# Purpose of differences in differences with multiple time dummies

I understand that the standard difference in differences with 2 groups, 2 time periods appears as follows \begin{align*} y = \beta_0 + \beta_1Tr + \beta_2Post + \beta_3 Tr \times Post + u \end{align*} Where $$Tr$$ is a dummy for being in the treatment group, $$Post$$ is a dummy for the post treatment time period. However, I have seen models utilizing multiple time dummies, formulated as follows

\begin{align*} y = \beta_0 + \beta_1Tr + \sum_{t = 1}^T\lambda_td_t + \beta_2D + u \end{align*} where each $$dt$$ is a dummy for the time period (among $$T$$ total time periods), and \begin{align*} D = \begin{cases} 1 \text{ if in treatment group and post-treatment}\\ 0 \text{ otherwise} \end{cases} \end{align*} I am confused as to the purpose of the second formulation, if ultimately we are still interested in the effect on the interaction term. If we say, have 10 time periods, what is the difference between including all 10 time dummies vs. just taking one pre- and one post- time period? Thank you.

• Welcome. In some settings, it doesn’t really matter how we model the time shocks. Is the “treatment” administered at a single point in time for all treated units? Mar 16, 2022 at 5:22
• @ThomasBilach thanks for the reply. Yes, the model I have in mind is for a singular point in time (e.g., if there are 10 months of data, and the treatment occurs between month 5 and 6) Mar 17, 2022 at 1:03

Because you're assessing a uniform exposure period, your choice of equation is immaterial, at least as it pertains to the estimate on your interaction term. Now it's quite possible that the year effects are of substantive interest. Maybe you actually want to inspect the coefficients on the individual year dummies; those effects represent the expected mean difference in $$y$$ in some year $$t$$ relative to a particular reference year, whatever that period may be. I suspect you want to assess how, absent the treatment, this change is evolving over time. In a setting with just 10 years, this is something worth exploring, or even reporting. However, once $$T$$ gets large, then I'd consider dropping the year fixed effects entirely; they're nuisance. You'll find most papers omit the year effects anyway.