Inference in Sparsity-Induced Weak Factor Models

In this paper, we consider statistical inference for high-dimensional approximate factor models. We posit a weak factor structure, in which the factor loading matrix can be sparse and the signal eigenvalues may diverge more slowly than the cross-sectional dimension, <i>N</i>. We propose a novel inferential procedure to decide whether each component of the factor loadings is zero or not, and prove that this controls the false discovery rate (FDR) below a pre-assigned level, while the power tends to unity. This “factor selection” procedure is primarily based on a debiased version of the SOFAR estimator of Uematsu and Yamagata (2021), but is also applicable to the principal component (PC) estimator. After the factor selection, the <i>re-sparsified</i> SOFAR and <i>sparsified</i> PC estimators are proposed and their consistency is established. Finite sample evidence supports the theoretical results. We apply our method to the FRED-MD dataset of macroeconomic variables and the monthly firm-level excess returns which constitute the S&amp;P 500 index. The results give very strong statistical evidence of sparse factor loadings under the identification restrictions and exhibit clear associations of factors and categories of the variables. Furthermore, our method uncovers a very weak but statistically significant factor in the residuals of Fama-French five factor regression.