Jean-Michel Zaköıan§

We consider tests for lack of fit in ARMA models with non independent innovations. In this framework, the standard Box-Pierce and Ljung-Box portmanteau tests can perform poorly. Specifically, the usual text book formulas for asymptotic distributions are based on strong assumptions and should not be applied without careful consideration. In this paper, we derive the asymptotic covariance matrix Σρ̂m of a vector of autocorrelations for residuals of ARMA models under weak assumptions on the noise. The asymptotic distribution of the portmanteau statistics follows. A consistent estimator of Σρ̂m, and a modification of the portmanteau tests are proposed. This allows to construct valid asymptotic significance limits for the residual autocorrelations, and (asymptotically) valid goodness-of-fit tests, when the underlying noise process is assumed to be non correlated rather than independent or a martingale difference. A set of Monte-Carlo experiments, and an application to the Standard & Poor’s 500 returns, illustrate the practical relevance of our theoretical results