multivariate regression problem - correlated residuals

this question asks for help for a statistical/econometric problem I can't get the clue:
I have a classic multivariate regression problem, i.e. dependent variables are stored in matrix Y having dimension n x p. So p observations come from the same respondent i and we need to expect correlated residuals. The explanatory variables are stored in X and as standard in the multivariate regression model, each of the p observations of respondent i has the same common X. Say Yi holds the income of a guy at 5 points in time, and it shall be explained by his gender and his education only, then Xi holds male and college as the explanatory variables for all of his Yi. As an output of the regression one gets vectors β1,...,β5, one vector β for each of the 5 points in time.

In my specific problem some of the Y have been constructed in such a way that the correlation of their errors is 1. So for these I would not need 5 vectors β1,...,β5, but because of the prefect correlation a reduced form would already contain all information, like β1=βα1, β2=βα2, ... Where each α is a skalar and not a vector.
I understand that the multivariate regression, i.e. estimating 5 vectors β1,...,β5, should give unbiased but inefficient results (true?).
Nevertheless, does anyone of you have an idea how to estimate the model from above without the redundant β-vectors?
Also running sureg does not work properly, so as an alternative I used mvreg in stata.
Is there a better solution?
Thanks for any hint!