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?


1 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.


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Not the answer you're looking for? Browse other questions tagged or ask your own question.