Partial identification of the distribution of treatment effects and its confidence sets

Fan, Yanqin and Park, Sang Soo (2009): Partial identification of the distribution of treatment effects and its confidence sets. Published in: Advances in Econometrics: Nonparametric Econometric Methods , Vol. 24, (2009): pp. 3-70.

Abstract

In this paper, we study partial identification of the distribution of treatment effects of a binary treatment for ideal randomized experiments, ideal randomized experiments with a known value of a dependence measure, and for data satisfying the selection-on-observables assumption respectively. For ideal randomized experiments, (i) we propose nonparametric estimators of the sharp bounds on the distribution of treatment effects and construct asymptotically valid confidence sets for the distribution of treatment effects; (ii) we propose bias-corrected estimators of the sharp bounds on the distribution of treatment effects; and (iii) we investigate finite sample performances of the proposed confidence sets and the bias-corrected estimators via simulation.

• Item Type: MPRA Paper
• Language: English
• Keywords: Confidence sets; partial identification; distribution; treatment effects
• Subjects: C - Mathematical and Quantitative Methods > C1 - Econometric and Statistical Methods and Methodology: General > C13 - Estimation: General; C - Mathematical and Quantitative Methods > C1 - Econometric and Statistical Methods and Methodology: General > C14 - Semiparametric and Nonparametric Methods: General; C - Mathematical and Quantitative Methods > C1 - Econometric and Statistical Methods and Methodology: General > C12 - Hypothesis Testing: General; C - Mathematical and Quantitative Methods > C1 - Econometric and Statistical Methods and Methodology: General > C15 - Statistical Simulation Methods: General; C - Mathematical and Quantitative Methods > C1 - Econometric and Statistical Methods and Methodology: General > C19 - Other; C - Mathematical and Quantitative Methods > C4 - Econometric and Statistical Methods: Special Topics > C49 - Other; C - Mathematical and Quantitative Methods > C2 - Single Equation Models ; Single Variables > C21 - Cross-Sectional Models ; Spatial Models ; Treatment Effect Models ; Quantile Regressions; C - Mathematical and Quantitative Methods > C0 - General > C01 - Econometrics

References

Aakvik, A. , Heckman, J. and Vytlacil, E. (2005), "Estimating Treatment Effects for Discrete Outcomes When Responses to Treatment Vary Among Observationally Identical Persons: An Application to Norwegian Vocational Rehabilitation Programs," Journal of Econometrics 125, 15-51.

Abadie, A., J. Angrist, and Imbens, G. (2002), "Instrumental Variables Estimation of Quantile Treatment Effects," Econometrica 70, 91-117.

Abbring, J. H. and Heckman, J. (2007), "Econometric Evaluation of Social Programs, Part III: Distributional Treatment Effects, Dynamic Treatment Effects, Dynamic Discrete Choice, and General Equilibrium Policy Evaluation," in The Handbook of Econometrics 6B, 5145-5301.

Alsina, C. (1981), "Some Functional Equations in the Space of Uniform Distribution Functions," Equationes Mathematicae 22, 153-164.

Andrews, D. W. K. (2000), "Inconsistency of the Bootstrap When a Parameter is on the Boundary of the Parameter Space," Econometrica 68, 399-405.

Andrews, D. W. K. and Guggenberger, P. (2005a), "The Limit of Finite-Sample Size and a Problem with Subsampling," Unpublished Manuscript, Cowles Foundation, Yale University.

Andrews, D. W. K. and Guggenberger, P. (2005b), "Hybrid and Size-corrected Subsampling Methods," Unpublished Manuscript, Cowles Foundation, Yale University.

Andrews, D. W. K. and Guggenberger, P. (2005c), "Applications of Subsampling, Hybrid, and Size-correction Methods," Cowles Foundation Discussion Paper No. 1608, Yale University.

Andrews, D. W. K. and Guggenberger, P. (2007), "Validity of Subsampling and `Plug-in Asymptotic' Inference for Parameters Defined by Moment Inequalities," Unpublished Manuscript, Cowles Foundation, Yale University.

Andrews, D. W. K. and Soares, G. (2007), "Inference for Parameters Defined by Moment Inequalities Using Generalized Moment Selection," Cowles Foundation Working Paper No. 1631, Yale University.

Biddle, J., Boden, L. and Reville, R. (2003), "A Method for Estimating the Full Distribution of a Treatment Effect, With Application to the Impact of Workfare Injury on Subsequent Earnings." Mimeo.

Bitler, M., Gelbach, J. and Hoynes, H. W. (2006), "What Mean Impacts Miss: Distributional Effects of Welfare Reform Experiments," American Economic Review 96, 988-1012.

Black, D. A., Smith, J. A., Berger, M. C. and Noel, B. J. (2003), "Is the Threat of Reemployment Services More Effective Than the Services Themselves? Experimental Evidence From the UI System." American Economic Review 93(3), 1313-1327.

Bugni, F. A. (2007), "Bootstrap Inference in Partially Identified Models," mimeo, Northwestern University.

Canay, I. A. (2007), "EL Inference for Partially Identified Models: Large Deviations Optimality and Bootstrap Validity," Manuscript, Unviersity of Wisconsin.

Cambanis, S., Simons, G. and Stout, W. (1976), "Inequalities for Ek(X,Y) when the Marginals are Fixed," Zeitschrift fur Wahrscheinlichkeitstheorie und verwandte Gebiete 36, 285-294.

Carneiro, P., Hansen, K. T. and Heckman, J. (2003), "Estimating Distributions of Treatment Effects With an Application to the Returns to Schooling and Measurement of the Effects of Uncertainty on College Choice," International Economic Review 44(2), 361-422.

Chernozhukov, V. and Hansen, C. (2005), "An IV Model of Quantile Treatment Effects," Econometrica 73, 245-261.

Chernozhukov, V., Hong, H. and Tamer, E. (2007), "Parameter Set Inference in a Class of Econometric Models," Econometrica 75, 1243-1284.

Davidson, R. and Mackinnon, J. G. (2004), Econometric Theory and Method. Oxford University Press.

Dehejia, R. (1997), "A Decision-theoretic Approach to Program Evaluation," Ph.D. Dissertation, Department of Economics, Harvard University.

Dehejia, R. and Wahba, S. (1999), "Causal Effects in Non-Experimental Studies: Re-Evaluating the Evaluation of Training Programs," Journal of the American Statistical Association 94, 1053-1062.

Denuit, M., Genest, C. and Marceau, E. (1999), "Stochastic Bounds on Sums of Dependent Risks," Insurance: Mathematics and Economics 25, 85-104.

Djebbari, H. and Smith, J. A. (2008), "Heterogeneous Impacts in PROGRESA," Journal of Econometrics 145, 64-80.

Doksum, K. (1974), "Empirical Probability Plots and Statistical Inference for Nonlinear Models in the Two-Sample Case," Annals of Statistics 2, 267-277.

Embrechts, P., Hoeing, A. and Juri, A. (2003), "Using Copulae to Bound the Value-at-Risk for Functions of Dependent Risks," Finance & Stochastics 7(2), 145-167.

Fan, Y. (2008), "Confidence Sets for Distributions of Treatment Effects With Covariates," Work in progress, Vanderbilt University.

Fan, Y. and Park, S. (2007a), "Sharp Bounds on the Distribution of Treatment Effects and Their Statistical Inference," Forthcoming in Econometric Theory.

Fan, Y. and Park, S. (2007b), "Confidence Sets for the Quantile of Treatment Effects," Manuscript, Vanderbilt University.

Fan, Y. and Park, S. (2007c), "Confidence Sets for Some Partially Identified Parameters," Manuscript, Vanderbilt University.

Fan, Y. and Wu, J. (2007), "Sharp Bounds on the Distribution of the Treatment Effect in Switching Regimes Models," Manuscript, Vanderbilt University.

Fan, Y. and Zhu, D. (2009), "Partial Identification and Confidence Sets for Parameters of the Joint Distribution of the Potential Outcomes," Work in progress, Vanderbilt University.

Firpo, S. (2007), "Efficient Semiparametric Estimation of Quantile Treatment Effects," Econometrica 75, 259-276.

Firpo, S. and Ridder, G. (2008), "Bounds on Functionals of the Distribution of Treatment Effects," Working Paper.

Frank, M. J., Nelsen, R. B. and Schweizer, B. (1987), "Best-Possible Bounds on the Distribution of a Sum - a Problem of Kolmogorov," Probability Theory and Related Fields 74, 199-211.

Galichon, A. and Henry, M. (2008), "A Test of Non-Identifying Restrictions and Confidence Regions for Partially Identified Parameters," Journal of Econometrica forthcoming.

Hahn, J. (1998), "On the Role of the Propensity Score in Efficient Semiparametric Estimation of Average Treatment Effects," Econometrica 66, 315-331.

Heckman, J. and Robb, R. (1985), "Alternative Methods for Evaluating the Impact of Interventions," in J. Heckman and B. Singer, eds., Longitudinal Analysis of Labor Market Data. New York: Cambridge University Press.

Heckman, J., Ichimura, H., Smith, J. and Todd, P. (1998), "Characterizing Selection Bias Using Experimental Data," Econometrica 66, 1017-1098.

Heckman, J. and Smith, J. (1993), "Assessing The Case For Randomized Evaluation of Social Programs," in Measuring Labour Market Measures: Evaluating the Effects of Active Labour Market Policies, ed. by K. Jensen and P. K. Madsen. Copenhagen: Danish Ministry of Labor, 35-96.

Heckman, J., Smith, J. and Clements, N. (1997), "Making The Most Out Of Programme Evaluations and Social Experiments: Accounting For Heterogeneity in Programme Impacts," Review of Economic Studies 64, 487-535.

Heckman, J. and Vytlacil, E. (2007a), "Econometric Evaluation of Social Programs. Part I: Causal Models, Structural Models and Econometric Policy Evaluation," in The Handbook of Econometrics 6B, 4779-4874.

Heckman, J. and Vytlacil, E. (2007b), "Econometric Evaluation of Social Programs. Part II: Using the Marginal Treatment Effect to Organize Alternative Econometric Estimators to Evaluate Social Programs, and to Forecast Their Effects in New Environments," in The Handbook of Econometrics 6B, 4875-5143.

Hirano, K., Imbens, G. W. and Ridder, G. (2003), "Efficient Estimation of Average Treatment Effects Using the Estimated Propensity Score," Econometrica 71, 1161-1189.

Horowitz, J. L. and Manski, C. F. (2000), "Nonparametric Analysis of Randomized Experiments with Missing Covariate and Outcome Data," Journal of the American Statistical Association 95, 77-84.

Imbens, G. W. and Manski, C. F. (2004), "Confidence Intervals For Partially Identified Parameters," Econometrica 72, 1845--1857.

Imbens, G. W. and Newey, W. (2005), "Identification and Estimation of Triangular Simultaneous Equations Models Without Additivity," Working Paper.

Imbens, G. W. and Rubin, D. B. (1997), "Estimating Outcome Distributions for Compliers in Instrumental Variables Models," Review of Economic Studies 64, 555-574.

Joe, H. (1997), Multivariate Models and Dependence Concepts. Chapman & Hall/CRC, London.

Lalonde, R. (1995), "The Promise of Public-Sector Sponsored Training Programs," Journal of Economic Perspectives 9, 149-168.

Lechner, M. (1999), "Earnings and Employment Effects of Continuous Off-the-job Training in East Germany After Unification," Journal of Business and Economic Statistics 17, 74-90.

Lee, L. F. (2002), "Correlation Bounds for Sample Selection Models with Mixed Continuous, Discrete and Count Data Variables," Manuscript, The Ohio State University.

Lee, M. J. (2005), Micro-Econometrics for Policy, Program, and Treatment Effects. Oxford University Press.

Lehmann, E. L. (1974), Nonparametrics: Statistical Methods Based on Ranks. Holden-Day Inc., San Francisco, California.

Makarov, G. D. (1981), "Estimates for the Distribution Function of a Sum of two Random Variables When the Marginal Distributions are Fixed," Theory of Probability and its Applications 26, 803-806.

Manski, C. F. (1990), "Non-parametric Bounds on Treatment Effects," American Economic Review, Papers and Proceedings 80, 319-323.

Manski, C. F. (1997a), "Monotone Treatment Effect," Econometrica 65, 1311-1334.

Manski, C. F. (1997b), "The Mixing Problem in Programme Evaluation," Review of Economic Studies 64, 537-553.

Manski, C. F. (2003), Partial Identification of Probability Distributions. Springer-Verlag, New York.

Moon, R. and Schorfheide, F. (2007), "A Bayesian Look at Partially-Identified Models," Manuscript, U.Penn.

Nelsen, R. B. (1999), An Introduction to Copulas. Springer, New York.

Nelsen, R. B., Quesada-Molina, J. J., Rodriguez-Lallena, J. A. and Ubeda-Flores, M. (2001), "Bounds on Bivariate Distribution Functions With Given Margins and Measures of Association," Communications in Statistics-Theory and Methods 30, 1155-1162.

Nelsen, R. B., Quesada-Molina, J. J., Rodriguez-Lallena, J. A. and Ubeda-Flores, M. (2004), "Best-Possible Bounds on Sets of Bivariate Distribution Functions," Journal of Multivariate Analysis 90, 348-358.

Nelsen, R. B. and Ubeda-Flores, M. (2004), "A Comparison of Bounds on Sets of Joint Distribution Functions Derived From Various Measures of Association," Communications in Statistics-Theory and Methods 33, 2299-2305.

Romano, J. and Shaikh, A. M. (2008), "Inference for Identifiable Parameters in Partially Identified Econometric Models," Journal of Statistical Planning and Inference forthcomming.

Rosen, A. (2008), "Confidence Sets for Partially Identified Parameters That Satisfy a Finite Number of Moment Inequalities," Journal of Econometrics 146, 107-117.

Rosenbaum, P. R. and Rubin, D. B. (1983a), "Assessing Sensitivity to an Unobserved Binary Covariate in an Observational Study with Binary Outcome," Journal of the Royal Statistical Society, Series B 45, 212-218.

Schweizer, B. and Sklar, A. (1983), Probabilistic Metric Spaces. North-Holland, New York.

Sklar, A. (1959), "Fonctions de reartition a n dimensions et leures marges," Publications de l'Institut de Statistique de L'Universite de Paris 8, 229-231.

Soares, G. (2006), "Inference for Partially Identified Models with Inequality Moment Constraints," Working Paper, Yale.

Stoye, J. (2008a), "More on Confidence Intervals for Partially Identified Parameters," Forthcoming in Econometrica.

Stoye, J. (2008b), "Partial Identification of Spread Parameters," Working paper, NYU.

Tchen, A. H. (1980), "Inequalities for Distributions with Given Marginals," Annals of Probability 8, 814-827.

Tesfatsion, L. (1976), "Stochastic Dominance and the Maximization of Expected Utility," Review of Economic Studies 43, 301-315.

Williamson, R. C. and Downs, T. (1990), "Probabilistic Arithmetic I: Numerical Methods for Calculating Convolutions and Dependency Bounds," International Journal of Approximate Reasoning 4, 89-158.