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## Nonparametric estimation of average treatment effects under exogeneity: a review (2004)

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Venue: | REVIEW OF ECONOMICS AND STATISTICS |

Citations: | 593 - 23 self |

### Citations

4955 | An Introduction to the Bootstrap
- Efron, Tibshirani
- 1994
(Show Context)
Citation Context ...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 ... |

1180 | Experimental and quasi-experimental designs for generalized causal inference - Shadish, Cook, et al. - 2002 |

1056 |
Matching as an Econometric Evaluation Estimator: Evidence from Evaluating a Job Training Programme
- Heckman, Ichimura, et al.
- 1997
(Show Context)
Citation Context ...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... |

699 |
Causal Effects in Nonexperimental Studies: Reevaluating the Evaluation of Training Programs
- Dehejia, Wahba
- 1999
(Show Context)
Citation Context ...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... |

692 | Characterizing Selection Bias Using Experimental Data - Heckman, Ichimura, et al. - 1998 |

682 | The Design of Experiments - Fisher - 1935 |

590 |
Jr, Alternative methods for evaluating the impact of interventions: An overview, journal of Econometrics 30
- Heckman, Robb
- 1985
(Show Context)
Citation Context ...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... |

533 | Evaluating the Econometric Evaluations of Training Programs with Experimental Data
- LaLonde
- 1996
(Show Context)
Citation Context ... 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 ... |

483 | Do Better Schools Matter? Parental Valuation of Elementary Education
- Black
- 1999
(Show Context)
Citation Context ...; 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... |

333 | The role of the propensity score in estimating dose-response functions - Imbens - 2000 |

322 | On the Application of Probability Theory to Agricultural Experiments. Essay on Principles - Neyman - 1923 |

291 | Using the Longitudinal Structure of Earnings to Estimate the Effects of Training Programs
- Ashenfelter, Card
- 1985
(Show Context)
Citation Context .... 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... |

266 | Empirical Strategies in Labor Economics - Angrist, Krueger - 1999 |

266 | Estimation of Average Treatment Effects Based on Propensity Scores - Ichino, Becker - 2002 |

247 | Implementing Matching Estimators for Average Treatment Effects in Stata
- Abadie, Drukker, et al.
- 2004
(Show Context)
Citation Context ... 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... |

244 |
Nonparametric bounds on treatment effects
- Manski
- 1990
(Show Context)
Citation Context ...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 ... |

237 | Making The Most Out Of Programme Evaluations and Social Experiments: Accounting For Heterogeneity - Heckman, Smith, et al. - 1997 |

192 | An Analysis of Sample Attrition in Panel Data: The Michigan Panel Study of Income Dynamics - Fitzgerald, Gottschalk, et al. - 1998 |

167 | An Empirical Framework for Testing Theories about Complementarity in Organizational Design," NBER Working Paper No - Athey, Stern - 1998 |

165 | Estimating the Labor Market Impact of Voluntary Military Service Using Social Security Data on Military Applicants
- Angrist
- 1998
(Show Context)
Citation Context ...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... |

165 |
Earnings and Employment Effects of Continuous Off-theJob Training in East Germany after Unification
- Lechner
- 1999
(Show Context)
Citation Context ...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... |

164 | Analysis of semiparametric regression models for repeated outcomes in the presence of missing data - Robins - 1995 |

156 | Semiparametric Difference-in-Differences Estimators - Abadie - 2005 |

145 | The Effect of Sample Selection and Initial Conditions in Duration Models: Evidence from Experimental Data on - Ham, Lalonde - 1996 |

140 | Semiparametric Instrumental Variable Estimation of Treatment Response Models - Abadie - 2003 |

133 | Simple and Bias-Corrected Matching Estimators for Average Treatment Effects,” Technical Working Paper T0283
- Abadie, Imbens
- 2002
(Show Context)
Citation Context ...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... |

122 | Measuring the Effect of Subsidized Training Programs on Movements
- Card, Sullivan
- 1988
(Show Context)
Citation Context ...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... |

120 | Statistics and Causal Inference (with discussion - Holland - 1986 |

100 |
Issues in the analysis of selectivity bias
- BARNOW, CAIN, et al.
- 1980
(Show Context)
Citation Context ...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... |

97 | Alternative Estimates of the Effect of Family Structure during Adolescence on High School Graduation - Manski, Sandefur, et al. - 1992 |

84 |
Comparison of multivariate matching methods: Structures, distances and algorithms
- Gu, Rosenbaum
- 1993
(Show Context)
Citation Context ...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 ... |

79 | Estimation of causal effects using propensity score weighting: An application to data on right heart catheterization - Hirano, Imbens - 2001 |

79 |
The Central Role of the Propensity Score
- Rosenbaum, Rubin
- 1983
(Show Context)
Citation Context ...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... |

70 | What Mean Impacts Miss: Distributional Effects of Welfare Reform Experiments - Bitler, Hoynes - 2006 |

70 | Empirical Probability Plots and Statistical Inference for Nonlinear Models in the Two-Sample Case,”Annals of Statistics - Doksum - 1974 |

69 | The Adequacy of Comparison Group Designs for Evaluations of Employment-Related Programs - Fraker, Maynard - 1987 |

62 |
Towards a Curse of Dimensionality Appropriate (CODA) Asymptotic Theory for Semi-parametric Models,”Statistics in Medicine 16
- Robins, Ritov
- 1997
(Show Context)
Citation Context ...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... |

57 | When to Control for Covariates? Panel-Asymptotic Results for Estimates of Treatment Effects,”forthcoming, Review of Economics and Statistics - Angrist, Hahn - 2004 |

52 | Controlling Bias in Observational Studies: A - Cochran, Rubin - 1973 |

51 | The Electoral Advantage to Incumbency and Voters’ Valuation of Politicians’ Experience: A Regression Discontinuity Analysis of Close Elections - Lee - 2001 |

43 | Estimating the Effect of Training Programs on Earnings’, The Review of Economics and Statistics 60.1: 47–57 - Ashenfelter - 1978 |

42 |
Alternative Methods for Evaluating the Impact of Training Programs”, (with discussion
- Heckman, Hotz
- 1989
(Show Context)
Citation Context ...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 ... |

42 |
Nonparametrics: Statistical Methods Based on Ranks
- Lehman
- 1974
(Show Context)
Citation Context ...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... |

42 |
Matching to Remove Bias
- Rubin
- 1973
(Show Context)
Citation Context ...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... |

41 | Causal inference for complex longitudinal data: the continuous case. The Annals of Statistics 29 - Gill, Robins - 2001 |

36 | Finite-sample Variance of Local Polynomials: Analysis and - Seifert, Gasser - 1996 |

34 | The Estimation of Wage Gains and - Björklund, Moffitt - 1987 |

34 | Asymptotic Expansions for some Semiparametric Program Evaluation Estimators”, (London School of Economics, mimeo - Ichimura, Linton - 2002 |

33 | Changes in the Distribution of Male and Female Wages Accounting for the Employment Composition,”unpublished paper
- Blundell, Gosling, et al.
- 2002
(Show Context)
Citation Context ...; 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;... |

32 |
Using Maimonides Rule to Estimate the Effect
- Angrist, Lavy
- 1999
(Show Context)
Citation Context ...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... |

32 |
Evaluating program evaluations: New evidence on commonly used nonexperimental methods
- Friedlander, Robins
- 1995
(Show Context)
Citation Context ...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... |

32 |
Identifi cation and Estimation of Local Average Treatment Effects.” Econometrica 62:467–75
- Imbens, Angrist
- 1994
(Show Context)
Citation Context ... 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... |

30 | The Long-Term Gains from GAIN: A - Hotz, Imbens, et al. - 2006 |

29 | Instrumental Variables Estimation of Quantile Treatment Effects,”Econometrica - Abadie, Angrist, et al. - 2002 |

29 | Estimation in the Regression Discontinuity Model
- Porter
- 2003
(Show Context)
Citation Context ... 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... |

26 | Mean-squared-error Calculations for Average Treatment Effects,”unpublished manuscript
- Imbens, Newey, et al.
- 2003
(Show Context)
Citation Context ...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... |

23 | E¢ cient Semiparametric Estimation of Quantile Treatment Effects - Firpo - 2007 |

22 | A Regression-Discontinuity Evaluation of the Effect - Klaauw - 2002 |

15 | Conditional Permutation Tests and the Propensity Score in Observational Studies - Rosenbaum - 1984 |

11 | Was there a Riverside Miracle? A Hierarchical Framework for Evaluating Programs with Grouped Data - Dehejia - 2002 |

9 |
Identifi cation of Causal Effects Using Instrumental Variables
- Angrist, Imbens, et al.
- 1996
(Show Context)
Citation Context ...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... |

5 | Treatment Evaluation: Matching versus Local Polynomial Regression,” Discussion paper 2000-17 - Frölich - 2000 |

4 | Handbook of Labor Economics vol - Card - 2000 |

3 | The Non-parametric Identi cation of Treatment Effects in Duration Models,” Free University of Amsterdam, unpublished manuscript - Abbring, Berg - 2002 |

3 |
Ef cient Estimation of Average Treatment Effects Using the Estimated Propensity Score,” Econometrica 71:4
- Hirano, Imbens, et al.
- 2003
(Show Context)
Citation Context ...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... |

3 | Predicting the Ef cacy of Future Training Programs Using Past Experiences.’ ’ Los Angeles: University of California—Los Angeles Working Paper - Hotz, Imbens, et al. - 1998 |

3 | Reconciling Con icting Evidence on the Performance - Smith, Todd - 2001 |

2 |
On the Role of the Propensity Score in Ef cient Semiparametric Estimation of Average Treatment Effects,” Econometrica 66:2
- Hahn
- 1998
(Show Context)
Citation Context ... 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... |

2 | der Klaauw, “Identi cation and Estimation of Treatment Effects with a Regression-Discontinuity Design,” Econometrica 69:1 - Hahn, Todd, et al. - 2000 |

2 |
Direct Estimation of Policy Effects
- Ichimura, Taber
- 2000
(Show Context)
Citation Context ...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; ... |

2 |
Subsampling (Springer-Verlag
- Politis, Romano
- 1999
(Show Context)
Citation Context ...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... |

2 |
psmatch: propensity score matching in STATA
- Sianesi
- 2001
(Show Context)
Citation Context ...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... |

1 |
Identi cation and Inference in Nonlinear Difference-in-Differences Models,” NBER technical working paper no
- Athey, Imbens
- 2002
(Show Context)
Citation Context ...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... |

1 |
An IV Model of Quantile Treatment Effects,” Department of Economics, MIT, unpublished working paper
- Chernozhukov, Hansen
- 2001
(Show Context)
Citation Context ... 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... |

1 | The Effectiveness of Adjustment by Subclassi cation - Cochran - 1968 |

1 |
The Bootstrap,” in James
- Horowitz
- 2002
(Show Context)
Citation Context ...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 ... |

1 | Identi cation and Estimation of Causal Effects of Multiple Treatments under the Conditional Independence Assumption - Lechner - 2001 |

1 | Nonparametric Analysis of Covariance by Matching,” Biometrics 38 - Quade - 1982 |

1 | Semiparametric Ef ciency in Multivariate Regression Models with Missing Data - Robins, Rotnitzky - 1995 |

1 | Af nely Invariant Matching Methods with Ellipsoidal Distributions, ” Annals of Statistics 20:2 - Rubin, Thomas - 1992 |

1 | Monica Costa-Dias, “Alternative Approaches to Evaluation in Empirical Microeconomics, ” Institute for Fiscal Studies, Cemmap working paper cwp10/02 - Blundell - 2002 |