August 2023 Bridging factor and sparse models
Jianqing Fan, Ricardo P. Masini, Marcelo C. Medeiros
Author Affiliations +
Ann. Statist. 51(4): 1692-1717 (August 2023). DOI: 10.1214/23-AOS2304


Factor and sparse models are widely used to impose a low-dimensional structure in high-dimensions. However, they are seemingly mutually exclusive. We propose a lifting method that combines the merits of these two models in a supervised learning methodology that allows for efficiently exploring all the information in high-dimensional datasets. The method is based on a flexible model for high-dimensional panel data with observable and/or latent common factors and idiosyncratic components. The model is called the factor-augmented regression model. It includes principal components and sparse regression as specific models, significantly weakens the cross-sectional dependence, and facilitates model selection and interpretability. The method consists of several steps and a novel test for (partial) covariance structure in high dimensions to infer the remaining cross-section dependence at each step. We develop the theory for the model and demonstrate the validity of the multiplier bootstrap for testing a high-dimensional (partial) covariance structure. A simulation study and applications support the theory.

Funding Statement

The first author was supported by ONR grant N00014-22-1-2340 and NSF grants DMS-2210833, DMS-2053832, and DMS-2052926.


The authors are grateful to Caio Almeida, Federico Bandi, Matteo Barigozzi, Gilberto Boareto, Gustavo Bulhões, Giuseppe Cavaliere, Frank Diebold, Bruno Ferman, Marcelo Fernandes, Claudio Flores, Conrado Garcia, Eric Ghysels, Alexander Giessing, Nathalie Gimenes, Marcelo J. Moreira, Henrique Pires, Yuri Saporito, and Rodrigo Targino for helpful comments. We also thank seminar participants at the SofiE online seminar series, Princeton University, the University of Amsterdam, the University of Pennsylvania, the University of Illinois at Urbana-Champaign, the Federal University of São Carlos, Rutgers University, the University of North Carolina at Chapel Hill, the University of Chicago, the University of California at Riverside, and Columbia University for several valuable comments. Finally, we are deeply grateful to Michele Lenza, Eduardo F. Mendes, and Michael Wolf for the careful reading of the paper and the many insightful discussions which led to a much-improved version of this manuscript. This manuscript has also been presented at many conferences, and we thank all the participants for their very useful comments.


Download Citation

Jianqing Fan. Ricardo P. Masini. Marcelo C. Medeiros. "Bridging factor and sparse models." Ann. Statist. 51 (4) 1692 - 1717, August 2023.


Received: 1 August 2022; Revised: 1 April 2023; Published: August 2023
First available in Project Euclid: 19 October 2023

Digital Object Identifier: 10.1214/23-AOS2304

Primary: 62F03 , 62H25

Keywords: (partial) covariance structure , factor models , high-dimensional inference , machine learning , penalized least-squares , prediction , supervised learning

Rights: Copyright © 2023 Institute of Mathematical Statistics


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Vol.51 • No. 4 • August 2023
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