Difference-in-Differences When Treatment Status Is Observed in Only One Period

This paper extends the difference-in-differences method to the setting of repeated cross-sections when treatment status is observed in either the pre- or the post-treatment period, but not in both time periods. The issue in such a setting is that the outcomes that are observed in the period when the treatment status is missing cannot be classified into outcomes for the treated and outcomes for the control. As such, the standard difference-in-differences\estimand cannot be calculated and the method cannot be used to point identify the average treatment effect on the treated. Our contribution is to show that the average treatment effect on the treated can still be point-identified by making use of the following observations (1) it is usually the case that the individual treatment status is observed in post-treatment household surveys even when it is not observed in pre-treatment surveys; (2) in most data sets, it is possible to find a proxy or a predictor for the treatment status; and (3) when samples are representative of the same population, the propensity score is stationary. Further, we show the point-identification of the distribution of outcomes conditional on the latent treatment status. We then propose a method of moments estimation strategy for the identified model. We illustrate our method on data from the Mexican conditional cash-transfer program PROGRESA, and we compare our results with those from a standard difference-in-differences analysis using the same data set.