Some Limit Theory for Autocovariances Whose Order Depends on Sample Size

In this paper we provide some weak convergence results for sample statistics of the product of a variable with its kth-order lag. We assume the variable is a stationary vector that can be represented by linear process, and the lag length k is allowed to be a function of the sample size. Employing the Beveridge–Nelson decomposition, we derive a new functional central limit theorem for this situation and establish related stochastic integral convergence results. We then consider the behavior of associated long-run variance estimators and also extend our analysis to the case where the sample statistics are based on regression residuals. We illustrate the potential range of application of these techniques in the context of (i) testing for I(0) versus I(1) behavior and (ii) estimation and testing in a heteroskedastically cointegrated regression model.We thank the co-editor and the referees for helpful comments on earlier drafts.