Difference-in-differences with long time horizon and repeated treatments
This question is related to a post I addressed on CrossValidated. The &quot;generalized&quot; difference-in-differences (DiD) estimator is amenable to settings with multiple groups and multiple exposure periods. Take the following specification:
$$
y_{it} = \gamma_{i} + \lambda_{t} + \delta T_{it} + \epsilon_{it},
$$
where $\gamma_{i}$ and $\lambda_{t}$ denote unit (i.e., individual, entity, county, state, etc.) and time (i.e., day, month, year, etc.) fixed effects, respectively. You may also see this referenced as a ‘two-way’ fixed effects estimator. The variable $T_{it}$ is a treatment dummy, indexing the $i$ units affected by the policy/intervention during periods $t$, 0 otherwise. In practice, $T_{it}$ can take on any pattern. Thus, it can more than handle intermittent exposure periods.
Your main policy dummy $T_{it}$ is your interaction term, but you must instantiate it manually to account for the different entities moving into and out of a treated status across time. Thus, $T_{it}$ 'turns on' (i.e., equals 1) if two conditions are met: (1) the entity is in the treatment group and it is in the post-treatment period. Be careful because $T_{it}$ does not demarcate a precise treatment group. Rather, it a discrete indictor for adopter entities and only during those adopter periods. Any 'off' period of treatment should be coded 0.
The problem in your setting is you have a large number of treated units with staggered adoption periods, making it exceedingly difficult to demonstrate a common trend across groups. Centering all units on their adoption date is insufficient as a subset of treated units have multiple treatment regimes. I wonder if you could work on subsets of your data where most treated entities enter into the treatment epoch at more or less the same time. However, this might be difficult with repeated, transient post-exposure periods. Pre-trends might also be violated in settings with irregular exposure periods and heterogeneous treatment effects, a point noted in a recent working paper. In sum, while the &quot;generalized&quot; DiD estimator will work in your setting, you must be cognizant of the downsides of using this approach.
I have previously evaluated policies that were introduced and repealed on several occasions over time. The only difference is the policy dummy turned 'on' and 'off' at the same time for all units. While a different subset of entities were treated in each iteration of the policy, the exposure period was standardized. Thus, I was able to focus on subsets of my panel and assess trend equivalence before each adoption period. I would be happy to share this research if it offers any further insight.
You might also find this working paper an interesting read.