The Multivariate Portmanteau Statistic

Box and Pierce have derived a goodness-of-fit test, the portmanteau test, for univariate autoregressive moving-average (ARMA) time series models. This test is here extended to multivariate ARMA models; the test statistic may be conveniently expressed as a function of the covariances between the residuals of the fitted model. A modified form of the statistic designed to have superior properties in small samples is derived, and the two forms of the statistic are compared via computer simulation.