  This paper studies the principal component (PC) method-based estimation of
weak factor models with sparse loadings. We uncover an intrinsic near-sparsity
preservation property for the PC estimators of loadings, which comes from the
approximately upper triangular (block) structure of the rotation matrix. It
implies an asymmetric relationship among factors: the rotated loadings for a
stronger factor can be contaminated by those from a weaker one, but the
loadings for a weaker factor is almost free of the impact of those from a
stronger one. More importantly, the finding implies that there is no need to
use complicated penalties to sparsify the loading estimators. Instead, we adopt
a simple screening method to recover the sparsity and construct estimators for
various factor strengths. In addition, for sparse weak factor models, we
provide a singular value thresholding-based approach to determine the number of
factors and establish uniform convergence rates for PC estimators, which
complement Bai and Ng (2023). The accuracy and efficiency of the proposed
estimators are investigated via Monte Carlo simulations. The application to the
FRED-QD dataset reveals the underlying factor strengths and loading sparsity as
well as their dynamic features.
