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All of the literature on synthetic control groups examines situations where 1 or more units receive a treatment and are compared to non-treated units. I am wondering if there is any statistical or mathematical reason why these techniques cant be applied to a situation where both treated and "control" units receive some treatment, but the treatment differs? For example, some units receive treatment A and the others receive treatment B. Is this viable?

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  • $\begingroup$ I suspect that this is a bit tricky since the SC method estimates the ATET. So if you treat Cleveland with A and San Diego with B, you get the effect of A for Cleveland and the effect of B for San Diego. These don't tell you if the effect would be larger in Cleveland had you applied B rather than A. $\endgroup$
    – dimitriy
    Dec 4, 2022 at 3:36
  • $\begingroup$ But there might be a way of doing this if you can construct two comparable treatment groups somehow and then use a binomial probability test or randomization inference to answer what is the probability that the effect would be larger in group A than group B in eight or more out of the ten periods of the test, under the null that they are the same. But I am uncertain that this is the case and hope to see more folks chime in. $\endgroup$
    – dimitriy
    Dec 4, 2022 at 3:36

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As usual, @dimitriy gives good advice. But I have a different approach: you can use the synthetic interventions estimator in this case. It was designed to solve this specific problem! I'm working on what I believe to be an improvement of the SI estimator, but this does as you want (what if treatment A happened in B, what if B happened in A...) for each unit.

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