A new statistical procedure for testing the presence of a significative correlation in the residuals of stable autoregressive processes

Frédéric Proia (1, 2)

1 IMB - Institut de Mathématiques de Bordeaux
2 ALEA - Advanced Learning Evolutionary Algorithms

The purpose of this paper is to investigate the asymptotic behavior of the Durbin-Watson statistic for the general stable $p-$order autoregressive process when the driven noise is 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 vector 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. Then, we prove the almost sure convergence and the asymptotic normality for the Durbin-Watson statistic. Finally, we propose a new bilateral statistical procedure for testing the presence of a significative first-order residual autocorrelation and we also explain how our procedure performs better than the commonly used Box-Pierce and Ljung-Box statistical tests for white noise applied to the stable autoregressive process, even on small-sized samples.

Mots clés: Durbin-Watson statistic; Stable autoregressive process; Residual autocorrelation; Statistical test for serial correlation

Domaines: Statistiques [math.ST]; Théorie [stat.TH]

Frédéric Proia. A new statistical procedure for testing the presence of a significative correlation in the residuals of stable autoregressive processes. 2012. ⟨hal-00677600⟩