On mixed portmanteau statistics for the diagnostic checking of time series models using Gaussian quasi-maximum likelihood approach

Abstract

This thesis aims at investigating different forms of residuals from a general time series model with conditional mean and conditional variance fitted by the Gaussian quasi-maximum likelihood method. We investigated the limiting distributions of autocorrelation and partial autocorrelation functions under different forms of residuals. Based on them we devised some individual portmanteau tests and two mixed portmanteau tests. We started by exploring the asymptotic normalities of the residual autocorrelation functions, the squared residual autocorrelation functions and absolute residual autocorrelation functions from the fitted time series model. This leads to three individual portmanteau tests. Then we generalized them to their counterparts of partial autocorrelation functions, and this results in another three individual portmanteau tests. We carried out simulations studies to compare the six individual portmanteau tests and find that some tests are sensitive to mis-specification in the conditional mean while other tests are effective to detect mis-specification in the conditional variance. However, for the case that the mis-specifications happen in both conditional mean and variance, none of these individual portmanteau tests present remarkable power. Based on this, we continued to investigate the joint limiting distributions of the residual autocorrelation functions and absolute residual autocorrelation functions of the fitted model since the former one is powerful to identify mis-specification in the conditional mean and the latter one works well when the conditional variance is mis-specified. This leads to two mixed portmanteau tests for diagnostic checking of the fitted model. Simulation studies are carried out to check the asymptotic theory and to assess the performance of the mixed portmanteau tests. It shows that the mixed portmanteau tests have the power to detect mis-specification in the conditional mean and conditional variance respectively while it is feasible to detect both of them.