Victor Chernozhukov, Christian Hansen, Martin Spindler
arXiv 14 Jan 2015 · Mathematics — Statistics Theory · publishedAnnual Review of Economics (2015) · 148 citations (OpenAlex)
arXiv:1501.03430 · PDF · DOI · OpenAlex · Extracted main text
Here we present an expository, general analysis of valid post-selection or post-regularization inference about a low-dimensional target parameter, $α$, in the presence of a very high-dimensional nuisance parameter, $η$, 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 $$M(α, η)=0$$ which identify $α$. Within this structure, we show that setting up such equations in a manner such that the orthogonality/immunization condition $$\partial_ηM(α, η) = 0$$ at the true parameter values is satisfied, coupled with plausible conditions on the smoothness of $M$ and the quality of the estimator $\hat η$, guarantees that inference on for the main parameter $α$ based on testing or point estimation methods discussed below will be regular despite selection or regularization biases occurring in estimation of $η$. In particular, the estimator of $α$ will often be uniformly consistent at the root-$n$ rate and uniformly asymptotically normal even though estimators $\hat η$ 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 $C(α)$ (orthogonal score) statistics.
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The works this paper leans on most, across its whole bibliography — not restricted to papers in our corpus. Ranked by composite intensity, which combines how often a work is mentioned, how many sections mention it, and how much of that falls in the main text rather than the appendix.
| Reference | Intensity | Mentions | Sections | Main text | |
|---|---|---|---|---|---|
| 1 | Belloni, Chernozhukov \ Kato (2013) `Uniform Post Selection Inference for LAD Regression Models and Other Z-estimation Problems', arXiv preprint arXiv:1304.0282 | 1.000 | 6 | 3 | 100% |
| 2 | Belloni, Chernozhukov \ Wei (2013) `Honest Confidence Regions for Logistic Regression with a Large Number of Controls', arXiv preprint arXiv:1304.3969 | 1.000 | 6 | 3 | 100% |
| 3 | Belloni, Chernozhukov, Fernández-Val \ Hansen (2013) `Program Evaluation with High-Dimensional Data', arXiv:1311.2645 | 0.928 | 4 | 3 | 100% |
| 4 | Belloni, Chernozhukov \ Hansen (2014) `Inference on Treatment Effects After Selection Amongst High-Dimensional Controls', Review of Economic Studies 81, 608–650 | 0.928 | 4 | 3 | 100% |
| 5 | Neyman (1959) Optimal asymptotic tests of composite statistical hypotheses, in U | 0.928 | 4 | 3 | 100% |
| 6 | Belloni, Chen, Chernozhukov \ Hansen (2012) `Sparse Models and Methods for Optimal Instruments with an Application to Eminent Domain', Econometrica 80, 2369–2429 self | 0.886 | 23 | 7 | 70% |
| 7 | Neyman (1979) `$C()$ tests and their use', Sankhya 41, 1–21 | 0.843 | 3 | 3 | 100% |
| 8 | Belloni, Chernozhukov, Hansen \ Kozbur (2014) `Inference in High Dimensional Panel Models with an Application to Gun Control', arXiv:1411.6507 | 0.737 | 4 | 3 | 50% |
| 9 | Belloni \ Chernozhukov (2013) `Least Squares After Model Selection in High-dimensional Sparse Models', Bernoulli 19(2), 521–547 | 0.737 | 3 | 3 | 67% |
| 10 | Berry, Levinsohn \ Pakes (1995) `Automobile Prices in Market Equilibrium', Econometrica 63, 841–890 | 0.693 | 5 | 1 | 100% |
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