<p><span class="math-container">$\beta_2$</span> is approximately the elasticity of <span class="math-container">$y$</span> with respect to <span class="math-container">$X_2$</span>, holding fixed <span class="math-container">$X_3$</span>.</p>
<p><span class="math-container">$$y = e^{\beta_1 +\beta_2 ln(X_2)+\beta_3X_3+u}$$</span>
<span class="math-container">$$\frac{\partial y}{\partial X_2} = \frac{\beta_2}{X_2}e^{\beta_1 +\beta_2 ln(X_2)+\beta_3X_3+u}$$</span>
We can plug in that <span class="math-container">$ y=e^{\beta_1 +\beta_2 ln(X_2)+\beta_3X_3+u}$</span>,
<span class="math-container">$$\frac{\partial y}{\partial X_2} = \frac{\beta_2}{X_2}y$$</span></p>
<p>By a linear approximation,</p>
<p><span class="math-container">$$\Delta y \approx \frac{\partial y}{\partial X_2} \Delta X_2$$</span>
We plug in the derivative.
<span class="math-container">$$\Delta y \approx \frac{\beta_2}{X_2}y \Delta X_2$$</span>
We rearrange,
<span class="math-container">$$\beta_2 \approx \frac{y}{X_2}\frac{\Delta X_2}{\Delta y}$$</span>
Which is an elasticity.</p>
<p>In terms of &quot;comparing&quot; <span class="math-container">$\beta_2$</span> and <span class="math-container">$\beta_3$</span>, 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 &quot;bigger&quot;? 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.</p>

CC BY-SA 4.0
2022-11-07T13:12:04.663Z
2022-11-07T13:12:04.663Z
Michael Gmeiner
<p>Let's say we have a model like:</p>
<p><span class="math-container">$$log(y) = \beta_{1} + \beta_{2} \cdot log(X_{2})+\beta_{3} \cdot X_{3} + u$$</span></p>
<p>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.</p>
<p>I know that normally (given a model like <span class="math-container">$y = \beta_{1} + \beta_{2} \cdot X_{2}+\beta_{3} \cdot X_{3} + u$</span>) we would just compute:</p>
<p><span class="math-container">$$BETA_{2} = \widehat{\beta}_{2} \cdot (S_{x_{2}}/S_{y})$$</span>
<span class="math-container">$$BETA_{3} = \widehat{\beta}_{3} \cdot (S_{x_{3}}/S_{y})$$</span></p>
<p>interpreting those BETA as standardized coefficients; the higher they are, the higher the impact of the independent variable over the dependent variable.</p>
<p>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 <span class="math-container">$X_{3}$</span> by one unit is associated with a <span class="math-container">$(100 \cdot \widehat{\beta}_{3})$</span>% increase in <span class="math-container">$y$</span>. Then, in order to standardize the coefficients and interpreting them, would</p>
<p><span class="math-container">$$BETA_{2} = \widehat{\beta}_{2} \cdot (S_{x_{2}}/S_{y})$$</span>
<span class="math-container">$$BETA_{3} = 100 \cdot \widehat{\beta}_{3} \cdot (S_{x_{3}}/S_{y})$$</span></p>
<p>be correct?</p>

<econometrics><regression>
How to interpret the standardized coefficients (BETA) in a mixed log-log and log-level regression
ParetoDeficient
# How to interpret the standardized coefficients (BETA) in a mixed log-log and log-level regression

(Asked by: username_0 on 2022-11-05)

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?

**Tags:** econometrics, regression


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## 1 answers

### Answer by username_1 on 2022-11-07. Score: 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.


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