Doubly Robust Instrumented Difference-in-Differences

Abstract

We study estimation of the local average treatment effect on the treated ($LATT$) in instrumented difference-in-differences (IDiD) designs with covariates and staggered instrument exposure. We derive the efficient influence function (EIF) of the target parameter in both panel and repeated cross-sections settings, allowing for two classes of control groups: never-exposed and not-yet-exposed. Building on the EIF, we construct doubly robust estimands and corresponding estimators from first principles. The resulting procedures are the IDiD analogues of the difference-in-differences (DiD) procedures in Callaway and Sant'Anna (2021), targeting $LATT$ rather than $ATT$. We further establish a Bloom-type result under one-sided compliance and absorbing treatment, linking $LATT$ to a convex combination of exposure-cohort-specific $ATT(g, t)$ parameters, making the connection between IDiD and DiD explicit. Asymptotic properties are established under conditions on the remainder term and either Donsker conditions or via cross-fitting. We also construct double machine learning (DML) estimators for the $LATT$ in both data settings and show their equivalence to cross-fitted estimators. Simulations assess the double robustness and finite-sample performance of the proposed methods. An implementation is available in the Python package \texttt{idid}.