  This paper studies the identification, estimation, and inference of long-term
(binary) treatment effect parameters when balanced panel data is not available,
or consists of only a subset of the available data. We develop a new estimator:
the chained difference-in-differences, which leverages the overlapping
structure of many unbalanced panel data sets. This approach consists in
aggregating a collection of short-term treatment effects estimated on multiple
incomplete panels. Our estimator accommodates (1) multiple time periods, (2)
variation in treatment timing, (3) treatment effect heterogeneity, (4) general
missing data patterns, and (5) sample selection on observables. We establish
the asymptotic properties of the proposed estimator and discuss identification
and efficiency gains in comparison to existing methods. Finally, we illustrate
its relevance through (i) numerical simulations, and (ii) an application about
the effects of an innovation policy in France.
