To avoid confusion, my question refers to multivariate regression as multiple dependent variables for the same set of independent variables.
As far as I understand (see e.g., this question), the coefficients and the SE for the independent variables of a multivariate regression are the same as if you would run separate regressions. Why are the SE of the independent variables not affected, given that multivariate takes into account the correlation between the dependent variables?
I know that multivariate regression makes it possible to compare the coefficient for the same variable across different dependent variables (F-test). What else is different between multivariate regression and running separate regressions?