The Multivariate General Linear Model

Ordinary linear regression predicts one outcome variable from a set of predictors. The multivariate general linear model (MGLM) extends this to predicting several outcome variables at once from the same predictors — say, regressing both a stock's return and its trading volume simultaneously on the same set of factor exposures — and, crucially, it models the outcomes jointly rather than as separate regressions run side by side, capturing the correlation between the outcomes' residuals rather than throwing that information away.

Fitting the model still uses the same least-squares logic as ordinary regression, just with a matrix of outcomes $Y$ instead of a single vector, producing a matrix of coefficients rather than a single coefficient vector. The real advantage shows up in hypothesis testing: because the residuals across outcomes are modeled jointly, you can test whether a predictor has any effect across the whole outcome vector at once (using multivariate test statistics like Wilks' lambda), which is both more statistically powerful and avoids the multiple-testing problem of running separate univariate tests on each outcome and hoping the significant ones aren't just false positives.

MANOVA — testing whether several groups differ on a set of outcome variables jointly — is a special case of the MGLM where the predictors are just group-membership indicators, which is why the two topics are almost always taught together.