Abstract. 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 firstorder 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.

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 clarify and to improve the well-known h-test suggested by Durbin. Finally, we briefly summarize our observations on simulated samples.

Abstract. The purpose of this paper is to investigate moderate deviations for the DurbinWatson statistic associated with the stable first-order autoregressive process where the driven noise is also given by a first-order autoregressive process. We first establish a moderate deviation principle for both the least squares estimator of the unknown parameter of the autoregressive process as well as for the serial correlation estimator associated with the driven noise. It enables us to provide a moderate deviation principle for the DurbinWatson statistic in the easy case where the driven noise is normally distributed and in the more general case where the driven noise satisfies a less restrictive Chen-Ledoux type condition.