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Valid post-selection and post-regularization inference: An elementary, general approach

Author

Listed:
  • Victor Chernozhukov

    () (Institute for Fiscal Studies and MIT)

  • Christian Hansen

    (Institute for Fiscal Studies and Chicago GSB)

  • Martin Spindler

    (Institute for Fiscal Studies)

Abstract

Here 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 which 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 estimation of the high-dimensional nuisance parameter. The results may be applied to establish uniform validity of post-selection or post-regularization inference procedures for low-dimensional target parameters over large classes of models. The high-level conditions allow one to clearly see the types of structure needed for achieving valid post-regularization inference and encompass many existing results. A key element of the structure we employ and discuss in detail is the use of orthogonal or "immunized" estimating equations that are locally insensitive to small mistakes in estimation of the high-dimensional nuisance parameter. As an illustration, we use the high-level conditions to provide readily veri able sucient conditions for a class of ane-quadratic models that include the usual linear model and linear instrumental variables model as special cases. As a further application and illustration, we use these results to provide an analysis of post-selection inference in 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 of the developments can be viewed as special cases of the general encompassing framework of orthogonal estimating equations provided in this paper.

Suggested Citation

  • Victor Chernozhukov & Christian Hansen & Martin Spindler, 2016. "Valid post-selection and post-regularization inference: An elementary, general approach," CeMMAP working papers CWP36/16, Centre for Microdata Methods and Practice, Institute for Fiscal Studies.
  • Handle: RePEc:ifs:cemmap:36/16
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    References listed on IDEAS

    as
    1. Carrasco, Marine, 2012. "A regularization approach to the many instruments problem," Journal of Econometrics, Elsevier, vol. 170(2), pages 383-398.
    2. 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.
    3. Eric Gautier & Alexandre Tsybakov, 2011. "High-Dimensional Instrumental Variables Regression and Confidence Sets," Working Papers 2011-13, Center for Research in Economics and Statistics.
    4. 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.
    5. 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.
    6. Chamberlain, Gary, 1987. "Asymptotic efficiency in estimation with conditional moment restrictions," Journal of Econometrics, Elsevier, vol. 34(3), pages 305-334, March.
    7. Okui, Ryo, 2011. "Instrumental variable estimation in the presence of many moment conditions," Journal of Econometrics, Elsevier, vol. 165(1), pages 70-86.
    8. 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.
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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. Christian Hansen & Damian Kozbur & Sanjog Misra, 2016. "Targeted undersmoothing," ECON - Working Papers 282, Department of Economics - University of Zurich, revised Apr 2018.
    4. repec:nbr:nberch:14009 is not listed on IDEAS
    5. 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.
    6. Victor Chernozhukov & Whitney K. Newey & James Robins, 2018. "Double/de-biased machine learning using regularized Riesz representers," CeMMAP working papers CWP15/18, Centre for Microdata Methods and Practice, Institute for Fiscal Studies.

    More about this item

    Keywords

    Neyman; orthogonalization; C( ) statistics; optimal instrument; optimal score; optimal moment; post-selection and post-regularization inference; eciency; 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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