Efficient Estimation of Average Treatment Effects Using the Estimated Propensity Score

49 Pages Posted: 16 May 2000 Last revised: 11 Mar 2023

See all articles by Keisuke Hirano

Keisuke Hirano

Pennsylvania State University, College of the Liberal Arts - Department of Economic

Guido W. Imbens

Stanford Graduate School of Business

Geert Ridder

University of Southern California

Multiple version iconThere are 2 versions of this paper

Date Written: March 2000

Abstract

We are interested in estimating the average effect of a binary treatment on a scalar outcome. If assignment to the treatment is independent of the potential outcomes given pretreatment variables, biases associated with simple treatment-control average comparisons can be removed by adjusting for differences in the pre-treatment variables. Rosenbaum and Rubin (1983, 1984) show that adjusting solely for differences between treated and control units in a scalar function of the pre-treatment, the propensity score, also removes the entire bias associated with differences in pre-treatment variables. Thus it is possible to obtain unbiased estimates of the treatment effect without conditioning on a possibly high-dimensional vector of pre-treatment variables. Although adjusting for the propensity score removes all the bias, this can come at the expense of efficiency. We show that weighting with the inverse of a nonparametric estimate of the propensity score, rather than the true propensity score, leads to efficient estimates of the various average treatment effects. This result holds whether the pre-treatment variables have discrete or continuous distributions. We provide intuition for this result in a number of ways. First we show that with discrete covariates, exact adjustment for the estimated propensity score is identical to adjustment for the pre-treatment variables. Second, we show that weighting by the inverse of the estimated propensity score can be interpreted as an empirical likelihood estimator that efficiently incorporates the information about the propensity score. Finally, we make a connection to results to other results on efficient estimation through weighting in the context of variable probability sampling.

Suggested Citation

Hirano, Keisuke and Imbens, Guido W. and Ridder, Geert, Efficient Estimation of Average Treatment Effects Using the Estimated Propensity Score (March 2000). NBER Working Paper No. t0251, Available at SSRN: https://ssrn.com/abstract=228061

Keisuke Hirano (Contact Author)

Pennsylvania State University, College of the Liberal Arts - Department of Economic ( email )

524 Kern Graduate Building
University Park, PA 16802-3306
United States

Guido W. Imbens

Stanford Graduate School of Business ( email )

655 Knight Way
Stanford, CA 94305-5015
United States

Geert Ridder

University of Southern California ( email )

Kaprielian Hall
Los Angeles, CA 90089
United States
213-740-2110 (Phone)
213-740-8543 (Fax)

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