  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 Pro\"ia 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 clarify and to
improve the well-known \textit{h-test} suggested by Durbin. Finally, we briefly
summarize our observations on simulated samples.
