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Efficient estimation of average treatment effects using the estimated propensity score. (English) Zbl 1152.62328
Summary: We are interested in estimating the average effect of a binary treatment on a scalar outcome. If assignment to the treatment is exogenous or 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. {\it P. R. Rosenbaum} and {\it D. B. Rubin} [Biometrika 70, 41--55 (1983; Zbl 0522.62091)] show that adjusting solely for differences between treated and control units in the propensity score removes all biases associated with differences in covariates. Although adjusting for differences in the propensity score removes all the bias, this can come at the expense of efficiency, as shown by {\it J. Hahn} [Econometrica 66, No. 2, 315--331 (1998; Zbl 1055.62572)], {\it J. Heckman, H. Ichimura} and {\it P. Todd} [Matching as an econometric evaluation estimator. Rev. Econ. Stud. 65, 261--294 (1998)], and {\it J. Robins, S. Mark} and {\it W. Newey} [Estimating exposure effects by modeling the expectation of exposures conditional on confounders. Biometrics, 48, 479--495 (1992)]. We show that weighting by the inverse of a nonparametric estimate of the propensity score, rather than the true propensity score, leads to an efficient estimate 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.

##### MSC:
 62G05 Nonparametric estimation 91B82 Statistical methods in economics; economic indices and measures
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