  The purpose of this paper is to provide a sharp analysis on the asymptotic
behavior of the Durbin-Watson statistic. We focus our attention on the
first-order autoregressive process where the driven noise is also given by a
first-order autoregressive process. We establish the almost sure convergence
and the asymptotic normality for both the least squares estimator of the
unknown parameter of the autoregressive process as well as for the serial
correlation estimator associated to the driven noise. In addition, the almost
sure rates of convergence of our estimates are also provided. It allows us to
establish the almost sure convergence and the asymptotic normality for the
Durbin-Watson statistic. Finally, we propose a new bilateral statistical test
for residual autocorrelation.
