Efficient Estimation Of Average Treatment Effects Using The Estimated Propensity Score

We are interested in estimating the average effect of a binary treatment on a scalar outcome. If assignment to the treatment is unconfounded, that is, independent of the potential outcomes given covariates, biases associated with simple treatment-control average comparisons can be removed by adjusting for differences in the covariates. Rosenbaum and Rubin (1983a) show that adjusting solely for differences between treated and control units in a scalar function of the covariates, the propensity score, also removes all biases associated with differences in covariates. Although adjusting for the propensity score removes all the bias, this can come at the expense of efficiency, as shown by Hahn (1998), Heckman, Ichimura, Todd (1998), and Rotnitzky and Robins (1995). We show that weighting by the inverse of a nonparametric estimate of the propensity score, rather than the true propensity score, leads to efficient estimates of the average treatment effect. We provide intuition for this result by showing that this estimator can be interpreted as an empirical likelihood estimator that efficiently incorporates the information about the propensity score.