Further results on theh- test of Durbin for stable autoregressive processes

Abstract

The purpose of this paper is to investigate the asymptotic behavior of the Durbin-Watson statistic for the stable p−order autoregressive process when the driven noise is given by a first-order autoregressive process. It is an extension of the previous work of Bercu and Proïa devoted to the particular case p = 1. 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 and we derive a two-sided statistical procedure for testing the presence of a significant first-order residual autocorrelation that appears to clarify and to improve the well-known h-test suggested by Durbin. Finally, we briefly summarize our observations on simulated samples.

Testing for residual correlation of any order in the autoregressive process

“…p (a result that may be found in [2]- [39]). In the last section, we present our statistical testing procedure.…”

“…Hence, Theorem 2.1 of [39] and Theorem 3.1 are clearly equivalent for q = 1. Similarly, one can see that Σ T in (3.11) becomes…”

“…One can cite for example the works of Maddala and Rao [29] in 1973, of Park [37] in 1975, of Inder [25]- [26] in the 1980s, of Durbin [14] in 1986, or of King and Wu [28] in 1991. Whereas the upper and lower bounds of the test were previously established by a Monte-Carlo study, the asymptotic normality of the Durbin-Watson statistic is suggested without proof in [13] under strong assumptions (such as gaussianity), and finally proved under less restrictive hypothesis in [2]- [39] for p ≥ 1 and q = 1, even under the alternative. We conclude the paper by giving a quick summary of some simulation results about the empirical power of our procedure, compared with the ones mentioned above.…”

A Durbin–Watson serial correlation test for ARX processes via excited adaptive tracking

“…For example, it is well known that for linear autoregressive models with autocorrelated residuals, the least squares estimator is asymptotically biased, see e.g. [5], [11], [14], [15], and therefore the estimated model is not the correct one. Consequently, to ensure a good interpretation of the results, it is necessary to have a powerful tool allowing to detect the possible autocorrelation of the residuals.…”

“…More precisely, we shall propose a bilateral statistical test allowing to decide between the null hypothesis H 0 : ρ = 0 which ensures that the driven noise is not correlated, and the alternative one H 1 : ρ = 0 which means that the residual process is effectively first-order autocorrelated. The choice of the Durbin-Watson statistic, instead of any other statistical tests, is governed by its efficienty for autoregressive processes without control, see [5], [11], [14], [15].…”