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We propose the Factor Augmented sparse linear Regression Model (FARM) thatnot only encompasses both the latent factor regression and sparse linearregression as special cases but also bridges dimension reduction and sparseregression together. We provide theoretical guarantees for the estimation ofour model under the existence of sub-Gaussian and heavy-tailed noises (withbounded (1+x)-th moment, for all x>0), respectively. In addition, the existingworks on supervised learning often assume the latent factor regression or thesparse linear regression is the true underlying model without justifying itsadequacy. To fill in such an important gap, we also leverage our model as thealternative model to test the sufficiency of the latent factor regression andthe sparse linear regression models. To accomplish these goals, we propose theFactor-Adjusted de-Biased Test (FabTest) and a two-stage ANOVA type testrespectively. We also conduct large-scale numerical experiments including bothsynthetic and FRED macroeconomics data to corroborate the theoreticalproperties of our methods. Numerical results illustrate the robustness andeffectiveness of our model against latent factor regression and sparse linearregression models.

Are Latent Factor Regression and Sparse Regression Adequate?