*If I have two very similar regressions where the only difference is the dependent variables, can you compare the influence of the independent variables between the two regressions. *

For example, if regression 1 : Y_1 = aX_1 + bX_2

and if regression 2 is : Y_2 = cX_1 + dX_2.

if c is greater that a, can you say that the influence of X_1 on Y_2 is greater that its influence on Y_1??

I have actually been thinking about this sort of question lately. I'm not confident that I can give you a definitive answer (sorry--I know that's why people usually post here) but maybe a bit of food for thought.

First, it depends on what you mean by "influence is greater" for the independent variable(s) in question. Do you mean the strength of the Beta coefficients across the two regressions, or the contribution to model fit?

If you mean the Betas, I would be very cautious. The two dep variables will differ, have different variation etc., as will the effect of the other ind variable(s). Other than a very loose heuristic statement, I don't think you could really make a defensible claim about relative strength here. (Unless, of course, the ind var(s) in question are significant in one regression but not in the other. Then it's a dichotomy--"var x has an effect on one dep var, but not the other.")

I think you might be on firmer ground with model fit. If you ran your two regressions with hierarchical entry of ind vars (like forward inclusion or stepwise) your program might allow you to see the relative R-squared contribution of the ind var(s) in question in both of your regression models. Since R-squared has a standard metric across the two regressions, this might be useful to know.

This makes sense to me, but again, I'm not suggesting this is a definitive answer. Hopefully someone more knowledgeable will weigh in and inform both of us.