  We propose the Factor Augmented sparse linear Regression Model (FARM) that
not only encompasses 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+x)-th moment, for all x>0), respectively. In addition, the existing
works on supervised learning often assume the latent factor regression or the
sparse linear regression is the true underlying model without justifying its
adequacy. To fill in such an important gap, 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 de-Biased 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.
