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I have a control group and treatment group and the treatment is introduced in the second phase after time point t. But the treatment can be different types at different time points after time point t. Let's suppose we have 3 different types of such treatment. Here are some examples of the data:

  • Subject a in the treatment group may receive type 1 treatment at time t+1, and type 3 treatment at time t+2, so on and so forth.
  • Subject b in the treatment group may receive type 1 and type 2 treatment at time t+1, and type 2 at time t+2, so on and so forth.
  • Subject c in the treatment group may receive type 1 and type 3 treatment at time t+1, and type 2 and type 3 at time t+2, so on and so forth.
  • Any subject in the control group will never receive any treatment.

In the dataset, I have variables

  • Treatment_Group: a dummy variable representing whether the subject is in the control or treatment group.
  • After: a dummy variable representing whether the subject is in the phase before or after the treatment is introduced.
  • Treatment_Type: a categorical variable representing different types of the treatment.

If I do not want to distinguish different types of treatment, I understand how to run a DID model with Treatment_Group and After variables. What if I want to further evaluate the effect of different types of treatment in this context? How shall I integreate Treatment_Type into the DID model? Or any other model would be more appropriate to identify the effect?

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  • $\begingroup$ There is some discussion of this in these questions. There is also a very nice survey of staggered DID with multiple treatments in Roth et al. (2022). There are several competing estimators, and the advice is to try all the ones that satisfy assumptions in your setting to check robustness to different methods. That is too much terrain to cover in a single answer, so I will leave these as a comment. $\endgroup$
    – dimitriy
    Commented Mar 3, 2023 at 6:08
  • $\begingroup$ If the treatments have permanent /lasting effects and interactions, there may not be a good solution if there are many permutations. $\endgroup$
    – dimitriy
    Commented Mar 3, 2023 at 6:11
  • $\begingroup$ @dimitriy thanks for your response. But I think my problem is different from a problem of treatment at different time periods. The treatment is introduced at the same time for all subjects, but the types of treatment will differ afterwards for different subjects. Put it another way, after time t, subjects in the treatment group will receive any type of treatment, and this is random. Say for subject a in the treatment group, it may receive treatment 1 at time t+1, then treatment 2 & 3 at time t+2, then treatment 2 at time t+3, so on and so forth. Do you suggest there is no good solution? $\endgroup$
    – ycenycute
    Commented Mar 3, 2023 at 8:03
  • $\begingroup$ I am not sure I understand the distinction. $\endgroup$
    – dimitriy
    Commented Mar 3, 2023 at 8:18
  • $\begingroup$ @dimitriy the different types of the treatment may not be permanent after the time of introduction. Consider that a platform introduce a new feature, and this feature can relate to different types. Type 1 may appear at time t+1, but disappear at time t + 2, and re-appear at some later time point. It is not like the traditional After variable where you have 0 before time t, and 1 after time t. Type variable can be 0 after time t as well, and this differs among different subjects in the treatment group. $\endgroup$
    – ycenycute
    Commented Mar 3, 2023 at 10:48

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