Non-gaussian (L-stable) Garch innovations

x <- seq(-2, 2)
pstable(x, 2, 0)-pnorm(x, 0, sqrt(2))

-----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; "Jos? 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

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]]
    >
    >
    >

-------------------------------------------------------------

-----------

    >
    >_______________________________________________
    > R-SIG-Finance at stat.math.ethz.ch

    <mailto:R-SIG-Finance at stat.math.ethz.ch> mailing list

    >https://stat.ethz.ch/mailman/listinfo/r-sig-finance
    >-- Subscriber-posting only.
    >-- If you want to post, subscribe first.
    >