Inference on Finite Population Treatment Effects Under Limited Overlap

This paper studies inference on finite population average and local average treatment effects under limited overlap, meaning some strata have a small proportion of treated or untreated units. We model limited overlap in an asymptotic framework sending the propensity score to zero (or one) with the sample size. We derive the asymptotic distribution of analog estimators of the treatment effects under two common randomization schemes: conditionally independent and stratified block randomization. Under either scheme, the limit distribution is the same and conventional standard error formulas remain asymptotically valid, but the rate of convergence is slower the faster the propensity score degenerates. The practical import of these results is twofold. When overlap is limited, standard methods can perform poorly in smaller samples, as asymptotic approximations are inadequate due to the slower rate of convergence. However, in larger samples, standard methods can work quite well even when the propensity score is small.