Weak convergence results for sample averages of nonlinear functions of (discrete-time) stochastic processes satisfying a functional central limit theorem (e.g., integrated processes) are given. These results substantially extend recent work by Park and Phillips (1999, Econometric Theory 15, 269–298) and de Jong (2002, working paper), in that a much wider class of functions is covered. For example, some of the results hold for the class of all locally integrable functions, thus avoiding any of the various regularity conditions imposed on the functions in Park and Phillips (1999) or de Jong (2002).I thank Robert de Jong for drawing my attention to this problem and Hannes Leeb for helpful comments. This paper was presented at the Econometric Society European Meeting 2002 in Venice.

Elliott, Rothenberg, and Stock (1996, Econometrica 64, 813–836) derive a class of point-optimal unit root tests in a time series model with Gaussian errors. Other authors have proposed “robust” tests that are not optimal for any model but perform well when the error distribution has thick tails. I derive a class of point-optimal tests for models with non-Gaussian errors. When the true error distribution is known and has thick tails, the point-optimal tests are generally more powerful than the tests of Elliott et al. (1996) and also than the robust tests. However, when the true error distribution is unknown and asymmetric, the point-optimal tests can behave very badly. Thus there is a trade-off between robustness to unknown error distributions and optimality with respect to the trend coefficients.This paper could not have been written without the encouragement of Thomas Rothenberg. This is based on my dissertation, which he supervised. I also thank Don Andrews, Jack Porter, Jim Stock, and seminar participants at the University of Pennsylvania, the University of Toronto, the University of Montreal, Princeton University, and the meetings of the Econometric Society in UCLA. Comments of three anonymous referees greatly improved the exposition of the paper. I owe special thanks to Gary Chamberlain for helping me to understand these results.

Two new stationarity tests are proposed. Both tests can be viewed as generalizations of existing stationarity tests and dominate these in terms of local asymptotic power. Improvements are achieved by accommodating stationary covariates. A Monte Carlo investigation of the small sample properties of the tests is conducted, and an empirical illustration from international finance is provided.This paper has benefited from the comments of Pentti Saikkonen (the co-editor), two anonymous referees, and seminar participants at University of Aarhus, Indiana University, Purdue University, Stanford University, UC Riverside, the 2001 Nordic Econometric Meeting, and the 2001 NBER Summer Institute. A MATLAB program that implements the tests proposed in this paper is available at http://elsa.Berkeley.EDU/users/mjansson.

In this paper we investigate the impact that starting values have upon the double differencing tests of Hasza and Fuller (1979, Annals of Statistics 7, 1106–1120) and Sen and Dickey (1987, Journal of Business and Economic Statistics 5, 463–473), based on ordinary least squares (OLS) and simple symmetric least squares (SSLS) estimation, respectively. We demonstrate that, contrary to what is observed for conventional unit root tests, when based on data that have been demeaned, either directly or by the inclusion of a constant in the test regression, these tests are not exact similar to the starting values of the process, except where they are fixed and equal. We show that such test statistics can also fail to be asymptotically pivotal, contrary to claims made previously in the literature. We demonstrate that OLS tests based on data that have been detrended, either directly or by the inclusion of a constant and linear time trend in the test regression, do provide exact similar inference. In the context of the SSLS-based approach, we demonstrate that one obtains exact similar tests only where direct detrending is used. We highlight another error that appears in the literature, demonstrating that the two demeaned OLS-based test statistics do not coincide, even asymptotically, and that the same holds for the OLS- and SSLS-based test statistics for detrended data. We use Monte Carlo methods to quantify the finite-sample dependence of these tests on the starting values. These results suggest that the SSLS-based tests are considerably more sensitive to the nature of the starting values than are the OLS-based tests.We are grateful to Pentti Saikkonen and two anonymous referees for their helpful and constructive comments on an earlier version of this paper. We are especially grateful to one of the referees for the suggestion implemented in Remark 3.1 of the paper.

Recent literature shows that embedding fractionally integrated time series models with spectral poles at the long-run and/or seasonal frequencies in autoregressive frameworks leads to estimators and test statistics with nonstandard limiting distributions. However, we demonstrate that when embedding such models in a general I(d) framework the resulting estimators and tests regain desirable properties from standard statistical analysis. We show the existence of a local time domain maximum likelihood estimator and its asymptotic normality—and under Gaussianity asymptotic efficiency. The Wald, likelihood ratio, and Lagrange multiplier tests are asymptotically equivalent and chi-squared distributed under local alternatives. With independent and identically distributed Gaussian errors and a scalar parameter, we show that the tests in addition achieve the asymptotic Gaussian power envelope of all invariant unbiased tests; i.e., they are asymptotically uniformly most powerful invariant unbiased against local alternatives. In a Monte Carlo study we document the finite sample superiority of the likelihood ratio test.I am grateful to Bent Jesper Christensen, Niels Haldrup, Pentti Saikkonen (the co-editor), and two anonymous referees for many useful comments and suggestions that significantly improved this paper. This work was done while the author was at the University of Aarhus, Denmark.

We establish the necessary and sufficient conditions for covariance stationarity of ARCH(∞), for both the levels and the squares. The result applies to any form of the conditional variance coefficients. This includes GARCH(p,q) and also specifications with hyperbolically decaying coefficients, such as the autoregressive coefficients of the autoregressive fractionally integrated moving average model. The covariance stationarity condition for the levels rules out long memory in the squares.I thank Peter M. Robinson for useful comments on previous versions of the paper. Also, I am grateful to the co-editor (Bruce E. Hansen) and an anonymous referee whose suggestions greatly improved the paper.

This paper presents results on the issue of weak convergence of a TVP-GQARCH-M(1,1) process. These suggest that the weak limit of this endogenous volatility model is an exogenous (stochastic) volatility continuous time process. Under the appropriate assumptions, we derive the invariant distributions at which the process converges in various cases. They reveal an approximate distributional relation between the GARCH or the asymmetric GQARCH variance processes and the continuous time Ornstein–Uhlenbeck models with respect to appropriate nonnegative Levy processes.I am grateful to Ritsa Panagiotou, Phyllis Alexander, Antonis Demos, Bruce Hansen, Nicholas Magginas, Nour Meddahi, Enrique Sentana, Paolo Zaffaroni, two anonymous referees, and the seminar participants at the Department of International and European Economic Studies for their valuable comments.

We study some methods of combining procedures for forecasting a continuous random variable. Statistical risk bounds under the square error loss are obtained under distributional assumptions on the future given the current outside information and the past observations. The risk bounds show that the combined forecast automatically achieves the best performance among the candidate procedures up to a constant factor and an additive penalty term. In terms of the rate of convergence, the combined forecast performs as well as if the best candidate forecasting procedure were known in advance.

Empirical studies suggest that combining procedures can sometimes improve forecasting accuracy over the original procedures. Risk bounds are derived to theoretically quantify the potential gain and price of linearly combining forecasts for improvement. The result supports the empirical finding that it is not automatically a good idea to combine forecasts. Indiscriminate combining can degrade performance dramatically as a result of the large variability in estimating the best combining weights. An automated combining method is shown in theory to achieve a balance between the potential gain and the complexity penalty (the price of combining), to take advantage (if any) of sparse combining, and to maintain the best performance (in rate) among the candidate forecasting procedures if linear or sparse combining does not help.This research was supported by U.S. National Security Agency Grant MDA9049910060 and U.S. National Science Foundation CAREER Grant DMS0094323. The author sincerely thanks three reviewers and Poti Giannakouros for their very valuable comments, which led to a substantial improvement of the paper.

A Hausman test based on the difference between fixed effects two-stage least squares and error components two-stage least squares.

Correcting for heteroskedasticity of unspecified form.

Deriving the observed information matrix in ordered probit and logit models using the complete-data likelihood function—solution.

Redundancy of lagged regressors in a conditionally heteroskedastic time series regressionAn excellent solution has been independently proposed by the S. Anatolyev, the poser of the problem.—solution.

Redundancy of lagged regressors in a conditionally heteroskedastic time series regression—solution.