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...eries, as parametric estimators, and calculate the variance this way. Under some conditions that will lead to valid standard errors and con dence intervals. A third approach is to use bootstrapping (=-=Efron and Tibshirani, 1993-=-; Horowitz, 2002). There is little formal evidence speci c for these estimators, but, given that the estimators are asymptotically linear, it is likely that bootstrapping will lead to valid standard ...

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...undedness or selection on observables, ranging from simple least squares analyses to matching on the propensity score (for example, Ashenfelter and Card, 1985; LaLonde, 1986; Card and Sullivan, 1988; =-=Heckman, Ichimura, and Todd, 1997-=-; Angrist, 1998; Dehejia and Wahba, 1999; Lechner, 1998; Friedlander and Robins, 1995; and many others). Here I focus primarily on two sets of analyses that can help researchers assess the value of th...

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...of ve blocks removes most of the bias associated with all the covariates. This has often been the starting point of empirical analyses using this estimator (for example, Rosenbaum and Rubin, 1983b; =-=Dehejia and Wahba, 1999-=-) and has been implemented in Stata by Becker and Ichino (2002).7 Often, however, researchers subsequently check the balance of the covariates within each block. If the true propensity score per block...

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...fects that do not rely on exogeneity assumptions. This includes approaches where selected observed covariates are not adjusted for, such as instrumental variables analyses (Björklund & Mof t, 1987; =-=Heckman & Robb, 1984-=-; Imbens & Angrist, 1994; Angrist, Imbens, & Rubin, 1996; Ichimura & Taber, 2000; Abadie, 2003a; Chernozhukov & Hansen, 2001). I will also not discuss methods exploiting the presence of additional dat...

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... many studies using some form of unconfoundedness or selection on observables, ranging from simple least squares analyses to matching on the propensity score (for example, Ashenfelter and Card, 1985; =-=LaLonde, 1986-=-; Card and Sullivan, 1988; Heckman, Ichimura, and Todd, 1997; Angrist, 1998; Dehejia and Wahba, 1999; Lechner, 1998; Friedlander and Robins, 1995; and many others). Here I focus primarily on two sets ...

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...; Blundell et al., 2002; Athey and Imbens, 2002) and regression discontinuity where the overlap assumption is violated (van der Klaauw, 2002; Hahn, Todd, & van der Klaauw, 2000; Angrist & Lavy, 1999; =-=Black, 1999-=-; Lee, 2001; Porter, 2003). I will also limit the discussion to binary treatments, excluding models with static multivalued treatments as in Imbens (2000) and Lechner (2001) and models with dynamic tr...

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.... VI. Applications There are many studies using some form of unconfoundedness or selection on observables, ranging from simple least squares analyses to matching on the propensity score (for example, =-=Ashenfelter and Card, 1985-=-; LaLonde, 1986; Card and Sullivan, 1988; Heckman, Ichimura, and Todd, 1997; Angrist, 1998; Dehejia and Wahba, 1999; Lechner, 1998; Friedlander and Robins, 1995; and many others). Here I focus primari...

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... where both treated and control units are (potentially) matched and matching is done with replacement (as in Dehejia & Wahba, 1999). The AbadieImbens estimator is implemented in Matlab and Stata (see =-=Abadie et al., 2003-=-).5 Formally, given a sample, {(Yi, X i, W i)} i51 N , let ,m(i) be the index l that satis es Wl Þ W i and O juWjÞWi 1$iX j 2 X ii # iX l 2 X ii% 5 m, where 1{ z } is the indicator function, equal to...

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...b). One may also wish to assess the responsiveness of the results to this assumption using a sensitivity analysis (Rosenbaum & Rubin, 1983b; Imbens, 2003), or, in its extreme form, a bounds analysis (=-=Manski, 1990-=-, 2003). The second assumption is that there exists appropriate overlap in the covariate distributions of the treated and control units. That is effectively an assumption on the joint distribution of ...

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...es, ranging from simple least squares analyses to matching on the propensity score (for example, Ashenfelter and Card, 1985; LaLonde, 1986; Card and Sullivan, 1988; Heckman, Ichimura, and Todd, 1997; =-=Angrist, 1998-=-; Dehejia and Wahba, 1999; Lechner, 1998; Friedlander and Robins, 1995; and many others). Here I focus primarily on two sets of analyses that can help researchers assess the value of the methods surve...

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...tion are referred to as unconfoundedness (Rosenbaum & Rubin, 1983a), selection on observables (Barnow, Cain, & Goldberger, 1980; Fitzgerald, Gottschalk, & Mof tt, 1998), or conditional independence (=-=Lechner, 1999-=-). In the remainder of this paper I will use the terms unconfoundedness and exogeneity interchangeably to denote the assumption that the receipt of treatment is independent of the potential outcomes w...

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...ing the unknown regression functions of the outcome on the covariates (Hahn, 1998; Heckman, Ichimura, & Todd, 1997, 1998; Imbens, Newey, & Ridder, 2003), (ii) matching on covariates (Rosenbaum, 1995; =-=Abadie and Imbens, 2002-=-) (iii) methods based on the propensity score, including blocking (Rosenbaum & Rubin, 1984) and weighting (Hirano, Imbens, & Ridder, 2003), (iv) combinations of these approaches, for example, weightin...

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...sing some form of unconfoundedness or selection on observables, ranging from simple least squares analyses to matching on the propensity score (for example, Ashenfelter and Card, 1985; LaLonde, 1986; =-=Card and Sullivan, 1988-=-; Heckman, Ichimura, and Todd, 1997; Angrist, 1998; Dehejia and Wahba, 1999; Lechner, 1998; Friedlander and Robins, 1995; and many others). Here I focus primarily on two sets of analyses that can help...

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...er the assumption that the treatment satis es some form of exogeneity. Different versions of this assumption are referred to as unconfoundedness (Rosenbaum & Rubin, 1983a), selection on observables (=-=Barnow, Cain, & Goldberger, 1980-=-; Fitzgerald, Gottschalk, & Mof tt, 1998), or conditional independence (Lechner, 1999). In the remainder of this paper I will use the terms unconfoundedness and exogeneity interchangeably to denote t...

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...ng bias is typically ignored in these studies. The literature has studied fast algorithms for matching the units, as fully ef cient matching methods are computationally cumbersome (see, for example, =-=Gu and Rosenbaum, 1993-=-; Rosenbaum, 1995). Note that in such matching schemes the order in which the units are matched may be important. Abadie and Imbens (2002) study both bias and variance in a more general setting where ...

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...ariates. The following result shows that all biases due to observable covariates can be removed by conditioning solely on the propensity score: Lemma 2.1 (Unconfoundedness Given the Propensity Score: =-=Rosenbaum and Rubin, 1983-=-a): Suppose that assumption 2.1 holds. Then ~Y~0!, Y~1!! \ W ue~X!. Proof: We will show that Pr(W 5 1uY(0), Y(1), e(X)) 5 Pr(W 5 1ue(X)) 5 e(X), implying independence of (Y(0), Y(1)) and W conditional...

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...ore AVERAGE TREATMENT EFFECTS 11 and the regression functions, can lead to an estimator that is consistent even if only one of the models is correctly speci ed (“doubly robust” in the terminology of =-=Robins & Ritov, 1997-=-). Finally, in the fth group I will discuss Bayesian approaches to inference for average treatment effects. Only some of the estimators discussed below achieve the semiparametric ef ciency bound, y...

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...erlap in the covariate distributions. The rst of these assumptions is in principle untestable. Nevertheless a number of approaches have been proposed that are useful for addressing its credibility (=-=Heckman and Hotz, 1989-=-; Rosenbaum, 1984b). One may also wish to assess the responsiveness of the results to this assumption using a sensitivity analysis (Rosenbaum & Rubin, 1983b; Imbens, 2003), or, in its extreme form, a ...

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...easures may mask important distributional changes. These can be captured more easily by focusing on quantiles of the distributions of potential outcomes, in the presence and absence of the treatment (=-=Lehman, 1974-=-; Docksum, 1974; Firpo, 2003). In section III, I will discuss in more detail some of the recently proposed semiparametric estimators for the average treatment effect, including those based on regressi...

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...s of smoothing parameters, and may explain some of its popularity. Matching estimators have been widely studied in practice and theory (for example, Gu & Rosenbaum, 1993; Rosenbaum, 1989, 1995, 2002; =-=Rubin, 1973-=-b, 1979; Heckman, Ichimura, & Todd, 1998; Dehejia & Wahba, 1999; Abadie & Imbens, 2002). Most often they have been applied in settings with the following two characteristics: (i) the interest is in th...

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...; Abadie, 2003a; Chernozhukov & Hansen, 2001). I will also not discuss methods exploiting the presence of additional data, such as difference in differences in repeated cross sections (Abadie, 2003b; =-=Blundell et al., 2002-=-; Athey and Imbens, 2002) and regression discontinuity where the overlap assumption is violated (van der Klaauw, 2002; Hahn, Todd, & van der Klaauw, 2000; Angrist & Lavy, 1999; Black, 1999; Lee, 2001;...

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...ections (Abadie, 2003b; Blundell et al., 2002; Athey and Imbens, 2002) and regression discontinuity where the overlap assumption is violated (van der Klaauw, 2002; Hahn, Todd, & van der Klaauw, 2000; =-=Angrist & Lavy, 1999-=-; Black, 1999; Lee, 2001; Porter, 2003). I will also limit the discussion to binary treatments, excluding models with static multivalued treatments as in Imbens (2000) and Lechner (2001) and models wi...

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...ing on the propensity score (for example, Ashenfelter and Card, 1985; LaLonde, 1986; Card and Sullivan, 1988; Heckman, Ichimura, and Todd, 1997; Angrist, 1998; Dehejia and Wahba, 1999; Lechner, 1998; =-=Friedlander and Robins, 1995-=-; and many others). Here I focus primarily on two sets of analyses that can help researchers assess the value of the methods surveyed in this paper: rst, studies attempting to assess the plausibilit...

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... on exogeneity assumptions. This includes approaches where selected observed covariates are not adjusted for, such as instrumental variables analyses (Björklund & Mof t, 1987; Heckman & Robb, 1984; =-=Imbens & Angrist, 1994-=-; Angrist, Imbens, & Rubin, 1996; Ichimura & Taber, 2000; Abadie, 2003a; Chernozhukov & Hansen, 2001). I will also not discuss methods exploiting the presence of additional data, such as difference in...

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... Athey and Imbens, 2002) and regression discontinuity where the overlap assumption is violated (van der Klaauw, 2002; Hahn, Todd, & van der Klaauw, 2000; Angrist & Lavy, 1999; Black, 1999; Lee, 2001; =-=Porter, 2003-=-). I will also limit the discussion to binary treatments, excluding models with static multivalued treatments as in Imbens (2000) and Lechner (2001) and models with dynamic treatment regimes as in Ham...

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...X], E[Y(1 2 W)uX], and E[W uX]) to estimate mw( x). Instead one can use series methods to directly estimate the two regression functions mw( x), eliminating the need to estimate the propensity score (=-=Imbens, Newey, and Ridder, 2003-=-). Heckman, Ichimura, and Todd (1997, 1998) and Heckman, Ichimura, Smith, and Todd (1998) consider kernel methods for estimating mw( x), in particular focusing on local linear approaches. The simple k...

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...ns. This includes approaches where selected observed covariates are not adjusted for, such as instrumental variables analyses (Björklund & Mof t, 1987; Heckman & Robb, 1984; Imbens & Angrist, 1994; =-=Angrist, Imbens, & Rubin, 1996-=-; Ichimura & Taber, 2000; Abadie, 2003a; Chernozhukov & Hansen, 2001). I will also not discuss methods exploiting the presence of additional data, such as difference in differences in repeated cross s...

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...he weight is proportional to the density of the covariates, whereas for the PATT the weight is proportional to the product of the density of the covariates and the propensity score (see, for example, =-=Hirano, Imbens, and Ridder, 2003-=-). Knowledge of the propensity score implies one does not need to estimate the weight function and thus improves precision. F. Ef ciency Bounds and Asymptotic Variances for Conditional and Sample Ave...

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... been proposed given these conditions. These estimation methods can be grouped into ve categories: (i) methods based on estimating the unknown regression functions of the outcome on the covariates (=-=Hahn, 1998-=-; Heckman, Ichimura, & Todd, 1997, 1998; Imbens, Newey, & Ridder, 2003), (ii) matching on covariates (Rosenbaum, 1995; Abadie and Imbens, 2002) (iii) methods based on the propensity score, including b...

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...re selected observed covariates are not adjusted for, such as instrumental variables analyses (Björklund & Mof t, 1987; Heckman & Robb, 1984; Imbens & Angrist, 1994; Angrist, Imbens, & Rubin, 1996; =-=Ichimura & Taber, 2000-=-; Abadie, 2003a; Chernozhukov & Hansen, 2001). I will also not discuss methods exploiting the presence of additional data, such as difference in differences in repeated cross sections (Abadie, 2003b; ...

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...score methods. Bootstrapping may be more complicated for matching estimators, as the process introduces discreteness in the distribution that will lead to ties in the matching algorithm. Subsampling (=-=Politis and Romano, 1999-=-) will still work in this setting. These rst three methods provide variance estimates for estimators of tP. As argued above, however, one may instead wish to estimate tS or t(X), in which case the a...

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...r of new estimators at her disposal. Challenges remain in making the new tools more easily applicable. Although software is available to implement some of the estimators (see Becker and Ichino, 2002; =-=Sianesi, 2001-=-; Abadie et al., 2003), many remain dif cult to apply. A particularly urgent task is therefore to provide fully implementable versions of the various estimators that do not require the applied resear...

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...zhukov & Hansen, 2001). I will also not discuss methods exploiting the presence of additional data, such as difference in differences in repeated cross sections (Abadie, 2003b; Blundell et al., 2002; =-=Athey and Imbens, 2002-=-) and regression discontinuity where the overlap assumption is violated (van der Klaauw, 2002; Hahn, Todd, & van der Klaauw, 2000; Angrist & Lavy, 1999; Black, 1999; Lee, 2001; Porter, 2003). I will a...

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... adjusted for, such as instrumental variables analyses (Björklund & Mof t, 1987; Heckman & Robb, 1984; Imbens & Angrist, 1994; Angrist, Imbens, & Rubin, 1996; Ichimura & Taber, 2000; Abadie, 2003a; =-=Chernozhukov & Hansen, 2001-=-). I will also not discuss methods exploiting the presence of additional data, such as difference in differences in repeated cross sections (Abadie, 2003b; Blundell et al., 2002; Athey and Imbens, 200...

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...ors, and calculate the variance this way. Under some conditions that will lead to valid standard errors and con dence intervals. A third approach is to use bootstrapping (Efron and Tibshirani, 1993; =-=Horowitz, 2002-=-). There is little formal evidence speci c for these estimators, but, given that the estimators are asymptotically linear, it is likely that bootstrapping will lead to valid standard errors and con ...