[Paper Review] Large-dimensional Factor Analysis without Moment Constraints

Yong He, Xinbing Kong+2 more|arXiv (Cornell University)|Aug 14, 2019

Spatial and Panel Data Analysis25 references4 citations

TL;DR

This paper proposes a robust two-step factor analysis method for large-dimensional data without moment constraints by using spatial Kendall’s tau matrix for factor space estimation and OLS regression for factor scores. Under elliptical distributions, the method achieves consistent estimation of factor loadings, scores, and common components even under heavy-tailed data, with established convergence rates and superior finite-sample performance over classical PCA.

ABSTRACT

Large-dimensional factor model has drawn much attention in the big-data era, in order to reduce the dimensionality and extract underlying features using a few latent common factors. Conventional methods for estimating the factor model typically requires finite fourth moment of the data, which ignores the effect of heavy-tailedness and thus may result in unrobust or even inconsistent estimation of the factor space and common components. In this paper, we propose to recover the factor space by performing principal component analysis to the spatial Kendall's tau matrix instead of the sample covariance matrix. In a second step, we estimate the factor scores by the ordinary least square (OLS) regression. Theoretically, we show that under the elliptical distribution framework the factor loadings and scores as well as the common components can be estimated consistently without any moment constraint. The convergence rates of the estimated factor loadings, scores and common components are provided. The finite sample performance of the proposed procedure is assessed through thorough simulations. An analysis of a financial data set of asset returns shows the superiority of the proposed method over the classical PCA method.

Motivation & Objective

• To address the inconsistency of classical factor analysis methods under heavy-tailed data, which rely on finite fourth-moment assumptions.
• To develop a factor estimation procedure that remains consistent and robust when data exhibit heavy-tailed behavior common in finance and genomics.
• To eliminate moment constraints on factors and idiosyncratic errors while maintaining consistency in estimating factor loadings, scores, and common components.
• To provide a theoretically grounded, robust alternative to classical PCA and OLS-based factor models in high-dimensional settings.

Proposed method

• Estimate the factor space by performing principal component analysis (PCA) on the spatial Kendall’s tau matrix instead of the sample covariance matrix.
• Use the spatial Kendall’s tau matrix, which preserves the eigenspace of the scatter matrix under elliptical distributions, enabling robustness to heavy tails.
• Apply ordinary least squares (OLS) regression in the second step to estimate factor scores using the estimated factor loadings from the first step.
• Leverage the polarization property of elliptical distributions to ensure consistency of factor scores up to orthogonal transformation.
• Establish convergence rates for estimated factor loadings, scores, and common components under the elliptical factor model framework.
• Use theoretical tools including Cauchy-Schwarz inequality and asymptotic properties of eigen-decompositions to prove consistency and convergence rates.

Experimental results

Research questions

• RQ1 Can factor space estimation remain consistent under heavy-tailed data when fourth moments are not assumed?
• RQ2 Does replacing the sample covariance matrix with the spatial Kendall’s tau matrix yield robust factor loadings and scores in large-dimensional factor models?
• RQ3 What are the convergence rates of the estimated factor loadings, scores, and common components under the elliptical factor model without moment constraints?
• RQ4 How does the proposed robust two-step procedure compare to classical PCA in finite samples under heavy-tailed distributions?
• RQ5 Can consistent estimation of common components be achieved without assuming bounded moments on factors and idiosyncratic errors?

Key findings

• The proposed robust two-step (RTS) procedure achieves consistent estimation of factor loadings and scores under the elliptical factor model without any moment constraints on the data.
• The convergence rate for estimated factor loadings is $ O_p(1/n + 1/p^2) $, and for factor scores it is $ O_p(1/n^2 + 1/p) $, demonstrating strong finite-sample performance.
• The common components are consistently estimated with a convergence rate of $ O_p(1/n + 1/p) $, confirming the robustness of the method.
• Simulations show that the RTS method significantly outperforms classical PCA in terms of bias and dispersion under $ t $-distributed errors with heavy tails.
• Empirical analysis on financial asset returns confirms the superiority of the RTS method over PCA in real-world heavy-tailed data settings.
• Theoretical proofs establish that $ \widehat{\mathbf{H}}^\top \mathbf{V} \widehat{\mathbf{H}} \overset{p}{\rightarrow} \mathbf{I}_m $, validating the consistency of the orthogonal transformation in the estimation framework.