Semiparametric Estimation of Average Treatment Effects underExogeneity: A Review∗Guido W. ImbensUC Berkeley, and NBERFirst Draft: July 2002This Draft: May 2003AbstractRecently there has been a surge in econometric work focusing on estimating averagetreatment effects under various sets of assumptions. One strand of this literature has de-veloped methods for estimating average treatment effects for a binary treatment under as-sumptions of exogeneity or unconfoundedness, also known as selection on observables. Thisassumption requires that given a set of covariates systematic differences in outcomes betweentreated units and control units with the same values for these covariates are attributable tothe treatment. The recent econometric literature has considered estimation and inferenceunder weaker assumptions than typically considered in the earlier literature, in particularlyby avoiding distributional and functional form assumptions, and has established efficiencybounds. Various approaches to semiparametric estimation have been proposed, includingestimating the unknown regression functions, matching, methods using the propensity scoreincluding weighting and blocking, and combinations of these approaches.In this paper I will review the state of this literature and discuss some of the unansweredquestions, focusing in particular on the practical implementation of these methods, theplausibility of this exogeneity assumption in economic applications, the relative performanceof the various semiparametric estimators when the key assumptions (unconfoundednessand overlap) are satisfied, alternative estimands such as quantile treatment effects, andalternative methods such as Bayesian inference.JEL Classification: C14, C21, C52Keywords: Average Treatment Effects, Unconfoundedness, Semiparametric Methods, Match-ing, Propensity Score, Exogeneity∗Department of Economics, and Department of Agricultural and Resource Economics, University of Californiaat Berkeley, 661 Evans Hall #3880, Berkeley, CA 94720-3880. I am grateful to Xiangyi Meng and Caroline Hoxbyand two referees for comments and to a number of collaborators, Alberto Abadie, Joshua Angrist, Susan Athey,Gary Chamberlain, Keisuke Hirano, V. Joseph Hotz, Julie Mortimer, Jack Porter, Whitney Newey, Geert Ridder,Paul Rosenbaum, and Donald Rubin, for many discussions on the topics of this paper. Financial supp ort forthis research was generously provided through NSF grants SBR 9818644 and SES 0136789 and the GianniniFoundation. Electronic correspondence: [email protected], http://elsa.berkeley.edu/users/imbens/.1 IntroductionSince the work by Ashenfelter and Card (1985), Card and Sullivan (1988), Heckman and Robb(1984), Lalonde (1986) and others there has been much interest in econometric methods forestimating the effects of active labor market programs such as job search assistance or classroomteaching programs. This interest has led to a surge in theoretical work focusing on estimatingaverage treatment effects under various sets of assumptions. See for general surveys of thisliterature Angrist and Krueger (2000), Heckman, Lalonde and Smith (2000), and Blundell andCosta-Dias (2002).One strand of this literature has developed methods for estimating the average effect ofreceiving or not receiving a binary treatment under the assumption that the treatment satisfiessome form of exogeneity. Different forms of this assumption are referred to as unconfounded-ness (Rosenbaum and Rubin, 1983a), selection on observables (Barnow, Cain, and Goldberger,1980; Fitzgerald, Gottschalk, and Moffitt, 1998), or the conditional independence assumption(Lechner, 1998). In the remainder of this paper I will use the terms unconfoundedness and exo-geneity interchangeably. The implication of these assumptions is that systematic (e.g., averageor distributional) differences in outcomes between treated units and control units with the samevalues for these covariates are attributable to the treatment. Much of the recent literature hasbuilt on the work in the statistical literature by Rubin (1973ab, 1977, 1978), Rosenbaum andRubin (1983ab, 1984), Holland (1986) and others. The recent literature considered estimationand inference without distributional and functional form assumptions. Hahn (1998) derivedefficiency bounds assuming only unconfoundedness and some regularity conditions. Various es-timators have been proposed under these conditions. These include (i) estimating the unknownregression functions of the outcome on the covariates (Hahn, 1998; Heckman, Ichimura, andTodd, 1997; Heckman, Ichimura, Smith and Todd, 1998), (ii) matching on covariates (Rosen-baum, 1995; Abadie and Imbens, 2002) (iii) methods based on the prop ensity score includingblocking (Rosenbaum and Rubin, 1984) and weighting (Hirano, Imbens, and Ridder, 2001),and (iv) combinations of these approaches, for example weighting and regression (Robins andRotnizky, 1995) or matching and regression (Abadie and Imbens, 2002).In this paper I will review the state of this literature, with a particular emphasis on im-plications for empirical work. In addition I will discuss some of the questions that are stilloutstanding. The organization of the paper is as follows. In Section 2 I will set up the notationand the basic issues. Here I will also discuss the difference between population versus sampleaverage treatment effects. The recent econometric literature, in contrast to some of the originalexperimental literature (Fisher, 1925, Neyman, 1923), has largely focused on estimation of thepopulation average treatment effect and its counterpart for the subpopulation of treated units.An alternative is to consider estimation of the average effect of the treatment for the unitsin the sample. Many of the estimators proposed can be interpreted as estimating either theaverage treatment effect for the sample at hand or the average treatment effect for the popu-latin. The focus, on either population or sample average treatment effects, matters, however,for the asymptotic variance, with the variance of the estimators for the sample average treat-ment effect in general smaller. This persp ective also has implications for the effiency bounds[1]and for the form of estimators for the asymptotic variance. In this section I will also discussalternative estimands. Almost the entire literature has focused on average treatment effects. Inmany cases such measures of typical effects may mask important distributional changes.