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

Abstract

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.

Moderate deviations for the Durbin–Watson statistic related to the first-order autoregressive process

“…Before starting the proof of Theorem 2.1, we need to introduce some technical tools. Denote by ℓ the almost sure limit of S n /n [2], given by…”

“…The first component R n (θ) is given in (4.21) while R n (ρ), whose definition may be found in the proof of Theorem 3.2 in [2], is made of isolated terms. Consequently, (4.25) and (4.29) are sufficient to ensure that…”

“…Durbin [9] proposed alternative tests to prevent this misuse, such as the h-test and the t-test, then substantial contributions were brought by Inder [14], King and Wu [15] and more recently Stocker [20]. Lately, a set of results have been established by Bercu and Proïa in [2], in particular a test procedure as powerful as the h-test, and they will be summarized thereafter as a basis for this paper.…”

“…First of all, we recall the results recently established by Bercu and Proïa [2]. In Section 2, we propose moderate deviation principles for the estimators of θ and ρ and for the Durbin-Watson statistic, given by (1.2), (1.4) and (1.5), under the normality assumption on the driven noise.…”

“…Then, we will extend our investigation to the more general framework where the driven noise satisfies a less restrictive Chen-Ledoux type condition [4], [16]. We are inspired by the recent paper of Bercu and Proïa [2], where the almost sure convergence and the central limit theorem are established for both the least squares estimators and the Durbin-Watson statistic. Our results are proved via an extensive use of the results of Dembo [5], Dembo and Zeitouni [6] and Worms [22], [23], [24] on the one hand, and of the paper of Puhalskii [19] and Djellout [7] on the other hand, about moderate deviations for martingales.…”