A sharp analysis on the asymptotic behavior of the Durbin-Watson statistic for the first-order autoregressive process

Bernard Bercu (1, 2) , Frédéric Proia (1, 2)

1 IMB - Institut de Mathématiques de Bordeaux
2 ALEA - Advanced Learning Evolutionary Algorithms

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.

Mots clés: Durbin-Watson statistic; Autoregressive process; Residual autocorrelation; Statistical test for serial correlation

Domaines: Probabilités [math.PR]

Bernard Bercu, Frédéric Proia. A sharp analysis on the asymptotic behavior of the Durbin-Watson statistic for the first-order autoregressive process. ESAIM: Probability and Statistics, 2013, 17, pp.500-530. ⟨10.1051/ps/2012005⟩. ⟨hal-00642634⟩