Moderate deviations for the Durbin-Watson statistic related to the first-order autoregressive process
ESAIM: Probability and Statistics, Tome 18 (2014) , pp. 308-331.

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.

DOI : https://doi.org/10.1051/ps/2013038
Classification : 60F10,  60G42,  62M10,  62G05
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://www.numdam.org/articles/10.1051/ps/2013038/}
}
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://www.numdam.org/articles/10.1051/ps/2013038/

[1] M.A. Arcones, The large deviation principle for stochastic processes I. Theory Probab. Appl. 47 (2003) 567-583. | MR 2001788 | Zbl 1069.60026

[2] M.A. Arcones, The large deviation principle for stochastic processes II. Theory Probab. Appl. 48 (2003) 19-44. | MR 2013408 | Zbl 1069.60027

[3] B. Bercu and F. Proïa, 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 3070889

[4] B. Bercu and A. Touati, Exponential inequalities for self-normalized martingales with applications. Ann. Appl. Probab. 18 (2008) 1848-1869. | MR 2462551 | Zbl 1152.60309

[5] X. Chen, Moderate deviations for m-dependent random variables with Banach space value. Stat. Probab. Lett. 35 (1998) 123-134. | MR 1483265 | Zbl 0887.60010

[6] A. Dembo, Moderate deviations for martingales with bounded jumps. Electron. Commun. Probab. 1 (1996) 11-17. | MR 1386290 | Zbl 0854.60027

[7] A. Dembo and O. Zeitouni, Large deviations techniques and applications, 2nd edition, vol. 38 of Appl. Math. Springer (1998). | MR 1619036 | Zbl 0896.60013

[8] H. Djellout, Moderate deviations for martingale differences and applications to φ-mixing sequences. Stoch. Stoch. Rep. 73 (2002) 37-63. | MR 1914978 | Zbl 1005.60044

[9] H. Djellout and A. Guillin, Moderate deviations for Markov chains with atom. Stochastic Process. Appl. 95 (2001) 203-217. | MR 1854025 | Zbl 1059.60029

[10] J. Durbin, Testing for serial correlation in least-squares regression when some of the regressors are lagged dependent variables. Econometrica 38 (1970) 410-421. | MR 269030 | Zbl 0042.38201

[11] J. Durbin and G.S. Watson, Testing for serial correlation in least squares regression I. Biometrika 37 (1950) 409-428. | MR 39210 | Zbl 0039.35803

[12] J. Durbin and G.S. Watson, Testing for serial correlation in least squares regression II. Biometrika 38 (1951) 159-178. | MR 42662 | Zbl 0042.38201

[13] J. Durbin and G.S. Watson, Testing for serial correlation in least squares regession III. Biometrika 58 (1971) 1-19. | MR 281300 | Zbl 0225.62112

[14] P. Eichelsbacher and M. Löwe, Moderate deviations for i.i.d. random variables. ESAIM: PS 7 (2003) 209-218. | Numdam | MR 1956079 | Zbl 1019.60021

[15] B.A. Inder, An approximation to the null distribution of the Durbin-Watson statistic in models containing lagged dependent variables. Econometric Theory 2 (1986) 413-428.

[16] M.L. King and P.X. Wu, Small-disturbance asymptotics and the Durbin-Watson and related tests in the dynamic regression model. J. Econometrics 47 (1991) 145-152. | MR 1087210

[17] M. Ledoux, Sur les déviations modérées des sommes de variables aléatoires vectorielles indépendantes de même loi. Ann. Inst. Henri-Poincaré 35 (1992) 123-134. | Numdam | Zbl 0751.60009

[18] E. Malinvaud, Estimation et prévision dans les modèles économiques autorégressifs. Rev. Int. Inst. Statis. 29 (1961) 1-32. | Zbl 0269.90007

[19] M. Nerlove and K.F. Wallis, Use of the Durbin-Watson statistic in inappropriate situations. Econometrica 34 (1966) 235-238. | MR 208790

[20] F. Proïa, Further results on the H-Test of Durbin for stable autoregressive processes. J. Multivariate. Anal. 118 (2013) 77-101. | MR 3054092

[21] A. Puhalskii, Large deviations of semimartingales: a maxingale problem approach I. Limits as solutions to a maxingale problem. Stoch. Stoch. Rep. 61 (1997) 141-243. | MR 1488137 | Zbl 0890.60025

[22] T. Stocker, On the asymptotic bias of OLS in dynamic regression models with autocorrelated errors. Statist. Papers 48 (2007) 81-93. | MR 2288173 | Zbl 1132.62348

[23] J. Worms, Moderate deviations for stable Markov chains and regression models. Electron. J. Probab. 4 (1999) 1-28. | MR 1684149 | Zbl 0980.62082

[24] J. Worms, Moderate deviations of some dependent variables I. Martingales. Math. Methods Statist. 10 (2001) 38-72. | MR 1841808 | Zbl 1007.60010

[25] J. Worms, Moderate deviations of some dependent variables II. Some kernel estimators. Math. Methods Statist. 10 (2001) 161-193. | MR 1851746 | Zbl 1007.60011