Multivariate versions of Bartlett’s formula

Abstract

This paper quantifies the form of the asymptotic covariance matrix of the sample autocovariances in a multivariate stationary time series—the classic Bartlett formula. Such quantification is useful in many statistical inferences involving autocovariances. While joint asymptotic normality of the sample autocovariances is well-known in univariate settings, explicit forms of the asymptotic covariances have not been investigated in the general multivariate non-Gaussian case. We fill this gap by providing such an analysis, bookkeeping all skewness terms. Additionally, following a recent univariate paper by Francq and Zakoian, we consider linear processes driven by non-independent errors, a feature that permits consideration of multivariate GARCH processes.