# How to specify a Diff-In-Diff Regression with multiple time periods?

I'm working on analysing experimental data for a thesis project. The data consists of subjects performing the same task over five rounds, and I'm interested in the difference in trends between subjects in two different treatments. The two treatments are identical until round 3.

I planned on using a diff-in-diff model to estimate the difference of Effort levels of subjects across these treatments. The problem is, I have five rounds, two of which are before-treatment and three of which are after-treatment. Currently, I'm using this specification, but I'm not sure if it's correct:

$$Effort_{it}=\beta _{0} + \beta_{1}Treatment_{i}+\sum_{n=2}^{5}\beta _{n}Roundn_t+\beta_6Treatment*After_{it}$$

Where treatment is a dummy for being in the treatment groub, Roundn is a dummy for being in Round N, and Treatment*After is an interaction dummy for being in the treatment group after round 2 (when the treatment "begins").

I'm confused mostly on what to do with the different time periods. Would it be best to use dummies for each round like above, or to just include a Round variable that is equal to the number of the round. Also should I just include one interaction term, or one for each round?

I am a bit confused because you said there are two treatments. I presume that one "treatment" group is the control group and the other is the treatment group. (I don't think you have a separate control group.) I am also confused because you said the treatment is identical until round 3 but then the treatment begins in round 2. I presume that there is no treatment in rounds 1 and 2, and the treatment group is treated differently thereafter in rounds 3, 4 and 5.

Your $Roundn$ dummy variables for $n=2,\ldots 5$ make sure that you handle the "round effects" (for the control group) properly in a nonparametric way. This looks fine to me. If you include only the $After$ dummy, it means that there is no trend within each of the "before" and "after" periods. You would not want that.

A single $round$ variable means that there is a linear trend in $Effort$ in the control group. You could try that, but I would wonder where the linearity belief comes from. Also, you lose only 3 more degrees of freedom by including the round dummies comparing to the linear trend model. That's not a big deal unless you have a really small sample. I would be happy with the full round dummies.

Your model assumes that the treatment effects (measured by diff-in-diff) are identical in all rounds 3, 4 and 5 (because you have only one interaction term). If you believe it is true, that's fine. If you believe otherwise, you can include three interaction terms $Treatment * Round3$, $Treatment * Round4$ and $Treatment * Round5$ instead of the single interaction term. If you want, you can test if those treatment effects are identical across rounds.

• Thanks for the help! Sorry for the wording issues, but you did interpret what I was saying correctly. One thing I have read was to specify it using fixed effects for each round, and then interaction terms for all rounds but one. Doing this, I wouldn't have to include the Treatment dummy correct? I think having interaction terms for each round would eliminate the need for that. – econra2017 Nov 13 '16 at 21:34
• Yes. Fixed effects for each round contain Treatment dummy, so you need not include the treatment dummy separately. But if you include interaction terms for all rounds, it means that there is a nonzero "treatment effect" in Period 2 as well, while there was no "treatment" in that period. – chan1142 Nov 15 '16 at 14:01