Hostname: page-component-78c5997874-m6dg7 Total loading time: 0 Render date: 2024-11-07T04:23:28.843Z Has data issue: false hasContentIssue false
  • English
  • Français

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

Published online by Cambridge University Press:  03 October 2014

S. Valère Bitseki Penda ,
Hacène Djellout  and
Frédéric Proïa
S. Valère Bitseki Penda
Affiliation:
Laboratoire de Mathématiques, CNRS UMR 6620, Université Blaise Pascal, Avenue des Landais, 63177 Aubière, France. [email protected]
Hacène Djellout
Affiliation:
Laboratoire de Mathématiques, CNRS UMR 6620, Université Blaise Pascal, Avenue des Landais, 63177 Aubière, France; [email protected]
Frédéric Proïa
Affiliation:
Université Bordeaux 1, Institut de Mathématiques de Bordeaux, UMR 5251, and INRIA Bordeaux, team ALEA, 200 Avenue de la Vieille Tour, 33405 Talence cedex, France; [email protected]
Get access

Abstract

The purpose of this paper is to investigate moderate deviations for the Durbin–Watsonstatistic associated with the stable first-order autoregressive process where the drivennoise is also given by a first-order autoregressive process. We first establish a moderatedeviation principle for both the least squares estimator of the unknown parameter of theautoregressive process as well as for the serial correlation estimator associated with thedriven noise. It enables us to provide a moderate deviation principle for theDurbin–Watson statistic in the case where the driven noise is normally distributed and inthe more general case where the driven noise satisfies a less restrictive Chen–Ledoux typecondition.

Keywords

Durbin–Watson statisticmoderate deviation principlefirst-order autoregressive processserial correlation
Type
Research Article
Information
ESAIM: Probability and Statistics , Volume 18 , 2014 , pp. 308 - 331
Copyright
© EDP Sciences, SMAI 2014

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Arcones, M.A., The large deviation principle for stochastic processes I. Theory Probab. Appl. 47 (2003) 567–583. Google Scholar
Arcones, M.A., The large deviation principle for stochastic processes II. Theory Probab. Appl. 48 (2003) 19–44. Google Scholar
Bercu, B. and Proïa, F., A sharp analysis on the asymptotic behavior of the Durbin-Watson statistic for the first-order autoregressive process. ESAIM: PS 17 (2013) 500–530. Google Scholar
Bercu, B. and Touati, A., Exponential inequalities for self-normalized martingales with applications. Ann. Appl. Probab. 18 (2008) 1848–1869. Google Scholar
Chen, X., Moderate deviations for m-dependent random variables with Banach space value. Stat. Probab. Lett. 35 (1998) 123–134. Google Scholar
Dembo, A., Moderate deviations for martingales with bounded jumps. Electron. Commun. Probab. 1 (1996) 11–17. Google Scholar
A. Dembo and O. Zeitouni, Large deviations techniques and applications, 2nd edition, vol. 38 of Appl. Math. Springer (1998).
Djellout, H., Moderate deviations for martingale differences and applications to φ-mixing sequences. Stoch. Stoch. Rep. 73 (2002) 37–63. Google Scholar
Djellout, H. and Guillin, A., Moderate deviations for Markov chains with atom. Stochastic Process. Appl. 95 (2001) 203–217. Google Scholar
Durbin, J., Testing for serial correlation in least-squares regression when some of the regressors are lagged dependent variables. Econometrica 38 (1970) 410–421. Google Scholar
Durbin, J. and Watson, G.S., Testing for serial correlation in least squares regression I. Biometrika 37 (1950) 409–428. Google ScholarPubMed
Durbin, J. and Watson, G.S., Testing for serial correlation in least squares regression II. Biometrika 38 (1951) 159–178. Google ScholarPubMed
Durbin, J. and Watson, G.S., Testing for serial correlation in least squares regession III. Biometrika 58 (1971) 1–19. Google Scholar
Eichelsbacher, P. and Löwe, M., Moderate deviations for i.i.d. random variables. ESAIM: PS 7 (2003) 209–218. Google Scholar
Inder, B.A., An approximation to the null distribution of the Durbin-Watson statistic in models containing lagged dependent variables. Econometric Theory 2 (1986) 413–428. Google Scholar
King, M.L. and Wu, P.X., Small-disturbance asymptotics and the Durbin-Watson and related tests in the dynamic regression model. J. Econometrics 47 (1991) 145–152. Google Scholar
Ledoux, M., Sur les déviations modérées des sommes de variables aléatoires vectorielles indépendantes de même loi. Ann. Inst. Henri-Poincaré 35 (1992) 123–134. Google Scholar
Malinvaud, E., Estimation et prévision dans les modèles économiques autorégressifs. Rev. Int. Inst. Statis. 29 (1961) 1–32. Google Scholar
Nerlove, M. and Wallis, K.F., Use of the Durbin-Watson statistic in inappropriate situations. Econometrica 34 (1966) 235–238. Google Scholar
Proïa, F., Further results on the H-Test of Durbin for stable autoregressive processes. J. Multivariate. Anal. 118 (2013) 77–101. Google Scholar
Puhalskii, A., Large deviations of semimartingales: a maxingale problem approach I. Limits as solutions to a maxingale problem. Stoch. Stoch. Rep. 61 (1997) 141–243. Google Scholar
Stocker, T., On the asymptotic bias of OLS in dynamic regression models with autocorrelated errors. Statist. Papers 48 (2007) 81–93. Google Scholar
Worms, J., Moderate deviations for stable Markov chains and regression models. Electron. J. Probab. 4 (1999) 1–28. Google Scholar
Worms, J., Moderate deviations of some dependent variables I. Martingales. Math. Methods Statist. 10 (2001) 38–72. Google Scholar
Worms, J., Moderate deviations of some dependent variables II. Some kernel estimators. Math. Methods Statist. 10 (2001) 161–193. Google Scholar