Inference in Sparsity-Induced Weak Factor Models

Abstract

Abstract: 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, N. 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 re-sparsified SOFAR and sparsified 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&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.

Keywords: Approximate Factor Models, Debiased SOFAR Estimator, Multiple Testing, FDR and Power, Re-Sparsification

JEL Classification: C12, C13, C23, C38, C58