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The influence function of semiparametric estimators

Author

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  • Hidehiko Ichimura

    (Institute for Fiscal Studies and University of Arizona, University of Tokyo)

  • Whitney K. Newey

    (Institute for Fiscal Studies and MIT)

Abstract

There are many economic parameters that depend on nonparametric first steps. Examples include games, dynamic discrete choice, average consumer surplus, and treatment effects. Often estimators of these parameters are asymptotically equivalent to a sample average of an object referred to as the influence function. The influence function is useful in formulating regularity conditions for asymptotic normality, for bias reduction, in efficiency comparisons, and for analyzing robustness. We show that the influence function of a semiparametric estimator is the limit of a Gateaux derivative with respect to a smooth deviation as the deviation approaches a point mass. This result generalizes the classic Von Mises (1947) and Hampel (1974) calculation to apply to estimators that depend on smooth nonparametic first steps. We characterize the influence function of M and GMM-estimators.We apply the Gateaux derivative to derive the influence function with a first step nonparametric two stage least squares estimator based on orthogonality conditions. We also use the influence function to analyze high level and primitive regularity conditions for asymptotic normality. We give primitive regularity conditions for linear functionals of series regression that are the weakest known, except for a log term, when the regression function is smooth enough.

Suggested Citation

  • Hidehiko Ichimura & Whitney K. Newey, 2017. "The influence function of semiparametric estimators," CeMMAP working papers CWP06/17, Centre for Microdata Methods and Practice, Institute for Fiscal Studies.
  • Handle: RePEc:ifs:cemmap:06/17
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    References listed on IDEAS

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    Citations

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    Cited by:

    1. Kyle Colangelo & Ying-Ying Lee, 2020. "Double Debiased Machine Learning Nonparametric Inference with Continuous Treatments," Papers 2004.03036, arXiv.org, revised Dec 2021.
    2. Kyle Colangelo & Ying-Ying Lee, 2019. "Double debiased machine learning nonparametric inference with continuous treatments," CeMMAP working papers CWP72/19, Centre for Microdata Methods and Practice, Institute for Fiscal Studies.
    3. Victor Chernozhukov & Denis Chetverikov & Mert Demirer & Esther Duflo & Christian Hansen & Whitney Newey & James Robins, 2018. "Double/debiased machine learning for treatment and structural parameters," Econometrics Journal, Royal Economic Society, vol. 21(1), pages 1-68, February.
    4. Isaiah Andrews & Matthew Gentzkow & Jesse M. Shapiro, 2020. "On the Informativeness of Descriptive Statistics for Structural Estimates," Econometrica, Econometric Society, vol. 88(6), pages 2231-2258, November.
    5. Santiago Pereda Fernández, 2019. "Identification and estimation of triangular models with a binary treatment," Temi di discussione (Economic working papers) 1210, Bank of Italy, Economic Research and International Relations Area.
    6. Ruoyao Shi, 2021. "An Averaging Estimator for Two Step M Estimation in Semiparametric Models," Working Papers 202105, University of California at Riverside, Department of Economics.
    7. Victor Chernozhukov & Denis Chetverikov & Mert Demirer & Esther Duflo & Christian Hansen & Whitney Newey & James Robins, 2016. "Double/Debiased Machine Learning for Treatment and Causal Parameters," Papers 1608.00060, arXiv.org, revised Dec 2017.
    8. AmirEmad Ghassami & Numair Sani & Yizhen Xu & Ilya Shpitser, 2021. "Multiply Robust Causal Mediation Analysis with Continuous Treatments," Papers 2105.09254, arXiv.org.
    9. Kyle Colangelo & Ying-Ying Lee, 2019. "Double debiased machine learning nonparametric inference with continuous treatments," CeMMAP working papers CWP54/19, Centre for Microdata Methods and Practice, Institute for Fiscal Studies.
    10. Victor Chernozhukov & Juan Carlos Escanciano & Hidehiko Ichimura & Whitney K. Newey, 2016. "Locally robust semiparametric estimation," CeMMAP working papers CWP31/16, Centre for Microdata Methods and Practice, Institute for Fiscal Studies.
    11. Vira Semenova, 2020. "Generalized Lee Bounds," Papers 2008.12720, arXiv.org, revised Nov 2021.
    12. Yuichi Kitamura, 2020. "A Comment on: “On the Informativeness of Descriptive Statistics for Structural Estimates” by Isaiah Andrews, Matthew Gentzkow, and Jesse M. Shapiro," Econometrica, Econometric Society, vol. 88(6), pages 2265-2269, November.

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    More about this item

    Keywords

    Influence function; semiparametric estimation; NPIV;
    All these keywords.

    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
    • C20 - Mathematical and Quantitative Methods - - Single Equation Models; Single Variables - - - General
    • C26 - Mathematical and Quantitative Methods - - Single Equation Models; Single Variables - - - Instrumental Variables (IV) Estimation
    • C36 - Mathematical and Quantitative Methods - - Multiple or Simultaneous Equation Models; Multiple Variables - - - Instrumental Variables (IV) Estimation

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