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Efficiency Bounds for Estimating Linear Functionals of Nonparametric Regression Models with Endogenous Regressors

Author

Listed:
  • Thomas A. Severini

    (Northwestern University)

  • Gautam Tripathi

    (University of Connecticut)

Abstract

Consider a nonparametric regression model Y=mu*(X) + e, where the explanatory variables X are endogenous and e satisfies the conditional moment restriction E[e|W]=0 w.p.1 for instrumental variables W. It is well known that in these models the structural parameter mu* is 'ill-posed' in the sense that the function mapping the data to mu* is not continuous. In this paper, we derive the efficiency bounds for estimating linear functionals E[p(X)mu*(X)] and int_{supp(X)}p(x)mu*(x)dx, where p is a known weight function and supp(X) the support of X, without assuming mu* to be well-posed or even identified.

Suggested Citation

  • Thomas A. Severini & Gautam Tripathi, 2007. "Efficiency Bounds for Estimating Linear Functionals of Nonparametric Regression Models with Endogenous Regressors," Working papers 2007-18, University of Connecticut, Department of Economics.
    • Handle: RePEc:uct:uconnp:2007-18
    Note: We thank Gary Chamberlain, Enno Mammen, Whitney Newey, and participants at several seminars for helpful suggestions and conversations. The first author also thanks the NSF for financial support.
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    Cited by:

    1. Michael Jansson & Demian Pouzo, 2017. "Towards a General Large Sample Theory for Regularized Estimators," Papers 1712.07248, arXiv.org, revised Jul 2020.
    2. Andrii Babii & Jean-Pierre Florens, 2017. "Is completeness necessary? Estimation in nonidentified linear models," Papers 1709.03473, arXiv.org, revised Jan 2025.
    3. Carolina Caetano & Juan Carlos Escaniano, 2015. "Identifying Multiple Marginal Effects with a Single Binary Instrument or by Regression Discontinuity," CAEPR Working Papers 2015-009, Center for Applied Economics and Policy Research, Department of Economics, Indiana University Bloomington.
    4. Chen, Qihui, 2021. "Robust and optimal estimation for partially linear instrumental variables models with partial identification," Journal of Econometrics, Elsevier, vol. 221(2), pages 368-380.
    5. Chen, Xiaohong & Pouzo, Demian & Powell, James L., 2019. "Penalized sieve GEL for weighted average derivatives of nonparametric quantile IV regressions," Journal of Econometrics, Elsevier, vol. 213(1), pages 30-53.
    6. Ben Deaner, 2019. "Nonparametric Instrumental Variables Estimation Under Misspecification," Papers 1901.01241, arXiv.org, revised Dec 2022.
    7. Hidehiko Ichimura & Whitney K. Newey, 2022. "The influence function of semiparametric estimators," Quantitative Economics, Econometric Society, vol. 13(1), pages 29-61, January.
    8. Andrew Bennett & Nathan Kallus & Xiaojie Mao & Whitney Newey & Vasilis Syrgkanis & Masatoshi Uehara, 2022. "Inference on Strongly Identified Functionals of Weakly Identified Functions," Papers 2208.08291, arXiv.org, revised Jun 2023.
    9. Claire-Océane Chevallier, 2017. "Empirical Investigation of the Effect of Bank Long Term Debt on Loans and Output in the Euro-zone," DEM Discussion Paper Series 17-04, Department of Economics at the University of Luxembourg.
    10. Chen, Xiaohong & Liao, Yuan & Wang, Weichen, 2025. "Inference on time series nonparametric conditional moment restrictions using nonlinear sieves," Journal of Econometrics, Elsevier, vol. 249(PA).
    11. Zhang, Jeffrey & Li, Wei & Miao, Wang & Tchetgen Tchetgen, Eric, 2023. "Proximal causal inference without uniqueness assumptions," Statistics & Probability Letters, Elsevier, vol. 198(C).
    12. Centorrino, Samuele & Florens, Jean-Pierre, 2021. "Nonparametric Instrumental Variable Estimation of Binary Response Models with Continuous Endogenous Regressors," Econometrics and Statistics, Elsevier, vol. 17(C), pages 35-63.
    13. V Chernozhukov & W K Newey & R Singh, 2023. "A simple and general debiased machine learning theorem with finite-sample guarantees," Biometrika, Biometrika Trust, vol. 110(1), pages 257-264.
    14. Andrew Bennett & Nathan Kallus & Xiaojie Mao & Whitney Newey & Vasilis Syrgkanis & Masatoshi Uehara, 2023. "Source Condition Double Robust Inference on Functionals of Inverse Problems," Papers 2307.13793, arXiv.org.
    15. Santos, Andres, 2011. "Instrumental variable methods for recovering continuous linear functionals," Journal of Econometrics, Elsevier, vol. 161(2), pages 129-146, April.
    16. Juan Carlos Escanciano & Wei Li, 2013. "On the identification of structural linear functionals," CeMMAP working papers CWP48/13, Centre for Microdata Methods and Practice, Institute for Fiscal Studies.
    17. Victor Chernozhukov & Juan Carlos Escanciano & Hidehiko Ichimura & Whitney K. Newey & James M. Robins, 2022. "Locally Robust Semiparametric Estimation," Econometrica, Econometric Society, vol. 90(4), pages 1501-1535, July.
    18. Laurent Davezies & Xavier d'Haultfoeuille, 2013. "Endogenous Attrition in Panels," Working Papers 2013-17, Center for Research in Economics and Statistics.
    19. Shuyuan Chen & Peng Zhang & Yifan Cui, 2025. "Identification and Debiased Learning of Causal Effects with General Instrumental Variables," Papers 2510.20404, arXiv.org.
    20. Stéphane Bonhomme & Martin Weidner, 2022. "Minimizing sensitivity to model misspecification," Quantitative Economics, Econometric Society, vol. 13(3), pages 907-954, July.
    21. Jiafeng Chen & Daniel L. Chen & Greg Lewis, 2020. "Mostly Harmless Machine Learning: Learning Optimal Instruments in Linear IV Models," Papers 2011.06158, arXiv.org, revised Jun 2021.
    22. Escanciano, Juan Carlos, 2023. "Irregular identification of structural models with nonparametric unobserved heterogeneity," Journal of Econometrics, Elsevier, vol. 234(1), pages 106-127.
    23. Egshiglen Batbayar & Christoph Breunig & Peter Haan & Boryana Ilieva, 2025. "Quantile Selection in the Gender Pay Gap," Papers 2511.16187, arXiv.org, revised Jan 2026.
    24. Escanciano, Juan Carlos & Li, Wei, 2021. "Optimal Linear Instrumental Variables Approximations," Journal of Econometrics, Elsevier, vol. 221(1), pages 223-246.
    25. Isaac Meza & Rahul Singh, 2021. "Nested Nonparametric Instrumental Variable Regression," Papers 2112.14249, arXiv.org, revised May 2025.

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    JEL classification:

    • C14 - Mathematical and Quantitative Methods - - Econometric and Statistical Methods and Methodology: General - - - Semiparametric and Nonparametric Methods: General

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