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

Bernard Bercu; Frédéric Proïa

ESAIM: Probability and Statistics (2013)

  • Volume: 17, page 500-530
  • ISSN: 1292-8100

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.

How to cite

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Bercu, Bernard, and Proïa, Frédéric. "A sharp analysis on the asymptotic behavior of the Durbin–Watson statistic for the first-order autoregressive process." ESAIM: Probability and Statistics 17 (2013): 500-530. <http://eudml.org/doc/273627>.

@article{Bercu2013,
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 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.},
author = {Bercu, Bernard, Proïa, Frédéric},
journal = {ESAIM: Probability and Statistics},
keywords = {Durbin–Watson statistic; autoregressive process; residual autocorrelation; statistical test for serial correlation; Durbin-Watson statistic},
language = {eng},
pages = {500-530},
publisher = {EDP-Sciences},
title = {A sharp analysis on the asymptotic behavior of the Durbin–Watson statistic for the first-order autoregressive process},
url = {http://eudml.org/doc/273627},
volume = {17},
year = {2013},
}

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
PB - EDP-Sciences
VL - 17
SP - 500
EP - 530
AB - 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.
LA - eng
KW - Durbin–Watson statistic; autoregressive process; residual autocorrelation; statistical test for serial correlation; Durbin-Watson statistic
UR - http://eudml.org/doc/273627
ER -

References

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