Instrumented Difference-in-Differences with Heterogeneous Treatment Effects

Many studies exploit variation in the timing of policy adoption across units as an instrument for treatment. This paper formalizes the underlying identification strategy as an instrumented difference-in-differences (DID-IV). In this design, a Wald-DID estimand, which scales the DID estimand of the outcome by the DID estimand of the treatment, captures the local average treatment effect on the treated (LATET). In contrast to Fuzzy DID design considered in de Chaisemartin and D'Haultfoeuille (2018), our DID-IV design does not ex ante require strong restrictions on treatment adoption behavior across units. Additionally, our target parameter, the LATET, is policy-relevant if the instrument is based on the policy change of interest to the researcher. We extend the canonical DID-IV design to multiple period settings with the staggered adoption of the instrument across units, calling it a staggered DID-IV design, and propose an estimation method that is robust to treatment effect heterogeneity. We illustrate our findings in the setting of Oreopoulos (2006), estimating returns to schooling in the United Kingdom. In this application, the two-way fixed effects instrumental variable regression, which is the conventional approach to implement a staggered DID-IV design, yields a negative estimate, whereas our estimation method indicates a substantial gain from schooling.