A Bayesian analysis of a nonidentified model is always possible if a proper prior on all the parameters is specified. There is, however, no Bayesian free lunch. The “price” is that there exist quantities about which the data are uninformative, i.e., their marginal prior and posterior distributions are identical. In the case of improper priors the analysis is problematic—resulting posteriors can be improper. This study investigates both proper and improper cases through a series of examples.

This paper derives the joint moment generating function of quadratic forms occurring in seasonal autoregressive models under stationary, unit root, and explosive specifications. The results are then used to investigate the impact of the seasonal periodicity parameter on various distributional results for both the normalized ordinary least squares coefficient and t-ratio and its effects on the asymptotic bias of parameter estimates.

This article analyzes the asymptotic behavior of the Dickey–Fuller unit root tests when the variable is generated under the breaking trend hypothesis. Our results show that the asymptotic behavior of these statistics allows for the rejection of the unit root hypothesis. This asymptotic finding contrasts with the results that can be found in the literature devoted to the analysis of the integration order of a variable in the presence of a structural break. However, some Monte Carlo exercises show that the argument of Perron (1989, Econometrica 57, 1361–1401) that the tests are biased in favor of nonrejection of the unit root hypothesis remains valid for sample sizes of practical interest.

This paper develops a test of autocorrelation in the presence of heteroskedasticity of unknown form in the nonlinear regression model. The test statistic is based on the sample autocovariance of the residuals standardized by a nonparametric kernel estimate of the unknown heteroskedasticity function. Under appropriate conditions, the test statistic is shown to have a limiting chi-square distribution. Local power and consistency results for the test are also established. Monte Carlo experiments show that the test has reasonable size performance and generally dominates some of the existing tests in terms of finite-sample power.

In this paper, we study the asymptotics of nonstationary fractional integrated time series, the long memory time series with d ≥ ½, with special attention focused on the cases when d = (2p + 1)/2 for integer n no less than 0. There is considerable empirical evidence showing long memory of this magnitude in many economic time series including the inflation rate and the stock market volatility. A study of the large-sample property is therefore both needed and useful. Also, we found the asymptotics of nonstationary fractional integrated time series useful in the study of the large-sample theory of the Kwiatkowski–Phillips–Schmidt–Shin test (1992, Journal of Econometrics 54, 159–178).

We obtain new central limit theorems (CLT's) and functional central limit theorems (FCLT's) for Hilbert-valued arrays near epoch dependent on mixing processes, and also new FCLT's for general Hilbert-valued adapted dependent heterogeneous arrays. These theorems are useful in delivering asymptotic distributions for parametric and nonparametric estimators and their functionals in time series econometrics. We give three significant applications for near epoch dependent observations: (1) A new CLT for any plug-in estimator of a cumulative distribution function (c.d.f.) (e.g., an empirical c.d.f., or a c.d.f. estimator based on a kernel density estimator), which can in turn deliver distribution results for many Von Mises functionals; (2) a new limiting distribution result for degenerate U-statistics, which delivers distribution results for Bierens's integrated conditional moment tests; (3) a new functional central limit result for Hilbert-valued stochastic approximation procedures, which delivers distribution results for nonparametric recursive generalized method of moment estimators, including nonparametric adaptive learning models.

Professor Tanaka has written a useful and unique book. Despite the main part of the title, the book does not seek to be a generalist work but rather has specific goals in mind and is focused on a single aspect of the large area of time series. This is not a criticism; there have been a large number of books that provide overviews of the field or components (Banerjee, Dolado, Galbraith, and Hendry, 1993; Hamilton, 1994; Hatanaka, 1996; Fuller, 1996). In directing his attention at a single subtopic, Tanaka has been able to do an excellent job of both covering the area in enough detail for the reader to obtain a solid foundation as a user of and potential contributor to this field and also to take time to include an extensive development of the basic skills necessary for the reader to understand the material at this level.

This paper proposes a nonparametric, kernel-based test of parametric quantile regression models. The test statistic has a limiting standard normal distribution if the parametric quantile model is correctly specified and diverges to infinity for any misspecification of the parametric model. Thus the test is consistent against any fixed alternative. The test also has asymptotic power 1 against local alternatives converging to the null at proper rates. A simulation study is provided to evaluate the finite-sample performance of the test.

“Notoriously, works on mathematical finance can be precise, and they can be comprehensible. Sadly, as Dr. Johnson might have put it, the ones which are precise are not necessarily comprehensible, and those comprehensible are not necessarily precise.” So starts the preface to Baxter and Rennie's recent treatise on financial calculus. The book attempts to give an introduction to modern continuous time finance in a precise and comprehensible fashion. Does it succeed? Yes, it is a very clear and precise little (233 pages) book, although the claim that the book is accessible to a reader with only “some knowledge of (classical) differential calculus and experience with symbolic notation” is exaggerated. Such a reader would find the material hard going indeed.

Efficiency scores of production units are measured by their distance to an estimated production frontier. Nonparametric data envelopment analysis estimators are based on a finite sample of observed production units, and radial distances are considered. We investigate the consistency and the speed of convergence of these estimated efficiency scores (or of the radial distances) in the very general setup of a multi-output and multi-input case. It is shown that the speed of convergence relies on the smoothness of the unknown frontier and on the number of inputs and outputs. Furthermore, one has to distinguish between the output- and the input-oriented cases.

When the disturbances of a regression model follow an I(1) process there is a tendency to estimate a break point in the middle of the sample, even though a break point does not actually exist. In this note, we provide a mathematical proof for this phenomenon.

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.

Since the notion of cointegration was established by Engel and Granger (1987), many statistical methods have been suggested to estimate and test cointegrated models. Undoubtedly the Gaussian likelihood–based method advocated by Johansen (1988, 1991) is one of the most popular choices among practitioners. In his 1988 paper, Johansen applied Anderson's (1951) maximum likelihood estimation procedure for reduced rank regression (RRR) models to isolate common stochastic trends in multiple time series. This was a remarkable breakthrough, which he and other authors have extended into various directions in the last decade. Johansen's approach is attractive in that it provides a unified set of tools of estimation, cointegration rank testing, and parametric hypothesis testing, based on the Gaussian likelihood for a vector autoregression (VAR). Although this book and other papers of Johansen are mostly concerned with reduced form models, the statistical information provided by his method is useful for applied econometricians, especially in fields where tractable dynamic structural models are not available. This book presents a concise yet comprehensive treatment of his methodology.

This paper presents a consistent specification test of conditional symmetry using a kernel method. The test statistic is shown to be asymptotically distributed as standard normal under the null hypothesis of conditional symmetry and consistent against any conditional asymmetric distribution. Power against local alternatives is also investigated. A Monte Carlo simulation is provided to evaluate the finite-sample performance of the test.

This book is a modern introduction to measure theoretic probability and statistical inference well targeted for graduate students in econometrics at top institutions. It would make an excellent textbook for first year graduate students who intend to specialize in econometrics or who have an advanced mathematical background, and it would also be a useful part of any graduate econometrics course. It is concise and intensely focused on the key conceptual points, thus counteracting the tendency toward long-windedness apparent in some recent econometric texts. Nevertheless, it provides many valuable insights into difficult material. In particular, the discussions of sigma fields and conditional expectation given a sigma field are very helpful. The coverage of multivariate concepts alongside univariate ones is particularly useful to econometricians and something that is missing from most comparable statistical texts. The author has a mature attitude to proof, providing complete and illuminating proofs of some results but making liberal use of simplifications provided by special cases, for example in Theorems 4.1 and 4.5 and Section 5.2.2, to shorten and focus the arguments. The proofs themselves are very clear and well presented. Carefully chosen diagrams are given throughout the book that nicely illustrate many of the key concepts. In addition, each chapter contains a long list of problems of varying complexity, which will be useful to instructors.

The abundance of high-frequency financial data and the rapid development of computer hardware have combined to transform financial economics into, arguably, the most empirically oriented field within the social sciences. At the same time, as a result of the difficulty of conducting genuine market experiments, empirical finance remains firmly grounded in the tradition of model-driven statistical inference that is characteristic of economics. Even so, the richness of data has often spurred a practical orientation that is more familiar in the natural sciences. The combination has proved fertile, leading to the classification of a set of loosely connected empirical topics as a distinct entity, financial econometrics.

Since the influential work by Nelson and Plosser (1982) and Engle and Granger (1987), many new and exciting developments have been made in the analysis of time series involving autoregressive unit roots. Hatanaka (1996; hereafter HT) reviewed the literature on unit roots and cointegration up to 1994 and provided new perspectives on this research area.