On diagnostic checking time series models with portmanteau test statistics based on generalized inverses and {2}-inverses

A class of univariate time series models is considered, which allows general specifications for the conditional mean and conditional variance functions. After deriving the asymptotic distributions of the residual autocorrelations based on the standardized residuals, portmanteau test statistics are studied. If the asymptotic covariance of a vector of fixed length of residual autocorrelations is non singular, portmanteau test statistics could be defined, following the approach advocated by Li (1992). However, assuming the invertibility of the asymptotic covariance of the residual autocorrelations may be restrictive, and, alternatively, the popular Box-Pierce-Ljung test statistic may be recommended. In our framework, that test statistic converges in distribution to a weighted sum of chi-square variables, and the critical values can be found using Imhof's (1961) algorithm. However, Imhof's algorithm may be time consuming. In view of this, we investigate in this article the use of generalized inverses and {2}-inverses, in order to propose new test statistics with asymptotic chi-square distributions, avoiding the need to implement Imhof's algorithm. In a small simulation study, the following test statistics are compared: Box-Pierce-Ljung test statistic, the test statistic based on the proposal of Li (1992), and the new test statistics relying on generalized inverses and {2}-inverses.