Mathematical and Statistical Methods for Actuarial Sciences and Finance

A skewed GARCH-type model for multivariate

financial time series

Cinzia Franceschini and Nicola Loperfido

Abstract.Skewness of a random vector can be evaluated via its third cumulant, i.e., a ma-

trix whose elements are central moments of order three. In the general case, modelling third

cumulants might require a large number of parameters, which can be substantially reduced if

skew-normality of theunderlying distribution is assumed. We propose a multivariate GARCH

model with constant conditional correlations and multivariate skew-normal random shocks.

The application deals with multivariate financial time series whose skewness is significantly

negative, according to the sign test for symmetry.

Key words:financial returns, skew-normal distribution, third cumulant

1 Introduction

Observed financial returns are often negatively skewed, i.e., the third central moment

is negative. This empirical finding is discussed in [3]. [7] conjectures that negative

skewness originates from asymmetric behaviour of financial markets with respect

to relevant news. [6] conclude that \Skewness should be taken intoaccount in the

estimation of stock returns".

Skewness of financial returns has been modelled in several ways. [12] reviews

previous literature on this topic. [5] models skewness as a direct consequence of the

feedback effect. [4] generalises the model to the multivariate case.

All the above authors deal with scalar measures of skewness, even when they

model multivariate returns. In this paper, we measure skewness of a random vector

using a matrix containing all its central moments of order three. More precisely, we

measure and model skewness of a random vector using its third cumulant and the

multivariate skew-normal distribution [2], respectively. It is structured as follows.

Sections 2 and 3 recall the definition and some basic properties of the multivariate

third moment and the multivariate skew-normal distribution. Section 4 introduces a

multivariate GARCH-type model with skew-normal errors. Section 5 introduces a

negatively skewed financial dataset. Section 6 applies the sign test for symmetry to

the same dataset. Section 7 contains some concluding remarks.

M. Corazza et al. (eds.), Mathematical and Statistical Methodsfor Actuarial Sciencesand Finance
© Springer-Verlag Italia 2010