  Multivariate linear regressions are widely used statistical tools in many
applications to model the associations between multiple related responses and a
set of predictors. To infer such associations, it is often of interest to test
the structure of the regression coefficients matrix, and the likelihood ratio
test (LRT) is one of the most popular approaches in practice. Despite its
popularity, it is known that the classical $\chi^2$ approximations for LRTs
often fail in high-dimensional settings, where the dimensions of responses and
predictors $(m,p)$ are allowed to grow with the sample size $n$. Though various
corrected LRTs and other test statistics have been proposed in the literature,
the fundamental question of when the classic LRT starts to fail is less
studied, an answer to which would provide insights for practitioners,
especially when analyzing data with $m/n$ and $p/n$ small but not negligible.
Moreover, the power performance of the LRT in high-dimensional data analysis
remains underexplored. To address these issues, the first part of this work
gives the asymptotic boundary where the classical LRT fails and develops the
corrected limiting distribution of the LRT for a general asymptotic regime. The
second part of this work further studies the test power of the LRT in the
high-dimensional setting. The result not only advances the current
understanding of asymptotic behavior of the LRT under alternative hypothesis,
but also motivates the development of a power-enhanced LRT. The third part of
this work considers the setting with $p>n$, where the LRT is not well-defined.
We propose a two-step testing procedure by first performing dimension reduction
and then applying the proposed LRT. Theoretical properties are developed to
ensure the validity of the proposed method. Numerical studies are also
presented to demonstrate its good performance.
