10.8 Multivariate t Distribution
Empirical analysis indicates that the multivariate Gaussian innovations used in the previous sections may fail to capture the kurtosis of asset returns. In this situation, a multivariate Student-t distribution might be useful. There are many versions of the multivariate Student-t distribution. We give a simple version here for volatility modeling.
A k-dimensional random vector 
has a multivariate Student-t distribution with v degrees of freedom and parameters u = 0 and 
(the identity matrix) if its probability density function (pdf) is
where Γ(y) is the gamma function; see Mardia, Kent, and Bibby (1979, p. 57). The variance of each component xi in Eq. (10.41) is v/(v − 2), and hence we define 
as the standardized multivariate Student-t distribution with v degrees of freedom. By transformation, the pdf of 
is
For volatility modeling, we write 
and assume that follows the multivariate Student-t distribution in Eq. (10.42). By transformation, the pdf of 
is
Furthermore, if we use the Cholesky decomposition of 
, then the pdf of the transformed shock 
becomes
where 
and gjj, t is the conditional variance of bjt. Because this pdf does not involve any matrix inversion, the conditional-likelihood function of the data is easy to evaluate.