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Inference for best linear approximations to set identified functions

Author

Listed:
  • Arun Chandrasekhar

    (Institute for Fiscal Studies)

  • Victor Chernozhukov

    () (Institute for Fiscal Studies and MIT)

  • Francesca Molinari

    () (Institute for Fiscal Studies and Cornell University)

  • Paul Schrimpf

    () (Institute for Fiscal Studies and The University of British Columbia)

Abstract

This paper provides inference methods for best linear approximations to functions which are known to lie within a band. It extends the partial identification literature by allowing the upper and lower functions defining the band to be any functions, including ones carrying an index, which can be estimated parametrically or non-parametrically. The identification region of the parameters of the best linear approximation is characterised via its support function, and limit theory is developed for the latter. We prove that the support function approximately converges to a Gaussian process, and validity of the Bayesian bootstrap is established. The paper nests as special cases the canonical examples in the literature: mean regression with interval valued outcome data and interval valued regressor data. Because the bounds may carry an index, the paper covers problems beyond mean regression; the framework is extremely versatile. Applications include quantile and distribution regression with interval valued data, sample selection problems, as well as mean, quantile and distribution treatment effects. Moreover, the framework can account for the availability of instruments. An application is carried out, studying female labor force participation along the lines of Mulligan and Rubinstein (2008).

Suggested Citation

  • Arun Chandrasekhar & Victor Chernozhukov & Francesca Molinari & Paul Schrimpf, 2012. "Inference for best linear approximations to set identified functions," CeMMAP working papers CWP43/12, Centre for Microdata Methods and Practice, Institute for Fiscal Studies.
  • Handle: RePEc:ifs:cemmap:43/12
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    File URL: http://www.cemmap.ac.uk/wps/cwp431212.pdf
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    References listed on IDEAS

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    Citations

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    Cited by:

    1. Liao, Yuan & Simoni, Anna, 2012. "Semi-parametric Bayesian Partially Identified Models based on Support Function," MPRA Paper 43262, University Library of Munich, Germany.
    2. Karun Adusumilli & Taisuke Otsu, 2017. "Empirical Likelihood for Random Sets," Journal of the American Statistical Association, Taylor & Francis Journals, vol. 112(519), pages 1064-1075, July.
    3. Vira Semenova, 2017. "Machine Learning for Set-Identified Linear Models," Papers 1712.10024, arXiv.org, revised Dec 2019.
    4. Otsu, Taisuke & Xu, Ke-Li & Matsushita, Yukitoshi, 2015. "Empirical likelihood for regression discontinuity design," Journal of Econometrics, Elsevier, vol. 186(1), pages 94-112.
    5. Lee, Ying-Ying & Bhattacharya, Debopam, 2019. "Applied welfare analysis for discrete choice with interval-data on income," Journal of Econometrics, Elsevier, vol. 211(2), pages 361-387.
    6. Kaido, Hiroaki, 2017. "Asymptotically Efficient Estimation Of Weighted Average Derivatives With An Interval Censored Variable," Econometric Theory, Cambridge University Press, vol. 33(5), pages 1218-1241, October.
    7. Hoshino, Tadao, 2013. "Partial identification in binary response models with nonignorable nonresponses," Economics Letters, Elsevier, vol. 121(1), pages 74-78.
    8. repec:cep:stiecm:/2014/574 is not listed on IDEAS
    9. Maasoumi, Esfandiar & Wang, Le, 2017. "What can we learn about the racial gap in the presence of sample selection?," Journal of Econometrics, Elsevier, vol. 199(2), pages 117-130.
    10. Juan Carlos Escanciano & Lin Zhu, 2013. "Set inferences and sensitivity analysis in semiparametric conditionally identified models," CeMMAP working papers CWP55/13, Centre for Microdata Methods and Practice, Institute for Fiscal Studies.
    11. Vira Semenova, 2020. "Better Lee Bounds," Papers 2008.12720, arXiv.org.
    12. Yuan Liao & Anna Simoni, 2016. "Bayesian Inference for Partially Identified Convex Models: Is it Valid for Frequentist Inference?," Departmental Working Papers 201607, Rutgers University, Department of Economics.

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    More about this item

    Keywords

    Set identified function; best linear approximation; partial identification; support function; bayesian bootstrap; convex set;
    All these keywords.

    JEL classification:

    • C13 - Mathematical and Quantitative Methods - - Econometric and Statistical Methods and Methodology: General - - - Estimation: General
    • C31 - Mathematical and Quantitative Methods - - Multiple or Simultaneous Equation Models; Multiple Variables - - - Cross-Sectional Models; Spatial Models; Treatment Effect Models; Quantile Regressions; Social Interaction Models

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