When I estimate a difference in differences model with two time periods, the equivalent regression model would be
a. $Y_{ist} = \alpha +\gamma_s*Treatment + \lambda d_t + \delta*(Treatment*d_t)+ \epsilon_{ist}$
- where $Treatment$ is a dummy which is equal to 1 if the observation is from the treatment group
- and $d$ is a dummy which is equal to 1 in the time period after the treatment occured
Thus the equation takes the following values.
- Control group, before treatment: $\alpha$
- Control group, after treatment: $\alpha +\lambda$
- Treatment group, before treatment: $\alpha +\gamma$
- Treatment group, after treatment: $\alpha+ \gamma+ \lambda+ \delta$
Hence, in a two period model the difference in differences estimate is $\delta$.
But what happens concerning $d_t$ if I have more than one pre and post treatment period? Do I still use a dummy that indicates whether a year is before or after the treatment?
Or do I add year dummies instead without specifying whether each year belong to the pre or post treatment period? Like this:
b. $Y_{ist} = \alpha +\gamma_s*Treatment + yeardummy + \delta*(Treatment*d_t)+ \epsilon_{ist}$
Or can I include both (i.e $yeardummy +\lambda d_t$)?
c. $Y_{ist} = \alpha +\gamma_s*Treatment + yeardummy + \lambda d_t + \delta*(Treatment*d_t)+ \epsilon_{ist}$
In conclusion, how do I specify a difference in differences model with multiple time periods (a,b or c)?