On the asymptotic behavior of the Durbin-Watson statistic for ARX processes in adaptive tracking

TL;DR

This paper investigates the asymptotic properties of the Durbin-Watson statistic in ARX models with adaptive control, providing theoretical results and a new statistical test for residual autocorrelation.

Contribution

It extends the understanding of the Durbin-Watson statistic to adaptive control settings in ARX models, including convergence, normality, and a new autocorrelation test.

Findings

Almost sure convergence of estimators and Durbin-Watson statistic

Asymptotic normality of estimators and Durbin-Watson statistic

Proposed bilateral test for residual autocorrelation

Abstract

A wide literature is available on the asymptotic behavior of the Durbin-Watson statistic for autoregressive models. However, it is impossible to find results on the Durbin-Watson statistic for autoregressive models with adaptive control. Our purpose is to fill the gap by establishing the asymptotic behavior of the Durbin Watson statistic for ARX models in adaptive tracking. On the one hand, we show the almost sure convergence as well as the asymptotic normality of the least squares estimators of the unknown parameters of the ARX models. On the other hand, we establish the almost sure convergence of the Durbin-Watson statistic and its asymptotic normality. Finally, we propose a bilateral statistical test for residual autocorrelation in adaptive tracking.