Optimal Tests of Noncorrelation Between Multivariate Time Series

Abstract

The problem of testing noncorrelation between two multivariate time series is considered. Assuming that the global process admits a joint vector autoregressive (VAR) representation, noncorrelation between the two component series is equivalent to the hypothesis that all off-diagonal blocks in the matrix coefficients and the innovation covariance of the joint VAR representation are zero. We establish an adequate local asymptotic normality (LAN) property for this VAR model in the vicinity of noncorrelation. This LAN structure allows construction of optimal pseudo-Gaussian tests—that is, tests that are locally and asymptotically optimal under Gaussian innovations, but remain valid under non-Gaussian ones—for the null hypothesis of noncorrelation and for comparing their local asymptotic powers with those of the heuristic tests (Haugh–El Himdi–Roy and Koch–Yang–Hallin–Saidi) proposed in the literature.