Dear Kofi,

It depends on the nature of the qualitative variable: If it can be ordered according to a criterion, which you think is relevant for its influence on y, then it may be possible to run regressions. But one should be aware, that the results cannot be very precise, but can always only be an indication of a direction. Example: If you have an ordering very good...5, good...4, average...3, bad...2, very bad...1, that does not mean that the differences between the states are the same between all 5 states. I think one should run regressions on y, with 5 dummies for the qualitiies, first (if possible) and one could do estimations for each of the 5 dummies (B1-B5). That could even be a way of tackling the problem, if one could not find a reasonable ordering of B.

One additional problem I see with your approach, that you have X in both the equations for y and B, which would cause identication problems in the y-equation.

Best wishes