IDEAS home Printed from https://ideas.repec.org/p/crs/wpaper/2017-26.html
   My bibliography  Save this paper

Pivotal Estimation Via Self-Normalization for High-Dimensional Linear Models with Errors in Variables

Author

Listed:
  • Alexandre Belloni

    (Duke’s Fuqua School of Business)

  • Victor Chernozhukov

    (MIT)

  • Abhishek Kaul

    (IBM)

  • Mathieu Rosenbaum

    (CREST; CEMAP; Polytechnique)

  • Alexandre B. Tsybakov

    (CREST; CMC-ENSAE)

Abstract

We propose a new estimator for the high-dimensional linear regression model with measurement error in the design where the number of coefficients is potentially larger than the sample size. The main novelty of our procedure is that the choice of penalty parameters is pivotal. The estimator is based on applying a self-normalization to the constraints that characterize the estimator. Importantly, we show how to cast the computation of the estimator as the solution of a convex program with second order cone constraints. This allows the use of algorithms with theoretical guarantees and enables reliable implementation. Under sparsity assumptions, we derive lq-rates of convergence and show that consistency can be achieved even if the number of regressors exceeds the sample size. We further provide a simple thresholded estimator that yields a provably sparse estimator with similar l2 and l1-rates of convergence.

Suggested Citation

  • Alexandre Belloni & Victor Chernozhukov & Abhishek Kaul & Mathieu Rosenbaum & Alexandre B. Tsybakov, 2017. "Pivotal Estimation Via Self-Normalization for High-Dimensional Linear Models with Errors in Variables," Working Papers 2017-26, Center for Research in Economics and Statistics.
  • Handle: RePEc:crs:wpaper:2017-26
    as

    Download full text from publisher

    File URL: http://crest.science/RePEc/wpstorage/2017-26.pdf
    File Function: CREST working paper version
    Download Restriction: no
    ---><---

    References listed on IDEAS

    as
    1. Alexandre Belloni & Victor Chernozhukov & Abhishek Kaul, 2017. "Confidence bands for coefficients in high dimensional linear models with error-in-variables," CeMMAP working papers 22/17, Institute for Fiscal Studies.
    2. A. Belloni & V. Chernozhukov & L. Wang, 2011. "Square-root lasso: pivotal recovery of sparse signals via conic programming," Biometrika, Biometrika Trust, vol. 98(4), pages 791-806.
    3. Alexandre Belloni & Mathieu Rosenbaum & Alexandre B. Tsybakov, 2017. "Linear and conic programming estimators in high dimensional errors-in-variables models," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 79(3), pages 939-956, June.
    Full references (including those not matched with items on IDEAS)

    Citations

    Citations are extracted by the CitEc Project, subscribe to its RSS feed for this item.
    as


    Cited by:

    1. Alexandre Belloni & Victor Chernozhukov & Denis Chetverikov & Christian Hansen & Kengo Kato, 2018. "High-dimensional econometrics and regularized GMM," CeMMAP working papers CWP35/18, Centre for Microdata Methods and Practice, Institute for Fiscal Studies.

    Most related items

    These are the items that most often cite the same works as this one and are cited by the same works as this one.
    1. Galea, Manuel & de Castro, Mário, 2017. "Robust inference in a linear functional model with replications using the t distribution," Journal of Multivariate Analysis, Elsevier, vol. 160(C), pages 134-145.
    2. Alexandre Belloni & Victor Chernozhukov & Denis Chetverikov & Christian Hansen & Kengo Kato, 2018. "High-Dimensional Econometrics and Regularized GMM," Papers 1806.01888, arXiv.org, revised Jun 2018.
    3. Li, Mengyan & Li, Runze & Ma, Yanyuan, 2021. "Inference in high dimensional linear measurement error models," Journal of Multivariate Analysis, Elsevier, vol. 184(C).
    4. 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.
    5. Zemin Zheng & Jie Zhang & Yang Li, 2022. "L 0 -Regularized Learning for High-Dimensional Additive Hazards Regression," INFORMS Journal on Computing, INFORMS, vol. 34(5), pages 2762-2775, September.
    6. Umberto Amato & Anestis Antoniadis & Italia De Feis & Irene Gijbels, 2021. "Penalised robust estimators for sparse and high-dimensional linear models," Statistical Methods & Applications, Springer;Società Italiana di Statistica, vol. 30(1), pages 1-48, March.
    7. Alexandre Belloni & Victor Chernozhukov & Ivan Fernandez-Val & Christian Hansen, 2013. "Program evaluation with high-dimensional data," CeMMAP working papers CWP77/13, Centre for Microdata Methods and Practice, Institute for Fiscal Studies.
    8. Victor Chernozhukov & Christian Hansen & Yuan Liao, 2015. "A lava attack on the recovery of sums of dense and sparse signals," CeMMAP working papers CWP56/15, Centre for Microdata Methods and Practice, Institute for Fiscal Studies.
    9. Victor Chernozhukov & Denis Chetverikov & Mert Demirer & Esther Duflo & Christian Hansen & Whitney K. Newey, 2016. "Double machine learning for treatment and causal parameters," CeMMAP working papers 49/16, Institute for Fiscal Studies.
    10. Yumou Qiu & Jing Tao & Xiao‐Hua Zhou, 2021. "Inference of heterogeneous treatment effects using observational data with high‐dimensional covariates," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 83(5), pages 1016-1043, November.
    11. Alexandre Belloni & Victor Chernozhukov & Christian Hansen, 2011. "Estimation of treatment effects with high-dimensional controls," CeMMAP working papers CWP42/11, Centre for Microdata Methods and Practice, Institute for Fiscal Studies.
    12. Ziwei Zhu & Tengyao Wang & Richard J. Samworth, 2022. "High‐dimensional principal component analysis with heterogeneous missingness," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 84(5), pages 2000-2031, November.
    13. Fan, Jianqing & Feng, Yang & Xia, Lucy, 2020. "A projection-based conditional dependence measure with applications to high-dimensional undirected graphical models," Journal of Econometrics, Elsevier, vol. 218(1), pages 119-139.
    14. Zanhua Yin, 2020. "Variable selection for sparse logistic regression," Metrika: International Journal for Theoretical and Applied Statistics, Springer, vol. 83(7), pages 821-836, October.
    15. 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.
    16. Anindya Bhadra & Jyotishka Datta & Nicholas G. Polson & Brandon T. Willard, 2020. "Global-Local Mixtures: A Unifying Framework," Sankhya A: The Indian Journal of Statistics, Springer;Indian Statistical Institute, vol. 82(2), pages 426-447, August.
    17. Luke Mosley & Idris A. Eckley & Alex Gibberd, 2022. "Sparse temporal disaggregation," Journal of the Royal Statistical Society Series A, Royal Statistical Society, vol. 185(4), pages 2203-2233, October.
    18. Xie, Fang & Xu, Lihu & Yang, Youcai, 2017. "Lasso for sparse linear regression with exponentially β-mixing errors," Statistics & Probability Letters, Elsevier, vol. 125(C), pages 64-70.
    19. Fan, Jinlin & Zhang, Yaowu & Zhu, Liping, 2022. "Independence tests in the presence of measurement errors: An invariance law," Journal of Multivariate Analysis, Elsevier, vol. 188(C).
    20. Timothy B. Armstrong & Michal Kolesár & Soonwoo Kwon, 2020. "Bias-Aware Inference in Regularized Regression Models," Working Papers 2020-2, Princeton University. Economics Department..

    Corrections

    All material on this site has been provided by the respective publishers and authors. You can help correct errors and omissions. When requesting a correction, please mention this item's handle: RePEc:crs:wpaper:2017-26. See general information about how to correct material in RePEc.

    If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

    If CitEc recognized a bibliographic reference but did not link an item in RePEc to it, you can help with this form .

    If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your RePEc Author Service profile, as there may be some citations waiting for confirmation.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: Secretariat General (email available below). General contact details of provider: https://edirc.repec.org/data/crestfr.html .

    Please note that corrections may take a couple of weeks to filter through the various RePEc services.

    IDEAS is a RePEc service. RePEc uses bibliographic data supplied by the respective publishers.