Asymptotic distribution of least square estimators for linear models with dependent errors

Emmanuel Caron (1, 2)

1 LMJL - Laboratoire de Mathématiques Jean Leray
2 ECN - École Centrale de Nantes

In this paper, we consider the usual linear regression model in the case where the error process is assumed strictly stationary. We use a result from Hannan (1973), who proved a Central Limit Theorem for the usual least squares estimator under general conditions on the design and on the error process. Whatever the design satisfying Hannan's conditions, we define an estimator of the covariance matrix and we prove its consistency under very mild conditions. As an application, we show how to modify the usual tests on the linear model in this dependent context, in such a way that the type-I error rate remains asymptotically correct, and we illustrate the performance of this procedure through different sets of simulations.

Mots clés: Stationary process; Linear regression model; Statistical Tests; Asymptotic normality; Spectral density estimates

Domaines: Théorie [stat.TH]; Applications [stat.AP]; Probabilités [math.PR]; Statistiques [math.ST]

Emmanuel Caron. Asymptotic distribution of least square estimators for linear models with dependent errors. Statistics, 2019, 53 (4), pp.885-902. ⟨10.1080/02331888.2019.1593987⟩. ⟨hal-01815253v2⟩