Estimation of Weak Factor Models

This paper investigates estimation of sparsity-induced weak factor (sWF) models, with large cross-sectional and time-series dimensions (N and T, respectively). It assumes that the kth largest eigenvalue of data covariance matrix grows proportionally to N^ak with unknown exponents 0 < ak <= 1 for k=1,...,r. Employing the same rotation of the principal component (PC) estimator, in the sWF models the growth rate ak is linked to the degree of sparsity of kth factor loadings. This is much weaker than the typical assumption on the recent factor models, in which all the r largest eigenvalues diverge proportionally to N. We apply the SOFAR method of Uematsu et al. (2019) to estimate the sWF models and derive the estimation error bound. Importantly, our method yields consistent estimation of ak's as well. A finite sample experiment shows that the performance of the new estimator uniformly dominates that of the PC estimator. We apply our method to forecasting bond yields and results demonstrate that our method outperforms that based on the PC. In another application we analyze S&P500 firm security returns and find that the first factor is consistently near strong while the others are indeed weak