基于随机矩阵理论决定多元GARCH模型最佳维度研究

摘要

基于随机矩阵理论(RMT)的降维技术能够通过去除噪声和只保留有用“信息”,而对相关矩阵估计中用来描述相关的主成分或因子的最佳使用数量做出确定.本文认为利用RMT对相关矩阵估计的降维操作来实现RMT对多元GARCH模型的有效降维是可能的.为说明基于RMT的降维技术用于多元GARCH模型的有效性,本文建立了两类将基于RMT的相关矩阵估计和波动率结合在一起的多元GARCH模型:滑动相关多元GARCH模型(SC-GARCH模型)和改进的O-GARCH模型(IO-GARCH模型).理论分析表明,这两类模型具有降维的相关结构,易于估计,并且利用RMT能确定出它们的理论最佳维度.实证研究中,本文建立了上海证券市场100只股票收益率的两类多元GARCH模型,并在马克维茨证券组合理论的框架下,考察了它们的协方差矩阵预测效果.结果表明这两类模型的预测效果很好.通过两类模型各个维度预测效果的比较可以看出.RMT能够为多元GARCH的降维提供有效的依据并且较准确地确定多元GARCH模型的最佳维度.理论和实证分析结果表明,基于RMT的降维技术是解决多元GARCH模型“维数灾祸”问题的有效手段.%Dimension reduction technique based on RMT can determine the optimal number of principal components or factors for the description of correlations in such a way that noise is eliminated and only statistically relevant information is used. We believe that using the dimension reduction operation of correlation estimation by RMT, the effective dimensionality reduction of Multi-GARCH models is possible. In order to demonstrate the effectiveness of dimensionality reduction technique based on RMT for Multi-GARCH modeling, we propose two classes of Multi-GARCH models combining the volatility process with the correlation estimates, Sliding Correlation GARCH Model (SC-GARCH) and Improved Orthogonal GARCH Model (IO-GARCH). Theoretical analysis makes it clear that with correlation structure of reduced dimensionality, the two classes of Multi-GARCH models are easy to estimate. Moreover their theoretical optimal dimensionality can be determined by RMT. In empirical study, we establish the two classes of Multi-GARCH models of 100 stocks in Shanghai stock market and test their covariance forecast results in the framework of Markowitz portfolio theory. It is proved that the two classes of Multi-GARCH models have good forecast performance. Forecast performance comparison of all dimensionalities of each class of model show that RMT provides an effective basis for dimensionality reduction of Multi-GARCH models and determines accurately the best dimension of Multi-GARCH models. Theoretical and empirical analyses demonstrate that dimension reduction technique based on RMT is an effective tool for resolving the ' curse of dimensionality ' of Multi-GARCH models.