Doubly Robust Instrumented Difference-in-Differences

TL;DR

This paper develops doubly robust estimators for local average treatment effects in instrumented difference-in-differences designs, extending DiD methods to handle covariates, staggered exposure, and different data settings.

Contribution

It derives the efficient influence function for the target parameter, constructs doubly robust estimands, and introduces double machine learning estimators for IDiD designs.

Findings

The proposed estimators are doubly robust and asymptotically normal.

Simulations show good finite-sample performance and robustness.

Implementation available in the Python package idid.

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}.