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 $Y_2$.
$Y_1= a_1X_1+a_2X_2+…$
$Y_2= b_1X_1+b_2X_2+…$
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 $Y_3$.
Best Answer
as ShannonC pointed out why not run regression y3=y1/y2 ~ x1,x2,...
however,if that is not possible(e.g. you don't have the original data) you can use taylor expansion to understand how y3 is influenced by x1,x2,...
let's re-state:
$$y_{k,i} = b_{k,0} + \sum_j b_{k,j} x_{j,i}$$ for k=1,2
let's write the taylor expansion around $(b_{1,0},b_{2,0})$ $$y3=f(y1,y2) = f(b_{1,0},b_{2,0}) + \frac{df(b_{1,0},b_{2,0})}{dy1} (y1-b_{1,0}) + \frac{df(b_{1,0},b_{2,0})}{dy2} (y2-b_{2,0}) $$
now, everything is expressed in terms of the $b_{k,j}$ coefficients. in your particular case:
$$ y_3 = \frac{b_{1,0}}{b_{2,0}} + \frac{1}{b_{2,0}} (\sum_j b_{1,j} x_{j,i}) - \frac{b_{1,0}}{(b_{2,0})^2} (\sum_j b_{2,j} x_{j,i}) \\ = \frac{b_{1,0}}{b_{2,0}} + \sum_j [\frac{1}{b_{2,0}} b_{1,j} - \frac{b_{1,0}}{(b_{2,0})^2} b_{2,j} ] x_{j,i} $$
i hope this makes sense and answers your question
few notes: 1)you can use any other type of regression not only linear and any other function of the $y$ variables 2) if 1st order approximation doesn't work use more derivatives 3) must be careful when $b_{2,0}=0$