# Difference in Difference with multiple period (pre, during, and post treatment)

I was wondering if someone can help me understand how I should formulate my difference in difference regression. Under a basic two period DiD, I would be estimating an equation similar to: $$y_{gt} = \beta_0 +\beta_1 Time_t + \beta_2 Group_g +\beta_3 Time\times Group_{gt} + \epsilon_{gt}$$ Where g denotes the group and t denotes the time.

However, I am trying to model a difference in difference, but I have time periods representing pre-treatment, during treatment, and post-treatment. I'm not sure how to capture the effects of the treatment. Originally I was thinking $$y_{gt}=\beta_0 +\beta_1Post_t +\beta_2Pre_t +\beta_3 Group_g+\beta_4Post\times Group_{gt}+\beta_5 Pre \times Group_{gt}+\epsilon_{gt}$$ But I'm uncertain if this would capture the effect of the treatment. Furthermore, I'm fairly new to Diff in Diff, so I'm no sure what the interaction beta's might imply and whether or not it captures the effect Can someone help?

• Possible duplicate of Specifying a difference in differences model with multiple time periods – Andy Jul 22 '16 at 18:09
• I looked at that one prior to posting this. The difference is that that post is looking at adding more year variables to either pre or post treatment. My issue is that I'm not sure how to organize my DiD since I have pre treatment obs, treatment obs, and post treatment obs. Not sure if it's the same as the solution in there or not – grad_student Jul 22 '16 at 18:21
• @grad_student I think this would depend on the type of treatment. Is there any reason why you could not include a $During$ dummy and a $During \times Treatment$ interaction? – Dimitriy V. Masterov Jul 22 '16 at 18:41
• @DimitriyV.Masterov include it into the last regression or let it replace the $Pre$ and $Pre\times Treatment$? – grad_student Jul 22 '16 at 19:03
• @grad_student In the first regression, but you would have dummies for both $During$ and $Post$ (instead of only $Post$), plus their interactions with $Treatment$. – Dimitriy V. Masterov Jul 22 '16 at 19:07

Let $t=1,2,3$ and $T=1$ if treated, zero otherwise. $\mathbf{I}$ is the indicator function. The expected value from the specification I suggested in the comments above is

$$E[Y \vert X] = \beta_0 + \beta_1 T + \beta_2 \mathbf{I}_{t=2}+\beta_3 \mathbf{I}_{t=3}+\beta_4 \mathbf{I}_{t=2} \cdot T + \beta_5 \cdot \mathbf{I}_{t=3} \cdot T.$$

The DID for the third (post) period would be: $$\left( E[Y \vert T=1,t=3]-E[Y \vert T=1,t=1] \right)-\left( E[Y \vert T=0,t=3]-E[Y \vert T=0,t=1] \right) = \beta_5.$$

The DID for the second (during) period would be: $$\left( E[Y \vert T=1,t=2]-E[Y \vert T=1,t=1] \right)-\left( E[Y \vert T=0,t=2]-E[Y \vert T=0,t=1] \right) = \beta_4.$$

If $\beta_5>0,$ it is plausible that $\beta_4<0$. One example might be graduate school, where the treated observations earn very little money while enrolled. You might consider adding the foregone income on the cost side of the programs.

In other settings, $\beta_5>\beta_4>0.$ This might be the case if the treatment is adoption of new technology, where there is a learning period before it becomes fully productive.

• @grad_student No need to thank me, but you can select my response as the answer if this cleared things up. – Dimitriy V. Masterov Jul 22 '16 at 21:07
• Well, if there's no green check mark it gets at least a +1. Good answer! – Andy Jul 22 '16 at 21:47
• @Andy Thanks for the kind words. I am glad when they don't turn out indifferent (pun intended). – Dimitriy V. Masterov Jul 22 '16 at 21:52