Randomization Tests in Observational Studies with Staggered Adoption of Treatment
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This paper studies inference in observational studies with time-varying adoption of treatment. The main restriction underlying our analysis is 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 observed characteristics, but restricts the probability of multiple units adopting treatment simultaneously to be zero. In this context, we study randomization tests of a "sharp" null hypothesis that there is no treatment effect for all units and all time periods. 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. We then establish that the feasible test that replaces these parameters with consistent estimators has limiting rejection probability no greater than the nominal level. These tests rely upon an implication of the Cox model providing a parametric expression for the probability that a particular unit is the first to adopt treatment conditional on both the observed characteristics and the time of first treatment. We provide an empirical application of our methodology using the synthetic control-based test statistic and tobacco legislation data in Abadie et al. (2010).
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