Mathematical and Statistical Methods for Actuarial Sciences and Finance

A skewed GARCH-type model for multivariate financial time series 151

Ta b le 2 .Tests statistics

Indices Frequency Statistic p-value

Dax30 0.542 3.579 < 0. 001

Ibex35 0.561 5.180 < 0. 001

S&PMib 0.543 3.673 < 0. 001

Product 0.544 3.767 < 0. 001

7 Conclusions

We considered the third cumulant of multivariate financial returns, motivated it
through a real data example and modeled it through the multivariate skew-normal
distribution. Preliminary studies hint that negative third cumulants might constitute
a stylised fact of multivariate financial returns [13], but more studies are needed to
confirm or disprove this conjecture. By proposition 2, testing for central symmetry
would be a natural way for doing it. [14] gives an excellent overview of the literature
on this topic. Multivariate GARCH-type models with skew-normal errors might be
helpful in keeping under control the number of parameters, but some caution is needed
when using maximum likelihood procedures, since it is well known that sometimes
they lead to frontier estimates.

Acknowledgement.Financially supported by the Ministero dell’Universita, dell’Istruzione e`
della Ricerca with grant PRIN No. 2006132978.

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