Multivariate portmanteau test for structural VARMA models with uncorrelated but non-independent error terms

We consider portmanteau tests for testing the adequacy of vector autoregressive moving-average (VARMA) models under the assumption that the errors are uncorrelated but not necessarily independent. We relax the standard independence assumption to extend the range of application of the VARMA models, and allow to cover linear representations of general nonlinear processes. We first study the joint distribution of the quasi-maximum likelihood estimator (QMLE) or the least squared estimator (LSE) and the noise empirical autocovariances. We then derive the asymptotic distribution of residual empirical autocovariances and autocorrelations under weak assumptions on the noise. We deduce the asymptotic distribution of the Ljung-Box (or Box-Pierce) portmanteau statistics for VARMA models with nonindependent innovations. In the standard framework (i.e. under iid assumptions on the noise), it is known that the asymptotic distribution of the portmanteau tests is that of a weighted sum of independent chi-squared random variables. The asymptotic distribution can be quite different when the independence assumption is relaxed. Consequently, the usual chi-squared distribution does not provide an adequate approximation to the distribution of the Box-Pierce goodness-of fit portmanteau test. Hence we propose a method to adjust the critical values of the portmanteau tests. Monte carlo experiments illustrate the finite sample performance of the modified portmanteau test.