An Introduction to Classical Econometric Theory.

By Paul A. Ruud

New York: Oxford Press, 2000. Pp. xxiv, 951. $85.00.

An Introduction to Classical Econometric Theory is one of several recent entries into the PhD-level econometrics market. [1] Ruud has distinguished his text from these competitors in a number of ways, the most notable of which are its organizing principles of mathematical projection and latent-variable models. The former is used to develop the geometry of ordinary least squares (OLS), which is extended to generate theoretical insights into the properties of a variety of other estimators. The latter is adopted as a unifying approach to model specification, emphasizing the dependence of observables on unobservables.

A successful graduate econometrics text will be a well-written and reasonable comprehensive guide to the current standard tools and techniques of econometric practice. As such, it will serve as both a primary course supplement and useful future reference. Of course, any judgement about a book will reflect the predilections of the judge. Thus, I should admit up front a preference for algebra over geometry when deriving the main results of OLS. I recognize that delving into the geometry can generate a depth of understanding that sometimes is elusive in the algebra. However, I suppose that, for most students in their first graduate econometrics course, the costs of pursuing that level of understanding outweigh the benefits. Clearly the textbook market thinks so, as there is only one other prominent book, Davidson and MacKinnon (1993), that places as much emphasis on least-squares geometry.

In addition, I prefer an introductory text to have a more extensive treatment of time-series issues than is offered by Ruud. For the most part, the time-series material is confined to the modeling of serially correlated errors. Absent is any systematic consideration of how estimation should proceed under dependent sampling and, in particular, how to deal with persistent series. An example of a book that strikes roughly the right balance is Wooldridge (2000), which, interestingly, is targeted at undergraduates.

With these qualifications in mind, there are many features of Ruud's book I like very much. First and foremost, the style is efficient but clear, rarely leaving the reader confused about the basics and usually providing reliable advice on the application of these results in real-world empirical settings. Further, most topics are introduced through an empirical example that motivates well the theoretical developments to follow, and end-of-chapter "Mathematical Notes" sections are wisely inserted to present the most technical arguments and proofs. I also appreciate that Ruud never relies on the fiction of fixed regressors, instead focusing on the estimation of E ([y.sub.n], \ X) (n = 1,..., N) under different assumptions about the conditional mean and variance of the model.

At a more detailed level, the book is comprised of 28 chapters, organized into four parts: I, Ordinary Least Squares (Chapters 1-5); II, Linear Regression (Chapters 6-12); III, Generalizations of the Linear Model (Chapters 13-23); and IV, Latent Variable Models (Chapters 24-28). The main text is supplemented by appendices containing the usual prerequisite material from matrix algebra, probability theory, and mathematical statistics, as well as a website that provides links to text figures, data used in examples, and solutions to the (excellent) end-of-chapter exercises. [2]

Part I sets forth the principle of mathematical projection and establishes the fundamental algebra and geometry of OLS. Ruud sums up these chapters like this: "Starting with the concepts of (1) a vector space, (2) linear dependence, (3) an inner product, length of a vector, and orthogonality, we have developed the idea of a projection as the solution to a minimum-distance problem" (p. 97).

The statistical treatment of the model begins in Part II, focusing on the small-sample properties of OLS and exact inference based on the OLS estimator under the usual assumptions of the classical model. At the outset of Chapter 6, the classical model begins to take shape with the statement of the conditional mean restriction, E([y.sub.n]\X)) = [[X.sup.1].sub.n] [beta]. Oddly, the conventional expression of the model [y.sub.n] = [[X.sup.1].sub.n][beta] + [[epsilon].sub.n], = with defined as a regression disturbance, is not introduced until section 2 of Chapter 20. There, the focus is on correlation between [X.sub.n] and the "latent" [[epsilon].sub.n]; that is, violations of the conditional mean assumption that underlies the material in Part II. While the emphasis on E([y.sub.n]\X) = [X'.sub.n][beta] is nice, this departure from convention is unwarranted and unnecessary.

The highlights of Part are Chapters 8 and 9, which clearly explain the sphericality restriction, the relationship between design (X) and precision (X'X) matrices, restricted least squares, and the Gauss Markov Theorem. Nevertheless, having arrived at this point, I wish there had been some prior consideration of the fundamental question of how we decide what is a good estimator. The student would be better served if the classical approach of searching for the best (most efficient) procedure within a certain class is made explicit, with an explanation of why we have to restrict the class of estimators to answer the question.

Part III addresses the consequences of departures from the classical assumptions, introducing maximum likelihood (ML), generalized least squares (GLS), instrumental variables (IV), and generalized method of moments (GMM) estimators along the way. These assumptions are relaxed in reverse order in which they were presented, allowing a sort of deconstruction of estimation theory into its fundamental components. This is a good strategy, but I think it would have been more successful if OLS asymptotics and ML estimation had been covered in Part II. As it turns out, the first generalization of the model--relaxing normality assumption in Chapter 13--is also the occasion for outlining the basics of asymptotic theory (a law of large numbers and a central limit theorem [CLT]). This is followed up in Chapters 14-17 by a thorough and rigorous presentation likelihood-based estimation and inference.

In the next two chapters, Ruud relaxes the sphericality assumption, first to admit heteroscedasticity and then autocorrelation. GLS and its feasible counterpart are introduced in these contexts, in contrast with the more typical approach of deriving these procedures as solutions to the estimation problem raised by the general linear model with an unrestricted error covariance matrix. Noteworthy in both chapters is the attention given to the necessary modifications required for CLT to hold in the presence of heteroscedasticity and serial correlation. Few texts make these clear, even though asymptotic claims depend on them.

Chapter 20 takes up violations of the conditional mean assumption, E([y.sub.n]\X) = [X'.sub.n][beta], cleverly motivating the statistical issues through issues that arise in an empirical Phillips curve equation. This example is used to introduce the IV estimator, which is shown in this instance to emerge from the computation of GLS. IV asymptotics are then discussed, with special consideration given to the question of optimal instruments and asymptotic efficiency in the special case of 2SLS, providing a natural transition to the material on GMM estimation and inference covered in Chapters 21 and 22. As with ML, the book proves all of the main asymptotic results for the GMM estimator, doing a particularly good job of explaining when GMM will be as efficient. After establishing the properties of the GMM estimator, classical testing principles are extended to the GMM case to obtain analogs of the likelihood ratio, Wald, and LM tests. Finally, the usual tests of overidentifying restrictions are developed, which are then exploited as a basis for understanding the Wu-Hausman specification test.

The last part of the book reviews the estimation strategies for basic panel data, ARMA, simultaneous equations, and limited dependent variable (LDV) models, relying on the latent-variable framework to unify the treatment of these disparate topics. This works best for the panel data material (Chapter 24), where the individual effects that characterize these models are commonly understood as latent variables, and with LDV models (Chapters 27-28), in which discrete realizations are viewed as the observable outcomes of latent indices.

Overall, these chapters offer the substantive coverage that should be expected in an introductory treatment of these topics. There are a couple of notable exceptions, however. On the positive side, the discussion of panel data methods extends beyond the usual fixed and random-effects models to GMM estimation of the well-known specifications of Chamberlain and Hausman and Taylor, as well as basic dynamic models. Less satisfactory is the treatment of ARMA models (Chapter 25), which are dealt with primarily as flexible set specifications for serially correlated errors, not as the foundation for time-series analysis in general. Only in a short section near the end of the chapter is the concession made that "One can apply the ARMA functional form directly to the observable {[y.sub.t]}." Then only a passing reference to multivariate time-series models is given.

In sum, I believe Ruud has produced a distinctive, well-written introductory graduate econometrics text. My predilections aside, the book will be suitable as the basis for any first PhD-level course in econometrics except those that emphasize time-series issues. Its distinctiveness ensures that it will also have great value as a reference for the basic estimation and inference results pertaining to the general linear model.

(1.) These include Baltagi (1998), Davidson (2000), Hayashi (2000), and Mittelhammer, Judge, and Miller (2000). Greene (2000), the fourth edition of this popular text, was also published this year.

(2.) The book's website URL is http://elsa.berkeley.edu/[sim]ruud/cet. Note that the exercise solutions are restricted to adopters of the book.

References

Baltagi, Bodi. 1998. Econometrics. Berlin: Springer.

Davidson, James. 2000. Econometric theory Oxford, UK: Blackwell Publishers.

Davidson, Russell and James MacKinnon 1993. Estimation and inference in econometrics. New York: Oxford University Press.

Greene, William 2000. Econometric analysis. 4th edition. Upper Saddle River, NJ: Prentice-Hall.

Hayashi, Fumio. 2000. Econometrics. Princeton, NJ: Princeton University Press.

Mittelhammer Ron, George Judge, and Douglas Miller. 2000. Econometric foundations. Cambridge, UK: Cambridge University Press.

Wooldridge, Jeffrey. 2000. Introductory econometrics A modern approach. Cincinnati, OH: South-Western College Publishing.