稳健高维协方差矩阵估计及其投资组合应用——基于中心正则化算法

Robust High-dimensional Covariance Matrix Estimation and the Application in Portfolio Selection——Based on the Central-Regularized Algorithm

高维协方差矩阵的估计问题现已成为大数据统计分析中的基本问题,传统方法要求数据满足正态分布假定且未考虑异常值影响,当前已无法满足应用需要,更加稳健的估计方法亟待被提出。针对高维协方差矩阵,一种稳健的基于子样本分组的均值-中位数估计方法被提出且简单易行,然而此方法估计的矩阵并不具备正定稀疏特性。基于此问题,本文引进一种中心正则化算法,弥补了原始方法的缺陷,通过在求解过程中对估计矩阵的非对角元素施加L1范数惩罚,使估计的矩阵具备正定稀疏的特性,显著提高了其应用价值。在数值模拟中,本文所提出的中心正则稳健估计有着更高的估计精度,同时更加贴近真实设定矩阵的稀疏结构。在后续的投资组合实证分析中,与传统样本协方差矩阵估计方法、均值-中位数估计方法和RA-LASSO方法相比,基于中心正则稳健估计构造的最小方差投资组合收益率有着更低的波动表现。

High-dimensional covariance matrix estimation has become a fundamental problem in big-data statistical analysis.Traditional methods require data to be normally distributed without considering the influence of outliers.At present,they cannot meet the needs of application.More robust methods need to be proposed.For high-dimensional covariance matrices,a robust mean-median estimation method based on sub-sample grouping is proposed and easy to use.However,the matrix estimated by this method is not positive-definite and sparse.Motivated by this problem,this paper introduces a central-regularized algorithm to avoid the shortcomings of the original method.By imposing L1 norm penalty on the off-diagonal elements of the estimated matrix,the estimated matrix could be positive-definite and sparse,thus greatly improving its application value.In the numerical simulation,the central-regularized estimation proposed in this paper has higher estimation accuracy,and is closer to the sparse structure of the real set matrix.In the subsequent empirical analysis of portfolios,the minimum variance portfolio based on central-regularized estimator has the lowest volatility,which outperforms traditional sample covariance matrix estimation method,mean-median estimation method and RALASSO method.