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A triangular treatment effect model with random coefficients in the selection equation

  • Eric Gautier

    (Institute for Fiscal Studies and CREST)

  • Stefan Hoderlein

    ()

    (Institute for Fiscal Studies and Boston College)

In this paper we study nonparametric estimation in a binary treatment model where the outcome equation is of unrestricted form, and the selection equation contains multiple unobservables that enter through a nonparametric random coefficients specification. This specification is flexible because it allows for complex unobserved heterogeneity of economic agents and non-monotone selection into treatment. We obtain conditions under which both the conditional distributions of Y0 and Y1, the outcome for the untreated, respectively treated, given first stage unobserved random coefficients, are identified. We can thus identify an average treatment effect, conditional on first stage unobservables called UCATE, which yields most treatment effects parameters that depend on averages, like ATE and TT. We provide sharp bounds on the variance, the joint distribution of (Y0 and Y1) and the distribution of treatment effects. In the particular case where the outcomes are continuously distributed, we provide novel and weak conditions that allow to point identify the join conditional distribution of Y0 and Y1, given the unobservables. This allows to derive every treatment effect parameter, e.g. the distribution of treatment effects and the proportion of individuals who benefit from treatment. We present estimators for the marginals, average and distribution of treatment effects, both conditional on unobservables and unconditional, as well as total population effects. The estimators use all data and discard tail values of the instruments when they are too unlikely. We provide their rates of convergence, and analyse their finite sample behaviour in a simulation study. Finally, we also discuss the situation where some of the instruments are discrete.

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File URL: http://www.cemmap.ac.uk/wps/cwp391212.pdf
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Paper provided by Centre for Microdata Methods and Practice, Institute for Fiscal Studies in its series CeMMAP working papers with number CWP39/12.

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Date of creation: Dec 2012
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Handle: RePEc:ifs:cemmap:39/12
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  1. Heckman, James & Singer, Burton, 1984. "A Method for Minimizing the Impact of Distributional Assumptions in Econometric Models for Duration Data," Econometrica, Econometric Society, vol. 52(2), pages 271-320, March.
  2. Lewbel, Arthur, 2000. "Semiparametric qualitative response model estimation with unknown heteroscedasticity or instrumental variables," Journal of Econometrics, Elsevier, vol. 97(1), pages 145-177, July.
  3. Bjorklund, Anders & Moffitt, Robert, 1987. "The Estimation of Wage Gains and Welfare Gains in Self-selection," The Review of Economics and Statistics, MIT Press, vol. 69(1), pages 42-49, February.
  4. Klein, Tobias J., 2007. "Heterogeneous Treatment Effects: Instrumental Variables without Monotonicity?," IZA Discussion Papers 2738, Institute for the Study of Labor (IZA).
  5. Eric Gautier & Yuichi Kitamura, 2009. "Nonparametric Estimation in Random Coefficients Binary Choice Models," Cowles Foundation Discussion Papers 1721, Cowles Foundation for Research in Economics, Yale University.
  6. Arthur Lewbel, 2000. "Endogenous Selection Or Treatment Model Estimation," Boston College Working Papers in Economics 462, Boston College Department of Economics, revised 13 Jun 2007.
  7. Hoderlein, Stefan & Klemelä, Jussi & Mammen, Enno, 2010. "Analyzing The Random Coefficient Model Nonparametrically," Econometric Theory, Cambridge University Press, vol. 26(03), pages 804-837, June.
  8. Angrist, Joshua D & Graddy, Kathryn & Imbens, Guido W, 2000. "The Interpretation of Instrumental Variables Estimators in Simultaneous Equations Models with an Application to the Demand for Fish," Review of Economic Studies, Wiley Blackwell, vol. 67(3), pages 499-527, July.
  9. Aakvik, Arild & Heckman, James J. & Vytlacil, Edward J., 2005. "Estimating treatment effects for discrete outcomes when responses to treatment vary: an application to Norwegian vocational rehabilitation programs," Journal of Econometrics, Elsevier, vol. 125(1-2), pages 15-51.
  10. Heckman, James J & Smith, Jeffrey, 1997. "Making the Most Out of Programme Evaluations and Social Experiments: Accounting for Heterogeneity in Programme Impacts," Review of Economic Studies, Wiley Blackwell, vol. 64(4), pages 487-535, October.
  11. James J. Heckman & Jeffrey A. Smith, 1998. "Evaluating the Welfare State," NBER Working Papers 6542, National Bureau of Economic Research, Inc.
  12. Eric Gautier & Erwann Le Pennec, 2011. "Adaptive Estimation in the Nonparametric Random Coefficients Binary Choice Model by Needlet Thresholding," Working Papers 2011-20, Centre de Recherche en Economie et Statistique.
  13. Fan, Yanqin & Park, Sang Soo, 2010. "Sharp Bounds On The Distribution Of Treatment Effects And Their Statistical Inference," Econometric Theory, Cambridge University Press, vol. 26(03), pages 931-951, June.
  14. F. Comte & C. Lacour, 2011. "Data‐driven density estimation in the presence of additive noise with unknown distribution," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 73(4), pages 601-627, 09.
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