Marginal Treatment Effects in Difference-in-Differences

Abstract

Difference-in-Differences (DiD) is a popular method used to evaluate the effect of a
treatment that exploits variation in treatment status that comes from the exposure to
a shock, usually in the form of a policy change. When there is imperfect compliance
towards the shock, the usual DiD estimand fails to recover relevant causal parameters.
This article presents an identification strategy in DiD settings with imperfect compliance that identifies Marginal treatment effects (MTE). We show how to combine and
modify standard instrumental variables (IV) and DiD assumptions to identify treatment effects in DiD settings where individuals enter into treatment with at least partial
knowledge of their unobservable gains. We propose two estimators for the MTE that
are consistent under different assumptions regarding the functional form of potential
outcomes and prove their asymptotically normality. Furthermore, we derive an estimator for the local average treatment effect (LATE) that is robust to misspecification of
the MTE model. We assert the desirable finite-sample properties through simulation
studies of a linear MTE model. Finally, we use our results to investigate heterogeneity
on the returns to primary education attendance in Indonesia.

Keywords: Marginal treatment effects, Heterogeneous effects, Difference-in-differences

JEL Classification: C01, C13, C21