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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)
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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: $$ \tilde \beta = \mathbb{E}(\ln(y_i)|D_i = 1) - \mathbb{E}(\ln(y_i)|D_i = 0). $$

  1. 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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