There are a few books out there on how to do multivariate regression, but of course the literature isn't as massive and detailed as univariate regression models.

So why bother with multivariate regression at all? Sure, what you gain from the ability to be able to model the strong correlation structure between the responses, you probably lose from your lack of ability to use the wide class of models and techniques available for univariate regressions, no?

So, my question is, if a multivariate response is highly correlated, is it really the end of the world if we just do separate but proper and highly informative univariate regressions?