Regression Interpretation conundrum I am running an OLS regression of the form
$$\log\left(Y\right)=x_0 + \left(\frac{x_1}{Y}\right)\beta_1+\log (x_2)\beta_2 + \epsilon$$
I have one covariate as $\left(\frac{x_1}{Y}\right)$ which is a fraction $\in [0,1]$ and it involves the dependent variable in the denominator. 
Let's say $\beta_1$ is 1.31. I was wondering how would I interpret this coefficient in terms of unit change in the independent variable causing a change in the dependent variable, as the dependent variable also appears in the denominator of one of the covariates? Any help is greatly appreciated.
 A: Ignoring any issues of model construction and fitting, let's answer the question: how does the predicted value of $Y$ change when $x_1$ is changed by a small amount?  To find this, treat only $x_1$ and $Y$ as variables, ignore the random components in the model, and differentiate:
$$\eqalign{\frac{dY}{Y}&=d\left(\log(Y)\right) \\
&= d\left(x_0 + \left(\frac{x_1}{Y}\right)\beta_1 + \log(x_2)\beta_2\right) \\
&= \beta_1 d\left(\frac{x_1}{Y}\right) \\
&=\beta_1 \left(\frac{dx_1}{Y} - \frac{x_1\,dY}{Y^2}\right).
}$$
The rules of arithmetic permit us to solve for $dY$ (knowing that $Y$ is necessarily positive, because the model involves its logarithm):
$$dY = \frac{\beta_1 Y}{Y + \beta_1 x_1} dx_1.$$
For instance, if $Y=100$ is annual GDP of a country in billions of dollars and $x_1=10$ is personal income in billions of dollars, then a change of $dx_1 = -1$ billion dollars corresponds to a change in GDP of
$$dY = \frac{\beta_1(100)}{100 + \beta_1(10)}(-1) = -\frac{\beta_1}{1 + \beta_1/10}$$
billions of dollars.
A: To use $Y$ in feature construction is a delicate thing as it almost always introduces endogeneity issues. Maybe you remember that in "classic" inference on the linear model, all regressors are considered to be fixed design points. This assumption will be invalid in your case quite certainly.
The interpretation of a beta of 1.31 is
A one percentage point (= 0.01) increase in $x_1/Y$ is associated with an increase of about 1.31% in the typical value of $Y$. By "typical" we mean with respect to the geomean (or, if the residuals are distributed quite symmetrically, the median).
