NONPARAMETRIC ESTIMATION OFAVERAGE TREATMENT EFFECTS UNDER EXOGENEITY: AREVIEW* Guido W.Imbens Abstract—Recently there has been asurge in econometric work focusing edness and exogeneity interchangeably to denote the as- on estimating average treatment effects under various sets of assumptions. One strand of this literature has developed methods for estimating average sumption that the receipt of treatmentis independent of the treatment effects for abinary treatment under assumptions variously potential outcomes with and without treatmentif certain described as exogeneity, unconfoundedness, or selection on observables. observable covariates areheld constant. The implication of The implication of these assumptions is that systematic (for example, average or distributional) differences in outcomes between treated and these assumptions is that systematic (forexample, average control units with the same values for the covariates are attributable to the or distributional) differences in outcomes between treated treatment. Recent analysis has considered estimation and inference for and control units with the samevalues forthese covariates average treatment effects under weaker assumptions than typical of the earlier literature by avoiding distributional and functional-form assump- areattributable to the treatment. tions. Various methods of semiparametric estimation have been proposed, Much of the recent work, building on the statistical including estimating the unknown regression functions, matching, meth- literatureby Cochran (1968), Cochran and Rubin (1973), ods using the propensity score such as weighting and blocking, and combinations of these approaches. In this paper Ireview the state of this Rubin (1973a, 1973b, 1977, 1978), Rosenbaum and Rubin literature and discuss some of its unanswered questions, focusing in (1983a, 1983b, 1984), Holland (1986), and others, considers particular on the practical implementation of these methods, the plausi- estimation and inference without distributional and func- bility of this exogeneity assumption in economic applications, the relative performance of the various semiparametric estimators when the key tional formassumptions. Hahn (1998) derived efciency assumptions (unconfoundedness and overlap) are satised, alternative bounds assuming only unconfoundedness and some regu- estimands such as quantile treatment effects, and alternate methods such as Bayesian inference. larity conditions and proposed an efcient estimator. Vari- ous alternative estimators have been proposed given these I.Introduction conditions. These estimation methods can be grouped into ve categories: (i)methods based on estimating the un- INCEthe work by Ashenfelter (1978), Card and Sulli- known regression functions of the outcome on the covari- Svan (1988), Heckman and Robb (1984), Lalonde ates (Hahn, 1998; Heckman, Ichimura, &Todd, 1997, 1998; (1986), and others, there has been much interest in econo- Imbens, Newey,&Ridder, 2003), (ii)matching on covari- metricmethods forestimating the effectsof active labor ates (Rosenbaum, 1995; Abadie and Imbens, 2002) (iii) marketprograms such asjob search assistance or classroom methods based on the propensity score, including blocking teaching programs. This interest has led to asurge in (Rosenbaum &Rubin, 1984) and weighting (Hirano, Im- theoretical work focusing on estimating average treatment bens, &Ridder, 2003), (iv)combinations of these ap- effectsunder various sets of assumptions. Seefor general proaches, forexample, weighting and regression (Robins & surveys of this literatureAngrist and Krueger (2000), Heck- Rotnizky,1995) or matching and regression (Abadie & man, LaLonde, and Smith (2000), and Blundell and Costa- Imbens, 2002), and (v)Bayesian methods, which have Dias (2002). found relatively little following since Rubin (1978). Inthis One strand of this literaturehas developed methods for paper Iwill review the state of this literature—with partic- estimating the average effectof receiving or not receiving a ular emphasis on implications forempirical work— and binary treatmentunder the assumption that the treatment discuss some of the remaining questions. satises some formof exogeneity.Differentversions of this The organization of the paper is as follows. In section II assumption arereferred to as unconfoundedness (Rosen- Iwill introduce the notation and the assumptions used for baum &Rubin, 1983a), selection on observables (Barnow, identication. Iwill also discuss the differencebetween Cain, &Goldberger, 1980; Fitzgerald, Gottschalk, &Mof- population- and sample-average treatmenteffects. The re- tt, 1998), or conditional independence (Lechner, 1999). In the remainderof this paper Iwill use the termsunconfound- cent econometric literaturehas largely focused on estima- tion of the population-average treatmenteffect and its coun- terpartfor the subpopulation of treated units. An alternative, Received for publication October 22, 2002. Revision accepted for publication June 4, 2003. following the early experimental literature(Fisher, 1925; *University of California at Berkeley and NBER Neyman, 1923), is to consider estimation of the average This paper was presented as an invited lecture at the Australian and European meetings of the Econometric Society in July and August 2003. effectof the treatmentfor the units in the sample. Many of Iam also grateful to Joshua Angrist, Jane Herr, Caroline Hoxby,Charles the estimators proposed can be interpreted as estimating Manski, Xiangyi Meng, Robert Moftt, and Barbara Sianesi, and two either the average treatmenteffect for the sample at hand, or referees for comments, and to anumber of collaborators, Alberto Abadie, Joshua Angrist, Susan Athey,Gary Chamberlain, Keisuke Hirano, V. the average treatmenteffect for the population. Although the Joseph Hotz, Charles Manski, Oscar Mitnik, Julie Mortimer, Jack Porter, choice of estimand may not affectthe formof the estimator, Whitney Newey,Geert Ridder, Paul Rosenbaum, and Donald Rubin for it has implications forthe efciency bounds and forthe many discussions on the topics of this paper. Financial support for this research was generously provided through NSFgrants SBR9818644 and formof estimators of the asymptotic variance; the variance SES0136789 and the Giannini Foundation. of estimators forthe sample average treatmenteffect are TheReview ofEconomicsand Statistics, February2004, 86(1): 4– 29 © 2004by the President and Fellows of HarvardCollege and the Massachusetts Instituteof Technology AVERAGE TREATMENTEFFECTS 5 generally smaller.In section II,I will also discuss alterna- createdeither to ful ll the unconfoundedness assumption or tive estimands. Almost the entire literaturehas focused on to failit aknown way —designed to compare the applica- average effects.However, in many cases such measures bility of the various treatmenteffect estimators in these may mask important distributional changes. These can be diverse settings. captured moreeasily by focusing on quantiles of the distri- This survey will not address alternatives forestimating butions of potential outcomes, in the presence and absence average treatmenteffects that do not rely on exogeneity of the treatment(Lehman, 1974; Docksum, 1974; Firpo, assumptions. This includes approaches whereselected ob- 2003). served covariates arenot adjusted for, such as instrumental In section III,I will discuss in moredetail some of the variables analyses (Bjo¨rklund & Mof t, 1987; Heckman & recently proposed semiparametricestimators forthe average Robb, 1984; Imbens &Angrist, 1994; Angrist, Imbens, & treatmenteffect, including those based on regression, Rubin, 1996; Ichimura& Taber, 2000; Abadie, 2003a; matching, and the propensity score. Iwill focus particularly Chernozhukov &Hansen, 2001). Iwill also not discuss on implementation, and compare the differentdecisions methods exploiting the presence of additional data, such as facedregarding smoothing parametersusing the various differencein differences in repeated cross sections (Abadie, estimators. 2003b; Blundell etal., 2002; Athey and Imbens, 2002) and Insection IV,Iwill discuss estimation of the variances of regression discontinuity wherethe overlap assumption is these average treatmenteffect estimators. For most of the violated (van der Klaauw, 2002; Hahn, Todd, &van der estimators introduced in the recentliterature, corresponding Klaauw, 2000; Angrist &Lavy,1999; Black, 1999; Lee, estimators forthe variance have also been proposed, typi- 2001; Porter, 2003). Iwill also limitthe discussion to binary cally requiring additional nonparametric regression. Inprac- treatments, excluding models with static multivalued treat- tice, however, researchersoften rely on bootstrapping, al- ments as inImbens (2000) and Lechner (2001) and models though this method has not been formallyjusti ed. In with dynamic treatmentregimes as in Hamand LaLonde addition, ifone is interested in the average treatmenteffect (1996), Gill and Robins (2001), and Abbring and van den forthe sample, bootstrapping isclearly inappropriate. Here Berg (2003). Reviews of many of these methods can be Idiscuss in moredetail asimple estimator forthe variance found in Shadish, Campbell, and Cook (2002), Angrist and formatching estimators, developed by Abadie and Imbens Krueger (2000), Heckman, LaLonde, and Smith (2000), and Blundell and Costa-Dias (2002). (2002), that does not require additional nonparametric esti- mation. Section Vdiscusses differentapproaches to assessing the II.Estimands, Identi cation, and Ef ciency Bounds plausibility