# Approximate Factor Models with Weaker Loadings

arXiv: https://arxiv.org/abs/2109.03773 · HTML: https://mlsanity.com/paper/2109.03773/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 8 Sep 2021.

## Built on by (most cited first, 2 of 2)

- [Enhancing Spectral Embedding through Robust and Flexible Knowledge Transfer in Electronic Health Records](https://mlsanity.com/paper/2606.11570/) (2026)
- [Unveiling Invariant and Transferable Latent Factors Across Heterogeneous Environments via ATLAS](https://mlsanity.com/paper/2607.18209/) (2026)

## Abstract (arXiv)

Pervasive cross-section dependence is increasingly recognized as a characteristic of economic data and the approximate factor model provides a useful framework for analysis. Assuming a strong factor structure where LopLo/Nα is positive definite in the limit when α=1, early work established convergence of the principal component estimates of the factors and loadings up to a rotation matrix. This paper shows that the estimates are still consistent and asymptotically normal when α∈(0,1] albeit at slower rates and under additional assumptions on the sample size. The results hold whether α is constant or varies across factor loadings. The framework developed for heterogeneous loadings and the simplified proofs that can be also used in strong factor analysis are of independent interest.