# [R] About the Skew Student distribution

Ben Bolker bolker at zoo.ufl.edu
Thu Sep 7 00:35:17 CEST 2006

```pierre clauss <pierreclauss <at> yahoo.fr> writes:

>
> Hello everybody,
> I need your help about the package SN and the skew student distribution. Il
will be very grateful if I have the solution.
>
> I construct a stochastic model with a white noise not gaussian but following a
skew student distribution. I
> fit the noise on monthly data to obtain the four parameters. The question is :
how to annualize the
> parameters to use my model for simulate daily data for example ?
>
> If the volatility is estimated to 3 for example, I need to multiply this by
sqrt(12) to have for the parameter
> of volatility of the skew student : 3*sqrt(12)*sqrt(dt) with "dt" the time
increment parameter (1/12 for
> monthly data, 1/261 for daily data, and so on). Do I do the same thing (and
what is the multiplicative factor
> ?) for the parameters of asymmetry and the degree of freedom ?
>

I'm not sure about your application, but I'll take a crack at it.
It sounds like you've got a continuous-time stochastic process (you
don't say explicitly).  For a Brownian motion, you would just do what
you suggest -- scale the variance by sqrt(dt) (I don't see exactly
why daily data have dt=1/261 -- although 365*5/7 = 261, so I guess
you're counting weekdays (trading days??) only).  Unfortunately, it's
not nearly as transparent (to me) what stochastic differential equation
would lead to aggregated data that were skew-Student.  The review
paper on Azzalini (SN's author)'s web site (
http://azzalini.stat.unipd.it/SN/review-web.ps ) cites some papers in
bet is probably to go back to those papers and see if they deal
with the effects of temporal aggregation.

VERY crudely, you can just experiment with this yourself
by simulating values from a particular skew-Student distribution,
aggregating them, and then looking at the properties of the
resulting distribution -- *do* the asymmetry and df change?
(I bet they do -- in some sense temporal aggregation must lead
to a distribution that is "more normal", larger df and smaller
skew -- but reversing this could be quite ugly).

good luck
Ben Bolker
finance

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