Estimation of Sparsity-Induced Weak Factor Models

Abstract

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 a data covariance matrix grows proportionally to N^alpha_k with unknown exponents 0 < alpha_k <= 1 for k=1,...,r. Employing the same rotation of the principal components (PC) estimator, the growth rate alpha_k 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 method of sparse orthogonal factor regression (SOFAR) by Uematsu et al (2018) to estimate the sWF models and derive the estimation error bound. Importantly, our method also yields consistent estimation of alpha_k. 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 the results demonstrate that our method outperforms that based on the PC. We also analyze S&P500 firm security returns and find that the first factor is consistently near strong while the others are weak.

Keywords: Sparsity-induced weak factor models, (Adaptive) SOFAR estimator, Estimation error bound, Estimating diverging exponents, Interpreting factors, Group factor structure.

JEL Classification: C13, C23, C38, C58