## Efficient Estimation of Average Treatment Effects Using the Estimated Propensity Score (2000)

Citations: | 247 - 28 self |

### BibTeX

@MISC{Hirano00efficientestimation,

author = {Keisuke Hirano and Guido W. Imbens and Keisuke Hirano Ucla and Guido W. Imbens Ucla and Geert Ridder and We Thank Gary Chamberlain and Jinyong Hahn and Donald Rubin and Seminar Participants},

title = {Efficient Estimation of Average Treatment Effects Using the Estimated Propensity Score},

year = {2000}

}

### Years of Citing Articles

### OpenURL

### Abstract

We are interested in estimating the average e#ect 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 di#erences in the pre-treatmentvariables. Rosenbaum and Rubin #1983, 1984# show that adjusting solely for di#erences between treated and control units in a scalar function of the pre-treatment variables, the propensity score, also removes the entire bias associated with di#erences in pre-treatment variables. Thus it is possible to obtain unbiased estimates of the treatment e#ect without conditioning on a possibly highdimensional vector of pre-treatment variables. Although adjusting for the propensity score removes all the bias, this can come at the expense of e#ciency. We show that weighting with the inverse of a nonparametric estimate of the propensity score, rather than the true propensity scor...

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Citation Context ... of the selection probability or propensity score on the covariates. An alternative interpretation of the estimated-weights estimator is based on a Generalized Method of Moments (GMM) representation (=-=Hansen, 1982-=-). Under the assumption that the selection probability is p(x) = 1/2, we can estimate β0 using the single moment restriction E[ψ1(Y, X, T, β0)] = 0, with ψ1(y, t, x, β) = y · t y · t − β = − β. p(x) 1... |

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Citation Context ...ation research is that for unit i we observe either Yi(0) or Yi(1), but never both. To solve the identification problem, we maintain throughout the paper the unconfoundedness assumption (Rubin, 1978; =-=Rosenbaum and Rubin, 1983-=-), also known as the selection–on–observables assumption (Barnow, Cain, and Goldberger, 1980), which asserts that conditional on the observed covariates, the treatment indicator is independent of the ... |

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Citation Context ...te) 2 + g(X)2 p ∗ (X) σ2 1(X) + g(X)2 1 − p ∗ (X) σ2 0(X) 4.3 Estimating the Average Treatment Effect for the Treated Under unconfoundedness the average treatment effect for the treated (Rubin, 1977; =-=Heckman and Robb, 1985-=-, Heckman, Ichimura and Todd, 1997, 1998) is a special case of the weighted average treatment effect, corresponding to the weighting function g(x) = p ∗ (x). To see this first note that under unconfou... |

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Citation Context ... using the moments ψ1(·) and ψ2(·) leads to a fully efficient estimator. 8 Here it is of particular interest to consider the empirical likelihood estimator (e.g., Qin and Lawless, 1994; Imbens, 1997; =-=Kitamura and Stutzer, 1997-=-; Imbens, Spady and Johnson, 1998), Empirical likelihood estimation is based on maximization, both over a nuisance parameter π = (π1, . . . , πN) and over the parameter of interest β, of the logarithm... |

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Citation Context ...wever, because p ∗ (x) is a probability such an approach has the unattractive feature that it approximates a probability by a linear function. We therefore estimate p ∗ (x) in a sieve approach (e.g., =-=Geman and Hwang, 1982-=-) by the Series Logit Estimator (SLE). For K = 1, 2, . . . , let R K (x) = (r1K(x), r2K(x) . . . , rKK(x)) ′ be a K−vector of functions. Although the theory is derived for general sequences of approxi... |

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Citation Context ...ents framework using the moments ψ1(·) and ψ2(·) leads to a fully efficient estimator. 8 Here it is of particular interest to consider the empirical likelihood estimator (e.g., Qin and Lawless, 1994; =-=Imbens, 1997-=-; Kitamura and Stutzer, 1997; Imbens, Spady and Johnson, 1998), Empirical likelihood estimation is based on maximization, both over a nuisance parameter π = (π1, . . . , πN) and over the parameter of ... |

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Citation Context ...istic argument for efficiency of estimated weights, deferring formal results to Section 4. The analog to the unconfoundedness assumption here is the assumption that the Yi are Missing At Random (MAR, =-=Rubin, 1976-=-), or � � T ⊥ Y � � X. The role of the propensity score is played here by the selection probability: p(x) = E[T |X = x] = P r(T = 1|X). First, we restrict our attention in this section to the case wit... |

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