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

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 (Central limit theorems for time series regression. Probab Theory Relat Fields. 1973;26(2):157–170), 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.