Sensitivity of propensity score methods to the specifications

Propensity score matching estimators have two advantages. One is that they overcome the curse of dimensionality of covariate matching, and the other is that they are nonparametric. However, the propensity score is usually unknown and needs to be estimated. If we estimate it nonparametrically, we are incurring the curse-of-dimensionality problem we are trying to avoid. If we estimate it parametrically, how sensitive the estimated treatment effects are to the specifications of the propensity score becomes an important question. In this paper, we study this issue. First, we use a Monte Carlo experimental method to investigate the sensitivity issue under the unconfoundedness assumption. We find that the estimates are not sensitive to the specifications. Next, we provide some theoretical justifications, using the insight from Rosenbaum and Rubin (1983) that any score finer than the propensity score is a balancing score. Then, we reconcile our finding with the finding in Smith and Todd (2005) that, if the unconfoundedness assumption fails, the matching results can be sensitive. However, failure of the unconfoundedness assumption will not necessarily result in sensitive estimates. Matching estimators can be speciously robust in the sense that the treatment effects are consistently overestimated or underestimated. Sensitivity checks applied in empirical studies are helpful in eliminating sensitive cases, but in general, it cannot help to solve the fundamental problem that the matching assumptions are inherently untestable. Last, our results suggest that including irrelevant variables in the propensity score will not bias the results, but overspecifying it (e.g., adding unnecessary nonlinear terms) probably will.