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_distribution).  
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

Jos? 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")

    Jos? 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.
    >
    >Jos? Augusto M de Andrade Jr
    >
    >       [[alternative HTML version deleted]]
    >
    >
    >

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