Triple Instrumented Difference-in-Differences

In this paper, we formalize a triple instrumented difference-in-differences (DID-IV). In this design, a triple Wald-DID estimand, which divides the difference-in-difference-in-differences (DDD) estimand of the outcome by the DDD estimand of the treatment, captures the local average treatment effect on the treated. The identifying assumptions mainly comprise a monotonicity assumption, and the common acceleration assumptions in the treatment and the outcome. We extend the canonical triple DID-IV design to staggered instrument cases. We also describe the estimation and inference in this design in practice.