Invariant Tests for Covariance Structures in Multivariate Linear Model

The null hypothesis that the error vectors in a multivariate linear model are independent is tested against the alternative hypothesis that they are dependent in some specified manner. This dependence is assumed to be due to common random components or autocorrelation over time. The testing problem is solved by classical invariance arguments under multinormality. The most powerful invariant test usually depends on the particular alternative and may even lack a closed form expression. Then the locally best test is derived. The power is maximized at the null hypothesis in the direction of some alternative. In most applications the direction where the maximization is performed does not enter the test. Then the locally uniformly best test exists. Several applications are outlined.