Volatility Martingale Difference Divergence Matrix and Its Application to Dimension Reduction for Multivariate Volatility

Abstract

In this article, we propose the so-called volatility martingale difference divergence matrix (VMDDM) to quantify the conditional variance dependence of a random vector $Y\in {\mathbb{R}}^p$ given $X\in {\mathbb{R}}^q$, 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.

KEY WORDS:

• Conditional heteroscedasticity
• Dimension reduction
• Nonlinear dependence
• Principal components.