Portmanteau Tests for Semiparametric Nonlinear Conditionally Heteroscedastic Time Series Models

A class of multivariate time series models is considered, with general parametric specifications for the conditional mean and variance. In this general framework, the usual Box–Pierce portmanteau test statistic, based on the sum of the squares of the first residual autocorrelations, cannot be accurately approximated by a parameter-free distribution. A first solution is to estimate from the data the complicated asymptotic distribution of the Box–Pierce statistic. The solution proposed by Li [23] consists of changing the test statistic by using a quadratic form of the residual autocorrelations which follows asymptotically a chi-square distribution. Katayama [21] proposed a distribution-free statistic based on a projection of the autocorrelation vector. The first aim of this paper is to show that the three methods, initially introduced for specific time series models, can be applied in our general framework. The second aim is to compare the three approaches. The comparison is made on (i) the mathematical assumptions required by the different methods and (ii) the computations of the Bahadur slopes (in some cases via Monte Carlo simulations).