高維度平均-變異數最佳化之共變異數矩陣估計:以台灣資料為例

Abstract

在平均-變異數投資組合最佳化的問題中，我們時常需要估計共變異數矩陣的反矩陣以計算投資組合的最適權重。當資產數量大且樣本數小的同時，共變異數矩陣在計算反矩陣上較為困難且複雜。常用來估計高維度共變異數矩陣的方法是在樣本共變異數上利用稀疏性假設，使高維度矩陣轉換為一個可逆矩陣。本研究提出一個統計框架，透過修正的Cholesky 分解法將高維度共變異數矩陣估計，轉換為迴歸係數估計之問題，並使用正交貪婪演算法(OGA)處理高維度迴歸模型之選擇。在模擬研究中，比較估計量與母體共變異數矩陣間的差異。此外，實證研究顯示在合理的參數假設下，OGA 估計結果優於Adaptive thresholding和linear shrinkage之方法。

The classical mean-variance portfolio optimization requires the estimation of an inverse covariance matrix. This is a challenging task given the large number of assets in the market and at the same time limited available historical data. Commonly used methods for estimating large covariance matrix exploit sparsity in the sample covariance matrix. In this study, I propose a statistical framework to estimate high-dimensional variance-covariance matrices under small sample size via the modified Cholesky decomposition with orthogonal greedy algorithm (OGA). This study transforms the covariance matrix estimation into a regression coefficient estimation problem, where the OGA is a fast stepwise regression method for the high-dimensional model selection and coefficient estimation. Therefore, I perform simulation studies to measure the difference between the estimators and the population covariance matrix. Moreover, empirical results show OGA estimators have better performance than adaptive thresholding and linear shrinkage approaches under reasonable parameter assumptions.