Estimating Number of Factors by Adjusted Eigenvalues Thresholding

TL;DR

This paper introduces an adjusted correlation thresholding method that accurately estimates the number of common factors in high-dimensional models by leveraging the correlation matrix and random matrix theory, overcoming eigenvalue scale issues.

Contribution

The paper proposes a novel adjusted correlation thresholding (ACT) method that improves factor number estimation by correcting biases in eigenvalues using random matrix theory.

Findings

ACT outperforms existing methods in simulations.

The method is optimal under certain signal strength conditions.

Eigenvalues greater than 1 of the correlation matrix indicate the number of factors.

Abstract

Determining the number of common factors is an important and practical topic in high dimensional factor models. The existing literatures are mainly based on the eigenvalues of the covariance matrix. Due to the incomparability of the eigenvalues of the covariance matrix caused by heterogeneous scales of observed variables, it is very difficult to give an accurate relationship between these eigenvalues and the number of common factors. To overcome this limitation, we appeal to the correlation matrix and show surprisingly that the number of eigenvalues greater than $1$ of population correlation matrix is the same as the number of common factors under some mild conditions. To utilize such a relationship, we study the random matrix theory based on the sample correlation matrix in order to correct the biases in estimating the top eigenvalues and to take into account of estimation errors in eigenvalue estimation. This leads us to propose adjusted correlation thresholding (ACT) for determining the number of common factors in high dimensional factor models, taking into account the sampling variabilities and biases of top sample eigenvalues. We also establish the optimality of the proposed methods in terms of minimal signal strength and optimal threshold. Simulation studies lend further support to our proposed method and show that our estimator outperforms other competing methods in most of our testing cases.