# Whether coefficient of a dummy variable can receive the value higher than 1 and using log for per-million variable?

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)

Consider a regression with a dummy variable: $$y_i = \alpha + \beta D_i + \varepsilon_i.$$ Then $$\beta$$ will be identified by: $$\mathbb{E}(y_i|D_i = 1) - \mathbb{E}(y_i| D_i = 0) = \beta$$
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: $$\tilde \beta = \mathbb{E}(\ln(y_i)|D_i = 1) - \mathbb{E}(\ln(y_i)|D_i = 0).$$
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.