Large-dimensional Factor Analysis with Weighted PCA

TL;DR

This paper introduces a weighted PCA approach that improves the consistency of large-dimensional factor analysis, especially under weak factors or complex noise structures, by adaptively selecting optimal weights.

Contribution

It proposes a novel weighting scheme for PCA that achieves consistent and asymptotically normal estimators under weaker conditions than standard PCA.

Findings

Weighted PCA outperforms traditional PCA in simulations.

The method achieves consistency under weaker assumptions.

Numerical results validate the theoretical advantages.

Abstract

Principal component analysis (PCA) is arguably the most widely used approach for large-dimensional factor analysis. While it is effective when the factors are sufficiently strong, it can be inconsistent when the factors are weak and/or the noise has complex dependence structure. We argue that the inconsistency often stems from bias and introduce a general approach to restore consistency. Specifically, we propose a general weighting scheme for PCA and show that with a suitable choice of weighting matrices, it is possible to deduce consistent and asymptotic normal estimators under much weaker conditions than the usual PCA. While the optimal weight matrix may require knowledge about the factors and covariance of the idiosyncratic noise that are not known a priori, we develop an agnostic approach to adaptively choose from a large class of weighting matrices that can be viewed as PCA for weighted linear combinations of auto-covariances among the observations. Theoretical and numerical results demonstrate the merits of our methodology over the usual PCA and other recently developed techniques for large-dimensional approximate factor models.