Volatility Martingale Difference Divergence Matrix and Its Application to Dimension Reduction for Multivariate Volatility
提出波动率鞅差散度矩阵（VMDDM）来量化随机向量Y在给定X下的条件方差依赖，并将其推广到时间序列中用于多变量波动率的降维，该方法计算简单、能捕捉非线性序列依赖且需用户选择的参数少。
In this article, we propose the so-called volatility martingale difference divergence matrix (VMDDM) to quantify the conditional variance dependence of a random vector Y∈Rp given X∈Rq, building on the recent work on martigale difference divergence matrix (MDDM) that measures the conditional mean dependence. We further generalize VMDDM to the time series context and apply it to do dimension reduction for multivariate volatility, following the recent work by Hu and Tsay and Li et al. Unlike the latter two papers, our metric is easy to compute, can fully capture nonlinear serial dependence and involves less user-chosen numbers. Furthermore, we propose a variant of VMDDM and apply it to the estimation of conditional uncorrelated components model (Fan, Wang, and Yao 2008). Simulation and data illustration show that our method can perform well in comparison with the existing ones with less computational time, and can outperform others in cases of strong nonlinear dependence.