How to interpret the standardized coefficients (BETA) in a mixed log-log and log-level regression

Question

Let's say we have a model like:

$$log(y) = \beta_{1} + \beta_{2} \cdot log(X_{2})+\beta_{3} \cdot X_{3} + u$$

After carrying an OLS, we are asked which independent variable has a higher impact on the dependent variable, for which we are interested in compute the BETA coefficients.

I know that normally (given a model like $y = \beta_{1} + \beta_{2} \cdot X_{2}+\beta_{3} \cdot X_{3} + u$) we would just compute:

$$BETA_{2} = \widehat{\beta}_{2} \cdot (S_{x_{2}}/S_{y})$$ $$BETA_{3} = \widehat{\beta}_{3} \cdot (S_{x_{3}}/S_{y})$$

interpreting those BETA as standardized coefficients; the higher they are, the higher the impact of the independent variable over the dependent variable.

But, dealing with log-log and log-level, I suspect that the comparison is not as straighforward, since, when interpreting semi-elasticities, we have to take into account that increasing $X_{3}$ by one unit is associated with a $(100 \cdot \widehat{\beta}_{3})$% increase in $y$. Then, in order to standardize the coefficients and interpreting them, would

$$BETA_{2} = \widehat{\beta}_{2} \cdot (S_{x_{2}}/S_{y})$$ $$BETA_{3} = 100 \cdot \widehat{\beta}_{3} \cdot (S_{x_{3}}/S_{y})$$

be correct?

Answer 1

$\beta_2$ is approximately the elasticity of $y$ with respect to $X_2$, holding fixed $X_3$.

$$y = e^{\beta_1 +\beta_2 ln(X_2)+\beta_3X_3+u}$$ $$\frac{\partial y}{\partial X_2} = \frac{\beta_2}{X_2}e^{\beta_1 +\beta_2 ln(X_2)+\beta_3X_3+u}$$ We can plug in that $ y=e^{\beta_1 +\beta_2 ln(X_2)+\beta_3X_3+u}$, $$\frac{\partial y}{\partial X_2} = \frac{\beta_2}{X_2}y$$

By a linear approximation,

$$\Delta y \approx \frac{\partial y}{\partial X_2} \Delta X_2$$ We plug in the derivative. $$\Delta y \approx \frac{\beta_2}{X_2}y \Delta X_2$$ We rearrange, $$\beta_2 \approx \frac{y}{X_2}\frac{\Delta X_2}{\Delta y}$$ Which is an elasticity.

In terms of "comparing" $\beta_2$ and $\beta_3$, I feel like we need to know what the goal is. Is the question statistical significance? Then we care about the t-stats. Is the goal which is "bigger"? I feel like the result depends on units and would need to be evaluated in context. Feel free to follow up and I can possibly edit this answer.