This paper studies a broad class of nonnegative ARCH(∞) models. Sufficient conditions for the existence of a stationary solution are established and an explicit representation of the solution as a Volterra type series is found. Under our assumptions, the covariance function can decay slowly like a power function, falling just short of the long memory structure. A moving average representation in martingale differences is established, and the central limit theorem is proved.

Some statistical properties of a vector autoregressive process with Markov-switching coefficients are considered. Sufficient conditions for this nonlinear process to be covariance stationary are given. The second moments of the process are derived under the conditions. The autocovariance matrix decays at exponential rate, permitting the application of the law of large numbers. Under the stationarity conditions, although sharing the “mean-reverting” property with conventional linear stationary processes, the process offers richer short-run dynamics such as conditional heteroskedasticity, asymmetric responses, and occasional nonstationary behavior.

We show the consistency of the log-periodogram regression estimate of the long memory parameter for long range dependent linear, not necessarily Gaussian, time series when we make a pooling of periodogram ordinates. Then, we study the asymptotic behavior of the tapered periodogram of long range dependent time series for frequencies near the origin, and we obtain the asymptotic distribution of the log-periodogram estimate for possibly non-Gaussian observation when the tapered periodogram is used. For these results we rely on higher order asymptotic properties of a vector of periodogram ordinates of the linear innovations. Finally, we assess the validity of the asymptotic results for finite samples via Monte Carlo simulation.

This paper considers a Bayesian analysis of the linear regression model under independent sampling from general scale mixtures of normals. Using a common reference prior, we investigate the validity of Bayesian inference and the existence of posterior moments of the regression and scale parameters. We find that whereas existence of the posterior distribution does not depend on the choice of the design matrix or the mixing distribution, both of them can crucially intervene in the existence of posterior moments. We identify some useful characteristics that allow for an easy verification of the existence of a wide range of moments. In addition, we provide full characterizations under sampling from finite mixtures of normals, Pearson VII, or certain modulated normal distributions. For empirical applications, a numerical implementation based on the Gibbs sampler is recommended.

This paper investigates Granger noncausality and the cointegrating relation between two time series in the Hilbert space framework. This framework allows us to analyze the relationship between cointegration and distance between two information sets. In particular, we prove that if two variables, X and Y, are cointegrated, then the distance between two information sets, concerning the differenced series ΔX and ΔY, must be less than the standard deviation of ΔX.

Professor Olav Reiersøl was born on 28 June 1908 in the countryside in Norway. He enrolled at Oslo University in 1928, where he studied chemistry, mathematics, physics, and botany. Choosing mathematics as his main subject, he received the cand. real. degree in 1935. As a graduate in mathematics and mathematical statistics he was employed as an assistant by Professor Ragnar Frisch.

Benedikt Pötscher and Ingmar Prucha are two exceptional econometricians who combine an extraordinary knowledge of the statistics and econometrics literature with great analytical skills. Both are excellent mathematicians, and the comment that can be heard among mathematicians that econometricians are “self-made mathematicians” definitely does not apply to them. Therefore, given the task of writing an overview of asymptotic theory for minimization estimators for dependent processes, one could hardly imagine a better choice of writers for such a task. The result is a remarkable book that deserves to receive attention from an audience that is broader than experts in the field only.

The accessibility of high-performance computing power has always influenced theoretical and applied econometrics. Gouriéroux and Monfort begin their recent offering, Simulation-Based Econometric Methods, with a stylized three-stage classification of the history of statistical econometrics. In the first stage, lasting through the 1960's, models and estimation methods were designed to produce closed-form expressions for the estimators. This spurred thorough investigation of the standard linear model, linear simultaneous equations with the associated instrumental variable techniques, and maximum likelihood estimation within the exponential family. During the 1970's and 1980's the development of powerful numerical optimization routines led to the exploration of procedures without closed-form solutions for the estimators. During this period the general theory of nonlinear statistical inference was developed, and nonlinear micro models such as limited dependent variable models and nonlinear time series models, e.g., ARCH, were explored. The associated estimation principles included maximum likelihood (beyond the exponential family), pseudo-maximum likelihood, nonlinear least squares, and generalized method of moments. Finally, the third stage considers problems without a tractable analytic criterion function. Such problems almost invariably arise from the need to evaluate high-dimensional integrals. The idea is to circumvent the associated numerical problems by a simulation-based approach. The main requirement is therefore that the model may be simulated given the parameters and the exogenous variables. The approach delivers simulated counterparts to standard estimation procedures and has inspired the development of entirely new procedures based on the principle of indirect inference.

The fourth edition of Econometric Methods by Jack Johnston and John DiNardo, is a rewrite of the venerable third edition by Johnston that sustained several generations of economists. As stated by the authors themselves, the reason for undertaking this major revision is to provide a comprehensive and accessible account of currently available econometric methodology, and in my opinion they have been successful in achieving their objective. The book has 13 chapters and runs to 531 pages. Each chapter ends with a selection of problems, several of which are new to this edition. Answers are not provided, although a solutions manual is available. Two appendices, one on matrix algebra and the other on statistical preliminaries, are intended to make the book as self-contained as possible. Not unexpectedly, the appendices are somewhat tersely worded, and the reader may wish to supplement them with additional reference material. Conforming to current practice, the book is accompanied by a data diskette containing several data sets, allowing the reader to replicate the applications given in the text.