  We propose a new class of estimators of the multivariate response linear
regression coefficient matrix that exploits the assumption that the response
and predictors have a joint multivariate Normal distribution. This allows us to
indirectly estimate the regression coefficient matrix through shrinkage
estimation of the parameters of the inverse regression, or the conditional
distribution of the predictors given the responses. We establish a convergence
rate bound for estimators in our class and we study two examples. The first
example estimator exploits an assumption that the inverse regression's
coefficient matrix is sparse. The second example estimator exploits an
assumption that the inverse regression's coefficient matrix is rank deficient.
These estimators do not require the popular assumption that the forward
regression coefficient matrix is sparse or has small Frobenius norm. Using
simulation studies, we show that our example estimators outperform relevant
competitors for some data generating models.
