Are Latent Factor Regression and Sparse Regression Adequate?

Abstract

We propose the Factor Augmented (sparse linear) Regression Model (FARM) that not only admits both the latent factor regression and sparse linear regression as special cases but also bridges dimension reduction and sparse regression together. We provide theoretical guarantees for the estimation of our model under the existence of sub-Gaussian and heavy-tailed noises (with bounded $(1+\vartheta )$ th moment, for all $\vartheta >0$), respectively. In addition, the existing works on supervised learning often assume the latent factor regression or sparse linear regression is the true underlying model without justifying its adequacy. To fill in such an important gap on high-dimensional inference, we also leverage our model as the alternative model to test the sufficiency of the latent factor regression and the sparse linear regression models. To accomplish these goals, we propose the Factor-Adjusted deBiased Test (FabTest) and a two-stage ANOVA type test, respectively. We also conduct large-scale numerical experiments including both synthetic and FRED macroeconomics data to corroborate the theoretical properties of our methods. Numerical results illustrate the robustness and effectiveness of our model against latent factor regression and sparse linear regression models. Supplementary materials for this article are available online.

KEYWORDS:

• Factor model
• High-dimensional inference
• Hypothesis
• Sparse linear regression
• Robustness

Supplementary Materials

The supplementary material contains additional simulation results, technical proofs, and relevant codes for implementing the methodology of this paper.

Additional information

Funding

The research is supported in part by the ONR grant N00014-22-1-2340, NSF grants DMS-2210833, DMS-2053832, DMS-2052926 and NIH grant 2R01-GM072611-16