Interesting paper. Thank you for calling my attention to it.
You will not face this problem if you are using matched pairs, because in the matched pair design, only untreated entities ever serve as controls in the analysis. The important things, however, is that your analysis must reflect the matched pairing as a level in the model. Whereas in the ordinary two-way fixed effects analysis you have repeated observations nested within firms (or jurisdictions, or whatever the entities are) , you now have a three way model where those entities are now nested in matched-pairs. This means that you cannot use a fixed-effects estimator, you must use a random effects estimator to capture this three-level structure. You will have to consider whether the problems associated with using a random-effects estimator are a reasonable trade-off in your circumstances compared to the problems of not having matched pairs and coping with the problems of the generalized DID estimator pointed out in the paper you cite.
You will not face this problem if you are using matched pairs, because in the matched pair design, only untreated entities ever serve as controls in the analysis. The important things, however, is that your analysis must reflect the matched pairing as a level in the model. Whereas in the ordinary two-way fixed effects analysis you have repeated observations nested within firms (or jurisdictions, or whatever the entities are) , you now have a three way model where those entities are now nested in matched-pairs. This means that you cannot use a fixed-effects estimator, you must use a random effects estimator to capture this three-level structure. You will have to consider whether the problems associated with using a random-effects estimator are a reasonable trade-off in your circumstances compared to the problems of not having matched pairs and coping with the problems of the generalized DID estimator pointed out in the paper you cite.
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