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

TL;DR

This paper investigates the asymptotic behavior of least squares estimators in linear models with dependent, stationary errors, demonstrating that under certain designs, their distribution resembles the i.i.d. case, and proposes a spectral density-based covariance estimator.

Contribution

It extends the understanding of least squares estimators' asymptotic distribution to dependent error processes and provides a consistent spectral density estimator under mild conditions.

Findings

Asymptotic covariance matrix simplifies to the i.i.d. case for many designs.

Spectral density estimator is consistent under mild conditions.

Central Limit Theorem applies to dependent stationary errors.

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, who proved a Central Limit Theorem for the usual least squares 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 independent and identically distributed case. We then estimate the covariance matrix using an estimator of the spectral density whose consistency is proved under very mild conditions.