  In this paper, we develop a systematic theory for high dimensional analysis
of variance in multivariate linear regression, where the dimension and the
number of coefficients can both grow with the sample size. We propose a new
\emph{U}~type test statistic to test linear hypotheses and establish a high
dimensional Gaussian approximation result under fairly mild moment assumptions.
Our general framework and theory can be applied to deal with the classical
one-way multivariate ANOVA and the nonparametric one-way MANOVA in high
dimensions. To implement the test procedure in practice, we introduce a
sample-splitting based estimator of the second moment of the error covariance
and discuss its properties. A simulation study shows that our proposed test
outperforms some existing tests in various settings.
