Purpose of differences in differences with multiple time dummies

Question

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.

Answer 1

The latter equation is just a generalization of the former.

You could model the time shocks with a single time indicator representing pre-/post-treatment, or you could model each period individually before and after the exposure. Assuming the exposure affects all treated units in the same year, then you'll find your estimate of the "treatment effect" is similar across the two specifications. Just be mindful that if you have multiple groups and different treatment adoption periods, then you must proceed with the latter specification.

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.