I'm currently doing two multiple linear regressions. Each of them with the same set of predictors (measurements for real estate quality) $X_1,...,X_n$, but with different dependent variables (one of them the purchasing price, the other one the yearly rent) $Y_1$ and $Y_2$.
What I am interested in is the influence of the independent variables on a third dependent variable $Y_3$ (a real estate investor's return assumption), which is approximately the quotient of the first two dependent variables $Y_1/Y_2$.
So what I want to do is to find out which of the independent variables could possibly have an influence on the third dependent variable. As predictors for the regression on $Y_3$ I only want to use the independent variables out of the original set, of which I think that they have influence on $Y_3$. To do so I want to compare the influence of the independent variables on $Y_1$ and $Y_2$. If the influence points in a different direction (e.g. coefficient $a_1$ is negative and $b_1$ is positive) it is obvious that this independent variable will probably have influence on $Y_3$. But what if the direction is the same? It could happen, that the independent variable $X_n$ has strong influence on both $Y_1$ and $Y_2$ but with the same magnitude, so that $Y_3$ is not determined by this independent variable.
So my question is, is there a way to find out (e.g. by comparing standardized regression coefficients?), how large the influence of one predictor is relatively on $Y_1$ and $Y_2$? So I can say for example, $X_n$ determines both $Y_1$ and $Y_2$ in a positive way, but $Y_1$ is determined stronger, so that the quotient $Y_1/Y_2$ and therefore probably $Y_3$ is influenced by $X_n$ in a positive way, that's why I use it as a predictor in the regression on $Y_3$."
I don't want to use the quotient of the predicted $Y_1$ and $Y_2$ as a estimator for $Y_3$, but do a fully new regression on $Y_3$.