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78
Nonparametric estimation of average treatment effects under exogeneity: a review
 REVIEW OF ECONOMICS AND STATISTICS
, 2004
"... Recently there has been a surge in econometric work focusing on estimating average treatment effects under various sets of assumptions. One strand of this literature has developed methods for estimating average treatment effects for a binary treatment under assumptions variously described as exogen ..."
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Cited by 597 (26 self)
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Recently there has been a surge in econometric work focusing on estimating average treatment effects under various sets of assumptions. One strand of this literature has developed methods for estimating average treatment effects for a binary treatment under assumptions variously described as exogeneity, unconfoundedness, or selection on observables. The implication of these assumptions is that systematic (for example, average or distributional) differences in outcomes between treated and control units with the same values for the covariates are attributable to the treatment. Recent analysis has considered estimation and inference for average treatment effects under weaker assumptions than typical of the earlier literature by avoiding distributional and functionalform assumptions. Various methods of semiparametric estimation have been proposed, including estimating the unknown regression functions, matching, methods using the propensity score such as weighting and blocking, and combinations of these approaches. In this paper I review the state of this
Efficient semiparametric estimation of quantile treatment effects
, 2003
"... This paper presents calculations of semiparametric efficiency bounds for quantile treatment effects parameters when selection to treatment is based on observable characteristics. The paper also presents three estimation procedures for these parameters, all of which have two steps: a nonparametric e ..."
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Cited by 120 (5 self)
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This paper presents calculations of semiparametric efficiency bounds for quantile treatment effects parameters when selection to treatment is based on observable characteristics. The paper also presents three estimation procedures for these parameters, all of which have two steps: a nonparametric estimation and a computation of the difference between the solutions of two distinct minimization problems. RootN consistency, asymptotic normality, and the achievement of the semiparametric efficiency bound is shown for one of the three estimators. In the final part of the paper, an empirical application to a job training program reveals the importance of heterogeneous treatment effects, showing that for this program the effects are concentrated in the upper quantiles of the earnings distribution.
An IV Model of Quantile Treatment Effects
 Econometrica
, 2001
"... Headnote.The ability of quantile regression models to characterize the heterogeneous impact of variables on different points of an outcome distribution makes them appealing in many economic applications. However, in observational studies, the variables of interest (e.g. education, prices) are ofte ..."
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Cited by 82 (4 self)
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Headnote.The ability of quantile regression models to characterize the heterogeneous impact of variables on different points of an outcome distribution makes them appealing in many economic applications. However, in observational studies, the variables of interest (e.g. education, prices) are often endogenous, making conventional quantile regression inconsistent and hence inappropriate for recovering the causal effects of these variables on the quantiles of economic outcomes. In order to address this problem, we develop a model of quantile treatment effects (QTE) in the presence of endogeneity and obtain conditions for identification of the QTE without functional form assumptions. The principal feature of the model is the imposition of conditions which restrict the evolution of ranks across treatment states. This feature allows us to overcome the endogeneity problem and recover the true QTE through the use of instrumental variables. The proposed model can also be equivalently viewed as a structural simultaneous equation model with nonadditive errors, where QTE can be interpreted as the structural quantile effects (SQE). Key Words: endogeneity, quantile regression, simultaneous equations, instrumental regression, identification, nonlinear model, monotone likelihood ratio, bounded completeness,
Instrumental variable quantile regression: A robust inference approach
 Journal of Econometrics
, 2008
"... Quantile regression is an increasingly important tool that estimates the conditional quantiles of a response Y given a vector of regressors D. It usefully generalizes Laplace’s median regression and can be used to measure the effect of covariates not only in the center of a distribution, but also in ..."
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Cited by 67 (10 self)
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Quantile regression is an increasingly important tool that estimates the conditional quantiles of a response Y given a vector of regressors D. It usefully generalizes Laplace’s median regression and can be used to measure the effect of covariates not only in the center of a distribution, but also in the upper and lower tails. For the linear quantile model defined by Y = D ′ γ(U) where D ′ γ(U) is strictly increasing in U and U is a standard uniform variable independent of D, quantile regression allows estimation of quantile specific covariate effects γ(τ) for τ ∈ (0, 1). In this paper, we propose an instrumental variable quantile regression estimator that appropriately modifies the conventional quantile regression and recovers quantilespecific covariate effects in an instrumental variables model defined by Y = D ′ α(U) where D ′ α(U) is strictly increasing in U and U is a uniform variable that may depend on D but is independent of a set of instrumental variables Z. The proposed estimator and inferential procedures are computationally convenient in typical applications and can be carried out using software available for conventional quantile regression. In addition, the proposed estimation procedure gives rise to a convenient inferential procedure that is naturally robust to weak identification. The use of the proposed estimator and testing procedure is illustrated through two empirical examples.
Inference on the Quantile Regression Process
, 2000
"... Tests based on the quantile regression process can be formulated like the classical KolmogorovSmirnov and CramervonMises tests of goodnessoffit employing the theory of Bessel processes as in?. However, it is frequently desirable to formulate hypotheses involving unknown nuisance parameters, the ..."
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Cited by 54 (2 self)
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Tests based on the quantile regression process can be formulated like the classical KolmogorovSmirnov and CramervonMises tests of goodnessoffit employing the theory of Bessel processes as in?. However, it is frequently desirable to formulate hypotheses involving unknown nuisance parameters, thereby jeopardizing the distribution free character of these tests. We characterize this situation as "the Durbin problem" since it was posed in?, for parametric empirical processes. In this paper we consider an approach to the Durbin problem involving a martingale transformation of the parametric empirical process suggested by? and show that it can be adapted to a wide variety of inference problems involving the quantile regression process. In particular, we suggest new tests of the location shift and locationscale shift models that underlie much of classical econometric inference. The methods are illustrated in some limited MonteCarlo experiments and with a reanalysis of data on unemployment durations from the Pennsylvania Reemployment Bonus Experiments. The Pennsylvania experiments, conducted in 198889, were designed to test the efficacy of cash bonuses paid for early reemployment in shortening the duration of insured unemployment spells.
Quantile Regression under Misspecification, with an Application to the U.S
 Wage Structure. Econometrica
, 2006
"... Quantile regression (QR) fits a linear model for conditional quantiles, just as ordinary least squares (OLS) fits a linear model for conditional means. An attractive feature of OLS is that it gives the minimum mean square error linear approximation to the conditional expectation function even when t ..."
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Cited by 47 (4 self)
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Quantile regression (QR) fits a linear model for conditional quantiles, just as ordinary least squares (OLS) fits a linear model for conditional means. An attractive feature of OLS is that it gives the minimum mean square error linear approximation to the conditional expectation function even when the linear model is misspecified. Empirical research using quantile regression with discrete covariates suggests that QR may have a similar property, but the exact nature of the linear approximation has remained elusive. In this paper, we show that QR minimizes a weighted meansquared error loss function for specification error. The weighting function is an average density of the dependent variable near the true conditional quantile. The weighted least squares interpretation of QR is used to derive an omitted variables bias formula and a partial quantile regression concept, similar to the relationship between partial regression and OLS. We also present asymptotic theory for the QR process under misspecification of the conditional quantile function. The approximation properties of QR are illustrated using wage data from the US census. These results point to major changes in inequality from 19902000.
2007): “Quantile and probability curves without crossing,”CeMMAP working papers CWP10/07, Centre for Microdata Methods and Practice, Institute for Fiscal Studies
"... Abstract. The commonly used approach in estimating conditional quantile curves is to fit typically a linear curve pointwise for each quantile. This is done for a number of reasons: linear models enjoy good approximation properties as well as have excellent computational properties. The resulting fit ..."
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Cited by 32 (6 self)
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Abstract. The commonly used approach in estimating conditional quantile curves is to fit typically a linear curve pointwise for each quantile. This is done for a number of reasons: linear models enjoy good approximation properties as well as have excellent computational properties. The resulting fits may not respect a logical monotonicity requirement – that the quantile curve, as a function of probability, should be monotone in that probability. This paper studies the natural monotonization of these empirical curves induced by sampling from the estimated nonmonotone model, and then taking the resulting conditional quantile curves, that by construction do not cross. This construction of monotone quantile curves may be seen as a semiparametric bootstrap and also as a monotonic rearrangement of the original nonmonotone function. These conditional quantile curves are monotone in the probability and we show that, under correct specification, these curves have the same asymptotic distribution as the original nonmonotone curves. Thus, the empirical nonmonotone curves can be rearranged to be monotone without changing their (first order) asymptotic distribution. However, this property does not hold under misspecification and the asymptotics of these curves partially differs from the asymptotics of the original nonmonotone curves. Towards establishing the result, we establish the compact (Hadamard) differentiability of the monotonized quantile and probability curves with respect to the original curves. In doing so, we establish the results on the compact differentiability of functions related to rearrangement operators. These results therefore generalize earlier results on the compact differentiability of the inverse (quantile) operators.
Nonparametric Tests for Treatment Effect Heterogeneity
 FORTHCOMING IN THE REVIEW OF ECONOMICS AND STATISTICS
, 2007
"... In this paper we develop two nonparametric tests of treatment effect heterogeneity. The first test is for the null hypothesis that the treatment has a zero average effect for all subpopulations defined by covariates. The second test is for the null hypothesis that the average effect conditional on t ..."
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Cited by 27 (6 self)
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In this paper we develop two nonparametric tests of treatment effect heterogeneity. The first test is for the null hypothesis that the treatment has a zero average effect for all subpopulations defined by covariates. The second test is for the null hypothesis that the average effect conditional on the covariates is identical for all subpopulations, i.e., that there is no heterogeneity in average treatment effects by covariates. We derive tests that are straightforward to implement and illustrate the use of these tests on data from two sets of experimental evaluations of the effects of welfaretowork programs.
The Effects of 401(k) Participation on the Wealth Distribution: An Instrumental Quantile Regression Analysis
 Review of Economics and Statistics 86 (2004) 735
"... Abstract—We use instrumental quantile regression approach to examine the effects of 401(k) plans on wealth using data from the Survey of Income and Program Participation. Using 401(k) eligibility as an instrument for 401(k) participation, we estimate the quantile treatment effects of participation i ..."
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Cited by 22 (2 self)
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Abstract—We use instrumental quantile regression approach to examine the effects of 401(k) plans on wealth using data from the Survey of Income and Program Participation. Using 401(k) eligibility as an instrument for 401(k) participation, we estimate the quantile treatment effects of participation in a 401(k) plan on several measures of wealth. The results show the effects of 401(k) participation on net financial assets are positive and significant over the entire range of the asset distribution, and that the increase in the low tail of the assets distribution appears to translate completely into an increase in wealth. However, there is significant evidence of substitution from other forms of wealth in the upper tail of the distribution.