[R-SIG-Finance] Standardization of GH distribution

[alexios ghalanos]

Hi Julien,

The GH distribution has a number of location and scale invariant 
parametrizations (this is the key thing to remember -- Prause 1999 is an 
excellent reference which you should read for more details).
The rugarch package uses \rho and \zeta, representing skew and shape 
respectively, in the estimation process. These are transformed, at each 
'iteration' of the solver, into the \alpha, \beta, \delta, \mu 
parameters of the standardized (0,1) GH distribution which is then used 
to compute the likelihood. The transformation is sequential as shown in
the document and proceeds with a number of substitutions. If you are 
worried you can check for yourself that the resulting parameters are
always the standardized (0,1) GH by looking at the transforming equation 
in rugarch:::.paramGH which was originally written by
Diethelm Wuertz.
The same goes for the GH Skew Student (GHST), but the transformation 
uses the \beta bar invariant parametrization which I wrote and included 
in the package in the latest version.

Here is an example:

library(rugarch)
library(fBasics)
rho = -0.9
zeta = 0.5
pars = rugarch:::.paramGH(rho = rho, zeta = zeta, -0.5)
ghMean(alpha=pars[1], beta=pars[2], delta=pars[3], mu=pars[4], lambda=-0.5)
ghVar(alpha=pars[1], beta=pars[2], delta=pars[3], mu=pars[4], lambda=-0.5)

Regards,

Alexios


On 05/05/2012 19:12, Julien Hambuckers wrote:
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> Hello guys,
> I'm actually writting a code to compute the optimal parameters (alpha, beta, delta, mu) of sample for a gen. hyperbolic function. I need to set up the mean and the variance of the fitted distribution to 0 and 1. Using the rugarch package, this is made easily, but in a more theoretical way, I do not understand how it is done. For the mean, the solution is easy (as explained p. 15 of http://cran.r-project.org/web/packages/rugarch/vignettes/Introduction_to_the_rugarch_package.pdf ) but for the variance, it needs to use estimation of zeta and rho. But how do you compute those estimators ?
> Thanks for your help,
> Julien
>
> Julien HAMBUCKERSHEC - Ecole de gestion de l'Université de LiègeUER OpérationRue Louvrex, 14 - Bâtiment 1 (310)tél: +32 (0)4 232 7271mail: jhambuckers at ulg.ac.be
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