# Compare influence of same set of independent variables on two different dependent variables

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 $Y_3$.

• Perhaps I'm missing the point of the question, but just curious--why not just run the regression on Y3 as well and see what those coefficients are in their own right? You'll quickly discover what magnitude and significance they do and don't hold. And selecting only variables that have a positive influence and significance for a new regression may be problematic--you might be missing important controls, have mis-specified variable forms, or have other problematic factors (although you can look at your R-squared or do a Ramsey RESET test to help with that). Sep 5 '14 at 15:03
• First of all, thank you very much everybody for your quick answers! As it's "only" my bachelor thesis and my real estate professor's work regularly is more qualitative than quantitative (no econometrics), I want to avoid to complicate it for him by logistic transformation. I already did the regression on Y_3 with all the predictors, but I wondered if you could "see" the probably significant predictors before running the regression. As this seems definitely too much for the purpose of my analysis, I will do what you said: Y_3~X_1+... and check for the magnitude and the significances. Thank you!
– Paul
Sep 5 '14 at 17:40

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:

$$y_3 = \frac{b_{1,0}}{b_{2,0}} + \frac{1}{b_{2,0}} (\sum_j b_{1,j} x_{j,i}) - \frac{b_{1,0}}{(b_{2,0})^2} (\sum_j b_{2,j} x_{j,i}) \\ = \frac{b_{1,0}}{b_{2,0}} + \sum_j [\frac{1}{b_{2,0}} b_{1,j} - \frac{b_{1,0}}{(b_{2,0})^2} b_{2,j} ] x_{j,i}$$

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 $b_{2,0}=0$
If you're willing to fit the models: $$\log Y_1= a_1X_1+a_2X_2+\ldots\\ \log Y_2= b_1X_1+b_2X_2+\ldots$$ then, noting that $\log c/d = \log c - \log d$, you can interpret the differences of coefficients, $a_j - b_j$, directly for their impact on $\log Y_1/Y_2$. Moreover, if you fit $(\log Y_, \log Y_2)$ as a multivariate response, you'll be able to estimate the standard errors of $a_j-b_j$ and hence test their contributions.