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The synthetic control method can be used to calculate the effect of an event when some units are affected by the event but others are not.

What method can I use if all units are affected by the same event, but at different times? For example, I have data on the productivity of different teams in a workplace, and also the number of people joining and leaving in each period (mostly 0 people leave each period, but sometimes someone leaves. The case is similar for joining.). I want to study the effect of someone leaving on team productivity.

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You need some untreated units to form the counterfactual. To the extent that there are none, or just a few, left to form the control, your ability to reweight them will be nonexistent or severely compromised.

Unfortunately there are no other methods that would work well in this setting that are also credible.

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