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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.

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    $\begingroup$ It's incorrect to use OLS to fit this model, because the $1/Y$ term on the right implicitly includes a (potentially enormous) random component that is completely dependent on $\epsilon$. $\endgroup$
    – whuber ♦
    May 20, 2016 at 13:01

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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.

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  • $\begingroup$ Thanks @whuber! This is helpful. I construct the covariate $\frac{x_1}{Y}$ before running the regression, so R/STATA won't even know that the covariate has $Y$ in it. The correlation between $\frac{x_1}{Y}$ and $\log(Y)$ is around 0.4 (so no collinearity issues). I was only unclear about the interpretation of the coefficient. What do you think in general about this kind of model construction? $\endgroup$
    – Blade Runner
    May 20, 2016 at 13:33
  • $\begingroup$ If some underlying theory suggests a model of this form, so be it. Otherwise, because it is challenging to interpret ($Y$ explicitly "depends" on $Y$) and can be very difficult to fit (you might try Maximum Likelihood), I would be inclined to look for alternative formulations at the outset. $\endgroup$
    – whuber ♦
    May 20, 2016 at 15:32
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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).

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