Multivariate Time Series Analysis and Applications

6Multivariate GARCH models

To study the heteroscedasticity problem, we can use the autoregressive conditional heteroscedasticity (ARCH) model and the generalized autoregressive conditional heteroscedasticity (GARCH) model, which have been introduced in many papers and books. These models can be applied to both univariate autoregressive integrated moving average (ARIMA) and vector autoregressive moving average (VARMA) models. In recent years, their multivariate extensions have become of great interest to researchers and many papers on this topic have been published. Great challenges in these extensions include the representations of the models and their estimation. In this chapter, we will introduce some useful representations of multivariate GARCH models and the estimation of these models.

6.1 Introduction

Let Z_t = μ_t + a_t be the univariate time series model, where μ_t = E_t − 1(Z_t) = E(Z_t|Ψ_t − 1) is the conditional expectation of Z_t given the past information set, Ψ_t − 1, up to time (t – 1), which corresponds to the terms related relevant covariates or a univariate ARMA structure, and a_t is the corresponding noise process. Engle (1982, 2002), Bollerslev (1986), and many other researchers introduced various conditional variance models to study the volatility phenomenon of Z_t through the following equation

(6.1)

where the e_t are i.i.d. random variables with mean 0 and variance ...