Testing the Number of Common Factors by Bootstrapped Sample Covariance Matrix in High-Dimensional Factor Models

This article studies the impact of bootstrap procedure on the eigenvalue distributions of the sample covariance matrix under a high-dimensional factor structure. We provide asymptotic distributions for the top eigenvalues of bootstrapped sample covariance matrix under mild conditions. After bootstrap, the spiked eigenvalues which are driven by common factors will converge weakly to Gaussian limits after proper scaling and centralization. However, the largest non-spiked eigenvalue is mainly determined by the order statistics of the bootstrap resampling weights, and follows extreme value distribution. Based on the disparate behavior of the spiked and non-spiked eigenvalues, we propose innovative methods to test the number of common factors. Indicated by extensive numerical and empirical studies, the proposed methods perform reliably and convincingly under the existence of both weak factors and cross-sectionally correlated errors. Our technical details contribute to random matrix theory on spiked covariance model with convexly decaying density and unbounded support, or with general elliptical distributions. Supplementary materials for this article are available online, including a standardized description of the materials available for reproducing the work.

本文研究了高维因子结构下，自助抽样法（bootstrap procedure）对样本协方差矩阵特征值分布的影响。本文在温和正则条件下，推导了自助抽样后样本协方差矩阵主导特征值的渐近分布。经自助抽样处理后，由公共因子驱动的尖峰特征值（spiked eigenvalues）在经过适当的标准化与中心化处理后，将弱收敛于高斯极限分布。与之相对，最大的非尖峰特征值（non-spiked eigenvalues）主要由自助重抽样权重的顺序统计量决定，并服从极值分布。基于尖峰与非尖峰特征值的迥异行为特征，本文提出了创新性的公共因子个数检验方法。通过大量数值模拟与实证研究验证，所提方法在同时存在弱因子与截面相关误差的场景下，均表现出可靠且令人信服的性能。本文的技术细节丰富了针对尖峰协方差模型（spiked covariance model）的随机矩阵理论（random matrix theory），该模型涵盖密度函数凸性衰减且支撑集无界的情形，以及一般椭圆分布的情形。本文的补充材料可在线获取，其中包含可用于复现研究成果的标准化材料说明。