Further results on the $h$-test of Durbin for stable autoregressive processes

Abstract

The purpose of this paper is to investigate the asymptotic behavior of the Durbin–Watson statistic for the stable $p$-order autoregressive process when the driven noise is given by a first-order autoregressive process. It is an extension of the previous work of Bercu and Proïa devoted to the particular case $p=1$. We establish the almost sure convergence and the asymptotic normality for both the least squares estimator of the unknown vector parameter of the autoregressive process as well as for the serial correlation estimator associated with the driven noise. In addition, the almost sure rates of convergence of our estimates are also provided. Then, we prove the almost sure convergence and the asymptotic normality for the Durbin–Watson statistic and we derive a two-sided statistical procedure for testing the presence of a significant first-order residual autocorrelation that appears to simplify and to improve the well-known h-test suggested by Durbin. Finally, we briefly summarize our observations on simulated samples.