High Dimensional Generalised Penalised Least Squares

TL;DR

This paper develops a generalized penalized least squares approach for high-dimensional linear models with serial correlation, improving inference accuracy and efficiency over traditional methods, especially under dependent and heavy-tailed data.

Contribution

It extends the asymptotic theory of Lasso to dependent, heavy-tailed high-dimensional data and introduces a debiased Lasso for uniform inference.

Findings

Sharper non-asymptotic bounds for dependent processes

Slower convergence rates under general conditions

Significant efficiency gains demonstrated in Monte Carlo simulations

Abstract

In this paper we develop inference for high dimensional linear models, with serially correlated errors. We examine Lasso under the assumption of strong mixing in the covariates and error process, allowing for fatter tails in their distribution. While the Lasso estimator performs poorly under such circumstances, we estimate via GLS Lasso the parameters of interest and extend the asymptotic properties of the Lasso under more general conditions. Our theoretical results indicate that the non-asymptotic bounds for stationary dependent processes are sharper, while the rate of Lasso under general conditions appears slower as $T,p\to \infty$. Further we employ the debiased Lasso to perform inference uniformly on the parameters of interest. Monte Carlo results support the proposed estimator, as it has significant efficiency gains over traditional methods.