This paper 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.

翻译：本文研究了在具有高维因子结构的框架下，自助法程序对样本协方差矩阵特征值分布的影响。我们在温和条件下给出了自助法样本协方差矩阵顶部特征值的渐近分布。经自助法处理后，由共同因子驱动的尖峰特征值在适当缩放和中心化后弱收敛于高斯极限。然而，最大非尖峰特征值主要由自助法重抽样权重的次序统计量决定，并服从极值分布。基于尖峰与非尖峰特征值行为上的差异，我们提出了检验共同因子个数的创新方法。广泛的数值模拟与实证研究表明，在存在弱因子及截面相关误差的情况下，所提方法能够可靠且具有说服力地运行。本文的技术细节为具有凸衰减密度且支撑集无界，或服从一般椭圆分布的尖峰协方差模型的随机矩阵理论做出了贡献。