Are Latent Factor Regression and Sparse Regression Adequate?
This work addresses the adequacy of common regression models for researchers and practitioners in statistics and machine learning, offering a hybrid approach and testing framework, but it is incremental as it builds on existing methods.
The authors tackled the problem of determining whether latent factor regression or sparse regression alone is sufficient for modeling, by proposing the Factor Augmented sparse linear Regression Model (FARM) that integrates both approaches. They developed tests to assess model adequacy and demonstrated robustness and effectiveness through numerical experiments on synthetic and macroeconomic data.
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.