Note on the Asymptotic Efficiency of Sample Covariances in Gaussian Vector Stationary Processes

In this note certain results obtained by Porat ( J. Time Ser. Anal. 8 (1987), 205–20) and Kakizawa and Taniguchi ( J. Time Ser. Anal. 15 (1994), 303–11) concerning the asymptotic efficiency of sample autocovariances of a zero-mean Gaussian stationary process are extended to the case of m -vector processes. It is shown that, for Gaussian vector AR( p ) processes, the sample autocovariance matrix at lag k is asymptotically efficient if 0 ≤ k ≤ p . Further, none of the sample autocovariance matrices is asymptotically efficient for Gaussian vector MA( q ) processes.