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

Published online by Cambridge University Press:  03 June 2013

Bernard Bercu and
Frédéric Proïa
Bernard Bercu
Affiliation:
Université Bordeaux 1, Institut de Mathématiques de Bordeaux, UMR 5251, and INRIA Bordeaux, team ALEA, 351 Cours de la Libération, 33405 Talence Cedex, France. Bernard.Bercu@math.u-bordeaux1.fr; Frederic.Proia@inria.fr
Frédéric Proïa
Affiliation:
Université Bordeaux 1, Institut de Mathématiques de Bordeaux, UMR 5251, and INRIA Bordeaux, team ALEA, 351 Cours de la Libération, 33405 Talence Cedex, France. Bernard.Bercu@math.u-bordeaux1.fr; Frederic.Proia@inria.fr

Abstract

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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 with 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. We show how our statistical test procedure performs better, from a theoretical and a practical point of view, than the commonly used Box–Pierce and Ljung–Box procedures, even on small-sized samples.

Keywords

Durbin–Watson statisticautoregressive processresidual autocorrelationstatistical test for serial correlation

Information

Type
Research Article
Information
ESAIM: Probability and Statistics , Volume 17 , 2013, pp. 500 - 530
Copyright
© EDP Sciences, SMAI, 2013

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