<p>As you can see from the plot, they are not the same. Infact, the 'rule of thumb', i.e. your first formula, only holds in approximation. The error gets larger for large coefficient values.</p>
<p><a href="https://i.sstatic.net/KuIdG.png" rel="nofollow noreferrer"><img src="https://i.sstatic.net/KuIdG.png" alt="plot" /></a></p>

CC BY-SA 4.0
2020-11-21T19:27:03.723Z
2020-11-21T19:27:03.723Z
E. Sommer
<p>Consider the model:</p>
<p><span class="math-container">$$y_i=\beta_0+\beta_1D_1+\beta_2D_2+u_i$$</span></p>
<p>Where <span class="math-container">$D_1=\{0,1\}$</span> and <span class="math-container">$D_2=\{0,1\}$</span> are binary (dummy) variables, and <span class="math-container">$y_i$</span> is a continuos variable in levels.</p>
<p>In this model, for example <span class="math-container">$\beta_1$</span> is interpreted as the gap between the category when <span class="math-container">$D_1=1$</span> and <span class="math-container">$D_1=0$</span>. Besides, this gap is in <em>absolute</em> terms, say in terms of the units of the <span class="math-container">$y_i$</span> variable.</p>
<p>Now, consider the similar model but dependent variable is in logs:</p>
<p><span class="math-container">$$\ln (y_i)=\beta_0+\beta_1D_1+\beta_2D_2+u_i$$</span></p>
<p>I would thought that, for example, <span class="math-container">$\beta_1$</span> is the gap between the mentioned categories but in <em>relative</em> terms, say a proportion or percentage. Say in average category <span class="math-container">$D_1=1$</span> receives <span class="math-container">$\beta_1*100$</span> percent more (or less) of <span class="math-container">$y_1$</span> than <span class="math-container">$D_1=0$</span> in average.</p>
<p>But I found another approximation in my class for this gap, and it takes the gap starting with the fact that the relative gap:</p>
<p><span class="math-container">$$Rel. gap\; between\; [D_1=1]\; and\; [D_1=0]=\frac{y_1-y_0}{y_0}*100\% = \left(\frac{y_1}{y_0}-1\right)*100\%$$</span></p>
<p>Where <span class="math-container">$y_1=y(D_1=1)$</span> and <span class="math-container">$y_0=y(D_1=0)$</span></p>
<p>Taking this into account we can get:</p>
<p><span class="math-container">$$\ln (y_1)-\ln(y_0)=\beta_1 \implies \frac{y_1}{y_0}=e^{\beta_1}$$</span></p>
<p>Replacing this into the relative gap equation specified earlier we get:</p>
<p><span class="math-container">$$Rel. gap\; between\; [D_1=1]\; and\; [D_1=0]=\left ( e^{\beta_1} -1 \right)*100\%$$</span></p>
<p>My question is: What is specifically the difference between those two approximations? And even if one of those perhaps is wrong, because the resulting gaps that you can obtain are way different. Moreover, which one should one use with academic purposes.</p>
<p>Thanks!</p>

CC BY-SA 4.0
2020-11-21T22:39:39.670Z
This would be easier to expalain and understand if you did not subtract $1$.  If $e^{\beta_1}=2.5$ then having $D_1=1$ multiplies the estimate for $y$ by $2.5$, while if $e^{\beta_2}=4$ then having $D_2=1$ multiplies the estimate for $y$ by $4$, and having both multiplies the estimate for $y$ by $10$.  So much simpler than saying that these increase the estimates for $y$ by $150\%$, $300\%$ and $900\%$ respectively
Henry
<econometrics><regression>
Difference in interpretation of log-lin coefficients with binary variable
manifold
# Difference in interpretation of log-lin coefficients with binary variable

(Asked by: username_0 on 2020-11-21)

Consider the model:

$$y\_i=\beta\_0+\beta\_1D\_1+\beta\_2D\_2+u\_i$$

Where $D\_1=\{0,1\}$ and $D\_2=\{0,1\}$ are binary (dummy) variables, and $y\_i$ is a continuos variable in levels.

In this model, for example $\beta\_1$ is interpreted as the gap between the category when $D\_1=1$ and $D\_1=0$. Besides, this gap is in *absolute* terms, say in terms of the units of the $y\_i$ variable.

Now, consider the similar model but dependent variable is in logs:

$$\ln (y\_i)=\beta\_0+\beta\_1D\_1+\beta\_2D\_2+u\_i$$

I would thought that, for example, $\beta\_1$ is the gap between the mentioned categories but in *relative* terms, say a proportion or percentage. Say in average category $D\_1=1$ receives $\beta\_1\*100$ percent more (or less) of $y\_1$ than $D\_1=0$ in average.

But I found another approximation in my class for this gap, and it takes the gap starting with the fact that the relative gap:

$$Rel. gap\; between\; [D\_1=1]\; and\; [D\_1=0]=\frac{y\_1-y\_0}{y\_0}\*100\% = \left(\frac{y\_1}{y\_0}-1\right)\*100\%$$

Where $y\_1=y(D\_1=1)$ and $y\_0=y(D\_1=0)$

Taking this into account we can get:

$$\ln (y\_1)-\ln(y\_0)=\beta\_1 \implies \frac{y\_1}{y\_0}=e^{\beta\_1}$$

Replacing this into the relative gap equation specified earlier we get:

$$Rel. gap\; between\; [D\_1=1]\; and\; [D\_1=0]=\left ( e^{\beta\_1} -1 \right)\*100\%$$

My question is: What is specifically the difference between those two approximations? And even if one of those perhaps is wrong, because the resulting gaps that you can obtain are way different. Moreover, which one should one use with academic purposes.

Thanks!

**Tags:** econometrics, regression

> **Comments:**
> **username_1**: This would be easier to expalain and understand if you did not subtract $1$.  If $e^{\beta_1}=2.5$ then having $D_1=1$ multiplies the estimate for $y$ by $2.5$, while if $e^{\beta_2}=4$ then having $D_2=1$ multiplies the estimate for $y$ by $4$, and having both multiplies the estimate for $y$ by $10$.  So much simpler than saying that these increase the estimates for $y$ by $150\%$, $300\%$ and $900\%$ respectively  

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## 1 answers

### Answer by username_2 on 2020-11-21. Score: 4 | Accepted answer

As you can see from the plot, they are not the same. Infact, the 'rule of thumb', i.e. your first formula, only holds in approximation. The error gets larger for large coefficient values.

[![plot](https://i.sstatic.net/KuIdG.png)](https://i.sstatic.net/KuIdG.png)


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