On portmanteau goodness-of-fit tests in robust time series modelling (Q1965960)

The problem of model adequacy checking is considered for the ARMA(p,q) models of the form \(\phi(B)X_t=\theta(B)a_t,\) where \(\phi(B)=1-\phi_1 B-\dots-\phi_p B^p\), \(\theta(B)=1-\theta_1 B-\dots -\theta_q B^q\), \(B\) is the backshift operator, and \(a_t\) is Gaussian white noise. The portmanteau statistic \(Q_m= n^2\sum_{k=1}^m\hat\rho_k^2/(n-k)\), where \(\hat\rho_k\) is the lag k residual autocorrelation and \(n\) is the number of observations, is commonly used for this purpose. The author proposes a robust version of \(Q_m\) in which an \(\alpha\)-trimmed residual autocorrelation function is substituted instead of \(\hat\rho_k\). It is demonstrated that this statistic has the asymptotic \(\chi^2\)-distribution with \(m-p-q\) degrees of freedom if the orders \((p,q)\) are correctly specified. In the simulation studies an AR(1) process is used with additive and innovative outliers. This modification of the portmanteau statistic is compared with the initial \(Q_m\) and some other robust modifications.