F-test and z-test for high-dimensional regression models with a factor structure

The classic F-test and z-test can fail for high-dimensional regression models. This paper addresses this problem, especially for the case where the covariates contain a latent factor structure. We first use a new technique, the cross-section averages (CSA) of covariates, to estimate the latent factors. We then develop two F-type tests, namely, the Wald test and the F-test, to assess the overall significance of covariates. If the covariates are tested jointly significant, we next carry out a CSA-based z-test to sequentially test the significance of covariates one at a time. Compared with the existing approaches in the literature, which often use principal component analysis (PCA) to estimate the latent factors, the new tests do not depend on the accurate estimation of the unknown degrees of freedom, or on the acquisition of unknown eigenvalues. Therefore, they can reduce the uncertainty arising from the estimation of unknown quantities. We show the power and model selection consistency of these tests and propose a follow-up ratio-type test to further control the model size. Numerical simulations and a real data analysis show the competitive performance of these CSA-based tests.