  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.
