Goodness-of-Fit Tests for SPARMA Models with Dependent Error Terms

In this paper we consider tests for lack of fit in a class of seasonal periodic autoregressive moving average (SPARMA) models under the assumption that the errors are uncorrelated but non-independent (i.e. weak SPARMA models). We derive the asymptotic distributions of residual and normalized residual empirical autocovariances and autocorrelations of these weak SPARMA models. We then deduce the modified portmanteau statistics. We establish the asymptotic behavior of the proposed statistics, which can be quite different from the usual chi-squared approximation used under independent and identically distributed (iid) assumptions on the noise. We also propose another test based on a self-normalization approach to cheek the adequacy of SPARMA models. A set of Monte Carlo experiments and an application to the daily returns of the SP500 are presented.