A sharp analysis on the asymptotic behavior of the Durbin-Watson statistic for the first-order autoregressive process

TL;DR

This paper provides a detailed asymptotic analysis of the Durbin-Watson statistic in first-order autoregressive models with autoregressive noise, establishing convergence, normality, and proposing a new test for residual autocorrelation.

Contribution

It offers the first comprehensive asymptotic analysis of the Durbin-Watson statistic in this specific autoregressive setting, including convergence rates and a novel bilateral test.

Findings

Almost sure convergence of estimators

Asymptotic normality of the Durbin-Watson statistic

New bilateral test for residual autocorrelation

Abstract

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 to 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.