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Uniform post selection inference for LAD regression and other Z-estimation problems

Author

Listed:
  • Alexandre Belloni

    (Institute for Fiscal Studies)

  • Victor Chernozhukov

    (Institute for Fiscal Studies and MIT)

  • Kengo Kato

    (Institute for Fiscal Studies)

Abstract

We develop uniformly valid confidence regions for regression coefficients in a high-dimensional sparse median regression model with homoscedastic errors. Our methods are based on a moment equation that is immunized against non-regular estimation of the nuisance part of the median regression function by using Neyman’s orthogonalization. We establish that the resulting instrumental median regression estimator of a target regression coefficient is asymptotically normally distributed uniformly with respect to the underlying sparse model and is semiparametrically efficient. We also generalize our method to a general non-smooth Z-estimation framework with the number of target parameters p1 being possibly much larger than the sample size n. We extend Huber’s results on asymptotic normality to this setting, demonstrating uniform asymptotic normality of the proposed estimators over p1-dimensional rectangles, constructing simultaneous confidence bands on all of the p1 target parameters, and establishing asymptotic validity of the bands uniformly over underlying approximately sparse models.

Suggested Citation

  • Alexandre Belloni & Victor Chernozhukov & Kengo Kato, 2014. "Uniform post selection inference for LAD regression and other Z-estimation problems," CeMMAP working papers CWP51/14, Centre for Microdata Methods and Practice, Institute for Fiscal Studies.
  • Handle: RePEc:ifs:cemmap:51/14
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    Cited by:

    1. Rahul Singh, 2021. "Kernel Ridge Riesz Representers: Generalization Error and Mis-specification," Papers 2102.11076, arXiv.org, revised May 2024.
    2. Alexandre Belloni & Victor Chernozhukov & Kengo Kato, 2019. "Valid Post-Selection Inference in High-Dimensional Approximately Sparse Quantile Regression Models," Journal of the American Statistical Association, Taylor & Francis Journals, vol. 114(526), pages 749-758, April.
    3. Alexandre Belloni & Victor Chernozhukov & Abhishek Kaul, 2017. "Confidence bands for coefficients in high dimensional linear models with error-in-variables," CeMMAP working papers CWP22/17, Centre for Microdata Methods and Practice, Institute for Fiscal Studies.
    4. Hansen, Christian & Liao, Yuan, 2019. "The Factor-Lasso And K-Step Bootstrap Approach For Inference In High-Dimensional Economic Applications," Econometric Theory, Cambridge University Press, vol. 35(3), pages 465-509, June.
    5. Yanqin Fan & Fang Han & Wei Li & Xiao-Hua Zhou, 2019. "On rank estimators in increasing dimensions," Papers 1908.05255, arXiv.org.
    6. Jiaying Gu & Stanislav Volgushev, 2018. "Panel Data Quantile Regression with Grouped Fixed Effects," Papers 1801.05041, arXiv.org, revised Aug 2018.
    7. Victor Chernozhukov & Wolfgang Härdle & Chen Huang & Weining Wang, 2018. "LASSO-driven inference in time and space," CeMMAP working papers CWP36/18, Centre for Microdata Methods and Practice, Institute for Fiscal Studies.
    8. Alexandre Belloni & Victor Chernozhukov & Christian Hansen, 2014. "High-Dimensional Methods and Inference on Structural and Treatment Effects," Journal of Economic Perspectives, American Economic Association, vol. 28(2), pages 29-50, Spring.
    9. Yuya Sasaki & Takuya Ura & Yichong Zhang, 2022. "Unconditional quantile regression with high‐dimensional data," Quantitative Economics, Econometric Society, vol. 13(3), pages 955-978, July.
    10. S Klaassen & J Kueck & M Spindler & V Chernozhukov, 2023. "Uniform inference in high-dimensional Gaussian graphical models," Biometrika, Biometrika Trust, vol. 110(1), pages 51-68.
    11. Fan, Yanqin & Han, Fang & Li, Wei & Zhou, Xiao-Hua, 2020. "On rank estimators in increasing dimensions," Journal of Econometrics, Elsevier, vol. 214(2), pages 379-412.
    12. Alexandre Belloni & Victor Chernozhukov & Lie Wang, 2013. "Pivotal estimation via square-root lasso in nonparametric regression," CeMMAP working papers CWP62/13, Centre for Microdata Methods and Practice, Institute for Fiscal Studies.
    13. Victor Chernozhukov & Christian Hansen & Martin Spindler, 2015. "Valid Post-Selection and Post-Regularization Inference: An Elementary, General Approach," Annual Review of Economics, Annual Reviews, vol. 7(1), pages 649-688, August.
    14. Doğan, Osman & Taşpınar, Süleyman & Bera, Anil K., 2021. "A Bayesian robust chi-squared test for testing simple hypotheses," Journal of Econometrics, Elsevier, vol. 222(2), pages 933-958.
    15. Alexandre Belloni & Victor Chernozhukov & Ying Wei, 2016. "Post-Selection Inference for Generalized Linear Models With Many Controls," Journal of Business & Economic Statistics, Taylor & Francis Journals, vol. 34(4), pages 606-619, October.
    16. Victor Chernozhukov & Christian Hansen & Martin Spindler, 2015. "Post-Selection and Post-Regularization Inference in Linear Models with Many Controls and Instruments," American Economic Review, American Economic Association, vol. 105(5), pages 486-490, May.
    17. Victor Chernozhukov & Chris Hansen & Martin Spindler, 2016. "High-Dimensional Metrics in R," Papers 1603.01700, arXiv.org, revised Aug 2016.
    18. Victor Chernozhukov & Whitney Newey & Rahul Singh & Vasilis Syrgkanis, 2020. "Adversarial Estimation of Riesz Representers," Papers 2101.00009, arXiv.org, revised Apr 2024.
    19. Victor Chernozhukov & Whitney K. Newey & Rahul Singh, 2022. "Automatic Debiased Machine Learning of Causal and Structural Effects," Econometrica, Econometric Society, vol. 90(3), pages 967-1027, May.
    20. Sven Klaassen & Jannis Kueck & Martin Spindler, 2017. "Transformation Models in High-Dimensions," Papers 1712.07364, arXiv.org.
    21. Jelena Bradic & Victor Chernozhukov & Whitney K. Newey & Yinchu Zhu, 2019. "Minimax Semiparametric Learning With Approximate Sparsity," Papers 1912.12213, arXiv.org, revised Aug 2022.
    22. Chen, Le-Yu & Lee, Sokbae, 2023. "Sparse quantile regression," Journal of Econometrics, Elsevier, vol. 235(2), pages 2195-2217.
    23. Zhentao Shi & Jingyi Huang, 2019. "Forward-Selected Panel Data Approach for Program Evaluation," Papers 1908.05894, arXiv.org, revised Apr 2021.
    24. Matthew Backus & Sida Peng, 2019. "On Testing Continuity and the Detection of Failures," NBER Working Papers 26016, National Bureau of Economic Research, Inc.

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    More about this item

    Keywords

    Instrument; post-selection inference; sparsity; Neyman's Orthogonal Score test; uniformly valid inference; Z-estimation;
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