Abstract: Multivariate linear regressions are widely used to model the associations between multiple related responses and a set of predictors. To infer such associations, researchers often test the structure of the regression coefficients matrix, usually using a likelihood ratio test (LRT). Despite their popularity, classical χ² approximations for LRTs are known to fail in high-dimensional settings, where the dimensions of the responses and the predictors (m, p) are allowed to grow with the sample size n. Although various corrected LRTs and other test statistics have been proposed, few studies have examined the important question of when the classic LRT starts to fail. An answer to this would provide insights for practitioners, especially when analyzing data in which m/n and p/n are small, but not negligible. Moreover, the power of the LRT in high-dimensional data analyses remains under-researched. To address these issues, the first part of this work determines the asymptotic boundary at which the classical LRT fails, and develops a corrected limiting distribution for the LRT with a general asymptotic regime. The second part of this work examines the power of the LRT in high-dimensional settings. In addition to advancing the current understanding of the asymptotic behavior of the LRT under an alternative hypothesis, these results motivate the development of a more powerful LRT. The third part of this work considers the setting in which p > n, where the LRT is not well defined. We propose a two-step testing procedure. First, we perform a dimension reduction, and then we apply the proposed LRT. Theoretical properties are developed to ensure the validity of the proposed method, and simulations demonstrate that the method performs well.

Key words and phrases: High dimension, likelihood ratio test, multivariate linear regression.