Analysis of Financial Time Series, Third Edition

10.5 Higher Dimensional Volatility Models

In this section, we make use of the sequential nature of Cholesky decomposition to suggest a strategy for building a high-dimensional volatility model. Again write the vector return series as . The mean equations for can be specified by using the methods of Chapter 8. A simple vector AR model is often sufficient. Here we focus on building a volatility model using the shock process .

Based on the discussion of Cholesky decomposition in Section 10.3, the orthogonal transformation from ait to bit only involves bjt for j < i. In addition, the time-varying volatility models built in Section 10.4 appear to be nested in the sense that the model for gii, t depends only on quantities related to bjt for j < i. Consequently, we consider the following sequential procedure to build a multivariate volatility model:

1. Select a market index or a stock return that is of major interest. Build a univariate volatility model for the selected return series.

2. Augment a second return series to the system, perform the orthogonal transformation on the shock process of this new return series, and build a bivariate volatility model for the system. The parameter estimates ...