Randomization Tests in Observational Studies with Staggered Adoption of Treatment

TL;DR

This paper develops Fisher-style randomization tests for observational studies with staggered treatment adoption, assuming a Cox model for adoption timing, and demonstrates their validity and robustness through theory, simulations, and an empirical example.

Contribution

It introduces a new inference method for staggered treatment adoption based on Cox models, ensuring valid tests even with model misspecification.

Findings

Feasible tests maintain correct size in finite samples.

The methods are robust to certain model misspecifications.

Application to tobacco legislation data demonstrates practical utility.

Abstract

This paper considers the problem of inference in observational studies with time-varying adoption of treatment. In addition to an unconfoundedness assumption that the potential outcomes are independent of the times at which units adopt treatment conditional on the units' observed characteristics, our analysis assumes that the time at which each unit adopts treatment follows a Cox proportional hazards model. This assumption permits the time at which each unit adopts treatment to depend on the observed characteristics of the unit, but imposes the restriction that the probability of multiple units adopting treatment at the same time is zero. In this context, we study Fisher-style randomization tests of a null hypothesis that specifies that there is no treatment effect for all units and all time periods in a distributional sense. We first show that an infeasible test that treats the parameters of the Cox model as known has rejection probability no greater than the nominal level in finite samples. We then establish that the feasible test that replaces these parameters with consistent estimators has limiting rejection probability no greater than the nominal level. In a simulation study, we examine the practical relevance of our theoretical results, including robustness to misspecification of the model for the time at which each unit adopts treatment. Finally, we provide an empirical application of our methodology using the synthetic control-based test statistic and tobacco legislation data found in Abadie et. al. (2010).