This paper investigates the effects of consistent and inconsistent long-run variance estimation on a test for a unit root, based on the generalization of the von Neumann ratio. The results from the Monte Carlo experiments suggest that the unit root tests based on an inconsistent estimator have less size distortion and more stability of size across different autocorrelation specifications as compared to the tests based on a consistent estimator. This improvement in size property, however, comes at the cost of a loss in power. The finite-sample power, in addition to the local asymptotic power, of the tests with an inconsistent estimator is shown to be much lower than that of conventional tests. This finding can be well generalized to the test for cointegration in a multivariate system. The paper also points out that combining consistent and inconsistent estimators in the long-run variance ratio test is one possibility of balancing the size and power.The authors thank two anonymous referees, Pentti Saikkonen, and participants of the 2004 Midwest Econometrics Group meetings, 2005 Spring Meetings of The Japanese Economic Association, and a workshop at Vanderbilt University for helpful comments and suggestions.

We propose several methods of on-line detection of a change in unconditional variance in a conditionally heteroskedastic time series. We follow the paradigm of Chu, Stinchcombe, and White (1996, Econometrica 64, 1045–1065) in which the first m observations are assumed to follow a stationary process and the monitoring scheme has asymptotically controlled probability of falsely rejecting the null hypothesis of no change. Our theory is applicable to broad classes of GARCH-type time series and relies on a strong invariance principle that holds for the squares of observations generated by such models. Practical implementation of the procedures, which uses a bandwidth selection procedure of Andrews (1991, Econometrica 59, 817–858), is proposed, and the performance of the methods is investigated by a simulation study.This research was partially supported by NSF grants INT-0223262 and DMS-0413653 and NATO grant PST.EAP.CLG 980599.

This paper develops an empirical likelihood approach for regular generalized autoregressive conditional heteroskedasticity (GARCH) models and GARCH models with unit roots. For regular GARCH models, it is shown that the log empirical likelihood ratio statistic asymptotically follows a χ2 distribution. For GARCH models with unit roots, two versions of the empirical likelihood methods, the least squares score and the maximum likelihood score functions, are considered. For both cases, the limiting distributions of the log empirical likelihood ratio statistics are established. These two statistics can be used to test for unit roots under the GARCH framework. Finite-sample performances are assessed through simulations for GARCH models with unit roots.This research was supported in part by Hong Kong Research grants Council Grants CUHK4043/02P and HKUST6273/03H. The authors thank two referees and the Co-Editor, Bruce Hansen, for insightful and helpful comments about the relationship between QMLE and MELE, which led to substantial improvement of the presentation. Computational assistance from Jerry Wong and Chun-Yip Yau is also gratefully acknowledged.

We consider the problem of hypothesis testing in a modified version of the stochastic integration and cointegration framework of Harris, McCabe, and Leybourne (2002, Journal of Econometrics 111, 363–384). This nonlinear setup allows for volatility in excess of that catered for by the standard integration/cointegration paradigm through the introduction of nonstationary heteroskedasticity. We propose a test for stochastic cointegration against the alternative of no cointegration and a secondary test for stationary cointegration against the heteroskedastic alternative. Asymptotic distributions of these tests under their respective null hypotheses are derived, and consistency under their respective alternatives is established. Monte Carlo evidence suggests that the tests will perform well in practice. An empirical application to the term structure of interest rates is also given.We are most grateful to the Associate Editor and two anonymous referees for providing helpful comments on earlier versions of this paper.

Testing goodness (or lack) of fit for distributions of observable and nonobservable random variables is one of the main topics in statistics. When they exist, the corresponding density functions and their shapes allow researchers to easily recognize the underlying distribution functions. The present paper is concerned with the densities of (unobservable) generalized autoregressive conditional heteroskedasticity (GARCH) innovations and also with developing goodness-of-fit tests for the densities. Specifically, we construct and investigate large-sample properties of a kernel-type density estimator for GARCH innovations based on (observable) residuals.The authors sincerely thank the Co-Editor Oliver Linton and three anonymous referees for constructive criticism and suggestions that helped us to prepare a much revised version of the original manuscript. The feedback by participants of the Conference on Statistical Models for Financial Data at the University of Graz in May 2004 is also greatly appreciated. The research of the first author was partially supported by NSF grant INT-0223262 and NATO grant PST.EAP.CLG 980599. The research of the second author was partially supported by a Discovery Research Grant from the Natural Sciences and Engineering Research Council (NSERC) of Canada.

This paper considers the estimation of the parameters of general systems of stochastic differential-difference equations in which the lag parameters themselves are treated as unknown and are not restricted to be integers and therefore form part of the parameter vector to be estimated. The asymptotic properties of an infeasible frequency domain maximum likelihood estimator are established in addition to those of a feasible version based on truncating an infinite series that arises in the computation of the spectral density function of the observed discrete time series. Precise conditions that the truncation parameter must satisfy for the asymptotic results to hold are provided.We are grateful to Gordon Kemp, Andrew Harvey, the Editor, and two anonymous referees for helpful comments on an earlier version of this paper. Any remaining errors are the sole responsibility of the authors. The first author thanks the Economic and Social Research Council for financial support under grant number R00429434216, and the second author thanks the Leverhulme Trust for financial support in the form of a Philip Leverhulme Prize.

This paper investigates a simple dynamic linear panel regression model with both fixed effects and time effects. Using “large n and large T” asymptotics, we approximate the distribution of the fixed effect estimator of the autoregressive parameter in the dynamic linear panel model and derive its asymptotic bias. We find that the same higher order bias correction approach proposed by Hahn and Kuersteiner (2002, Econometrica 70, 1639–1659) can be applied to the dynamic linear panel model even when time specific effects are present.We thank Peter Phillips and three anonymous referees for helpful comments. The first author gratefully acknowledges financial support from NSF grant SES-0313651. The second author appreciates the Faculty Development Awards of USC for research support.

This paper studies robust inference methods for nonlinear moment restriction models with weakly identified parameters in time series contexts. Our methods are based on generalized empirical likelihood with kernel smoothing. The proposed test statistics, which follow the standard χ2 limiting distributions, are robust to weak identification and dependent data.The author is deeply grateful to Bruce Hansen, John Kennan, and Gautam Tripathi for their guidance and time. Comments from a coeditor and two anonymous referees substantially helped this revision. The author also thanks Allan Gregory, Patrik Guggenberger, Philip Haile, Hiroyuki Kasahara, Matthew Kim, Yuichi Kitamura, and seminar participants at Queen's University, University of Wisconsin, and the 2003 North America Summer Meeting of the Econometric Society for helpful discussions and suggestions. Financial support from the Alice Gengler Wisconsin Distinguished Graduate Fellowship and Wisconsin Alumni Research Foundation Dissertation Fellowship is gratefully acknowledged.

Necessary and sufficient conditions are established for the second subsample to be partially redundant given the first subsample for the best linear unbiased estimator of a subset of the coefficient vector of a general linear regression model.We thank the Co-editor for his very helpful comments on an earlier draft.

In most cases where estimation with averaged data is performed, interest lies on the parameters of a model at the individual level, but grouped data are used because disaggregate data are not observed. In this note we study the conditions under which it is possible to consistently estimate the parameters of the individual data model using averaged data, giving particular attention to the case of endogenous selection into groups.We are grateful to an anonymous referee for helpful comments and suggestions. We also thank Pedro Duarte Neves, Les Godfrey, Paulo Parente, Maximiano Pinheiro, and Pedro Portugal for helpful discussions. The usual disclaimer applies. Machado is consultant for the Research Department of Banco de Portugal. Santos Silva is grateful for the hospitality, working conditions, and financial support provided by Banco de Portugal, which made this work possible. The authors also gratefully acknowledge partial financial support from Fundação para a Ciência e Tecnologia, program POCTI, partially funded by FEDER.