Comparing Regression Coefficients from a "log-log" to an Alternative De-meaning Procedure

Question

Consider two regression models:

$log(y_i) = \log(x_i)\alpha + \epsilon_i \,\,\,\,\,$ (Model 1),

$log(y_i) = (\frac{x_i}{\overline{x}})\beta + \varepsilon_i \,\,\,\,\,\,\,\,$ (Model 2),

where $\overline{x}$ is the sample average of $x_i$.

Both of these models transform the variable $x_i$, the first with a log, the second by dividing by the sample average.

In short, why aren't $\alpha$ and $\beta$ equal?

I am confused because, it is my understanding that:

$\alpha = \frac{\partial \log(y)}{\partial \log(x)} = \frac{\partial \log(y)}{\partial y}\frac{\partial y}{\partial x} \frac{\partial x}{\partial \log(x)} = \frac{\partial y}{\partial x}\frac{x}{y}$

and

$\beta = \frac{\partial \log(y)}{\partial (\frac{x_i}{\overline{x}})} = \frac{\partial \log(y)}{\partial y}\frac{\partial y}{\partial x} \frac{\partial x}{\partial (\frac{x_i}{\overline{x}})} = \frac{\partial y}{\partial x}\frac{x}{y}$

However, in simulations, these two regression coefficients do not exactly equal each other. Is there an approximation going on somewhere in my definitions that I am ignoring? Is there some kind of small-sample bias that is relevant in practice that is missed here?

Answer 1

You made a small mistake when taking the derivative of Model 2. $$log(y) = \Bigl(\frac{x_i}{\bar{x}}\Bigr)\beta + e_i$$ If you are differentiating with respect to $x_i$ then $\bar{x}$ and $\beta$ are treated as constants exactly the same way if you were differentiating $x*5*2$, the derivative of which would be just 10. Therefore:

$$\frac{1}{y}\frac{dy}{dx_i}= \frac{\beta}{\bar{x}}$$ $$\beta =\frac{dy}{dx_i}\frac{\bar{x}}{y} $$

And hence the two are not equal. On an intuitive level you can think, that since the original two models apply different transformations to the $x$ variable it is logical that the OLS parameters $\alpha$ in Model 1 and $\beta$ in Model 2 will give different results.

On another important note, if you are trying to estimate the regression coefficient, you are differentiating the wrong functions with respect to the wrong variables. The OLS regression estimates regression coefficients by minimizing the sum of squared residual, and hence the funciton that you need to take the derivative of is: $$\sum_{i=1}^{n}{\hat{u}_i}^2 = \sum_{i=1}^{n}{(y_i - \hat{y}_i)^2} =\sum_{i=1}^{n}{(y_i-\hat{\beta}x_i)^2} $$ You can see the full derivation of the estimation here: https://are.berkeley.edu/courses/EEP118/current/derive_ols.pdf

For a multivariate case where you have many different variables such that $y = \beta_1 x_1 + \beta_2 x_2 + \beta_3 x_3 ... +e$ see this:
https://pdfs.semanticscholar.org/7aa9/77cea941df656739bf428a73cb62da6c0e74.pdf