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Identification In Triangular Systems Using Control Functions


  • Kasy, Maximilian


This note discusses identification in nonparametric, continuous triangular systems. It provides conditions that are both necessary and sufficient for the existence of control functions satisfying conditional independence and support requirements. Confirming a commonly noticed pattern, these conditions restrict the admissible dimensionality of unobserved heterogeneity in the first-stage structural relation, or more generally the dimensionality of the family of conditional distributions of second-stage heterogeneity given explanatory variables and instruments. These conditions imply that no such control function exists without assumptions that seem hard to justify in most applications. In particular, none exists in the context of a generic random coefficient model.

Suggested Citation

  • Kasy, Maximilian, 2011. "Identification In Triangular Systems Using Control Functions," Econometric Theory, Cambridge University Press, vol. 27(3), pages 663-671, June.
  • Handle: RePEc:cup:etheor:v:27:y:2011:i:03:p:663-671_00

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    Cited by:

    1. Kasy, Maximilian, "undated". "Instrumental variables with unrestricted heterogeneity and continuous treatment - DON'T CITE! SEE ERRATUM BELOW," Working Paper 33257, Harvard University OpenScholar.
    2. Sam Cosaert & Thomas Demuynck, 2018. "Nonparametric Welfare and Demand Analysis with Unobserved Individual Heterogeneity," The Review of Economics and Statistics, MIT Press, vol. 100(2), pages 349-361, May.
    3. Denis Chetverikov & Daniel Wilhelm, 2017. "Nonparametric Instrumental Variable Estimation Under Monotonicity," Econometrica, Econometric Society, vol. 85, pages 1303-1320, July.
    4. Mourifié, Ismael, 2015. "Sharp bounds on treatment effects in a binary triangular system," Journal of Econometrics, Elsevier, vol. 187(1), pages 74-81.
    5. Carlson, Alyssa, 2023. "Relaxing conditional independence in an endogenous binary response model," Journal of Econometrics, Elsevier, vol. 232(2), pages 490-500.
    6. Hoderlein, Stefan & Holzmann, Hajo & Meister, Alexander, 2017. "The triangular model with random coefficients," Journal of Econometrics, Elsevier, vol. 201(1), pages 144-169.
    7. 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.
    8. Ming Li, 2021. "A Time-Varying Endogenous Random Coefficient Model with an Application to Production Functions," Papers 2110.00982,
    9. Hoderlein, Stefan & Su, Liangjun & White, Halbert & Yang, Thomas Tao, 2016. "Testing for monotonicity in unobservables under unconfoundedness," Journal of Econometrics, Elsevier, vol. 193(1), pages 183-202.
    10. Gautier, Eric & Hoderlein, Stefan, 2011. "A triangular treatment effect model with random coefficients in the selection equation," TSE Working Papers 15-598, Toulouse School of Economics (TSE), revised 25 Aug 2015.
    11. 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.
    12. Pereda-Fernández, Santiago, 2023. "Identification and estimation of triangular models with a binary treatment," Journal of Econometrics, Elsevier, vol. 234(2), pages 585-623.
    13. Jun, Sung Jae & Pinkse, Joris, 2020. "Counterfactual prediction in complete information games: Point prediction under partial identification," Journal of Econometrics, Elsevier, vol. 216(2), pages 394-429.
    14. Kasy, Maximilian, 2022. "Who wins, who loses? Identification of conditional causal effects, and the welfare impact of changing wages," Journal of Econometrics, Elsevier, vol. 226(1), pages 155-170.
    15. Tadao Hoshino, 2021. "Estimating a Continuous Treatment Model with Spillovers: A Control Function Approach," Papers 2112.15114,, revised Jan 2023.
    16. Amaresh K Tiwari, 2021. "A Control Function Approach to Estimate Panel Data Binary Response Model," Papers 2102.12927,, revised Sep 2021.
    17. Tsionas, Mike G., 2022. "Convex non-parametric least squares, causal structures and productivity," European Journal of Operational Research, Elsevier, vol. 303(1), pages 370-387.
    18. Caetano, Carolina & Rothe, Christoph & Yıldız, Neşe, 2016. "A discontinuity test for identification in triangular nonseparable models," Journal of Econometrics, Elsevier, vol. 193(1), pages 113-122.
    19. Stefan Hoderlein & Hajo Holzmann & Alexander Meister, 2015. "The triangular model with random coefficients," CeMMAP working papers 33/15, Institute for Fiscal Studies.
    20. Denis Chetverikov & Daniel Wilhelm, 2015. "Nonparametric instrumental variable estimation under monotonicity," CeMMAP working papers 39/15, Institute for Fiscal Studies.
    21. David Powell, 2020. "Quantile Treatment Effects in the Presence of Covariates," The Review of Economics and Statistics, MIT Press, vol. 102(5), pages 994-1005, December.
    22. Kim Kyoo il & Petrin Amil, 2022. "A Generalized Non-Parametric Instrumental Variable-Control Function Approach to Estimation in Nonlinear Settings," Journal of Econometric Methods, De Gruyter, vol. 11(1), pages 91-125, January.
    23. D’Haultfœuille, Xavier & Hoderlein, Stefan & Sasaki, Yuya, 2023. "Nonparametric difference-in-differences in repeated cross-sections with continuous treatments," Journal of Econometrics, Elsevier, vol. 234(2), pages 664-690.
    24. Denis Chetverikov & Daniel Wilhelm, 2016. "Nonparametric instrumental variable estimation under monotonicity," CeMMAP working papers 48/16, Institute for Fiscal Studies.

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