Difference-in-Differences and Changes-in-Changes with Sample Selection

Abstract: Sample selection arises endogenously in causal research when the treatment affects whether certain units are observed. It is a common pitfall in longitudinal studies, particularly in settings where treatment assignment is confounded. In this paper, I highlight the drawbacks of one of the most popular identification strategies in such settings: Difference-in-Differences (DiD). Specifically, I employ principal stratification analysis to show that the conventional ATT estimand may not be well defined, and the DiD estimand cannot be interpreted causally without additional assumptions. To address these issues, I develop an identification strategy to partially identify causal effects on the subset of units with well-defined and observed outcomes under both treatment regimes. I adapt Lee bounds to the Changes-in-Changes (CiC) setting (Athey & Imbens, 2006), leveraging the time dimension of the data to relax the unconfoundedness assumption in the original trimming strategy of Lee (2009). This setting has the DiD identification strategy as a particular case, which I also implement in the paper. Additionally, I explore how to leverage multiple sources of sample selection to relax the monotonicity assumption in Lee (2009), which may be of independent interest. Alongside the identification strategy, I present estimators and inference results. I illustrate the relevance of the proposed methodology by analyzing a job training program in Colombia.