Placebo inference on treatment effects when the number of clusters is small

Abstract

I introduce a general, Fisher-style randomization testing framework to conduct nearly exact inference about the lack of effect of a binary treatment in the presence of very few, large clusters when the treatment effect is identified across clusters. The proposed randomization test formalizes and extends the intuitive notion of generating null distributions by assigning placebo treatments to untreated clusters. I show that under simple and easily verifiable conditions, the placebo test leads to asymptotically valid inference in a very large class of empirically relevant models. Examples discussed explicitly are (i) least squares regression with cluster-level treatment, (ii) difference-in-differences estimation, and (iii) binary choice models with cluster-level treatment. A simulation study and an empirical example are provided. The proposed inference procedure is easy to implement and performs well with as few as three treated and three untreated clusters.