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Gujarati: Basic Econometrics, Fourth Edition
Front Matter Preface © The McGraw−Hill Companies, 2004
xxv
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 intro- duction 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 sophisti- cated and user-friendly statistical packages, such as Eviews, Limdep, Microfit, Minitab, PcGive, SAS, Shazam, and Stata, it is now possible to dis- cuss several econometric techniques that could not be included in the pre- vious 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 eco- nomics and business students but also by students and researchers in sev- eral other disciplines, such as politics, international relations, agriculture, and health sciences. Students in these disciplines will find the expanded dis- cussion 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 tra- ditional 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 eco- nomic variables. It is important to know whether the variables are ratio
Gujarati: Basic Econometrics, Fourth Edition
Front Matter Preface © The McGraw−Hill Companies, 2004
xxvi 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 ac- count 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
- 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 cau- tionary advice to the practitioner about econometric theory and economet- ric 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 esti- mated by iterative methods. This chapter discusses and illustrates some comparatively simple methods of estimating nonlinear-in-parameter regres- sion models. 12. Chapter 15, on qualitative response regression models, which re- places 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
Gujarati: Basic Econometrics, Fourth Edition
Front Matter Preface © The McGraw−Hill Companies, 2004
xxviii 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 Eviews Student Ver- sion 3.1 on a CD along with all of the data from the text. This software is available from the publisher packaged with the text (ISBN: 0072565705). Eviews Student Version is available separately from QMS. Go to http://www.eviews.com for further information.
Web Site
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
Gujarati: Basic Econometrics, Fourth Edition
Front Matter Preface © The McGraw−Hill Companies, 2004
PREFACE xxix
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 Col- lege, 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 sup- port 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 as- pects 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 encour- agement in the preparation of this and the previous editions. Damodar N. Gujarati
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(^5) E. Malinvaud, Statistical Methods of Econometrics, Rand McNally, Chicago, 1966, p. 514. (^6) Adrian C. Darnell and J. Lynne Evans, The Limits of Econometrics, Edward Elgar Publish- ing, Hants, England, 1990, p. 54. (^7) T. Haavelmo, “The Probability Approach in Econometrics,” Supplement to Econometrica, vol. 12, 1944, preface p. iii.
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 eco- nomic theory and actual measurements, using the theory and technique of statis- tical inference as a bridge pier.^7
I.2 WHY A SEPARATE DISCIPLINE?
As the preceding definitions suggest, econometrics is an amalgam of eco- nomic theory, mathematical economics, economic statistics, and mathe- matical 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 quali- tative in nature. For example, microeconomic theory states that, other things remaining the same, a reduction in the price of a commodity is ex- pected to increase the quantity demanded of that commodity. Thus, eco- nomic theory postulates a negative or inverse relationship between the price and quantity demanded of a commodity. But the theory itself does not pro- vide 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 econome- trician to provide such numerical estimates. Stated differently, economet- rics 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 pro- posed 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
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INTRODUCTION 3
(^8) Aris Spanos, Probability Theory and Statistical Inference: Econometric Modeling with Obser- vational 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.
jobs of the economic statistician. It is he or she who is primarily responsible for collecting data on gross national product (GNP), employment, unem- ployment, 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 na- ture of most economic data, namely, that the data are not generated as the result of a controlled experiment. The econometrician, like the meteorolo- gist, 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 eco- nomics and other social and behavioral sciences.^9 Broadly speaking, traditional econometric methodology proceeds along the following lines:
1. Statement of theory or hypothesis. 2. Specification of the mathematical model of the theory 3. Specification of the statistical, or econometric, model 4. Obtaining the data 5. Estimation of the parameters of the econometric model 6. Hypothesis testing 7. Forecasting or prediction 8. Using the model for control or policy purposes.
To illustrate the preceding steps, let us consider the well-known Keynesian theory of consumption.
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INTRODUCTION 5
early related to income, is an example of a mathematical model of the rela- tionship 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 depen- dent 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 ex- ample, 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 = β 1 + β 2 X + u (I.3.2)
where u , known as the disturbance, or error, term, is a random (stochas- tic) 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 techni- cally, 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 ex- planatory 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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Consumption expenditure
X
Y
Income
u
FIGURE I.2 Econometric model of the Keynesian consumption function.
4. Obtaining Data
To estimate the econometric model given in (I.3.2), that is, to obtain the numerical values of β 1 and β 2 , 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 Y X
1982 3081.5 4620. 1983 3240.6 4803. 1984 3407.6 5140. 1985 3566.5 5323. 1986 3708.7 5487. 1987 3822.3 5649. 1988 3972.7 5865. 1989 4064.6 6062. 1990 4132.2 6136. 1991 4105.8 6079. 1992 4219.8 6244. 1993 4343.6 6389. 1994 4486.0 6610. 1995 4595.3 6742. 1996 4714.1 6928. Source: Economic Report of the President, 1998, Table B–2, p. 282.
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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 con- sumption 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, ac- cording 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 esti- mates 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 empiri- cal 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 en- quire 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 statis- tical 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 consid- eration, 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 expen- diture 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 Eco- nomics, 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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INTRODUCTION 9
this GDP figure on the right-hand side of (I.3.3), we obtain:
Y^ ˆ 1997 = − 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 dol- lars. 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 Presi- dent decides to propose a reduction in the income tax. What will be the ef- fect 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 ex- penditure increases. What will be the effect on the economy? As macroeco- nomic 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 − 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 even- tually 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 in- formation 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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INTRODUCTION 11
(^15) 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.
other consumption model (theory) might equally fit the data as well? For ex- ample, 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 hypoth- esis 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 qual- ity 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 func- tion, which is basically a relationship between output and inputs (say, capi- tal 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 pro- duction 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.
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Econometrics
Theoretical
Classical Bayesian
Applied
Classical Bayesian FIGURE I.5 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 clas- sical or Bayesian tradition. In this book the emphasis is on the classical approach. For the Bayesian approach, the reader may consult the refer- ences given at the end of the chapter. Theoretical econometrics is concerned with the development of appro- priate methods for measuring economic relationships specified by econo- metric 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 produc- tion 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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balanced discussion of the various methodological approaches to economet- rics, 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 perspec- tive 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 , Univer- sity 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 Statis- tics 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.
Gujarati: Basic Econometrics, Fourth Edition
I. Single−Equation Regression Models
Introduction © The McGraw−Hill Companies, 2004
15
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 interpreta- tion of the term regression and illustrate the difference between the two in- terpretations 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 intro- duce what is known as the classical linear regression model, a model that makes several simplifying assumptions. With these assumptions, we intro- duce 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 re- gression 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 proper- ties than the nonnormal classical linear regression model—properties that enable us to engage in statistical inference, namely, hypothesis testing.
