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# 11. Multivariate GARCH Processes

1. Christian Francq1 and
2. Jean-Michel Zakoïan1,2

Published Online: 14 JUL 2010

DOI: 10.1002/9780470670057.ch11

## GARCH Models: Structure, Statistical Inference and Financial Applications

#### How to Cite

Francq, C. and Zakoïan, J.-M. (2010) Multivariate GARCH Processes, in GARCH Models: Structure, Statistical Inference and Financial Applications, John Wiley & Sons, Ltd, Chichester, UK. doi: 10.1002/9780470670057.ch11

#### Author Information

1. 1

University Lille 3, Lille, France

2. 2

CREST, Paris, France

#### Publication History

1. Published Online: 14 JUL 2010
2. Published Print: 23 JUL 2010

#### ISBN Information

Print ISBN: 9780470683910

Online ISBN: 9780470670057

## SEARCH

### Keywords:

• BEKK model;
• CCC model;
• multivariate GARCH models;
• vector ARMA (VARMA);
• vector autoregressive (VAR) models

### Summary

The standard linear modeling of real time series has a natural multivariate extension through the framework of the vector ARMA (VARMA) models. In particular, the subclass of vector autoregressive (VAR) models is widely studied in the econometric literature. This chapter reviews the main concepts for the analysis of the multivariate time series. It considers a vector process (Xt)t ∈Z of dimension m, Xt = (X1t, … , Xtm ) . The chapter discusses the difficulty of establishing stationarity conditions, or the existence of moments, for multivariate GARCH models. For the general vector model, in particular for the BEKK model, there exist sufficient stationarity conditions. The stationary solution being nonexplicit, proposes an algorithm that converges, under certain assumptions, to the stationary solution. It then views that the problem is much simpler for the CCC model. The chapter estimates the m-dimensional CCC-GARCH (p, q) model by the quasimaximum likelihood method.

#### Controlled Vocabulary Terms

ARMA model; autoregressive conditional heteroskedasticity; autoregressive models; generalized autoregressive conditional heteroskedasticity