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

Emmanuel Caron (1, 2) , Sophie Dede (3)

1 LMJL - Laboratoire de Mathématiques Jean Leray
2 ECN - École Centrale de Nantes
3 Lycée Stanislas, Paris

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 [13], who proved a Central Limit Theorem for the usual least square estimator under general conditions on the design and on the error process. We show that for a large class of designs, the asymptotic covariance matrix is as simple as the i.i.d. 1 case. We then estimate the covariance matrix using an estimator of the spectral density whose consistency is proved under very mild conditions. As an application, we show how to modify the usual Fisher tests 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.

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

Emmanuel Caron, Sophie Dede. Asymptotic distribution of least square estimators for linear models with dependent errors : regular designs. 2017. ⟨hal-01617479v1⟩