This book, Topics in Advanced Econometrics, is written primarily as a textbook for an advanced graduate econometrics course. The topics covered include consistent model specification testing, unit roots and cointegration, and nonparametric regression estimation; they are mainly the topics in which Professor Bierens has made significant contributions to the literature over the last 15 years. This book is unusual as a textbook in the sense that it treats both cross-sectional and time series (i.e., both subscript i and t) issues in econometrics at an advanced level. Most of the results given, other than those available in standard econometrics or statistics textbooks, are drawn from the published work of the author. The book is very useful because it puts together a number of important current issues that have been treated separately in the literature and presents them systematically using a well-organized set of statistical tools. Another advantage of this book is that the materials given are almost self-contained, making this book suitable for self-tuition. This book thus ideally suits students who need tools for independent research especially in the area of nonlinear and nonparametric models and time series analysis. It will also be useful to more advanced researchers who are interested in a thorough understanding of some of the author's original and influential work in these areas.

This paper studies the uniform convergence rates of Li and Vuong’s (1998, Journal of Multivariate Analysis 65, 139–165; hereafter LV) nonparametric deconvolution estimator and its regularized version by Comte and Kappus (2015, Journal of Multivariate Analysis 140, 31–46) for the classical measurement error model, where repeated noisy measurements on the error-free variable of interest are available. In contrast to LV, our assumptions allow unbounded supports for the error-free variable and measurement errors. Compared to Bonhomme and Robin (2010, Review of Economic Studies 77, 491–533) specialized to the measurement error model, our assumptions do not require existence of the moment generating functions of the square and product of repeated measurements. Furthermore, by utilizing a maximal inequality for the multivariate normalized empirical characteristic function process, we derive uniform convergence rates that are faster than the ones derived in these papers under such weaker conditions.

Phoebus J. Dhrymes is one of the best known econometricians of the last 40 years. He has made substantial contributions to econometric theory through articles in leading journals and by way of a series of outstanding texts on the foundations and methods of econometrics. His early research began with an applied econometric focus on problems of production and investment. His later contributions concentrated on the foundations of econometric methodology, including systems of simultaneous equations. Throughout the econometrics community, Dhrymes is well known for his influential textbooks, some of which have been translated into several languages. His 1970 book Econometrics: Statistical Foundations and Applications provided an accessible and rigorous foundation for both students and teachers of econometrics. His subsequent books have continued to treat foundational issues and have tracked new areas of econometric interest through to his 1998 book Time Series, Unit Roots, and Cointegration. Reading his books reveals Dhrymes as a teacher, synthesizer, and master expositor. As he says in the interview that follows, “my books are not typical textbooks. I perceive them more as books that bridge the gap between ordinary textbooks and journal articles and as filters that distill and synthesize the wisdom of many contributors to the subject. On this score I was influenced in my writing by the way I learn when studying by myself.”

In this comment we identify a lacuna in a proof in the paper by M. Caner published in 1997 in Econometric Theory concerning the weak limit behavior of various expressions involving heavy-tailed multivariate vectors and the convergence of stochastic integrals. In a later paper (Caner, 1998) the results for these limit relations are used to formulate tests for cointegration with infinite variance errors.

There are some errors in the paper by Victoria Zinde-Walsh, “Some Exact Formulae for Autoregressive Moving Average Processes,” Econometric Theory 4 (1988): 384–402. The following is a list of corrections.

On page 393 the τ * τ matrices El,E2 should be replaced by their transposes ,, respectively, throughout. In (3.21) the expression (S3 – λλA3) should be replaced by (S3 – λλA3) in the definitions of both T1, and T3. In (3.8), (3.21), (5.1), and (5.2) the entry in the bottom right-hand corner should be pre- and post-multiplied by λ. In (3.6) the matrix is obtained as the negative of the matrix denoted in the previous section 3.1. In (3.19) the top bar was omitted in Ē2. The top bar was also omitted in Ē1 on the second to last line of p. 392. In (5.1) D′−1 rather than D−l should appear in all the matrix entries in the second column; D′−1 should premultiply the parentheses in the top left and bottom right corners instead of D−1 post-multiplying them. In (5.2) transposes were omitted on D in the second entries on the first and last lines; in the second line of the second column, the second time A2 appears it should be transposed.

Let {X(t)} be a multivariate Gaussian stationary process with the spectral density matrix f0(ω), where θ is an unknown parameter vector. Using a quasi-maximum likelihood estimator of θ, we estimate the spectral density matrix f0(ω) by f(ω). Then we derive asymptotic expansions of the distributions of functions of f(ω). Also asymptotic expansions for the distributions of functions of the eigenvalues of f(ω) are given. These results can be applied to many fundamental statistics in multivariate time series analysis. As an example, we take the reduced form of the cobweb model which is expressed as a two-dimensional vector autoregressive process of order 1 (AR(1) process) and show the asymptotic distribution of , the estimated coherency, and contribution ratio in the principal component analysis based on in the model, up to the second-order terms. Although our general formulas seem very involved, we can show that they are tractable by using REDUCE 3.

A Sharpening Of Nonuniform bounds of the Berry-Esseen type initially obtained by Esseen and later generalized by Kolodjažnyĭ–who also proved that they are, in some sense, optimal–is proposed. Further, the corresponding inequalities are shown to provide uniformly improved Chebyshev bounds for the tail probabilities of the distribution functions to be approximated. In contrast with most results on Berry–Esseen bounds, which emphasize rates of convergence to normality, the bounds proposed are sufficiently explicit to allow the computation of numerical bounds on a distribution function. For example, they can be applied to the sum of a small number of independent random variables. The bounds are easy to compute and can be used in confidence estimation as well as in testing problems. Applications include signed-rank tests, permutation tests, and the chi-square approximation to Bartlett's test statistic for the homogeneity of several variances.

This paper is devoted to a detailed examination of the exact sampling properties of the instrumental variables (IV) estimator of the vector of coefficients on the exogenous variables in a single structural equation. The first two moments of a linear combination of the elements of this estimator and the joint distribution of these elements are considered. Estimable bounds for the first moment that can readily be incorporated into any IV estimation package are provided. The results obtained are in terms of the same special functions as those that characterize other results for this model, allowing a unified treatment of the model.

Alternative definitions of the concentration ellipsoid of a random vector are surveyed, and an extension of the concentration ellipsoid of Darmois is suggested as being the most convenient and natural definition. The advantage of the proposed definition in providing substantially simplified proofs of results in (linear) estimation theory is discussed, and is illustrated by new and short proofs of two key results. A not-so-well-known, but elementary, extremal representation of a nonnegative definite quadratic form, together with the corresponding Cauchy-Schwarẓ-type inequality, is seen to play a crucial role in these proofs.

Under normality, we obtain higher-order approximations to the distributions of the periodogram and related statistics. Our approach is based on the theorem which decomposes the periodogram into the sum of two independent random variables. It is seen that this decomposition enables us to study fairly closely the higher-order properties of not only the periodogram, but also periodogram-based statistics such as the estimators of the spectrum and prediction error variance. Some of the approximation results are graphically presented together with the exact results based on simulations.