Tests for Autocorrelation and Randomness in Multiple Time Series

Abstract

Abstract There exist several tests for autocorrelation and randomness in multiple time series. Unfortunately, the exact distributions of these statistics are unknown and the asymptotic distributions that are known do not provide adequate approximation to the exact ones in small samples. In this article, the test statistics are modified. Asymptotically, these modified statistics are equivalent to their original counterparts; however, it is found that the asymptotic distributions of these statistics provide adequate approximation to the exact ones in relatively small samples and possibly when the time series are nonnormal. The adequacy of the approximations is examined by simulation experiments. The original test statistics are based on sample lag cross-covariances, autocovariances, cross-correlations, and autocorrelations standardized by their asymptotic means and covariances. The modified statistics are obtained when the asymptotic means and covariances in the standardization are replaced by the exact means and covariances. The expressions for these moments are derived on the assumption that the time series is Gaussian. These moments (both asymptotic and exact) involve nuisance parameters. In constructing the test statistics, these nuisance parameters are replaced by their sample counterparts, which are consistent estimates of the parameters.