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Valid Post-Selection and Post-Regularization Inference: An Elementary, General Approach

Author

Listed:
  • Victor Chernozhukov

    () (Department of Economics, Massachusetts Institute of Technology, Cambridge, Massachusetts 02142)

  • Christian Hansen

    () (University of Chicago Booth School of Business, Chicago, Illinois 60637)

  • Martin Spindler

    () (Munich Center for the Economics of Aging, 80799 Munich, Germany)

Abstract

We present an expository, general analysis of valid post-selection or post-regularization inference about a low-dimensional target parameter in the presence of a very high-dimensional nuisance parameter that is estimated using selection or regularization methods. Our analysis provides a set of high-level conditions under which inference for the low-dimensional parameter based on testing or point estimation methods will be regular despite selection or regularization biases occurring in the estimation of the high-dimensional nuisance parameter. A key element is the use of so-called immunized or orthogonal estimating equations that are locally insensitive to small mistakes in the estimation of the high-dimensional nuisance parameter. As an illustration, we analyze affine-quadratic models and specialize these results to a linear instrumental variables model with many regressors and many instruments. We conclude with a review of other developments in post-selection inference and note that many can be viewed as special cases of the general encompassing framework of orthogonal estimating equations provided in this article.

Suggested Citation

  • 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.
  • Handle: RePEc:anr:reveco:v:7:y:2015:p:649-688
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    References listed on IDEAS

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    1. 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.
    2. Chao, John C. & Swanson, Norman R. & Hausman, Jerry A. & Newey, Whitney K. & Woutersen, Tiemen, 2012. "Asymptotic Distribution Of Jive In A Heteroskedastic Iv Regression With Many Instruments," Econometric Theory, Cambridge University Press, vol. 28(01), pages 42-86, February.
    3. Carrasco, Marine, 2012. "A regularization approach to the many instruments problem," Journal of Econometrics, Elsevier, vol. 170(2), pages 383-398.
    4. Chamberlain, Gary, 1987. "Asymptotic efficiency in estimation with conditional moment restrictions," Journal of Econometrics, Elsevier, vol. 34(3), pages 305-334, March.
    5. Hansen, Christian & Kozbur, Damian, 2014. "Instrumental variables estimation with many weak instruments using regularized JIVE," Journal of Econometrics, Elsevier, vol. 182(2), pages 290-308.
    6. Okui, Ryo, 2011. "Instrumental variable estimation in the presence of many moment conditions," Journal of Econometrics, Elsevier, vol. 165(1), pages 70-86.
    7. Victor Chernozhukov & Denis Chetverikov & Kengo Kato, 2012. "Gaussian approximations and multiplier bootstrap for maxima of sums of high-dimensional random vectors," Papers 1212.6906, arXiv.org, revised Jan 2018.
    8. Eric Gautier & Alexandre Tsybakov, 2011. "High-Dimensional Instrumental Variables Regression and Confidence Sets," Working Papers 2011-13, Center for Research in Economics and Statistics.
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    Citations

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    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.
    2. Victor Chernozhukov & Juan Carlos Escanciano & Hidehiko Ichimura & Whitney K. Newey, 2016. "Locally robust semiparametric estimation," CeMMAP working papers CWP31/16, Centre for Microdata Methods and Practice, Institute for Fiscal Studies.
    3. Victor Chernozhukov & Whitney Newey & James Robins, 2018. "Double/De-Biased Machine Learning Using Regularized Riesz Representers," Papers 1802.08667, arXiv.org, revised Mar 2018.
    4. Tom Boot & Didier Nibbering, 2017. "Inference in high-dimensional linear regression models," Tinbergen Institute Discussion Papers 17-032/III, Tinbergen Institute, revised 05 Jul 2017.
    5. Christian Hansen & Damian Kozbur & Sanjog Misra, 2016. "Targeted undersmoothing," ECON - Working Papers 282, Department of Economics - University of Zurich, revised Apr 2018.
    6. repec:nbr:nberch:14009 is not listed on IDEAS

    More about this item

    Keywords

    Neyman; orthogonalization; C (α) statistics; optimal instrument; optimal score; optimal moment; efficiency; optimality;

    JEL classification:

    • C18 - Mathematical and Quantitative Methods - - Econometric and Statistical Methods and Methodology: General - - - Methodolical Issues: General
    • C55 - Mathematical and Quantitative Methods - - Econometric Modeling - - - Large Data Sets: Modeling and Analysis
    • C26 - Mathematical and Quantitative Methods - - Single Equation Models; Single Variables - - - Instrumental Variables (IV) Estimation

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