Sparse Multivariate Linear Regression with Strongly Associated Response Variables

TL;DR

This paper introduces new sparse multivariate linear regression methods tailored for cases with dense error covariance matrices, especially under equicorrelation assumptions, and demonstrates their effectiveness through extensive experiments.

Contribution

The paper presents novel procedures for sparse multivariate regression with equicorrelated error structures, including high-dimensional approximations and a new tuning parameter selection method.

Findings

Procedures outperform relevant competitors in various scenarios.

Methods are robust to moderate model misspecification.

Effective in high-dimensional settings with dense error covariance.

Abstract

We propose new methods for multivariate linear regression when the regression coefficient matrix is sparse and the error covariance matrix is dense. We assume that the error covariance matrix has equicorrelation across the response variables. Two procedures are proposed: one is based on constant marginal response variance (compound symmetry), and the other is based on general varying marginal response variance. Two approximate procedures are also developed for high dimensions. We propose an approximation to the Gaussian validation likelihood for tuning parameter selection. Extensive numerical experiments illustrate when our procedures outperform relevant competitors as well as their robustness to a moderate degree of model misspecification.