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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.
@article{PS_2013__17__500_0,
author = {Bercu, Bernard and Pro{\"\i}a, Fr\'ed\'eric},
title = {A sharp analysis on the asymptotic behavior of the {Durbin-Watson} statistic for the first-order autoregressive process},
journal = {ESAIM: Probability and Statistics},
pages = {500--530},
publisher = {EDP-Sciences},
volume = {17},
year = {2013},
doi = {10.1051/ps/2012005},
mrnumber = {3070889},
language = {en},
url = {http://geodesic.mathdoc.fr/articles/10.1051/ps/2012005/}
}
TY - JOUR AU - Bercu, Bernard AU - Proïa, Frédéric TI - A sharp analysis on the asymptotic behavior of the Durbin-Watson statistic for the first-order autoregressive process JO - ESAIM: Probability and Statistics PY - 2013 SP - 500 EP - 530 VL - 17 PB - EDP-Sciences UR - http://geodesic.mathdoc.fr/articles/10.1051/ps/2012005/ DO - 10.1051/ps/2012005 LA - en ID - PS_2013__17__500_0 ER -
%0 Journal Article %A Bercu, Bernard %A Proïa, Frédéric %T A sharp analysis on the asymptotic behavior of the Durbin-Watson statistic for the first-order autoregressive process %J ESAIM: Probability and Statistics %D 2013 %P 500-530 %V 17 %I EDP-Sciences %U http://geodesic.mathdoc.fr/articles/10.1051/ps/2012005/ %R 10.1051/ps/2012005 %G en %F PS_2013__17__500_0
Bercu, Bernard; Proïa, Frédéric. A sharp analysis on the asymptotic behavior of the Durbin-Watson statistic for the first-order autoregressive process. ESAIM: Probability and Statistics, Tome 17 (2013), pp. 500-530. doi: 10.1051/ps/2012005
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