”Inference in Weak factor models”

Abstract:

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 de-sparsified (or debiased) version of the

WF-SOFAR estimator of Uematsu and Yamagata (2020), but is also applicable to the

principal component (PC) estimator. After the factor selection, the re-sparsified WFSOFAR

and sparsified PC estimators are proposed and their consistency is established.

Finite sample evidence supports the theoretical results. We apply our procedure to the

FRED-MD macroeconomic and financial data, consisting of 128 series from June 1999

to May 2019. The results strongly suggest the existence of sparse factor loadings and

exhibit a clear association of each of the extracted factors with a group of macroeconomic

variables.