高维因子模型两类“主成分”估计的比较——以S & P 500股票数据分析为例
Comparison of Two Types of Principal Component Estimation in High-Dimensional Factor Model—A Case Study of S & P 500 Stock Data Analysis

高维因子模型在超高维度的大型数据集降维处理中发挥了重要作用。目前，高维因子模型有两种主成分估计方法，分别是基于协方差的主成分估计PCE和基于滞后自协方差的主成分估计LPCE。本文以S & P 500公司股票数据的高维因子建模为例，比较了PCE和LPCE在高维股票数据降维中的实际表现，其中因子个数通过信息准则法和特征值比值估计法确定。结果表明，在高维非平稳序列因子模型中，PCE的均方根误差和预测误差都比LPCE小，PCE得到的因子也比LPCE更能捕捉高维非平稳序列变化特征。在高维平稳序列因子模型中，PCE和LPCE的估计误差相同，两者的估计因子均能还原高维平稳序列的变化特征。此外，在确定因子个数时，信息准则倾向于高估因子个数，表现出严重的过拟合。特征值比值估计法的估计结果相对更准确和稳定，在PCE中倾向于放弃相对弱势的主成分，在LPCE中则倾向于将弱势的主成分视为因子。
High-dimensional factor models play a crucial role in dimensionality reduction of large datasets with ultra-high dimensions. Currently, there are two principal component methods for estimating high-dimensional factor models: Principal Component Estimation (PCE) based on covariance and Lagged Principal Component Estimation (LPCE) based on lagged autocovariance. This paper utilizes high-dimensional factor modeling of S & P 500 company stock data as a case study to compare the practical performance of PCE and LPCE in dimensionality reduction of high-dimensional stock data, where the number of factors is determined through the information criterion method and eigenvalue ratio method. Results indicate that in high-dimensional non-stationary sequence factor models, both the root mean square error and prediction error of PCE are smaller than LPCE. Additionally, factors obtained from PCE are more effective in capturing the characteristics of high-dimensional non-stationary sequence changes compared to LPCE. In high-dimensional stationary sequence factor models, the estimation errors of PCE and LPCE are identical, and both estimation methods can effectively capture the changing characteristics of high-dimensional stationary sequences. Furthermore, when determining the number of factors, the information criterion tends to overestimate the number of factors, indicating severe overfitting. The eigenvalue ratio method provides relatively more accurate and stable estimation results, with PCE tending to discard relatively weaker principal components, while LPCE tends to treat weaker principal components as factors.