Multivariate portmanteau tests for weak multiplicative seasonal VARMA models

Numerous multivariate time series encountered in real applications display seasonal behavior. In this paper we consider portmanteau tests for testing the adequacy of structural multiplicative seasonal vector autoregressive moving-average (SVARMA) models under the assumption that the errors are uncorrelated but not necessarily independent (i.e. weak SVARMA). We study the asymptotic distributions of residual autocorrelations at seasonal lags of multiple of the length of the seasonal period under weak assumptions on the noise. We deduce the asymptotic distribution of the proposed multivariate portmanteau statistics, which can be quite different from the usual chi-squared approximation used under independent and identically distributed (iid) assumptions on the noise. A set of Monte Carlo experiments and an application of U.S. monthly housing starts and housing sold are presented.