Inference for low-rank models
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Published version
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Author(s)
Chernozhukov, Victor
Hansen, Christian
Liao, Yuan
Zhu, Yinchu
Date Issued
June 2023
Journal
The Annals of Statistics
Publisher
Institute of Mathematical Statistics
Citation
Victor Chernozhukov. Christian Hansen. Yuan Liao. Yinchu Zhu. "Inference for low-rank models." Ann. Statist. 51 (3) 1309 - 1330, June 2023.
Version
Final published version
Abstract
This paper studies inference in linear models with a high-dimensional parameter matrix that can be well approximated by a “spiked low-rank matrix.” A spiked low-rank matrix has rank that grows slowly compared to its dimensions and nonzero singular values that diverge to infinity. We show that this framework covers a broad class of models of latent variables, which can accommodate matrix completion problems, factor models, varying coefficient models and heterogeneous treatment effects. For inference, we apply a procedure that relies on an initial nuclear-norm penalized estimation step followed by two ordinary least squares regressions. We consider the framework of estimating incoherent eigenvectors and use a rotation argument to argue that the eigenspace estimation is asymptotically unbiased. Using this framework, we show that our procedure provides asymptotically normal inference and achieves the semiparametric efficiency bound. We illustrate our framework by providing low-level conditions for its application in a treatment effects context where treatment assignment might be strongly dependent.
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DOI of Published Version
https://doi.org/10.1214/23-aos2293
