I have a simple question but I do not manage to be sure! I would be very grateful if you can confirm me! Do we have the possibility to estimate the following model : $$\frac{y}{x}= \alpha+\beta x+\varepsilon$$ Is there any problem because the endogeneous variable contains already the explanatory variable $x$? Thanks a lot!
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Algebraically, there is no problem. Just define $z=(z_1,\ldots,z_n)$ with $z_t=y_t/x_t$. Regressing $z$ on an interecept and $x$ can be done by usual formulas. – yurnero Jun 01 '16 at 21:46
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As yumero commented, there is no problem from an algebraic point of view.
However, from a regression point of view, this is problematic since what is measured is $y$ and not $\frac y x$. I would definitely suggest that you set the regression as $$y= \alpha x+\beta x^2+\varepsilon$$ which is just a multilinear regression without intercept.
For sure, if the errors are marginal, this will not make much difference.
Claude Leibovici
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