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Pivotal Estimation in High-Dimensional Regression via Linear Programming

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  • Eric Gautier

    (CREST)

  • Alexandre B, Tsybakov

    (CREST)

Abstract

We propose a new method of estimation in high-dimensional linear regression model. It allows for very weak distributional assumptions including heteroscedasticity, and does not require the knowledge of the variance of random errors. The method is based on linear programming only, so that its numerical implementation is faster than for previously known techniques using conic programs, and it allows one to deal with higher dimensional models. We provide upper bounds for estimation and prediction errors of the proposed estimator showing that it achieves the same rate as in the more restrictive situation of fixed design and i.i.d. Gaussian errors with known variance. Following Gautier and Tsybakov (2011), we obtain the results under weaker sensitivity assumptions than the restricted eigenvalue or assimilated conditions

Suggested Citation

  • Eric Gautier & Alexandre B, Tsybakov, 2013. "Pivotal Estimation in High-Dimensional Regression via Linear Programming," Working Papers 2013-40, Center for Research in Economics and Statistics.
    • Handle: RePEc:crs:wpaper:2013-40
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    References listed on IDEAS

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    1. Eric Gautier & Alexandre Tsybakov, 2011. "High-Dimensional Instrumental Variables Regression and Confidence Sets," Working Papers 2011-13, Center for Research in Economics and Statistics.
    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. A. Belloni & D. Chen & V. Chernozhukov & C. Hansen, 2012. "Sparse Models and Methods for Optimal Instruments With an Application to Eminent Domain," Econometrica, Econometric Society, vol. 80(6), pages 2369-2429, November.
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    Cited by:

    1. 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.
    2. Shi, Zhentao, 2016. "Econometric estimation with high-dimensional moment equalities," Journal of Econometrics, Elsevier, vol. 195(1), pages 104-119.
    3. Zhentao Shi, 2016. "Estimation of Sparse Structural Parameters with Many Endogenous Variables," Econometric Reviews, Taylor & Francis Journals, vol. 35(8-10), pages 1582-1608, December.
    4. Belloni, Alexandre & Hansen, Christian & Newey, Whitney, 2022. "High-dimensional linear models with many endogenous variables," Journal of Econometrics, Elsevier, vol. 228(1), pages 4-26.
    5. Alexandre Belloni & Mathieu Rosenbaum & Alexandre Tsybakov, 2016. "An {l1, l2, l-infinity} Regularization Approach to High-Dimensional Errors-in-variables Models," Working Papers 2016-12, Center for Research in Economics and Statistics.
    6. Alexandre Belloni & Mathieu Rosenbaum & Alexandre B. Tsybakov, 2014. "Linear and Conic Programming Estimators in High-Dimensional Errors-in-variables Models," Working Papers 2014-34, Center for Research in Economics and Statistics.

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