Inference in High-dimensional Linear Regression

TL;DR

This paper proposes a novel inference method for high-dimensional linear regression that avoids penalization, retains interpretability, and uses sparsity and marginal analysis to handle more variables than observations.

Contribution

It introduces an optimal interest-respecting transformation approach that exploits sparsity in the Fisher information matrix, enabling inference without regularization.

Findings

Avoids penalization like lasso, preserving coefficient interpretability.

Provides an analytic solution for the transformation, simplifying analysis.

Facilitates inference in high-dimensional settings with more variables than samples.

Abstract

This paper develops an approach to inference in a linear regression model when the number of potential explanatory variables is larger than the sample size. The approach treats each regression coefficient in turn as the interest parameter, the remaining coefficients being nuisance parameters, and seeks an optimal interest-respecting transformation, inducing sparsity on the relevant blocks of the notional Fisher information matrix. The induced sparsity is exploited through a marginal least squares analysis for each variable, as in a factorial experiment, thereby avoiding penalization. One parameterization of the problem is found to be particularly convenient, both computationally and mathematically. In particular, it permits an analytic solution to the optimal transformation problem, facilitating theoretical analysis and comparison to other work. In contrast to regularized regression such as the lasso and its extensions, neither adjustment for selection nor rescaling of the explanatory variables is needed, ensuring the physical interpretation of regression coefficients is retained. Recommended usage is within a broader set of inferential statements, so as to reflect uncertainty over the model as well as over the parameters. The considerations involved in extending the work to other regression models are briefly discussed.