The exact mean integrated squared error (MISE) of the nonparametric kernel density estimator is derived for the asymptotically optimal smooth polynomial kernels of Müller (1984, Annals of Statistics 12, 766–774) and the trapezoid kernel of Politis and Romano (1999, Journal of Multivariate Analysis 68, 1–25) and is used to contrast their finite-sample efficiency with the higher order Gaussian kernels of Wand and Schucany (1990Canadian Journal of Statistics 18, 197–204). We find that these three kernels have similar finite-sample efficiency. Of greater importance is the choice of kernel order, as we find that kernel order can have a major impact on finite-sample MISE, even in small samples, but the optimal kernel order depends on the unknown density function. We propose selecting the kernel order by the criterion of minimax regret, where the regret (the loss relative to the infeasible optimum) is maximized over the class of two-component mixture-normal density functions. This minimax regret rule produces a kernel that is a function of sample size only and uniformly bounds the regret below 12% over this density class.

The paper also provides new analytic results for the smooth polynomial kernels, including their characteristic function.This research was supported in part by the National Science Foundation. I thank Oliver Linton and a referee for helpful comments and suggestions that improved the paper.

This paper provides two main new results: the first shows theoretically that large biases and variances can arise when the quasi-maximum likelihood (QML) estimation method is employed in a simple bivariate structure under the assumption of conditional heteroskedasticity; and the second demonstrates how these analytical theoretical results can be used to improve the finite-sample performance of a test for multivariate autoregressive conditional heteroskedastic (ARCH) effects, suggesting an alternative to a traditional Bartlett-type correction. We analyze two models: one proposed in Wong and Li (1997, Biometrika 84, 111–123) and another proposed by Engle and Kroner (1995, Econometric Theory 11, 122–150) and Liu and Polasek (1999, Modelling and Decisions in Economics; 2000, working paper, University of Basel). We prove theoretically that a relatively large difference between the intercepts in the two conditional variance equations, which leads to the two series having correspondingly different volatilities in the restricted case, may produce very large variances in some QML estimators in the first model and very severe biases in some QML estimators in the second. Later we use our bias expressions to propose an LM-type test of multivariate ARCH effects and show through simulations that small-sample improvements are possible, especially in relation to the size, when we bias correct the estimators and use the expected hessian version of the test.Both authors thank H. Wong for providing us with the Gauss program to simulate the Wong and Li (1997) model. We also thank three anonymous referees for extremely helpful comments, and we are grateful for the comments received at seminars given at Cardiff University, Michigan State University, Queen Mary London, University of Exeter, and University of Montreal. We acknowledge gratefully also the financial support from an ESRC grant (award number T026 27 1238). A previous version of this paper appeared as IVIE Working paper 2004-09.

The sampling window method of Hall, Jing, and Lahiri (1998, Statistica Sinica 8, 1189–1204) is known to consistently estimate the distribution of the sample mean for a class of long-range dependent processes, generated by transformations of Gaussian time series. This paper shows that the same nonparametric subsampling method is also valid for an entirely different category of long-range dependent series that are linear with possibly non-Gaussian innovations. For these strongly dependent time processes, subsampling confidence intervals allow inference on the process mean without knowledge of the underlying innovation distribution or the long-memory parameter. The finite-sample coverage accuracy of the subsampling method is examined through a numerical study.The authors thank two referees for comments and suggestions that greatly improved an earlier draft of the paper. This research was partially supported by U.S. National Science Foundation grants DMS 00-72571 and DMS 03-06574 and by the Deutsche Forschungsgemeinschaft (SFB 475).

In this paper we analyze the effects of a very general class of time-varying variances on well-known “stationarity” tests of the I(0) null hypothesis. Our setup allows, among other things, for both single and multiple breaks in variance, smooth transition variance breaks, and (piecewise-) linear trending variances. We derive representations for the limiting distributions of the test statistics under variance breaks in the errors of I(0), I(1), and near-I(1) data generating processes, demonstrating the dependence of these representations on the precise pattern followed by the variance processes. Monte Carlo methods are used to quantify the effects of fixed and smooth transition single breaks and trending variances on the size and power properties of the tests. Finally, bootstrap versions of the tests are proposed that provide a solution to the inference problem.We are grateful to Peter Phillips, a co-editor, and two anonymous referees whose comments on an earlier draft have led to a considerable improvement in the paper.

A new first-order asymptotic theory for heteroskedasticity-autocorrelation (HAC) robust tests based on nonparametric covariance matrix estimators is developed. The bandwidth of the covariance matrix estimator is modeled as a fixed proportion of the sample size. This leads to a distribution theory for HAC robust tests that explicitly captures the choice of bandwidth and kernel. This contrasts with the traditional asymptotics (where the bandwidth increases more slowly than the sample size) where the asymptotic distributions of HAC robust tests do not depend on the bandwidth or kernel. Finite-sample simulations show that the new approach is more accurate than the traditional asymptotics. The impact of bandwidth and kernel choice on size and power of t-tests is analyzed. Smaller bandwidths lead to tests with higher power but greater size distortions, and large bandwidths lead to tests with lower power but smaller size distortions. Size distortions across bandwidths increase as the serial correlation in the data becomes stronger. Overall, the results clearly indicate that for bandwidth and kernel choice there is a trade-off between size distortions and power. Finite-sample performance using the new asymptotics is comparable to the bootstrap, which suggests that the asymptotic theory in this paper could be useful in understanding the theoretical properties of the bootstrap when applied to HAC robust tests.We thank an editor and a referee for constructive comments on a previous version of the paper. Helpful comments provided by Cliff Hurvich, Andy Levin, Jeff Simonoff, and seminar participants at NYU (Statistics), U. Texas Austin, Yale, U. Montreal, UCSD, UC Riverside, UC Berkeley, U. of Pittsburgh, SUNY Albany, U. Aarhus, Brown U., NBER/NSF Time Series Conference, and 2003 Winter Meetings of the Econometrics Society are gratefully acknowledged. We gratefully acknowledge financial support from the National Science Foundation through grant SES-0095211. We thank the Center for Analytic Economics at Cornell University.

Conditions ensuring a central limit theorem for strongly mixing triangular arrays are given. Larger samples can show longer range dependence than shorter samples. The result is obtained by constraining the rate growth of dependence as a function of the sample size, with the usual trade-off of memory and moment conditions. An application to heteroskedasticity and autocorrelation consistent estimators is proposed.

In this note we show that, when the true data generating process is a stationary one around a constant term with a break, the stationarity test of Kim (2000, Journal of Econometrics 95, 97–116) against the alternative hypothesis of change of persistence rejects the null of stationarity asymptotically with probability one.I am grateful to an anonymous referee for his useful comments, which have helped to improve the content and presentation of this note. I acknowledge financial support from Ministerio de Ciencia y Tecnología, project SEC2003-09205.