We study the convergence in distribution of M-estimators over a convex kernel. Under convexity, the limit distribution of M-estimators can be obtained under minimal assumptions. We consider the case when the limit is arbitrary, not necessarily normal. If some Taylor expansions hold, the limit distribution is stable. As an application, we examine the limit distribution of M-estimators for the multivariate linear regression model. We obtain the distributional convergence of M-estimators for the multivariate linear regression model for a wide range of sequences of regressors and different types of conditions on the sequence of errors.

A semiparametric likelihood method is proposed for the estimation of sample selection models. The method is a two-step semiparametric scoring estimation procedure based on an index restriction and kernel estimation. Under some regularity conditions, the estimator is square-root n-consistent and asymptotically normal. The estimator is also asymptotically efficient in the sense that its asymptotic covariance matrix attains the semiparametric efficiency bound under the index restriction. For the binary choice sample selection model, it also attains the efficiency bound under the independence assumption. This method can be applied to the estimation of general sample selection models.

We consider a linear regression model with regression parameters (θ1,...,θp) and error variance parameter σ2. Our aim is to find a confidence interval with minimum coverage probability 1 − α for a parameter of interest θ1 in the presence of nuisance parameters (θ2,...,θp,σ2). We consider two confidence intervals, the first of which is the standard confidence interval for θ1 with coverage probability 1 − α. The second confidence interval for θ1 is obtained after a variable selection procedure has been applied to θp. This interval is chosen to be as short as possible subject to the constraint that it has minimum coverage probability 1 − α. The confidence intervals are compared using a risk function that is defined as a scaled version of the expected length of the confidence interval. We show that, subject to certain conditions including that [(dimension of response vector) − p] is small, the second confidence interval is preferable to the first when we anticipate (without being certain) that |θp|/σ is small. This comparison of confidence intervals is shown to be mathematically equivalent to a corresponding comparison of prediction intervals.

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.

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.

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.