Diagnostic checking in ARMA Models with Uncorrelated Errors

We consider tests for lack of fit in ARMA models with nonindependent 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 article we derive the asymptotic covariance matrix ρ̂m 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 ρ̂m, and a modification of the portmanteau tests are proposed. This allows us 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 noncorrelated rather than independent or a martingale difference. A set of Monte Carlo experiments, and an application to the Standard & Poor 500 returns, illustrate the practical relevance of our theoretical results