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

TL;DR

This paper analyzes the asymptotic behavior of least squares estimators in linear models with dependent, stationary errors, proposing a consistent covariance estimator and adjusted tests to maintain correct error rates.

Contribution

It extends classical linear regression theory to dependent errors by providing a consistent covariance estimator and modified testing procedures under mild conditions.

Findings

The covariance matrix estimator is consistent under mild conditions.

Modified tests maintain asymptotic correctness of type-I error.

Simulation results demonstrate the effectiveness of the proposed methods.

Abstract

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 square 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.