Average Treatment effect on the Treated in Repeated Cross Sections

Abstract: The difference-in-differences (DiD) estimator recovers the average treatment effect on the treated (ATT) under standard assumptions of no anticipation and parallel trends when panel data are available. However, researchers frequently work with repeated cross-sections where the exact treatment date is not always observed: individuals observed as untreated in a given period may receive treatment later or never be treated. A recent approach in the literature on child penalty constructs pseudo-panels by synthetically matching treated units to younger untreated units observed in earlier periods. We establish that a natural conditional reformulation of the identifying assumptions underlying this estimator implies a conditional missing at random (CMAR) assumption. We show that under standard DiD assumptions alone, the ATT is only partially identified, and we derive a tractable outer set for the ATT along with a sensitivity analysis framework for departures from CMAR. We apply these results to analyze child penalties using three Korean datasets.