We develop a new method to fit the multivariate response linear regression model that exploits a parametric link between the regression coefficient matrix and the error covariance matrix. Specifically, we assume that the correlations between entries in the multivariate error random vector are proportional to the cosines of the angles between their corresponding regression coefficient matrix columns, so as the angle between two regression coefficient matrix columns decreases, the correlation between the corresponding errors increases. We highlight two models under which this parameterization arises: a latent variable reduced-rank regression model and the errors-in-variables regression model. We propose a novel nonconvex weighted residual sum of squares criterion which exploits this parameterization and admits a new class of penalized estimators. The optimization is solved with an accelerated proximal gradient descent algorithm. Our method is used to study the association between microRNA expression and cancer drug activity measured on the NCI-60 cell lines. An R package implementing our method, MCMVR, is available online.