[R-SIG-Finance] Non-gaussian (L-stable) Garch innovations

[Christopher G. Green (L)]

The alpha-stable distribution for \alpha = 2, "scale" parameter \gamma = 1
and location parameter \delta = 0 is a normal distribution with mean 0 and
variance 2:

> x <- seq(-2, 2)
> pstable(x, 2, 0)-pnorm(x, 0, sqrt(2))
[1] 0 0 0 0 0
attr(,"control")
   dist alpha beta gamma delta pm
 stable     2    0     1     0  0



cg
________________________________

Christopher G. Green (cggreen AT stat.washington.edu) 
Doctoral Candidate
Department of Statistics, Box 354322, Seattle, WA, 98195-4322, U.S.A.
http://www.stat.washington.edu/cggreen/




> -----Original Message-----
> From: r-sig-finance-bounces at stat.math.ethz.ch 
> [mailto:r-sig-finance-bounces at stat.math.ethz.ch] On Behalf Of 
> Spencer Graves
> Sent: Monday, December 24, 2007 1:06 PM
> To: Patrick Burns
> Cc: r-sig-finance at stat.math.ethz.ch; "José Augusto M. de 
> Andrade Junior"
> Subject: Re: [R-SIG-Finance] Non-gaussian (L-stable) Garch innovations
> 
> Hi, Patrick, et al.: 
> 
> 
> IS NORMAL STABLE? 
> 
>       I'm confused:  According to Wikipedia, a normal 
> distribution is a stable distribution with parameters alpha = 
> 2 and beta = 0 
> (http://en.wikipedia.org/wiki/L%C3%A9vy_skew_alpha-stable_dist
> ribution).  
> However, I get large discrepancies between 'pstable{fBasics}' 
> and pnorm: 
> 
>  > library(fBasics)
>  > x <- seq(-2, 2)
>  > pstable(x, 2, 0)-pnorm(x
> + )
> [1]  0.05589947  0.08109481  0.00000000 -0.08109481 -0.05589947
> attr(,"control")
>    dist alpha beta gamma delta pm
>  stable     2    0     1     0  0
> 
>       What am I doing wrong? 
> 
> 
> ASYMPTOTICS 
> 
>       What about the maximum likelihood estimates of garch 
> parameters?  
> Don't they follow the standard asymptotic normal distribution 
> with mean and variance of the approximating normal 
> distribution = the true but unknown parameters and the 
> inverse of the information  matrix (Fisher or observed, take 
> your pick)? 
> 
>       My favorite example for this is logistic regression, 
> where no moments exist for the MLEs, because the MLEs are 
> Infinite for some possible outcomes.  However, the standard 
> normal approximation still works great.  Moreover, the 
> probability of observing Infinite MLEs at a rate proportional 
> to 2^(-N), if my memory is correct. 
> 
> 
> DISTRIBUTION OF RESIDUALS
> 
>       What can be said about the distribution of the whitened 
> residuals?  If N gets large faster than the number of 
> parameters estimated, won't the distribution of the whitened 
> residuals converge to the actual parent distribution, more or 
> less whatever it is? 
> 
>       Best Wishes,
>       Spencer
> 
> Patrick Burns wrote:
> > Yes, you are wrong.  Stable distributions DO have a 
> constant variance: 
> > infinity.
> >
> > Pat
> >
> > José Augusto M. de Andrade Junior wrote:
> >
> >   
> >> Hi Patrick,
> >>  
> >> Thanks for the explanation.
> >>
> >> I want to discuss the infinite variance of stable distributions 
> >> (except normal). I understand that infinite variance means 
> only that 
> >> this distributions does not have a constant variance, that the 
> >> integral does not converge to a finite constant value.
> >>  
> >> When someone uses GARCH to model the variance he is indeed 
> recogning 
> >> the same fact: the varince is not constant and should not 
> converge, 
> >> as with stable distributions also occur.
> >>  
> >> Am i wrong?
> >>  
> >> 2007/12/24, Patrick Burns <patrick at burns-stat.com
> >> <mailto:patrick at burns-stat.com>>:
> >>
> >>     Given the model parameters and the starting volatility state,
> >>     the procedure (which you can use a 'for' loop to do) is:
> >>
> >>     * select the next random innovation.
> >>
> >>     * multiply by the volatility at that time point to get 
> the simulated
> >>     return for that period.
> >>
> >>     * use the return to get the next period's variance 
> using the garch
> >>     equation.
> >>
> >>     So there are two series that are being produced: the return
> >>     series and the variance series.
> >>
> >>
> >>     I'm not exactly objecting, but I hope you realize that 
> garch models
> >>     variances while stable distributions (except the Gaussian) have
> >>     infinite
> >>     variance.  Hence a garch model with a stable 
> distribution is at least
> >>     a bit nonsensical.
> >>
> >>     Patrick Burns
> >>     patrick at burns-stat.com <mailto:patrick at burns-stat.com>
> >>     +44 (0)20 8525 0696
> >>     http://www.burns-stat.com
> >>     (home of S Poetry and "A Guide for the Unwilling S User")
> >>
> >>     José Augusto M. de Andrade Junior wrote:
> >>
> >>     >Hi,
> >>     >
> >>     >Could someone give an example on how to simulate 
> paths (forecast)
> >>     of a Garch
> >>     >process with Levy stable innovations (by using rstable random
> >>     deviates, for
> >>     >example)?
> >>     >
> >>     >Thanks in advance.
> >>     >
> >>     >José Augusto M de Andrade Jr
> >>     >
> >>     >       [[alternative HTML version deleted]]
> >>     >
> >>     >
> >>     >
> >>     
> >-------------------------------------------------------------
> -----------
> >>     >
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