This paper studies the asymptotic properties of a nonstationary partially linear regression model. In particular, we allow for covariates to enter the unit root (or near unit root) model in a nonparametric fashion, so that our model is an extension of the semiparametric model analyzed in Robinson (1988, Econometrica 56, 931–954). It is proved that the autoregressive parameter can be estimated at rate N even though part of the model is estimated nonparametrically. Unit root tests based on the semiparametric estimate of the autoregressive parameter have a limiting distribution that is a mixture of a standard normal and the Dickey–Fuller distribution. A Monte Carlo experiment is conducted to evaluate the performance of the tests for various linear and nonlinear specifications.We thank Bruce Hansen, Roger Koenker, Helmut Lütkepohl, Peter Phillips, three referees, and participants of the 8th World Congress of the Econometric Society and the 10th Midwest Econometrics Group Meeting for helpful comments on an earlier version of this paper. This investigation was supported by the University of Kansas General Research Fund allocation 2301789-003.

This paper develops an asymptotic theory for integrated and near-integrated time series whose range is constrained in some ways. Such a framework arises when integration and cointegration analyses are applied to time series that are bounded either by construction or because they are subject to control. The asymptotic properties of some commonly used integration tests are discussed; the bounded unit root distribution is introduced to describe the limiting distribution of the sample first-order autoregressive coefficient of a random walk under range constraints. The theoretical results show that the presence of such constraints can lead to drastically different asymptotics. Because deviations from the standard unit root theory are measured through two noncentrality parameters that can be consistently estimated, simple measures of the impact of range constraints on the asymptotic distributions are obtained. Generalizations of standard unit root tests that are robust to the presence of range constraints are also provided. Finally, it is shown that the proposed asymptotic framework provides an adequate approximation to the finite-sample properties of the unit root statistics under range constraints.Partial financial support from Italian MIUR grants is gratefully acknowledged. I thank, without implicating, Pentti Saikkonen (the co-editor), an anonymous referee, Attilio Gardini, Martin Jacobsen, Robert de Jong, Paolo Paruolo, Anders Rahbek, and participants at the 58th European Meeting of the Econometric Society, Stockholm, August 21–24, 2003, for helpful comments. I also thank the Bank of International Settlements for providing the European monetary system exchange rate data.

Strong consistency and asymptotic normality are established for the quasi-maximum likelihood estimator for a class of ARCH(q) models. The conditions are that the ARCH process is geometrically ergodic with a moment of arbitrarily small order. Furthermore for consistency, we assume that the second-order moment exists for the nondegenerate rescaled errors and, similarly, that the fourth-order moment exists for asymptotic normality to hold. Contrary to existing literature on (G)ARCH models the parameter space is not assumed to be compact; we only impose a lower bound for the constant term in our parameterization of the conditional variance. It is demonstrated that the general conditions are satisfied for a range of specific models.We are grateful to the editor and the referees for their very helpful and detailed suggestions, which, we believe, improved the paper substantially. We thank Søren T. Jensen for stimulating discussions and Jonathan Dennis for helpful research assistance. Rahbek acknowledges continuing financial support from the Danish Social Sciences Research Council. Kristensen received funding from the Danish Research Agency and the Financial Markets Group, LSE, during this research.

This paper develops optimal tests for model selection between two nested models in the presence of underlying parameter instability. These are joint tests for both parameter instability and a null hypothesis on a subset of the parameters. They modify the existing tests for parameter instability to allow the parameter vector to be unknown. These test statistics are useful if one is interested in testing a null hypothesis on some parameters but is worried about the possibility that the parameters may be time varying. The paper provides the asymptotic distributions of this class of test statistics and their critical values for some interesting cases.I thank M. Watson for the idea of this paper and for numerous discussions, suggestions, comments, and teaching. I am grateful to T. Clark, G. Chow, R. Gallant, F. Sowell, N. Swanson, E. Tamer, and A. Tarozzi and also to the co-editor and two referees and to seminar participants at the University of Virginia, ECARES Université Libre de Brussels, the 2001 Triangle Econometrics Conference, the 2002 NBER Summer Institute, the 2003 Summer Meetings of the Econometric Society, and the 2003 EC2 Conference for comments. Financial support from IFS Summer Research, Princeton University, is gratefully acknowledged. All mistakes are mine.

This paper introduces a test for the comparison of multiple misspecified conditional interval models, for the case of dependent observations. Model accuracy is measured using a distributional analog of mean square error, in which the approximation error associated with a given model, say, model i, for a given interval, is measured by the expected squared difference between the conditional confidence interval under model i and the “true” one.

When comparing more than two models, a “benchmark” model is specified, and the test is constructed along the lines of the “reality check” of White (2000, Econometrica 68, 1097–1126). Valid asymptotic critical values are obtained via a version of the block bootstrap that properly captures the effect of parameter estimation error. The results of a small Monte Carlo experiment indicate that the test does not have unreasonable finite sample properties, given small samples of 60 and 120 observations, although the results do suggest that larger samples should likely be used in empirical applications of the test.The authors express their gratitude to Don Andrews and an anonymous referee for providing numerous useful suggestions, all of which we feel have been instrumental in improving earlier drafts of this paper. The authors also thank Russell Davidson, Clive Granger, Lutz Kilian, Christelle Viaroux, and seminar participants at the 2002 UK Econometrics Group meeting in Bristol, the 2002 European Econometric Society meetings, the 2002 University of Pennsylvania NSF-NBER time series conference, the 2002 EC2 Conference in Bologna, Cornell University, the State University of New York at Stony Brook, and the University of California at Davis for many helpful comments and suggestions on previous versions of this paper.

We investigate the issue of the uniqueness of the cross-validation selected smoothing parameters in kernel estimation of multivariate nonparametric regression or conditional probability functions. When the covariates are all continuous variables, we provide a necessary and sufficient condition, and when the covariates are a mixture of categorical and continuous variables, we provide a simple sufficient condition that guarantees asymptotically the uniqueness of the cross-validation selected smoothing parameters.We thank a referee for the constructive comments.

This problem shows how the key ignorability-of-treatment assumption used in estimating treatment effects can be violated when certain factors are included among the covariates. The case considered is when there are J + 1 treatment levels, treatment is randomized with respect to potential outcomes, but the distribution of included covariates differs across treatment levels.

On May 18, 2005, Francesc Marmol died of cancer in Madrid, Spain, at the age of 38. Francesc (Paco) was born in Barcelona to a family of modest means. He graduated from the Universitat Autònoma de Barcelona with a B.Sc. in Economics. He later went on to obtain a Ph.D. in the International Doctorate in Economic Analysis, where he defended a thesis on asymptotic theory for integrated processes in 1996. After University, Paco took a position at the Universidad Carlos III de Madrid, where he was Professor at the Economics Department.