Estimating Number of Factors by Adjusted Eigenvalues Thresholding

Determining the number of common factors is an important and practical topic in high-dimensional factor models. The existing literature is mainly based on the eigenvalues of the covariance matrix. Owing to the incomparability of the eigenvalues of the covariance matrix caused by the heterogeneous scales of the observed variables, it is not easy to find an accurate relationship between these eigenvalues and the number of common factors. To overcome this limitation, we appeal to the correlation matrix and demonstrate, surprisingly, that the number of eigenvalues greater than 1 of the population correlation matrix is the same as the number of common factors under certain mild conditions. To use such a relationship, we study random matrix theory based on the sample correlation matrix to correct biases in estimating the top eigenvalues and to take into account of estimation errors in eigenvalue estimation. Thus, we propose a tuning-free scale-invariant adjusted correlation thresholding (ACT) method 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 ACT method in terms of minimal signal strength and the optimal threshold. Simulation studies lend further support to our proposed method and show that our estimator outperforms competing methods in most test cases. Supplementary materials for this article are available online.

高维因子模型中，公共因子个数的确定是一项兼具重要性与实用性的研究课题。现有相关研究主要基于协方差矩阵的特征值展开。由于观测变量的量纲存在异质性，导致协方差矩阵的特征值不具备可比性，因此难以建立特征值与公共因子个数之间的精准关联。为克服这一局限，本文转而采用相关矩阵进行分析，并通过严谨推导得出一项意外结论：在若干温和正则条件下，总体相关矩阵中大于1的特征值个数与公共因子个数完全一致。为利用这一关联关系，本文基于样本相关矩阵展开随机矩阵理论分析，以校正顶部特征值的估计偏差，并充分考虑特征值估计过程中的抽样误差。据此，本文提出一种无调参且尺度不变的校正相关阈值（adjusted correlation thresholding, ACT）方法，用于高维因子模型的公共因子个数确定，该方法充分考量了样本顶部特征值的抽样变异与估计偏差。同时，本文还从最小信号强度与最优阈值两个维度，证明了所提ACT方法的最优性。模拟实验进一步验证了所提方法的有效性，结果表明，在绝大多数测试场景下，本文提出的估计器性能优于现有同类竞争方法。本文的补充材料可在线获取。