1 2.3 will be negative, whereas if they are of the opposite signs, it will be positive. An example will make this point clear. Let Y = crop yield, X2 = rainfall, and X3 = temperature. Assume r1 2 = 0, that is, no association between crop yield and rainfall. Assume further that r1 3 is positive and r2 3 is negative. Then, as (7.11.1) shows, r1 2.3 will be positive; that is, holding temperature constant, there is a positive association between yield and rainfall. This seemingly paradoxical result, however, is not surprising. Since temperature X3 affects both yield Y and rainfall X2, in order to ﬁnd out the net relationship between crop yield and rainfall, we need to remove the inﬂuence of the “nuisance” variable temperature. This example shows how one might be misled by the simple coefﬁcient of correlation. 3. The terms r1 2.3 and r1 2 (and similar comparisons) need not have the same sign. 4. In the two-variable case we have seen that r2 lies between 0 and 1. The same property holds true of the squared partial correlation coefﬁcients. Using this fact, the reader should verify that one can obtain the following expression from (7.11.1):

2 2 2 0 ≤ r1 2 + r1 3 + r2 3 − 2r1 2r1 3r2 3 ≤ 1

(7.11.4)

which gives the interrelationships among the three zero-order correlation coefﬁcients. Similar expressions can be derived from Eqs. (7.9.3) and (7.9.4). 5. Suppose that r1 3 = r2 3 = 0. Does this mean that r1 2 is also zero? The answer is obvious from (7.11.4). The fact that Y and X3 and X2 and X3 are uncorrelated does not mean that Y and X2 are uncorrelated. 2 In passing, note that the expression r1 2.3 may be called the coefﬁcient of partial determination and may be interpreted as the proportion of the variation in Y not explained by the variable X3 that has been explained

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by the inclusion of X2 into the model (see exercise 7.5). Conceptually it is similar to R2. Before moving on, note the following relationships between R2, simple correlation coefﬁcients, and partial correlation coefﬁcients: R2 =

2 2 r1 2 + r1 3 − 2r1 2r1 3r2 3 2 1 − r2 3

(7.11.5) (7.11.6) (7.11.7)

2 2 2 R2 = r1 2 + 1 − r1 2 r1 3.2 2 2 2 R2 = r1 3 + 1 − r1 3 r1 2.3

In concluding this section, consider the following: It was stated previously that R2 will not decrease if an additional explanatory variable is introduced into the model, which can be seen clearly from (7.11.6). This equation states that the proportion of the variation in Y explained by X2 and X3 jointly is the 2 sum of two parts: the part explained by X2 alone ( = r1 2 ) and the part not ex2 plained by X2 ( = 1 − r1 2 ) times the proportion that is explained by X3 after 2 2 holding the inﬂuence of X2 constant. Now R2 > r1 2 so long as r1 3.2 > 0. At 2 2 2 worst, r1 3.2 will be zero, in which case R = r1 2 .

7.12

SUMMARY AND CONCLUSIONS

1. This chapter introduced the simplest possible multiple linear regression model, namely, the three-variable regression model. It is understood that the term linear refers to linearity in the parameters and not necessarily in the variables. 2. Although a three-variable regression model is in many ways an extension of the two-variable model, there are some new concepts involved, such as partial regression coefﬁcients, partial correlation coefﬁcients, multiple correlation coefﬁcient, adjusted and unadjusted (for degrees of freedom) R2, multicollinearity, and speciﬁcation bias. 3. This chapter also considered the functional form of the multiple regression model, such as the Cobb–Douglas production function and the polynomial regression model. 4. Although R2 and adjusted R2 are overall measures of how the chosen model ﬁts a given set of data, their importance should not be overplayed. What is critical is the underlying theoretical expectations about the model in terms of a priori signs of the coefﬁcients of the variables entering the model and, as it is shown in the following chapter, their statistical signiﬁcance. 5. The results presented in this chapter can be easily generalized to a multiple linear regression model involving any number of regressors. But the algebra becomes very tedious. This tedium can be avoided by resorting to matrix algebra. For the interested reader, the extension to the k-variable

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regression model using matrix algebra is presented in Appendix C, which is optional. But the general reader can read the remainder of the text without knowing much of matrix algebra.

EXERCISES

Questions 7.1. Consider the data in Table 7.5.

TABLE 7.5 Y 1 3 8 X2 1 2 3 X3 2 1 −3

Based on these data, estimate the following regressions:

Yi = α1 + α2 X 2i + u1i Yi = λ1 + λ3 X 3i + u2i Yi = β1 + β2 X 2i + β3 X 3i + ui

(1) (2) (3)

Note: Estimate only the coefﬁcients and not the standard errors. a. Is α2 = β2 ? Why or why not? b. Is λ3 = β3 ? Why or why not? What important conclusion do you draw from this exercise? 7.2. From the following data estimate the partial regression coefﬁcients, their standard errors, and the adjusted and unadjusted R2 values:

¯ Y = 367.693 ¯ (Yi − Y)2 = 66042 .269 ¯ ( X 3i − X 3 )2 = 280.000 ¯ ¯ (Yi − Y)( X 3i − X 3 ) = 4250 .900 ¯ X 2 = 402.760 ¯ X 3 = 8.0

¯ ( X 2i − X 2 )2 = 84855 .096 ¯ ¯ (Yi − Y)( X 2i − X 2 ) = 74778 .346 ¯ ¯ ( X 2i − X 2 )( X 3i − X 3 ) = 4796 .000 n = 15

7.3. Show that Eq. (7.4.7) can also be expressed as

ˆ β2 = = yi (x2i − b2 3 x3i ) (x2i − b2 3 x3i )2 net (of x3 ) covariation between y and x2 net (of x3 ) variation in x2

where b2 3 is the slope coefﬁcient in the regression of X2 on X3. (Hint: Re2 x2i x3i / x3i .) call that b2 3 =

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7.4. In a multiple regression model you are told that the error term ui has the following probability distribution, namely, ui ∼ N(0, 4). How would you set up a Monte Carlo experiment to verify that the true variance is in fact 4? 2 2 2 7.5. Show that r 1 2.3 = ( R2 − r 1 3 )/(1 − r 1 3 ) and interpret the equation. 7.6. If the relation α1 X 1 + α2 X 2 + α3 X 3 = 0 holds true for all values of X1, X2, and X3, ﬁnd the values of the three partial correlation coefﬁcients. 7.7. Is it possible to obtain the following from a set of data? a. r 2 3 = 0.9, r 1 3 = −0.2, r 1 2 = 0.8 b. r 1 2 = 0.6, r 2 3 = −0.9, r 3 1 = −0.5 c. r 2 1 = 0.01, r 1 3 = 0.66, r 2 3 = −0.7 7.8. Consider the following model:

Yi = β1 + β2 Education i + β2 Years of experience + ui

7.9.

7.10.

7.11. 7.12.

Suppose you leave out the years of experience variable. What kinds of problems or biases would you expect? Explain verbally. Show that β2 and β3 in (7.9.2) do, in fact, give output elasticities of labor and capital. (This question can be answered without using calculus; just recall the deﬁnition of the elasticity coefﬁcient and remember that a change in the logarithm of a variable is a relative change, assuming the changes are rather small.) Consider the three-variable linear regression model discussed in this chapter. a. Suppose you multiply all the X2 values by 2. What will be the effect of this rescaling, if any, on the estimates of the parameters and their standard errors? b. Now instead of a, suppose you multiply all the Y values by 2. What will be the effect of this, if any, on the estimated parameters and their standard errors? 2 2 In general R2 = r 1 2 + r 1 3 , but it is so only if r 2 3 = 0. Comment and point out the signiﬁcance of this ﬁnding. [Hint: See Eq. (7.11.5).] Consider the following models.*

Model A: Model B: Yt = α1 + α2 X 2t + α3 X 3t + u1t (Yt − X 2t ) = β1 + β2 X 2t + β3 X 3t + u2t

a. Will OLS estimates of α1 and β1 be the same? Why? b. Will OLS estimates of α3 and β3 be the same? Why? c. What is the relationship between α2 and β2 ? d. Can you compare the R 2 terms of the two models? Why or why not? 7.13. Suppose you estimate the consumption function†

Yi = α1 + α2 X i + u1i

and the savings function

Z i = β1 + β2 X i + u2i

*

Adapted from Wojciech W. Charemza and Derek F. Deadman, Econometric Practice: General to Speciﬁc Modelling, Cointegration and Vector Autogression, Edward Elgar, Brookﬁeld, Vermont, 1992, p. 18. † Adapted from Peter Kennedy, A Guide to Econometrics, 3d ed., The MIT Press, Cambridge, Massachusetts, 1992, p. 308, Question #9.

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where Y = consumption, Z = savings, X = income, and X = Y + Z, that is, income is equal to consumption plus savings. a. What is the relationship, if any, between α2 and β2 ? Show your calculations. b. Will the residual sum of squares, RSS, be the same for the two models? Explain. c. Can you compare the R2 terms of the two models? Why or why not? 7.14. Suppose you express the Cobb–Douglas model given in (7.9.1) as follows:

Yi = β1 X 2i2 X 3i3 ui β β

If you take the log-transform of this model, you will have ln ui as the disturbance term on the right-hand side. a. What probabilistic assumptions do you have to make about ln ui to be able to apply the classical normal linear regression model (CNLRM)? How would you test this with the data given in Table 7.3? b. Do the same assumptions apply to ui ? Why or why not? 7.15. Regression through the origin. Consider the following regression through the origin:

ˆ ˆ Yi = β2 X 2i + β3 X 3i + ui ˆ

a. How would you go about estimating the unknowns? ˆ b. Will ui be zero for this model? Why or why not? ˆ ui X 3i = 0 for this model? ˆ c. Will ui X 2i = d. When would you use such a model? e. Can you generalize your results to the k-variable model? (Hint: Follow the discussion for the two-variable case given in Chapter 6.) Problems 7.16. The demand for roses.* Table 7.6 gives quarterly data on these variables: Y = quantity of roses sold, dozens X2 = average wholesale price of roses, $/dozen X3 = average wholesale price of carnations, $/dozen X4 = average weekly family disposable income, $/week X5 = the trend variable taking values of 1, 2, and so on, for the period 1971–III to 1975–II in the Detroit metropolitan area You are asked to consider the following demand functions:

Yt = α1 + α2 X 2t + α3 X 3t + α4 X 4t + α5 X 5t + ut ln Yt = β1 + β2 ln X 2t + β3 ln X 3t + β4 ln X 4t + β5 X 5t + ut

a. Estimate the parameters of the linear model and interpret the results. b. Estimate the parameters of the log-linear model and interpret the results. c. β2 , β3 , and β4 give, respectively, the own-price, cross-price, and income elasticities of demand. What are their a priori signs? Do the results concur with the a priori expectations?

* I am indebted to Joe Walsh for collecting these data from a major wholesaler in the Detroit metropolitan area and subsequently processing them.

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TABLE 7.6 Year and quarter 1971–III –IV 1972–I –II –III –IV 1973–I –II –III –IV 1974–I –II –III –IV 1975–I –II Y 11,484 9,348 8,429 10,079 9,240 8,862 6,216 8,253 8,038 7,476 5,911 7,950 6,134 5,868 3,160 5,872 X2 2.26 2.54 3.07 2.91 2.73 2.77 3.59 3.23 2.60 2.89 3.77 3.64 2.82 2.96 4.24 3.69 X3 3.49 2.85 4.06 3.64 3.21 3.66 3.76 3.49 3.13 3.20 3.65 3.60 2.94 3.12 3.58 3.53 X4 158.11 173.36 165.26 172.92 178.46 198.62 186.28 188.98 180.49 183.33 181.87 185.00 184.00 188.20 175.67 188.00 X5 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16

d. How would you compute the own-price, cross-price, and income elasticities for the linear model? e. On the basis of your analysis, which model, if either, would you choose and why? 7.17. Wildcat activity. Wildcats are wells drilled to ﬁnd and produce oil and/or gas in an improved area or to ﬁnd a new reservoir in a ﬁeld previously found to be productive of oil or gas or to extend the limit of a known oil or gas reservoir. Table 7.7 gives data on these variables*: Y = the number of wildcats drilled X2 = price at the wellhead in the previous period (in constant dollars, 1972 = 100) X3 = domestic output X4 = GNP constant dollars (1972 = 100) X5 = trend variable, 1948 = 1, 1949 = 2, . . . , 1978 = 31 See if the following model ﬁts the data:

Yt = β1 + β2 X 2t + β3 ln X 3t + β4 X 4t + β5 X 5t + ut

a. Can you offer an a priori rationale to this model? b. Assuming the model is acceptable, estimate the parameters of the ¯ model and their standard errors, and obtain R2 and R2 . c. Comment on your results in view of your prior expectations. d. What other speciﬁcation would you suggest to explain wildcat activity? Why?

*

I am indebted to Raymond Savino for collecting and processing the data.

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TABLE 7.7 Domestic output (millions of barrels per day), (X3) 5.52 5.05 5.41 6.16 6.26 6.34 6.81 7.15 7.17 6.71 7.05 7.04 7.18 7.33 7.54 7.61 7.80 8.30 8.81 8.66 8.78 9.18 9.03 9.00 8.78 8.38 8.01 7.78 7.88 7.88 8.67

Thousands of wildcats, (Y ) 8.01 9.06 10.31 11.76 12.43 13.31 13.10 14.94 16.17 14.71 13.20 13.19 11.70 10.99 10.80 10.66 10.75 9.47 10.31 8.88 8.88 9.70 7.69 6.92 7.54 7.47 8.63 9.21 9.23 9.96 10.78

Per barrel price, constant $, (X2) 4.89 4.83 4.68 4.42 4.36 4.55 4.66 4.54 4.44 4.75 4.56 4.29 4.19 4.17 4.11 4.04 3.96 3.85 3.75 3.69 3.56 3.56 3.48 3.53 3.39 3.68 5.92 6.03 6.12 6.05 5.89

GNP, constant $ billions, (X4) 487.67 490.59 533.55 576.57 598.62 621.77 613.67 654.80 668.84 681.02 679.53 720.53 736.86 755.34 799.15 830.70 874.29 925.86 980.98 1,007.72 1,051.83 1,078.76 1,075.31 1,107.48 1,171.10 1,234.97 1,217.81 1,202.36 1,271.01 1,332.67 1,385.10

Time, (X5) 1948 = 1 1949 = 2 1950 = 3 1951 = 4 1952 = 5 1953 = 6 1954 = 7 1955 = 8 1956 = 9 1957 = 10 1958 = 11 1959 = 12 1960 = 13 1961 = 14 1962 = 15 1963 = 16 1964 = 17 1965 = 18 1966 = 19 1967 = 20 1968 = 21 1969 = 22 1970 = 23 1971 = 24 1972 = 25 1973 = 26 1974 = 27 1975 = 28 1976 = 29 1977 = 30 1978 = 31

Source: Energy Information Administration, 1978 Report to Congress.

7.18. U.S. defense budget outlays, 1962–1981. In order to explain the U.S. defense budget, you are asked to consider the following model:

Yt = β1 + β2 X 2t + β3 X 3t + β4 X 4t + β5 X 5t + ut

where Yt = defense budget-outlay for year t, $ billions X 2t = GNP for year t, $ billions X 3t = U.S. military sales/assistance in year t, $ billions X 4t = aerospace industry sales, $ billions X 5t = military conﬂicts involving more than 100,000 troops. This variable takes a value of 1 when 100,000 or more troops are

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involved but is equal to zero when that number is under 100,000. To test this model, you are given the data in Table 7.8. a. Estimate the parameters of this model and their standard errors and ¯ obtain R2 , modiﬁed R2 , and R2 . b. Comment on the results, taking into account any prior expectations you have about the relationship between Y and the various X variables. c. What other variable(s) might you want to include in the model and why? 7.19. The demand for chicken in the United States, 1960–1982. To study the per capita consumption of chicken in the United States, you are given the data in Table 7.9, where Y = per capita consumption of chickens, lb X 2 = real disposable income per capita, $ X 3 = real retail price of chicken per lb, ¢ X 4 = real retail price of pork per lb, ¢ X 5 = real retail price of beef per lb, ¢ X 6 = composite real price of chicken substitutes per lb, ¢, which is a weighted average of the real retail prices per lb of pork and beef, the weights being the relative consumptions of beef and pork in total beef and pork consumption

TABLE 7.8 Defense budget outlays, Y 51.1 52.3 53.6 49.6 56.8 70.1 80.5 81.2 80.3 77.7 78.3 74.5 77.8 85.6 89.4 97.5 105.2 117.7 135.9 162.1 U.S. military sales/ assistance, X3 0.6 0.9 1.1 1.4 1.6 1.0 0.8 1.5 1.0 1.5 2.95 4.8 10.3 16.0 14.7 8.3 11.0 13.0 15.3 18.0 Aerospace industry sales, X4 16.0 16.4 16.7 17.0 20.2 23.4 25.6 24.6 24.8 21.7 21.5 24.3 26.8 29.5 30.4 33.3 38.0 46.2 57.6 68.9 Conﬂicts 100,000+, X5 0 0 0 1 1 1 1 1 1 1 1 0 0 0 0 0 0 0 0 0

Year 1962 1963 1964 1965 1966 1967 1968 1969 1970 1971 1972 1973 1974 1975 1976 1977 1978 1979 1980 1981

GNP, X2 560.3 590.5 632.4 684.9 749.9 793.9 865.0 931.4 992.7 1,077.6 1,185.9 1,326.4 1,434.2 1,549.2 1,718.0 1,918.3 2,163.9 2,417.8 2,633.1 2,937.7

Source: The data were collected by Albert Lucchino from various government publications.

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TABLE 7.9

Year 1960 1961 1962 1963 1964 1965 1966 1967 1968 1969 1970 1971 1972 1973 1974 1975 1976 1977 1978 1979 1980 1981 1982

Y 27.8 29.9 29.8 30.8 31.2 33.3 35.6 36.4 36.7 38.4 40.4 40.3 41.8 40.4 40.7 40.1 42.7 44.1 46.7 50.6 50.1 51.7 52.9

X2 397.5 413.3 439.2 459.7 492.9 528.6 560.3 624.6 666.4 717.8 768.2 843.3 911.6 931.1 1,021.5 1,165.9 1,349.6 1,449.4 1,575.5 1,759.1 1,994.2 2,258.1 2,478.7

X3 42.2 38.1 40.3 39.5 37.3 38.1 39.3 37.8 38.4 40.1 38.6 39.8 39.7 52.1 48.9 58.3 57.9 56.5 63.7 61.6 58.9 66.4 70.4

X4 50.7 52.0 54.0 55.3 54.7 63.7 69.8 65.9 64.5 70.0 73.2 67.8 79.1 95.4 94.2 123.5 129.9 117.6 130.9 129.8 128.0 141.0 168.2

X5 78.3 79.2 79.2 79.2 77.4 80.2 80.4 83.9 85.5 93.7 106.1 104.8 114.0 124.1 127.6 142.9 143.6 139.2 165.5 203.3 219.6 221.6 232.6

X6 65.8 66.9 67.8 69.6 68.7 73.6 76.3 77.2 78.1 84.7 93.3 89.7 100.7 113.5 115.3 136.7 139.2 132.0 132.1 154.4 174.9 180.8 189.4

Source: Data on Y are from Citibase and on X2 through X6 are from the U.S. Department of Agriculture. I am indebted to Robert J. Fisher for collecting the data and for the statistical analysis. Note: The real prices were obtained by dividing the nominal prices by the Consumer Price Index for food.

Now consider the following demand functions: ln Yt = α1 + α2 ln X 2t + α3 ln X 3t + ut ln Yt = γ1 + γ2 ln X 2t + γ3 ln X 3t + γ4 ln X 4t + ut ln Yt = λ1 + λ2 ln X 2t + λ3 ln X 3t + λ4 ln X 5t + ut ln Yt = θ1 + θ2 ln X 2t + θ3 ln X 3t + θ4 ln X 4t + θ5 ln X 5t + ut ln Yt = β1 + β2 ln X 2t + β3 ln X 3t + β4 ln X 6t + ut

(1) (2) (3) (4) (5)

From microeconomic theory it is known that the demand for a commodity generally depends on the real income of the consumer, the real price of the commodity, and the real prices of competing or complementary commodities. In view of these considerations, answer the following questions. a. Which demand function among the ones given here would you choose, and why? b. How would you interpret the coefﬁcients of ln X 2t and ln X 3t in these models? c. What is the difference between speciﬁcations (2) and (4)? d. What problems do you foresee if you adopt speciﬁcation (4)? (Hint: Prices of both pork and beef are included along with the price of chicken.)

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e. Since speciﬁcation (5) includes the composite price of beef and pork, would you prefer the demand function (5) to the function (4)? Why? f. Are pork and/or beef competing or substitute products to chicken? How do you know? g. Assume function (5) is the “correct” demand function. Estimate the ¯ parameters of this model, obtain their standard errors, and R2 , R2 , and modiﬁed R2 . Interpret your results. h. Now suppose you run the “incorrect” model (2). Assess the consequences of this mis-speciﬁcation by considering the values of γ2 and γ3 in relation to β2 and β3 , respectively. (Hint: Pay attention to the discussion in Section 7.7.) 7.20. In a study of turnover in the labor market, James F. Ragan, Jr., obtained the following results for the U.S. economy for the period of 1950–I to 1979–IV.* (Figures in the parentheses are the estimated t statistics.) ln Yt = 4.47 −

0.34 ln X2t + 1.22 ln X3t + 1.22 ln X4t (−5.31) (−3.09) (3.64) 0.0055 X6t (3.10)

¯ R2 = 0.5370

(4.28) (1.10)

+ 0.80 ln X5t −

Note: We will discuss the t statistics in the next chapter. where Y = quit rate in manufacturing, deﬁned as number of people leaving jobs voluntarily per 100 employees X 2 = an instrumental or proxy variable for adult male unemployment rate X 3 = percentage of employees younger than 25 X 4 = Nt−1 /Nt−4 = ratio of manufacturing employment in quarter (t − 1) to that in quarter (t − 4) X 5 = percentage of women employees X 6 = time trend (1950–I = 1) a. Interpret the foregoing results. b. Is the observed negative relationship between the logs of Y and X2 justiﬁable a priori? c. Why is the coefﬁcient of ln X3 positive? d. Since the trend coefﬁcient is negative, there is a secular decline of what percent in the quit rate and why is there such a decline? ¯ e. Is the R2 “too” low? f. Can you estimate the standard errors of the regression coefﬁcients from the given data? Why or why not? 7.21. Consider the following demand function for money in the United States for the period 1980–1998:

Mt = β1 Yt 2 r t 3 e ut β β

* Source: See Ragan’s article, “Turnover in the Labor Market: A Study of Quit and Layoff Rates,” Economic Review, Federal Reserve Bank of Kansas City, May 1981, pp. 13–22.

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where M = real money demand, using the M2 deﬁnition of money Y = real GDP r = interest rate To estimate the above demand for money function, you are given the data in Table 7.10. Note: To convert nominal quantities into real quantities, divide M and GDP by CPI. There is no need to divide the interest rate variable by CPI. Also, note that we have given two interest rates, a short-term rate as measured by the 3-month treasury bill rate and the long-term rate as measured by the yield on 30-year treasury bond, as prior empirical studies have used both types of interest rates. a. Given the data, estimate the above demand function. What are the income and interest rate elasticities of demand for money? b. Instead of estimating the above demand function, suppose you were to ﬁt the function (M/Y)t = α1 r tα2 e ut. How would you interpret the results? Show the necessary calculations. c. How will you decide which is a better speciﬁcation? (Note: A formal statistical test will be given in Chapter 8.) 7.22. Table 7.11 gives data for the manufacturing sector of the Greek economy for the period 1961–1987.

TABLE 7.10

DEMAND FOR MONEY IN THE UNITED STATES, 1980–1998 Observation 1980 1981 1982 1983 1984 1985 1986 1987 1988 1989 1990 1991 1992 1993 1994 1995 1996 1997 1998 GDP 2795.6 3131.3 3259.2 3534.9 3932.7 4213.0 4452.9 4742.5 5108.3 5489.1 5803.2 5986.2 6318.9 6642.3 7054.3 7400.5 7813.2 8300.8 8759.9 M2 1600.4 1756.1 1911.2 2127.8 2311.7 2497.4 2734.0 2832.8 2995.8 3159.9 3279.1 3379.8 3434.1 3487.5 3502.2 3649.3 3824.2 4046.7 4401.4 CPI 82.4 90.9 96.5 99.6 103.9 107.6 109.6 113.6 118.3 124.0 130.7 136.2 140.3 144.5 148.2 152.4 156.9 160.5 163.0 LTRATE 11.27 13.45 12.76 11.18 12.41 10.79 7.78 8.59 8.96 8.45 8.61 8.14 7.67 6.59 7.37 6.88 6.71 6.61 5.58 TBRATE 11.506 14.029 10.686 8.630 9.580 7.480 5.980 5.820 6.690 8.120 7.510 5.420 3.450 3.020 4.290 5.510 5.020 5.070 4.810

Notes: GDP: gross domestic product ($ billions) M2: M2 money supply. CPI: Consumer Price Index (1982–1984 = 100). LTRATE: long-term interest rate (30-year Treasury bond). TBRATE: three-month Treasury bill rate (% per annum). Source: Economic Report of the President, 2000, Tables B-1, B-58, B-67, B-71.

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TABLE 7.11

GREEK INDUSTRIAL SECTOR Observation 1961 1962 1963 1964 1965 1966 1967 1968 1969 1970 1971 1972 1973 1974 1975 1976 1977 1978 1979 1980 1981 1982 1983 1984 1985 1986 1987 Output* 35.858 37.504 40.378 46.147 51.047 53.871 56.834 65.439 74.939 80.976 90.802 101.955 114.367 101.823 107.572 117.600 123.224 130.971 138.842 135.486 133.441 130.388 130.615 132.244 137.318 137.468 135.750 Capital 59.600 64.200 68.800 75.500 84.400 91.800 99.900 109.100 120.700 132.000 146.600 162.700 180.600 197.100 209.600 221.900 232.500 243.500 257.700 274.400 289.500 301.900 314.900 327.700 339.400 349.492 358.231 Labor† 637.0 643.2 651.0 685.7 710.7 724.3 735.2 760.3 777.6 780.8 825.8 864.1 894.2 891.2 887.5 892.3 930.1 969.9 1006.9 1020.9 1017.1 1016.1 1008.1 985.1 977.1 1007.2 1000.0 Capital-to-labor ratio 0.0936 0.0998 0.1057 0.1101 0.1188 0.1267 0.1359 0.1435 0.1552 0.1691 0.1775 0.1883 0.2020 0.2212 0.2362 0.2487 0.2500 0.2511 0.2559 0.2688 0.2846 0.2971 0.3124 0.3327 0.3474 0.3470 0.3582

*Billions of Drachmas at constant 1970 prices † Thousands of workers per year. Source: I am indebted to George K. Zestos of Christopher Newport University, Virginia, for the data.

a. See if the Cobb–Douglas production function ﬁts the data given in the table and interpret the results. What general conclusion do you draw? b. Now consider the following model: Output/labor = A(K/L)β e u where the regressand represents labor productivity and the regressor represents the capital labor ratio. What is the economic signiﬁcance of such a relationship, if any? Estimate the parameters of this model and interpret your results. 7.23. Refer to Example 3.3 and the data given in Table 2.6. Now consider the following models: a. ln (hwage )i = β1 + β2 ln (education )i + β3 (ln education )2 + ui where ln = natural log. How would you interpret this model? Estimate this model, obtaining the usual statistics and comment on your results. b. Now consider the following model: ln (hwage) = β1 + β2 ln (education ) + β3 ln (education ) + ui

2

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If you try to estimate this model, what problem(s) would you encounter? Try to estimate this model and see if your software package can estimate this model. 7.24. Monte Carlo experiment: Consider the following model:

Yi = β1 + β2 X 2i + β3 X 3i + ui

You are told that β1 = 262, β2 = −0.006, β3 = −2.4, σ 2 = 42, and ui ∼ N(0, 42) . Generate 10 sets of 64 observations on ui from the given normal distribution and use the 64 observations given in Table 6.4, where Y = CM, X 2 = PGNP, and X 3 = FLR to generate 10 sets of the estimated β coefﬁcients (each set will have the three estimated parameters). Take the averages of each of the estimated β coefﬁcients and relate them to the true values of these coefﬁcients given above. What overall conclusion do you draw?

APPENDIX 7A

7A.1 DERIVATION OF OLS ESTIMATORS GIVEN IN EQUATIONS (7.4.3) TO (7.4.5)

Differentiating the equation ui = ˆ2 ˆ ˆ ˆ (Yi − β1 − β2 X2i − β3 X3i )2 (7.4.2)

partially with respect to the three unknowns and setting the resulting equations to zero, we obtain

∂ ∂ ∂ ui ˆ2 ˆ ∂ β1 ui ˆ2 ˆ ∂ β2 ui ˆ2 ˆ ∂ β3 =2 =2 =2 ˆ ˆ ˆ (Yi − β1 − β2 X 2i − β3 X 3i )(−1) = 0 ˆ ˆ ˆ (Yi − β1 − β2 X 2i − β3 X 3i )(− X 2i ) = 0 ˆ ˆ ˆ (Yi − β1 − β2 X 2i − β3 X 3i )(− X 3i ) = 0

Simplifying these, we obtain Eqs. (7.4.3) to (7.4.5). In passing, note that the three preceding equations can also be written as ui = 0 ˆ ui X2i = 0 ˆ ui X3i = 0 ˆ which show the properties of the least-squares ﬁt, namely, that the residuals sum to zero and that they are uncorrelated with the explanatory variables X2 and X3. (Why?)

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Incidentally, notice that to obtain the OLS estimators of the k-variable linear regression model (7.4.20) we proceed analogously. Thus, we ﬁrst write ui = ˆ2 ˆ ˆ ˆ (Yi − β1 − β2 X2i − · · · − βk Xki )2

Differentiating this expression partially with respect to each of the k unknowns, setting the resulting equations equal to zero, and rearranging, we obtain the following k normal equations in the k unknowns: ˆ ˆ Yi = nβ1 + β2 ˆ Yi X2i = β1 ˆ Yi X3i = β1 ˆ Yi Xki = β1 ˆ X2i + β3 ˆ X3i + · · · + βk Xki X2i Xki X3i Xki

2 Xki

ˆ X2i + β2 ˆ X3i + β2 ˆ Xki + β2

2 ˆ X2i + β3

ˆ X2i X3i + · · · + βk

2 ˆ X3i + · · · + βk

ˆ X2i X3i + β3 ˆ X2i Xki + β3

........................................................ ˆ X3i Xki + · · · + βk

Or, switching to small letters, these equations can be expressed as ˆ yi x2i = β2 ˆ yi x3i = β2 ˆ yi xki = β2

2 ˆ x2i + β3

ˆ x2i x3i + · · · + βk

2 ˆ x3i + · · · + βk

x2i xki x3i xki

2 xki

ˆ x2i x3i + β3 ˆ x2i xki + β3

........................................... ˆ x3i xki + · · · + βk

It should further be noted that the k-variable model also satisﬁes these equations: ui = 0 ˆ ui X2i = ˆ ui X3i = · · · = ˆ ui Xki = 0 ˆ

7A.2 EQUALITY BETWEEN THE COEFFICIENTS OF PGNP IN (7.3.5) AND (7.6.2)

Letting Y = CM, X2 = PGNP, and X3 = FLR and using the deviation form, write yi = b1 3 x3i + u1i ˆ x2i = b2 3 x3i + u2i ˆ ˆ ˆ Now regress u1 on u2 to obtain: a1 = u1i u2i ˆ ˆ = −0.0056 u2 ˆ 2i (for our example) (3) (1) (2)

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ˆ Note that because the u’s are residuals, their mean values are zero. Using (1) and (2), we can write (3) as a1 = (yi − b1 3 x3i )(x2i − b2 3 x3i ) (x2i − b2 3 x3i )2 (4)

Expand the preceding expression, and note that b2 3 = and b1 3 = yi x3i 2 x3i (6) x2i x3i 2 x3i (5)

Making these substitutions into (4), we get ˆ β2 = yi x2i

2 x2i 2 x3i − 2 x3i

yi x3i − x2i x3i

x2i x3i

2

(7.4.7)

= −0.0056

7A.3 DERIVATION OF EQUATION (7.4.19)

(for our example)

Recall that ˆ ˆ ˆ ui = Yi − β1 − β2 X2i − β3 X3i ˆ which can also be written as ˆ ˆ ui = yi − β2 x2i − β3 x3i ˆ where small letters, as usual, indicate deviations from mean values. Now ui = ˆ2 = = (ui ui ) ˆ ˆ ˆ ˆ ui (yi − β2 x2i − β3 x3i ) ˆ ui yi ˆ ui x2i = ˆ ui x3i = 0. (Why?) Also ˆ

where use is made of the fact that ui yi = ˆ that is, ui = ˆ2 which is the required result. ˆ yi2 − β2 yi ui = ˆ

ˆ ˆ yi (yi − β2 x2i − β3 x3i ) ˆ yi x2i − β3

yi x3i

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7A.4 MAXIMUM LIKELIHOOD ESTIMATION OF THE MULTIPLE REGRESSION MODEL

Extending the ideas introduced in Chapter 4, Appendix 4A, we can write the log-likelihood function for the k-variable linear regression model (7.4.20) as n n 1 ln L = − ln σ 2 − ln (2π) − 2 2 2 (Yi − β1 − β2 X2i − · · · − βk Xki )2 σ2

Differentiating this function partially with respect to β1 , β2 , . . . , βk and σ 2 , we obtain the following (K + 1) equations: ∂ ln L 1 =− 2 ∂β1 σ (Yi − β1 − β2 X2i − · · · − βk Xki )(−1) (1) (2) (K) (K + 1)

∂ ln L 1 =− 2 (Yi − β1 − β2 X2i − · · · − βk Xki )(−X2i ) ∂β2 σ ............................................. ∂ ln L 1 =− 2 (Yi − β1 − β2 X2i − · · · − βk Xki )(−Xki ) ∂βk σ ∂ ln L n 1 =− 2 + 2 ∂σ 2σ 2σ 4 (Yi − β1 − β2 X2i − · · · − βk Xki )2

Setting these equations equal to zero (the ﬁrst-order condition for optimiza˜ ˜ ˜ ˜ tion) and letting β1 , β2 , . . . , βk and σ 2 denote the ML estimators, we obtain, after simple algebraic manipulations, ˜ ˜ Yi = nβ1 + β2 ˜ X2i + · · · + βk Xki

2 ˜ ˜ ˜ Yi X2i = β1 X2i + β2 X2i + · · · + βk X2i Xki ............................................

˜ Yi Xki = β1

˜ Xki + β2

˜ X2i Xki + · · · + βk

2 Xki

which are precisely the normal equations of the least-squares theory, as can be seen from Appendix 7A, Section 7A.1. Therefore, the ML estimators, the ˜ ˆ β’s, are the same as the OLS estimators, the β’s, given previously. But as noted in Chapter 4, Appendix 4A, this equality is not accidental. Substituting the ML ( = OLS) estimators into the (K + 1)st equation just given, we obtain, after simpliﬁcation, the ML estimator of σ 2 as 1 ˜ ˜ ˜ σ2 = ˜ (Yi − β1 − β2 X2i − · · · − βk Xki )2 n 1 = ui ˆ2 n ˆ As noted in the text, this estimator differs from the OLS estimator σ 2 = ui /(n − k). And since the latter is an unbiased estimator of σ 2 , this conˆ2 ˜ clusion implies that the ML estimator σ 2 is a biased estimator. But, as can ˜ be readily veriﬁed, asymptotically, σ 2 is unbiased too.

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7A.5

SAS OUTPUT OF THE COBB–DOUGLAS PRODUCTION FUNCTION (7.9.4)

DEP VARIABLE: Y1 SUM OF SQUARES 0.538038 0.067153 0.605196 0.074810 10.096535 0.7409469 PARAMETER ESTIMATE −3.338455 1.498767 0.489858 STANDARD ERROR 2.449508 0.539803 0.102043 MEAN SQUARE 0.269019 0.005596531 PROB > F 0.0001

SOURCE MODEL ERROR C TOTAL

DF 2 12 14 ROOT MSE DEP MEAN C.V.

F VALUE 48.069

R-SQUARE ADJ R-SQ

0.8890 0.8705

VARIABLE INTERCEP Y2 Y3

DF 1 1 1

T FOR HO: PARAMETER = 0 −1.363 2.777 4.800

PROB > |T| 0.1979 0.0168 0.0004

COVARIANCE OF ESTIMATES COVB INTERCEP Y2 Y3 Y 16607.7 17511.3 20171.2 20932.9 20406.0 20831.6 24806.3 26465.8 27403.0 28628.7 29904.5 27508.2 29035.5 29281.5 31535.8 X2 275.5 274.4 269.7 267.0 267.8 275.0 283.0 300.7 307.5 303.7 304.7 298.6 295.5 299.0 288.1 INTERCEP 6.000091 −1.26056 0.01121951 X3 17803.7 18096.8 18271.8 19167.3 19647.6 20803.5 22076.6 23445.2 24939.0 26713.7 29957.8 31585.9 33474.5 34821.8 41794.3 Y1 9.7176 9.7706 9.9120 9.9491 9.9236 9.9442 10.1189 10.1836 10.2184 10.2622 10.3058 10.2222 10.2763 10.2847 10.3589 PORTION INDEX 1.000 89.383 351.925 Y2 −1.26056 0.2913868 −0.0384272 Y2 5.61859 5.61459 5.59731 5.58725 5.59024 5.61677 5.64545 5.70611 5.72848 5.71604 5.71933 5.69910 5.68867 5.70044 5.66331 Y3 9.7872 9.8035 9.8131 9.8610 9.8857 9.9429 10.0023 10.0624 10.1242 10.1929 10.3075 10.3605 10.4185 10.4580 10.6405 Y3 0.1121951 −0.0384272 0.01041288 Y1HAT 9.8768 9.8788 9.8576 9.8660 9.8826 9.9504 10.0225 10.1428 10.2066 10.2217 10.2827 10.2783 10.2911 10.3281 10.3619 PORTION Y2 0.0000 0.0069 0.0031 Y1RESID −0.15920 −0.10822 0.05437 0.08307 0.04097 −0.00615 0.09640 0.04077 0.01180 0.04051 0.02304 −0.05610 −0.01487 −0.04341 −0.00299

COLLINEARITY DIAGNOSTICS NUMBER 1 2 3 CONDITION EIGENVALUE 3.000 .000375451 .000024219

VARIANCE PROPORTIONS PORTION INTERCEP 0.0000 0.0491 0.9509 0.891

0.366

Y3 0.0000 0.5959 0.4040

DURBIN-WATSON d

1ST ORDER AUTOCORRELATION

Notes: Y1 = ln Y; Y2 = ln X2; Y3 = ln X3. The numbers under the heading PROB > |T| represent p values. See Chapter 10 for a discussion of collinearity diagnostics.

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MULTIPLE REGRESSION ANALYSIS: THE PROBLEM OF INFERENCE

This chapter, a continuation of Chapter 5, extends the ideas of interval estimation and hypothesis testing developed there to models involving three or more variables. Although in many ways the concepts developed in Chapter 5 can be applied straightforwardly to the multiple regression model, a few additional features are unique to such models, and it is these features that will receive more attention in this chapter.

8.1 THE NORMALITY ASSUMPTION ONCE AGAIN

We know by now that if our sole objective is point estimation of the parameters of the regression models, the method of ordinary least squares (OLS), which does not make any assumption about the probability distribution of the disturbances ui , will sufﬁce. But if our objective is estimation as well as inference, then, as argued in Chapters 4 and 5, we need to assume that the ui follow some probability distribution. For reasons already clearly spelled out, we assumed that the ui follow the normal distribution with zero mean and constant variance σ 2 . We continue to make the same assumption for multiple regression models. With the normality assumption and following the discussion of Chapters 4 and 7, we ﬁnd that the OLS estimators of the partial regression coefﬁcients, which are identical with the maximum likelihood (ML) estimators, are best linear unˆ ˆ ˆ biased estimators (BLUE).1 Moreover, the estimators β2 , β3 , and β1 are

1 ˆ ˆ ˆ With the normality assumption, the OLS estimators β2 , β3 , and β1 are minimum-variance estimators in the entire class of unbiased estimators, whether linear or not. In short, they are BUE (best unbiased estimators). See C. R. Rao, Linear Statistical Inference and Its Applications, John Wiley & Sons, New York, 1965, p. 258.

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themselves normally distributed with means equal to true β2 , β3 , and β1 and ˆ the variances given in Chapter 7. Furthermore, (n − 3)σ 2 /σ 2 follows the χ 2 distribution with n − 3 df, and the three OLS estimators are distributed inˆ dependently of σ 2 . The proofs follow the two-variable case discussed in Appendix 3. As a result and following Chapter 5, one can show that, upon ˆ replacing σ 2 by its unbiased estimator σ 2 in the computation of the standard errors, each of the following variables ˆ β1 − β1 ˆ se (β1 ) ˆ β2 − β2 ˆ se (β2 ) ˆ β3 − β3 ˆ se (β3 )

t= t= t=

(8.1.1) (8.1.2) (8.1.3)

follows the t distribution with n − 3 df. ui and hence σ 2 ˆ2 ˆ Note that the df are now n − 3 because in computing we ﬁrst need to estimate the three partial regression coefﬁcients, which therefore put three restrictions on the residual sum of squares (RSS) (following this logic in the four-variable case there will be n − 4 df, and so on). Therefore, the t distribution can be used to establish conﬁdence intervals as well as test statistical hypotheses about the true population partial regression coefﬁcients. Similarly, the χ 2 distribution can be used to test hypotheses about the true σ 2 . To demonstrate the actual mechanics, we use the following illustrative example.

8.2

EXAMPLE 8.1: CHILD MORTALITY EXAMPLE REVISITED

In Chapter 7 we regressed child mortality (CM) on per capita GNP (PGNP) and the female literacy rate (FLR) for a sample of 64 countries. The regression results given in (7.6.2) are reproduced below with some additional information: CMi = 263.6416 − se = (11.5932) t = (22.7411) p value = (0.0000)* 0.0056 PGNPi − (0.0019) (−2.8187) (0.0065) R2 = 0.7077 where * denotes extremely low value. 2.2316 FLRi (0.2099) (−10.6293) (0.0000)* ¯ R2 = 0.6981 (8.2.1)

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In Eq. (8.2.1) we have followed the format ﬁrst introduced in Eq. (5.11.1), where the ﬁgures in the ﬁrst set of parentheses are the estimated standard errors, those in the second set are the t values under the null hypothesis that the relevant population coefﬁcient has a value of zero, and those in the third are the estimated p values. Also given are R2 and adjusted R2 values. We have already interpreted this regression in Example 7.1. What about the statistical signiﬁcance of the observed results? Consider, for example, the coefﬁcient of PGNP of −0.0056. Is this coefﬁcient statistically signiﬁcant, that is, statistically different from zero? Likewise, is the coefﬁcient of FLR of −2.2316 statistically signiﬁcant? Are both coefﬁcients statistically signiﬁcant? To answer this and related questions, let us ﬁrst consider the kinds of hypothesis testing that one may encounter in the context of a multiple regression model.

8.3 HYPOTHESIS TESTING IN MULTIPLE REGRESSION: GENERAL COMMENTS

Once we go beyond the simple world of the two-variable linear regression model, hypothesis testing assumes several interesting forms, such as the following: 1. Testing hypotheses about an individual partial regression coefﬁcient (Section 8.4) 2. Testing the overall signiﬁcance of the estimated multiple regression model, that is, ﬁnding out if all the partial slope coefﬁcients are simultaneously equal to zero (Section 8.5) 3. Testing that two or more coefﬁcients are equal to one another (Section 8.6) 4. Testing that the partial regression coefﬁcients satisfy certain restrictions (Section 8.7) 5. Testing the stability of the estimated regression model over time or in different cross-sectional units (Section 8.8) 6. Testing the functional form of regression models (Section 8.9) Since testing of one or more of these types occurs so commonly in empirical analysis, we devote a section to each type.

8.4 HYPOTHESIS TESTING ABOUT INDIVIDUAL REGRESSION COEFFICIENTS

If we invoke the assumption that ui ∼ N(0, σ 2 ), then, as noted in Section 8.1, we can use the t test to test a hypothesis about any individual partial regression coefﬁcient. To illustrate the mechanics, consider the child mortality regression, (8.2.1). Let us postulate that H0 : β2 = 0 and H1 : β2 = 0

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The null hypothesis states that, with X3 (female literacy rate) held constant, X2 (PGNP) has no (linear) inﬂuence on Y (child mortality).2 To test the null hypothesis, we use the t test given in (8.1.2). Following Chapter 5 (see Table 5.1), if the computed t value exceeds the critical t value at the chosen level of signiﬁcance, we may reject the null hypothesis; otherwise, we may not reject it. For our illustrative example, using (8.1.2) and noting that β2 = 0 under the null hypothesis, we obtain t= −0.0056 = −2.8187 0.0020 (8.4.1)

as shown in Eq. (8.2.1). Notice that we have 64 observations. Therefore, the degrees of freedom in this example are 61 (why?). If you refer to the t table given in Appendix D, we do not have data corresponding to 61 df. The closest we have are for 60 df. If we use these df, and assume α, the level of signiﬁcance (i.e., the probability of committing a Type I error) of 5 percent, the critical t value is 2.0 for a two-tail test (look up tα/2 for 60 df) or 1.671 for a one-tail test (look up tα for 60 df). For our example, the alternative hypothesis is two-sided. Therefore, we use the two-tail t value. Since the computed t value of 2.8187 (in absolute terms) exceeds the critical t value of 2, we can reject the null hypothesis that PGNP has no effect on child mortality. To put it more positively, with the female literacy rate held constant, per capita GNP has a signiﬁcant (negative) effect on child mortality, as one would expect a priori. Graphically, the situation is as shown in Figure 8.1. In practice, one does not have to assume a particular value of α to conduct hypothesis testing. One can simply use the p value given in (8.2.2), f (t)

Density

t = –2.82 Critical region, 2.5% –2.0 95% Region of acceptance Critical region, 2.5% t

0

+2.0

FIGURE 8.1

The 95% conﬁdence interval for t (60 df).

2 In most empirical investigations the null hypothesis is stated in this form, that is, taking the extreme position (a kind of straw man) that there is no relationship between the dependent variable and the explanatory variable under consideration. The idea here is to ﬁnd out whether the relationship between the two is a trivial one to begin with.

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which in the present case is 0.0065. The interpretation of this p value (i.e., the exact level of signiﬁcance) is that if the null hypothesis were true, the probability of obtaining a t value of as much as 2.8187 or greater (in absolute terms) is only 0.0065 or 0.65 percent, which is indeed a small probability, much smaller than the artiﬁcially adopted value of α = 5%. This example provides us an opportunity to decide whether we want to use a one-tail or a two-tail t test. Since a priori child mortality and per capita GNP are expected to be negatively related (why?), we should use the one-tail test. That is, our null and alternative hypothesis should be: H0 : β2 < 0 and H1 : β2 ≥ 0

As the reader knows by now, we can reject the null hypothesis on the basis of the one-tail t test in the present instance. In Chapter 5 we saw the intimate connection between hypothesis testing and conﬁdence interval estimation. For our example, the 95% conﬁdence interval for β2 is: ˆ ˆ ˆ ˆ β2 − tα/2 se (β2 ) ≤ β2 ≤ β2 + tα/2 se (β2 ) which in our example becomes −0.0056 − 2(0.0020) ≤ β2 ≤ −0.0056 + 2(0.0020) that is, −0.0096 ≤ β2 ≤ −0.0016 (8.4.2)

that is, the interval, −0.0096 to −0.0016 includes the true β2 coefﬁcient with 95% conﬁdence coefﬁcient. Thus, if 100 samples of size 64 are selected and 100 conﬁdence intervals like (8.4.2) are constructed, we expect 95 of them to contain the true population parameter β2 . Since the interval (8.4.2) does not include the null-hypothesized value of zero, we can reject the null hypothesis that the true β2 is zero with 95% conﬁdence. Thus, whether we use the t test of signiﬁcance as in (8.4.1) or the conﬁdence interval estimation as in (8.4.2), we reach the same conclusion. However, this should not be surprising in view of the close connection between conﬁdence interval estimation and hypothesis testing. Following the procedure just described, we can test hypotheses about the other parameters of our child mortality regression model. The necessary data are already provided in Eq. (8.2.1). For example, suppose we want to test the hypothesis that, with the inﬂuence of PGNP held constant, the female literacy rate has no effect whatsoever on child mortality. We can conﬁdently reject this hypothesis, for under this null hypothesis the p value of obtaining an absolute t value of as much as 10.6 or greater is practically zero. Before moving on, remember that the t-testing procedure is based on the assumption that the error term ui follows the normal distribution. Although ˆ we cannot directly observe ui , we can observe their proxy, the ui , that is, the

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10

Series: Residuals Sample 1 64 Observations 64 Mean Median Maximum Minimum Std. dev. Skewness Kurtosis Jarque–Bera Probability –4.95 x 10–14 0.709227 96.80276 –84.26686 41.07980 0.227575 2.948855 0.559405 0.756009

8

6

4

2

0

– 80

– 40

0

40

80

FIGURE 8.2

Histogram of residuals from regression (8.2.1).

residuals. For our mortality regression, the histogram of the residuals is as shown in Figure 8.2. From the histogram it seems that the residuals are normally distributed. We can also compute the Jarque–Bera (JB) test of normality, as shown in Eq. (5.12.1). In our case the JB value is 0.5594 with a p value 0.76.3 Therefore, it seems that the error term in our example follows the normal distribution. Of course, keep in mind that the JB test is a large-sample test and our sample of 64 observations may not be necessarily large.

8.5 TESTING THE OVERALL SIGNIFICANCE OF THE SAMPLE REGRESSION

Throughout the previous section we were concerned with testing the significance of the estimated partial regression coefﬁcients individually, that is, under the separate hypothesis that each true population partial regression coefﬁcient was zero. But now consider the following hypothesis: H0 : β2 = β3 = 0 (8.5.1)

This null hypothesis is a joint hypothesis that β2 and β3 are jointly or simultaneously equal to zero. A test of such a hypothesis is called a test of the overall signiﬁcance of the observed or estimated regression line, that is, whether Y is linearly related to both X2 and X3 . Can the joint hypothesis in (8.5.1) be tested by testing the signiﬁcance of ˆ ˆ β2 and β3 individually as in Section 8.4? The answer is no, and the reasoning is as follows.

3 For our example, the skewness value is 0.2276 and the kurtosis value is 2.9488. Recall that for a normally distributed variable the skewness and kurtosis values are, respectively, 0 and 3.

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In testing the individual signiﬁcance of an observed partial regression coefﬁcient in Section 8.4, we assumed implicitly that each test of signiﬁcance was based on a different (i.e., independent) sample. Thus, in testing ˆ the signiﬁcance of β2 under the hypothesis that β2 = 0, it was assumed tacitly that the testing was based on a different sample from the one used in ˆ testing the signiﬁcance of β3 under the null hypothesis that β3 = 0. But to test the joint hypothesis of (8.5.1), if we use the same sample data, we shall be violating the assumption underlying the test procedure.4 The matter can be put differently: In (8.4.2) we established a 95% conﬁdence interval for β2 . But if we use the same sample data to establish a conﬁdence interval for β3 , say, with a conﬁdence coefﬁcient of 95%, we cannot assert that both β2 and β3 lie in their respective conﬁdence intervals with a probability of (1 − α)(1 − α) = (0.95)(0.95). In other words, although the statements ˆ ˆ ˆ ˆ Pr [β2 − tα/2 se (β2 ) ≤ β2 ≤ β2 + tα/2 se (β2 )] = 1 − α ˆ ˆ ˆ ˆ Pr [β3 − tα/2 se (β3 ) ≤ β3 ≤ β3 + tα/2 se (β3 )] = 1 − α are individually true, it is not true that the probability that the intervals ˆ ˆ ˆ ˆ [β2 ± tα/2 se (β2 ), β3 ± tα/2 se (β3 )] simultaneously include β2 and β3 is (1 − α)2 , because the intervals may not be independent when the same data are used to derive them. To state the matter differently,

. . . testing a series of single [individual] hypotheses is not equivalent to testing those same hypotheses jointly. The intuitive reason for this is that in a joint test of several hypotheses any single hypothesis is “affected’’ by the information in the other hypotheses.5

The upshot of the preceding argument is that for a given example (sample) only one conﬁdence interval or only one test of signiﬁcance can be obtained. How, then, does one test the simultaneous null hypothesis that β2 = β3 = 0? The answer follows.

The Analysis of Variance Approach to Testing the Overall Signiﬁcance of an Observed Multiple Regression: The F Test

For reasons just explained, we cannot use the usual t test to test the joint hypothesis that the true partial slope coefﬁcients are zero simultaneously. However, this joint hypothesis can be tested by the analysis of variance (ANOVA) technique ﬁrst introduced in Section 5.9, which can be demonstrated as follows.

4 ˆ ˆ ˆ ˆ In any given sample the cov (β2 , β3 ) may not be zero; that is, β2 and β3 may be correlated. See (7.4.17). 5 Thomas B. Fomby, R. Carter Hill, and Stanley R. Johnson, Advanced Econometric Methods, Springer-Verlag, New York, 1984, p. 37.

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TABLE 8.1

ANOVA TABLE FOR THE THREE-VARIABLE REGRESSION Source of variation Due to regression (ESS) Due to residual (RSS) Total ˆ β2 ui2 ˆ yi2 SS ˆ yi x2i + β3 yi x3i 2 n−3 n−1 df ˆ β2 MSS ˆ yi x2i + β3 2 u i2 ˆ 2 = σ ˆ n−3 yi x3i

Recall the identity ˆ yi2 = β2 ˆ yi x2i + β3 TSS = yi x3i + ESS ui ˆ2 + RSS (8.5.2)

TSS has, as usual, n − 1 df and RSS has n − 3 df for reasons already disˆ ˆ cussed. ESS has 2 df since it is a function of β2 and β3 . Therefore, following the ANOVA procedure discussed in Section 5.9, we can set up Table 8.1. Now it can be shown6 that, under the assumption of normal distribution for ui and the null hypothesis β2 = β3 = 0, the variable F= ˆ β2 ˆ yi x2i + β3 ui ˆ2 yi x3i 2 (n − 3) = ESS/df RSS/df (8.5.3)

is distributed as the F distribution with 2 and n − 3 df. What use can be made of the preceding F ratio? It can be proved7 that under the assumption that the ui ∼ N(0, σ 2 ), E ˆ2 ui = E(σ 2 ) = σ 2 ˆ n− 3 (8.5.4)

With the additional assumption that β2 = β3 = 0, it can be shown that ˆ E β2 ˆ yi x2i + β3 2 yi x3i = σ2 (8.5.5)

Therefore, if the null hypothesis is true, both (8.5.4) and (8.5.5) give identical estimates of true σ 2 . This statement should not be surprising because if there is a trivial relationship between Y and X2 and X3 , the sole source of variation in Y is due to the random forces represented by ui . If, however, the null hypothesis is false, that is, X2 and X3 deﬁnitely inﬂuence Y, the equality between (8.5.4) and (8.5.5) will not hold. In this case, the ESS will be

6 See K. A. Brownlee, Statistical Theory and Methodology in Science and Engineering, John Wiley & Sons, New York, 1960, pp. 278–280. 7 Ibid.

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relatively larger than the RSS, taking due account of their respective df. Therefore, the F value of (8.5.3) provides a test of the null hypothesis that the true slope coefﬁcients are simultaneously zero. If the F value computed from (8.5.3) exceeds the critical F value from the F table at the α percent level of signiﬁcance, we reject H0; otherwise we do not reject it. Alternatively, if the p value of the observed F is sufﬁciently low, we can reject H0. Table 8.2 summarizes the F test. Turning to our illustrative example, we obtain the ANOVA table, as shown in Table 8.3. Using (8.5.3), we obtain F= 128,681.2 = 73.8325 1742.88 (8.5.6)

The p value of obtaining an F value of as much as 73.8325 or greater is almost zero, leading to the rejection of the hypothesis that together PGNP and FLR have no effect on child mortality. If you were to use the conventional 5 percent level-of-signiﬁcance value, the critical F value for 2 df in the numerator and 60 df in the denominator (the actual df, however, are 61) is about 3.15 or about 4.98 if you were to use the 1 percent level of signiﬁcance. Obviously, the observed F of about 74 far exceeds any of these critical F values.

TABLE 8.2 A SUMMARY OF THE F STATISTIC Null hypothesis H0

2 2 σ1 = σ2 2 2 σ1 = σ2

Alternative hypothesis H1

2 2 σ1 > σ2 2 2 σ1 = σ2

Critical region Reject H0 if

2 S1 2 S2 2 S1

> Fα,ndf,ddf

> Fα/2,ndf,ddf 2 S2 or < F(1−α/2),ndf,ddf

Notes: 2 2 1. σ1 and σ2 are the two population variances. 2 2 2. S1 and S2 are the two sample variances. 3. ndf and ddf denote, respectively, the numerator and denominator df. 4. In computing the F ratio, put the larger S 2 value in the numerator. 5. The critical F values are given in the last column. The ﬁrst subscript of F is the level of signiﬁcance and the second subscript is the numerator and denominator df. 6. Note that F(1−α/2),n df,d df = 1/Fα/2,ddf,ndf.

TABLE 8.3

ANOVA TABLE FOR THE CHILD MORTALITY EXAMPLE Source of variation Due to regression Due to residuals Total SS 257,362.4 106,315.6 363,678 df 2 61 63 MSS 128,681.2 1742.88

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We can generalize the preceding F-testing procedure as follows.

Testing the Overall Signiﬁcance of a Multiple Regression: The F Test

Decision Rule. Given the k-variable regression model: Yi = β1 + β2 X2i + β3 X3i + · · · + βk Xki + ui To test the hypothesis H0 : β2 = β3 = · · · = βk = 0 (i.e., all slope coefﬁcients are simultaneously zero) versus H1: Not all slope coefﬁcients are simultaneously zero compute F= ESS/df ESS/(k − 1) = RSS/df RSS/(n − k) (8.5.7)

If F > Fα (k − 1, n − k), reject H0 ; otherwise you do not reject it, where Fα (k − 1, n − k) is the critical F value at the α level of signiﬁcance and (k − 1) numerator df and (n − k) denominator df. Alternatively, if the p value of F obtained from (8.5.7) is sufﬁciently low, one can reject H0 . Needless to say, in the three-variable case (Y and X2, X3) k is 3, in the fourvariable case k is 4, and so on. In passing, note that most regression packages routinely calculate the F value (given in the analysis of variance table) along with the usual regression output, such as the estimated coefﬁcients, their standard errors, t values, etc. The null hypothesis for the t computation is usually assumed to be βi = 0. Individual versus Joint Testing of Hypotheses. In Section 8.4 we discussed the test of signiﬁcance of a single regression coefﬁcient and in Section 8.5 we have discussed the joint or overall test of signiﬁcance of the estimated regression (i.e., all slope coefﬁcients are simultaneously equal to zero). We reiterate that these tests are different. Thus, on the basis of the t test or conﬁdence interval (of Section 8.4) it is possible to accept the hypothesis that a particular slope coefﬁcient, βk , is zero, and yet reject the joint hypothesis that all slope coefﬁcients are zero.

The lesson to be learned is that the joint “message’’ of individual conﬁdence intervals is no substitute for a joint conﬁdence region [implied by the F test] in performing joint tests of hypotheses and making joint conﬁdence statements.8

8

Fomby et al., op. cit., p. 42.

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An Important Relationship between R 2 and F

There is an intimate relationship between the coefﬁcient of determination R2 and the F test used in the analysis of variance. Assuming the normal distribution for the disturbances ui and the null hypothesis that β2 = β3 = 0, we have seen that F= ESS/2 RSS/(n − 3) (8.5.8)

is distributed as the F distribution with 2 and n − 3 df. More generally, in the k-variable case (including intercept), if we assume that the disturbances are normally distributed and that the null hypothesis is H0 : β2 = β3 = · · · = βk = 0 then it follows that F= ESS/(k − 1) RSS/(n − k) (8.5.7) = (8.5.10) (8.5.9)

follows the F distribution with k − 1 and n − k df. (Note: The total number of parameters to be estimated is k, of which one is the intercept term.) Let us manipulate (8.5.10) as follows: n − k ESS k − 1 RSS n− k ESS k − 1 TSS − ESS n − k ESS/TSS k − 1 1 − (ESS/TSS) n − k R2 k − 1 1 − R2 R2 /(k − 1) (1 − R2 )/(n − k) (8.5.11)

F= = = = =

where use is made of the deﬁnition R2 = ESS/TSS. Equation (8.5.11) shows how F and R2 are related. These two vary directly. When R2 = 0, F is zero ipso facto. The larger the R2, the greater the F value. In the limit, when R2 = 1, F is inﬁnite. Thus the F test, which is a measure of the overall signiﬁcance of the estimated regression, is also a test of signiﬁcance of R2. In other words, testing the null hypothesis (8.5.9) is equivalent to testing the null hypothesis that (the population) R2 is zero.

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TABLE 8.4

ANOVA TABLE IN TERMS OF R 2 Source of variation Due to regression Due to residuals Total R 2( SS y i2 ) y i2 ) 2 n−3 n−1 df R 2( MSS* y i2 )/2 y i2 )/(n − 3)

(1 − R 2)( y i2

(1 − R 2)(

y i2 because it drops out, as shown in (8.5.12).

*Note that in computing the F value there is no need to multiply R 2 and (1 − R 2) by

For the three-variable case (8.5.11) becomes F= R2 /2 (1 − R2 )/(n − 3) (8.5.12)

By virtue of the close connection between F and R2, the ANOVA Table 8.1 can be recast as Table 8.4. For our illustrative example, using (8.5.12) we obtain: F= 0.7077/2 = 73.8726 (1 − 0.7077)/61

which is about the same as obtained before, except for the rounding errors. One advantage of the F test expressed in terms of R2 is its ease of computation: All that one needs to know is the R2 value. Therefore, the overall F test of signiﬁcance given in (8.5.7) can be recast in terms of R2 as shown in Table 8.4.

Testing the Overall Signiﬁcance of a Multiple Regression in Terms of R 2

Decision Rule. Testing the overall signiﬁcance of a regression in terms of R2: Alternative but equivalent test to (8.5.7). Given the k-variable regression model: Yi = βi + β2 X2i + β3 X3i + · · · + βx Xki + ui To test the hypothesis H0 : β2 = β3 = · · · = βk = 0 versus H1: Not all slope coefﬁcients are simultaneously zero compute F= R2 /(k − 1) (1 − R2 )/(n − k) (8.5.13)

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If F > Fα(k−1,n−k) , reject H0 ; otherwise you may accept H0 where Fα(k−1,n−k) is the critical F value at the α level of signiﬁcance and (k − 1) numerator df and (n − k) denominator df. Alternatively, if the p value of F obtained from (8.5.13) is sufﬁciently low, reject H0. Before moving on, return to Example 7.5 in Chapter 7. From regression (7.10.7) we observe that RGDP (relative per capita GDP) and RGDP squared explain only about 5.3 percent of the variation in GDPG (GDP growth rate) in a sample of 119 countries. This R2 of 0.053 seems a “low” value. Is it really statistically different from zero? How do we ﬁnd that out? Recall our earlier discussion in “An Important Relationship between R2 and F” about the relationship between R2 and the F value as given in (8.5.11) or (8.5.12) for the speciﬁc case of two regressors. As noted, if R2 is zero, then F is zero ipso facto, which will be the case if the regressors have no impact whatsoever on the regressand. Therefore, if we insert R2 = 0.053 into formula (8.5.12), we obtain F= 0.053/2 = 3.2475 (1 − 0.053)/116 (8.5.13)

Under the null hypothesis that R2 = 0, the preceding F value follows the F distribution with 2 and 116 df in the numerator, respectively. (Note: There are 119 observations and two regressors.) From the F table we see that this F value is signiﬁcant at about the 5 percent level; the p value is actually 0.0425. Therefore, we can reject the null hypothesis that the two regressors have no impact on the regressand, notwithstanding the fact that the R2 is only 0.053. This example brings out an important empirical observation that in cross-sectional data involving several observations, one generally obtains low R2 because of the diversity of the cross-sectional units. Therefore, one should not be surprised or worried about ﬁnding low R2’s in cross-sectional regressions. What is relevant is that the model is correctly speciﬁed, that the regressors have the correct (i.e., theoretically expected) signs, and that (hopefully) the regression coefﬁcients are statistically signiﬁcant. The reader should check that individually both the regressors in (7.10.7) are statistically signiﬁcant at the 5 percent or better level (i.e., lower than 5 percent).

The “Incremental” or “Marginal” Contribution of an Explanatory Variable

In Chapter 7 we stated that generally we cannot allocate the R2 value among the various regressors. In our child mortality example we found that the R2 was 0.7077 but we cannot say what part of this value is due to the regressor PGNP and what part is due to female literacy rate (FLR) because of possible correlation between the two regressors in the sample at hand. We can shed more light on this using the analysis of covariance technique.

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For our illustrative example we found that individually X2 (PGNP) and X3 (FLR) were statistically signiﬁcant on the basis of (separate) t tests. We have also found that on the basis of the F test collectively both the regressors have a signiﬁcant effect on the regressand Y (child mortality). Now suppose we introduce PGNP and FLR sequentially; that is, we ﬁrst regress child mortality on PGNP and assess its signiﬁcance and then add FLR to the model to ﬁnd out whether it contributes anything (of course, the order in which PGNP and FLR enter can be reversed). By contribution we mean whether the addition of the variable to the model increases ESS (and hence R2) “signiﬁcantly” in relation to the RSS. This contribution may appropriately be called the incremental, or marginal, contribution of an explanatory variable. The topic of incremental contribution is an important one in practice. In most empirical investigations the researcher may not be completely sure whether it is worth adding an X variable to the model knowing that several other X variables are already present in the model. One does not wish to include variable(s) that contribute very little toward ESS. By the same token, one does not want to exclude variable(s) that substantially increase ESS. But how does one decide whether an X variable signiﬁcantly reduces RSS? The analysis of variance technique can be easily extended to answer this question. Suppose we ﬁrst regress child mortality on PGNP and obtain the following regression: CMi = 157.4244 − t = (15.9894) p value = (0.0000) 0.0114 PGNP (−3.5156) (0.0008) r = 0.1662

2

(8.5.14)

adj r 2 = 0.1528

As these results show, PGNP has a signiﬁcant effect on CM. The ANOVA table corresponding to the preceding regression is given in Table 8.5. Assuming the disturbances ui are normally distributed and the hypothesis that PGNP has no effect on CM, we obtain the F value of F= 60,449.5 = 12.3598 4890.7822 (8.5.15)

TABLE 8.5

ANOVA TABLE FOR REGRESSION (8.5.14) Source of variation ESS (due to PGNP) RSS Total SS 60,449.5 303,228.5 363,678 df 1 62 63 MSS 604,495.1 4890.7822

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which follows the F distribution with 1 and 62 df. This F value is highly signiﬁcant, as the computed p value is 0.0008. Thus, as before, we reject the hypothesis that PGNP has no effect on CM. Incidentally, note that t2 = (−3.5156)2 = 12.3594, which is approximately the same as the F value of (8.5.15), where the t value is obtained from (8.5.14). But this should not be surprising in view of the fact that the square of the t statistic with n df is equal to the F value with 1 df in the numerator and n df in the denominator, a relationship ﬁrst established in Chapter 5. Note that in the present example, n = 64. Having run the regression (8.5.14), let us suppose we decide to add FLR to the model and obtain the multiple regression (8.2.1). The questions we want to answer are: 1. What is the marginal, or incremental, contribution of FLR, knowing that PGNP is already in the model and that it is signiﬁcantly related to CM? 2. Is the incremental contribution of FLR statistically signiﬁcant? 3. What is the criterion for adding variables to the model? The preceding questions can be answered by the ANOVA technique. To see this, let us construct Table 8.6. In this table X2 refers to PGNP and X3 refers to FLR. To assess the incremental contribution of X3 after allowing for the contribution of X2, we form F= = = Q2 /df Q4 /df ESSnew − ESSold /number of new regressors RSSnew /df ( = n − number of parameters in the new model) Q2 /1 for our example Q4 /12 (8.5.16)

TABLE 8.6

ANOVA TABLE TO ASSESS INCREMENTAL CONTRIBUTION OF A VARIABLE(S) Source of variation ESS due to X2 alone ESS due to the addition of X 3 ESS due to both X 2, X 3 RSS Total ˆ2 Q1 = β1 2 SS x2 2 1 1 yi x3i 2 n−3 n−1 df MSS Q1 1 Q2 1 Q3 2 Q4 n−3

Q2 = Q3 − Q1 ˆ Q3 = β2 ˆ yi x2i + β3

Q4 = Q5 − Q3 Q5 = y i2

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TABLE 8.7

ANOVA TABLE FOR THE ILLUSTRATIVE EXAMPLE: INCREMENTAL ANALYSIS Source of variation ESS due to PGNP ESS due to the addition of FLR ESS due to PGNP and FLR RSS Total SS 60,449.5 196,912.9 257,362.4 106,315.6 363,678 df 1 1 2 61 63 MSS 60,449.5 196,912.9 128,681.2 1742.8786

where ESSnew = ESS under the new model (i.e., after adding the new regressors = Q3), ESSold = ESS under the old model ( = Q1 ) and RSSnew = RSS under the new model (i.e., after taking into account all the regressors = Q4). For our illustrative example the results are as shown in Table 8.7. Now applying (8.5.16), we obtain: F= 196,912.9 = 112.9814 1742.8786 (8.5.17)

Under the usual assumptions, this F value follows the F distribution with 1 and 62 df. The reader should check that this F value is highly signiﬁcant, suggesting that addition of FLR to the model signiﬁcantly increases ESS and hence the R2 value. Therefore, FLR should be added to the model. Again, note that if you square the value of the FLR coefﬁcient in the multiple regression (8.2.1), which is (−10.6293)2, you will obtain the F value of (8.5.17), save for the rounding errors. Incidentally, the F ratio of (8.5.16) can be recast by using the R2 values only, as we did in (8.5.13). As exercise 8.2 shows, the F ratio of (8.5.16) is equivalent to the following F ratio:9 F= =

2 R2 − Rold new

df df

2 R2 − Rold new

1− 1−

R2 new

number of new regressors

R2 new

df ( = n − number of parameters in the new model) (8.5.18)

This F ratio follows the F distribution with the appropriate numerator and denominator df, 1 and 61 in our illustrative example. 2 For our example, R2 = 0.7077 [from Eq. (8.2.1)] and Rold = 0.1662 [from new Eq. (8.5.14)]. Therefore, F= (0.7077 − 0.1662)/1 = 113.05 (1 − 0.7077)/61 (8.5.19)

9 The following F test is a special case of the more general F test given in (8.7.9) or (8.7.10) in Sec. 8.7.

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which is about the same as that obtained from (8.5.17), except for the rounding errors. This F is highly signiﬁcant, reinforcing our earlier ﬁnding that the variable FLR belongs in the model. A cautionary note: If you use the R2 version of the F test given in (8.5.11), make sure that the dependent variable in the new and the old models is the same. If they are different, use the F test given in (8.5.16). When to Add a New Variable. The F-test procedure just outlined provides a formal method of deciding whether a variable should be added to a regression model. Often researchers are faced with the task of choosing from several competing models involving the same dependent variable but with different explanatory variables. As a matter of ad hoc choice (because very often the theoretical foundation of the analysis is weak), these researchers frequently choose the model that gives the highest adjusted R2 . ¯ Therefore, if the inclusion of a variable increases R2 , it is retained in the model although it does not reduce RSS signiﬁcantly in the statistical sense. The question then becomes: When does the adjusted R2 increase? It can be ¯ shown that R2 will increase if the t value of the coefﬁcient of the newly added variable is larger than 1 in absolute value, where the t value is computed under the hypothesis that the population value of the said coefﬁcient is zero [i.e., the t value computed from (5.3.2) under the hypothesis that the true β ¯ value is zero].10 The preceding criterion can also be stated differently: R2 will increase with the addition of an extra explanatory variable only if the F( = t 2 ) value of that variable exceeds 1. Applying either criterion, the FLR variable in our child mortality example ¯ with a t value of −10.6293 or an F value of 112.9814 should increase R2 , ¯ 2 increases from which indeed it does—when FLR is added to the model, R 0.1528 to 0.6981. When to Add a Group of Variables. Can we develop a similar rule for deciding whether it is worth adding (or dropping) a group of variables from a model? The answer should be apparent from (8.5.18): If adding (dropping) a group of variables to the model gives an F value greater (less) than 1, R2 will increase (decrease). Of course, from (8.5.18) one can easily ﬁnd out whether the addition (subtraction) of a group of variables signiﬁcantly increases (decreases) the explanatory power of a regression model.

8.6

TESTING THE EQUALITY OF TWO REGRESSION COEFFICIENTS

Suppose in the multiple regression Yi = β1 + β2 X2i + β3 X3i + β4 X4i + ui (8.6.1)

10 For proof, see Dennis J. Aigner, Basic Econometrics, Prentice Hall, Englewood Cliffs, N.J., 1971, pp. 91–92.

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we want to test the hypotheses H0 : β3 = β4 H1 : β3 = β4 or or (β3 − β4 ) = 0 (β3 − β4 ) = 0 (8.6.2)

that is, the two slope coefﬁcients β3 and β4 are equal. Such a null hypothesis is of practical importance. For example, let (8.6.1) represent the demand function for a commodity where Y = amount of a commodity demanded, X2 = price of the commodity, X3 = income of the consumer, and X4 = wealth of the consumer. The null hypothesis in this case means that the income and wealth coefﬁcients are the same. Or, if Yi and the X’s are expressed in logarithmic form, the null hypothesis in (8.6.2) implies that the income and wealth elasticities of consumption are the same. (Why?) How do we test such a null hypothesis? Under the classical assumptions, it can be shown that t= ˆ ˆ (β3 − β4 ) − (β3 − β4 ) ˆ ˆ se (β3 − β4 ) (8.6.3)

follows the t distribution with (n − 4) df because (8.6.1) is a four-variable model or, more generally, with (n − k) df, where k is the total number of paˆ ˆ rameters estimated, including the constant term. The se (β3 − β4 ) is obtained from the following well-known formula (see Appendix A for details): ˆ ˆ se (β3 − β4 ) = ˆ ˆ ˆ ˆ var (β3 ) + var (β4 ) − 2 cov (β3 , β4 ) (8.6.4)

ˆ ˆ If we substitute the null hypothesis and the expression for the se (β3 − β4 ) into (8.6.3), our test statistic becomes t= ˆ ˆ β3 − β4 ˆ ˆ ˆ ˆ var (β3 ) + var (β4 ) − 2 cov (β3 , β4 ) (8.6.5)

Now the testing procedure involves the following steps: ˆ ˆ 1. Estimate β3 and β4 . Any standard computer package can do that. 2. Most standard computer packages routinely compute the variances and covariances of the estimated parameters.11 From these estimates the standard error in the denominator of (8.6.5) can be easily obtained. 3. Obtain the t ratio from (8.6.5). Note the null hypothesis in the present case is (β3 − β4 ) = 0. 4. If the t variable computed from (8.6.5) exceeds the critical t value at the designated level of signiﬁcance for given df, then you can reject the null hypothesis; otherwise, you do not reject it. Alternatively, if the p value of the

11 The algebraic expression for the covariance formula is rather involved. Appendix C provides a compact expression for it, however, using matrix notation.

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t statistic from (8.6.5) is reasonably low, one can reject the null hypothesis. Note that the lower the p value, the greater the evidence against the null hypothesis. Therefore, when we say that a p value is low or reasonably low, we mean that it is less than the signiﬁcance level, such as 10, 5, or 1 percent. Some personal judgment is involved in this decision.

EXAMPLE 8.2 THE CUBIC COST FUNCTION REVISITED Recall the cubic total cost function estimated in Section 7.10, which for convenience is reproduced below: ˆ Yi = 141.7667 + 63.4777Xi − 12.9615X i2 + 0.9396Xi3 se = (6.3753) (4.7786) (0.9857) R = 0.9983

2

(0.0591)

(7.10.6)

ˆ ˆ cov ( β3 , β4 ) = −0.0576;

where Y is total cost and X is output, and where the ﬁgures in parentheses are the estimated standard errors. Suppose we want to test the hypothesis that the coefﬁcients of the X 2 and X 3 terms in the cubic cost function are the same, that is, β3 = β4 or (β3 − β4 ) = 0. In the regression (7.10.6) we have all the necessary output to conduct the t test of (8.6.5). The actual mechanics are as follows: ˆ ˆ β3 − β4 t= ˆ3 ) + var ( β4 ) − 2 cov ( β3 , β4 ) ˆ ˆ ˆ var ( β = = −12.9615 − 0.9396 (0.9867) 2 + (0.0591) 2 − 2(−0.0576) −13.9011 = −13.3130 1.0442 (8.6.6)

The reader can verify that for 6 df (why?) the observed t value exceeds the critical t value even at the 0.002 (or 0.2 percent) level of signiﬁcance (two-tail test); the p value is extremely small, 0.000006. Hence we can reject the hypothesis that the coefﬁcients of X 2 and X 3 in the cubic cost function are identical.

8.7 RESTRICTED LEAST SQUARES: TESTING LINEAR EQUALITY RESTRICTIONS

There are occasions where economic theory may suggest that the coefﬁcients in a regression model satisfy some linear equality restrictions. For instance, consider the Cobb–Douglas production function: Yi = β1 X2i2 X3i3 eui β β

(7.9.1) = (8.7.1)

where Y = output, X2 = labor input, and X3 = capital input. Written in log form, the equation becomes ln Yi = β0 + β2 ln X2i + β3 ln X3i + ui where β0 = ln β1 . (8.7.2)

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Now if there are constant returns to scale (equiproportional change in output for an equiproportional change in the inputs), economic theory would suggest that β2 + β3 = 1 (8.7.3) which is an example of a linear equality restriction.12 How does one ﬁnd out if there are constant returns to scale, that is, if the restriction (8.7.3) is valid? There are two approaches.

The t-Test Approach

The simplest procedure is to estimate (8.7.2) in the usual manner without taking into account the restriction (8.7.3) explicitly. This is called the unrestricted or unconstrained regression. Having estimated β2 and β3 (say, by OLS method), a test of the hypothesis or restriction (8.7.3) can be conducted by the t test of (8.6.3), namely, t= = ˆ ˆ (β2 + β3 ) − (β2 + β3 ) ˆ ˆ se (β2 + β3 ) ˆ ˆ (β2 + β3 ) − 1 ˆ ˆ ˆ ˆ var (β2 ) + var (β3 ) + 2 cov (β2 β3 )

(8.7.4)

where (β2 + β3 ) = 1 under the null hypothesis and where the denominator is ˆ ˆ the standard error of (β2 + β3 ). Then following Section 8.6, if the t value computed from (8.7.4) exceeds the critical t value at the chosen level of signiﬁcance, we reject the hypothesis of constant returns to scale; otherwise we do not reject it.

The F-Test Approach: Restricted Least Squares

The preceding t test is a kind of postmortem examination because we try to ﬁnd out whether the linear restriction is satisﬁed after estimating the “unrestricted’’ regression. A direct approach would be to incorporate the restriction (8.7.3) into the estimating procedure at the outset. In the present example, this procedure can be done easily. From (8.7.3) we see that β2 = 1 − β3 or β3 = 1 − β2 (8.7.6) Therefore, using either of these equalities, we can eliminate one of the β coefﬁcients in (8.7.2) and estimate the resulting equation. Thus, if we use (8.7.5), we can write the Cobb–Douglas production function as ln Yi = β0 + (1 − β3 ) ln X2i + β3 ln X3i + ui = β0 + ln X2i + β3 (ln X3i − ln X2i ) + ui

12 If we had β2 + β3 < 1, this relation would be an example of a linear inequality restriction. To handle such restrictions, one needs to use mathematical programming techniques.

(8.7.5)

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or (ln Yi − ln X2i ) = β0 + β3 (ln X3i − ln X2i ) + ui or ln (Yi /X2i ) = β0 + β3 ln (X3i /X2i ) + ui (8.7.8) where (Yi /X2i ) = output/labor ratio and (X3i /X2i ) = capital labor ratio, quantities of great economic importance. Notice how the original equation (8.7.2) is transformed. Once we estimate β3 from (8.7.7) or (8.7.8), β2 can be easily estimated from the relation (8.7.5). Needless to say, this procedure will guarantee that the sum of the estimated coefﬁcients of the two inputs will equal 1. The procedure outlined in (8.7.7) or (8.7.8) is known as restricted least squares (RLS). This procedure can be generalized to models containing any number of explanatory variables and more than one linear equality restriction. The generalization can be found in Theil.13 (See also general F testing below.) How do we compare the unrestricted and restricted least-squares regressions? In other words, how do we know that, say, the restriction (8.7.3) is valid? This question can be answered by applying the F test as follows. Let u2 = RSS of the unrestricted regression (8.7.2) ˆ UR u2 = RSS of the restricted regression (8.7.7) ˆR m = number of linear restrictions (1 in the present example) k = number of parameters in the unrestricted regression n = number of observations Then, F= = (RSSR − RSSUR )/m RSSUR /(n − k) u2 − ˆR u2 ˆ UR m u2 (n − k) ˆ UR (8.7.7)

(8.7.9)

follows the F distribution with m, (n − k) df. (Note: UR and R stand for unrestricted and restricted, respectively.) The F test above can also be expressed in terms of R2 as follows: F=

2 2 RUR − RR

m

1−

2 RUR

(n − k)

(8.7.10)

2 2 where RUR and RR are, respectively, the R2 values obtained from the unrestricted and restricted regressions, that is, from the regressions (8.7.2) and

13

Henri Theil, Principles of Econometrics, John Wiley & Sons, New York, 1971, pp. 43–45.

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(8.7.7). It should be noted that R2 ≥ R2 UR R and u2 ≤ ˆ UR u2 ˆR (8.7.12) (8.7.11)

In exercise 8.4 you are asked to justify these statements. A Cautionary Note: In using (8.7.10) keep in mind that if the dependent variable in the restricted and unrestricted models is not the same, R2 and UR R2 are not directly comparable. In that case, use the procedure described in R Chapter 7 to render the two R2 values comparable (see Example 8.3 below) or use the F test given in (8.7.9).

EXAMPLE 8.3 THE COBB–DOUGLAS PRODUCTION FUNCTION FOR THE MEXICAN ECONOMY, 1955–1974 By way of illustrating the preceding discussion consider the data given in Table 8.8. Attempting to ﬁt the Cobb–Douglas production function to these data, yielded the following results: ln GDPt = −1.6524 + 0.3397 ln Labort + 0.8460 ln Capitalt t = (−2.7259) p value = (0.0144) (1.8295) (0.0849)

2

(8.7.13)

(9.0625) (0.0000) RSSUR = 0.0136

R = 0.9951

where RSSUR is the unrestricted RSS, as we have put no restrictions on estimating (8.7.13). We have already seen in Chapter 7 how to interpret the coefﬁcients of the Cobb– Douglas production function. As you can see, the output/labor elasticity is about 0.34 and the output/capital elasticity is about 0.85. If we add these coefﬁcients, we obtain 1.19, suggesting that perhaps the Mexican economy during the stated time period was experiencing increasing returns to scale. Of course, we do not know if 1.19 is statistically different from 1. To see if that is the case, let us impose the restriction of constant returns to scale, which gives the following regression: ln (GDP/Labor)t = −0.4947 + t = (−4.0612) p value = (0.0007) 1.0153 ln (Capital/Labor)t (28.1056) (0.0000)

2 R R = 0.9777

(8.7.14)

RSSR = 0.0166

where RSSR is the restricted RSS, for we have imposed the restriction that there are constant returns to scale. (Continued)

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EXAMPLE 9.3 (Continued) Since the dependent variable in the preceding two regressions is different, we have to use the F test given in (8.7.9). We have the necessary data to obtain the F value. F= = (RSSR − RSSUR )/m RSSUR /(n − k) (0.0166 − 0.0136)/1 (0.0136)/(20 − 3)

= 3.75 Note in the present case m = 1, as we have imposed only one restriction and (n − k) is 17, since we have 20 observations and three parameters in the unrestricted regression. This F value follows the F distribution with 1 df in the numerator and 17 df in the denominator. The reader can easily check that this F value is not signiﬁcant at the 5% level. (See Appendix D, Table D.3.) The conclusion then is that the Mexican economy was probably characterized by constant returns to scale over the sample period and therefore there may be no harm in using the restricted regression given in (8.7.14). As this regression shows, if capital/labor ratio increased by 1 percent, on average, labor productivity went up by about 1 percent. TABLE 8.8 REAL GDP, EMPLOYMENT, AND REAL FIXED CAPITAL—MEXICO Year 1955 1956 1957 1958 1959 1960 1961 1962 1963 1964 1965 1966 1967 1968 1969 1970 1971 1972 1973 1974 GDP* 114043 120410 129187 134705 139960 150511 157897 165286 178491 199457 212323 226977 241194 260881 277498 296530 306712 329030 354057 374977 Employment† 8310 8529 8738 8952 9171 9569 9527 9662 10334 10981 11746 11521 11540 12066 12297 12955 13338 13738 15924 14154 Fixed capital‡ 182113 193749 205192 215130 225021 237026 248897 260661 275466 295378 315715 337642 363599 391847 422382 455049 484677 520553 561531 609825

*Millions of 1960 pesos; † Thousands of people; ‡ Millions of 1960 pesos. Source: Victor J. Elias, Sources of Growth: A Study of Seven Latin American Economies, International Center for Economic Growth, ICS Press, San Francisco, 1992. Data from Tables E5, E12, and E14.

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General F Testing14

The F test given in (8.7.10) or its equivalent (8.7.9) provides a general method of testing hypotheses about one or more parameters of the kvariable regression model: Yi = β1 + β2 X2i + β3 X3i + · · · + βk Xki + ui (8.7.15)

The F test of (8.5.16) or the t test of (8.6.3) is but a speciﬁc application of (8.7.10). Thus, hypotheses such as H0 : β2 = β3 H0 : β3 + β4 + β5 = 3 (8.7.16) (8.7.17)

which involve some linear restrictions on the parameters of the k-variable model, or hypotheses such as H0 : β3 = β4 = β5 = β6 = 0 (8.7.18)

which imply that some regressors are absent from the model, can all be tested by the F test of (8.7.10). From the discussion in Sections 8.5 and 8.7, the reader will have noticed that the general strategy of F testing is this: There is a larger model, the unconstrained model (8.7.15), and then there is a smaller model, the constrained or restricted model, which is obtained from the larger model by deleting some variables from it, e.g., (8.7.18), or by putting some linear restrictions on one or more coefﬁcients of the larger model, e.g., (8.7.16) or (8.7.17). We then ﬁt the unconstrained and constrained models to the data and 2 2 obtain the respective coefﬁcients of determination, namely, RUR and RR . We note the df in the unconstrained model ( = n − k) and also note the df in the constrained model ( = m), m being the number of linear restriction [e.g., 1 in (8.7.16) or (8.7.18)] or the number of regressors omitted from the model [e.g., m = 4 if (8.7.18) holds, since four regressors are assumed to be absent from the model]. We then compute the F ratio as indicated in (8.7.9) or (8.7.10) and use this Decision Rule: If the computed F exceeds Fα (m, n − k), where Fα (m, n − k) is the critical F at the α level of signiﬁcance, we reject the null hypothesis: otherwise we do not reject it.

214 If one is using the maximum likelihood approach to estimation, then a test similar to the one discussed shortly is the likelihood ratio test, which is slightly involved and is therefore discussed in the appendix to the chapter. For further discussion, see Theil, op. cit., pp. 179–184.

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Let us illustrate:

EXAMPLE 8.4 THE DEMAND FOR CHICKEN IN THE UNITED STATES, 1960–1982 In exercise 7.19, among other things, you were asked to consider the following demand function for chicken: ln Yt = β1 + β2 ln X2t + β3 ln X3t + β4 ln X4t + β5 ln X5t + ui (8.7.19)

where Y = per capita consumption of chicken, lb, X2 = real disposable per capita income, $, X3 = real retail price of chicken per lb, ¢, X4 = real retail price of pork per lb, ¢, and X5 = real retail price of beef per lb, ¢. In this model β2, β3, β4, and β5 are, respectively, the income, own-price, cross-price (pork), and cross-price (beef) elasticities. (Why?) According to economic theory, β2 > 0 β3 < 0 β4 > 0, < 0, = 0, β5 > 0, < 0, = 0, if chicken and pork are competing products if chicken and pork are complementary products if chicken and pork are unrelated products if chicken and beef are competing products if they are complementary products if they are unrelated products (8.7.20)

Suppose someone maintains that chicken and pork and beef are unrelated products in the sense that chicken consumption is not affected by the prices of pork and beef. In short, H0: β4 = β5 = 0 Therefore, the constrained regression becomes lnYt = β1 + β2 ln X2t + β3 ln X3t + ut Equation (8.7.19) is of course the unconstrained regression. Using the data given in exercise 7.19, we obtain the following: Unconstrained regression lnYt = 2.1898 + 0.3425 ln X2t − 0.5046 ln X3t + 0.1485 ln X4t + 0.0911 ln X5t (0.1557) (0.0833) (0.1109) (0.0997)

2 R UR

(8.7.21)

(8.7.22)

(0.1007) = 0.9823 (8.7.23)

Constrained regression lnYt = 2.0328 + 0.4515 ln X2t − 0.3772 ln X3t (0.1162) (0.0247) (0.0635)

2 RR

(8.7.24)

= 0.9801 (Continued)

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EXAMPLE 8.4 (Continued) where the ﬁgures in parentheses are the estimated standard errors. Note: The R 2 values of (8.7.23) and (8.7.24) are comparable since the dependent variable in the two models is the same. Now the F ratio to test the hypothesis (8.7.21) is F=

2 2 RUR − RR /m 2 1 − RUR /(n − k)

(8.7.10)

The value of m in the present case is 2, since there are two restrictions involved: β4 = 0 and β5 = 0. The denominator df, (n − k), is 18, since n = 23 and k = 5 (5 β coefﬁcients). Therefore, the F ratio is F= (0.9823 − 0.9801)/2 (1 − 0.9823)/18

(8.7.25)

= 1.1224 which has the F distribution with 2 and 18 df. At 5 percent, clearly this F value is not statistically signiﬁcant [F0.5 (2,18) = 3.55]. The p value is 0.3472. Therefore, there is no reason to reject the null hypothesis—the demand for chicken does not depend on pork and beef prices. In short, we can accept the constrained regression (8.7.24) as representing the demand function for chicken. Notice that the demand function satisﬁes a priori economic expectations in that the ownprice elasticity is negative and that the income elasticity is positive. However, the estimated price elasticity, in absolute value, is statistically less than unity, implying that the demand for chicken is price inelastic. (Why?) Also, the income elasticity, although positive, is also statistically less than unity, suggesting that chicken is not a luxury item; by convention, an item is said to be a luxury item if its income elasticity is greater than one.

8.8 TESTING FOR STRUCTURAL OR PARAMETER STABILITY OF REGRESSION MODELS: THE CHOW TEST

When we use a regression model involving time series data, it may happen that there is a structural change in the relationship between the regressand Y and the regressors. By structural change, we mean that the values of the parameters of the model do not remain the same through the entire time period. Sometime the structural change may be due to external forces (e.g., the oil embargoes imposed by the OPEC oil cartel in 1973 and 1979 or the Gulf War of 1990–1991), or due to policy changes (such as the switch from a ﬁxed exchange-rate system to a ﬂexible exchange-rate system around 1973) or action taken by Congress (e.g., the tax changes initiated by President Reagan in his two terms in ofﬁce or changes in the minimum wage rate) or to a variety of other causes. How do we ﬁnd out that a structural change has in fact occurred? To be speciﬁc, consider the data given in Table 8.9. This table gives data on disposable personal income and personal savings, in billions of dollars, for the United States for the period 1970–1995. Suppose we want to estimate a

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TABLE 8.9

SAVINGS AND PERSONAL DISPOSABLE INCOME (BILLIONS OF DOLLARS), UNITED STATES, 1970–1995 Observation 1970 1971 1972 1973 1974 1975 1976 1977 1978 1979 1980 1981 1982 Savings 61.0 68.6 63.6 89.6 97.6 104.4 96.4 92.5 112.6 130.1 161.8 199.1 205.5 Income 727.1 790.2 855.3 965.0 1054.2 1159.2 1273.0 1401.4 1580.1 1769.5 1973.3 2200.2 2347.3 Observation 1983 1984 1985 1986 1987 1988 1989 1990 1991 1992 1993 1994 1995 Savings 167.0 235.7 206.2 196.5 168.4 189.1 187.8 208.7 246.4 272.6 214.4 189.4 249.3 Income 2522.4 2810.0 3002.0 3187.6 3363.1 3640.8 3894.5 4166.8 4343.7 4613.7 4790.2 5021.7 5320.8

Source: Economic Report of the President, 1997, Table B-28, p. 332.

simple savings function that relates savings (Y) to disposable personal income DPI (X). Since we have the data, we can obtain an OLS regression of Y on X. But if we do that, we are maintaining that the relationship between savings and DPI has not changed much over the span of 26 years. That may be a tall assumption. For example, it is well known that in 1982 the United States suffered its worst peacetime recession. The civilian unemployment rate that year reached 9.7 percent, the highest since 1948. An event such as this might disturb the relationship between savings and DPI. To see if this happened, let us divide our sample data into two time periods: 1970–1981 and 1982–1995, the pre- and post-1982 recession periods. Now we have three possible regressions: Time period 1970–1981: Yt = λ1 + λ2 Xt + u1t Time period 1982–1995: Yt = γ1 + γ2 Xt + u2t Time period 1970–1995: Yt = α1 + α2 Xt + ut n1 = 12 n2 = 14 (8.8.1) (8.8.2)

n = (n1 + n2 ) = 26 (8.8.3)

Regression (8.8.3) assumes that there is no difference between the two time periods and therefore estimates the relationship between savings and DPI for the entire time period consisting of 26 observations. In other words, this regression assumes that the intercept as well as the slope coefﬁcient remains the same over the entire period; that is, there is no structural change. If this is in fact the situation, then α1 = λ1 = γ1 and α2 = λ2 = γ2 . Regressions (8.8.1) and (8.8.2) assume that the regressions in the two time periods are different; that is, the intercept and the slope coefﬁcients are different, as indicated by the subscripted parameters. In the preceding regressions, the u’s represent the error terms and the n’s represent the number of observations.

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For the data given in Table 8.9, the empirical counterparts of the preceding three regressions are as follows: ˆ Yt = 1.0161 + 0.0803 Xt t = (0.0873) R = 0.9021

2

(9.6015) df = 10

(8.8.1a)

RSS1 = 1785.032

ˆ Yt = 153.4947 + 0.0148Xt t = (4.6922) R = 0.2971

2

(1.7707) df = 12

(8.8.2a)

RSS2 = 10,005.22

ˆ Yt = 62.4226 + 0.0376 Xt + · · · t = (4.8917) R = 0.7672

2

(8.8937) + · · · df = 24

(8.8.3a)

RSS3 = 23,248.30

In the preceding regressions, RSS denotes the residual sum of squares, and the ﬁgures in parentheses are the estimated t values. A look at the estimated regressions suggests that the relationship between savings and DPI is not the same in the two subperiods. The slope in the preceding savings-income regressions represents the marginal propensity to save (MPS), that is, the (mean) change in savings as a result of a dollar’s increase in disposable personal income. In the period 1970–1981 the MPS was about 0.08, whereas in the period 1982–1995 it was about 0.02. Whether this change was due to the economic policies pursued by President Reagan is hard to say. This further suggests that the pooled regression (8.8.3a)—that is, the one that pools all the 26 observations and runs a common regression, disregarding possible differences in the two subperiods may not be appropriate. Of course, the preceding statements need to be supported by appropriate statistical test(s). Incidentally, the scattergrams and the estimated regression lines are as shown in Figure 8.3. Now the possible differences, that is, structural changes, may be caused by differences in the intercept or the slope coefﬁcient or both. How do we ﬁnd that out? A visual feeling about this can be obtained as shown in Figure 8.2. But it would be useful to have a formal test. This is where the Chow test comes in handy.15 This test assumes that: 1. u1t ∼ N(0, σ 2 ) and u2t ∼ N(0, σ 2 ). That is, the error terms in the subperiod regressions are normally distributed with the same (homoscedastic) variance σ 2 .

15 Gregory C. Chow, “Tests of Equality Between Sets of Coefﬁcients in Two Linear Regressions,” Econometrica, vol. 28, no. 3, 1960, pp. 591–605.

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1970–1981 250 280 260 200 240 Savings 150 Savings 220 200 100 180 50 500 1000 1500 2000 Income 2500 160 2000 3000

1982–1995

4000 5000 Income

6000

FIGURE 8.3

2. The two error terms u1t and u2t are independently distributed. The mechanics of the Chow test are as follows: 1. Estimate regression (8.8.3), which is appropriate if there is no parameter instability, and obtain RSS3 with df = (n1 + n2 − k), where k is the number of parameters estimated, 2 in the present case. For our example RSS3 = 23,248.30. We call RSS3 the restricted residual sum of squares (RSSR) because it is obtained by imposing the restrictions that λ1 = γ1 and λ2 = γ2 , that is, the subperiod regressions are not different. 2. Estimate (8.8.1) and obtain its residual sum of squares, RSS1, with df = (n1 − k). In our example, RSS1 = 1785.032 and df = 10. 3. Estimate (8.8.2) and obtain its residual sum of squares, RSS2, with df = (n2 − k). In our example, RSS2 = 10,005.22 with df = 12. 4. Since the two sets of samples are deemed independent, we can add RSS1 and RSS2 to obtain what may be called the unrestricted residual sum of squares (RSSUR), that is, obtain: RSSUR = RSS1 + RSS2 In the present case, RSSUR = (1785.032 + 10,005.22) = 11,790.252 5. Now the idea behind the Chow test is that if in fact there is no structural change [i.e., regressions (8.8.1) and (8.8.2) are essentially the same], then the RSSR and RSSUR should not be statistically different. Therefore, if we form the following ratio: (RSSR − RSSUR )/k F= ∼ F[k,(n1 +n2 −2k)] (8.8.4) (RSSUR )/(n1 + n2 − 2k) with df = (n1 + n2 − 2k)

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then Chow has shown that under the null hypothesis the regressions (8.8.1) and (8.8.2) are (statistically) the same (i.e., no structural change or break) and the F ratio given above follows the F distribution with k and (n1 + n2 − 2k) df in the numerator and denominator, respectively. 6. Therefore, we do not reject the null hypothesis of parameter stability (i.e., no structural change) if the computed F value in an application does not exceed the critical F value obtained from the F table at the chosen level of signiﬁcance (or the p value). In this case we may be justiﬁed in using the pooled (restricted?) regression (8.8.3). Contrarily, if the computed F value exceeds the critical F value, we reject the hypothesis of parameter stability and conclude that the regressions (8.8.1) and (8.8.2) are different, in which case the pooled regression (8.8.3) is of dubious value, to say the least. Returning to our example, we ﬁnd that F= (23,248.30 − 11,790.252)/2 (11,790.252)/22 = 10.69

(8.8.5)

From the F tables, we ﬁnd that for 2 and 22 df the 1 percent critical F value is 5.72. Therefore, the probability of obtaining an F value of as much as or greater than 10.69 is much smaller than 1 percent; actually the p value is only 0.00057. The Chow test therefore seems to support our earlier hunch that the savings–income relation has undergone a structural change in the United States over the period 1970–1995, assuming that the assumptions underlying the test are fulﬁlled. We will have more to say about this shortly. Incidentally, note that the Chow test can be easily generalized to handle cases of more than one structural break. For example, if we believe that the savings–income relation changed after President Clinton took ofﬁce in January 1992, we could divide our sample into three periods: 1970–1981, 1982–1991, 1992–1995, and carry out the Chow test. Of course, we will have four RSS terms, one for each subperiod and one for the pooled data. But the logic of the test remains the same. Data through 2001 are now available to extend the last period to 2001. There are some caveats about the Chow test that must be kept in mind: 1. The assumptions underlying the test must be fulﬁlled. For example, one should ﬁnd out if the error variances in the regressions (8.8.1) and (8.8.2) are the same. We will discuss this point shortly. 2. The Chow test will tell us only if the two regressions (8.8.1) and (8.8.2) are different, without telling us whether the difference is on account of the intercepts, or the slopes, or both. But in Chapter 9, on dummy variables, we will see how we can answer this question. 3. The Chow test assumes that we know the point(s) of structural break. In our example, we assumed it to be in 1982. However, if it is not possible to

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determine when the structural change actually took place, we may have to use other methods.16 Before we leave the Chow test and our savings–income regression, let us examine one of the assumptions underlying the Chow test, namely, that the error variances in the two periods are the same. Since we cannot observe the true error variances, we can obtain their estimates from the RSS given in the regressions (8.8.1a) and (8.8.2a), namely, σ1 = ˆ2 σ2 = ˆ2 RSS1 1785.032 = = 178.5032 n1 − 2 10 RSS2 10,005.22 = = 833.7683 n2 − 2 14 − 2 (8.8.6)

(8.8.7)

Notice that, since there are two parameters estimated in each equation, we deduct 2 from the number of observations to obtain the df. Given the asˆ2 ˆ2 sumptions underlying the Chow test, σ1 and σ2 are unbiased estimators of the true variances in the two subperiods. As a result, it can be shown that if 2 2 σ1 = σ2 —that is, the variances in the two subpopulations are the same (as assumed by the Chow test)—then it can be shown that σ1 σ1 ˆ2 2 σ1 σ2 ˆ2 2 ∼ F(n1 −k),(n2 −k) (8.8.8)

follows the F distribution with (n1 − k) and (n2 − k) df in the numerator and the denominator, respectively, in our example k = 2, since there are only two parameters in each subregression. 2 2 Of course, σ1 = σ2 , the preceding F test reduces to computing F= σ1 ˆ2 σ2 ˆ2 (8.8.9)

Note: By convention we put the larger of the two estimated variances in the numerator. (See Appendix A for the details of the F and other probability distributions.) Computing this F in an application and comparing it with the critical F value with the appropriate df, one can decide to reject or not reject the null hypothesis that the variances in the two subpopulations are the same. If the null hypothesis is not rejected, then one can use the Chow test. Returning to our savings–income regression, we obtain the following result: 833.7683 F= = 4.6701 (8.8.10) 178.5032

16 For a detailed discussion, see William H. Greene, Econometric Analysis, 4th ed., Prentice Hall, Englewood Cliffs, N.J., 2000, pp. 293–297.

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Under the null hypothesis of equality of variances in the two subpopulations, this F value follows the F distribution with 12 and 10 df, in the numerator and denominator, respectively. (Note: We have put the larger of the two estimated variances in the numerator.) From the F tables in Appendix D, we see that the 5 and 1 percent critical F values for 12 and 10 df are 2.91 and 4.71, respectively. The computed F value is signiﬁcant at the 5 percent level and is almost signiﬁcant at the 1 percent level. Thus, our conclusion would be that the two subpopulation variances are not the same and, therefore, strictly speaking we should not use the Chow test. Our purpose here has been to demonstrate the mechanics of the Chow test, which is used popularly in applied work. If the error variances in the two subpopulations are heteroscedastic, the Chow test can be modiﬁed. But the procedure is beyond the scope of this book.17 Another point we made earlier was that the Chow test is sensitive to the choice of the time at which the regression parameters might have changed. In our example, we assumed that the change probably took place in the recession year of 1982. If we had assumed it to be 1981, when Ronald Reagan began his presidency, we might ﬁnd that the computed F value is different. As a matter of fact, in exercise 8.34 the reader is asked to check this out. If we do not want to choose the point at which the break in the underlying relationship might have occurred, we could choose alternative methods, such as the recursive residual test. We will take this topic up in Chapter 13, the chapter on model speciﬁcation analysis.

8.9 PREDICTION WITH MULTIPLE REGRESSION

In Section 5.10 we showed how the estimated two-variable regression model can be used for (1) mean prediction, that is, predicting the point on the population regression function (PRF), as well as for (2) individual prediction, that is, predicting an individual value of Y given the value of the regressor X = X0, where X0 is the speciﬁed numerical value of X. The estimated multiple regression too can be used for similar purposes, and the procedure for doing that is a straightforward extension of the twovariable case, except the formulas for estimating the variances and standard errors of the forecast value [comparable to (5.10.2) and (5.10.6) of the twovariable model] are rather involved and are better handled by the matrix methods discussed in Appendix C. Of course, most standard regression packages can do this routinely, so there is no need to look up the matrix formulation. It is given in Appendix C for the beneﬁt of the mathematically inclined students. This appendix also gives a fully worked out example.

17 For a discussion of the Chow test under heteroscedasticity, see William H. Greene, Econometric Analysis, 4th ed., Prentice Hall, Englewood Cliffs, N.J., 2000, pp. 292–293, and Adrian C. Darnell, A Dictionary of Econometrics, Edward Elgar, U.K., 1994, p. 51.

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*8.10 THE TROIKA OF HYPOTHESIS TESTS: THE LIKELIHOOD RATIO (LR), WALD (W), AND LAGRANGE MULTIPLIER (LM) TESTS18

In this and the previous chapters we have, by and large, used the t, F, and chi-square tests to test a variety of hypotheses in the context of linear (in-parameter) regression models. But once we go beyond the somewhat comfortable world of linear regression models, we need method(s) to test hypotheses that can handle regression models, linear or not. The well-known trinity of likelihood, Wald, and Lagrange multiplier tests can accomplish this purpose. The interesting thing to note is that asymptotically (i.e., in large samples) all three tests are equivalent in that the test statistic associated with each of these tests follows the chi-square distribution. Although we will discuss the likelihood ratio test in the appendix to this chapter, in general we will not use these tests in this textbook for the pragmatic reason that in small, or ﬁnite, samples, which is unfortunately what most researchers deal with, the F test that we have used so far will sufﬁce. As Davidson and MacKinnon note:

For linear regression models, with or without normal errors, there is of course no need to look at LM, W and LR at all, since no information is gained from doing so over and above what is already contained in F.19

*8.11 TESTING THE FUNCTIONAL FORM OF REGRESSION: CHOOSING BETWEEN LINEAR AND LOG–LINEAR REGRESSION MODELS

The choice between a linear regression model (the regressand is a linear function of the regressors) or a log–linear regression model (the log of the regressand is a function of the logs of the regressors) is a perennial question in empirical analysis. We can use a test proposed by MacKinnon, White, and Davidson, which for brevity we call the MWD test to choose between the two models.20 To illustrate this test, assume the following H0: Linear Model: Y is a linear function of regressors, the X’s. H1: Log–Linear Model: ln Y is a linear function of logs of regressors, the logs of X’s. where, as usual, H0 and H1 denote the null and alternative hypotheses.

*Optional. 18 For an accessible discussion, see A. Buse, “The Likelihood Ratio, Wald and Lagrange Multiplier Tests: An Expository Note,’’ American Statistician, vol. 36, 1982, pp. 153–157. 19 Russell Davidson and James G. MacKinnon, Estimation and Inference in Econometrics, Oxford University Press, New York, 1993, p. 456. 20 J. MacKinnon, H. White, and R. Davidson, “Tests for Model Speciﬁcation in the Presence of Alternative Hypothesis; Some Further Results.” Journal of Econometrics, vol. 21, 1983, pp. 53–70. A similar test is proposed in A. K. Bera and C. M. Jarque, “Model Speciﬁcation Tests: A Simultaneous Approach,” Journal of Econometrics, vol. 20, 1982, pp. 59–82.

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The MWD test involves the following steps21: Step I: Estimate the linear model and obtain the estimated Y values. Call ˆ them Yf (i.e., Y ). Step: II: Estimate the log–linear model and obtain the estimated ln Y values; call them ln f (i.e., ln Y). Step III: Obtain Z1 = (ln Y f − ln f ). Step IV: Regress Y on X’s and Z1 obtained in Step III. Reject H0 if the coﬁcient of Z1 is statistically signiﬁcant by the usual t test. Step V: Obtain Z2 = (antilog of ln f − Y f ). Step VI: Regress log of Y on the logs of X’s and Z2. Reject H1 if the coefﬁcient of Z2 is statistically signiﬁcant by the usual t test. Although the MWD test seems involved, the logic of the test is quite simple. If the linear model is in fact the correct model, the constructed variable Z1 should not be statistically signiﬁcant in Step IV, for in that case the estimated Y values from the linear model and those estimated from the log–linear model (after taking their antilog values for comparative purposes) should not be different. The same comment applies to the alternative hypothesis H1.

EXAMPLE 8.5 THE DEMAND FOR ROSES Refer to exercise 7.16 where we have presented data on the demand for roses in the Detroit metropolitan area for the period 1971–II to 1975–II. For illustrative purposes, we will consider the demand for roses as a function only of the prices of roses and carnations, leaving out the income variable for the time being. Now we consider the following models: Linear model: Log–linear model: Yt = α1 + α2X2t + α3X3t + ut lnYt = β1 + β2 lnX2t + β3 lnX3t + ut (8.11.1) (8.11.2)

where Y is the quantity of roses in dozens, X2 is the average wholesale price of roses ($/dozen), and X3 is the average wholesale price of carnations ($/dozen). A priori, α2 and β2 are expected to be negative (why?), and α3 and β3 are expected to be positive (why?). As we know, the slope coefﬁcients in the log–linear model are elasticity coefﬁcients. The regression results are as follows: ˆ Y t = 9734.2176 − 3782.1956X2t + 2815.2515X3t t= (3.3705) (−6.6069) (2.9712) F = 21.84 ln Yt = 9.2278 − 1.7607 lnX2t + 1.3398 lnX3t (−5.9044) (2.5407) F = 17.50

2

(8.11.3)

2

R = 0.77096

t = (16.2349)

(8.11.4) R = 0.7292 (Continued)

21 This discussion is based on William H. Greene, ET. The Econometrics Toolkit Version 3, Econometric Software, Bellport, New York, 1992, pp. 245–246.

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EXAMPLE 8.5 (Continued) As these results show, both the linear and the log–linear models seem to ﬁt the data reasonably well: The parameters have the expected signs and the t and R 2 values are statistically signiﬁcant. To decide between these models on the basis of the MWD test, we ﬁrst test the hypothesis that the true model is linear. Then, following Step IV of the test, we obtain the following regression: ˆ Y t = 9727.5685 − 3783.0623X2t + 2817.7157X3t + 85.2319Z1t t= (3.2178) (−6.3337) F = 13.44

2

(2.8366) R = 0.7707

(0.0207)

(8.11.5)

Since the coefﬁcient of Z1 is not statistically signiﬁcant (the p value of the estimated t is 0.98), we do not reject the hypothesis that the true model is linear. Suppose we switch gears and assume that the true model is log–linear. Following step VI of the MWD test, we obtain the following regression results: ln Y t = 9.1486 − 1.9699 ln Xt + 1.5891 ln X2t − (−6.4189) F = 14.17 (3.0728) R = 0.7798

2

0.0013Z2t (−1.6612) (8.11.6)

t = (17.0825)

The coefﬁcient of Z 2 is statistically signiﬁcant at about the 12 percent level (p value is 0.1225). Therefore, we can reject the hypothesis that the true model is log–linear at this level of signiﬁcance. Of course, if one sticks to the conventional 1 or 5 percent signiﬁcance levels, then one cannot reject the hypothesis that the true model is log–linear. As this example shows, it is quite possible that in a given situation we cannot reject either of the speciﬁcations.

8.12

SUMMARY AND CONCLUSIONS

1. This chapter extended and reﬁned the ideas of interval estimation and hypothesis testing ﬁrst introduced in Chapter 5 in the context of the two-variable linear regression model. 2. In a multiple regression, testing the individual signiﬁcance of a partial regression coefﬁcient (using the t test) and testing the overall signiﬁcance of the regression (i.e., H0: all partial slope coefﬁcients are zero or R2 = 0) are not the same thing. 3. In particular, the ﬁnding that one or more partial regression coefﬁcients are statistically insigniﬁcant on the basis of the individual t test does not mean that all partial regression coefﬁcients are also (collectively) statistically insigniﬁcant. The latter hypothesis can be tested only by the F test. 4. The F test is versatile in that it can test a variety of hypotheses, such as whether (1) an individual regression coefﬁcient is statistically signiﬁcant, (2) all partial slope coefﬁcients are zero, (3) two or more coefﬁcients are statistically equal, (4) the coefﬁcients satisfy some linear restrictions, and (5) there is structural stability of the regression model. 5. As in the two-variable case, the multiple regression model can be used for the purpose of mean and or individual prediction.

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EXERCISES

Questions 8.1. Suppose you want to study the behavior of sales of a product, say, automobiles over a number of years and suppose someone suggests you try the following models:

Yt = β0 + β1 t Yt = α0 + α1 t + α2 t 2

8.2. 8.3. 8.4. 8.5.

where Yt = sales at time t and t = time, measured in years. The ﬁrst model postulates that sales is a linear function of time, whereas the second model states that it is a quadratic function of time. a. Discuss the properties of these models. b. How would you decide between the two models? c. In what situations will the quadratic model be useful? d. Try to obtain data on automobile sales in the United States over the past 20 years and see which of the models ﬁts the data better. Show that the F ratio of (8.5.16) is equal to the F ratio of (8.5.18). (Hint: ESS/TSS = R2.) Show that F tests of (8.5.18) and (8.7.10) are equivalent. Establish statements (8.7.11) and (8.7.12). Consider the Cobb–Douglas production function

Y = β1 L β2 K β3

(1)

where Y = output, L = labor input, and K = capital input. Dividing (1) through by K, we get

(Y/K ) = β1 (L/K )β2 K β2 +β3 −1

(2)

Taking the natural log of (2) and adding the error term, we obtain ln (Y/K ) = β0 + β2 ln (L/K ) + (β2 + β3 − 1) ln K + ui

(3)

where β0 = ln β1 . a. Suppose you had data to run the regression (3). How would you test the hypothesis that there are constant returns to scale, i.e., (β2 + β3) = 1? b. If there are constant returns to scale, how would you interpret regression (3)? c. Does it make any difference whether we divide (1) by L rather than by K? 8.6. Critical values of R2 when true R2 = 0. Equation (8.5.11) gave the relationship between F and R2 under the hypothesis that all partial slope coefﬁcients are simultaneously equal to zero (i.e., R2 = 0). Just as we can ﬁnd the critical F value at the α level of signiﬁcance from the F table, we can ﬁnd the critical R2 value from the following relation:

R2 = (k − 1)F (k − 1)F + (n − k)

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where k is the number of parameters in the regression model including the intercept and where F is the critical F value at the α level of signiﬁcance. If the observed R2 exceeds the critical R2 obtained from the preceding formula, we can reject the hypothesis that the true R2 is zero. Establish the preceding formula and ﬁnd out the critical R2 value (at α = 5 percent) for the regression (8.2.1). 8.7. From annual data for the years 1968–1987, the following regression results were obtained:

ˆ Yt = −859.92 + 0.6470 X 2t − 23.195 X 3t ˆ Yt = −261.09 + 0.2452 X 2t R2 = 0.9776 R2 = 0.9388

(1) (2)

where Y = U.S. expenditure on imported goods, billions of 1982 dollars, X2 = personal disposable income, billions of 1982 dollars, and X3 = trend variable. True or false: The standard error of X3 in (1) is 4.2750. Show your calculations. (Hint: Use the relationship between R2, F, and t.) 8.8. Suppose in the regression ln (Yi / X 2i ) = α1 + α2 ln X 2i + α3 ln X 3i + ui

the values of the regression coefﬁcients and their standard errors are known.* From this knowledge, how would you estimate the parameters and standard errors of the following regression model? ln Yi = β1 + β2 ln X 2i + β3 ln X 3i + ui

8.9. Assume the following:

Yi = β1 + β2 X 2i + β3 X 3i + β4 X 2i X 3i + ui

where Y is personal consumption expenditure, X 2 is personal income, and X 3 is personal wealth.† The term ( X 2i X 3i ) is known as the interaction term. What is meant by this expression? How would you test the hypothesis that the marginal propensity to consume (MPC) (i.e., β2 ) is independent of the wealth of the consumer? 8.10. You are given the following regression results:

ˆ Yt = 16,899 t= t=

− 2978.5X2t (−4.7280) − 3782.2X2t + (−6.6070) 2815X3t (2.9712)

R2 = 0.6149 R2 = 0.7706

(8.5152)

ˆ Yt = 9734.2

(3.3705)

Can you ﬁnd out the sample size underlying these results? (Hint: Recall the relationship between R2, F, and t values.)

* Adapted from Peter Kennedy, A Guide to Econometrics, the MIT Press, 3d ed., Cambridge, Mass., 1992, p. 310. † Ibid., p. 327.

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8.11. Based on our discussion of individual and joint tests of hypothesis based, respectively, on the t and F tests, which of the following situations are likely? 1. Reject the joint null on the basis of the F statistic, but do not reject each separate null on the basis of the individual t tests. 2. Reject the joint null on the basis of the F statistic, reject one individual hypothesis on the basis of the t test, and do not reject the other individual hypotheses on the basis of the t test. 3. Reject the joint null hypothesis on the basis of the F statistic, and reject each separate null hypothesis on the basis of the individual t tests. 4. Do not reject the joint null on the basis of the F statistic, and do not reject each separate null on the basis of individual t tests. 5. Do not reject the joint null on the basis of the F statistic, reject one individual hypothesis on the basis of a t test, and do not reject the other individual hypotheses on the basis of the t test. 6. Do not reject the joint null on the basis of the F statistic, but reject each separate null on the basis of individual t tests.* Problems 8.12. Refer to exercise 7.21. a. What are the real income and interest rate elasticities of real cash balances? b. Are the preceding elasticities statistically signiﬁcant individually? c. Test the overall signiﬁcance of the estimated regression. d. Is the income elasticity of demand for real cash balances signiﬁcantly different from unity? e. Should the interest rate variable be retained in the model? Why? 8.13. From the data for 46 states in the United States for 1992, Baltagi obtained the following regression results†: log C = 4.30 − 1.34 log P + 0.17 log Y se = (0.91) (0.32) (0.20)

¯ R2 = 0.27

where C = cigarette consumption, packs per year P = real price per pack Y = real disposable income per capita a. What is the elasticity of demand for cigarettes with respect to price? Is it statistically signiﬁcant? If so, is it statistically different from one? b. What is the income elasticity of demand for cigarettes? Is it statistically signiﬁcant? If not, what might be the reasons for it? c. How would you retrieve R2 from the adjusted R2 given above?

* Quoted from Ernst R. Berndt, The Practice of Econometrics: Classic and Contemporary, Addison-Wesley, Reading, Mass., 1991, p. 79. † See Badi H. Baltagi, Econometrics, Springer-Verlag, New York, 1998, p. 111.

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8.14. From a sample of 209 ﬁrms, Wooldridge obtained the following regression results*: log (salary ) = 4.32 + 0.280 log (sales) + 0.0174 roe + 0.00024 ros se = (0.32) (0.035) (0.0041) (0.00054) R2 = 0.283 where salary = salary of CEO sales = annual ﬁrm sales roe = return on equity in percent ros = return on ﬁrm’s stock and where ﬁgures in the parentheses are the estimated standard errors. a. Interpret the preceding regression taking into account any prior expectations that you may have about the signs of the various coefﬁcients. b. Which of the coefﬁcients are individually statistically signiﬁcant at the 5 percent level? c. What is the overall signiﬁcance of the regression? Which test do you use? And why? d. Can you interpret the coefﬁcients of roe and ros as elasticity coefﬁcients? Why or why not? 8.15. Assuming that Y and X 2 , X 3 , . . . , X k are jointly normally distributed and assuming that the null hypothesis is that the population partial correlations are individually equal to zero, R. A. Fisher has shown that t= √ r 1 2.3 4...k n − k − 2

2 1 − r 1 2.3 4...k

follows the t distribution with n − k − 2 df, where k is the kth-order partial correlation coefﬁcient and where n is the total number of observations. (Note: r1 2.3 is a ﬁrst-order partial correlation coefﬁcient, r1 2.3 4 is a secondorder partial correlation coefﬁcient, and so on.) Refer to exercise 7.2. Assuming Y and X 2 and X 3 to be jointly normally distributed, compute the three partial correlations r1 2.3, r1 3.2, and r2 3.1 and test their signiﬁcance under the hypothesis that the corresponding population correlations are individually equal to zero. 8.16. In studying the demand for farm tractors in the United States for the periods 1921–1941 and 1948–1957, Griliches† obtained the following results: logYt = constant − 0.519 log X2t − 4.933 log X3t

R2 = 0.793

(0.231)

(0.477)

where Yt = value of stock of tractors on farms as of January 1, in 1935–1939 dollars, X 2 = index of prices paid for tractors divided by an

* See Jeffrey M. Wooldridge, Introductory Econometrics, South-Western Publishing Co., 2000, pp. 154–155. † Z. Griliches, “The Demand for a Durable Input: Farm Tractors in the United States, 1921–1957,” in The Demand for Durable Goods, Arnold C. Harberger (ed.), The University of Chicago Press, Chicago, 1960, Table 1, p. 192.

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index of prices received for all crops at time t − 1, X 3 = interest rate prevailing in year t − 1 , and the estimated standard errors are given in the parentheses. a. Interpret the preceding regression. b. Are the estimated slope coefﬁcients individually statistically signiﬁcant? Are they signiﬁcantly different from unity? c. Use the analysis of variance technique to test the signiﬁcance of the overall regression. Hint: Use the R2 variant of the ANOVA technique. d. How would you compute the interest-rate elasticity of demand for farm tractors? e. How would you test the signiﬁcance of estimated R2? 8.17. Consider the following wage-determination equation for the British economy* for the period 1950–1969:

ˆ Wt = 8.582 + 0.364(PF)t + 0.004(PF)t−1 − 2.560Ut

(1.129)

(0.080)

(0.072) R = 0.873

2

(0.658) df = 15

where W = wages and salaries per employee PF = prices of ﬁnal output at factor cost U = unemployment in Great Britain as a percentage of the total number of employees of Great Britain t = time (The ﬁgures in the parentheses are the estimated standard errors.) a. Interpret the preceding equation. b. Are the estimated coefﬁcients individually signiﬁcant? c. What is the rationale for the introduction of (PF)t−1? d. Should the variable (PF)t−1 be dropped from the model? Why? e. How would you compute the elasticity of wages and salaries per employee with respect to the unemployment rate U? 8.18. A variation of the wage-determination equation given in exercise 8.17 is as follows†:

ˆ Wt = 1.073 + 5.288Vt − 0.116Xt + 0.054Mt + 0.046Mt−1

(0.797) where

(0.812)

(0.111)

(0.022) R = 0.934

2

(0.019) df = 14

W = wages and salaries per employee V = unﬁlled job vacancies in Great Britain as a percentage of the total number of employees in Great Britain X = gross domestic product per person employed M = import prices Mt−1 = import prices in the previous (or lagged) year

(The estimated standard errors are given in the parentheses.)

* Taken from Prices and Earnings in 1951–1969: An Econometric Assessment, Dept. of Employment, HMSO, 1971, Eq. (19), p. 35. † Ibid., Eq. (67), p. 37.

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a. Interpret the preceding equation. b. Which of the estimated coefﬁcients are individually statistically signiﬁcant? c. What is the rationale for the introduction of the X variable? A priori, is the sign of X expected to be negative? d. What is the purpose of introducing both Mt and Mt−1 in the model? e. Which of the variables may be dropped from the model? Why? f. Test the overall signiﬁcance of the observed regression. 8.19. For the demand for chicken function estimated in (8.7.24), is the estimated income elasticity equal to 1? Is the price elasticity equal to −1? 8.20. For the demand function (8.7.24) how would you test the hypothesis that the income elasticity is equal in value but opposite in sign to the price elasticity of demand? Show the necessary calculations. [Note:

ˆ ˆ cov (β2 , β3 ) = −0.00142.]

8.21. Refer to the demand for roses function of exercise 7.16. Conﬁning your considerations to the logarithmic speciﬁcation, a. What is the estimated own-price elasticity of demand (i.e., elasticity with respect to the price of roses)? b. Is it statistically signiﬁcant? c. If so, is it signiﬁcantly different from unity? d. A priori, what are the expected signs of X 3 (price of carnations) and X 4 (income)? Are the empirical results in accord with these expectations? e. If the coefﬁcients of X 3 and X 4 are statistically insigniﬁcant, what may be the reasons? 8.22. Refer to exercise 7.17 relating to wildcat activity. a. Is each of the estimated slope coefﬁcients individually statistically signiﬁcant at the 5 percent level? b. Would you reject the hypothesis that R2 = 0 ? c. What is the instantaneous rate of growth of wildcat activity over the period 1948–1978? The corresponding compound rate of growth? 8.23. Refer to the U.S. defense budget outlay regression estimated in exercise 7.18. a. Comment generally on the estimated regression results. b. Set up the ANOVA table and test the hypothesis that all the partial slope coefﬁcients are zero. 8.24. The following is known as the transcendental production function (TPF), a generalization of the well-known Cobb–Douglas production function:

Yi = β1 L β2 kβ3 e β4 L+β5 K

where Y = output, L = labor input, and K = capital input. After taking logarithms and adding the stochastic disturbance term, we obtain the stochastic TPF as ln Yi = β0 + β2 ln L i + β3 ln K i + β4 L i + β5 K i + ui

where β0 = ln β1 . a. What are the properties of this function? b. For the TPF to reduce to the Cobb–Douglas production function, what must be the values of β4 and β5?

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c. If you had the data, how would you go about ﬁnding out whether the TPF reduces to the Cobb–Douglas production function? What testing procedure would you use? d. See if the TPF ﬁts the data given in Table 8.8. Show your calculations. 8.25. Energy prices and capital formation: United States, 1948–1978. To test the hypothesis that a rise in the price of energy relative to output leads to a decline in the productivity of existing capital and labor resources, John A. Tatom estimated the following production function for the United States for the quarterly period 1948–I to 1978–II*: ln (y/k) = 1.5492 + (16.33) + 0.7135 ln (h/k) − (21.69) 0.0045t (15.86) where y = real output in the private business sector k = a measure of the ﬂow of capital services h = person hours in the private business sector Pe = producer price index for fuel and related products P = private business sector price deﬂator t = time. The numbers in parentheses are t statistics. a. Do the results support the author’s hypothesis? b. Between 1972 and 1977 the relative price of energy, (Pe/P), increased by 60 percent. From the estimated regression, what is the loss in productivity? c. After allowing for the changes in (h/k) and (Pe/P), what has been the trend rate of growth of productivity over the sample period? d. How would you interpret the coefﬁcient value of 0.7135? e. Does the fact that each estimated partial slope coefﬁcient is individually statistically signiﬁcant (why?) mean we can reject the hypothesis that R2 = 0? Why or why not? 8.26. The demand for cable. Table 8.10 gives data used by a telephone cable manufacturer to predict sales to a major customer for the period 1968–1983.† The variables in the table are deﬁned as follows: Y = annual sales in MPF, million paired feet X2 = gross national product (GNP), $, billions X3 = housing starts, thousands of units X4 = unemployment rate, % X5 = prime rate lagged 6 months X6 = Customer line gains, %

* See his “Energy Prices and Capital Formation: 1972–1977,” Review, Federal Reserve Bank of St. Louis, vol. 61, no. 5, May 1979, p. 4. † I am indebted to Daniel J. Reardon for collecting and processing the data.

0.1081 ln (Pe/P) (−6.42)

R = 0.98

2

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TABLE 8.10

REGRESSION VARIABLES X2, GNP 1051.8 1078.8 1075.3 1107.5 1171.1 1235.0 1217.8 1202.3 1271.0 1332.7 1399.2 1431.6 1480.7 1510.3 1492.2 1535.4 X3, housing starts 1503.6 1486.7 1434.8 2035.6 2360.8 2043.9 1331.9 1160.0 1535.0 1961.8 2009.3 1721.9 1298.0 1100.0 1039.0 1200.0 X4, unemployment, % 3.6 3.5 5.0 6.0 5.6 4.9 5.6 8.5 7.7 7.0 6.0 6.0 7.2 7.6 9.2 8.8 X5, prime rate lag, 6 mos. 5.8 6.7 8.4 6.2 5.4 5.9 9.4 9.4 7.2 6.6 7.6 10.6 14.9 16.6 17.5 16.0 X6, customer line gains, % 5.9 4.5 4.2 4.2 4.9 5.0 4.1 3.4 4.2 4.5 3.9 4.4 3.9 3.1 0.6 1.5 Y, total plastic purchases (MPF) 5873 7852 8189 7497 8534 8688 7270 5020 6035 7425 9400 9350 6540 7675 7419 7923

Year 1968 1969 1970 1971 1972 1973 1974 1975 1976 1977 1978 1979 1980 1981 1982 1983

You are to consider the following model:

Yi = β1 + β2 X 2t + β3 X 3t + β4 X 4t + β5 X 5t + β6 X 6t + ut

Estimate the preceding regression. What are the expected signs of the coefﬁcients of this model? Are the empirical results in accordance with prior expectations? Are the estimated partial regression coefﬁcients individually statistically signiﬁcant at the 5 percent level of signiﬁcance? e. Suppose you ﬁrst regress Y on X2, X3, and X4 only and then decide to add the variables X5 and X6. How would you ﬁnd out if it is worth adding the variables X5 and X6? Which test do you use? Show the necessary calculations. 8.27. Marc Nerlove has estimated the following cost function for electricity generation*:

Y = AX β P α 1 P α 2 P α 3 u

a. b. c. d.

(1)

where Y = total cost of production X = output in kilowatt hours P1 = price of labor input P2 = price of capital input P3 = price of fuel u = disturbance term

* Marc Nerlove, “Returns to Scale in Electric Supply,’’ in Carl Christ, ed., Measurement in Economics, Stanford University Press, Palo Alto, Calif., 1963. The notation has been changed.

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Theoretically, the sum of the price elasticities is expected to be unity, i.e.,

(α1 + α2 + α3 ) = 1. By imposing this restriction, the preceding cost func-

tion can be written as

(Y/ P3 ) = AX β ( P1 / P3 )α1 ( P2 / P3 )α 2 u

(2)

In other words, (1) is an unrestricted and (2) is the restricted cost function. On the basis of a sample of 29 medium-sized ﬁrms, and after logarithmic transformation, Nerlove obtained the following regression results ln Yi = −4.93

+ 0.94 ln Xi + 0.31 ln P1 (0.11) (0.07) (0.23) RSS = 0.336

se = (1.96) (0.29)

(3)

−0.26 ln P2 + 0.44 ln P3 ln (Y/ P3 ) = −6.55 + 0.91 ln X + 0.51 ln (P1/P3) + 0.09 ln (P2/P3)

se = (0.16)

(0.11)

(0.19)

(0.16)

RSS = 0.364 (4)

a. Interpret Eqs. (3) and (4). b. How would you ﬁnd out if the restriction (α1 + α2 + α3 ) = 1 is valid? Show your calculations. 8.28. Estimating the capital asset pricing model (CAPM). In Section 6.1 we considered brieﬂy the well-known capital asset pricing model of modern portfolio theory. In empirical analysis, the CAPM is estimated in two stages. Stage I (Time-series regression). For each of the N securities included in the sample, we run the following regression over time:

ˆ R it = αi + βi R mt + eit ˆ

(1)

where Rit and Rmt are the rates of return on the ith security and on the market portfolio (say, the S&P 500) in year t; βi , as noted elsewhere, is the Beta or market volatility coefﬁcient of the ith security, and eit are the residuals. In all there are N such regressions, one for each security, giving therefore N estimates of βi . Stage II (Cross-section regression). In this stage we run the following regression over the N securities:

¯ R i = γ1 + γ2 βi + ui ˆ ˆ ˆ

(2)

¯ where R i is the average or mean rate of return for security i computed ˆ over the sample period covered by Stage I, βi is the estimated beta coefﬁcient from the ﬁrst-stage regression, and ui is the residual term. Comparing the second-stage regression (2) with the CAPM Eq. (6.1.2), written as ER i = r f + βi (ER m − r f )

(3)

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ˆ where rf is the risk-free rate of return, we see that γ1 is an estimate of rf ˆ and γ2 is an estimate of (ER m − r f ), the market risk premium. ¯ ˆ Thus, in the empirical testing of CAPM, R i and βi are used as estimators of ERi and βi , respectively. Now if CAPM holds, statistically, γ1 = r f ˆ γ2 = Rm − r f , the estimator of (ERm − rf) ˆ

Next consider an alternative model:

¯ R i = γ1 + γ2 βi + γ3 sei + ui ˆ ˆ ˆ ˆ 2

(4)

2 where sei is the residual variance of the ith security from the ﬁrst-stage reˆ gression. Then, if CAPM is valid, γ3 should not be signiﬁcantly different from zero. To test the CAPM, Levy ran regressions (2) and (4) on a sample of 101 stocks for the period 1948–1968 and obtained the following results*:

ˆ ¯ Ri =

0.109 + 0.037βi (0.009) (0.008) (5.1) R = 0.21

2

(2)

t = (12.0)

ˆ ¯ Ri =

ˆ 0.106 + 0.0024βi + 0.201s2i

(0.008) t = (13.2)

(0.007) (3.3)

(0.038) (5.3) R = 0.39

2

(4)

a. Are these results supportive of the CAPM? 2 b. Is it worth adding the variable sei to the model? How do you know? ˆ c. If the CAPM holds, γ1 in (2) should approximate the average value of the risk-free rate, r f . The estimated value is 10.9 percent. Does this seem a reasonable estimate of the risk-free rate of return during the observation period, 1948–1968? (You may consider the rate of return on Treasury bills or a similar comparatively risk-free asset.) ¯ d. If the CAPM holds, the market risk premium ( R m − r f ) from (2) is ¯ about 3.7 percent. If r f is assumed to be 10.9 percent, this implies Rm for the sample period was about 14.6 percent. Does this sound a reasonable estimate? e. What can you say about the CAPM generally? 8.29. Refer to exercise 7.21c. Now that you have the necessary tools, which test(s) would you use to choose between the two models. Show the necessary computations. Note that the dependent variables in the two models are different. 8.30. Refer to Example 8.3. Use the t test as shown in (8.7.4) to ﬁnd out if there were constant returns to scale in the Mexican economy for the period of the study.

* H. Levy, “Equilibrium in an Imperfect Market: A constraint on the number of securities in the portfolio,” American Economic Review, vol. 68, no. 4, September 1978, pp. 643–658.

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8.31. Return to the child mortality example that we have discussed several times. In regression (7.6.2) we regressed child mortality (CM) on per capita GNP (PGNP) and female literacy rate (FLR). Now we extend this model by including total fertility rate (TFR). The data on all these variables are already given in Table 6.4. We reproduce regression (7.6.2) and give results of the extended regression model below: 1. CM i = 263.6416 − 0.0056 PGNPi − 2.2316 FLRi se = (11.5932) (0.0019) (0.2099) R = 0.7077

2

(7.6.2)

2. CM i = 168.3067 − 0.0055 PGNPi − 1.7680 FLRi + 12.8686TFRi se = (32.8916) (0.0018) (0.2480) (?) R2 = 0.7474 a. How would you interpret the coefﬁcient of TFR? A priori, would you expect a positive or negative relationship between CM and TFR? Justify your answer. b. Have the coefﬁcient values of PGNP and FR changed between the two equations? If so, what may be the reason(s) for such a change? Is the observed difference statistically signiﬁcant? Which test do you use and why? c. How would you choose between models 1 and 2? Which statistical test would you use to answer this question? Show the necessary calculations. d. We have not given the standard error of the coefﬁcient of TFR. Can you ﬁnd it out? (Hint: Recall the relationship between the t and F distributions.) 8.32. Return to exercise 1.7, which gave data on advertising impressions retained and advertising expenditure for a sample of 21 ﬁrms. In exercise 5.11 you were asked to plot these data and decide on an appropriate model about the relationship between impressions and advertising expenditure. Letting Y represent impressions retained and X the advertising expenditure, the following regressions were obtained:

ˆ Model I: Yi = 22.163 + 0.3631Xi

se = (7.089)

(0.0971)

r 2 = 0.424

ˆ Model II: Yi = 7.059 + 1.0847Xi − 0.0040 X i2

se = (9.986)

(0.3699)

(0.0019)

R2 = 0.53

a. Interpret both models. b. Which is a better model? Why? c. Which statistical test(s) would you use to choose between the two models? d. Are there “diminishing returns” to advertising expenditure, that is, after a certain level of advertising expenditure (the saturation level) it does not pay to advertise? Can you ﬁnd out what that level of expenditure might be? Show the necessary calculations.

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8.33. In regression (7.9.4), we presented the results of the Cobb–Douglas production function ﬁtted to the Taiwanese agricultural sector for the years 1958–1972. On the basis of that regression, ﬁnd out if there are constant returns to scale in that sector, using a. The t test given in (8.7.4). You are told that the covariance between the two slope estimators is −0.03843. b. The F test given in (8.7.9). c. Is there a difference in the two test results? And what is your conclusion regarding the returns to scale in the agriculture sector of Taiwan over the sample period? 8.34 Reconsider the savings–income regression in Section 8.8. Suppose we divide the sample into two periods as 1970–1982 and 1983–1995. Using the Chow test, decide if there is a structural change in the savings–income regression in the two periods. Comparing your results with those given in Section 8.8, what overall conclusion do you draw about the sensitivity of the Chow test to the choice of the break point that divides the sample into two (or more) periods?

*APPENDIX 8A Likelihood Ratio (LR) Test

The LR test is based on the maximum likelihood (ML) principle discussed in Appendix 4A, where we showed how one obtains the ML estimators of the two-variable regression model. The principle can be straightforwardly extended to the multiple regression model. Under the assumption that the disturbances ui are normally distributed, we showed that, for the twovariable regression model, the OLS and ML estimators of the regression coefﬁcients are identical, but the estimated error variances are different. The ˆ ˆ2 OLS estimator of σ 2 is ui2 /(n − 2) but the ML estimator is ui /n, the former being unbiased and the latter biased, although in large samples the bias tends to disappear. The same is true in the multiple regression case. To illustrate, consider the three-variable regression model: Yi = β1 + β2 X2i + β3 X3i + ui (1)

Corresponding to Eq. (5) of Appendix 4A, the log-likelihood function for the model (1) can be written as: n n 1 ln LF = − σ 2 − ln (2π) − 2 2 2 (Yi − β1 − β2 X2i − β3 X3i )2 (2)

*

Optional.

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As shown in Appendix 4A, differentiating this function with respect to β1 , β2 , β3 , and σ 2 , setting the resulting expressions to zero, and solving, we obtain the ML estimators of these estimators. The ML estimators of β1 , β2 , and β3 will be identical to OLS estimators, which are already given in Eqs. (7.4.6) to (7.4.8), but the error variance will be different in that the residual sum of squares (RSS) will be divided by n rather than by (n − 3), as in the case of OLS. Now let us suppose that our null hypothesis H0 is that β3 , the coefﬁcient of X3 , is zero. In this case, log LF given in (2) will become n n 1 ln LF = − σ 2 − ln (2π) − 2 2 2 (Yi − β1 − β2 X2i )2 (3)

Equation (3) is known as the restricted log-likelihood function (RLLF) because it is estimated with the restriction that a priori β3 is zero, whereas Eq. (1) is known as the unrestricted log LF (ULLF) because a priori there are no restrictions put on the parameters. To test the validity of the a priori restriction that β3 is zero, the LR test obtains the following test statistic: λ = 2(ULLF − RLLF) (4)*

where ULLF and RLLF are, respectively, the unrestricted log-likelihood function [Eq. (1)] and the restricted log-likelihood function [Eq. (3)]. If the sample size is large, it can be shown that the test statistic λ given in (4) follows the chi-square (χ 2 ) distribution with df equal to the number of restrictions imposed by the null hypothesis, 1 in the present case. The basic idea behind the LR test is simple: If the a priori restriction(s) are valid, the restricted and unrestricted (log) LF should not be different, in which case λ in (4) will be zero. But if that is not the case, the two LFs will diverge. And since in a large sample we know that λ follows the chi-square distribution, we can ﬁnd out if the divergence is statistically signiﬁcant, say, at a 1 or 5 percent level of signiﬁcance. Or else, we can ﬁnd out the p value of the estimated λ. Let us illustrate the LR test with our child mortality example. If we regress child mortality (CM) on per capita GNP (PGNP) and female literacy rate (FLR) as we did in (8.2.1), we obtain ULLF of −328.1012, but if we regress CM on PGNP only, we obtain the RLLF of −361.6396. In absolute value (i.e., disregarding the sign), the former is smaller than the latter, which makes sense since we have an additional variable in the former model. The question now is whether it is worth adding the FLR variable. If it is not, the restricted and unrestricted LLF should not differ much, but if it is, the LLFs will be different. To see if this difference is statistically signiﬁcant,

*

This expression can also be expressed as −2(RLLF − ULLF) or as −2 ln (RLF/ULF).

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we now use the LR test given in (4), which gives: λ = 2[−328.1012 − (−361.6396)] = 67.0768 Asymptotically, this is distributed as the chi-square distribution with 1 df (because we have only one restriction imposed when we omitted the FLR variable from the full model). The p value of obtaining such a chi-square value for 1 df is almost zero, leading to the conclusion that the FLR variable should not be excluded from the model. In other words, the restricted regression in the present instance is not valid. Because of the mathematical complexity of the Wald and LM tests, we will not discuss them here. But as noted in the text, asymptotically, the LR, Wald, and LM tests give identical answers, the choice of the test depending on computational convenience.

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PART

TWO

RELAXING THE ASSUMPTIONS OF THE CLASSICAL MODEL

In Part I we considered at length the classical normal linear regression model and showed how it can be used to handle the twin problems of statistical inference, namely, estimation and hypothesis testing, as well as the problem of prediction. But recall that this model is based on several simplifying assumptions, which are as follows. Assumption 1. The regression model is linear in the parameters. Assumption 2. The values of the regressors, the X’s, are ﬁxed in repeated sampling. Assumption 3. For given X’s, the mean value of the disturbance ui is zero. Assumption 4. For given X’s, the variance of ui is constant or homoscedastic. Assumption 5. For given X’s, there is no autocorrelation in the disturbances. Assumption 6. If the X’s are stochastic, the disturbance term and the (stochastic) X’s are independent or at least uncorrelated. Assumption 7. The number of observations must be greater than the number of regressors. Assumption 8. There must be sufﬁcient variability in the values taken by the regressors. Assumption 9. The regression model is correctly speciﬁed. Assumption 10. There is no exact linear relationship (i.e., multicollinearity) in the regressors. Assumption 11. The stochastic (disturbance) term ui is normally distributed.

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Before proceeding further, let us note that most textbooks list fewer than 11 assumptions. For example, assumptions 7 and 8 are taken for granted rather than spelled out explicitly. We decided to state them explicitly because distinguishing between the assumptions required for OLS to have desirable statistical properties (such as BLUE) and the conditions required for OLS to be useful seems sensible. For example, OLS estimators are BLUE even if assumption 8 is not satisﬁed. But in that case the standard errors of the OLS estimators will be large relative to their coefﬁcients (i.e., the t ratios will be small), thereby making it difﬁcult to assess the contribution of one or more regressors to the explained sum of squares. As Wetherill notes, in practice two major types of problems arise in applying the classical linear regression model: (1) those due to assumptions about the speciﬁcation of the model and about the disturbances ui and (2) those due to assumptions about the data.1 In the ﬁrst category are Assumptions 1, 2, 3, 4, 5, 9, and 11. Those in the second category include Assumptions 6, 7, 8, and 10. In addition, data problems, such as outliers (unusual or untypical observations) and errors of measurement in the data, also fall into the second category. With respect to problems arising from the assumptions about disturbances and model speciﬁcations, three major questions arise: (1) How severe must the departure be from a particular assumption before it really matters? For example, if ui are not exactly normally distributed, what level of departure from this assumption can one accept before the BLUE property of the OLS estimators is destroyed? (2) How do we ﬁnd out whether a particular assumption is in fact violated in a concrete case? Thus, how does one ﬁnd out if the disturbances are normally distributed in a given application? We have already discussed the Anderson-Darling, chi-square, and Jarque–Bera tests of normality. (3) What remedial measures can we take if one or more of the assumptions are false? For example, if the assumption of homoscedasticity is found to be false in an application, what do we do then? With regard to problems attributable to assumptions about the data, we also face similar questions. (1) How serious is a particular problem? For example, is multicollinearity so severe that it makes estimation and inference very difﬁcult? (2) How do we ﬁnd out the severity of the data problem? For example, how do we decide whether the inclusion or exclusion of an observation or observations that may represent outliers will make a tremendous difference in the analysis? (3) Can some of the data problems be easily remedied? For example, can one have access to the original data to ﬁnd out the sources of errors of measurement in the data? Unfortunately, satisfactory answers cannot be given to all these questions. What we will do in the rest of Part II is to look at some of the assumptions more critically, but not all will receive full scrutiny. In particular, we will not discuss in depth the following: Assumptions 2, 3, 6, and 11. The

1 G. Barrie Wetherill, Regression Analysis with Applications, Chapman and Hall, New York, 1986, pp. 14–15.

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reasons are as follows: Assumptions 2 and 6: Fixed versus stochastic regressors. Remember that our regression analysis is based on the assumption that the regressors are nonstochastic and assume ﬁxed values in repeated sampling. There is a good reason for this strategy. Unlike scientists in the physical sciences, as noted in Chapter 1, economists generally have no control over the data they use. More often than not, economists depend on secondary data, that is, data collected by someone else, such as the government and private organizations. Therefore, the practical strategy to follow is to assume that for the problem at hand the values of the explanatory variables are given even though the variables themselves may be intrinsically stochastic or random. Hence, the results of the regression analysis are conditional upon these given values. But suppose that we cannot regard the X’s as truly nonstochastic or ﬁxed. This is the case of random or stochastic regressors. Now the situation is rather involved. The ui, by assumption, are stochastic. If the X’s too are stochastic, then we must specify how the X’s and ui are distributed. If we are willing to make Assumption 6 (i.e., the X’s, although random, are distributed independently of, or at least uncorrelated with, ui), then for all practical purposes we can continue to operate as if the X’s were nonstochastic. As Kmenta notes:

Thus, relaxing the assumption that X is nonstochastic and replacing it by the assumption that X is stochastic but independent of [u] does not change the desirable properties and feasibility of least squares estimation.2

Therefore, we will retain Assumption 2 or Assumption 6 until we come to deal with simultaneous equations models in Part IV.3 Assumption 3: Zero mean value of ui. Recall the k-variable linear regression model: Yi = β1 + β2 X2i + β3 X3i + · · · + βk Xki + ui Let us now assume that E(ui |X2i , X3i , . . . , Xki ) = w (2) (1)

where w is a constant; note in the standard model w = 0, but now we let it be any constant.

2 Jan Kmenta, Elements of Econometrics, 2d ed., Macmillan, New York, 1986, p. 338. (Emphasis in the original.) 3 A technical point may be noted here. Instead of the strong assumption that the X’s and u are independent, we may use the weaker assumption that the values of X variables and u are uncorrelated contemporaneously (i.e., at the same point in time). In this case OLS estimators may be biased but they are consistent, that is, as the sample size increases indeﬁnitely, the estimators converge on their true values. If, however, the X’s and u are contemporaneously correlated, the OLS estimators are biased as well as inconsistent. In Chap. 17 we will show how the method of instrumental variables can sometimes be used to obtain consistent estimators in this situation.

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Taking the conditional expectation of (1), we obtain E(Yi |X2i , X3i , . . . , Xki ) = β1 + β2 X2i + β3 X3i + · · · + βk Xki + w = (β1 + w) + β2 X2i + β3 X3i + · · · + βk Xki = α + β2 X2i + β3 X3i + · · · + βk Xki where α = (β1 + w) and where in taking the expectations one should note that the X’s are treated as constants. (Why?) Therefore, if Assumption 3 is not fulﬁlled, we see that we cannot estimate the original intercept β1 ; what we obtain is α, which contains β1 and E(ui) = w. In short, we obtain a biased estimate of β1 . But as we have noted on many occasions, in many practical situations the intercept term, β1 , is of little importance; the more meaningful quantities are the slope coefﬁcients, which remain unaffected even if Assumption 3 is violated.4 Besides, in many applications the intercept term has no physical interpretation. Assumption 11: Normality of u. This assumption is not essential if our objective is estimation only. As noted in Chapter 3, the OLS estimators are BLUE regardless of whether the ui are normally distributed or not. With the normality assumption, however, we were able to establish that the OLS estimators of the regression coefﬁcients follow the normal distribution, that (n − k)σ 2 /σ 2 has the χ 2 distribution, and that one could use the t and F tests ˆ to test various statistical hypotheses regardless of the sample size. But what happens if the ui are not normally distributed? We then rely on the following extension of the central limit theorem; recall that it was the central limit theorem we invoked to justify the normality assumption in the ﬁrst place:

If the disturbances [ui] are independently and identically distributed with zero mean and [constant] variance σ 2 and if the explanatory variables are constant in repeated samples, the [O]LS coefﬁcient estimators are asymptotically normally distributed with means equal to the corresponding β’s.5

(3)

Therefore, the usual test procedures—the t and F tests—are still valid asymptotically, that is, in the large sample, but not in the ﬁnite or small samples. The fact that if the disturbances are not normally distributed the OLS estimators are still normally distributed asymptotically (under the assumption of homoscedastic variance and ﬁxed X’s) is of little comfort to practicing

4 It is very important to note that this statement is true only if E(ui) = w for each i. However, if E(ui) = wi, that is, a different constant for each i, the partial slope coefﬁcients may be biased as well as inconsistent. In this case violation of Assumption 3 will be critical. For proof and further details, see Peter Schmidt, Econometrics, Marcel Dekker, New York, 1976, pp. 36–39. 5 Henri Theil, Introduction to Econometrics, Prentice-Hall, Englewood Cliffs, N.J., 1978, p. 240. It must be noted the assumptions of ﬁxed X ’s and constant σ 2 are crucial for this result.

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economists, who often do not have the luxury of large-sample data. Therefore, the normality assumption becomes extremely important for the purposes of hypothesis testing and prediction. Hence, with the twin problems of estimation and hypothesis testing in mind, and given the fact that small samples are the rule rather than the exception in most economic analyses, we shall continue to use the normality assumption.6 Of course, this means that when we deal with a ﬁnite sample, we must explicitly test for the normality assumption. We have already considered the Anderson-Darling and the Jarque-Bera tests of normality. The reader is strongly urged to apply these or other tests of normality to regression residuals. Keep in mind that in ﬁnite samples without the normality assumption the usual t and F statistics may not follow the t and F distributions. We are left with Assumptions 1, 4, 5, 7, 8, 9, 10. Assumptions 7, 8, and 10 are closely related and are discussed in the chapter on multicollinearity (Chapter 10). Assumption 4 is discussed in the chapter on heteroscedasticity (Chapter 11). Assumption 5 is discussed in the chapter on autocorrelation (Chapter 12). Assumption 9 is discussed in the chapter on model speciﬁcation and diagnostic testing (Chapter 13). Because of its specialized nature and mathematical demands, Assumption 1 is discussed as a special topic in Part III (Chapter 14). For pedagogical reasons, in each of these chapters we follow a common format, namely, (1) identify the nature of the problem, (2) examine its consequences, (3) suggest methods of detecting it, and (4) consider remedial measures so that they may lead to estimators that possess the desirable statistical properties discussed in Part I. A cautionary note is in order: As noted earlier, satisfactory answers to all the problems arising out of the violation of the assumptions of the CLRM do not exist. Moreover, there may be more than one solution to a particular problem, and often it is not clear which method is best. Besides, in a particular application more than one violation of the CLRM may be involved. Thus, speciﬁcation bias, multicollinearity, and heteroscedasticity may coexist in an application, and there is no single omnipotent test that will solve all the problems simultaneously.7 Furthermore, a particular test that was popular at one time may not be in vogue later because somebody found a ﬂaw in the earlier test. But this is how science progresses. Econometrics is no exception.

6 In passing, note that the effects of departure from normality and related topics are often discussed under the topic of robust estimation in the literature, a topic beyond the scope of this book. 7 This is not for lack of trying. See A. K. Bera and C. M. Jarque, “Efﬁcient Tests for Normality, Homoscedasticity and Serial Independence of Regression Residuals: Monte Carlo Evidence,” Economic Letters, vol. 7, 1981, pp. 313–318.

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9

DUMMY VARIABLE REGRESSION MODELS

In Chapter 1 we discussed brieﬂy the four types of variables that one generally encounters in empirical analysis: These are: ratio scale, interval scale, ordinal scale, and nominal scale. The types of variables that we have encountered in the preceding chapters were essentially ratio scale. But this should not give the impression that regression models can deal only with ratio scale variables. Regression models can also handle other types of variables mentioned previously. In this chapter, we consider models that may involve not only ratio scale variables but also nominal scale variables. Such variables are also known as indicator variables, categorical variables, qualitative variables, or dummy variables.1

9.1 THE NATURE OF DUMMY VARIABLES

In regression analysis the dependent variable, or regressand, is frequently inﬂuenced not only by ratio scale variables (e.g., income, output, prices, costs, height, temperature) but also by variables that are essentially qualitative, or nominal scale, in nature, such as sex, race, color, religion, nationality, geographical region, political upheavals, and party afﬁliation. For example, holding all other factors constant, female workers are found to earn less than their male counterparts or nonwhite workers are found to earn less than whites.2 This pattern may result from sex or racial discrimination, but whatever the reason, qualitative variables such as sex and race seem to

We will discuss ordinal scale variables in Chap. 15. For a review of the evidence on this subject, see Bruce E. Kaufman and Julie L. Hotchkiss, The Economics of Labor Market, 5th ed., Dryden Press, New York, 2000.

2 1

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inﬂuence the regressand and clearly should be included among the explanatory variables, or the regressors. Since such variables usually indicate the presence or absence of a “quality” or an attribute, such as male or female, black or white, Catholic or non-Catholic, Democrat or Republican, they are essentially nominal scale variables. One way we could “quantify” such attributes is by constructing artiﬁcial variables that take on values of 1 or 0, 1 indicating the presence (or possession) of that attribute and 0 indicating the absence of that attribute. For example 1 may indicate that a person is a female and 0 may designate a male; or 1 may indicate that a person is a college graduate, and 0 that the person is not, and so on. Variables that assume such 0 and 1 values are called dummy variables.3 Such variables are thus essentially a device to classify data into mutually exclusive categories such as male or female. Dummy variables can be incorporated in regression models just as easily as quantitative variables. As a matter of fact, a regression model may contain regressors that are all exclusively dummy, or qualitative, in nature. Such models are called Analysis of Variance (ANOVA) models.4

9.2 ANOVA MODELS

To illustrate the ANOVA models, consider the following example.

EXAMPLE 9.1 PUBLIC SCHOOL TEACHERS’ SALARIES BY GEOGRAPHICAL REGION Table 9.1 gives data on average salary (in dollars) of public school teachers in 50 states and the District of Columbia for the year 1985. These 51 areas are classiﬁed into three geographical regions: (1) Northeast and North Central (21 states in all), (2) South (17 states in all), and (3) West (13 states in all). For the time being, do not worry about the format of the table and the other data given in the table. Suppose we want to ﬁnd out if the average annual salary (AAS) of public school teachers differs among the three geographical regions of the country. If you take the simple arithmetic average of the average salaries of the teachers in the three regions, you will ﬁnd that these averages for the three regions are as follows: $24,424.14 (Northeast and North Central), $22,894 (South), and $26,158.62 (West). These numbers look different, but are they (Continued)

3 It is not absolutely essential that dummy variables take the values of 0 and 1. The pair (0,1) can be transformed into any other pair by a linear function such that Z = a + bD (b = 0), where a and b are constants and where D = 1 or 0. When D = 1, we have Z = a + b, and when D = 0, we have Z = a. Thus the pair (0, 1) becomes (a, a + b). For example, if a = 1 and b = 2, the dummy variables will be (1, 3). This expression shows that qualitative, or dummy, variables do not have a natural scale of measurement. That is why they are described as nominal scale variables. 4 ANOVA models are used to assess the statistical signiﬁcance of the relationship between a quantitative regressand and qualitative or dummy regressors. They are often used to compare the differences in the mean values of two or more groups or categories, and are therefore more general than the t test which can be used to compare the means of two groups or categories only.

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EXAMPLE 9.1 (Continued) TABLE 9.1 AVERAGE SALARY OF PUBLIC SCHOOL TEACHERS, BY STATE, 1986 Salary 19,583 20,263 20,325 26,800 29,470 26,610 30,678 27,170 25,853 24,500 24,274 27,170 30,168 26,525 27,360 21,690 21,974 20,816 18,095 20,939 22,644 24,624 27,186 33,990 23,382 20,627 Spending 3346 3114 3554 4642 4669 4888 5710 5536 4168 3547 3159 3621 3782 4247 3982 3568 3155 3059 2967 3285 3914 4517 4349 5020 3594 2821 D2 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0 0 0 0 0 D3 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 1 Salary 22,795 21,570 22,080 22,250 20,940 21,800 22,934 18,443 19,538 20,460 21,419 25,160 22,482 20,969 27,224 25,892 22,644 24,640 22,341 25,610 26,015 25,788 29,132 41,480 25,845 Spending 3366 2920 2980 3731 2853 2533 2729 2305 2642 3124 2752 3429 3947 2509 5440 4042 3402 2829 2297 2932 3705 4123 3608 8349 3766 D2 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 D3 1 1 1 1 1 1 1 1 1 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0

Note: D2 = 1 for states in the Northeast and North Central; 0 otherwise. D3 = 1 for states in the South; 0 otherwise. Source: National Educational Association, as reported by Albuquerque Tribune, Nov. 7, 1986.

statistically different from one another? There are various statistical techniques to compare two or more mean values, which generally go by the name of analysis of variance.5 But the same objective can be accomplished within the framework of regression analysis. To see this, consider the following model: Yi = β1 + β2D2i + β3i D3i + ui where Yi = (average) salary of public school teacher in state i D2i = 1 if the state is in the Northeast or North Central = 0 otherwise (i.e., in other regions of the country) D3i = 1 if the state is in the South = 0 otherwise (i.e., in other regions of the country) (Continued) (9.2.1)

5 For an applied treatment, see John Fox, Applied Regression Analysis, Linear Models, and Related Methods, Sage Publications, 1997, Chap. 8.

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EXAMPLE 9.1 (Continued) Note that (9.2.1) is like any multiple regression model considered previously, except that, instead of quantitative regressors, we have only qualitative, or dummy, regressors, taking the value of 1 if the observation belongs to a particular category and 0 if it does not belong to that category or group. Hereafter, we shall designate all dummy variables by the letter D. Table 9.1 shows the dummy variables thus constructed. What does the model (9.2.1) tell us? Assuming that the error term satisﬁes the usual OLS assumptions, on taking expectation of (9.2.1) on both sides, we obtain: Mean salary of public school teachers in the Northeast and North Central: E(Yi | D2i = 1, D3i = 0) = β1 + β2 Mean salary of public school teachers in the South: E(Yi | D2i = 0, D3i = 1) = β1 + β3 (9.2.3) (9.2.2)

You might wonder how we ﬁnd out the mean salary of teachers in the West. If you guessed that this is equal to β1 , you would be absolutely right, for Mean salary of public school teachers in the West: E(Yi | D2i = 0, D3i = 0) = β1 (9.2.4)

In other words, the mean salary of public school teachers in the West is given by the intercept, β1, in the multiple regression (9.2.1), and the “slope” coefﬁcients β2 and β3 tell by how much the mean salaries of teachers in the Northeast and North Central and in the South differ from the mean salary of teachers in the West. But how do we know if these differences are statistically signiﬁcant? Before we answer this question, let us present the results based on the regression (9.2.1). Using the data given in Table 9.1, we obtain the following results: ˆ Yi = 26,158.62 se = (1128.523) t= (23.1759) (0.0000)* − 1734.473D2i − 3264.615D3i (1435.953) (−1.2078) (0.2330)* (1499.615) (−2.1776) (0.0349)* R 2 = 0.0901 (9.2.5)

where * indicates the p values. As these regression results show, the mean salary of teachers in the West is about $26,158, that of teachers in the Northeast and North Central is lower by about $1734, and that of teachers in the South is lower by about $3265. The actual mean salaries in the last two regions can be easily obtained by adding these differential salaries to the mean salary of teachers in the West, as shown in Eqs. (9.2.3) and (9.2.4). Doing this, we will ﬁnd that the mean salaries in the latter two regions are about $24,424 and $22,894. But how do we know that these mean salaries are statistically different from the mean salary of teachers in the West, the comparison category? That is easy enough. All we have to do is to ﬁnd out if each of the “slope” coefﬁcients in (9.2.5) is statistically signiﬁcant. As can be seen from this regression, the estimated slope coefﬁcient for Northeast and North Central is not statistically signiﬁcant, as its p value is 23 percent, whereas that of the South is statistically signiﬁcant, as the p value is only about 3.5 percent. Therefore, the overall conclusion is that statistically the mean salaries of public school teachers in the West and the Northeast and North Central are about the same but the mean salary of teachers in the South is statistically signiﬁcantly lower by about $3265. Diagrammatically, the situation is shown in Figure 9.1. A caution is in order in interpreting these differences. The dummy variables will simply point out the differences, if they exist, but they do not suggest the reasons for the differences. (Continued)

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EXAMPLE 9.1 (Continued)

β1 = $26,158

$24,424 ( β1 + β2) $22,894 ( β1 + β3)

West

Northeast and North Central

South

FIGURE 9.1 Average salary (in dollars) of public school teachers in three regions.

Differences in educational levels, in cost of living indexes, in gender and race may all have some effect on the observed differences. Therefore, unless we take into account all the other variables that may affect a teacher’s salary, we will not be able to pin down the cause(s) of the differences. From the preceding discussion, it is clear that all one has to do is see if the coefﬁcients attached to the various dummy variables are individually statistically signiﬁcant. This example also shows how easy it is to incorporate qualitative, or dummy, regressors in the regression models.

Caution in the Use of Dummy Variables

Although they are easy to incorporate in the regression models, one must use the dummy variables carefully. In particular, consider the following aspects: 1. In Example 9.1, to distinguish the three regions, we used only two dummy variables, D2 and D3. Why did we not use three dummies to distinguish the three regions? Suppose we do that and write the model (9.2.1) as: Yi = α + β1 D1i + β2 D2i + β3 D3i + ui (9.2.6)

where D1i takes a value of 1 for states in the West and 0 otherwise. Thus, we now have a dummy variable for each of the three geographical regions. Using the data in Table 9.1, if you were to run the regression (9.2.6), the computer will “refuse” to run the regression (try it).6 Why? The reason is that in

6

Actually you will get a message saying that the data matrix is singular.

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the setup of (9.2.6) where you have a dummy variable for each category or group and also an intercept, you have a case of perfect collinearity, that is, exact linear relationships among the variables. Why? Refer to Table 9.1. Imagine that now we add the D1 column, taking the value of 1 whenever a state is in the West and 0 otherwise. Now if you add the three D columns horizontally, you will obtain a column that has 51 ones in it. But since the value of the intercept α is (implicitly) 1 for each observation, you will have a column that also contains 51 ones. In other words, the sum of the three D columns will simply reproduce the intercept column, thus leading to perfect collinearity. In this case, estimation of the model (9.2.6) is impossible. The message here is: If a qualitative variable has m categories, introduce only (m − 1) dummy variables. In our example, since the qualitative variable “region” has three categories, we introduced only two dummies. If you do not follow this rule, you will fall into what is called the dummy variable trap, that is, the situation of perfect collinearity or perfect multicollinearity, if there is more than one exact relationship among the variables. This rule also applies if we have more than one qualitative variable in the model, an example of which is presented later. Thus we should restate the preceding rule as: For each qualitative regressor the number of dummy variables introduced must be one less than the categories of that variable. Thus, if in Example 9.1 we had information about the gender of the teacher, we would use an additional dummy variable (but not two) taking a value of 1 for female and 0 for male or vice versa. 2. The category for which no dummy variable is assigned is known as the base, benchmark, control, comparison, reference, or omitted category. And all comparisons are made in relation to the benchmark category. 3. The intercept value (β1) represents the mean value of the benchmark category. In Example 9.1, the benchmark category is the Western region. Hence, in the regression (9.2.5) the intercept value of about 26,159 represents the mean salary of teachers in the Western states. 4. The coefﬁcients attached to the dummy variables in (9.2.1) are known as the differential intercept coefﬁcients because they tell by how much the value of the intercept that receives the value of 1 differs from the intercept coefﬁcient of the benchmark category. For example, in (9.2.5), the value of about −1734 tells us that the mean salary of teachers in the Northeast or North Central is smaller by about $1734 than the mean salary of about $26,159 for the benchmark category, the West. 5. If a qualitative variable has more than one category, as in our illustrative example, the choice of the benchmark category is strictly up to the researcher. Sometimes the choice of the benchmark is dictated by the particular problem at hand. In our illustrative example, we could have chosen the South as the benchmark category. In that case the regression results given in (9.2.5) will change, because now all comparisons are made in relation to the South. Of course, this will not change the overall conclusion of our example (why?). In this case, the intercept value will be about $22,894, which is the mean salary of teachers in the South.

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6. We warned above about the dummy variable trap. There is a way to circumvent this trap by introducing as many dummy variables as the number of categories of that variable, provided we do not introduce the intercept in such a model. Thus, if we drop the intercept term from (9.2.6), and consider the following model, Yi = β1 D1i + β2 D2i + β3 D3i + ui (9.2.7)

we do not fall into the dummy variable trap, as there is no longer perfect collinearity. But make sure that when you run this regression, you use the nointercept option in your regression package. How do we interpret regression (9.2.7)? If you take the expectation of (9.2.7), you will ﬁnd that: β1 = mean salary of teachers in the West β2 = mean salary of teachers in the Northeast and North Central. β3 = mean salary of teachers in the South. In other words, with the intercept suppressed, and allowing a dummy variable for each category, we obtain directly the mean values of the various categories. The results of (9.2.7) for our illustrative example are as follows: ˆ Yi = 26,158.62D1i + 24,424.14D2i + 22,894D3i se = (1128.523) t= (23.1795)* (887.9170) (27.5072)* (986.8645) (23.1987)* R2 = 0.0901 where * indicates that the p values of these t ratios are very small. As you can see, the dummy coefﬁcients give directly the mean (salary) values in the three regions, West, Northeast and North Central, and South. 7. Which is a better method of introducing a dummy variable: (1) introduce a dummy for each category and omit the intercept term or (2) include the intercept term and introduce only (m − 1) dummies, where m is the number of categories of the dummy variable? As Kennedy notes:

Most researchers ﬁnd the equation with an intercept more convenient because it allows them to address more easily the questions in which they usually have the most interest, namely, whether or not the categorization makes a difference, and if so, by how much. If the categorization does make a difference, by how much is measured directly by the dummy variable coefﬁcient estimates. Testing whether or not the categorization is relevant can be done by running a t test of a dummy variable coefﬁcient against zero (or, to be more general, an F test on the appropriate set of dummy variable coefﬁcient estimates).7

7

(9.2.8)

Peter Kennedy, A Guide to Econometrics, 4th ed., MIT Press, Cambridge, Mass., 1998, p. 223.

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9.3

ANOVA MODELS WITH TWO QUALITATIVE VARIABLES

In the previous section we considered an ANOVA model with one qualitative variable with three categories. In this section we consider another ANOVA model, but with two qualitative variables, and bring out some additional points about dummy variables.

EXAMPLE 9.2 HOURLY WAGES IN RELATION TO MARITAL STATUS AND REGION OF RESIDENCE From a sample of 528 persons in May 1985, the following regression results were obtained8: ˆ Yi = 8.8148 + 1.0997D2i − (0.4642) (2.3688) (0.0182)* 1.6729D3i (0.4854) (−3.4462) (0.0006)* R 2 = 0.0322 where Y = hourly wage ($) D2 = married status, 1 = married, 0 = otherwise D3 = region of residence; 1 = South, 0 = otherwise and * denotes the p values. In this example we have two qualitative regressors, each with two categories. Hence we have assigned a single dummy variable for each category. (9.3.1)

se = (0.4015) t = (21.9528) (0.0000)*

Which is the benchmark category here? Obviously, it is unmarried, non-South residence. In other words, unmarried persons who do not live in the South are the omitted category. Therefore, all comparisons are made in relation to this group. The mean hourly wage in this benchmark is about $8.81. Compared with this, the average hourly wage of those who are married is higher by about $1.10, for an actual average wage of $9.91 ( = 8.81 + 1.10). By contrast, for those who live in the South, the average hourly wage is lower by about $1.67, for an actual average hourly wage of $7.14. Are the preceding average hourly wages statistically different compared to the base category? They are, for all the differential intercepts are statistically signiﬁcant, as their p values are quite low. The point to note about this example is this: Once you go beyond one qualitative variable, you have to pay close attention to the category that is treated as the base category, since all comparisons are made in relation to that category. This is especially important when you have several qualitative regressors, each with several categories. But the mechanics of introducing several qualitative variables should be clear by now.

9.4 REGRESSION WITH A MIXTURE OF QUANTITATIVE AND QUALITATIVE REGRESSORS: THE ANCOVA MODELS

ANOVA models of the type discussed in the preceding two sections, although common in ﬁelds such as sociology, psychology, education, and market research, are not that common in economics. Typically, in most economic research a regression model contains some explanatory variables that are quantitative and some that are qualitative. Regression models containing an admixture of quantitative and qualitative variables are called analysis of covariance (ANCOVA) models. ANCOVA models are an extension of the ANOVA models in that they provide a method of statistically controlling the effects of quantitative regressors, called covariates or control

8 The data are obtained from the data disk in Arthur S. Goldberger, Introductory Econometrics, Harvard University Press, Cambridge, Mass., 1998. We have already considered these data in Chap. 2.

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variables, in a model that includes both quantitative and qualitative, or dummy, regressors. We now illustrate the ANCOVA models. To motivate the analysis, let us reconsider Example 9.1 by maintaining that the average salary of public school teachers may not be different in the three regions if we take into account any variables that cannot be standardized across the regions. Consider, for example, the variable expenditure on public schools by local authorities, as public education is primarily a local and state question. To see if this is the case, we develop the following model: Yi = β1 + β2 D2i + β3 D3i + β4 Xi + ui (9.4.1)

where Yi = average annual salary of public school teachers in state ($) Xi = spending on public school per pupil ($) D2i = 1, if the state is in the Northeast or North Central = 0, otherwise D3i = 1, if the state is in the South = 0, otherwise The data on X are given in Table 9.1. Keep in mind that we are treating the West as the benchmark category. Also, note that besides the two qualitative regressors, we have a quantitative variable, X, which in the context of the ANCOVA models is known as a covariate, as noted earlier.

EXAMPLE 9.3 TEACHER’S SALARY IN RELATION TO REGION AND SPENDING ON PUBLIC SCHOOL PER PUPIL From the data in Table 9.1, the results of the model (9.4.1) are as follows: ˆ Yi = 13,269.11 se = (1395.056) t= (9.5115)* − 1673.514D2i − 1144.157D3i + (801.1703) (−2.0889)* (861.1182) (−1.3286)** 3.2889Xi (0.3176) (9.4.2) (10.3539)* R 2 = 0.7266 where * indicates p values less than 5 percent, and ** indicates p values greater than 5 percent. As these results suggest, ceteris paribus: as public expenditure goes up by a dollar, on average, a public school teacher’s salary goes up by about $3.29. Controlling for spending on education, we now see that the differential intercept coefﬁcient is signiﬁcant for the Northeast and North-Central region, but not for the South. These results are different from those of (9.2.5). But this should not be surprising, for in (9.2.5) we did not account for the covariate, differences in per pupil public spending on education. Diagrammatically, we have the situation shown in Figure 9.2. Note that although we have shown three regression lines for the three regions, statistically the regression lines are the same for the West and the South. Also note that the three regression lines are drawn parallel (why?). (Continued)

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EXAMPLE 9.3 (Continued) Y

t Wes

3.29

1 1 13,269 12,125 11,595 1

d st an thea ntral 3.29 Nor e th C Nor 3.29

Sou

th

X FIGURE 9.2 Public school teacher’s salary (Y ) in relation to per pupil expenditure on education (X ).

9.5

THE DUMMY VARIABLE ALTERNATIVE TO THE CHOW TEST9

In Section 8.8 we discussed the Chow test to examine the structural stability of a regression model. The example we discussed there related to the relationship between savings and income in the United States over the period 1970–1995. We divided the sample period into two, 1970–1981 and 1982–1995, and showed on the basis of the Chow test that there was a difference in the regression of savings on income between the two periods. However, we could not tell whether the difference in the two regressions was because of differences in the intercept terms or the slope coefﬁcients or both. Very often this knowledge itself is very useful. Referring to Eqs. (8.8.1) and (8.8.2), we see that there are four possibilities, which we illustrate in Figure 9.3. 1. Both the intercept and the slope coefﬁcients are the same in the two regressions. This, the case of coincident regressions, is shown in Figure 9.3a. 2. Only the intercepts in the two regressions are different but the slopes are the same. This is the case of parallel regressions, which is shown in Figure 9.3b.

9 The material in this section draws on the author’s articles, “Use of Dummy Variables in Testing for Equality between Sets of Coefﬁcients in Two Linear Regressions: A Note,” and “Use of Dummy Variables . . . A Generalization,” both published in the American Statistician, vol. 24, nos. 1 and 5, 1970, pp. 50–52 and 18–21.

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Savings

Savings

1

γ 2 = λ2 γ 2 = λ2

γ 2 = λ2

1 γ1 = λ 1 Income (a) Coincident regressions

γ1 λ1 1

Income (b) Parallel regressions

Savings

Savings

γ2

1 1

γ2 γ

1 1

λ2

λ2 λ1 y γ1

Income Income (d) Dissimilar regressions

γ1 = λ 1

(c) Concurrent regressions

FIGURE 9.3

Plausible savings–income regressions.

3. The intercepts in the two regressions are the same, but the slopes are different. This is the situation of concurrent regressions (Figure 9.3c). 4. Both the intercepts and slopes in the two regressions are different. This is the case of dissimilar regressions, which is shown in Figure 9.3d. The multistep Chow test procedure discussed in Section 8.8, as noted earlier, tells us only if two (or more) regressions are different without telling us what is the source of the difference. The source of difference, if any, can be pinned down by pooling all the observations (26 in all) and running just one multiple regression as shown below10: Yt = α1 + α2 Dt + β1 Xt + β2 (Dt Xt ) + ut where Y X t D = = = = = savings income time 1 for observations in 1982–1995 0, otherwise (i.e., for observations in 1970–1981) (9.5.1)

10

2 2 As in the Chow test, the pooling technique assumes homoscedasticity, that is, σ1 = σ2 = σ 2 .

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TABLE 9.2

SAVINGS AND INCOME DATA, UNITED STATES, 1970–1995 Observation 1970 1971 1972 1973 1974 1975 1976 1977 1978 1979 1980 1981 1982 1983 1984 1985 1986 1987 1988 1989 1990 1991 1992 1993 1994 1995 Savings 61 68.6 63.6 89.6 97.6 104.4 96.4 92.5 112.6 130.1 161.8 199.1 205.5 167 235.7 206.2 196.5 168.4 189.1 187.8 208.7 246.4 272.6 214.4 189.4 249.3 Income 727.1 790.2 855.3 965 1054.2 1159.2 1273 1401.4 1580.1 1769.5 1973.3 2200.2 2347.3 2522.4 2810 3002 3187.6 3363.1 3640.8 3894.5 4166.8 4343.7 4613.7 4790.2 5021.7 5320.8 Dum 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1

Note: Dum = 1 for observations beginning in 1982; 0 otherwise. Savings and income ﬁgures are in billions of dollars. Source: Economic Report of the President, 1997, Table B-28, p. 332.

Table 9.2 shows the structure of the data matrix. To see the implications of (9.5.1), and, assuming, as usual, that E(ui ) = 0, we obtain: Mean savings function for 1970–1981: E(Yt | Dt = 0, Xt ) = α1 + β1 Xt Mean savings function for 1982–1995: E(Yt | Dt = 1, Xt ) = (α1 + α2 ) + (β1 + β2 )Xt (9.5.3) (9.5.2)

The reader will notice that these are the same functions as (8.8.1) and (8.8.2), with λ1 = α1 , λ2 = β1 , γ1 = (α1 + α2 ), and γ2 = (β1 + β2 ). Therefore, estimating (9.5.1) is equivalent to estimating the two individual savings functions (8.8.1) and (8.8.2). In (9.5.1), α2 is the differential intercept, as previously, and β2 is the differential slope coefﬁcient (also called the slope drifter), indicating by how much the slope coefﬁcient of the second period’s savings function (the

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category that receives the dummy value of 1) differs from that of the ﬁrst period. Notice how the introduction of the dummy variable D in the interactive, or multiplicative, form (D multiplied by X) enables us to differentiate between slope coefﬁcients of the two periods, just as the introduction of the dummy variable in the additive form enabled us to distinguish between the intercepts of the two periods.

EXAMPLE 9.4 STRUCTURAL DIFFERENCES IN THE U.S. SAVINGS–INCOME REGRESSION, THE DUMMY VARIABLE APPROACH Before we proceed further, let us ﬁrst present the regression results of model (9.5.1) applied to the U.S. savings–income data. ˆ Yt = 1.0161 + 152.4786Dt + 0.0803Xt − (33.0824) (4.6090)* (0.0144) (5.5413)* 0.0655(DtXt) (0.0159) (−4.0963)* R 2 = 0.8819 where * indicates p values less than 5 percent and ** indicates p values greater than 5 percent. As these regression results show, both the differential intercept and slope coefﬁcients are statistically signiﬁcant, strongly suggesting that the savings–income regressions for the two time periods are different, as in Figure 9.3d. From (9.5.4), we can derive equations (9.5.2) and (9.5.3), which are: Savings–income regression, 1970–1981: ˆ Y t = 1.0161 + 0.0803Xt Savings–income regression, 1982–1995: ˆ Yt = (1.0161 + 152.4786) + (0.0803 − 0.0655)Xt 0.0148Xt (9.5.6) (9.5.5) (9.5.4)

se = (20.1648) t = (0.0504)**

= 153.4947 +

These are precisely the results we obtained in (8.8.1a) and (8.8.2a), which should not be surprising. These regressions are already shown in Figure 8.3. The advantages of the dummy variable technique [i.e., estimating (9.5.1)] over the Chow test [i.e., estimating the three regressions (8.8.1), (8.8.2), and (8.8.3)] can now be seen readily: 1. We need to run only a single regression because the individual regressions can easily be derived from it in the manner indicated by equations (9.5.2) and (9.5.3). 2. The single regression (9.5.1) can be used to test a variety of hypotheses. Thus if the differential intercept coefﬁcient α2 is statistically insigniﬁcant, we may accept the hypothesis that the two regressions have the same intercept, that is, the two regressions are concurrent (see Figure 9.3c). Similarly, if the differential slope coefﬁcient β2 is statistically insigniﬁcant but α2 is signiﬁcant, we may not reject the hypothesis that the two regressions have the same slope, that is, the two regression lines are parallel (cf. Figure 9.3b). The test of the stability of the entire regression (i.e., α2 = β2 = 0, simultaneously) can be made by the usual F test (recall the restricted least-squares F test). If this hypothesis is not rejected, the regression lines will be coincident, as shown in Figure 9.3a. (Continued)

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EXAMPLE 9.4 (Continued) 3. The Chow test does not explicitly tell us which coefﬁcient, intercept, or slope is different, or whether (as in this example) both are different in the two periods. That is, one can obtain a signiﬁcant Chow test because the slope only is different or the intercept only is different, or both are different. In other words, we cannot tell, via the Chow test, which one of the four possibilities depicted in Figure 9.2 exists in a given instance. In this respect, the dummy variable approach has a distinct advantage, for it not only tells if the two are different but also pinpoints the source(s) of the difference—whether it is due to the intercept or the slope or both. In practice, the knowledge that two regressions differ in this or that coefﬁcient is as important as, if not more than, the plain knowledge that they are different. 4. Finally, since pooling (i.e., including all the observations in one regression) increases the degrees of freedom, it may improve the relative precision of the estimated parameters. Of course, keep in mind that every addition of a dummy variable will consume one degree of freedom.

9.6

INTERACTION EFFECTS USING DUMMY VARIABLES

Dummy variables are a ﬂexible tool that can handle a variety of interesting problems. To see this, consider the following model: Yi = α1 + α2 D2i + α3 D3i + β Xi + ui where Y X D2 D3 = hourly wage in dollars = education (years of schooling) = 1 if female, 0 otherwise = 1 if nonwhite and non-Hispanic, 0 otherwise (9.6.1)

In this model gender and race are qualitative regressors and education is a quantitative regressor.11 Implicit in this model is the assumption that the differential effect of the gender dummy D2 is constant across the two categories of race and the differential effect of the race dummy D3 is also constant across the two sexes. That is to say, if the mean salary is higher for males than for females, this is so whether they are nonwhite/non-Hispanic or not. Likewise, if, say, nonwhite/non-Hispanics have lower mean wages, this is so whether they are females or males. In many applications such an assumption may be untenable. A female nonwhite/non-Hispanic may earn lower wages than a male nonwhite/nonHispanic. In other words, there may be interaction between the two qualitative variables D2 and D3. Therefore their effect on mean Y may not be simply additive as in (9.6.1) but multiplicative as well, as in the following model. ˆ Yi = α1 + α2 D2i + α3 D3i + α4 (D2i D3i ) + β Xi + ui where the variables are as deﬁned for model (9.6.1). From (9.6.2), we obtain: E(Yi | D2i = 1, D3i = 1, Xi ) = (α1 + α2 + α3 + α4 ) + β Xi (9.6.3) (9.6.2)

11 If we were to deﬁne education as less than high school, high school, and more than high school, we could then use two dummies to represent the three classes.

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which is the mean hourly wage function for female nonwhite/non-Hispanic workers. Observe that α2 = differential effect of being a female α3 = differential effect of being a nonwhite/non-Hispanic α4 = differential effect of being a female nonwhite/non-Hispanic which shows that the mean hourly wages of female nonwhite/non-Hispanics is different (by α4) from the mean hourly wages of females or nonwhite/nonHispanics. If, for instance, all the three differential dummy coefﬁcients are negative, this would imply that female nonwhite/non-Hispanic workers earn much lower mean hourly wages than female or nonwhite/non-Hispanic workers as compared with the base category, which in the present example is male white or Hispanic. Now the reader can see how the interaction dummy (i.e., the product of two qualitative or dummy variables) modiﬁes the effect of the two attributes considered individually (i.e., additively).

EXAMPLE 9.5 AVERAGE HOURLY EARNINGS IN RELATION TO EDUCATION, GENDER, AND RACE Let us ﬁrst present the regression results based on model (9.6.1). Using the data that were used to estimate regression (9.3.1), we obtained the following results: ˆ Y i = −0.2610 t = (−0.2357)** − 2.3606D2i − (−5.4873)* 1.7327D3i + 0.8028Xi (−2.1803)*

2

(9.9094)* n = 528

(9.6.4)

R = 0.2032

where * indicates p values less than 5 percent and ** indicates p values greater than 5 percent. The reader can check that the differential intercept coefﬁcients are statistically signiﬁcant, that they have the expected signs (why?), and that education has a strong positive effect on hourly wage, an unsurprising ﬁnding. As (9.6.4) shows, ceteris paribus, the average hourly earnings of females are lower by about $2.36, and the average hourly earnings of nonwhite non-Hispanic workers are also lower by about $1.73. We now consider the results of model (9.6.2), which includes the interaction dummy. ˆ Y i = −0.26100 − t = (−0.2357)** 2.3606D2i − (−5.4873)* 1.7327D3i + 2.1289D2iD3i + 0.8028Xi (−2.1803)* (1.7420)** R = 0.2032

2

(9.9095)** n = 528

(9.6.5)

where * indicates p values less than 5 percent and ** indicates p values greater than 5 percent. As you can see, the two additive dummies are still statistically signiﬁcant, but the interactive dummy is not at the conventional 5 percent level; the actual p value of the interaction dummy is about the 8 percent level. If you think this is a low enough probability, then the results of (9.6.5) can be interpreted as follows: Holding the level of education constant, if you add the three dummy coefﬁcients you will obtain: −1.964 ( = −2.3605 − 1.7327 + 2.1289), which means that mean hourly wages of nonwhite/non-Hispanic female workers is lower by about $1.96, which is between the value of −2.3605 (gender difference alone) and −1.7327 (race difference alone).

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The preceding example clearly reveals the role of interaction dummies when two or more qualitative regressors are included in the model. It is important to note that in the model (9.6.5) we are assuming that the rate of increase of hourly earnings with respect to education (of about 80 cents per additional year of schooling) remains constant across gender and race. But this may not be the case. If you want to test for this, you will have to introduce differential slope coefﬁcients (see exercise 9.25)

9.7 THE USE OF DUMMY VARIABLES IN SEASONAL ANALYSIS

Many economic time series based on monthly or quarterly data exhibit seasonal patterns (regular oscillatory movements). Examples are sales of department stores at Christmas and other major holiday times, demand for money (or cash balances) by households at holiday times, demand for ice cream and soft drinks during summer, prices of crops right after harvesting season, demand for air travel, etc. Often it is desirable to remove the seasonal factor, or component, from a time series so that one can concentrate on the other components, such as the trend.12 The process of removing the seasonal component from a time series is known as deseasonalization or seasonal adjustment, and the time series thus obtained is called the deseasonalized, or seasonally adjusted, time series. Important economic time series, such as the unemployment rate, the consumer price index (CPI), the producer’s price index (PPI), and the index of industrial production, are usually published in seasonally adjusted form. There are several methods of deseasonalizing a time series, but we will consider only one of these methods, namely, the method of dummy variables.13 To illustrate how the dummy variables can be used to deseasonalize economic time series, consider the data given in Table 9.3. This table gives quarterly data for the years 1978–1995 on the sale of four major appliances, dishwashers, garbage disposers, refrigerators, and washing machines, all data in thousands of units. The table also gives data on durable goods expenditure in 1982 billions of dollars. To illustrate the dummy technique, we will consider only the sales of refrigerators over the sample period. But ﬁrst let us look at the data, which is shown in Figure 9.4. This ﬁgure suggests that perhaps there is a seasonal pattern in the data associated with the various quarters. To see if this is the case, consider the following model: Yt = α1 D1t + α2 D2t + α3t D3t + α4 D4t + ut (9.7.1)

where Yt = sales of refrigerators (in thousands) and the D’s are the dummies, taking a value of 1 in the relevant quarter and 0 otherwise. Note that

12 A time series may contain four components: a seasonal, a cyclical, a trend, and one that is strictly random. 13 For the various methods of seasonal adjustment, see, for instance, Francis X. Diebod, Elements of Forecasting, 2d ed., South-Western Publishers, 2001, Chap. 5.

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TABLE 9.3

QUARTERLY DATA ON APPLIANCE SALES (IN THOUSANDS) AND EXPENDITURE ON DURABLE GOODS (1978-I TO 1985-IV) DISH 841 957 999 960 894 851 863 878 792 589 657 699 675 652 628 529 DISP 798 837 821 858 837 838 832 818 868 623 662 822 871 791 759 734 FRIG 1317 1615 1662 1295 1271 1555 1639 1238 1277 1258 1417 1185 1196 1410 1417 919 WASH 1271 1295 1313 1150 1289 1245 1270 1103 1273 1031 1143 1101 1181 1116 1190 1125 DUR 252.6 272.4 270.9 273.9 268.9 262.9 270.9 263.4 260.6 231.9 242.7 248.6 258.7 248.4 255.5 240.4 DISH 480 530 557 602 658 749 827 858 808 840 893 950 838 884 905 909 DISP 706 582 659 837 867 860 918 1017 1063 955 973 1096 1086 990 1028 1003 FRIG 943 1175 1269 973 1102 1344 1641 1225 1429 1699 1749 1117 1242 1684 1764 1328 WASH 1036 1019 1047 918 1137 1167 1230 1081 1326 1228 1297 1198 1292 1342 1323 1274 DUR 247.7 249.1 251.8 262 263.3 280 288.5 300.5 312.6 322.5 324.3 333.1 344.8 350.3 369.1 356.4

Note: DISH = dishwashers; DISP = garbage disposers; FRIG = refrigerators; WASH = dishwashers; DUR = durable goods expenditure, billions of 1992 dollars. Source: Business Statistics and Survey of Current Business, Department of Commerce (various issues).

1800 1600 1400 1200 1000 800

Thousands of units

78

79

80

81

82 Year

83

84

85

86

FIGURE 9.4

Sales of refrigerators 1978–1985 (quarterly).

to avoid the dummy variable trap, we are assigning a dummy to each quarter of the year, but omitting the intercept term. If there is any seasonal effect in a given quarter, that will be indicated by a statistically signiﬁcant t value of the dummy coefﬁcient for that quarter.14

14 Note a technical point. This method of assigning a dummy to each quarter assumes that the seasonal factor, if present, is deterministic and not stochastic. We will revisit this topic when we discuss time series econometrics in Part V of this book.

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Notice that in (9.7.1) we are regressing Y effectively on an intercept, except that we allow for a different intercept in each season (i.e., quarter). As a result, the dummy coefﬁcient of each quarter will give us the mean refrigerator sales in each quarter or season (why?).

EXAMPLE 9.6 SEASONALITY IN REFRIGERATOR SALES From the data on refrigerator sales given in Table 9.3, we obtain the following regression results: ˆ Yt = 1222.125D1t + 1467.500D2t + 1569.750D3t + 1160.000D4t t= (20.3720) (24.4622) (26.1666) (19.3364) R = 0.5317

2

(9.7.2)

Note: We have not given the standard errors of the estimated coefﬁcients, as each standard error is equal to 59.9904, because all the dummies take only a value of 1 or zero. The estimated α coefﬁcients in (9.7.2) represent the average, or mean, sales of refrigerators (in thousands of units) in each season (i.e., quarter). Thus, the average sale of refrigerators in the ﬁrst quarter, in thousands of units, is about 1222, that in the second quarter about 1468, that in the third quarter about 1570, and that in the fourth quarter about 1160. TABLE 9.4 U.S. REFRIGERATOR SALES (THOUSANDS),1978–1995 (QUARTERLY) FRIG 1317 1615 1662 1295 1271 1555 1639 1238 1277 1258 1417 1185 1196 1410 1417 919 DUR 252.6 272.4 270.9 273.9 268.9 262.9 270.9 263.4 260.6 231.9 242.7 248.6 258.7 248.4 255.5 240.4 D2 0 1 0 0 0 1 0 0 0 1 0 0 0 1 0 0 D3 0 0 1 0 0 0 1 0 0 0 1 0 0 0 1 0 D4 0 0 0 1 0 0 0 1 0 0 0 1 0 0 0 1 FRIG 943 1175 1269 973 1102 1344 1641 1225 1429 1699 1749 1117 1242 1684 1764 1328 DUR 247.7 249.1 251.8 262.0 263.3 280.0 288.5 300.5 312.6 322.5 324.3 333.1 344.8 350.3 369.1 356.4 D2 0 1 0 0 0 1 0 0 0 1 0 0 0 1 0 0 D3 0 0 1 0 0 0 1 0 0 0 1 0 0 0 1 0 D4 0 0 0 1 0 0 0 1 0 0 0 1 0 0 0 1

Note: FRIG = refrigerator sales, thousands DUR = durable goods expenditure, billions of 1992 dollars D2 = 1 in the second quarter, 0 otherwise D3 = 1 in the third quarter, 0 otherwise D4 = 1 in the fourth quarter, 0 otherwise Source: Business Statistics and Survey of Current Business, Department of Commerce (various issues).

(Continued)

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EXAMPLE 9.6 (Continued) Incidentally, instead of assigning a dummy for each quarter and suppressing the intercept term to avoid the dummy variable trap, we could assign only three dummies and include the intercept term. Suppose we treat the ﬁrst quarter as the reference quarter and assign dummies to the second, third, and fourth quarters. This produces the following regression results (see Table 9.4 for the data setup): ˆ Yt = 1222.1250 + 245.3750D2t + 347.6250D3t − 62.1250D4t t= (20.3720)* (2.8922)* (4.0974)* (−0.7322)** R 2 = 0.5318 where * indicates p values less than 5 percent and ** indicates p values greater than 5 percent. Since we are treating the ﬁrst quarter as the benchmark, the coefﬁcients attached to the various dummies are now differential intercepts, showing by how much the average value of Y in the quarter that receives a dummy value of 1 differs from that of the benchmark quarter. Put differently, the coefﬁcients on the seasonal dummies will give the seasonal increase or decrease in the average value of Y relative to the base season. If you add the various differential intercept values to the benchmark average value of 1222.125, you will get the average value for the various quarters. Doing so, you will reproduce exactly Eq. (9.7.2), except for the rounding errors. But now you will see the value of treating one quarter as the benchmark quarter, for (9.7.3) shows that the average value of Y for the fourth quarter is not statistically different from the average value for the ﬁrst quarter, as the dummy coefﬁcient for the fourth quarter is not statistically signiﬁcant. Of course, your answer will change, depending on which quarter you treat as the benchmark quarter, but the overall conclusion will not change. How do we obtain the deseasonalized time series of refrigerator sales? This can be done easily. You estimate the values of Y from model (9.7.2) [or (9.7.3)] for each observation and ˆ subtract them from the actual values of Y, that is, you obtain (Yt −Yt ) which are simply the residuals from the regression (9.7.2). We show them in Table 9.5.15 What do these residuals represent? They represent the remaining components of the refrigerator time series, namely, the trend, cycle, and random components (but see the caution given in footnote 15). Since models (9.7.2) and (9.7.3) do not contain any covariates, will the picture change if we bring in a quantitative regressor in the model? Since expenditure on durable goods has an important factor inﬂuence on the demand for refrigerators, let us expand our model (9.7.3) by bringing in this variable. The data for durable goods expenditure in billions of 1982 dollars are already given in Table 9.3. This is our (quantitative) X variable in the model. The regression results are as follows ˆ Yt = 456.2440 + 242.4976D2t + 325.2643D3t − 86.0804D4t + 2.7734Xt t= (2.5593)* (3.6951)* (4.9421)* (−1.3073)** (4.4496)* R = 0.7298

2

(9.7.3)

(9.7.4)

where * indicates p values less than 5 percent and ** indicates p values greater than 5 percent. (Continued)

15 Of course, this assumes that the dummy variables technique is an appropriate method of deseasonalizing a time series and that a time series (TS) can be represented as: TS = s + c + t + u, where s represents the seasonal, t the trend, c the cyclical, and u the random component. However, if the time series is of the form, TS = (s)(c)(t)(u), where the four components enter multiplicatively, the preceding method of deseasonalization is inappropriate, for that method assumes that the four components of a time series are additive. But we will have more to say about this topic in the chapters on time series econometrics.

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EXAMPLE 9.6 (Continued) TABLE 9.5 REFRIGERATOR SALES REGRESSION: ACTUAL, FITTED, AND RESIDUAL VALUES (EQ. 9.7.3) Actual 1978-I 1978-II 1978-III 1978-IV 1979-I 1979-II 1979-III 1979-IV 1980-I 1980-II 1980-III 1980-IV 1981-I 1981-II 1981-III 1981-IV 1982-I 1982-II 1982-III 1982-IV 1983-I 1983-II 1983-III 1983-IV 1984-I 1984-II 1984-III 1984-IV 1985-I 1985-II 1985-III 1985-IV 1317 1615 1662 1295 1271 1555 1639 1238 1277 1258 1417 1185 1196 1410 1417 919 943 1175 1269 973 1102 1344 1641 1225 1429 1699 1749 1117 1242 1684 1764 1328 Fitted 1222.12 1467.50 1569.75 1160.00 1222.12 1467.50 1569.75 1160.00 1222.12 1467.50 1569.75 1160.00 1222.12 1467.50 1569.75 1160.00 1222.12 1467.50 1569.75 1160.00 1222.12 1467.50 1569.75 1160.00 1222.12 1467.50 1569.75 1160.00 1222.12 1467.50 1569.75 1160.00 Residuals 94.875 147.500 92.250 135.000 48.875 87.500 69.250 78.000 54.875 −209.500 −152.750 25.000 −26.125 −57.500 −152.750 −241.000 −279.125 −292.500 −300.750 −187.000 −120.125 −123.500 71.250 65.000 206.875 231.500 179.250 −43.000 19.875 216.500 194.250 168.000 Residual graph 0 . . . . . . . . . *. . * . . * . * .* *. * . * . * . *. . * .* . . . . . . * . . . . − *. *. *. *. * * * * . . . .

* . . . * . . . . . . . . . . . * . * . .* . * .* . * . .* .* * 0 +

Again, keep in mind that we are treating the ﬁrst quarter as our base. As in (9.7.3), we see that the differential intercept coefﬁcients for the second and third quarters are statistically different from that of the ﬁrst quarter, but the intercepts of the fourth quarter and the ﬁrst quarter are statistically about the same. The coefﬁcient of X (durable goods expenditure) of about 2.77 tells us that, allowing for seasonal effects, if expenditure on durable goods goes up by a dollar, on average, sales of refrigerators go up by about 2.77 units, that is, approximately 3 units; bear in mind that refrigerators are in thousands of units and X is in (1982) billions of dollars. (Continued)

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EXAMPLE 9.6 (Continued) An interesting question here is: Just as sales of refrigerators exhibit seasonal patterns, would not expenditure on durable goods also exhibit seasonal patterns? How then do we take into account seasonality in X? The interesting thing about (9.7.4) is that the dummy variables in that model not only remove the seasonality in Y but also the seasonality, if any, in X. (This follows from a well-known theorem in statistics, known as the Frisch–Waugh theorem.16) So to speak, we kill (deseasonalize) two birds (two series) with one stone (the dummy technique). If you want an informal proof of the preceding statement, just follow these steps: (1) Run the regression of Y on the dummies as in (9.7.2) or (9.7.3) and save the residuals, say, S 1; these residuals represent deseasonalized Y. (2) Run a similar regression for X and obtain the residuals from this regression, say, S 2; these residuals represent deseasonalized X. (3) Regress S1 on S 2. You will ﬁnd that the slope coefﬁcient in this regression is precisely the coefﬁcient of X in the regression (9.7.4).

9.8

PIECEWISE LINEAR REGRESSION

To illustrate yet another use of dummy variables, consider Figure 9.5, which shows how a hypothetical company remunerates its sales representatives. It pays commissions based on sales in such a manner that up to a certain level, the target, or threshold, level X*, there is one (stochastic) commission structure and beyond that level another. (Note: Besides sales, other factors affect sales commission. Assume that these other factors are represented

Y

Sales commission

II

I

X* FIGURE 9.5

X (sales)

Hypothetical relationship between sales commission and sales volume. (Note: The intercept on the Y axis denotes minimum guaranteed commission.)

16 For proof, see Adrian C. Darnell, A Dictionary of Econometrics, Edward Elgar, Lyme, U.K., 1995, pp. 150–152.

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by the stochastic disturbance term.) More speciﬁcally, it is assumed that sales commission increases linearly with sales until the threshold level X*, after which also it increases linearly with sales but at a much steeper rate. Thus, we have a piecewise linear regression consisting of two linear pieces or segments, which are labeled I and II in Figure 9.5, and the commission function changes its slope at the threshold value. Given the data on commission, sales, and the value of the threshold level X*, the technique of dummy variables can be used to estimate the (differing) slopes of the two segments of the piecewise linear regression shown in Figure 9.5. We proceed as follows: Yi = α1 + β1 Xi + β2 (Xi − X * )Di + ui (9.8.1)

where Yi = sales commission Xi = volume of sales generated by the sales person X* = threshold value of sales also known as a knot (known in advance)17 D = 1 if Xi > X * = 0 if Xi < X * Assuming E(ui ) = 0, we see at once that E(Yi | Di = 0, Xi , X * ) = α1 + β1 Xi (9.8.2)

which gives the mean sales commission up to the target level X * and E(Yi | Di = 1, Xi , X * ) = α1 − β2 X * + (β1 + β2 )Xi (9.8.3)

which gives the mean sales commission beyond the target level X * . Thus, β1 gives the slope of the regression line in segment I, and β1 + β2 gives the slope of the regression line in segment II of the piecewise linear regression shown in Figure 9.5. A test of the hypothesis that there is no break in the regression at the threshold value X * can be conducted easily by noting the statistical signiﬁcance of the estimated differential slope coefﬁˆ cient β2 (see Figure 9.6). Incidentally, the piecewise linear regression we have just discussed is an example of a more general class of functions known as spline functions.18

17 The threshold value may not always be apparent, however. An ad hoc approach is to plot the dependent variable against the explanatory variable(s) and observe if there seems to be a sharp change in the relation after a given value of X (i.e., X*). An analytical approach to ﬁnding the break point can be found in the so-called switching regression models. But this is an advanced topic and a textbook discussion may be found in Thomas Fomby, R. Carter Hill, and Stanley Johnson, Advanced Econometric Methods, Springer-Verlag, New York, 1984, Chap. 14. 18 For an accessible discussion on splines (i.e., piecewise polynomials of order k), see Douglas C. Montgomery, Elizabeth A. Peck, and G. Geoffrey Vining, Introduction to Linear Regression Analysis, John Wiley & Sons, 3d ed., New York, 2001, pp. 228–230.

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Y

Sales commission

β1 + β2 b b

1

1

β1 b

α1 a

X* a b α1 – β 2 X* X (sales)

FIGURE 9.6

Parameters of the piecewise linear regression.

EXAMPLE 9.7 TOTAL COST IN RELATION TO OUTPUT As an example of the application of the piecewise linear regression, consider the hypothetical total cost–total output data given in Table 9.6. We are told that the total cost may change its slope at the output level of 5500 units. Letting Y in (9.8.4) represent total cost and X total output, we obtain the following results:

TABLE 9.6 HYPOTHETICAL DATA ON OUTPUT AND TOTAL COST Total cost, dollars 256 414 634 778 1,003 1,839 2,081 2,423 2,734 2,914 Output, units 1,000 2,000 3,000 4,000 5,000 6,000 7,000 8,000 9,000 10,000

ˆ Yi = −145.72 t= (−0.8245)

+ 0.2791Xi + 0.0945(Xi − X * )Di i (6.0669) R 2 = 0.9737 (1.1447) X * = 5500 (9.8.4)

As these results show, the marginal cost of production is about 28 cents per unit and although it is about 37 cents (28 + 9) for output over 5500 units, the difference between the two is not statistically signiﬁcant because the dummy variable is not signiﬁcant at, say, the

5 percent level. For all practical purposes, then, one can regress total cost on total output, dropping the dummy variable.

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9.9

PANEL DATA REGRESSION MODELS

Recall that in Chapter 1 we discussed a variety of data that are available for empirical analysis, such as cross-section, time series, pooled (combination of time series and cross-section data), and panel data. The technique of dummy variable can be easily extended to pooled and panel data. Since the use of panel data is becoming increasingly common in applied work, we will consider this topic in some detail in Chapter 16.

9.10 SOME TECHNICAL ASPECTS OF THE DUMMY VARIABLE TECHNIQUE The Interpretation of Dummy Variables in Semilogarithmic Regressions

In Chapter 6 we discussed the log–lin models, where the regressand is logarithmic and the regressors are linear. In such a model, the slope coefﬁcients of the regressors give the semielasticity, that is, the percentage change in the regressand for a unit change in the regressor. This is only so if the regressor is quantitative. What happens if a regressor is a dummy variable? To be speciﬁc, consider the following model: ln Yi = β1 + β2 Di + ui (9.10.1)

where Y = hourly wage rate ($) and D = 1 for female and 0 for male. How do we interpret such a model? Assuming E(ui ) = 0, we obtain: Wage function for male workers: E(ln Yi | Di = 0) = β1 Wage function for female workers: E(ln Yi | Di = 1) = β1 + β2 (9.10.3) (9.10.2)

Therefore, the intercept β1 gives the mean log hourly earnings and the “slope” coefﬁcient gives the difference in the mean log hourly earnings of male and females. This is a rather awkward way of stating things. But if we take the antilog of β1 , what we obtain is not the mean hourly wages of male workers, but their median wages. As you know, mean, median, and mode are the three measures of central tendency of a random variable. And if we take the antilog of (β1 + β2 ), we obtain the median hourly wages of female workers.

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EXAMPLE 9.8 LOGARITHM OF HOURLY WAGES IN RELATION TO GENDER To illustrate (9.10.1), we use the data that underlie Example 9.2. The regression results based on 528 observations are as follows: ln Yi = 2.1763 − 0.2437Di (−5.5048)* R = 0.0544

2

t = (72.2943)*

(9.10.4)

where * indicates p values are practically zero. Taking the antilog of 2.1763, we ﬁnd 8.8136 ($), which is the median hourly earnings of male workers, and taking the antilog of [(2.1763 − 0.2437) = 1.92857],

we obtain 6.8796 ($), which is the median hourly earnings of female workers. Thus, the female workers’ median hourly earnings is lower by about 21.94 percent compared to their male counterparts [(8.8136 − 6.8796)/ 8.8136]. Interestingly, we can obtain semielasticity for a dummy regressor directly by the device suggested by Halvorsen and Palmquist.19 Take the antilog (to base e) of the estimated dummy coefﬁcient and subtract 1 from it and multiply the difference by 100. (For the underlying logic, see Appendix 9.A.1.) Therefore, if you take the antilog of −0.2437, you will obtain 0.78366. Subtracting 1 from this gives −0.2163, after multiplying this by 100, we get −21.63 percent, suggesting that a female worker’s (D = 1) median salary is lower than that of her male counterpart by about 21.63 percent, the same as we obtained previously, save the rounding errors.

Dummy Variables and Heteroscedasticity

Let us revisit our savings–income regression for the United States for the periods 1970–1981 and 1982–1995 and for the entire period 1970–1995. In testing for structural stability using the dummy technique, we assumed that the error var (u1i ) = var (u2i ) = σ 2 , that is, the error variances in the two periods were the same. This was also the assumption underlying the Chow test. If this assumption is not valid—that is, the error variances in the two subperiods are different—it is quite possible to draw misleading conclusions. Therefore, one must ﬁrst check on the equality of variances in the subperiod, using suitable statistical techniques. Although we will discuss this topic more thoroughly in the chapter on heteroscedasticity, in Chapter 8 we showed how the F test can be used for this purpose.20 (See our discussion of the Chow test in that chapter.) As we showed there, it seems the error variances in the two periods are not the same. Hence, the results of both the Chow test and the dummy variable technique presented before may not be entirely reliable. Of course, our purpose here is to illustrate the various techniques that one can use to handle a problem (e.g., the problem of structural stability). In any particular application, these techniques may not be valid. But that is par for most statistical techniques. Of course, one can take appropriate remedial actions to resolve the problem, as we will do in the chapter on heteroscedasticity later (however, see exercise 9.28).

19 Robert Halvorsen and Raymond Palmquist, “The Interpretation of Dummy Variables in Semilogarithmic Equations,” American Economic Review, vol. 70, no. 3, pp. 474–475. 20 The Chow test procedure can be performed even in the presence of heteroscedasticity, but then one will have to use the Wald test. The mathematics involved behind the test is somewhat involved. But in the chapter on heteroscedasticity, we will revisit this topic.

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Dummy Variables and Autocorrelation

Besides homoscedasticity, the classical linear regression model assumes that the error term in the regression models is uncorrelated. But what happens if that is not the case, especially in models involving dummy regressors? Since we will discuss the topic of autocorrelation in depth in the chapter on autocorrelation, we will defer the answer to this question until then.

What Happens if the Dependent Variable Is a Dummy Variable?

So far we have considered models in which the regressand is quantitative and the regressors are quantitative or qualitative or both. But there are occasions where the regressand can also be qualitative or dummy. Consider, for example, the decision of a worker to participate in the labor force. The decision to participate is of the yes or no type, yes if the person decides to participate and no otherwise. Thus, the labor force participation variable is a dummy variable. Of course, the decision to participate in the labor force depends on several factors, such as the starting wage rate, education, and conditions in the labor market (as measured by the unemployment rate). Can we still use OLS to estimate regression models where the regressand is dummy? Yes, mechanically, we can do so. But there are several statistical problems that one faces in such models. And since there are alternatives to OLS estimation that do not face these problems, we will discuss this topic in a later chapter (see Chapter 15 on logit and probit models). In that chapter we will also discuss models in which the regressand has more than two categories; for example, the decision to travel to work by car, bus, or train, or the decision to work part-time, full time, or not work at all. Such models are called polytomous dependent variable models in contrast to dichotomous dependent variable models in which the dependent variable has only two categories.

9.11 TOPICS FOR FURTHER STUDY

Several topics related to dummy variables are discussed in the literature that are rather advanced, including (1) random, or varying, parameters models, (2) switching regression models, and (3) disequilibrium models. In the regression models considered in this text it is assumed that the parameters, the β’s, are unknown but ﬁxed entities. The random coefﬁcient models—and there are several versions of them—assume the β’s can be random too. A major reference work in this area is by Swamy.21 In the dummy variable model using both differential intercepts and slopes, it is implicitly assumed that we know the point of break. Thus, in our savings–income example for 1970–1995, we divided the period into

21 P. A.V. B. Swamy, Statistical Inference in Random Coefﬁcient Regression Models, SpringerVerlag, Berlin, 1971.

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1970–1981 and 1982–1995, the pre- and postrecession periods, under the belief that the recession in 1982 changed the relation between savings and income. Sometimes it is not easy to pinpoint when the break took place. The technique of switching regression models (SRM) is developed for such situations. SRM treats the breakpoint as a random variable and through an iterative process determines when the break might have actually taken place. The seminal work in this area is by Goldfeld and Quandt.22 Special estimation techniques are required to deal with what are known as disequilibrium situations, that is, situations where markets do not clear (i.e., demand is not equal to supply). The classic example is that of demand for and supply of a commodity. The demand for a commodity is a function of its price and other variables, and the supply of the commodity is a function of its price and other variables, some of which are different from those entering the demand function. Now the quantity actually bought and sold of the commodity may not necessarily be equal to the one obtained by equating the demand to supply, thus leading to disequilibrium. For a thorough discussion of disequilibrium models, the reader may refer to Quandt.23

9.12 SUMMARY AND CONCLUSIONS

1. Dummy variables, taking values of 1 and zero (or their linear transforms), are a means of introducing qualitative regressors in regression models. 2. Dummy variables are a data-classifying device in that they divide a sample into various subgroups based on qualities or attributes (gender, marital status, race, religion, etc. ) and implicitly allow one to run individual regressions for each subgroup. If there are differences in the response of the regressand to the variation in the qualitative variables in the various subgroups, they will be reﬂected in the differences in the intercepts or slope coefﬁcients, or both, of the various subgroup regressions. 3. Although a versatile tool, the dummy variable technique needs to be handled carefully. First, if the regression contains a constant term, the number of dummy variables must be one less than the number of classiﬁcations of each qualitative variable. Second, the coefﬁcient attached to the dummy variables must always be interpreted in relation to the base, or reference, group—that is, the group that receives the value of zero. The base chosen will depend on the purpose of research at hand. Finally, if a model has several qualitative variables with several classes, introduction of dummy variables can consume a large number of degrees of freedom. Therefore, one should always weigh the number of dummy variables to be introduced against the total number of observations available for analysis.

22 S. Goldfeld and R. Quandt, Nonlinear Methods in Econometrics, North Holland, Amsterdam, 1972. 23 Richard E. Quandt, The Econometrics of Disequilibrium, Basil Blackwell, New York, 1988.

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4. Among its various applications, this chapter considered but a few. These included (1) comparing two (or more) regressions, (2) deseasonalizing time series data, (3) interactive dummies, (4) interpretation of dummies in semilog models, and (4) piecewise linear regression models. 5. We also sounded cautionary notes in the use of dummy variables in situations of heteroscedasticity and autocorrelation. But since we will cover these topics fully in subsequent chapters, we will revisit these topics then. EXERCISES

Questions 9.1. If you have monthly data over a number of years, how many dummy variables will you introduce to test the following hypotheses: a. All the 12 months of the year exhibit seasonal patterns. b. Only February, April, June, August, October, and December exhibit seasonal patterns. 9.2. Consider the following regression results (t ratios are in parentheses)*:

ˆ Yi = 1286

+ 104.97X2i − (3.70) (6.94)

0.026X3i + (−3.80)

1.20X4i + (0.24) (−3.04)

0.69X5i (0.08) (−6.14) n = 1543

t=

(4.67) (−0.40)

−19.47X6i + 266.06X7i

− 118.64X8i − 110.61X9i R = 0.383

2

where Y = wife’s annual desired hours of work, calculated as usual hours of work per year plus weeks looking for work X2 = after-tax real average hourly earnings of wife X3 = husband’s previous year after-tax real annual earnings X4 = wife’s age in years X5 = years of schooling completed by wife X6 = attitude variable, 1 = if respondent felt that it was all right for a woman to work if she desired and her husband agrees, 0 = otherwise X7 = attitude variable, 1 = if the respondent’s husband favored his wife’s working, 0 = otherwise X8 = number of children less than 6 years of age X9 = number of children in age groups 6 to 13 a. Do the signs of the coefﬁcients of the various nondummy regressors make economic sense? Justify your answer. b. How would you interpret the dummy variables, X6 and X7? Are these dummies statistically signiﬁcant? Since the sample is quite large, you may use the “2-t” rule of thumb to answer the question.

* Jane Leuthold, “The Effect of Taxation on the Hours Worked by Married Women,” Industrial and Labor Relations Review, no. 4, July 1978, pp. 520–526 (notation changed to suit our format).

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c. Why do you think that age and education variables are not signiﬁcant factors in a woman’s labor force participation decision in this study? 9.3. Consider the following regression results.* (The actual data are in Table 9.7.)

UN t =

2.7491 + 1.1507Dt − (3.6288)

1.5294Vt − (−12.5552)

0.8511(DtVt) (−1.9819) R2 = 0.9128

t = (26.896)

TABLE 9.7

DATA MATRIX FOR REGRESSION, IN EXERCISE 9.3 Year and quarter 1958-IV 1959-I -II -III -IV 1960-I -II -III -IV 1961-I -II -III -IV 1962-I -II -III -IV 1963-I -II -III -IV 1964-I -II -III -IV Unemployment rate UN, % 1.915 1.876 1.842 1.750 1.648 1.450 1.393 1.322 1.260 1.171 1.182 1.221 1.340 1.411 1.600 1.780 1.941 2.178 2.067 1.942 1.764 1.532 1.455 1.409 1.296 Job vacancy rate V, % 0.510 0.541 0.541 0.690 0.771 0.836 0.908 0.968 0.998 0.968 0.964 0.952 0.849 0.748 0.658 0.562 0.510 0.510 0.544 0.568 0.677 0.794 0.838 0.885 0.978 Year and quarter 1965-I -II -III -IV 1966-I -II -III -IV 1967-I -II -III -IV 1968-I -II -III -IV 1969-I -II -III -IV 1970-I -II -III -IV 1971-I -II Unemployment rate UN, % 1.201 1.192 1.259 1.192 1.089 1.101 1.243 1.623 1.821 1.990 2.114 2.115 2.150 2.141 2.167 2.107 2.104 2.056 2.170 2.161 2.225 2.241 2.366 2.324 2.516* 2.909* Job vacancy rate V, % 0.997 1.035 1.040 1.086 1.101 1.058 0.987 0.819 0.740 0.661 0.660 0.698 0.695 0.732 0.749 0.800 0.783 0.800 0.794 0.790 0.757 0.746 0.739 0.707 0.583* o.524*

D 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

DV 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

D 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0 0 0 0

DV

0 0 0 0.819 0.740 0.661 0.660 0.698 0.695 0.732 0.749 0.800 0.783 0.800 0.794 0.790 0.757 0.746 0.739 0.707 0.583* 0.524*

*Preliminary estimates. Source: Damodar Gujarati, “The Behaviour of Unemployment and Unﬁlled Vacancies: Great Britain, 1958–1971,” The Economic Journal, vol. 82, March 1972, p. 202.

* Damodar Gujarati, “The Behaviour of Unemployment and Unﬁlled Vacancies: Great Britain, 1958–1971,” The Economic Journal, vol. 82, March 1972, pp. 195–202.

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where UN = unemployment rate, % V = job vacancy rate, % D = 1, for period beginning in 1966–IV = 0, for period before 1966–IV t = time, measured in quarters Note: In the fourth quarter of 1966, the then Labor government liberalized the National Insurance Act by replacing the ﬂat-rate system of shortterm unemployment beneﬁts by a mixed system of ﬂat-rate and (previous) earnings-related beneﬁts, which increased the level of unemployment beneﬁts. a. What are your prior expectations about the relationship between the unemployment and vacancy rates? b. Holding the job vacancy rate constant, what is the average unemployment rate in the period beginning in the fourth quarter of 1966? Is it statistically different from the period before 1966 fourth quarter? How do you know? c. Are the slopes in the pre- and post-1966 fourth quarter statistically different? How do you know? d. Is it safe to conclude from this study that generous unemployment beneﬁts lead to higher unemployment rates? Does this make economic sense? 9.4. From annual data for 1972–1979, William Nordhaus estimated the following model to explain the OPEC’s oil price behavior (standard errors in parentheses).* yt = 0.3x1t + 5.22x2t ˆ se = (0.03) (0.50) where y = difference between current and previous year’s price (dollars per barrel) x1 = difference between current year’s spot price and OPEC’s price in the previous year x2 = 1 for 1974 and 0 otherwise Interpret this result and show the results graphically. What do these results suggest about OPEC’s monopoly power? 9.5. Consider the following model

Yi = α1 + α2 Di + β X i + ui

where Y = annual salary of a college professor X = years of teaching experience D = dummy for gender Consider three ways of deﬁning the dummy variable. a. D = 1 for male, 0 for female. b. D = 1 for female, 2 for male. c. D = 1 for female, −1 for male. Interpret the preceding regression model for each dummy assignment. Is one method preferable to another? Justify your answer.

* “Oil and Economic Performance in Industrial Countries,” Brookings Papers on Economic Activity, 1980, pp. 341–388.

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9.6. Refer to regression (9.7.3). How would you test the hypothesis that the coefﬁcients of D2 and D3 are the same? And that the coefﬁcients of D2 and D4 are the same? If the coefﬁcient of D3 is statistically different from that of D2 and the coefﬁcient of D4 is different from that of D2, does that mean that the coefﬁcients D3 and D4 are also different? Hint: var ( A ± B) = var (A) + var (B) ± 2 cov (A, B) 9.7. Refer to the U.S. savings–income example discussed in the chapter. a. How would you obtain the standard errors of the regression coefﬁcients given in (9.5.5) and (9.5.6), which were obtained from the pooled regression (9.5.4)? b. To obtain numerical answers, what additional information, if any, is required? 9.8. In his study on the labor hours spent by the FDIC (Federal Deposit Insurance Corporation) on 91 bank examinations, R. J. Miller estimated the following function*: ln Y = 2.41 + 0.3674 ln X1 +

0.2217 ln X2 + 0.0803 ln X3 (0.0628) −0.1755D1 (0.2905) (0.0287) + 0.2799D2 + 0.5634D3 − 0.2572D4 (0.1044) (0.1657) (0.0787) R2 = 0.766

(0.0477)

where Y X1 X2 X3 D1 D2 D3 D4

= = = = = = = =

FDIC examiner labor hours total assets of bank total number of ofﬁces in bank ratio of classiﬁed loans to total loans for bank 1 if management rating was “good” 1 if management rating was “fair” 1 if management rating was “satisfactory” 1 if examination was conducted jointly with the state

The ﬁgures in parentheses are the estimated standard errors. a. Interpret these results. b. Is there any problem in interpreting the dummy variables in this model since Y is in the log form? c. How would you interpret the dummy coefﬁcients? 9.9. To assess the effect of the Fed’s policy of deregulating interest rates beginning in July 1979, Sidney Langer, a student of mine, estimated the following model for the quarterly period of 1975–III to 1983–II.†

ˆ Yt = 8.5871 − 0.1328Pt − 0.7102Unt − 0.2389Mt

se (1.9563)

(0.0992)

(0.1909) (0.1036)

(0.0727) R2 = 0.9156 (0.7549)

+ 0.6592Yt−1 + 2.5831Dumt

* “Examination of Man-Hour Cost for Independent, Joint, and Divided Examination Programs,” Journal of Bank Research, vol. 11, 1980, pp. 28–35. Note: The notations have been altered to conform with our notations. † Sidney Langer, “Interest Rate Deregulation and Short-Term Interest Rates,” unpublished term paper.

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where

Y P Un M Dum

= = = = =

3-month Treasury bill rate expected rate of inﬂation seasonally adjusted unemployment rate changes in the monetary base dummy, taking value of 1 for observations beginning July 1, 1979

a. Interpret these results. b. What has been the effect of rate deregulation? Do the results make economic sense? c. The coefﬁcients of Pt , Unt, and Mt are negative. Can you offer an economic rationale? 9.10. Refer to the piecewise regression discussed in the text. Suppose there not only is a change in the slope coefﬁcient at X* but also the regression line jumps, as shown in Figure 9.7. How would you modify (9.8.1) to take into account the jump in the regression line at X*? 9.11. Determinants of price per ounce of cola. Cathy Schaefer, a student of mine, estimated the following regression from cross-sectional data of 77 observations*:

Pi = β0 + β1 D1i + β2 D2i + β3 D3i + µ i

where Pi = price per ounce of cola D1i = 001 if discount store = 010 if chain store = 100 if convenience store D2i = 10 if branded good = 01 if unbranded good D3i = 0001 if 67.6 ounce (2 liter) bottle = 0010 if 28–33.8 ounce bottles (Note: 33.8 oz = 1 liter) = 0100 if 16-ounce bottle = 1000 if 12-ounce can

Y

FIGURE 9.7

Discontinuous piecewise linear regression.

X*

X

* Cathy Schaefer, “Price Per Ounce of Cola Beverage as a Function of Place of Purchase, Size of Container, and Branded or Unbranded Product,” unpublished term project.

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The results were as follows:

ˆ Pi = 0.0143 −

0.000004D1i + 0.0090D2i + 0.00001D3i (0.00001) (−0.3837) (0.00011) (8.3927) (0.00000) (5.8125) R2 = 0.6033

Se = t=

Note: The standard errors are shown only to ﬁve decimal places. a. Comment on the way the dummies have been introduced in the model. b. Assuming the dummy setup is acceptable, how would you interpret the results? c. The coefﬁcient of D3 is positive and statistically signiﬁcant. How do you rationalize this result? 9.12. From data for 101 countries on per capita income in dollars (X) and life expectancy in years (Y ) in the early 1970s, Sen and Srivastava obtained the following regression results*:

ˆ Yi = −2.40 + 9.39 ln Xi − 3.36 [Di (ln Xi − 7)]

se = (4.73)

(0.859)

(2.42)

R2 = 0.752

where Di = 1 if ln X i > 7 , and Di = 0 otherwise. Note: When ln X i = 7, X = $1097 (approximately). a. What might be the reason(s) for introducing the income variable in the log form? b. How would you interpret the coefﬁcient 9.39 of ln X i ? c. What might be the reason for introducing the regressor Di (ln X i − 7)? How do you explain this regressor verbally? And how do you interpret the coefﬁcient −3.36 of this regressor (Hint: linear piecewise regression)? d. Assuming per capita income of $1097 as the dividing line between poorer and richer countries, how would you derive the regression for countries whose per capita is less than $1097 and the regression for countries whose per capita income is greater than $1097? e. What general conclusions do you draw from the regression result presented in this problem? 9.13. Consider the following model:

Yi = β1 + β2 Di + ui

where Di = 0 for the ﬁrst 20 observations and Di = 1 for the remaining 30 observations. You are also told that var (ui2 ) = 300 . a. How would you interpret β1 and β2? b. What are the mean values of the two groups? ˆ ˆ c. How would you compute the variance of (β1 + β2 ) ? Note: You are given ˆ1 , β2 ) = −15 . ˆ that the cov (β

* Ashish Sen and Muni Srivastava, Regression Analysis: Theory, Methods, and Applications, Springer-Verlag, New York, 1990, p. 92. Notation changed.

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9.14. To assess the effect of state right-to-work laws (which do not require membership in the union as a precondition of employment) on union membership, the following regression results were obtained, from the data for 50 states in the United States for 1982*:

PVTi = 19.8066 −

9.3917 RTWi (−5.1086) r2 = 0.3522

t = (17.0352)

where PVT = percentage of private sector employees in unions, 1982, and RTW = 1 if right-to-work law exists, 0 otherwise. Note: In 1982, twenty states had right-to-work laws. a. A priori, what is the expected relationship between PVT and RTW? b. Do the regression results support the prior expectations? c. Interpret the regression results. d. What was the average percent of private sector employees in unions in the states that did not have the right-to-work laws? 9.15. In the following regression model:

Yi = β1 + β2 Di + ui

Y represents hourly wage in dollars and D is the dummy variable, taking a value of 1 for a college graduate and a value of 0 for a high-school gradu¯ ˆ ate. Using the OLS formulas given in Chapter 3, show that β1 = Yhg and ¯ ¯ ˆ β2 = Ycg − Yhg , where the subscripts have the following meanings: hg = high-school graduate, cg = college graduate. In all, there are n1 high-school graduates and n2 college graduates, for a total sample of n = n1 + n2. 9.16. To study the rate of growth of population in Belize over the period 1970–1992, Mukherjee et al. estimated the following models†: Model I: ln (Pop)t = 4.73 + 4.77 + 0.024t (54.71) 0.015t − (34.01) 0.075Dt + (−17.03) 0.011(Dtt) (25.54) t = (781.25) Model II: ln (Pop)t = t = (2477.92)

where Pop = population in millions, t = trend variable, Dt = 1 for observations beginning in 1978 and 0 before 1978, and ln stands for natural logarithm. a. In Model I, what is the rate of growth of Belize’s population over the sample period? b. Are the population growth rates statistically different pre- and post1978? How do you know? If they are different, what are the growth rates for 1972–1977 and 1978–1992?

* The data used in the regression results were obtained from N. M. Meltz, “Interstate and Interprovincial Differences in Union Density,” Industrial Relations, vol. 28, no. 2, 1989, pp. 142–158. † Chandan Mukherjee, Howard White, and Marc Wuyts, Econometrics and Data Analysis for Developing Countries, Routledge, London, 1998, pp. 372–375. Notations adapted.

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Problems 9.17. Using the data given in Table 9.7, test the hypothesis that the error variances in the two subperiods 1958–IV to 1966–III and 1966–IV to 1971–II are the same. 9.18. Using the methodology discussed in Chapter 8, compare the unrestricted and restricted regressions (9.7.3) and (9.7.4); that is, test for the validity of the imposed restrictions. 9.19. In the U.S. savings–income regression (9.5.4) discussed in the chapter, suppose that instead of using 1 and 0 values for the dummy variable you use Z i = a + bDi , where Di = 1 and 0, a = 2, and b = 3. Compare your results. 9.20. Continuing with the savings–income regression (9.5.4), suppose you were to assign Di = 0 to observations in the second period and Di = 1 to observations in the ﬁrst period. How would the results shown in (9.5.4) change? 9.21. Use the data given in Table 9.2 and consider the following model: ln Savingsi = β1 + β2 ln Incomei + β3 ln Di + ui where ln stands for natural log and where Di = 1 for 1970–1981 and 10 for 1982–1995. a. What is the rationale behind assigning dummy values as suggested? b. Estimate the preceding model and interpret your results. c. What are the intercept values of the savings function in the two subperiods and how do you interpret them? 9.22. Refer to the quarterly appliance sales data given in Table 9.3. Consider the following model:

Salesi = α1 + α2 D2i + α3 D3i + α4 D4i + ui where the D’s are dummies taking 1 and 0 values for quarters II through IV. a. Estimate the preceding model for dishwashers, disposers, and washing machines individually. b. How would you interpret the estimated slope coefﬁcients? c. How would you use the estimated α’s to deseasonalize the sales data for individual appliances? 9.23. Reestimate the model in exercise 9.22 by adding the regressor, expenditure on durable goods. a. Is there a difference in the regression results you obtained in exercise 9.22 and in this exercise? If so, what explains the difference? b. If there is seasonality in the durable goods expenditure data, how would you account for it? 9.24. Table 9.8 gives data on quadrennial presidential elections in the United States from 1916 to 1996.* a. Using the data given in Table 9.6, develop a suitable model to predict the Democratic share of the two-party presidential vote. b. How would you use this model to predict the outcome of a presidential election?

* These data were originally compiled by Ray Fair of Yale University, who has been predicting the outcome of presidential elections for several years. The data are reproduced from Samprit Chatterjee, Ali S. Hadi, and Petram Price, Regression Analysis by Example, 3d ed., John Wiley & Sons, New York, 2000, pp. 150–151.

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TABLE 9.8

DATA ON U.S. PRESIDENTIAL ELECTIONS, 1916–1996 Year 1916 1920 1924 1928 1932 1936 1940 1944 1948 1952 1956 1960 1964 1968 1972 1976 1980 1984 1988 1992 1996

Notes: Year V I D W G P N

V 0.5168 0.3612 0.4176 0.4118 0.5916 0.6246 0.5500 0.5377 0.5237 0.4460 0.4224 0.5009 0.6134 0.4960 0.3821 0.5105 0.4470 0.4083 0.4610 0.5345 0.5474

W 0 1 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 0 0 0 0

D 1 0 −1 0 −1 1 1 1 1 0 −1 0 1 0 −1 0 1 −1 0 −1 1

G 2.229 −11.463 −3.872 4.623 −14.901 11.921 3.708 4.119 1.849 0.627 −1.527 0.114 5.054 4.836 6.278 3.663 −3.789 5.387 2.068 2.293 2.918

I 1 1 −1 −1 −1 1 1 1 1 1 −1 −1 1 1 −1 −1 1 −1 −1 −1 1

N 3 5 10 7 4 9 8 14 5 6 5 5 10 7 4 4 5 7 6 1 3

P 4.252 16.535 5.161 0.183 7.069 2.362 0.028 5.678 8.722 2.288 1.936 1.932 1.247 3.215 4.766 7.657 8.093 5.403 3.272 3.692 2.268

Election year Democratic share of the two-party presidential vote Indicator variable (1 if there is a Democratic incumbent at the time of the election and −1 if there is a Republican incumbent) Indicator variable (1 if a Democratic incumbent is running for election, −1 if a Republican incumbent is running for election, and 0 otherwise) Indicator variable (1 for the elections of 1920, 1944, and 1948, and 0 otherwise) Growth rate of real per capita GDP in the ﬁrst three quarters of the election year Absolute value of the growth rate of the GDP deﬂator in the ﬁrst 15 quarters of the administration Number of quarters in the ﬁrst 15 quarters of the administration in which the growth rate of real per capita GDP is greater than 3.2%

c. Chatterjee et al. suggested considering the following model as a trial model to predict presidential elections:

V = β0 + β1 I + β2 D + β3 W + β4 (G I) + β5 P + β6 N + u

Estimate this model and comment on the results in relation to the results of the model you have chosen. 9.25. Refer to regression (9.6.4). Test the hypothesis that the rate of increase of average hourly earnings with respect to education differs by gender and race. (Hint: Use multiplicative dummies.) 9.26. Refer to the regression (9.3.1). How would you modify the model to ﬁnd out if there is any interaction between the gender and the region of residence dummies? Present the results based on this model and compare them with those given in (9.3.1). 9.27. In the model Yi = β1 + β2 Di + ui , let Di = 0 for the ﬁrst 40 observations and Di = 1 for the remaining 60 observations. You are told that ui has zero

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mean and a variance of 100. What are the mean values and variances of the two sets of observations?* 9.28. Refer to the U.S. savings–income regression discussed in the chapter. As an alternative to (9.5.1), consider the following model: ln Yt = β1 + β2 Dt + β3 X t + β4 ( Dt X t ) + ut

where Y is savings and X is income. a. Estimate the preceding model and compare the results with those given in (9.5.4). Which is a better model? b. How would you interpret the dummy coefﬁcient in this model? c. As we will see in the chapter on heteroscedasticity, very often a log transformation of the dependent variable reduces heteroscedasticity in the data. See if this is the case in the present example by running the regression of log of Y on X for the two periods and see if the estimated error variances in the two periods are statistically the same. If they are, the Chow test can be used to pool the data in the manner indicated in the chapter.

APPENDIX 9A Semilogarithmic Regression with Dummy Regressor

In Section 9.10 we noted that in models of the type ln Yi = β1 + β2 Di (1)

the relative change in Y (i.e., semielasticity), with respect to the dummy regressor taking values of 1 or 0, can be obtained as (antilog of estimated β2) − 1 times 100, that is, as eβ2 − 1 × 100

ˆ

(2)

The proof is as follows: Since ln and exp (= e) are inverse functions, we can write (1) as: ln Yi = β1 + ln eβ2 Di (3) Now when D = 0, eβ2 Di = 1 and when D = 1, eβ2 Di = eβ2 . Therefore, in going from state 0 to state 1, ln Yi changes by (eβ2 − 1). But a change in the log of a variable is a relative change, which after multiplication by 100 becomes a percentage change. Hence the percentage change is (eβ2 − 1) × 100, as claimed. (Note: ln e e = 1, that is, the log of e to base e is 1, just as the log of 10 to base 10 is 1. Recall that log to base e is called the natural log and that log to base 10 is called the common log.)

* This example is adapted from Peter Kennedy, A Guide to Econometrics, 4th ed., MIT Press, Cambridge, Mass., 1998, p. 347.

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MULTICOLLINEARITY: WHAT HAPPENS IF THE REGRESSORS ARE CORRELATED?

There is no pair of words that is more misused both in econometrics texts and in the applied literature than the pair “multi-collinearity problem.” That many of our explanatory variables are highly collinear is a fact of life. And it is completely clear that there are experimental designs X X [i.e., data matrix] which would be much preferred to the designs the natural experiment has provided us [i.e., the sample at hand]. But a complaint about the apparent malevolence of nature is not at all constructive, and the ad hoc cures for a bad design, such as stepwise regression or ridge regression, can be disastrously inappropriate. Better that we should rightly accept the fact that our non-experiments [i.e., data not collected by designed experiments] are sometimes not very informative about parameters of interest.1

Assumption 10 of the classical linear regression model (CLRM) is that there is no multicollinearity among the regressors included in the regression model. In this chapter we take a critical look at this assumption by seeking answers to the following questions: 1. What is the nature of multicollinearity? 2. Is multicollinearity really a problem? 3. What are its practical consequences? 4. How does one detect it? 5. What remedial measures can be taken to alleviate the problem of multicollinearity?

1 Edward E. Leamer, “Model Choice and Speciﬁcation Analysis,” in Zvi Griliches and Michael D. Intriligator, eds., Handbook of Econometrics, vol. I, North Holland Publishing Company, Amsterdam, 1983, pp. 300–301.

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In this chapter we also discuss Assumption 7 of the CLRM, namely, that the number of observations in the sample must be greater than the number of regressors, and Assumption 8, which requires that there be sufﬁcient variability in the values of the regressors, for they are intimately related to the assumption of no multicollinearity. Arthur Goldberger has christened Assumption 7 as the problem of micronumerosity,2 which simply means small sample size.

10.1 THE NATURE OF MULTICOLLINEARITY

The term multicollinearity is due to Ragnar Frisch.3 Originally it meant the existence of a “perfect,” or exact, linear relationship among some or all explanatory variables of a regression model.4 For the k-variable regression involving explanatory variable X1 , X2 , . . . , Xk (where X1 = 1 for all observations to allow for the intercept term), an exact linear relationship is said to exist if the following condition is satisﬁed: λ1 X1 + λ2 X2 + · · · + λk Xk = 0 (10.1.1)

where λ1 , λ2 , . . . , λk are constants such that not all of them are zero simultaneously.5 Today, however, the term multicollinearity is used in a broader sense to include the case of perfect multicollinearity, as shown by (10.1.1), as well as the case where the X variables are intercorrelated but not perfectly so, as follows6: λ1 X1 + λ2 X2 + · · · + λ2 Xk + vi = 0 (10.1.2)

where vi is a stochastic error term. To see the difference between perfect and less than perfect multicollinearity, assume, for example, that λ2 = 0. Then, (10.1.1) can be written as X2i = − λ1 λ3 λk X1i − X3i − · · · − Xki λ2 λ2 λ2 (10.1.3)

2 See his A Course in Econometrics, Harvard University Press, Cambridge, Mass., 1991, p. 249. 3 Ragnar Frisch, Statistical Conﬂuence Analysis by Means of Complete Regression Systems, Institute of Economics, Oslo University, publ. no. 5, 1934. 4 Strictly speaking, multicollinearity refers to the existence of more than one exact linear relationship, and collinearity refers to the existence of a single linear relationship. But this distinction is rarely maintained in practice, and multicollinearity refers to both cases. 5 The chances of one’s obtaining a sample of values where the regressors are related in this fashion are indeed very small in practice except by design when, for example, the number of observations is smaller than the number of regressors or if one falls into the “dummy variable trap” as discussed in Chap. 9. See exercise 10.2. 6 If there are only two explanatory variables, intercorrelation can be measured by the zeroorder or simple correlation coefﬁcient. But if there are more than two X variables, intercorrelation can be measured by the partial correlation coefﬁcients or by the multiple correlation coefﬁcient R of one X variable with all other X variables taken together.

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which shows how X2 is exactly linearly related to other variables or how it can be derived from a linear combination of other X variables. In this situation, the coefﬁcient of correlation between the variable X2 and the linear combination on the right side of (10.1.3) is bound to be unity. Similarly, if λ2 = 0, Eq. (10.1.2) can be written as X2i = − λ1 λ3 λk 1 X1i − X3i − · · · − Xki − vi λ2 λ2 λ2 λ2 (10.1.4)

which shows that X2 is not an exact linear combination of other X’s because it is also determined by the stochastic error term vi. As a numerical example, consider the following hypothetical data:

X2 10 15 18 24 30 X3 50 75 90 120 150 * X3 52 75 97 129 152

It is apparent that X3i = 5X2i . Therefore, there is perfect collinearity between X2 and X3 since the coefﬁcient of correlation r23 is unity. The variable X* was created from X3 by simply adding to it the following numbers, which 3 were taken from a table of random numbers: 2, 0, 7, 9, 2. Now there is no longer perfect collinearity between X2 and X*. However, the two variables 3 are highly correlated because calculations will show that the coefﬁcient of correlation between them is 0.9959. The preceding algebraic approach to multicollinearity can be portrayed succinctly by the Ballentine (recall Figure 3.9, reproduced in Figure 10.1). In this ﬁgure the circles Y, X2, and X3 represent, respectively, the variations in Y (the dependent variable) and X2 and X3 (the explanatory variables). The degree of collinearity can be measured by the extent of the overlap (shaded area) of the X2 and X3 circles. In Figure 10.1a there is no overlap between X2 and X3, and hence no collinearity. In Figure 10.1b through 10.1e there is a “low” to “high” degree of collinearity—the greater the overlap between X2 and X3 (i.e., the larger the shaded area), the higher the degree of collinearity. In the extreme, if X2 and X3 were to overlap completely (or if X2 were completely inside X3, or vice versa), collinearity would be perfect. In passing, note that multicollinearity, as we have deﬁned it, refers only to linear relationships among the X variables. It does not rule out nonlinear relationships among them. For example, consider the following regression model: Yi = β0 + β1 Xi + β2 Xi2 + β3 Xi3 + ui (10.1.5)

where, say, Y = total cost of production and X = output. The variables Xi2 (output squared) and Xi3 (output cubed) are obviously functionally related

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Y Y

X3 X2

X2 X3

(a) No collinearity

(b) Low collinearity

Y Y Y

X2

X3

X2

X3

X2

X3

(c) Moderate collinearity FIGURE 10.1 The Ballentine view of multicollinearity.

(d ) High collinearity

(e) Very high collinearity

to Xi , but the relationship is nonlinear. Strictly, therefore, models such as (10.1.5) do not violate the assumption of no multicollinearity. However, in concrete applications, the conventionally measured correlation coefﬁcient will show Xi , Xi2 , and Xi3 to be highly correlated, which, as we shall show, will make it difﬁcult to estimate the parameters of (10.1.5) with greater precision (i.e., with smaller standard errors). Why does the classical linear regression model assume that there is no multicollinearity among the X’s? The reasoning is this: If multicollinearity is perfect in the sense of (10.1.1), the regression coefﬁcients of the X variables are indeterminate and their standard errors are inﬁnite. If multicollinearity is less than perfect, as in (10.1.2), the regression coefﬁcients, although determinate, possess large standard errors (in relation to the coefﬁcients themselves), which means the coefﬁcients cannot be estimated with great precision or accuracy. The proofs of these statements are given in the following sections.

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There are several sources of multicollinearity. As Montgomery and Peck note, multicollinearity may be due to the following factors7: 1. The data collection method employed, for example, sampling over a limited range of the values taken by the regressors in the population. 2. Constraints on the model or in the population being sampled. For example, in the regression of electricity consumption on income (X2) and house size (X3) there is a physical constraint in the population in that families with higher incomes generally have larger homes than families with lower incomes. 3. Model speciﬁcation, for example, adding polynomial terms to a regression model, especially when the range of the X variable is small. 4. An overdetermined model. This happens when the model has more explanatory variables than the number of observations. This could happen in medical research where there may be a small number of patients about whom information is collected on a large number of variables. An additional reason for multicollinearity, especially in time series data, may be that the regressors included in the model share a common trend, that is, they all increase or decrease over time. Thus, in the regression of consumption expenditure on income, wealth, and population, the regressors income, wealth, and population may all be growing over time at more or less the same rate, leading to collinearity among these variables.

10.2 ESTIMATION IN THE PRESENCE OF PERFECT MULTICOLLINEARITY

It was stated previously that in the case of perfect multicollinearity the regression coefﬁcients remain indeterminate and their standard errors are inﬁnite. This fact can be demonstrated readily in terms of the three-variable regression model. Using the deviation form, where all the variables are expressed as deviations from their sample means, we can write the threevariable regression model as ˆ ˆ yi = β2 x2i + β3 x3i + ui ˆ Now from Chapter 7 we obtain ˆ β2 = yi x2i

2 x2i 2 x3i − 2 x3i

(10.2.1)

yi x3i − x2i x3i

x2i x3i

2

(7.4.7)

7 Douglas Montgomery and Elizabeth Peck, Introduction to Linear Regression Analysis, John Wiley & Sons, New York, 1982, pp. 289–290. See also R. L. Mason, R. F. Gunst, and J. T. Webster, “Regression Analysis and Problems of Multicollinearity,” Communications in Statistics A, vol. 4, no. 3, 1975, pp. 277–292; R. F. Gunst, and R. L. Mason, “Advantages of Examining Multicollinearities in Regression Analysis,” Biometrics, vol. 33, 1977, pp. 249–260.

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ˆ β3 =

yi x3i

2 x2i

2 x2i − 2 x3i

yi x2i − x2i x3i

x2i x3i

2

(7.4.8)

Assume that X3i = λX2i , where λ is a nonzero constant (e.g., 2, 4, 1.8, etc.). Substituting this into (7.4.7), we obtain ˆ β2 = 0 = 0 ˆ which is an indeterminate expression. The reader can verify that β3 is also 8 indeterminate. ˆ Why do we obtain the result shown in (10.2.2)? Recall the meaning of β2 : It gives the rate of change in the average value of Y as X2 changes by a unit, holding X3 constant. But if X3 and X2 are perfectly collinear, there is no way X3 can be kept constant: As X2 changes, so does X3 by the factor λ. What it means, then, is that there is no way of disentangling the separate inﬂuences of X2 and X3 from the given sample: For practical purposes X2 and X3 are indistinguishable. In applied econometrics this problem is most damaging since the entire intent is to separate the partial effects of each X upon the dependent variable. To see this differently, let us substitute X3i = λX2i into (10.2.1) and obtain the following [see also (7.1.9)]: ˆ ˆ yi = β2 x2i + β3 (λx2i) + ui ˆ ˆ ˆ = (β2 + λβ3 )x2i + ui ˆ = αx2i + ui ˆ ˆ where ˆ ˆ α = (β2 + λβ3 ) ˆ (10.2.4) (10.2.3) yi x2i λ2

2 x2i − λ

yi x2i λ

2 x2i 2

2 x2i

2 x2i λ2

2 x2i − λ2

(10.2.2)

Applying the usual OLS formula to (10.2.3), we get ˆ ˆ α = (β2 + λβ3 ) = ˆ x2i yi 2 x2i (10.2.5)

Therefore, although we can estimate α uniquely, there is no way to estimate β2 and β3 uniquely; mathematically ˆ ˆ α = β2 + λβ3 ˆ

8

(10.2.6)

Another way of seeing this is as follows: By deﬁnition, the coefﬁcient of correlation 2 2 2 x2i x3i . If r2 3 = 1, i.e., perfect collinearity between X2 between X2 and X3, r2 3, is x2i x3i / and X3, the denominator of (7.4.7) will be zero, making estimation of β2 (or of β3) impossible.

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gives us only one equation in two unknowns (note λ is given) and there is an ˆ inﬁnity of solutions to (10.2.6) for given values of α and λ. To put this idea ˆ in concrete terms, let α = 0.8 and λ = 2. Then we have ˆ ˆ 0.8 = β2 + 2β3 or ˆ ˆ β2 = 0.8 − 2β3 (10.2.8) (10.2.7)

ˆ ˆ Now choose a value of β3 arbitrarily, and we will have a solution for β2 . ˆ3 , and we will have another solution for β2 . No ˆ Choose another value for β ˆ matter how hard we try, there is no unique value for β2 . The upshot of the preceding discussion is that in the case of perfect multicollinearity one cannot get a unique solution for the individual regression coefﬁcients. But notice that one can get a unique solution for linear combinations of these coefﬁcients. The linear combination (β2 + λβ3 ) is uniquely estimated by α, given the value of λ.9 In passing, note that in the case of perfect multicollinearity the variˆ ˆ ances and standard errors of β2 and β3 individually are inﬁnite. (See exercise 10.21.)

10.3 ESTIMATION IN THE PRESENCE OF “HIGH” BUT “IMPERFECT” MULTICOLLINEARITY

The perfect multicollinearity situation is a pathological extreme. Generally, there is no exact linear relationship among the X variables, especially in data involving economic time series. Thus, turning to the three-variable model in the deviation form given in (10.2.1), instead of exact multicollinearity, we may have x3i = λx2i + vi (10.3.1)

x2i vi = 0. where λ = 0 and where vi is a stochastic error term such that (Why?) Incidentally, the Ballentines shown in Figure 10.1b to 10.1e represent cases of imperfect collinearity. In this case, estimation of regression coefﬁcients β2 and β3 may be possible. For example, substituting (10.3.1) into (7.4.7), we obtain ˆ β2 = (yi x2i ) λ2

2 x2i + 2 x2i λ2 2 vi − λ 2 x2i +

yi x2i +

yi vi λ

2 x2i 2

2 x2i

2 vi − λ

(10.3.2)

where use is made of

9

ˆ x2i vi = 0. A similar expression can be derived for β3 .

In econometric literature, a function such as (β2 + λβ3) is known as an estimable function.

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Now, unlike (10.2.2), there is no reason to believe a priori that (10.3.2) cannot be estimated. Of course, if vi is sufﬁciently small, say, very close to zero, (10.3.1) will indicate almost perfect collinearity and we shall be back to the indeterminate case of (10.2.2).

10.4 MULTICOLLINEARITY: MUCH ADO ABOUT NOTHING? THEORETICAL CONSEQUENCES OF MULTICOLLINEARITY

Recall that if the assumptions of the classical model are satisﬁed, the OLS estimators of the regression estimators are BLUE (or BUE, if the normality assumption is added). Now it can be shown that even if multicollinearity is very high, as in the case of near multicollinearity, the OLS estimators still retain the property of BLUE.10 Then what is the multicollinearity fuss all about? As Christopher Achen remarks (note also the Leamer quote at the beginning of this chapter):

Beginning students of methodology occasionally worry that their independent variables are correlated—the so-called multicollinearity problem. But multicollinearity violates no regression assumptions. Unbiased, consistent estimates will occur, and their standard errors will be correctly estimated. The only effect of multicollinearity is to make it hard to get coefﬁcient estimates with small standard error. But having a small number of observations also has that effect, as does having independent variables with small variances. (In fact, at a theoretical level, multicollinearity, few observations and small variances on the independent variables are essentially all the same problem.) Thus “What should I do about multicollinearity?” is a question like “What should I do if I don’t have many observations?” No statistical answer can be given.11

To drive home the importance of sample size, Goldberger coined the term micronumerosity, to counter the exotic polysyllabic name multicollinearity. According to Goldberger, exact micronumerosity (the counterpart of exact multicollinearity) arises when n, the sample size, is zero, in which case any kind of estimation is impossible. Near micronumerosity, like near multicollinearity, arises when the number of observations barely exceeds the number of parameters to be estimated. Leamer, Achen, and Goldberger are right in bemoaning the lack of attention given to the sample size problem and the undue attention to the multicollinearity problem. Unfortunately, in applied work involving secondary data (i.e., data collected by some agency, such as the GNP data collected by the government), an individual researcher may not be able to do much about the size of the sample data and may have to face “estimating problems

10 Since near multicollinearity per se does not violate the other assumptions listed in Chap. 7, the OLS estimators are BLUE as indicated there. 11 Christopher H. Achen, Interpreting and Using Regression, Sage Publications, Beverly Hills, Calif., 1982, pp. 82–83.

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important enough to warrant our treating it [i.e., multicollinearity] as a violation of the CLR [classical linear regression] model.”12 First, it is true that even in the case of near multicollinearity the OLS estimators are unbiased. But unbiasedness is a multisample or repeated sampling property. What it means is that, keeping the values of the X variables ﬁxed, if one obtains repeated samples and computes the OLS estimators for each of these samples, the average of the sample values will converge to the true population values of the estimators as the number of samples increases. But this says nothing about the properties of estimators in any given sample. Second, it is also true that collinearity does not destroy the property of minimum variance: In the class of all linear unbiased estimators, the OLS estimators have minimum variance; that is, they are efﬁcient. But this does not mean that the variance of an OLS estimator will necessarily be small (in relation to the value of the estimator) in any given sample, as we shall demonstrate shortly. Third, multicollinearity is essentially a sample (regression) phenomenon in the sense that even if the X variables are not linearly related in the population, they may be so related in the particular sample at hand: When we postulate the theoretical or population regression function (PRF), we believe that all the X variables included in the model have a separate or independent inﬂuence on the dependent variable Y. But it may happen that in any given sample that is used to test the PRF some or all of the X variables are so highly collinear that we cannot isolate their individual inﬂuence on Y. So to speak, our sample lets us down, although the theory says that all the X’s are important. In short, our sample may not be “rich” enough to accommodate all X variables in the analysis. As an illustration, reconsider the consumption–income example of Chapter 3. Economists theorize that, besides income, the wealth of the consumer is also an important determinant of consumption expenditure. Thus, we may write Consumptioni = β1 + β2 Incomei + β3 Wealthi + ui Now it may happen that when we obtain data on income and wealth, the two variables may be highly, if not perfectly, correlated: Wealthier people generally tend to have higher incomes. Thus, although in theory income and wealth are logical candidates to explain the behavior of consumption expenditure, in practice (i.e., in the sample) it may be difﬁcult to disentangle the separate inﬂuences of income and wealth on consumption expenditure. Ideally, to assess the individual effects of wealth and income on consumption expenditure we need a sufﬁcient number of sample observations of wealthy individuals with low income, and high-income individuals with

12 Peter Kennedy, A Guide to Econometrics, 3d ed., The MIT Press, Cambridge, Mass., 1992, p. 177.

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low wealth (recall Assumption 8). Although this may be possible in crosssectional studies (by increasing the sample size), it is very difﬁcult to achieve in aggregate time series work. For all these reasons, the fact that the OLS estimators are BLUE despite multicollinearity is of little consolation in practice. We must see what happens or is likely to happen in any given sample, a topic discussed in the following section.

10.5 PRACTICAL CONSEQUENCES OF MULTICOLLINEARITY

In cases of near or high multicollinearity, one is likely to encounter the following consequences: 1. Although BLUE, the OLS estimators have large variances and covariances, making precise estimation difﬁcult. 2. Because of consequence 1, the conﬁdence intervals tend to be much wider, leading to the acceptance of the “zero null hypothesis” (i.e., the true population coefﬁcient is zero) more readily. 3. Also because of consequence 1, the t ratio of one or more coefﬁcients tends to be statistically insigniﬁcant. 4. Although the t ratio of one or more coefﬁcients is statistically insigniﬁcant, R2, the overall measure of goodness of ﬁt, can be very high. 5. The OLS estimators and their standard errors can be sensitive to small changes in the data. The preceding consequences can be demonstrated as follows.

Large Variances and Covariances of OLS Estimators

To see large variances and covariances, recall that for the model (10.2.1) the ˆ ˆ variances and covariances of β2 and β3 are given by ˆ var (β2 ) = ˆ var (β3 ) = ˆ ˆ cov (β2 , β3 ) = σ2 2 2 x2i 1 − r2 3

2 x3i

(7.4.12)

σ2 2 1 − r2 3 −r2 3 σ 2

(7.4.15)

2 1 − r2 3

(7.4.17)

2 x3i

2 x2i

where r2 3 is the coefﬁcient of correlation between X2 and X3. It is apparent from (7.4.12) and (7.4.15) that as r2 3 tends toward 1, that is, as collinearity increases, the variances of the two estimators increase and in the limit when r2 3 = 1, they are inﬁnite. It is equally clear from (7.4.17) that as r2 3 increases toward 1, the covariance of the two estimators also inˆ ˆ ˆ ˆ creases in absolute value. [Note: cov (β2 , β3 ) ≡ cov (β3 , β2 ).]

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The speed with which variances and covariances increase can be seen with the variance-inﬂating factor (VIF), which is deﬁned as VIF = 1 2 1 − r2 3 (10.5.1)

VIF shows how the variance of an estimator is inﬂated by the presence of 2 multicollinearity. As r2 3 approaches 1, the VIF approaches inﬁnity. That is, as the extent of collinearity increases, the variance of an estimator increases, and in the limit it can become inﬁnite. As can be readily seen, if there is no collinearity between X2 and X3, VIF will be 1. Using this deﬁnition, we can express (7.4.12) and (7.4.15) as ˆ var (β2 ) = ˆ var (β3 ) = σ2 VIF 2 x2i σ2 VIF 2 x3i (10.5.2)

(10.5.3)

ˆ ˆ which show that the variances of β2 and β3 are directly proportional to the VIF. To give some idea about how fast the variances and covariances increase as r2 3 increases, consider Table 10.1, which gives these variances and covariances for selected values of r2 3. As this table shows, increases in r2 3

TABLE 10.1 ˆ ˆ ˆ THE EFFECT OF INCREASING r 2 3 ON VAR ( β2 ) AND COV ( β2 , β3 ) ˆ var ( β2 ) (3)* σ2

2 x 2i

Value of r 2 3 (1) 0.00 0.50 0.70 0.80 0.90 0.95 0.97 0.99 0.995 0.999

Note: A = B= σ2 x2 2i −σ 2 x2 2i

VIF (2) 1.00 1.33 1.96 2.78 5.76 10.26 16.92 50.25 100.00 500.00

ˆ var ( β2 )(r 2 3 = 0) ˆ2 )(r 2 3 = 0) var ( β (4) — 1.33 1.96 2.78 5.26 10.26 16.92 50.25 100.00 500.00

ˆ ˆ cov ( β2 , β3 ) (5) 0 0.67 × B 1.37 × B 2.22 × B 4.73 × B 9.74 × B 16.41 × B 49.75 × B 99.50 × B 499.50 × B

=A

1.33 × A 1.96 × A 2.78 × A 5.26 × A 10.26 × A 16.92 × A 50.25 × A 100.00 × A 500.00 × A

x3 3i x 2 when r 2 3 = 0, but the variance 3i

× = times ˆ *To ﬁnd out the effect of increasing r 2 3 on var ( β3 ), note that A = σ 2 / and covariance magnifying factors remain the same.

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var ( β 2) A=

σ2 2 Σ x2i

5.26A

1.33A A 0 FIGURE 10.2 0.5 0.8 0.9 1.0

r2 3

ˆ The behavior of var ( β2 ) as a function of r 2 3.

have a dramatic effect on the estimated variances and covariances of the ˆ OLS estimators. When r2 3 = 0.50, the var (β2 ) is 1.33 times the variance when r2 3 is zero, but by the time r2 3 reaches 0.95 it is about 10 times as high as when there is no collinearity. And lo and behold, an increase of r2 3 from 0.95 to 0.995 makes the estimated variance 100 times that when collinearity is zero. The same dramatic effect is seen on the estimated covariance. All this can be seen in Figure 10.2. The results just discussed can be easily extended to the k-variable model. In such a model, the variance of the kth coefﬁcient, as noted in (7.5.6), can be expressed as: σ2 ˆ var (β j ) = xj2 where 1 1 − R2 j (7.5.6)

ˆ β j = (estimated) partial regression coefﬁcient of regressor Xj R2 = R2 in the regression of Xj on the remaining (k − 2) regressions j [Note: There are (k − 1) regressors in the k-variable regression model.] ¯ xj2 = (X j − X j )2

We can also write (7.5.6) as σ2 ˆ var (β j ) = VIF j xj2 (10.5.4)

ˆ As you can see from this expression, var (β j ) is proportional to σ 2 and VIF 2 ˆ xj . Thus, whether var (β j ) is large or small but inversely proportional to

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xj2 . The last will depend on the three ingredients: (1) σ 2 , (2) VIF, and (3) one, which ties in with Assumption 8 of the classical model, states that the larger the variability in a regressor, the smaller the variance of the coefﬁcient of that regressor, assuming the other two ingredients are constant, and therefore the greater the precision with which that coefﬁcient can be estimated. Before proceeding further, it may be noted that the inverse of the VIF is called tolerance (TOL). That is, TOLj = 1 = 1 − R2 j VIFj (10.5.5)

When R2 = 1 (i.e., perfect collinearity), TOLj = 0 and when R2 = 0 (i.e., no j j collinearity whatsoever), TOL j is 1. Because of the intimate connection between VIF and TOL, one can use them interchangeably.

Wider Conﬁdence Intervals

Because of the large standard errors, the conﬁdence intervals for the relevant population parameters tend to be larger, as can be seen from Table 10.2. For example, when r2 3 = 0.95, the conﬁdence interval for β2 is larger than when √ r2 3 = 0 by a factor of 10.26, or about 3. Therefore, in cases of high multicollinearity, the sample data may be compatible with a diverse set of hypotheses. Hence, the probability of accepting a false hypothesis (i.e., type II error) increases.

TABLE 10.2 THE EFFECT OF INCREASING COLLINEARITY ON THE 95% CONFIDENCE INTERVAL FOR ˆ ˆ β2 : β2 ± 1.96 se ( β2 ) Value of r 2 3 0.00 0.50 0.95 0.995 0.999 95% conﬁdence interval for β2 ˆ β2 ± 1.96 σ2 x2 2i σ2 x2 2i σ2 x2 2i σ2 x2 2i σ2 x2 2i

ˆ β2 ± 1.96 (1.33) ˆ β2 ± 1.96 (10.26) ˆ β2 ± 1.96 (100) ˆ β2 ± 1.96 (500)

Note: We are using the normal distribution because σ 2 is assumed for convenience to be known. Hence the use of 1.96, the 95% conﬁdence factor for the normal distribution. The standard errors corresponding to the various r2 3 values are obtained from Table 10.1.

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“Insigniﬁcant” t Ratios

Recall that to test the null hypothesis that, say, β2 = 0, we use the t ratio, ˆ ˆ that is, β2 /se (β2 ), and compare the estimated t value with the critical t value from the t table. But as we have seen, in cases of high collinearity the estimated standard errors increase dramatically, thereby making the t values smaller. Therefore, in such cases, one will increasingly accept the null hypothesis that the relevant true population value is zero.13

A High R 2 but Few Signiﬁcant t Ratios

Consider the k-variable linear regression model: Yi = β1 + β2 X2i + β3 X3i + · · · + βk Xki + ui In cases of high collinearity, it is possible to ﬁnd, as we have just noted, that one or more of the partial slope coefﬁcients are individually statistically insigniﬁcant on the basis of the t test. Yet the R2 in such situations may be so high, say, in excess of 0.9, that on the basis of the F test one can convincingly reject the hypothesis that β2 = β3 = · · · = βk = 0. Indeed, this is one of the signals of multicollinearity—insigniﬁcant t values but a high overall R2 (and a signiﬁcant F value)! We shall demonstrate this signal in the next section, but this outcome should not be surprising in view of our discussion on individual vs. joint testing in Chapter 8. As you may recall, the real problem here is the covariances between the estimators, which, as formula (7.4.17) indicates, are related to the correlations between the regressors.

Sensitivity of OLS Estimators and Their Standard Errors to Small Changes in Data

As long as multicollinearity is not perfect, estimation of the regression coefﬁcients is possible but the estimates and their standard errors become very sensitive to even the slightest change in the data. To see this, consider Table 10.3. Based on these data, we obtain the following multiple regression: ˆ Yi = 1.1939 + 0.4463X2i + 0.0030X3i (0.7737) t = (1.5431) (0.1848) (2.4151) (0.0851) (0.0358) r2 3 = 0.5523 df = 2 (10.5.6)

R2 = 0.8101

ˆ ˆ cov (β2 , β3 ) = −0.00868

13 In terms of the conﬁdence intervals, β2 = 0 value will lie increasingly in the acceptance region as the degree of collinearity increases.

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TABLE 10.3 HYPOTHETICAL DATA ON Y, X2, AND X3 Y 1 2 3 4 5 X2 2 0 4 6 8 X3 4 2 12 0 16

TABLE 10.4 HYPOTHETICAL DATA ON Y, X2, AND X3 Y 1 2 3 4 5 X2 2 0 4 6 8 X3 4 2 0 12 16

Regression (10.5.6) shows that none of the regression coefﬁcients is individually signiﬁcant at the conventional 1 or 5 percent levels of signiﬁcance, alˆ though β2 is signiﬁcant at the 10 percent level on the basis of a one-tail t test. Now consider Table 10.4. The only difference between Tables 10.3 and 10.4 is that the third and fourth values of X3 are interchanged. Using the data of Table 10.4, we now obtain ˆ Yi = 1.2108 + 0.4014X2i + 0.0270X3i (0.7480) t = (1.6187) (0.2721) (1.4752) (0.1252) (0.2158) (10.5.7)

r2 3 = 0.8285 ˆ2 , β3 ) = −0.0282 ˆ cov (β df = 2 ˆ As a result of a slight change in the data, we see that β2 , which was statistically signiﬁcant before at the 10 percent level of signiﬁcance, is no longer ˆ ˆ signiﬁcant even at that level. Also note that in (10.5.6) cov (β2 , β3 ) = −0.00868 whereas in (10.5.7) it is −0.0282, a more than threefold increase. All these changes may be attributable to increased multicollinearity: In (10.5.6) r2 3 = ˆ 0.5523, whereas in (10.5.7) it is 0.8285. Similarly, the standard errors of β2 ˆ3 increase between the two regressions, a usual symptom of collinearity. and β We noted earlier that in the presence of high collinearity one cannot estimate the individual regression coefﬁcients precisely but that linear combinations of these coefﬁcients may be estimated more precisely. This fact can be substantiated from the regressions (10.5.6) and (10.5.7). In the ﬁrst regression the sum of the two partial slope coefﬁcients is 0.4493 and in the second it is 0.4284, practically the same. Not only that, their standard errors are practically the same, 0.1550 vs. 0.1823.14 Note, however, the coefﬁcient of X3 has changed dramatically, from 0.003 to 0.027.

14

R2 = 0.8143

These standard errors are obtained from the formula ˆ ˆ se (β2 + β3 ) = ˆ ˆ ˆ ˆ var (β2 ) + var (β3 ) + 2 cov (β2 , β3 )

ˆ ˆ Note that increasing collinearity increases the variances of β2 and β3 , but these variances may be offset if there is high negative covariance between the two, as our results clearly point out.

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Consequences of Micronumerosity

In a parody of the consequences of multicollinearity, and in a tongue-incheek manner, Goldberger cites exactly similar consequences of micronumerosity, that is, analysis based on small sample size.15 The reader is advised to read Goldberger’s analysis to see why he regards micronumerosity as being as important as multicollinearity.

10.6 AN ILLUSTRATIVE EXAMPLE: CONSUMPTION EXPENDITURE IN RELATION TO INCOME AND WEALTH

To illustrate the various points made thus far, let us reconsider the consumption–income example of Chapter 3. In Table 10.5 we reproduce the data of Table 3.2 and add to it data on wealth of the consumer. If we assume that consumption expenditure is linearly related to income and wealth, then, from Table 10.5 we obtain the following regression: ˆ Yi = 24.7747 + 0.9415X2i − (6.7525) t = (3.6690) (0.8229) (1.1442) 0.0424X3i (0.0807) (−0.5261) df = 7 (10.6.1)

R2 = 0.9635

¯ R2 = 0.9531

Regression (10.6.1) shows that income and wealth together explain about 96 percent of the variation in consumption expenditure, and yet neither of the slope coefﬁcients is individually statistically signiﬁcant. Moreover, not only is the wealth variable statistically insigniﬁcant but also it has the wrong

TABLE 10.5 HYPOTHETICAL DATA ON CONSUMPTION EXPENDITURE Y, INCOME X2, AND WEALTH X3 Y, $ 70 65 90 95 110 115 120 140 155 150 X2, $ 80 100 120 140 160 180 200 220 240 260 X3, $ 810 1009 1273 1425 1633 1876 2052 2201 2435 2686

15

Goldberger, op. cit., pp. 248–250.

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TABLE 10.6

ANOVA TABLE FOR THE CONSUMPTION–INCOME–WEALTH EXAMPLE Source of variation Due to regression Due to residual SS 8,565.5541 324.4459 df 2 7 MSS 4,282.7770 46.3494

sign. A priori, one would expect a positive relationship between consumpˆ ˆ tion and wealth. Although β2 and β3 are individually statistically insigniﬁcant, if we test the hypothesis that β2 = β3 = 0 simultaneously, this hypothesis can be rejected, as Table 10.6 shows. Under the usual assumption we obtain F= 4282.7770 = 92.4019 46.3494 (10.6.2)

This F value is obviously highly signiﬁcant. It is interesting to look at this result geometrically. (See Figure 10.3.) Based on the regression (10.6.1), we have established the individual 95% conﬁdence intervals for β2 and β3 following the usual procedure discussed in Chapter 8. As these intervals show, individually each of them includes the value of zero. Therefore, individually we can accept the hypothesis that the β3 0.1484 95% confidence interval for β3

Joint 95% confidence interval for β2 and β3

–1.004

0

2.887

β2

95% confidence interval for β2

– 0.2332

FIGURE 10.3

Individual conﬁdence intervals for β2 and β3 and joint conﬁdence interval (ellipse) for β2 and β3.

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two partial slopes are zero. But, when we establish the joint conﬁdence interval to test the hypothesis that β2 = β3 = 0, that hypothesis cannot be accepted since the joint conﬁdence interval, actually an ellipse, does not include the origin.16 As already pointed out, when collinearity is high, tests on individual regressors are not reliable; in such cases it is the overall F test that will show if Y is related to the various regressors. Our example shows dramatically what multicollinearity does. The fact that the F test is signiﬁcant but the t values of X2 and X3 are individually insigniﬁcant means that the two variables are so highly correlated that it is impossible to isolate the individual impact of either income or wealth on consumption. As a matter of fact, if we regress X3 on X2, we obtain ˆ X3i = 7.5454 + 10.1909X2i (29.4758) t = (0.2560) (0.1643) (62.0405) R = 0.9979

2

(10.6.3)

which shows that there is almost perfect collinearity between X3 and X2. Now let us see what happens if we regress Y on X2 only: ˆ Yi = 24.4545 + (6.4138) t = (3.8128) 0.5091X2i (0.0357) (14.2432) R2 = 0.9621 (10.6.4)

In (10.6.1) the income variable was statistically insigniﬁcant, whereas now it is highly signiﬁcant. If instead of regressing Y on X2, we regress it on X3, we obtain ˆ Yi = 24.411 + (6.874) t = (3.551) 0.0498X3i (0.0037) (13.29) R = 0.9567

2

(10.6.5)

We see that wealth has now a signiﬁcant impact on consumption expenditure, whereas in (10.6.1) it had no effect on consumption expenditure. Regressions (10.6.4) and (10.6.5) show very clearly that in situations of extreme multicollinearity dropping the highly collinear variable will often make the other X variable statistically signiﬁcant. This result would suggest that a way out of extreme collinearity is to drop the collinear variable, but we shall have more to say about it in Section 10.8.

16 As noted in Sec. 5.3, the topic of joint conﬁdence interval is rather involved. The interested reader may consult the reference cited there.

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10.7

DETECTION OF MULTICOLLINEARITY

Having studied the nature and consequences of multicollinearity, the natural question is: How does one know that collinearity is present in any given situation, especially in models involving more than two explanatory variables? Here it is useful to bear in mind Kmenta’s warning:

1. Multicollinearity is a question of degree and not of kind. The meaningful distinction is not between the presence and the absence of multicollinearity, but between its various degrees. 2. Since multicollinearity refers to the condition of the explanatory variables that are assumed to be nonstochastic, it is a feature of the sample and not of the population. Therefore, we do not “test for multicollinearity” but can, if we wish, measure its degree in any particular sample.17

Since multicollinearity is essentially a sample phenomenon, arising out of the largely nonexperimental data collected in most social sciences, we do not have one unique method of detecting it or measuring its strength. What we have are some rules of thumb, some informal and some formal, but rules of thumb all the same. We now consider some of these rules. 1. High R2 but few signiﬁcant t ratios. As noted, this is the “classic” symptom of multicollinearity. If R2 is high, say, in excess of 0.8, the F test in most cases will reject the hypothesis that the partial slope coefﬁcients are simultaneously equal to zero, but the individual t tests will show that none or very few of the partial slope coefﬁcients are statistically different from zero. This fact was clearly demonstrated by our consumption–income–wealth example. Although this diagnostic is sensible, its disadvantage is that “it is too strong in the sense that multicollinearity is considered as harmful only when all of the inﬂuences of the explanatory variables on Y cannot be disentangled.”18 2. High pair-wise correlations among regressors. Another suggested rule of thumb is that if the pair-wise or zero-order correlation coefﬁcient between two regressors is high, say, in excess of 0.8, then multicollinearity is a serious problem. The problem with this criterion is that, although high zero-order correlations may suggest collinearity, it is not necessary that they be high to have collinearity in any speciﬁc case. To put the matter somewhat technically, high zero-order correlations are a sufﬁcient but not a necessary condition for the existence of multicollinearity because it can exist even though the zero-order or simple correlations are comparatively low (say, less than 0.50). To see this relationship, suppose we have a four-variable model: Yi = β1 + β2 X2i + β3 X3i + β4 X4i + ui

17 18

Jan Kmenta, Elements of Econometrics, 2d ed., Macmillan, New York, 1986, p. 431. Ibid., p. 439.

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and suppose that X4i = λ2 X2i + λ3 X3i where λ2 and λ3 are constants, not both zero. Obviously, X4 is an exact lin2 ear combination of X2 and X3, giving R4.23 = 1, the coefﬁcient of determination in the regression of X4 on X2 and X3. Now recalling the formula (7.11.5) from Chapter 7, we can write

2 R4.2 3 = 2 2 r4 2 + r4 3 − 2r4 2r4 3r2 3 2 1 − r2 3

(10.7.1)

2 But since R4.2 3 = 1 because of perfect collinearity, we obtain

1=

2 2 r4 2 + r4 3 − 2r4 2r4 3r2 3 2 1 − r2 3

(10.7.2)

It is not difﬁcult to see that (10.7.2) is satisﬁed by r4 2 = 0.5, r4 3 = 0.5, and r2 3 = −0.5, which are not very high values. Therefore, in models involving more than two explanatory variables, the simple or zero-order correlation will not provide an infallible guide to the presence of multicollinearity. Of course, if there are only two explanatory variables, the zero-order correlations will sufﬁce. 3. Examination of partial correlations. Because of the problem just mentioned in relying on zero-order correlations, Farrar and Glauber have suggested that one should look at the partial correlation coefﬁcients.19 Thus, 2 in the regression of Y on X2, X3, and X4, a ﬁnding that R1.2 3 4 is very high but 2 2 2 r1 2.3 4 , r1 3.2 4 , and r1 4.2 3 are comparatively low may suggest that the variables X2, X3, and X4 are highly intercorrelated and that at least one of these variables is superﬂuous. Although a study of the partial correlations may be useful, there is no guarantee that they will provide an infallible guide to multicollinearity, for it may happen that both R2 and all the partial correlations are sufﬁciently high. But more importantly, C. Robert Wichers has shown20 that the FarrarGlauber partial correlation test is ineffective in that a given partial correlation may be compatible with different multicollinearity patterns. The Farrar–Glauber test has also been severely criticized by T. Krishna Kumar21 and John O’Hagan and Brendan McCabe.22

19 D. E. Farrar and R. R. Glauber, “Multicollinearity in Regression Analysis: The Problem Revisited,” Review of Economics and Statistics, vol. 49, 1967, pp. 92–107. 20 “The Detection of Multicollinearity: A Comment,” Review of Economics and Statistics, vol. 57, 1975, pp. 365–366. 21 “Multicollinearity in Regression Analysis,” Review of Economics and Statistics, vol. 57, 1975, pp. 366–368. 22 “Tests for the Severity of Multicollinearity in Regression Analysis: A Comment,” Review of Economics and Statistics, vol. 57, 1975, pp. 368–370.

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4. Auxiliary regressions. Since multicollinearity arises because one or more of the regressors are exact or approximately linear combinations of the other regressors, one way of ﬁnding out which X variable is related to other X variables is to regress each Xi on the remaining X variables and compute the corresponding R2, which we designate as R2 ; each one of these rei gressions is called an auxiliary regression, auxiliary to the main regression of Y on the X’s. Then, following the relationship between F and R2 established in (8.5.11), the variable Fi = R2i ·x2 x3 ···xk (k − 2) x 1 − R2i ·x2 x3 ···xk x (n − k + 1) (10.7.3)

follows the F distribution with k − 2 and n − k + 1 df. In Eq. (10.7.3) n stands for the sample size, k stands for the number of explanatory variables including the intercept term, and R2i ·x2 x3 ···xk is the coefﬁcient of determination x in the regression of variable Xi on the remaining X variables.23 If the computed F exceeds the critical Fi at the chosen level of signiﬁcance, it is taken to mean that the particular Xi is collinear with other X’s; if it does not exceed the critical Fi, we say that it is not collinear with other X’s, in which case we may retain that variable in the model. If Fi is statistically signiﬁcant, we will still have to decide whether the particular Xi should be dropped from the model. This question will be taken up in Section 10.8. But this method is not without its drawbacks, for

. . . if the multicollinearity involves only a few variables so that the auxiliary regressions do not suffer from extensive multicollinearity, the estimated coefﬁcients may reveal the nature of the linear dependence among the regressors. Unfortunately, if there are several complex linear associations, this curve ﬁtting exercise may not prove to be of much value as it will be difﬁcult to identify the separate interrelationships.24

Instead of formally testing all auxiliary R2 values, one may adopt Klien’s rule of thumb, which suggests that multicollinearity may be a troublesome problem only if the R2 obtained from an auxiliary regression is greater than the overall R2, that is, that obtained from the regression of Y on all the regressors.25 Of course, like all other rules of thumb, this one should be used judiciously. 5. Eigenvalues and condition index. If you examine the SAS output of the Cobb–Douglas production function given in Appendix 7A.5 you will see

23 2 For example, Rx2 can be obtained by regressing X2i as follows: X2i = a1 + a3 X3i + a4 X4i + · · · + ak Xki + ui . ˆ 24 George G. Judge, R. Carter Hill, William E. Grifﬁths, Helmut Lütkepohl, and Tsoung-Chao Lee, Introduction to the Theory and Practice of Econometrics, John Wiley & Sons, New York, 1982, p. 621. 25 Lawrence R. Klien, An Introduction to Econometrics, Prentice-Hall, Englewood Cliffs, N.J., 1962, p. 101.

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that SAS uses eigenvalues and the condition index to diagnose multicollinearity. We will not discuss eigenvalues here, for that would take us into topics in matrix algebra that are beyond the scope of this book. From these eigenvalues, however, we can derive what is known as the condition number k deﬁned as Maximum eigenvalue k= Minimum eigenvalue and the condition index (CI) deﬁned as CI = Maximum eigenvalue √ = k Minimum eigenvalue

Then we have this rule of thumb. If k is between 100 and 1000 there is moderate to strong multicollinearity and if√ exceeds 1000 there is severe it multicollinearity. Alternatively, if the CI ( = k) is between 10 and 30, there is moderate to strong multicollinearity and if it exceeds 30 there is severe multicollinearity. For the illustrative example, k = 3.0/0.00002422 or about 123,864, and √ CI = 123, 864 = about 352; both k and the CI therefore suggest severe multicollinearity. Of course, k and CI can be calculated between the maximum eigenvalue and any other eigenvalue, as is done in the printout. (Note: The printout does not explicitly compute k, but that is simply the square of CI.) Incidentally, note that a low eigenvalue (in relation to the maximum eigenvalue) is generally an indication of near-linear dependencies in the data. Some authors believe that the condition index is the best available multicollinearity diagnostic. But this opinion is not shared widely. For us, then, the CI is just a rule of thumb, a bit more sophisticated perhaps. But for further details, the reader may consult the references.26 6. Tolerance and variance inﬂation factor. We have already introduced TOL and VIF. As R2 , the coefﬁcient of determination in the regression j of regressor Xj on the remaining regressors in the model, increases toward unity, that is, as the collinearity of Xj with the other regressors increases, VIF also increases and in the limit it can be inﬁnite. Some authors therefore use the VIF as an indicator of multicollinearity. The larger the value of VIFj, the more “troublesome” or collinear the variable Xj. As a rule of thumb, if the VIF of a variable exceeds 10, which will happen if R2 exceeds 0.90, that variable is said be highly collinear.27 j Of course, one could use TOLj as a measure of multicollinearity in view of its intimate connection with VIFj. The closer is TOLj to zero, the greater the degree of collinearity of that variable with the other regressors. On the

26 See especially D. A. Belsley, E. Kuh, and R. E. Welsch, Regression Diagnostics: Identifying Inﬂuential Data and Sources of Collinearity, John Wiley & Sons, New York, 1980, Chap. 3. However, this book is not for the beginner. 27 See David G. Kleinbaum, Lawrence L. Kupper, and Keith E. Muller, Applied Regression Analysis and other Multivariate Methods, 2d ed., PWS-Kent, Boston, Mass., 1988, p. 210.

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other hand, the closer TOLj is to 1, the greater the evidence that Xj is not collinear with the other regressors. VIF (or tolerance) as a measure of collinearity is not free of criticism. As ˆ (10.5.4) shows, var (β j ) depends on three factors: σ 2 , xj2 , and VIFj. A high VIF can be counterbalanced by a low σ 2 or a high xj2 . To put it differently, a high VIF is neither necessary nor sufﬁcient to get high variances and high standard errors. Therefore, high multicollinearity, as measured by a high VIF, may not necessarily cause high standard errors. In all this discussion, the terms high and low are used in a relative sense. To conclude our discussion of detecting multicollinearity, we stress that the various methods we have discussed are essentially in the nature of “ﬁshing expeditions,” for we cannot tell which of these methods will work in any particular application. Alas, not much can be done about it, for multicollinearity is speciﬁc to a given sample over which the researcher may not have much control, especially if the data are nonexperimental in nature— the usual fate of researchers in the social sciences. Again as a parody of multicollinearity, Goldberger cites numerous ways of detecting micronumerosity, such as developing critical values of the sample size, n*, such that micronumerosity is a problem only if the actual sample size, n, is smaller than n*. The point of Goldberger’s parody is to emphasize that small sample size and lack of variability in the explanatory variables may cause problems that are at least as serious as those due to multicollinearity.

10.8 REMEDIAL MEASURES

What can be done if multicollinearity is serious? We have two choices: (1) do nothing or (2) follow some rules of thumb.

Do Nothing

The “do nothing” school of thought is expressed by Blanchard as follows28:

When students run their ﬁrst ordinary least squares (OLS) regression, the ﬁrst problem that they usually encounter is that of multicollinearity. Many of them conclude that there is something wrong with OLS; some resort to new and often creative techniques to get around the problem. But, we tell them, this is wrong. Multicollinearity is God’s will, not a problem with OLS or statistical technique in general.

What Blanchard is saying is that multicollinearity is essentially a data deﬁciency problem (micronumerosity, again) and some times we have no choice over the data we have available for empirical analysis. Also, it is not that all the coefﬁcients in a regression model are statistically insigniﬁcant. Moreover, even if we cannot estimate one or more regression coefﬁcients with greater precision, a linear combination of them (i.e., estimable function) can be estimated relatively efﬁciently. As we saw in

28 Blanchard, O. J., Comment, Journal of Business and Economic Statistics, vol. 5, 1967, pp. 449–451. The quote is reproduced from Peter Kennedy, A Guide to Econometrics, 4th ed., MIT Press, Cambridge, Mass., 1998, p. 190.

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(10.2.3), we can estimate α uniquely, even if we cannot estimate its two components given there individually. Sometimes this is the best we can do with a given set of data.29

Rule-of-Thumb Procedures

One can try the following rules of thumb to address the problem of multicollinearity, the success depending on the severity of the collinearity problem. 1. A priori information. Suppose we consider the model Yi = β1 + β2 X2i + β3 X3i + ui where Y = consumption, X2 = income, and X3 = wealth. As noted before, income and wealth variables tend to be highly collinear. But suppose a priori we believe that β3 = 0.10β2 ; that is, the rate of change of consumption with respect to wealth is one-tenth the corresponding rate with respect to income. We can then run the following regression: Yi = β1 + β2 X2i + 0.10β2 X3i + ui = β1 + β2 Xi + ui ˆ ˆ where Xi = X2i + 0.1X3i . Once we obtain β2 , we can estimate β3 from the β2 and β3 . postulated relationship between How does one obtain a priori information? It could come from previous empirical work in which the collinearity problem happens to be less serious or from the relevant theory underlying the ﬁeld of study. For example, in the Cobb–Douglas–type production function (7.9.1), if one expects constant returns to scale to prevail, then (β2 + β3 ) = 1, in which case we could run the regression (8.7.14), regressing the output-labor ratio on the capital-labor ratio. If there is collinearity between labor and capital, as generally is the case in most sample data, such a transformation may reduce or eliminate the collinearity problem. But a warning is in order here regarding imposing such a priori restrictions, “. . . since in general we will want to test economic theory’s a priori predictions rather than simply impose them on data for which they may not be true.”30 However, we know from Section 8.7 how to test for the validity of such restrictions explicitly. 2. Combining cross-sectional and time series data. A variant of the extraneous or a priori information technique is the combination of crosssectional and time-series data, known as pooling the data. Suppose we want

29 For an interesting discussion on this, see Conlisk, J., “When Collinearity is Desirable,” Western Economic Journal, vol. 9, 1971, pp. 393–407. 30 Mark B. Stewart and Kenneth F. Wallis, Introductory Econometrics, 2d ed., John Wiley & Sons, A Halstead Press Book, New York, 1981, p. 154.

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to study the demand for automobiles in the United States and assume we have time series data on the number of cars sold, average price of the car, and consumer income. Suppose also that ln Yt = β1 + β2 ln Pt + β3 ln It + ut where Y = number of cars sold, P = average price, I = income, and t = time. Out objective is to estimate the price elasticity β2 and income elasticity β3 . In time series data the price and income variables generally tend to be highly collinear. Therefore, if we run the preceding regression, we shall be faced with the usual multicollinearity problem. A way out of this has been suggested by Tobin.31 He says that if we have cross-sectional data (for example, data generated by consumer panels, or budget studies conducted by various private and governmental agencies), we can obtain a fairly reliable estimate of the income elasticity β3 because in such data, which are at a point in time, the prices do not vary much. Let the cross-sectionally estiˆ mated income elasticity be β3 . Using this estimate, we may write the preceding time series regression as Yt* = β1 + β2 ln Pt + ut ˆ where Y * = ln Y − β3 ln I, that is, Y * represents that value of Y after removing from it the effect of income. We can now obtain an estimate of the price elasticity β2 from the preceding regression. Although it is an appealing technique, pooling the time series and crosssectional data in the manner just suggested may create problems of interpretation, because we are assuming implicitly that the cross-sectionally estimated income elasticity is the same thing as that which would be obtained from a pure time series analysis.32 Nonetheless, the technique has been used in many applications and is worthy of consideration in situations where the cross-sectional estimates do not vary substantially from one cross section to another. An example of this technique is provided in exercise 10.26. 3. Dropping a variable(s) and speciﬁcation bias. When faced with severe multicollinearity, one of the “simplest” things to do is to drop one of the collinear variables. Thus, in our consumption–income–wealth illustration, when we drop the wealth variable, we obtain regression (10.6.4), which shows that, whereas in the original model the income variable was statistically insigniﬁcant, it is now “highly” signiﬁcant. But in dropping a variable from the model we may be committing a speciﬁcation bias or speciﬁcation error. Speciﬁcation bias arises from

31 J. Tobin, “A Statistical Demand Function for Food in the U.S.A.,” Journal of the Royal Statistical Society, Ser. A, 1950, pp. 113–141. 32 For a thorough discussion and application of the pooling technique, see Edwin Kuh, Capital Stock Growth: A Micro-Econometric Approach, North-Holland Publishing Company, Amsterdam, 1963, Chaps. 5 and 6.

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incorrect speciﬁcation of the model used in the analysis. Thus, if economic theory says that income and wealth should both be included in the model explaining the consumption expenditure, dropping the wealth variable would constitute speciﬁcation bias. Although we will discuss the topic of speciﬁcation bias in Chapter 13, we caught a glimpse of it in Section 7.7. If, for example, the true model is Yi = β1 + β2 X2i + β3 X3i + ui but we mistakenly ﬁt the model Yi = b1 + b1 2 X2i + ui ˆ then it can be shown that (see Appendix 13A.1) E(b1 2 ) = β2 + β3 b3 2 (10.8.2) (10.8.1)

where b3 2 = slope coefﬁcient in the regression of X3 on X2. Therefore, it is obvious from (10.8.2) that b12 will be a biased estimate of β2 as long as b3 2 is different from zero (it is assumed that β3 is different from zero; otherwise there is no sense in including X3 in the original model).33 Of course, if b3 2 is zero, we have no multicollinearity problem to begin with. It is also clear from (10.8.2) that if both b3 2 and β3 are positive (or both are negative), E(b1 2) will be greater than β2 ; hence, on the average b1 2 will overestimate β2 , leading to a positive bias. Similarly, if the product b3 2β3 is negative, on the average b1 2 will underestimate β2 , leading to a negative bias. From the preceding discussion it is clear that dropping a variable from the model to alleviate the problem of multicollinearity may lead to the speciﬁcation bias. Hence the remedy may be worse than the disease in some situations because, whereas multicollinearity may prevent precise estimation of the parameters of the model, omitting a variable may seriously mislead us as to the true values of the parameters. Recall that OLS estimators are BLUE despite near collinearity. 4. Transformation of variables. Suppose we have time series data on consumption expenditure, income, and wealth. One reason for high multicollinearity between income and wealth in such data is that over time both the variables tend to move in the same direction. One way of minimizing this dependence is to proceed as follows. If the relation Yt = β1 + β2 X2t + β3 X3t + ut (10.8.3)

33 Note further that if b3 2 does not approach zero as the sample size is increased indeﬁnitely, then b1 2 will be not only biased but also inconsistent.

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holds at time t, it must also hold at time t − 1 because the origin of time is arbitrary anyway. Therefore, we have Yt−1 = β1 + β2 X2,t−1 + β3 X3,t−1 + ut−1 If we subtract (10.8.4) from (10.8.3), we obtain Yt − Yt−1 = β2 (X2t − X2,t−1 ) + β3 (X3t − X3,t−1 ) + vt (10.8.5) (10.8.4)

where vt = ut − ut−1 . Equation (10.8.5) is known as the ﬁrst difference form because we run the regression, not on the original variables, but on the differences of successive values of the variables. The ﬁrst difference regression model often reduces the severity of multicollinearity because, although the levels of X2 and X3 may be highly correlated, there is no a priori reason to believe that their differences will also be highly correlated. As we shall see in the chapters on time series econometrics, an incidental advantage of the ﬁrst-difference transformation is that it may make a nonstationary time series stationary. In those chapters we will see the importance of stationary time series. As noted in Chapter 1, loosely speaking, a time series, say, Yt, is stationary if its mean and variance do not change systematically over time. Another commonly used transformation in practice is the ratio transformation. Consider the model: Yt = β1 + β2 X2t + β3 X3t + ut (10.8.6)

where Y is consumption expenditure in real dollars, X2 is GDP, and X3 is total population. Since GDP and population grow over time, they are likely to be correlated. One “solution” to this problem is to express the model on a per capita basis, that is, by dividing (10.8.4) by X3, to obtain: Yt = β1 X3t 1 X3t + β2 X2t X3t + β3 + ut X3t (10.8.7)

Such a transformation may reduce collinearity in the original variables. But the ﬁrst-difference or ratio transformations are not without problems. For instance, the error term vt in (10.8.5) may not satisfy one of the assumptions of the classical linear regression model, namely, that the disturbances are serially uncorrelated. As we will see in Chapter 12, if the original disturbance term ut is serially uncorrelated, the error term vt obtained previously will in most cases be serially correlated. Therefore, the remedy may be worse than the disease. Moreover, there is a loss of one observation due to the differencing procedure, and therefore the degrees of freedom are

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reduced by one. In a small sample, this could be a factor one would wish at least to take into consideration. Furthermore, the ﬁrst-differencing procedure may not be appropriate in cross-sectional data where there is no logical ordering of the observations. Similarly, in the ratio model (10.8.7), the error term ut X3t will be heteroscedastic, if the original error term ut is homoscedastic, as we shall see in Chapter 11. Again, the remedy may be worse than the disease of collinearity. In short, one should be careful in using the ﬁrst difference or ratio method of transforming the data to resolve the problem of multicollinearity. 5. Additional or new data. Since multicollinearity is a sample feature, it is possible that in another sample involving the same variables collinearity may not be so serious as in the ﬁrst sample. Sometimes simply increasing the size of the sample (if possible) may attenuate the collinearity problem. For example, in the three-variable model we saw that ˆ var (β2 ) = σ2 2 2 x2i 1 − r2 3

2 x2i will generally increase. (Why?) Now as the sample size increases, ˆ Therefore, for any given r2 3, the variance of β2 will decrease, thus decreasing the standard error, which will enable us to estimate β2 more precisely. As an illustration, consider the following regression of consumption expenditure Y on income X2 and wealth X3 based on 10 observations34:

ˆ Yi = 24.377 + 0.8716X2i − t = (3.875) (2.7726)

0.0349X3i (−1.1595) R2 = 0.9682

(10.8.8)

The wealth coefﬁcient in this regression not only has the wrong sign but is also statistically insigniﬁcant at the 5 percent level. But when the sample size was increased to 40 observations (micronumerosity?), the following results were obtained: ˆ Yi = 2.0907 + 0.7299X2i + 0.0605X3i t = (0.8713) (6.0014) (2.0014) R2 = 0.9672 (10.8.9)

Now the wealth coefﬁcient not only has the correct sign but also is statistically signiﬁcant at the 5 percent level.

34

I am indebted to Albert Zucker for providing the results given in the following regressions.

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Obtaining additional or “better” data is not always that easy, for as Judge et al. note:

Unfortunately, economists seldom can obtain additional data without bearing large costs, much less choose the values of the explanatory variables they desire. In addition, when adding new variables in situations that are not controlled, we must be aware of adding observations that were generated by a process other than that associated with the original data set; that is, we must be sure that the economic structure associated with the new observations is the same as the original structure.35

6. Reducing collinearity in polynomial regressions. In Section 7.10 we discussed polynomial regression models. A special feature of these models is that the explanatory variable(s) appear with various powers. Thus, in the total cubic cost function involving the regression of total cost on output, (output)2, and (output)3, as in (7.10.4), the various output terms are going to be correlated, making it difﬁcult to estimate the various slope coefﬁcients precisely.36 In practice though, it has been found that if the explanatory variable(s) are expressed in the deviation form (i.e., deviation from the mean value), multicollinearity is substantially reduced. But even then the problem may persist,37 in which case one may want to consider techniques such as orthogonal polynomials.38 7. Other methods of remedying multicollinearity. Multivariate statistical techniques such as factor analysis and principal components or techniques such as ridge regression are often employed to “solve” the problem of multicollinearity. Unfortunately, these techniques are beyond the scope of this book, for they cannot be discussed competently without resorting to matrix algebra.39

10.9 IS MULTICOLLINEARITY NECESSARILY BAD? MAYBE NOT IF THE OBJECTIVE IS PREDICTION ONLY

It has been said that if the sole purpose of regression analysis is prediction or forecasting, then multicollinearity is not a serious problem because the higher the R2, the better the prediction.40 But this may be so “. . . as long as

Judge et al., op. cit., p. 625. See also Sec. 10.9. As noted, since the relationship between X, X 2, and X3 is nonlinear, polynomial regressions do not violate the assumption of no multicollinearity of the classical model, strictly speaking. 37 See R. A. Bradley and S. S. Srivastava, “Correlation and Polynomial Regression,” American Statistician, vol. 33, 1979, pp. 11–14. 38 See Norman Draper and Harry Smith, Applied Regression Analysis, 2d ed., John Wiley & Sons, New York, 1981, pp. 266–274. 39 A readable account of these techniques from an applied viewpoint can be found in Samprit Chatterjee and Bertram Price, Regression Analysis by Example, John Wiley & Sons, New York, 1977, Chaps. 7 and 8. See also H. D. Vinod, “A Survey of Ridge Regression and Related Techniques for Improvements over Ordinary Least Squares,” Review of Economics and Statistics, vol. 60, February 1978, pp. 121–131. 40 See R. C. Geary, “Some Results about Relations between Stochastic Variables: A Discussion Document,” Review of International Statistical Institute, vol. 31, 1963, pp. 163–181.

36 35

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the values of the explanatory variables for which predictions are desired obey the same near-exact linear dependencies as the original design [data] matrix X.”41 Thus, if in an estimated regression it was found that X2 = 2X3 approximately, then in a future sample used to forecast Y, X2 should also be approximately equal to 2X3, a condition difﬁcult to meet in practice (see footnote 35), in which case prediction will become increasingly uncertain.42 Moreover, if the objective of the analysis is not only prediction but also reliable estimation of the parameters, serious multicollinearity will be a problem because we have seen that it leads to large standard errors of the estimators. In one situation, however, multicollinearity may not pose a serious problem. This is the case when R2 is high and the regression coefﬁcients are individually signiﬁcant as revealed by the higher t values. Yet, multicollinearity diagnostics, say, the condition index, indicate that there is serious collinearity in the data. When can such a situation arise? As Johnston notes:

This can arise if individual coefﬁcients happen to be numerically well in excess of the true value, so that the effect still shows up in spite of the inﬂated standard error and/or because the true value itself is so large that even an estimate on the downside still shows up as signiﬁcant.43

10.10

AN EXTENDED EXAMPLE: THE LONGLEY DATA

We conclude this chapter by analyzing the data collected by Longley.44 Although originally collected to assess the computational accuracy of least-squares estimates in several computer programs, the Longley data has become the workhorse to illustrate several econometric problems, including multicollinearity. The data are reproduced in Table 10.7. The data are time series for the years 1947–1962 and pertain to Y = number of people employed, in thousands; X1 = GNP implicit price deﬂator; X2 = GNP, millions of dollars; X3 = number of people unemployed in thousands, X4 = number of people in the armed forces, X5 = noninstitutionalized population over 14 years of age; and X6 = year, equal to 1 in 1947, 2 in 1948, and 16 in 1962.

41 Judge et al., op. cit., p. 619. You will also ﬁnd on this page proof of why, despite collinearity, one can obtain better mean predictions if the existing collinearity structure also continues in the future samples. 42 For an excellent discussion, see E. Malinvaud, Statistical Methods of Econometrics, 2d ed., North-Holland Publishing Company, Amsterdam, 1970, pp. 220–221. 43 J. Johnston, Econometric Methods, 3d ed., McGraw-Hill, New York, 1984, p. 249. 44 Longley, J. “An Appraisal of Least-Squares Programs from the Point of the User,” Journal of the American Statistical Association, vol. 62, 1967, pp. 819–841.

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TABLE 10.7

LONGLEY DATA Observation 1947 1948 1949 1950 1951 1952 1953 1954 1955 1956 1957 1958 1959 1960 1961 1962 y 60,323 61,122 60,171 61,187 63,221 63,639 64,989 63,761 66,019 67,857 68,169 66,513 68,655 69,564 69,331 70,551 X1 830 885 882 895 962 981 990 1000 1012 1046 1084 1108 1126 1142 1157 1169 X2 234,289 259,426 258,054 284,599 328,975 346,999 365,385 363,112 397,469 419,180 442,769 444,546 482,704 502,601 518,173 554,894 X3 2356 2325 3682 3351 2099 1932 1870 3578 2904 2822 2936 4681 3813 3931 4806 4007 X4 1590 1456 1616 1650 3099 3594 3547 3350 3048 2857 2798 2637 2552 2514 2572 2827 X5 107,608 108,632 109,773 110,929 112,075 113,270 115,094 116,219 117,388 118,734 120,445 121,950 123,366 125,368 127,852 130,081 Time 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16

Source: See footnote 44.

Assume that our objective is to predict Y on the basis of the six X variables. Using Eviews3, we obtain the following regression results:

Dependent Variable: Y Sample: 1947–1962 Variable C X1 X2 X3 X4 X5 X6 Coefficient -3482259. 15.06187 -0.035819 -2.020230 -1.033227 -0.051104 1829.151 0.995479 0.992465 304.8541 836424.1 -109.6174 2.559488 Std. Error 890420.4 84.91493 0.033491 0.488400 0.214274 0.226073 455.4785 t-Statistic -3.910803 0.177376 -1.069516 -4.136427 -4.821985 -0.226051 4.015890 Prob. 0.0036 0.8631 0.3127 0.0025 0.0009 0.8262 0.0030 65317.00 3511.968 14.57718 14.91519 330.2853 0.000000

R-squared Adjusted R-squared S.E. of regression Sum squared resid Log likelihood Durbin-Watson stat

Mean dependent var S.D. dependent var Akaike info criterion Schwarz criterion F-statistic Prob(F-statistic)

A glance at these results would suggest that we have the collinearity problem, for the R2 value is very high, but quite a few variables are statistically insigniﬁcant (X1, X2, and X5), a classic symptom of multicollinearity. To shed more light on this, we show in Table 10.8 the intercorrelations among the six regressors.

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TABLE 10.8

INTERCORRELATIONS X1 X1 X2 X3 X4 X5 X6 1.000000 0.991589 0.620633 0.464744 0.979163 0.991149 X2 0.991589 1.000000 0.604261 0.446437 0.991090 0.995273 X3 0.620633 0.604261 1.000000 −0.177421 0.686552 0.668257 X4 0.464744 0.446437 −0.177421 1.000000 0.364416 0.417245 X5 0.979163 0.991090 0.686552 0.364416 1.000000 0.993953 X6 0.991149 0.995273 0.668257 0.417245 0.993953 1.000000

This table gives what is called the correlation matrix. In this table the entries on the main diagonal (those running from the upper left-hand corner to the lower right-hand corner) give the correlation of one variable with itself, which is always 1 by deﬁnition, and the entries off the main diagonal are the pair-wise correlations among the X variables. If you take the ﬁrst row of this table, this gives the correlation of X1 with the other X variables. For example, 0.991589 is the correlation between X1 and X2, 0.620633 is the correlation between X1 and X3, and so on. As you can see, several of these pair-wise correlations are quite high, suggesting that there may be a severe collinearity problem. Of course, remember the warning given earlier that such pair-wise correlations may be a sufﬁcient but not a necessary condition for the existence of multicollinearity. To shed further light on the nature of the multicollinearity problem, let us run the auxiliary regressions, that is the regression of each X variable on the remaining X variables. To save space, we will present only the R2 values obtained from these regressions, which are given in Table 10.9. Since the R2 values in the auxiliary regressions are very high (with the possible exception of the regression of X4) on the remaining X variables, it seems that we do have a serious collinearity problem. The same information is obtained from the tolerance factors. As noted previously, the closer the tolerance factor is to zero, the greater is the evidence of collinearity. Applying Klein’s rule of thumb, we see that the R2 values obtained from the auxiliary regressions exceed the overall R2 value (that is the one obtained from the regression of Y on all the X variables) of 0.9954 in 3 out of

TABLE 10.9

R 2 VALUES FROM THE AUXILIARY REGRESSIONS Dependent variable X1 X2 X3 X4 X5 X6 R 2 value 0.9926 0.9994 0.9702 0.7213 0.9970 0.9986 Tolerance (TOL) = 1 − R 2 0.0074 0.0006 0.0298 0.2787 0.0030 0.0014

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6 auxiliary regressions, again suggesting that indeed the Longley data are plagued by the multicollinearity problem. Incidentally, applying the F test given in (10.7.3) the reader should verify that the R2 values given in the preceding tables are all statistically signiﬁcantly different from zero. We noted earlier that the OLS estimators and their standard errors are sensitive to small changes in the data. In exercise 10.32 the reader is asked to rerun the regression of Y on all the six X variables but drop the last data observations, that is, run the regression for the period 1947–1961. You will see how the regression results change by dropping just a single year’s observations. Now that we have established that we have the multicollinearity problem, what “remedial” actions can we take? Let us reconsider our original model. First of all, we could express GNP not in nominal terms, but in real terms, which we can do by dividing nominal GNP by the implicit price deﬂator. Second, since noninstitutional population over 14 years of age grows over time because of natural population growth, it will be highly correlated with time, the variable X6 in our model. Therefore, instead of keeping both these variables, we will keep the variable X5 and drop X6. Third, there is no compelling reason to include X3, the number of people unemployed; perhaps the unemployment rate would have been a better measure of labor market conditions. But we have no data on the latter. So, we will drop the variable X3. Making these changes, we obtain the following regression results (RGNP = real GNP)45:

Dependent Variable: Y Sample: 1947–1962 Variable C RGNP X4 X5 Coefficient 65720.37 9.736496 -0.687966 -0.299537 0.981404 0.976755 535.4492 3440470. -120.9313 1.654069 Std. Error 10624.81 1.791552 0.322238 0.141761 t-Statistic 6.185558 5.434671 -2.134965 -2.112965 Prob. 0.0000 0.0002 0.0541 0.0562 65317.00 3511.968 15.61641 15.80955 211.0972 0.000000

R-squared Adjusted R-squared S.E. of regression Sum squared resid Log likelihood Durbin-Watson stat

Mean dependent var S.D. dependent var Akaike info criterion Schwarz criterion F-statistic Prob(F-statistic)

Although the R2 value has declined slightly compared with the original R2, it is still very high. Now all the estimated coefﬁcients are signiﬁcant and the signs of the coefﬁcients make economic sense.

45 The coefﬁcient of correlation between X5 and X6 is about 0.9939, a very high correlation indeed.

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We leave it for the reader to devise alternative models and see how the results change. Also keep in mind the warning sounded earlier about using the ratio method of transforming the data to alleviate the problem of collinearity. We will revisit this question in Chapter 11.

10.11 SUMMARY AND CONCLUSIONS

1. One of the assumptions of the classical linear regression model is that there is no multicollinearity among the explanatory variables, the X’s. Broadly interpreted, multicollinearity refers to the situation where there is either an exact or approximately exact linear relationship among the X variables. 2. The consequences of multicollinearity are as follows: If there is perfect collinearity among the X’s, their regression coefﬁcients are indeterminate and their standard errors are not deﬁned. If collinearity is high but not perfect, estimation of regression coefﬁcients is possible but their standard errors tend to be large. As a result, the population values of the coefﬁcients cannot be estimated precisely. However, if the objective is to estimate linear combinations of these coefﬁcients, the estimable functions, this can be done even in the presence of perfect multicollinearity. 3. Although there are no sure methods of detecting collinearity, there are several indicators of it, which are as follows: (a) The clearest sign of multicollinearity is when R2 is very high but none of the regression coefﬁcients is statistically signiﬁcant on the basis of the conventional t test. This case is, of course, extreme. (b) In models involving just two explanatory variables, a fairly good idea of collinearity can be obtained by examining the zero-order, or simple, correlation coefﬁcient between the two variables. If this correlation is high, multicollinearity is generally the culprit. (c) However, the zero-order correlation coefﬁcients can be misleading in models involving more than two X variables since it is possible to have low zero-order correlations and yet ﬁnd high multicollinearity. In situations like these, one may need to examine the partial correlation coefﬁcients. (d) If R2 is high but the partial correlations are low, multicollinearity is a possibility. Here one or more variables may be superﬂuous. But if R2 is high and the partial correlations are also high, multicollinearity may not be readily detectable. Also, as pointed out by C. Robert, Krishna Kumar, John O’Hagan, and Brendan McCabe, there are some statistical problems with the partial correlation test suggested by Farrar and Glauber. (e) Therefore, one may regress each of the Xi variables on the remaining X variables in the model and ﬁnd out the corresponding 2 2 coefﬁcients of determination R i . A high R i would suggest that Xi

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is highly correlated with the rest of the X’s. Thus, one may drop that Xi from the model, provided it does not lead to serious speciﬁcation bias. 4. Detection of multicollinearity is half the battle. The other half is concerned with how to get rid of the problem. Again there are no sure methods, only a few rules of thumb. Some of these rules are as follows: (1) using extraneous or prior information, (2) combining cross-sectional and time series data, (3) omitting a highly collinear variable, (4) transforming data, and (5) obtaining additional or new data. Of course, which of these rules will work in practice will depend on the nature of the data and severity of the collinearity problem. 5. We noted the role of multicollinearity in prediction and pointed out that unless the collinearity structure continues in the future sample it is hazardous to use the estimated regression that has been plagued by multicollinearity for the purpose of forecasting. 6. Although multicollinearity has received extensive (some would say excessive) attention in the literature, an equally important problem encountered in empirical research is that of micronumerosity, smallness of sample size. According to Goldberger, “When a research article complains about multicollinearity, readers ought to see whether the complaints would be convincing if “micronumerosity” were substituted for “multicollinearity.”46 He suggests that the reader ought to decide how small n, the number of observations, is before deciding that one has a small-sample problem, just as one decides how high an R2 value is in an auxiliary regression before declaring that the collinearity problem is very severe.

EXERCISES

Questions 10.1. In the k-variable linear regression model there are k normal equations to estimate the k unknowns. These normal equations are given in Appendix C. Assume that Xk is a perfect linear combination of the remaining X variables. How would you show that in this case it is impossible to estimate the k regression coefﬁcients? 10.2. Consider the set of hypothetical data in Table 10.10. Suppose you want to ﬁt the model

Yi = β1 + β2 X 2i + β3 X 3i + ui

to the data. a. Can you estimate the three unknowns? Why or why not? b. If not, what linear functions of these parameters, the estimable functions, can you estimate? Show the necessary calculations.

46

Goldberger, op. cit., p. 250.

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TABLE 10.10

Y −10 −8 −6 −4 −2 0 2 4 6 8 10

X2 1 2 3 4 5 6 7 8 9 10 11

X3 1 3 5 7 9 11 13 15 17 19 21

10.3. Refer to the child mortality example discussed in Chapter 8. The example there involved the regression of child mortality (CM) rate on per capita GNP (PGNP) and female literacy rate (FLR). Now suppose we add the variable, total fertility rate (TFR). This gives the following regression results.

Dependent Variable: CM Variable C PGNP FLR TFR Coefficient 168.3067 -0.005511 -1.768029 12.86864 0.747372 0.734740 39.13127 91875.38 -323.4298 2.170318 Std. Error 32.89165 0.001878 0.248017 4.190533 t-Statistic 5.117003 -2.934275 -7.128663 3.070883 Prob. 0.0000 0.0047 0.0000 0.0032 141.5000 75.97807 10.23218 10.36711 59.16767 0.000000

R-squared Adjusted R-squared S.E. of regression Sum squared resid Log likelihood Durbin-Watson stat

Mean dependent var S.D. dependent var Akaike info criterion Schwarz criterion F-statistic Prob(F-statistic)

a. Compare these regression results with those given in Eq. (8.2.1). What changes do you see? And how do you account for them? b. Is it worth adding the variable TFR to the model? Why? c. Since all the individual t coefﬁcients are statistically signiﬁcant, can we say that we do not have a collinearity problem in the present case? 10.4. If the relation λ1 X 1i + λ2 X 2i + λ3 X 3i = 0 holds true for all values of λ1 , λ2 , 2 2 2 and λ3 , estimate r 1 2.3 , r 1 3.2 , and r 2 3.1 . Also ﬁnd R1.2 3 , R2.1 3 , and R3.12 . 2 What is the degree of multicollinearity in this situation? Note: R1.2 3 is the coefﬁcient of determination in the regression of Y on X2 and X3. Other R2 values are to be interpreted similarly. 10.5. Consider the following model:

Yt = β1 + β2 X t + β3 X t−1 + β4 X t−2 + β5 X t−3 + β6 X t−4 + ut

where Y = consumption, X = income, and t = time. The preceding model postulates that consumption expenditure at time t is a function not only

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of income at time t but also of income through previous periods. Thus, consumption expenditure in the ﬁrst quarter of 2000 is a function of income in that quarter and the four quarters of 1999. Such models are called distributed lag models, and we shall discuss them in a later chapter. a. Would you expect multicollinearity in such models and why? b. If collinearity is expected, how would you resolve the problem? 10.6. Consider the illustrative example of Section 10.6. How would you reconcile the difference in the marginal propensity to consume obtained from (10.6.1) and (10.6.4)? 10.7. In data involving economic time series such as GNP, money supply, prices, income, unemployment, etc., multicollinearity is usually suspected. Why? 10.8. Suppose in the model

Yi = β1 + β2 X 2i + β3 X 3i + ui

that r2 3 , the coefﬁcient of correlation between X2 and X3, is zero. Therefore, someone suggests that you run the following regressions:

Yi = α1 + α2 X 2i + u1i Yi = γ1 + γ3 X 3i + u2i ˆ ˆ ˆ ˆ a. Will α2 = β2 and γ3 = β3 ? Why? ˆ ˆ ˆ b. Will β1 equal α1 or γ1 or some combination thereof? ˆ ˆ ˆ ˆ c. Will var (β2 ) = var (α2 ) and var (β3 ) = var (γ3 ) ? 10.9. Refer to the illustrative example of Chapter 7 where we ﬁtted the Cobb–Douglas production function to the Taiwanese agricultural sector. The results of the regression given in (7.9.4) show that both the labor and capital coefﬁcients are individually statistically signiﬁcant. a. Find out whether the variables labor and capital are highly correlated. b. If your answer to (a) is afﬁrmative, would you drop, say, the labor variable from the model and regress the output variable on capital input only? c. If you do so, what kind of speciﬁcation bias is committed? Find out the nature of this bias. 10.10. Refer to Example 7.4. For this problem the correlation matrix is as follows:

Xi Xi Xi2 Xi3 1 Xi2 0.9742 1.0 Xi3 0.9284 0.9872 1.0

a. “Since the zero-order correlations are very high, there must be serious multicollinearity.” Comment. b. Would you drop variables X i2 and X i3 from the model? c. If you drop them, what will happen to the value of the coefﬁcient of X i ?

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10.11. Stepwise regression. In deciding on the “best” set of explanatory variables for a regression model, researchers often follow the method of stepwise regression. In this method one proceeds either by introducing the X variables one at a time (stepwise forward regression) or by including all the possible X variables in one multiple regression and rejecting them one at a time (stepwise backward regression). The decision to add or drop a variable is usually made on the basis of the contribution of that variable to the ESS, as judged by the F test. Knowing what you do now about multicollinearity, would you recommend either procedure? Why or why not?* 10.12. State with reason whether the following statements are true, false, or uncertain: a. Despite perfect multicollinearity, OLS estimators are BLUE. b. In cases of high multicollinearity, it is not possible to assess the individual signiﬁcance of one or more partial regression coefﬁcients. c. If an auxiliary regression shows that a particular R2 is high, there is i deﬁnite evidence of high collinearity. d. High pair-wise correlations do not suggest that there is high multicollinearity. e. Multicollinearity is harmless if the objective of the analysis is prediction only. f. Ceteris paribus, the higher the VIF is, the larger the variances of OLS estimators. g. The tolerance (TOL) is a better measure of multicollinearity than the VIF. h. You will not obtain a high R2 value in a multiple regression if all the partial slope coefﬁcients are individually statistically insigniﬁcant on the basis of the usual t test. i. In the regression of Y on X2 and X3, suppose there is little variability ˆ in the values of X3. This would increase var (β3 ) . In the extreme, if all ˆ X3 are identical, var (β3 ) is inﬁnite. 10.13. a. Show that if r1i = 0 for i = 2, 3, . . . , k then

R1.2 3. . . k = 0

b. What is the importance of this ﬁnding for the regression of variable X1(= Y) on X2, X3, . . . , Xk? 10.14. Suppose all the zero-order correlation coefﬁcients of X1(= Y), X2, . . . , Xk are equal to r. 2 a. What is the value of R1.2 3 . . . k ? b. What are the values of the ﬁrst-order correlation coefﬁcients? ** 10.15. In matrix notation it can be shown (see Appendix C) that

ˆ β = (X X)−1 X y ˆ a. What happens to β when there is perfect collinearity among the X’s? b. How would you know if perfect collinearity exists?

* See if your reasoning agrees with that of Arthur S. Goldberg and D. B. Jochems, “Note on Stepwise Least-Squares,” Journal of the American Statistical Association, vol. 56, March 1961, pp. 105–110. ** Optional.

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*

10.16. Using matrix notation, it can be shown var–cov ( ˆ ) = σ 2 (X X)−1

What happens to this var–cov matrix: a. When there is perfect multicollinearity? b. When collinearity is high but not perfect? * 10.17. Consider the following correlation matrix:

X2 1 X2 R = X3 r3 2 ··· X k r k2 X3 r2 3 1 ··· r k3 ··· ··· ··· ··· ··· Xk r 2k r 3k 1

How would you ﬁnd out from the correlation matrix whether (a) there is perfect collinearity, (b) there is less than perfect collinearity, and (c) the X’s are uncorrelated. Hint: You may use |R| to answer these questions, where |R| denotes the determinant of R. * 10.18. Orthogonal explanatory variables. Suppose in the model

Yi = β1 + β2 X 2i + β3 X 3i + · · · + βk X ki + ui

X2 to Xk are all uncorrelated. Such variables are called orthogonal variables. If this is the case: a. What will be the structure of the (X X) matrix? b. How would you obtain ˆ = (X X)−1X y? c. What will be the nature of the var–cov matrix of ˆ ? d. Suppose you have run the regression and afterward you want to introduce another orthogonal variable, say, X k+1 into the model. Do ˆ ˆ you have to recompute all the previous coefﬁcients β1 to βk ? Why or why not? 10.19. Consider the following model:

GNP t = β1 + β2 M t + β3 M t−1 + β4 (M t − M t−1 ) + ut

where GNPt = GNP at time t, M t = money supply at time t, M t−1 = money supply at time (t − 1), and (M t − M t−1 ) = change in the money supply between time t and time (t − 1). This model thus postulates that the level of GNP at time t is a function of the money supply at time t and time (t − 1) as well as the change in the money supply between these time periods. a. Assuming you have the data to estimate the preceding model, would you succeed in estimating all the coefﬁcients of this model? Why or why not? b. If not, what coefﬁcients can be estimated?

*

Optional.

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c. Suppose that the β3 M t−1 term were absent from the model. Would your answer to (a) be the same? d. Repeat (c), assuming that the term β2 M t were absent from the model. 10.20. Show that (7.4.7) and (7.4.8) can also be expressed as

ˆ β2 = ˆ β3 = yi x2i

2 x3i − 2 x2i 2 x3i

yi x3i 1− yi x2i

2 r2 3

x2i x3i

yi x3i

2 x2i − 2 x2i 2 x3i

x2i x3i

2 r2 3

1−

where r 2 3 is the coefﬁcient of correlation between X2 and X3. 10.21. Using (7.4.12) and (7.4.15), show that when there is perfect collinearity, ˆ ˆ the variances of β2 and β3 are inﬁnite. 10.22. Verify that the standard errors of the sums of the slope coefﬁcients estimated from (10.5.6) and (10.5.7) are, respectively, 0.1549 and 0.1825. (See Section 10.5.) 10.23. For the k-variable regression model, it can be shown that the variance of the kth (k = 2, 3, . . . , K ) partial regression coefﬁcient given in (7.5.6) can also be expressed as*

ˆ var (βk ) =

2 1 σy 2 n − k σk

1 − R2

2 1 − Rk

2 2 where σ y = variance of Y, σk = variance of the kth explanatory variable, 2 Rk = R2 from the regression of X k on the remaining X variables, and R2 = coefﬁcient of determination from the multiple regression, that is, regression of Y on all the X variables. 2 ˆ a. Other things the same, if σk increases, what happens to var (βk )? What are the implications for the multicollinearity problem? b. What happens to the preceding formula when collinearity is perfect? ˆ c. True or false: “The variance of βk decreases as R2 rises, so that the ef2 fect of a high Rk can be offset by a high R2 .” 10.24. From the annual data for the U.S. manufacturing sector for 1899–1922, Dougherty obtained the following regression results†:

log Y = 2.81 − 0.53 log K + 0.91 log L + 0.047t

se = (1.38)

(0.34)

(0.14) R = 0.97

2

(0.021) F = 189.8

(1)

where Y = index of real output, K = index of real capital input, L = index of real labor input, t = time or trend.

* This formula is given by R. Stone, “The Analysis of Market Demand,” Journal of the Royal Statistical Society, vol. B7, 1945, p. 297. Also recall (7.5.6). For further discussion, see Peter Kennedy, A Guide to Econometrics, 2d ed., The MIT Press, Cambridge, Mass., 1985, p. 156. † Christopher Dougherty, Introduction to Econometrics, Oxford University Press, New York, 1992, pp. 159–160.

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Using the same data, he also obtained the following regression: log (Y/L) = −0.11 + 0.11 log (K/ L) + 0.006t

se = (0.03)

(0.15)

2

(0.006) R = 0.65 F = 19.5

(2)

a. Is there multicollinearity in regression (1)? How do you know? b. In regression (1), what is the a priori sign of log K? Do the results conform to this expectation? Why or why not? c. How would you justify the functional form of regression (1)? (Hint: Cobb–Douglas production function.) d. Interpret regression (1). What is the role of the trend variable in this regression? e. What is the logic behind estimating regression (2)? f. If there was multicollinearity in regression (1), has that been reduced by regression (2)? How do you know? g. If regression (2) is a restricted version of regression (1), what restriction is imposed by the author? (Hint: returns to scale.) How do you know if this restriction is valid? Which test do you use? Show all your calculations. h. Are the R2 values of the two regressions comparable? Why or why not? How would you make them comparable, if they are not comparable in the present form? 10.25. Critically evaluate the following statements: a. “In fact, multicollinearity is not a modeling error. It is a condition of deﬁcient data.”* b. “If it is not feasible to obtain more data, then one must accept the fact that the data one has contain a limited amount of information and must simplify the model accordingly. Trying to estimate models that are too complicated is one of the most common mistakes among inexperienced applied econometricians.”† c. “It is common for researchers to claim that multicollinearity is at work whenever their hypothesized signs are not found in the regression results, when variables that they know a priori to be important have insigniﬁcant t values, or when various regression results are changed substantively whenever an explanatory variable is deleted. Unfortunately, none of these conditions is either necessary or sufﬁcient for the existence of collinearity, and furthermore none provides any useful suggestions as to what kind of extra information might be required to solve the estimation problem they present.”‡

* Samprit Chatterjee, Ali S. Hadi, and Betram Price, Regression Analysis by Example, 3d ed., John Wiley & Sons, New York, 2000, p. 226. † Russel Davidson and James G. MacKinnon, Estimation and Inference in Econometrics, Oxford University Press, New York, 1993, p. 186. ‡ Peter Kennedy, A Guide to Econometrics, 4th ed., MIT Press, Cambridge, Mass., 1998, p. 187.

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d. “. . . any time series regression containing more than four independent variables results in garbage.”* Problems

10.26. Klein and Goldberger attempted to ﬁt the following regression model to the U.S. economy:

Yi = β1 + β2 X 2i + β3 X 3i + β4 X 4i + ui

where Y = consumption, X2 = wage income, X3 = nonwage, nonfarm income, and X4 = farm income. But since X2, X3, and X4 are expected to be highly collinear, they obtained estimates of β3 and β4 from crosssectional analysis as follows: β3 = 0.75β2 and β4 = 0.625β2 . Using these estimates, they reformulated their consumption function as follows:

Yi = β1 + β2 ( X 2i + 0.75 X 3i + 0.625 X 4i ) + ui = β1 + β2 Z i + ui

where Z i = X 2i + 0.75 X 3i + 0.625 X 4i . a. Fit the modiﬁed model to the data in Table 10.11 and obtain estimates of β1 to β4 . b. How would you interpret the variable Z? 10.27. Table 10.12 gives data on imports, GDP, and the Consumer Price Index (CPI) for the United States over the period 1970–1998. You are asked to consider the following model: ln Imports t = β1 + β2 ln GDPt + β3 ln CPI t + ut

a. Estimate the parameters of this model using the data given in the table. b. Do you suspect that there is multicollinearity in the data?

TABLE 10.11

Year 1936 1937 1938 1939 1940 1941 1945*

Y 62.8 65.0 63.9 67.5 71.3 76.6 86.3

X2 43.41 46.44 44.35 47.82 51.02 58.71 87.69

X3 17.10 18.65 17.09 19.28 23.24 28.11 30.29

X4 3.96 5.48 4.37 4.51 4.88 6.37 8.96

Year 1946 1947 1948 1949 1950 1951 1952

Y 95.7 98.3 100.3 103.2 108.9 108.5 111.4

X2 76.73 75.91 77.62 78.01 83.57 90.59 95.47

X3 28.26 27.91 32.30 31.39 35.61 37.58 35.17

X4 9.76 9.31 9.85 7.21 7.39 7.98 7.42

*The data for the war years 1942–1944 are missing. The data for other years are billions of 1939 dollars. Source: L. R. Klein and A. S. Goldberger, An Economic Model of the United States, 1929–1952, North Holland Publishing Company, Amsterdam, 1964, p. 131.

* This quote attributed to the late econometrician Zvi Griliches, is obtained from Ernst R. Berndt, The Practice of Econometrics: Classic and Contemporary, Addison Wesley, Reading, Mass., 1991, p. 224.

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TABLE 10.12

U.S. IMPORTS, GDP, AND CPI, 1970–1998 Observation 1970 1971 1972 1973 1974 1975 1976 1977 1978 1979 1980 1981 1982 1983 1984 CPI 38.8 40.5 41.8 44.4 49.3 53.8 56.9 60.6 65.2 72.6 82.4 90.9 96.5 99.6 103.9 GDP 1039.7 1128.6 1240.4 1385.5 1501.0 1635.2 1823.9 2031.4 2295.9 2566.4 2795.0 3131.3 3259.2 3534.9 3932.7 Imports 39,866 45,579 55,797 70,499 103,811 98,185 124,228 151,907 176,002 212,007 249,750 265,067 247,642 268,901 332,418 Observation 1985 1986 1987 1988 1989 1990 1991 1992 1993 1994 1995 1996 1997 1998 CPI 107.6 109.6 113.6 118.3 124.0 130.7 136.2 140.3 144.5 148.2 152.4 156.9 160.5 163.0 GDP 4213.0 4452.9 4742.5 5108.3 5489.1 5803.2 5986.2 6318.9 6642.3 7054.3 7400.5 7813.2 8300.8 8759.9 Imports 338,088 368,425 409,765 447,189 477,365 498,337 490,981 536,458 589,441 668,590 749,574 803,327 876,366 917,178

c. Regress: (1) ln Importst = A1 + A2 ln GDPt (2) ln Importst = B1 + B2 ln CPIt (3) ln GDPt = C1 + C2 ln CPIt On the basis of these regressions, what can you say about the nature of multicollinearity in the data? ˆ ˆ d. Suppose there is multicollinearity in the data but β2 and β3 are individually signiﬁcant at the 5 percent level and the overall F test is also signiﬁcant. In this case should we worry about the collinearity problem? 10.28. Refer to Exercise 7.19 about the demand function for chicken in the United States. a. Using the log-linear, or double-log, model, estimate the various auxiliary regressions. How many are there? b. From these auxiliary regressions, how do you decide which of the regressor(s) are highly collinear? Which test do you use? Show the details of your calculations. c. If there is signiﬁcant collinearity in the data, which variable(s) would you drop to reduce the severity of the collinearity problem? If you do that, what econometric problems do you face? d. Do you have any suggestions, other than dropping variables, to ameliorate the collinearity problem? Explain. 10.29. Table 10.13 gives data on new passenger cars sold in the United States as a function of several variables. a. Develop a suitable linear or log-linear model to estimate a demand function for automobiles in the United States. b. If you decide to include all the regressors given in the table as explanatory variables, do you expect to face the multicollinearity problem? Why? c. If you do, how would you go about resolving the problem? State your assumptions clearly and show all the calculations explicitly.

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TABLE 10.13

Year 1971 1972 1973 1974 1975 1976 1977 1978 1979 1980 1981 1982 1983 1984 1985 1986

Y 10,227 10,872 11,350 8,775 8,539 9,994 11,046 11,164 10,559 8,979 8,535 7,980 9,179 10,394 11,039 11,450

X2 112.0 111.0 111.1 117.5 127.6 135.7 142.9 153.8 166.0 179.3 190.2 197.6 202.6 208.5 215.2 224.4

X3 121.3 125.3 133.1 147.7 161.2 170.5 181.5 195.3 217.7 247.0 272.3 286.6 297.4 307.6 318.5 323.4

X4 776.8 839.6 949.8 1,038.4 1,142.8 1,252.6 1,379.3 1,551.2 1,729.3 1,918.0 2,127.6 2,261.4 2,428.1 2,670.6 2,841.1 3,022.1

X5 4.89 4.55 7.38 8.61 6.16 5.22 5.50 7.78 10.25 11.28 13.73 11.20 8.69 9.65 7.75 6.31

X6 79,367 82,153 85,064 86,794 85,846 88,752 92,017 96,048 98,824 99,303 100,397 99,526 100,834 105,005 107,150 109,597

Y = new passenger cars sold (thousands), seasonally unadjusted X2 = new cars, Consumer Price Index, 1967 = 100, seasonally unadjusted X3 = Consumer Price Index, all items, all urban consumers, 1967 = 100, seasonally unadjusted X4 = the personal disposable income (PDI), billions of dollars, unadjusted for seasonal variation X5 = the interest rate, percent, ﬁnance company paper placed directly X6 = the employed civilian labor force (thousands), unadjusted for seasonal variation Source: Business Statistics, 1986, A Supplement to the Current Survey of Business, U.S. Department of Commerce.

10.30. To assess the feasibility of a guaranteed annual wage (negative income tax), the Rand Corporation conducted a study to assess the response of labor supply (average hours of work) to increasing hourly wages.* The data for this study were drawn from a national sample of 6000 households with a male head earnings less than $15,000 annually. The data were divided into 39 demographic groups for analysis. These data are given in Table 10.14. Because data for four demographic groups were missing for some variables, the data given in the table refer to only 35 demographic groups. The deﬁnitions of the various variables used in the analysis are given at the end of the table. a. Regress average hours worked during the year on the variables given in the table and interpret your regression. b. Is there evidence of multicollinearity in the data? How do you know? c. Compute the variance inﬂation factors (VIF) and TOL measures for the various regressors. d. If there is the multicollinearity problem, what remedial action, if any, would you take? e. What does this study tell about the feasibility of a negative income tax? 10.31. Table 10.15 gives data on the crime rate in 47 states in the United States for 1960. Try to develop a suitable model to explain the crime rate in relation to the 14 socioeconomic variables given in the table. Pay particular attention to the collinearity problem in developing your model.

* D. H. Greenberg and M. Kosters, Income Guarantees and the Working Poor, Rand Corporation, R-579-OEO, December 1970.

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TABLE 10.14

HOURS OF WORK AND OTHER DATA FOR 35 GROUPS Observation 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 Hours 2157 2174 2062 2111 2134 2185 2210 2105 2267 2205 2121 2109 2108 2047 2174 2067 2159 2257 1985 2184 2084 2051 2127 2102 2098 2042 2181 2186 2188 2077 2196 2093 2173 2179 2200 Rate 2.905 2.970 2.350 2.511 2.791 3.040 3.222 2.493 2.838 2.356 2.922 2.499 2.796 2.453 3.582 2.909 2.511 2.516 1.423 3.636 2.983 2.573 3.262 3.234 2.280 2.304 2.912 3.015 3.010 1.901 3.009 1.899 2.959 2.971 2.980 ERSP 1121 1128 1214 1203 1013 1135 1100 1180 1298 885 1251 1207 1036 1213 1141 1805 1075 1093 553 1091 1327 1194 1226 1188 973 1085 1072 1122 990 350 947 342 1116 1128 1126 ERNO 291 301 326 49 594 287 295 310 252 264 328 347 300 297 414 290 289 176 381 291 331 279 314 414 364 328 304 30 366 209 294 311 296 312 204 NEIN 380 398 185 117 730 382 474 255 431 373 312 271 259 139 498 239 308 392 146 560 296 172 408 352 272 140 383 352 374 95 342 120 387 397 393 Assets 7250 7744 3068 1632 12710 7706 9338 4730 8317 6789 5907 5069 4614 1987 10239 4439 5621 7293 1866 11240 5653 2806 8042 7557 4400 1739 7340 7292 7325 1370 6888 1425 7625 7779 7885 Age 38.5 39.3 40.1 22.4 57.7 38.6 39.0 39.9 38.9 38.8 39.8 39.7 38.2 40.3 40.0 39.1 39.3 37.9 40.6 39.1 39.8 40.0 39.5 39.8 40.6 41.8 39.0 37.2 38.4 37.4 37.5 37.5 39.2 39.4 39.2 DEP 2.340 2.335 2.851 1.159 1.229 2.602 2.187 2.616 2.024 2.662 2.287 3.193 2.040 2.545 2.064 2.301 2.486 2.042 3.833 2.328 2.208 2.362 2.259 2.019 2.661 2.444 2.337 2.046 2.847 4.158 3.047 4.512 2.342 2.341 2.341 School 10.5 10.5 8.9 11.5 8.8 10.7 11.2 9.3 11.1 9.5 10.3 8.9 9.2 9.1 11.7 10.5 9.5 10.1 6.6 11.6 10.2 9.1 10.8 10.7 8.4 8.2 10.2 10.9 10.6 8.2 10.6 8.1 10.5 10.5 10.6

Notes: Hours = average hours worked during the year Rate = average hourly wage (dollars) ERSP = average yearly earnings of spouse (dollars) ERNO = average yearly earnings of other family members (dollars) NEIN = average yearly nonearned income Assets = average family asset holdings (bank account, etc.) (dollars) Age = average age of respondent Dep = average number of dependents School = average highest grade of school completed Source: D. H. Greenberg and M. Kosters, Income Guarantees and the Working Poor, The Rand Corporation, R-579-OEO, December 1970.

10.32. Refer to the Longley data given in Section 10.10. Repeat the regression given in the table there by omitting the data for 1962; that is, run the regression for the period 1947–1961. Compare the two regressions. What general conclusion can you draw from this exercise?

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TABLE 10.15 Observation 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47

U.S. CRIME DATA FOR 47 STATES IN 1960 R 79.1 163.5 57.8 196.9 123.4 68.2 96.3 155.5 85.6 70.5 167.4 84.9 51.1 66.4 79.8 94.6 53.9 92.9 75.0 122.5 74.2 43.9 121.6 96.8 52.3 199.3 34.2 121.6 104.3 69.6 37.3 75.4 107.2 92.3 65.3 127.2 83.1 56.6 82.6 115.1 88.0 54.2 82.3 103.0 45.5 50.8 84.9 Age 151 143 142 136 141 121 127 131 157 140 124 134 128 135 152 142 143 135 130 125 126 157 132 131 130 131 135 152 119 166 140 125 147 126 123 150 177 133 149 145 148 141 162 136 139 126 130 S 1 0 1 0 0 0 1 1 1 0 0 0 0 0 1 1 0 1 0 0 0 1 0 0 0 0 0 0 0 1 0 0 1 0 0 0 1 0 1 1 0 0 1 0 1 0 0 ED 91 113 89 121 121 110 111 109 90 118 105 108 113 117 87 88 110 104 116 108 108 89 96 116 116 121 109 112 107 89 93 109 104 118 102 100 87 104 88 104 122 109 99 121 88 104 121 EX0 58 103 45 149 109 118 82 115 65 71 121 75 67 62 57 81 66 123 128 113 74 47 87 78 63 160 69 82 166 58 55 90 63 97 97 109 58 51 61 82 72 56 75 95 46 106 90 EX1 56 95 44 141 101 115 79 109 62 68 116 71 60 61 53 77 63 115 128 105 67 44 83 73 57 143 71 76 157 54 54 81 64 97 87 98 56 47 54 74 66 54 70 96 41 97 91 LF 510 583 533 577 591 547 519 542 553 632 580 595 624 595 530 497 537 537 536 567 602 512 564 574 641 631 540 571 521 521 535 586 560 542 526 531 638 599 515 560 601 523 522 574 480 599 623 M 950 1012 969 994 985 964 982 969 955 1029 966 972 972 986 986 956 977 978 934 985 984 962 953 1038 984 1071 965 1018 938 973 1045 964 972 990 948 964 974 1024 953 981 998 968 996 1012 968 989 1049 N 33 13 18 157 18 25 4 50 39 7 101 47 28 22 30 33 10 31 51 78 34 22 43 7 14 3 6 10 168 46 6 97 23 18 113 9 24 7 36 96 9 4 40 29 19 40 3 NW 301 102 219 80 30 44 139 179 286 15 106 59 10 46 72 321 6 170 24 94 12 423 92 36 26 77 4 79 89 254 20 82 95 21 76 24 349 40 165 126 19 2 208 36 49 24 22 U1 108 96 94 102 91 84 97 79 81 100 77 83 77 77 92 116 114 89 78 130 102 97 83 142 70 102 80 103 92 72 135 105 76 102 124 87 76 99 86 88 84 107 73 111 135 78 113 U2 41 36 33 39 20 29 38 35 28 24 35 31 25 27 43 47 35 34 34 58 33 34 32 42 21 41 22 28 36 26 40 43 24 35 50 38 28 27 35 31 20 37 27 37 53 25 40 W 394 557 318 673 578 689 620 472 421 526 657 580 507 529 405 427 487 631 627 626 557 288 513 540 486 674 564 537 637 396 453 617 462 589 572 559 382 425 395 488 590 489 496 622 457 593 588 X 261 194 250 167 174 126 168 206 239 174 170 172 206 190 264 247 166 165 135 166 195 276 227 176 196 152 139 215 154 237 200 163 233 166 158 153 254 225 251 228 144 170 224 162 249 171 160

Deﬁnitions of variables: R = crime rate, number of offenses reported to police per million population Age = number of males of age 14–24 per 1000 population S = indicator variable for southern states (0 = no, 1 = yes) ED = mean number of years of schooling times 10 for persons age 25 or older. EX0 = 1960 per capita expenditure on police by state and local government EX1 = 1959 per capita expenditure on police by state and local government LF = labor force participation rate per 1000 civilian urban males age 14–24 M = number of males per 1000 females N = state population size in hundred thousands NW = number of nonwhites per 1000 population U1 = unemployment rate of urban males per 1000 of age 14–24 U2 = unemployment rate of urban males per 1000 of age 35–39 W = median value of transferable goods and assets or family income in tens of dollars X = the number of families per 1000 earnings 1/2 the median income Observation = state (47 states for the year 1960) Source: W. Vandaele, “Participation in Illegitimate Activities: Erlich Revisted,” in A. Blumstein, J. Cohen, and Nagin, D., eds., Deterrence and Incapacitation, National Academy of Sciences, 1978, pp. 270–335.

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HETEROSCEDASTICITY: WHAT HAPPENS IF THE ERROR VARIANCE IS NONCONSTANT?

An important assumption of the classical linear regression model (Assumption 4) is that the disturbances ui appearing in the population regression function are homoscedastic; that is, they all have the same variance. In this chapter we examine the validity of this assumption and ﬁnd out what happens if this assumption is not fulﬁlled. As in Chapter 10, we seek answers to the following questions: 1. 2. 3. 4. What is the nature of heteroscedasticity? What are its consequences? How does one detect it? What are the remedial measures?

11.1

THE NATURE OF HETEROSCEDASTICITY

As noted in Chapter 3, one of the important assumptions of the classical linear regression model is that the variance of each disturbance term ui , conditional on the chosen values of the explanatory variables, is some constant number equal to σ 2 . This is the assumption of homoscedasticity, or equal (homo) spread (scedasticity), that is, equal variance. Symbolically,

2 E ui = σ 2

i = 1, 2, . . . , n

(11.1.1)

Diagrammatically, in the two-variable regression model homoscedasticity can be shown as in Figure 3.4, which, for convenience, is reproduced as

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Density

Savings

Y

β 1 + β 2 Xi

Inco me

X

FIGURE 11.1

Homoscedastic disturbances.

Density

Savings

Y

β 1 + β 2 Xi

Inco me

X

FIGURE 11.2

Heteroscedastic disturbances.

Figure 11.1. As Figure 11.1 shows, the conditional variance of Y i (which is equal to that of ui ), conditional upon the given Xi , remains the same regardless of the values taken by the variable X. In contrast, consider Figure 11.2, which shows that the conditional variance of Yi increases as X increases. Here, the variances of Yi are not the same. Hence, there is heteroscedasticity. Symbolically,

2 E ui = σi2

(11.1.2)

Notice the subscript of σ 2 , which reminds us that the conditional variances of ui ( = conditional variances of Yi ) are no longer constant. To make the difference between homoscedasticity and heteroscedasticity clear, assume that in the two-variable model Yi = β1 + β2 Xi + ui , Y represents savings and X represents income. Figures 11.1 and 11.2 show that as income increases, savings on the average also increase. But in Figure 11.1

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the variance of savings remains the same at all levels of income, whereas in Figure 11.2 it increases with income. It seems that in Figure 11.2 the higherincome families on the average save more than the lower-income families, but there is also more variability in their savings. There are several reasons why the variances of ui may be variable, some of which are as follows.1 1. Following the error-learning models, as people learn, their errors of behavior become smaller over time. In this case, σi2 is expected to decrease. As an example, consider Figure 11.3, which relates the number of typing errors made in a given time period on a test to the hours put in typing practice. As Figure 11.3 shows, as the number of hours of typing practice increases, the average number of typing errors as well as their variances decreases. 2. As incomes grow, people have more discretionary income2 and hence more scope for choice about the disposition of their income. Hence, σi2 is likely to increase with income. Thus in the regression of savings on income one is likely to ﬁnd σi2 increasing with income (as in Figure 11.2) because people have more choices about their savings behavior. Similarly, companies with larger proﬁts are generally expected to show greater variability in their dividend policies than companies with lower proﬁts. Also, growthoriented companies are likely to show more variability in their dividend payout ratio than established companies. 3. As data collecting techniques improve, σi2 is likely to decrease. Thus, banks that have sophisticated data processing equipment are likely to

Density

Typing er

rors

Y

Hou

rs of

typin

β 1 + β 2 Xi g pra ctice

X

FIGURE 11.3

Illustration of heteroscedasticity.

See Stefan Valavanis, Econometrics, McGraw-Hill, New York, 1959, p. 48. As Valavanis puts it, “Income grows, and people now barely discern dollars whereas previously they discerned dimes,’’ ibid., p. 48.

2

1

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commit fewer errors in the monthly or quarterly statements of their customers than banks without such facilities. 4. Heteroscedasticity can also arise as a result of the presence of outliers. An outlying observation, or outlier, is an observation that is much different (either very small or very large) in relation to the observations in the sample. More precisely, an outlier is an observation from a different population to that generating the remaining sample observations.3 The inclusion or exclusion of such an observation, especially if the sample size is small, can substantially alter the results of regression analysis. As an example, consider the scattergram given in Figure 11.4. Based on the data given in exercise 11.22, this ﬁgure plots percent rate of change of stock prices (Y) and consumer prices (X) for the post–World War II period through 1969 for 20 countries. In this ﬁgure the observation on Y and X for Chile can be regarded as an outlier because the given Y and X values are much larger than for the rest of the countries. In situations such as this, it would be hard to maintain the assumption of homoscedasticity. In exercise 11.22, you are asked to ﬁnd out what happens to the regression results if the observations for Chile are dropped from the analysis.

25

Chile

15 10

Stock prices (% change)

9 8 7 6 5 4 3 2 1 1 2 3 4 5 6 7 8 9 10 26

Consumer prices (% change) FIGURE 11.4 The relationship between stock prices and consumer prices.

3

I am indebted to Michael McAleer for pointing this out to me.

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5. Another source of heteroscedasticity arises from violating Assumption 9 of CLRM, namely, that the regression model is correctly speciﬁed. Although we will discuss the topic of speciﬁcation errors more fully in Chapter 13, very often what looks like heteroscedasticity may be due to the fact that some important variables are omitted from the model. Thus, in the demand function for a commodity, if we do not include the prices of commodities complementary to or competing with the commodity in question (the omitted variable bias), the residuals obtained from the regression may give the distinct impression that the error variance may not be constant. But if the omitted variables are included in the model, that impression may disappear. As a concrete example, recall our study of advertising impressions retained (Y) in relation to advertising expenditure (X). (See exercise 8.32.) If you regress Y on X only and observe the residuals from this regression, you will see one pattern, but if you regress Y on X and X 2, you will see another pattern, which can be seen clearly from Figure 11.5. We have already seen that X 2 belongs in the model. (See exercise 8.32.) 6. Another source of heteroscedasticity is skewness in the distribution of one or more regressors included in the model. Examples are economic variables such as income, wealth, and education. It is well known that the distribution of income and wealth in most societies is uneven, with the bulk of the income and wealth being owned by a few at the top. 7. Other sources of heteroscedasticity: As David Hendry notes, heteroscedasticity can also arise because of (1) incorrect data transformation (e.g., ratio or ﬁrst difference transformations) and (2) incorrect functional form (e.g., linear versus log–linear models).4 Note that the problem of heteroscedasticity is likely to be more common in cross-sectional than in time series data. In cross-sectional data, one

60 40 20 20 0 –20 –20 –40 –60 2 4 6 8 10 12 14 16 18 20 22 (a) FIGURE 11.5 –40 2 4 6 8 10 12 14 16 18 20 22 (b) 0 40

Residuals from the regression of (a) impressions of advertising expenditure and (b) impression on Adexp and Adexp2.

4

David F. Hendry, Dynamic Econometrics, Oxford University Press, 1995, p. 45.

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usually deals with members of a population at a given point in time, such as individual consumers or their families, ﬁrms, industries, or geographical subdivisions such as state, country, city, etc. Moreover, these members may be of different sizes, such as small, medium, or large ﬁrms or low, medium, or high income. In time series data, on the other hand, the variables tend to be of similar orders of magnitude because one generally collects the data for the same entity over a period of time. Examples are GNP, consumption expenditure, savings, or employment in the United States, say, for the period 1950 to 2000. As an illustration of heteroscedasticity likely to be encountered in crosssectional analysis, consider Table 11.1. This table gives data on compensation per employee in 10 nondurable goods manufacturing industries, classiﬁed by the employment size of the ﬁrm or the establishment for the year 1958. Also given in the table are average productivity ﬁgures for nine employment classes. Although the industries differ in their output composition, Table 11.1 shows clearly that on the average large ﬁrms pay more than the small ﬁrms.

TABLE 11.1 COMPENSATION PER EMPLOYEE ($) IN NONDURABLE MANUFACTURING INDUSTRIES ACCORDING TO EMPLOYMENT SIZE OF ESTABLISHMENT, 1958 Employment size (average number of employees) Industry Food and kindred products Tobacco products Textile mill products Apparel and related products Paper and allied products Printing and publishing Chemicals and allied products Petroleum and coal products Rubber and plastic products Leather and leather products Average compensation Standard deviation Average productivity 1–4 5–9 10–19 20–49 50–99 100–249 250–499 500–999 1000–2499

2994 1721 3600 3494 3498 3611 3875 4616 3538 3016 3396 742.2 9355

3295 2057 3657 3787 3847 4206 4660 5181 3984 3196 3787 851.4 8584

3565 3336 3674 3533 3913 4695 4930 5317 4014 3149 4013 727.8 7962

3907 3320 3437 3215 4135 5083 5005 5337 4287 3317 4104 805.06 8275

4189 2980 3340 3030 4445 5301 5114 5421 4221 3414 4146 929.9 8389

4486 2848 3334 2834 4885 5269 5248 5710 4539 3254 4241 1080.6 9418

4676 3072 3225 2750 5132 5182 5630 6316 4721 3177 4388 1241.2 9795

4968 2969 3163 2967 5342 5395 5870 6455 4905 3346 4538 1307.7 10,281

5342 3822 3168 3453 5326 5552 5876 6347 5481 4067 4843 1110.5 11,750

Source: The Census of Manufacturers, U.S. Department of Commerce, 1958 (computed by author).

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1400

1200 Standard deviation

1000

800

600 3000

3500 4000 4500 Mean compensation

5000

FIGURE 11.6

Standard deviation of compensation and mean compensation.

As an example, ﬁrms employing one to four employees paid on the average about $3396, whereas those employing 1000 to 2499 employees on the average paid about $4843. But notice that there is considerable variability in earning among various employment classes as indicated by the estimated standard deviations of earnings. This can be seen also from Figure 11.6, which plots the standard deviation of compensation and average compensation in each employment class. As can be seen clearly, on average, the standard deviation of compensation increases with the average value of compensation.

11.2 OLS ESTIMATION IN THE PRESENCE OF HETEROSCEDASTICITY

What happens to OLS estimators and their variances if we introduce het2 eroscedasticity by letting E(ui ) = σi2 but retain all other assumptions of the classical model? To answer this question, let us revert to the two-variable model: Yi = β1 + β2 Xi + ui Applying the usual formula, the OLS estimator of β2 is ˆ β2 = = n n xi yi xi2 Xi Yi − Xi Yi Xi2 − ( Xi )2 (11.2.1)

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but its variance is now given by the following expression (see Appendix 11A, Section 11A.1): ˆ var (β2 ) = xi2 σi2 xi2

2

(11.2.2)

which is obviously different from the usual variance formula obtained under the assumption of homoscedasticity, namely, σ2 ˆ var (β2 ) = xi2 (11.2.3)

Of course, if σi2 = σ 2 for each i, the two formulas will be identical. (Why?) ˆ Recall that β2 is best linear unbiased estimator (BLUE) if the assumptions of the classical model, including homoscedasticity, hold. Is it still BLUE when we drop only the homoscedasticity assumption and replace it with the ˆ assumption of heteroscedasticity? It is easy to prove that β2 is still linear and unbiased. As a matter of fact, as shown in Appendix 3A, Section 3A.2, to ˆ establish the unbiasedness of β2 it is not necessary that the disturbances (ui ) be homoscedastic. In fact, the variance of ui , homoscedastic or heteroscedastic, plays no part in the determination of the unbiasedness propˆ erty. Recall that in Appendix 3A, Section 3A.7, we showed that β2 is a consistent estimator under the assumptions of the classical linear regression ˆ model. Although we will not prove it, it can be shown that β2 is a consistent estimator despite heteroscedasticity; that is, as the sample size increases indeﬁnitely, the estimated β2 converges to its true value. Furthermore, it can ˆ also be shown that under certain conditions (called regularity conditions), β2 ˆ2 is asymptotically normally distributed. Of course, what we have said about β also holds true of other parameters of a multiple regression model. ˆ Granted that β2 is still linear unbiased and consistent, is it “efﬁcient” or “best”; that is, does it have minimum variance in the class of unbiased estimators? And is that minimum variance given by Eq. (11.2.2)? The answer is ˆ no to both the questions: β2 is no longer best and the minimum variance is not given by (11.2.2). Then what is BLUE in the presence of heteroscedasticity? The answer is given in the following section.

11.3 THE METHOD OF GENERALIZED LEAST SQUARES (GLS)

Why is the usual OLS estimator of β2 given in (11.2.1) not best, although it is still unbiased? Intuitively, we can see the reason from Table 11.1. As the table shows, there is considerable variability in the earnings between employment classes. If we were to regress per-employee compensation on the size of employment, we would like to make use of the knowledge that there is considerable interclass variability in earnings. Ideally, we would like to devise

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the estimating scheme in such a manner that observations coming from populations with greater variability are given less weight than those coming from populations with smaller variability. Examining Table 11.1, we would like to weight observations coming from employment classes 10–19 and 20–49 more heavily than those coming from employment classes like 5–9 and 250–499, for the former are more closely clustered around their mean values than the latter, thereby enabling us to estimate the PRF more accurately. Unfortunately, the usual OLS method does not follow this strategy and therefore does not make use of the “information” contained in the unequal variability of the dependent variable Y, say, employee compensation of Table 11.1: It assigns equal weight or importance to each observation. But a method of estimation, known as generalized least squares (GLS), takes such information into account explicitly and is therefore capable of producing estimators that are BLUE. To see how this is accomplished, let us continue with the now-familiar two-variable model: Yi = β1 + β2 Xi + ui which for ease of algebraic manipulation we write as Yi = β1 X0i + β2 Xi + ui (11.3.2) (11.3.1)

where X0i = 1 for each i. The reader can see that these two formulations are identical. Now assume that the heteroscedastic variances σi2 are known. Divide (11.3.2) through by σi to obtain Yi = β1 σi X0i σi + β2 Xi σi + ui σi (11.3.3)

which for ease of exposition we write as

* * * * Yi* = β1 X0i + β2 Xi* + ui

(11.3.4)

where the starred, or transformed, variables are the original variables divided ∗ ∗ by (the known) σi . We use the notation β1 and β2 , the parameters of the transformed model, to distinguish them from the usual OLS parameters β1 and β2 . What is the purpose of transforming the original model? To see this, ∗ notice the following feature of the transformed error term ui :

* * var (ui ) = E(ui )2 = E

ui σi

2

= =

1 2 E ui σi2 1 2 σi σi2

since σi2 is known

2 since E ui = σi2

(11.3.5)

=1

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which is a constant. That is, the variance of the transformed disturbance * term ui is now homoscedastic. Since we are still retaining the other assumptions of the classical model, the ﬁnding that it is u* that is homoscedastic suggests that if we apply OLS to the transformed model ∗ (11.3.3) it will produce estimators that are BLUE. In short, the estimated β1 * ˆ1 and β2 . ˆ and β2 are now BLUE and not the OLS estimators β This procedure of transforming the original variables in such a way that the transformed variables satisfy the assumptions of the classical model and then applying OLS to them is known as the method of generalized least squares (GLS). In short, GLS is OLS on the transformed variables that satisfy the standard least-squares assumptions. The estimators thus obtained are known as GLS estimators, and it is these estimators that are BLUE. * * The actual mechanics of estimating β1 and β2 are as follows. First, we write down the SRF of (11.3.3) Yi ˆ* = β1 σi or ˆ* * ˆ* Yi* = β1 X0i + β2 Xi* + ui ˆ* Now, to obtain the GLS estimators, we minimize ui = ˆ 2* that is, ui ˆ σi

2

X0i σi

ˆ* + β2

Xi σi

+

ui ˆ σi (11.3.6)

ˆ* * ˆ* (Yi* − β1 X0i − β2 Xi* )2

=

Yi σi

ˆ* − β1

X0i σi

ˆ* − β2

Xi σi

2

(11.3.7)

The actual mechanics of minimizing (11.3.7) follow the standard calculus techniques and are given in Appendix 11A, Section 11A.2. As shown there, * the GLS estimator of β2 is ˆ* β2 = wi wi Xi Yi − wi wi Xi2 − wi Xi wi Xi wi Yi

2

(11.3.8)

and its variance is given by ˆ* var (β2 ) = where wi = 1/σi2 . wi wi wi Xi2 − wi Xi

2

(11.3.9)

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Difference between OLS and GLS

Recall from Chapter 3 that in OLS we minimize ui = ˆ2 ˆ ˆ (Yi − β1 − β2 Xi )2 (11.3.10)

but in GLS we minimize the expression (11.3.7), which can also be written as wi ui = ˆ2 ˆ* ˆ* wi (Yi − β1 X0i − β2 Xi )2 (11.3.11)

where wi = 1/σi2 [verify that (11.3.11) and (11.3.7) are identical]. Thus, in GLS we minimize a weighted sum of residual squares with wi = 1/σi2 acting as the weights, but in OLS we minimize an unweighted or (what amounts to the same thing) equally weighted RSS. As (11.3.7) shows, in GLS the weight assigned to each observation is inversely proportional to its σi , that is, observations coming from a population with larger σi will get relatively smaller weight and those from a population with smaller σi will get proportionately larger weight in minimizing the RSS (11.3.11). To see the difference between OLS and GLS clearly, consider the hypothetical scattergram given in Figure 11.7. ˆ2 In the (unweighted) OLS, each ui associated with points A, B, and C will receive the same weight in minimizing the RSS. Obviously, in this case ˆ2 the ui associated with point C will dominate the RSS. But in GLS the extreme observation C will get relatively smaller weight than the other two observations. As noted earlier, this is the right strategy, for in estimating the

Y C Yi = β1 + β2 Xi

u

u

{A

u B

0 FIGURE 11.7 Hypothetical scattergram.

X

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PART TWO:

RELAXING THE ASSUMPTIONS OF THE CLASSICAL MODEL

population regression function (PRF) more reliably we would like to give more weight to observations that are closely clustered around their (population) mean than to those that are widely scattered about. Since (11.3.11) minimizes a weighted RSS, it is appropriately known as weighted least squares (WLS), and the estimators thus obtained and given in (11.3.8) and (11.3.9) are known as WLS estimators. But WLS is just a special case of the more general estimating technique, GLS. In the context of heteroscedasticity, one can treat the two terms WLS and GLS interchangeably. In later chapters we will come across other special cases of GLS. ˆ* ˆ In passing, note that if wi = w, a constant for all i, β2 is identical with β2 ˆ ˆ * ) is identical with the usual (i.e., homoscedastic) var (β2 ) given in and var (β2 (11.2.3), which should not be surprising. (Why?) (See exercise 11.8.)

11.4 CONSEQUENCES OF USING OLS IN THE PRESENCE OF HETEROSCEDASTICITY

ˆ* ˆ As we have seen, both β2 and β2 are (linear) unbiased estimators: In reˆ* ˆ peated sampling, on the average, β2 and β2 will equal the true β2 ; that is, ˆ* they are both unbiased estimators. But we know that it is β2 that is efﬁcient, that is, has the smallest variance. What happens to our conﬁdence interval, hypotheses testing, and other procedures if we continue to use the OLS ˆ estimator β2 ? We distinguish two cases.

OLS Estimation Allowing for Heteroscedasticity

ˆ Suppose we use β2 and use the variance formula given in (11.2.2), which takes into account heteroscedasticity explicitly. Using this variance, and assuming σi2 are known, can we establish conﬁdence intervals and test hypotheses with the usual t and F tests? The answer generally is no because ˆ* ˆ it can be shown that var (β2 ) ≤ var (β2 ), 5 which means that conﬁdence intervals based on the latter will be unnecessarily larger. As a result, the t and ˆ F tests are likely to give us inaccurate results in that var (β2 ) is overly large and what appears to be a statistically insigniﬁcant coefﬁcient (because the t value is smaller than what is appropriate) may in fact be signiﬁcant if the correct conﬁdence intervals were established on the basis of the GLS procedure.

OLS Estimation Disregarding Heteroscedasticity

ˆ The situation can become serious if we not only use β2 but also continue to use the usual (homoscedastic) variance formula given in (11.2.3) even if heteroscedasticity is present or suspected: Note that this is the more likely

5 A formal proof can be found in Phoebus J. Dhrymes, Introductory Econometrics, Springerˆ Verlag, New York, 1978, pp. 110–111. In passing, note that the loss of efﬁciency of β2 [i.e., by ˆ* ˆ how much var (β2 ) exceeds var (β2 )] depends on the sample values of the X variables and the value of σi2 .

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case of the two we discuss here, because running a standard OLS regression package and ignoring (or being ignorant of) heteroscedasticity will yield ˆ ˆ variance of β2 as given in (11.2.3). First of all, var (β2 ) given in (11.2.3) is a ˆ2 ) given in (11.2.2), that is, on the average it overbiased estimator of var (β estimates or underestimates the latter, and in general we cannot tell whether the bias is positive (overestimation) or negative (underestimation) because it depends on the nature of the relationship between σi2 and the values taken by the explanatory variable X, as can be seen clearly from (11.2.2) (see exerˆ cise 11.9). The bias arises from the fact that σ 2 , the conventional estimator ui /(n − 2) is no longer an unbiased estimator of the latter ˆ2 of σ 2 , namely, when heteroscedasticity is present (see Appendix 11A.3). As a result, we can no longer rely on the conventionally computed conﬁdence intervals and the conventionally employed t and F tests.6 In short, if we persist in using the usual testing procedures despite heteroscedasticity, whatever conclusions we draw or inferences we make may be very misleading. To throw more light on this topic, we refer to a Monte Carlo study conducted by Davidson and MacKinnon.7 They consider the following simple model, which in our notation is Yi = β1 + β2 Xi + ui Xiα ). (11.4.1) They assume that β1 = 1, β2 = 1, and ui ∼ N(0, As the last expression shows, they assume that the error variance is heteroscedastic and is related to the value of the regressor X with power α. If, for example, α = 1, the error variance is proportional to the value of X; if α = 2, the error variance is proportional to the square of the value of X, and so on. In Section 11.6 we will consider the logic behind such a procedure. Based on 20,000 replications and allowing for various values for α, they obtain the standard errors of the two regression coefﬁcients using OLS [see Eq. (11.2.3)], OLS allowing for heteroscedasticity [see Eq. (11.2.2)], and GLS [see Eq. (11.3.9)]. We quote their results for selected values of α:

ˆ Standard error of β 1 Value of α 0.5 1.0 2.0 3.0 4.0 OLS 0.164 0.142 0.116 0.100 0.089 OLShet 0.134 0.101 0.074 0.064 0.059 GLS 0.110 0.048 0.0073 0.0013 0.0003 OLS 0.285 0.246 0.200 0.173 0.154 ˆ Standard error of β 2 OLShet 0.277 0.247 0.220 0.206 0.195 GLS 0.243 0.173 0.109 0.056 0.017

Note: OLShet means OLS allowing for heteroscedasticity.

6 ˆ ˆ From (5.3.6) we know that the 100(1 − α)% conﬁdence interval for β2 is [β2 ± tα/2 se (β2 )]. ˆ But if se (β2 ) cannot be estimated unbiasedly, what trust can we put in the conventionally computed conﬁdence interval? 7 Russell Davidson and James G. MacKinnon, Estimation and Inference in Econometrics, Oxford University Press, New York, 1993, pp. 549–550.

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