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.

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.

DOI : https://doi.org/10.1051/ps/2012005
Classification : 60F05,  60G42,  62F05,  62G05,  62M10
Mots clés : Durbin-Watson statistic, autoregressive process, residual autocorrelation, statistical test for serial correlation
@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://www.numdam.org/articles/10.1051/ps/2012005/}
}
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. http://www.numdam.org/articles/10.1051/ps/2012005/

[1] B. Bercu, On the convergence of moments in the almost sure central limit theorem for martingales with statistical applications. Stoch. Process. Appl. 11 (2004) 157-173. | MR 2049573 | Zbl 1076.62066

[2] B. Bercu, P. Cenac and G. Fayolle, On the almost sure central limit theorem for vector martingales: convergence of moments and statistical applications. J. Appl. Probab. 46 (2009) 151-169. | MR 2508511 | Zbl 1175.60009

[3] V. Bitseki Penda, H. Djellout and F. Proïa, Moderate deviations for the Durbin-Watson statistic related to the first-order autoregressive process. Submitted for publication, arXiv:1201.3579 (2012).

[4] G. Box and G. Ljung, On a measure of a lack of fit in time series models. Biometrika 65 (1978) 297-303. | Zbl 0386.62079

[5] G. Box and D. Pierce, Distribution of residual autocorrelations in autoregressive-integrated moving average time series models. Amer. Statist. Assn. J. 65 (1970) 1509-1526. | MR 273762 | Zbl 0224.62041

[6] T. Breusch, Testing for autocorrelation in dynamic linear models. Austral. Econ. Papers. 17 (1978) 334-355.

[7] M. Duflo, Random iterative models, Appl. Math., vol. 34. Springer-Verlag, Berlin (1997). | MR 1485774 | Zbl 0868.62069

[8] 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

[9] J. Durbin, Approximate distributions of student's t-statistics for autoregressive coefficients calculated from regression residuals. J. Appl. Probab. 23A (1986) 173-185. | MR 803171 | Zbl 0581.62021

[10] 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

[11] 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

[12] 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

[13] L. Godfrey, Testing against general autoregressive and moving average error models when the regressors include lagged dependent variables. Econometrica 46 (1978) 1293-1302. | Zbl 0395.62062

[14] P. Hall and C.C. Heyde, Martingale limit theory and its application, Probability and Mathematical Statistics. Academic Press Inc., New York (1980). | MR 624435 | Zbl 0462.60045

[15] B.A. Inder, Finite-sample power of tests for autocorrelation in models containing lagged dependent variables. Econom. Lett. 14 (1984) 179-185. | Zbl 1273.62268

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

[17] 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

[18] G.S. Maddala and A.S. Rao, Tests for serial correlation in regression models with lagged dependent variables and serially correlated errors. Econometrica 41 (1973) 761-774. | Zbl 0332.62050

[19] E. Malinvaud, Estimation et prévision dans les modèles économiques autorégressifs. Review of the International Institute of Statistics 29 (1961) 1-32. | Zbl 0269.90007

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

[21] S.B. Park, On the small-sample power of Durbin's h-test. J. Amer. Stat. Assoc. 70 (1975) 60-63. | Zbl 0309.62016

[22] F. Proïa, A new statistical procedure for testing the presence of a significative correlation in the residuals of stable autoregressive processes. Submitted for publication, arXiv:1203.1871 (2012).

[23] 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

[24] W.F. Stout, A martingale analogue of Kolmogorov's law of the iterated logarithm. Z. Wahrscheinlichkeitstheorie und Verw. Gebiete 15 (1970) 279-290. | MR 293701 | Zbl 0209.49004

[25] W.F. Stout, Almost sure convergence, Probab. Math. Statist. Academic Press, New York, London 24 (1974). | MR 455094 | Zbl 0321.60022

[26] J.A. Tillman, The power of the Durbin-Watson test. Econometrica 43 (1975) 959-974. | MR 440768 | Zbl 0322.62027

[27] C. Wei and J. Winnicki, Estimation on the means in the branching process with immigration. Ann. Statist. 18 (1990) 1757-1773. | MR 1074433 | Zbl 0736.62071