The purpose of this paper is to investigate moderate deviations for the Durbin-Watson statistic associated with the stable first-order autoregressive process where the driven noise is also given by a first-order autoregressive process. We first establish a moderate deviation principle 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. It enables us to provide a moderate deviation principle for the Durbin-Watson statistic in the case where the driven noise is normally distributed and in the more general case where the driven noise satisfies a less restrictive Chen-Ledoux type condition.
Mots-clés : Durbin-Watson statistic, moderate deviation principle, first-order autoregressive process, serial correlation
@article{PS_2014__18__308_0,
author = {Bitseki Penda, S. Val\`ere and Djellout, Hac\`ene and Pro{\"\i}a, Fr\'ed\'eric},
title = {Moderate deviations for the {Durbin-Watson} statistic related to the first-order autoregressive process},
journal = {ESAIM: Probability and Statistics},
pages = {308--331},
publisher = {EDP-Sciences},
volume = {18},
year = {2014},
doi = {10.1051/ps/2013038},
language = {en},
url = {http://archive.numdam.org/articles/10.1051/ps/2013038/}
}
TY - JOUR AU - Bitseki Penda, S. Valère AU - Djellout, Hacène AU - Proïa, Frédéric TI - Moderate deviations for the Durbin-Watson statistic related to the first-order autoregressive process JO - ESAIM: Probability and Statistics PY - 2014 SP - 308 EP - 331 VL - 18 PB - EDP-Sciences UR - http://archive.numdam.org/articles/10.1051/ps/2013038/ DO - 10.1051/ps/2013038 LA - en ID - PS_2014__18__308_0 ER -
%0 Journal Article %A Bitseki Penda, S. Valère %A Djellout, Hacène %A Proïa, Frédéric %T Moderate deviations for the Durbin-Watson statistic related to the first-order autoregressive process %J ESAIM: Probability and Statistics %D 2014 %P 308-331 %V 18 %I EDP-Sciences %U http://archive.numdam.org/articles/10.1051/ps/2013038/ %R 10.1051/ps/2013038 %G en %F PS_2014__18__308_0
Bitseki Penda, S. Valère; Djellout, Hacène; Proïa, Frédéric. Moderate deviations for the Durbin-Watson statistic related to the first-order autoregressive process. ESAIM: Probability and Statistics, Tome 18 (2014), pp. 308-331. doi : 10.1051/ps/2013038. http://archive.numdam.org/articles/10.1051/ps/2013038/
[1] , The large deviation principle for stochastic processes I. Theory Probab. Appl. 47 (2003) 567-583. | MR | Zbl
[2] , The large deviation principle for stochastic processes II. Theory Probab. Appl. 48 (2003) 19-44. | MR | Zbl
[3] and , A sharp analysis on the asymptotic behavior of the Durbin-Watson statistic for the first-order autoregressive process. ESAIM: PS 17 (2013) 500-530. | Numdam | MR
[4] and , Exponential inequalities for self-normalized martingales with applications. Ann. Appl. Probab. 18 (2008) 1848-1869. | MR | Zbl
[5] , Moderate deviations for m-dependent random variables with Banach space value. Stat. Probab. Lett. 35 (1998) 123-134. | MR | Zbl
[6] , Moderate deviations for martingales with bounded jumps. Electron. Commun. Probab. 1 (1996) 11-17. | MR | Zbl
[7] and , Large deviations techniques and applications, 2nd edition, vol. 38 of Appl. Math. Springer (1998). | MR | Zbl
[8] , Moderate deviations for martingale differences and applications to φ-mixing sequences. Stoch. Stoch. Rep. 73 (2002) 37-63. | MR | Zbl
[9] and , Moderate deviations for Markov chains with atom. Stochastic Process. Appl. 95 (2001) 203-217. | MR | Zbl
[10] , Testing for serial correlation in least-squares regression when some of the regressors are lagged dependent variables. Econometrica 38 (1970) 410-421. | MR | Zbl
[11] and , Testing for serial correlation in least squares regression I. Biometrika 37 (1950) 409-428. | MR | Zbl
[12] and , Testing for serial correlation in least squares regression II. Biometrika 38 (1951) 159-178. | MR | Zbl
[13] and , Testing for serial correlation in least squares regession III. Biometrika 58 (1971) 1-19. | MR | Zbl
[14] and , Moderate deviations for i.i.d. random variables. ESAIM: PS 7 (2003) 209-218. | Numdam | MR | Zbl
[15] , An approximation to the null distribution of the Durbin-Watson statistic in models containing lagged dependent variables. Econometric Theory 2 (1986) 413-428.
[16] and , Small-disturbance asymptotics and the Durbin-Watson and related tests in the dynamic regression model. J. Econometrics 47 (1991) 145-152. | MR
[17] , Sur les déviations modérées des sommes de variables aléatoires vectorielles indépendantes de même loi. Ann. Inst. Henri-Poincaré 35 (1992) 123-134. | Numdam | Zbl
[18] , Estimation et prévision dans les modèles économiques autorégressifs. Rev. Int. Inst. Statis. 29 (1961) 1-32. | Zbl
[19] and , Use of the Durbin-Watson statistic in inappropriate situations. Econometrica 34 (1966) 235-238. | MR
[20] , Further results on the H-Test of Durbin for stable autoregressive processes. J. Multivariate. Anal. 118 (2013) 77-101. | MR
[21] , Large deviations of semimartingales: a maxingale problem approach I. Limits as solutions to a maxingale problem. Stoch. Stoch. Rep. 61 (1997) 141-243. | MR | Zbl
[22] , On the asymptotic bias of OLS in dynamic regression models with autocorrelated errors. Statist. Papers 48 (2007) 81-93. | MR | Zbl
[23] , Moderate deviations for stable Markov chains and regression models. Electron. J. Probab. 4 (1999) 1-28. | MR | Zbl
[24] , Moderate deviations of some dependent variables I. Martingales. Math. Methods Statist. 10 (2001) 38-72. | MR | Zbl
[25] , Moderate deviations of some dependent variables II. Some kernel estimators. Math. Methods Statist. 10 (2001) 161-193. | MR | Zbl
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