Likelihood-Based Inference with Separable Correlation Matrices

TL;DR

This paper introduces efficient likelihood-based inference methods for multivariate regressions with separable correlation matrices, significantly improving computational speed and accuracy over existing approaches.

Contribution

It develops a block-coordinate ascent algorithm with closed-form updates for separable correlation matrices, enabling practical inference and hypothesis testing.

Findings

Algorithm is 300-2500 times faster than general solvers.

Estimator has lower error than traditional methods when the model holds.

Bootstrap tests maintain nominal size where asymptotic tests fail.

Abstract

This paper proposes methods for likelihood-based inference in multivariate linear regressions when the correlation matrix of the responses is separable; that is, it has a Kronecker product structure, but the variances are unrestricted. The methods are enabled by a block-coordinate ascent-like algorithm with closed-form updates that strictly increases the likelihood at every iteration until convergence. In the numerical experiments, the proposed algorithm is 300--2500 times faster than a general-purpose solver, making parametric bootstrap tests of correlation and covariance separability practical. Parameters are identifiable, and standard errors can therefore be obtained from the expected Fisher information, which can be computed efficiently using the Kronecker product structure. Simulations show that the proposed estimator has lower error than both separable covariance and unrestricted estimators when the model holds, and that bootstrap tests maintain nominal size where asymptotic tests fail. An application to dissolved oxygen data from the Mississippi River demonstrates that separable correlation captures location-specific variance patterns that separable covariance cannot.