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Adaptive Testing on a Regression Function at a Point

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Abstract

We consider the problem of inference on a regression function at a point when the entire function satisfies a sign or shape restriction under the null. We propose a test that achieves the optimal minimax rate adaptively over a range of Holder classes, up to a log log n term, which we show to be necessary for adaptation. We apply the results to adaptive one-sided tests for the regression discontinuity parameter under a monotonicity restriction, the value of a monotone regression function at the boundary, and the proportion of true null hypotheses in a multiple testing problem.

Suggested Citation

  • Timothy B. Armstrong, 2014. "Adaptive Testing on a Regression Function at a Point," Cowles Foundation Discussion Papers 1957R, Cowles Foundation for Research in Economics, Yale University, revised Feb 2015.
    • Handle: RePEc:cwl:cwldpp:1957r
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    1. Imbens, Guido W. & Lemieux, Thomas, 2008. "Regression discontinuity designs: A guide to practice," Journal of Econometrics, Elsevier, vol. 142(2), pages 615-635, February.
    2. Chamberlain, Gary, 1986. "Asymptotic efficiency in semi-parametric models with censoring," Journal of Econometrics, Elsevier, vol. 32(2), pages 189-218, July.
    3. Donald W. K. Andrews & Marcia M. A. Schafgans, 1998. "Semiparametric Estimation of the Intercept of a Sample Selection Model," The Review of Economic Studies, Review of Economic Studies Ltd, vol. 65(3), pages 497-517.
    4. Hahn, Jinyong & Todd, Petra & Van der Klaauw, Wilbert, 2001. "Identification and Estimation of Treatment Effects with a Regression-Discontinuity Design," Econometrica, Econometric Society, vol. 69(1), pages 201-209, January.
    5. John D. Storey, 2002. "A direct approach to false discovery rates," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 64(3), pages 479-498, August.
    6. Shakeeb Khan & Elie Tamer, 2010. "Irregular Identification, Support Conditions, and Inverse Weight Estimation," Econometrica, Econometric Society, vol. 78(6), pages 2021-2042, November.
    7. Heckman, James J, 1990. "Varieties of Selection Bias," American Economic Review, American Economic Association, vol. 80(2), pages 313-318, May.
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    Cited by:

    1. Patrick Kline & Evan K Rose & Christopher R Walters, 2022. "Systemic Discrimination Among Large U.S. Employers [“Teachers and Student Achievement in the Chicago Public High Schools,”]," The Quarterly Journal of Economics, Oxford University Press, vol. 137(4), pages 1963-2036.
    2. Byunghoon Kang, 2017. "Inference in Nonparametric Series Estimation with Data-Dependent Undersmoothing," Working Papers 170712442, Lancaster University Management School, Economics Department.
    3. Timothy B Armstrong & Michal Kolesár, 2018. "A Simple Adjustment for Bandwidth Snooping," The Review of Economic Studies, Review of Economic Studies Ltd, vol. 85(2), pages 732-765.
    4. Susanne M Schennach, 2020. "A Bias Bound Approach to Non-parametric Inference," Review of Economic Studies, Oxford University Press, vol. 87(5), pages 2439-2472.
    5. Xu, Ke-Li, 2020. "Inference of local regression in the presence of nuisance parameters," Journal of Econometrics, Elsevier, vol. 218(2), pages 532-560.
    6. Babii, Andrii & Kumar, Rohit, 2023. "Isotonic regression discontinuity designs," Journal of Econometrics, Elsevier, vol. 234(2), pages 371-393.
    7. Philipp Ketz & Adam McCloskey, 2021. "Short and Simple Confidence Intervals when the Directions of Some Effects are Known," Papers 2109.08222, arXiv.org.
    8. Koohyun Kwon & Soonwoo Kwon, 2020. "Adaptive Inference in Multivariate Nonparametric Regression Models Under Monotonicity," Papers 2011.14219, arXiv.org.
    9. Victor Chernozhukov & Whitney K. Newey & Andres Santos, 2023. "Constrained Conditional Moment Restriction Models," Econometrica, Econometric Society, vol. 91(2), pages 709-736, March.
    10. Patrick Kline & Christopher Walters, 2019. "Audits as Evidence: Experiments, Ensembles, and Enforcement," Papers 1907.06622, arXiv.org, revised Jul 2019.
    11. Patrick Kline & Evan K Rose & Christopher R Walters, 2023. "Systemic Discrimination Among Large U.S. Employers," The Quarterly Journal of Economics, President and Fellows of Harvard College, vol. 137(4), pages 1963-2036.
    12. Timothy B. Armstrong & Michal Kolesár, 2018. "Optimal Inference in a Class of Regression Models," Econometrica, Econometric Society, vol. 86(2), pages 655-683, March.
    13. Patrick Kline & Evan K Rose & Christopher R Walters, 2023. "Systemic Discrimination Among Large U.S. Employers," Journal of Economic Geography, Oxford University Press, vol. 137(4), pages 1963-2036.
    14. Koohyun Kwon & Soonwoo Kwon, 2020. "Inference in Regression Discontinuity Designs under Monotonicity," Papers 2011.14216, arXiv.org.
    15. Zheng Fang & Juwon Seo, 2019. "A Projection Framework for Testing Shape Restrictions That Form Convex Cones," Papers 1910.07689, arXiv.org, revised Sep 2021.
    16. Harold D. Chiang & Kengo Kato & Yuya Sasaki & Takuya Ura, 2021. "Linear programming approach to nonparametric inference under shape restrictions: with an application to regression kink designs," Papers 2102.06586, arXiv.org.

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    More about this item

    Keywords

    Adaptive inference; Regression discontinuity; Identification at infinity;
    All these keywords.

    JEL classification:

    • C14 - Mathematical and Quantitative Methods - - Econometric and Statistical Methods and Methodology: General - - - Semiparametric and Nonparametric Methods: General
    • C12 - Mathematical and Quantitative Methods - - Econometric and Statistical Methods and Methodology: General - - - Hypothesis Testing: General

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