Variance of the Sample Space-Time Correlation Function of Contemporaneously Correlated Variables

An important part of the identification and diagnostic checking of space-time ARMA models is the evaluation of the significance of the autocorrelations of the observations and residuals, respectively. Since such tests are based on the calculated space-time autocorrelation function, it is necessary too know the variance of these correlations when in fact the underlying process is temporally independent. Previously the variance of the sample space-time autocorrelation function was developed for the case when the observed process consists of T independent observations of a vector random variable with mean zero and spherical variance covariance matrix, $\sigma ^2 I$ (Pfeifer and Deutsch, J. Roy. Stat. Soc. B, 42 (1980)). This paper extends these results to the case of contemporaneously correlated variables. In this instance, G, the error covariance matrix of the observations, is nonspherical.