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

Victor Chernozhukov, Christian Hansen and Martin Spindler

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Abstract: 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.

Date: 2015-01, Revised 2015-08
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Citations: View citations in EconPapers (61)

Published in Annual Review of Economics, Vol. 7: 649-688 (August 2015)

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Related works:
Working Paper: Valid post-selection and post-regularization inference: An elementary, general approach (2016)
Working Paper: Valid post-selection and post-regularization inference: An elementary, general approach (2016)
Journal Article: Valid Post-Selection and Post-Regularization Inference: An Elementary, General Approach (2015)
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Persistent link: https://EconPapers.repec.org/RePEc:arx:papers:1501.03430

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