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


  • Victor Chernozhukov
  • Christian Hansen
  • Martin Spindler


Here we present an expository, general analysis of valid post-selection or post-regularization inference about a low-dimensional target parameter, $\alpha$, in the presence of a very high-dimensional nuisance parameter, $\eta$, which is estimated using modern selection or regularization methods. Our analysis relies on high-level, easy-to-interpret conditions that allow one to clearly see the structures needed for achieving valid post-regularization inference. Simple, readily verifiable sufficient conditions are provided for a class of affine-quadratic models. We focus our discussion on estimation and inference procedures based on using the empirical analog of theoretical equations $$M(\alpha, \eta)=0$$ which identify $\alpha$. Within this structure, we show that setting up such equations in a manner such that the orthogonality/immunization condition $$\partial_\eta M(\alpha, \eta) = 0$$ at the true parameter values is satisfied, coupled with plausible conditions on the smoothness of $M$ and the quality of the estimator $\hat \eta$, guarantees that inference on for the main parameter $\alpha$ based on testing or point estimation methods discussed below will be regular despite selection or regularization biases occurring in estimation of $\eta$. In particular, the estimator of $\alpha$ will often be uniformly consistent at the root-$n$ rate and uniformly asymptotically normal even though estimators $\hat \eta$ will generally not be asymptotically linear and regular. The uniformity holds over large classes of models that do not impose highly implausible "beta-min" conditions. We also show that inference can be carried out by inverting tests formed from Neyman's $C(\alpha)$ (orthogonal score) statistics.

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  • Victor Chernozhukov & Christian Hansen & Martin Spindler, 2015. "Valid Post-Selection and Post-Regularization Inference: An Elementary, General Approach," Papers 1501.03430,, revised Aug 2015.
  • Handle: RePEc:arx:papers:1501.03430

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    1. repec:eme:aecozz:s0731-905320140000034020 is not listed on IDEAS
    2. Eric Gautier & Alexandre Tsybakov, 2011. "High-Dimensional Instrumental Variables Regression and Confidence Sets," Working Papers 2011-13, Center for Research in Economics and Statistics.
    3. 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.
    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. Carrasco, Marine, 2012. "A regularization approach to the many instruments problem," Journal of Econometrics, Elsevier, vol. 170(2), pages 383-398.
    6. Chamberlain, Gary, 1987. "Asymptotic efficiency in estimation with conditional moment restrictions," Journal of Econometrics, Elsevier, vol. 34(3), pages 305-334, March.
    7. 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.
    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,, revised Jan 2018.
    9. Okui, Ryo, 2011. "Instrumental variable estimation in the presence of many moment conditions," Journal of Econometrics, Elsevier, vol. 165(1), pages 70-86.
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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. repec:oup:biomet:v:105:y:2018:i:3:p:645-664. is not listed on IDEAS
    4. Neng-Chieh Chang, 2018. "Semiparametric Difference-in-Differences with Potentially Many Control Variables," Papers 1812.10846,, revised Jan 2019.
    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
    7. 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.
    8. 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.
    9. Yanqin Fan & Fang Han & Wei Li & Xiao-Hua Zhou, 2019. "On rank estimators in increasing dimensions," Papers 1908.05255,
    10. Victor Chernozhukov & Whitney Newey & Vira Semenova, 2019. "Inference on weighted average value function in high-dimensional state space," Papers 1908.09173,
    11. Victor Chernozhukov & Whitney Newey & James Robins & Rahul Singh, 2018. "Double/De-Biased Machine Learning of Global and Local Parameters Using Regularized Riesz Representers," Papers 1802.08667,, revised Sep 2019.
    12. repec:eee:ecosta:v:9:y:2019:i:c:p:1-16 is not listed on IDEAS

    More about this item

    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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