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

Author

Listed:
  • Eric Gautier

    (Institute for Fiscal Studies and CREST)

  • Stefan Hoderlein

    (Institute for Fiscal Studies and Boston College)

Abstract
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.

Suggested Citation

  • Eric Gautier & Stefan Hoderlein, 2012. "A triangular treatment effect model with random coefficients in the selection equation," CeMMAP working papers CWP39/12, Centre for Microdata Methods and Practice, Institute for Fiscal Studies.
    • Handle: RePEc:ifs:cemmap:39/12
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    Citations

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    Cited by:

    1. Fabian Dunker & Konstantin Eckle & Katharina Proksch & Johannes Schmidt-Hieber, 2017. "Tests for qualitative features in the random coefficients model," Papers 1704.01066, arXiv.org, revised Mar 2018.
    2. Fabian Dunker & Stefan Hoderlein & Hiroaki Kaido, 2023. "Nonparametric identification of random coefficients in aggregate demand models for differentiated products," The Econometrics Journal, Royal Economic Society, vol. 26(2), pages 279-306.
    3. Christoph Breunig & Stefan Hoderlein, 2018. "Specification testing in random coefficient models," Quantitative Economics, Econometric Society, vol. 9(3), pages 1371-1417, November.
    4. Sokbae Lee & Bernard Salanié, 2018. "Identifying Effects of Multivalued Treatments," Econometrica, Econometric Society, vol. 86(6), pages 1939-1963, November.
    5. Kasy, Maximilian, "undated". "Instrumental variables with unrestricted heterogeneity and continuous treatment - DON'T CITE! SEE ERRATUM BELOW," Working Paper 33257, Harvard University OpenScholar.
    6. Eric Gautier & Erwann Le Pennec, 2011. "Adaptive Estimation in the Nonparametric Random Coefficients Binary Choice Model by Needlet Thresholding," Working Papers 2011-20, Center for Research in Economics and Statistics.
    7. Stefan Hoderlein & Hajo Holzmann & Maximilian Kasy & Alexander Meister, 2015. "Erratum regarding “Instrumental variables with unrestricted heterogeneity and continuous treatment”," Boston College Working Papers in Economics 896, Boston College Department of Economics, revised 01 Feb 2016.
    8. Nail Kashaev, 2018. "Identification and estimation of multinomial choice models with latent special covariates," Papers 1811.05555, arXiv.org, revised Mar 2022.
    9. Éric Gautier, 2021. "Relaxing Monotonicity in Endogenous Selection Models and Application to Surveys," Post-Print hal-03306234, HAL.
    10. Zhan Gao & M. Hashem Pesaran, 2023. "Identification and estimation of categorical random coefficient models," Empirical Economics, Springer, vol. 64(6), pages 2543-2588, June.
    11. Tymon Sloczynski, 2021. "When Should We (Not) Interpret Linear IV Estimands as LATE?," CESifo Working Paper Series 9064, CESifo.
    12. Christoph Breunig & Stefan Hoderlein, 2016. "Nonparametric Specification Testing in Random Parameter Models," Boston College Working Papers in Economics 897, Boston College Department of Economics.
    13. Gautier, Eric & Gaillac, Christophe, 2019. "Adaptive estimation in the linear random coefficients model when regressors have limited variation," TSE Working Papers 19-1026, Toulouse School of Economics (TSE).
    14. D’Haultfœuille, Xavier & Hoderlein, Stefan & Sasaki, Yuya, 2024. "Testing and relaxing the exclusion restriction in the control function approach," Journal of Econometrics, Elsevier, vol. 240(2).
    15. Maximilian Kasy, 2014. "Instrumental Variables with Unrestricted Heterogeneity and Continuous Treatment," The Review of Economic Studies, Review of Economic Studies Ltd, vol. 81(4), pages 1614-1636.
    16. repec:hum:wpaper:sfb649dp2015-053 is not listed on IDEAS
    17. Gaurab Aryal & Federico Zincenko, 2014. "Identification and Estimation of Multidimensional Screening," Papers 1411.6250, arXiv.org, revised Oct 2024.
    18. Arthur Lewbel & Thomas Tao Yang, 2013. "Identifying the Average Treatment Effect in a Two Threshold Model," Boston College Working Papers in Economics 825, Boston College Department of Economics.

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    More about this item

    JEL classification:

    • C13 - Mathematical and Quantitative Methods - - Econometric and Statistical Methods and Methodology: General - - - Estimation: General
    • C14 - Mathematical and Quantitative Methods - - Econometric and Statistical Methods and Methodology: General - - - Semiparametric and Nonparametric Methods: General
    • C21 - Mathematical and Quantitative Methods - - Single Equation Models; Single Variables - - - Cross-Sectional Models; Spatial Models; Treatment Effect Models
    • C26 - Mathematical and Quantitative Methods - - Single Equation Models; Single Variables - - - Instrumental Variables (IV) Estimation

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