Recently, increasing interest in the issue of fractional cointegration has emerged from theoretical and empirical viewpoints. Here, as opposed to the traditional prescription of unit root observables with weak dependent cointegrating errors, the orders of integration of these series are allowed to take real values, but, as in the traditional framework, equality of the orders of at least two observable series is necessary for cointegration. This assumption, in view of the real-valued nature of these orders, could pose some difficulties, and in the present paper we explore some ideas related to this issue in a simple bivariate framework. First, in a situation of “near-cointegration,” where the only difference with respect to the “usual” fractional cointegration is that the orders of the two observable series differ in an asymptotically negligible way, we analyze properties of standard estimates of the cointegrating parameter. Second, we discuss the estimation of the cointegrating parameter in a situation where the orders of integration of the two observables are truly different but their corresponding balanced versions (with same order of integration) are cointegrated in the usual sense. A Monte Carlo study of finite-sample performance and simulated series is included.I thank Adrian Pagan, James Davidson, and seminar participants at the 2004 Econometric Society European Meeting and the 2004 Simposio de Análisis Económico for helpful comments. I also thank two referees and a co-editor whose comments led to improvements of the paper. This research was supported by the Spanish Ministerio de Educación y Ciencia through a contract Juan de la Cierva and ref. SEJ2005-07657/ECON, and also by the Universidad de Navarra, ref. 16037001.

We consider general, and possibly nonparametric, GARCH(1,1) processes. First we give conditions for the existence and the uniqueness of stationary ergodic solutions. Then we identify additional conditions for geometric ergodicity. These conditions consist of mild restrictions on the distribution of the latent independent process. No moment assumption is made on the generalized autoregressive conditionally heteroskedastic (GARCH) process. Applications to the asymptotic behavior of sample autocorrelations and to unit-root tests are proposed.This work was supported by INTAS (research project 03-51-3714). The authors gratefully acknowledge the quick and careful reading of the manuscript by Bruce Hansen and three referees. Their detailed comments led to a greatly improved presentation.

We propose a portmanteau test for serial correlation of the error term in a fixed effects model. The test is derived as a Lagrange multiplier test, but it also has a straightforward Wald test interpretation. In Monte Carlo experiments, the test displays good size and power properties.The authors thank the co-editor, the referee, David Drukker, Christian Hansen, and Jeffrey Wooldridge for their helpful comments.

In this paper, some structural properties of a family of generalized autoregressive conditionally heteroskedastic (GARCH) processes are considered. First, a sufficient and necessary condition for the strict stationarity of this family of GARCH processes is given. Second, some simple conditions for the existence of the moments of the family of GARCH processes are also derived. Finally, we describe the tail of the marginal distribution of the family of GARCH processes.I am grateful to Bruce E. Hansen and an anonymous referee for helpful comments on the initial version.

The bias-reduced log-periodogram estimator of Andrews and Guggenberger (2003, Econometrica 71, 675–712) for the long-memory parameter d in a stationary long-memory time series reduces the asymptotic bias of the original log-periodogram estimator of Geweke and Porter-Hudak (1983) by an order of magnitude but inflates the asymptotic variance by a multiplicative constant cr, for example, c1 = 2.25 and c2 = 3.52. In this paper, we introduce a new, computationally attractive estimator by taking a weighted average of estimators over different bandwidths. We show that, for each fixed r ≥ 0, the new estimator can be designed to have the same asymptotic bias properties as but its asymptotic variance is changed by a constant cr* that can be chosen to be as small as desired, in particular smaller than cr. The same idea is also applied to the local-polynomial Whittle estimator in Andrews and Sun (2004, Econometrica 72, 569–614) leading to the weighted estimator . We establish the asymptotic bias, variance, and mean-squared error of the weighted estimators and show their asymptotic normality. Furthermore, we introduce a data-dependent adaptive procedure for selecting r and the bandwidth m and show that up to a logarithmic factor, the resulting adaptive weighted estimator achieves the optimal rate of convergence. A Monte Carlo study shows that the adaptive weighted estimator compares very favorably to several other adaptive estimators.We thank a co-editor and three referees for very helpful suggestions. We are grateful for the constructive comments offered by Marc Henry, Javier Hidalgo, and especially Katsumi Shimotsu.

We revisit the exact properties of two-stage least squares and limited information maximum likelihood estimators in a structural equation/instrumental variables regression under Gaussian assumptions. Simple derivations based on conditioning serve both to demystify the apparently complicated formulas, and to isolate the key quantities that determine the properties of the estimators. Some recent results obtained under weak-instrument asymptotics are sharpened and clarified by the exact analysis.Thanks to Peter Phillips and several anonymous referees for helpful comments that improved the paper considerably.

We investigate the possible bimodality of the density of the two-stage least squares (TSLS) estimator in a just-identified/overidentified linear structural equation. By studying the interaction between weakness of instruments, degree of endogeneity, and degree of overidentification we are able to identify conditions for its existence.I thank Grant Hillier, Patrick Marsh, Don Poskitt, the editor Peter Phillips, and two anonymous referees for useful and encouraging comments.

In a simple model composed of a structural equation and identity, the finite-sample distribution of the instrumental variable/limited information maximum likelihood (IV/LIML) estimator is always bimodal, and this is most apparent when the concentration parameter is small. Weak instrumentation is the energy that feeds the secondary mode, and the coefficient in the structural identity provides a point of compression in the density that gives rise to it. The IV limit distribution can be normal, bimodal, or inverse normal depending on the behavior of the concentration parameter and the weakness of the instruments. The limit distribution of the ordinary least squares (OLS) estimator is normal in all cases and has a much faster rate of convergence under very weak instrumentation. The IV estimator is therefore more resistant to the attractive effect of the identity than OLS. Some of these limit results differ from conventional weak instrument asymptotics, including convergence to a constant in very weak instrument cases and limit distributions that are inverse normal.My thanks to Richard Smith and two referees for comments on an earlier version. Section 2 of the paper is based on lectures given to students over the 1970s and 1980s at Essex, Birmingham, and Yale. Partial support is acknowledged from a Kelly Fellowship at the University of Auckland School of Business and the NSF under Grant SES 04-142254.

Vector time series analysis has become a standard tool in macroeconometrics and in empirical finance since the early 1980s. Over the last two decades, Helmut Lütkepohl has published extensively on virtually all topics covered in this book, which puts him in an excellent position for writing an authoritative textbook on this subject.

This book, Matrix Algebra, is the first volume of the Econometric Exercises series. Other volumes of the series (Statistics [vol. 2], Econometric Theory I [vol. 3], and Empirical Applications [vol. 4]) are to appear in the near future. A variety of planned titles will also follow (e.g., Time Series, Microeconometrics, and Financial Econometrics). Although there have been a few worked-exercise books in econometrics (e.g., Phillips and Wickens, 1978), this series is the first comprehensive exercise series in modern econometrics that covers the materials for advanced undergraduate and graduate econometrics courses. According to the preface, each volume follows the same format. Chapters start with a brief overview, and then exercises and solutions follow. Solutions are provided immediately after the exercise is posed. There is a variety of exercises from proofs and applications of theorems to numerical examples. In this review I discuss only the first volume in the Econometric Exercises series and find it very useful and successful. I expect succeeding volumes in the series to be of similar quality and usefulness, given the overall objectives of the series.

This note considers a panel data regression model with spatial autoregressive disturbances and random effects where the weight matrix is normalized and has equal elements. This is motivated by Kelejian, Prucha, and Yuzefovich (2005, Journal of Regional Science, forthcoming), who argue that such a weighting matrix, having blocks of equal elements, might be considered when units are equally distant within certain neighborhoods but unrelated between neighborhoods. We derive a simple weighted least squares transformation that obtains generalized least squares (GLS) on this model as a simple ordinary least squares (OLS). For the special case of a spatial panel model with no random effects, we obtain two sufficient conditions where GLS on this model is equivalent to OLS. Finally, we show that these results, for the equal weight matrix, hold whether we use the spatial autoregressive specification, the spatial moving average specification, the spatial error components specification, or the Kapoor, Kelejian, and Prucha (2005, Journal of Econometrics, forthcoming) alternative to modeling panel data with spatially correlated error components.I thank Paolo Paruolo and an anonymous referee for helpful comments and suggestions.

We consider a family of GARCH(1,1) processes introduced in He and Teräsvirta (1999a, Journal of Econometrics 92, 173–192). This family contains various popular generalized autoregressive conditional heteroskedasticity (GARCH) models as special cases. A necessary and sufficient condition for the existence of a strictly stationary solution is given.This research was financially supported by the Jan Wallander's and Tom Hedelius' Foundation, Grant J03–41. The author thanks the editor, an anonymous referee, Pentti Saikkonen, and Timo Teräsvirta for useful comments.