5.2 Multivariate Student’s t-distri

5.2 Multivariate Student’s t-distributed errors
There exists many candidates for the multivariate generalization of the univariate Student’s t-distribution. In this theis the most commonly used distribution is considered. When the standardized errors, zt, are multivariate Student’s t-distributed, the joint density of z1,...,zT is:
is the Gamma function. Again, by using the transformation rule, the likelihood function of at = H1/2 t zt is:
where θ denotes the parameters of the model. The log-likelihood is obtained by taking the logarithm and substituting Ht = DtRtDt:
As for the Gaussian, standardized errors in Section5.1 θ is divided in two groups;(φ,ψ)= (φ1,...,φn,ψ),where φi =(α0i,α1i,...,αqi,β1i,...,βpi) are the parameters of the univariate GARCH model for the ith asset series, i =1,...,n and ψ =(a,b,ν). The optimization of(35) is difficult. Hence,also in this case the parameters are obtained in two steps. In the first step, the parameter φ is estimated assuming that the standardized errors are Gaussian distributed, while the parameter ψ is estimated in the second step using the correct log-likelihood in 35, given the parameter φ.
5.2.1 Step one
Several authors have shown that the change of the error distribution does not virtually affect the parameters, see e.g. [6] and [19]. Hence the parameters φ = φ1, ...,φn of the univariate GARCH models are fitted using the pseudo-maximum-likelihood; assuming the errors to be Gaussian distributed. Assuming Gaussian distributed errors, the first stage quasi-likelihood is the same as in (33):
T X t=1"ln(hit)+ a2 it hit#+constant! and the parameter set φi,i = 1,...,n are estimated assuming univariate GARCH models with Gaussian distributed errors. The parameters that remain to be estimated are a, b and ν. These are estimated in the second step.