<p>Consider a regression with a dummy variable:
<span class="math-container">$$
y_i = \alpha + \beta D_i + \varepsilon_i.
$$</span>
Then <span class="math-container">$\beta$</span> will be identified by:
<span class="math-container">$$
\mathbb{E}(y_i|D_i = 1) - \mathbb{E}(y_i| D_i = 0) = \beta
$$</span></p>
<blockquote>
<ol>
<li>Whether I should use log for this outcome variable because I am not sure it is a ratio or actual value (ratio to me normally percent, not per million like that)?</li>
</ol>
</blockquote>
<p>It depends what you want. If you measure as is, you get
<span class="math-container">$$
\beta = \mathbb{E}(y_i | D_i = 1) - \mathbb{E}(y_i|D_i = 0).
$$</span>
If you measure in logs, you are estimating:
<span class="math-container">$$
\tilde \beta = \mathbb{E}(\ln(y_i)|D_i = 1) - \mathbb{E}(\ln(y_i)|D_i = 0).
$$</span></p>
<blockquote>
<ol start="2">
<li>I run the regression with the outcome above and the coefficient of a dummy variable is 2 (higher than 1), is it abnormal? (The dummy here only receive the values of 0 and 1)</li>
</ol>
</blockquote>
<p>Yes, <span class="math-container">$\beta$</span> can be higher than 1 if the range of <span class="math-container">$y_i$</span> is goes beyond the unit interval. On the other hand, if <span class="math-container">$y_i$</span> is bounded between zero and 1, then <span class="math-container">$\beta$</span> should normally also be between <span class="math-container">$0$</span> and <span class="math-container">$1$</span> as then <span class="math-container">$0 \le \mathbb{E}(y_i|D_i = 1), \mathbb{E}(y_i|D_i = 0) \le 1$</span>.</p>
<p>In your case, <span class="math-container">$y_i$</span> is between 0 and 1 million, so these bounds are not satisfied.</p>

CC BY-SA 4.0
2021-09-20T11:53:34.203Z
2021-09-20T11:53:34.203Z
tdm
<p>Today when I run a regression (in specific Difference-in-Differences but I think it does not matter here). My outcome variable is a ratio (lower than 1 million, it is about a number of people per million).</p>
<p>I have two questions here:</p>
<ol>
<li>Whether I should use log for this outcome variable because I am not sure it is a ratio or actual value (ratio to me normally percent, not per million like that)?</li>
<li>I run the regression with the outcome above and the coefficient of a dummy variable is 2 (higher than 1), is it abnormal? (The dummy here only receive the values of 0 and 1)</li>
</ol>

<econometrics><nonexpert>
Whether coefficient of a dummy variable can receive the value higher than 1 and using log for per-million variable?
Phil Nguyen
# Whether coefficient of a dummy variable can receive the value higher than 1 and using log for per-million variable?

(Asked by: username_0 on 2021-09-20)

Today when I run a regression (in specific Difference-in-Differences but I think it does not matter here). My outcome variable is a ratio (lower than 1 million, it is about a number of people per million).

I have two questions here:

1. Whether I should use log for this outcome variable because I am not sure it is a ratio or actual value (ratio to me normally percent, not per million like that)?
2. I run the regression with the outcome above and the coefficient of a dummy variable is 2 (higher than 1), is it abnormal? (The dummy here only receive the values of 0 and 1)

**Tags:** econometrics, nonexpert

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## 1 answers

### Answer by username_1 on 2021-09-20. Score: 2 | Accepted answer

Consider a regression with a dummy variable:
$$
y\_i = \alpha + \beta D\_i + \varepsilon\_i.
$$
Then $\beta$ will be identified by:
$$
\mathbb{E}(y\_i|D\_i = 1) - \mathbb{E}(y\_i| D\_i = 0) = \beta
$$

> 1. Whether I should use log for this outcome variable because I am not sure it is a ratio or actual value (ratio to me normally percent, not per million like that)?

It depends what you want. If you measure as is, you get
$$
\beta = \mathbb{E}(y\_i | D\_i = 1) - \mathbb{E}(y\_i|D\_i = 0).
$$
If you measure in logs, you are estimating:
$$
\tilde \beta = \mathbb{E}(\ln(y\_i)|D\_i = 1) - \mathbb{E}(\ln(y\_i)|D\_i = 0).
$$

> 2. I run the regression with the outcome above and the coefficient of a dummy variable is 2 (higher than 1), is it abnormal? (The dummy here only receive the values of 0 and 1)

Yes, $\beta$ can be higher than 1 if the range of $y\_i$ is goes beyond the unit interval. On the other hand, if $y\_i$ is bounded between zero and 1, then $\beta$ should normally also be between $0$ and $1$ as then $0 \le \mathbb{E}(y\_i|D\_i = 1), \mathbb{E}(y\_i|D\_i = 0) \le 1$.

In your case, $y\_i$ is between 0 and 1 million, so these bounds are not satisfied.


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