A Test for Serial Correlation in Multivariate Data

1. Introduction. A common assumption in multivariate analysis is that the data (or the residuals after removing the mean) are a sequence of random vectors that are independent and normally distributed with zero means and identical but unknown covariance matrices. In the test presented here, this is the null hypothesis. The alternative hypothesis for which this test is designed is that the data are a stationary Gaussian multiple time series in which observations at different times are correlated. The usual approach to detecting serial correlation is graphical presentation of either spectral-matrix estimates or covariance function estimates. For multiple time series, this approach has several difficulties including the need for several different graphs including ones that show dependence among the component series and the choice of spectral resolution. Thus, a way to test simultaneously for all the types of serial correlation that a stationary Gaussian time series can exhibit is needed. A univariate time series can be tested for serial correlation by computing the periodogram, dividing the frequency band into two parts, and comparing the sum of the periodogram over the lower part with the sum over the whole band. The sums are compared for all divisions of the frequency band using KolmogorovSmirnov limits (Bartlett (1966), Durbin (1969)). The test presented here is a generalization of this. As with the univariate case, the multidimensional periodogramn is computed, the frequency band is divided, and the sum over the lower part is compared with the siim over the whole band. These estimates are compared using the largest and smallest eigenvalues of one estimate with respect to the other. Some but not all divisions of the frequency band are considered simultaneously. As shown in Section 3, the cornputation of the distribution under the null hypothesis also has similarities to Kolmogorov-Smirnov tests.