Multivariate Volatility Models

Abstract

Multivariate volatility models are widely used in finance to capture both volatility clustering and contemporaneous correlation of asset return vectors. Here, we focus on multivariate GARCH models. In this common model class, it is assumed that the covariance of the error distribution follows a time dependent process conditional on information which is generated by the history of the process. To provide a particular example, we consider a system of exchange rates of two currencies measured against the US Dollar (USD), namely the Deutsche Mark (DEM) and the British Pound Sterling (GBP). For this process, we compare the dynamic properties of the bivariate model with univariate GARCH specifications where cross sectional dependencies are ignored. Moreover, we illustrate the scope of the bivariate model by ex-ante forecasts of bivariate exchange rate densities.

Appendix: Software Packages

This section gives a brief overview of BEKK model implementations for the numerical programming languages and environments R, MATLAB and Stata. Built-in functions and external packages for estimating univariate and further multivariate volatility models are briefly reviewed in Chap. 1 Appendix.

There exist two publicly available R packages which attempt to implement the BEKK approach. Both implementations are in early stages and, therefore, computed results need to be critically reviewed by the user. The package mgarchBEKK Schmidbauer et al. (2016) might be used for simulating, estimating and predicting BEKK models. The estimation of simulated data returns plausible results. In contrast, the package MTS by Tsay (2015) contains a single function BEKK11 for estimating two- or three-dimensional BEKK(1,1) models only.

MATLAB offers methods to assess univariate GARCH-type models by means of its Econometrics Toolbox. However, there is no official MATLAB Toolbox that implements the BEKK model. As described in Chap. 1 Appendix, the MFE Toolbox tries to fill the gap of assessing of multivariate volatility models in MATLAB. It is the direct successor to the UCSD Toolbox by Kevin Sheppard which is not being further developed. The codebase might help getting insights into the technical details of the BEKK approach. Because the toolbox is still under development, an optimized, error-free use can not be guaranteed.

Currently, Stata supports only the analysis of univariate volatility models, diagonal half-vec models, which are restricted versions of the half-vec model in (2.2), and conditional correlation models. It seems that there exists no publicly available extension to estimate a BEKK model. As an alternative, users might employ the tools of the independent software package JMulTi,¹ which is closely related to Lütkepohl and Krätzig (2004), for BEKK model estimation and investigation in combination with Stata.