2. CHAPTER-BY-CHAPTER REVIEW

The opening chapter of ETM is called “Regression Models.” The book opens with the definition of a simple linear regression,

This first chapter gives a good sense of the flavor of the book, both in terms of strengths and in noting practical areas that require some supplementation.

The opening pages talk through equation (2.1) sufficiently clearly that after only three pages a reader has a basic understanding of what you get out of a regression. On the other hand, the simple regression is illustrated with equation (2.1) but without the traditional scatter diagram showing points in (y,x). It is hard to imagine that a student who has never seen a regression can learn from the equation without a picture.

Major negative? Not necessarily. First, students starting a graduate econometrics course almost certainly bring with them a rough idea of what a regression is. Second, it's not difficult to show the traditional scatter diagram in class. On the other hand, the absence of a scatter diagram surely makes the concept of a residual less intuitive. (For comparison, Greene does provide a scatter diagram; Hayashi does not.)

ETM announces that equation (2.1) might be a consumption function relating household consumption in year t to measured disposable income; β₂ is the marginal propensity to consume; and β₁ is autonomous consumption. It is not that difficult to teach graduate students to regress y on x and to then give an interpretation of the resulting computer output. It is extraordinarily difficult to dissuade students from running regressions with endogenous variables on the right-hand side. (I find it somewhat effective to have my labor economist wife make a guest appearance during lecture to threaten mayhem upon any student who runs such a regression. But this option may not be available to most econometricians.) Page 1 of a text is not the place to bring up endogeneity, but couldn't we have an example that doesn't leave something to unlearn? Mind you, both Greene and Hayashi open with the Keynesian consumption function as the first example.

Writing style is probably less important than content. On the other hand, students are going to—it is hoped—spend many, many hours with the text. Davidson and MacKinnon write delightfully and are unafraid to say what they mean in a straightforward way. I find their use of language engaging. Here is one example taken from a footnote:

Chapter 2 presents the geometry of linear regression. A thorough review of the geometry of vector spaces is followed by everything you ever wanted students to know about the geometry of least squares. Emphasis is given to the use of projection matrices because these play an important role in the algebra of regressions. The Frisch–Waugh–Lovell (FWL) theorem is proved and discussed at length. FWL states that β₁ in the regression y = X₁ β₁ + X₂ β₂ + u can be estimated by forming the residuals from a regression of y on X₂ and the residuals from a regression of X₁ on X₂ and regressing the former on the latter. Three applications of FWL are given: sweeping out the dummies, removing a deterministic time trend from a regression, and the relation between several measures of goodness of fit. The final section discusses influential observations and leverage.

The third chapter turns to the statistical properties of ordinary least squares (OLS). Least squares is shown to be unbiased if the right-hand-side variables are exogenous, implying E(u|X) = 0, but not necessarily if the right-hand-side variables are merely predetermined, E(u_t|X_t) = 0. As an algebraic illustration of the latter point the authors consider the regression of y_t on y_t−1 given independent and identically distributed (i.i.d.) errors, pointing out that least squares is not unbiased. This is true but not helpful. Endogeneity is an important source of bias in least squares. Measurement error is an important source of bias. Selection is an important source of bias. Lagged dependent variables with serially correlated errors are an important source of bias. The presence of a lagged dependent variable with i.i.d. errors is not an important source of bias. Although mathematically an estimator either is unbiased or it isn't, the untidy part of econometrics involves learning when biases are likely to be large or small. Some other example would have been better here.

Continuing, the third chapter explains probability limits and consistency, gives the covariance of OLS, proves the Gauss–Markov theorem, discusses the relation between error terms and residuals, gives a good discussion of misspecification, and concludes with defining R² and discussing goodness of fit.

The fourth chapter covers hypothesis testing. The chapter opens with a discussion of the principles of hypothesis testing and proceeds to work through t- and F-tests. The discussion is both thorough and succinct. As might be expected from the authors the bootstrap is introduced as a standard tool, rather than exotica relegated to the back of the book.

Chapter 5 covers confidence intervals. In addition to the obvious material, this chapter discusses bootstrap confidence regions and confidence regions in multiple dimensions. Heteroskedasticity-consistent covariance matrices are explained. So is the delta method for nonlinear hypothesis tests.

The sixth chapter takes up nonlinear regression. This is earlier than is traditional. But the early placement has two advantages. First, with modern software nonlinear regression is not much harder to execute than is linear regression. Second, this permits introduction of the Gauss–Newton regression, a tool that the authors use later in the book for computing both linear and nonlinear tests. The chapter concentrates on estimation and testing but gives little in the way of examples of useful nonlinear functional forms or in the way of guidance in the interpretation of nonlinear models. The chapter explores several basic optimization algorithms. Such algorithms are useful not only in econometrics but in the growing areas of economics that employ significant computation. The discussion of optimization is terrific, as one would expect from the authors. They make it look almost too easy, so an additional word of warning that numerical computation is a difficult field might have been in order.

Generalized least squares is introduced in chapter 7. Along with the discussion, autoregressive (AR) and moving average (MA) processes are introduced. Several tests for serial correlation are given. It is noteworthy that the Durbin–Watson (DW) statistic is explained clearly, identified as the most popular test for serial correlation, and then it is explained why the DW is now obsolete and should not be used. Panel data are briefly introduced in this chapter, and the fixed effect and random effect models are explained.

The eighth chapter covers instrumental variable estimation, beginning with brief discussions of errors-in-variables and simultaneous equation systems. (The latter is discussed in depth in chapter 12.) The discussion covers finite-sample properties, test of overidentifying restrictions, Durbin–Wu–Hausman tests, and instrumental variable estimation of nonlinear models. Chapter 9 moves to generalized method of moments (GMM). The discussion begins with using GMM for the linear regression model, which is a good way to build intuition. Careful discussion is given to alternative formulations of the GMM estimator and to heteroskedasticity and autocorrelation consistent estimation. A variety of tests are discussed. The chapter concludes with an introduction to the method of simulated moments.

Up to this point in the text the underlying estimation theory is based on method of moments. Chapter 10 turns to maximum likelihood estimation (MLE). In addition to explaining MLE, the authors use this opportunity to introduce the likelihood ratio, Wald, and Lagrange multiplier trio. Chapter 11 begins with logit and probit models. (It would be nice if a little more time were spent on interpreting results, something students frequently—and even experts sometimes—have trouble with.) Multinomial, ordered, and nested models are then presented. Models for count data, censored and truncated models, selection models, and duration models are all introduced briefly and clearly.

Multivariate regression models are covered in chapter 12. The chapter begins with seemingly unrelated regression, including nonlinear systems. Simultaneous equation models are then covered. GMM, three-stage least squares (3SLS), full information maximum likelihood (FIML), and limited information maximum likelihood (LIML) are discussed. Structural and reduced forms are given, although there is no discussion of the interpretation of the reduced form.

Chapter 13 goes over methods for stationary time series. The chapter opens with autoregressive moving average (ARMA) models. Estimation is discussed, including some of the practical difficulties of MA estimation. Distributed lag models come next, including partial adjustment, autoregressive distributed lag, and error-correction models. Neither polynomial distributed lag nor adaptive expectations models are discussed. An unusually thorough discussion of seasonal adjustment is given. Basic autoregressive conditional heteroskedasticity (ARCH) and generalized autoregressive conditional heteroskedasticity (GARCH) models are given. VARs are discussed extremely briefly, without mention of identification issues, impulse response functions, or variance decompositions.

Chapter 14 turns to unit roots and cointegration. The chapter opens by building the idea of a random walk. The subject is then motivated with a discussion of spurious regressions. Unit root tests are discussed, followed by a discussion of cointegration and tests for cointegration.

Chapter 15 concludes the text with an examination of specification testing. A number of tests based on artificial regressions are presented. There are then a section on nonnested hypothesis tests, a very brief discussion of model selection based on the Akaike information criterion (AIC) and Bayesian information criterion (BIC), and an introduction to nonparametric estimation.