Are Latent Factor Regression and Sparse Regression Adequate?
提出因子增强回归模型（FARM），统一了潜在因子回归和稀疏线性回归，并设计了检验方法来判断这两种模型是否充分，适用于高维数据场景。
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+ϑ) th moment, for all ϑ>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.