Difference-in-Differences Estimates under Selective Migration

Difference-in-differences designs often study place-based treatments that can trigger migration in and out of treated areas. When treatment changes who is observed, aggregate and within-individual DiD no longer retrieve the average treatment effect on the treated. This paper uses principal stratification to characterize three estimands under treatment-induced migration: a locality-level treatment effect, a stayer average treatment effect, and an individual ATT for the pre-treatment population. Aggregate DiD identifies the locality-level effect under aggregate parallel trends, while within-individual DiD identifies the stayer effect under stayer parallel trends. Together, the two assumptions imply parallel trends for escapees, a non-trivial restriction on the group whose departure drives compositional change. With panel data, the stayer effect and the compositional term are point-identified, and the ATT lies in a one-parameter sensitivity region. With repeated cross-sections, Lee-type bounds apply in the one-sided exit case. An appendix extends these results to natural turnover and treated-control interference.