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

[Spencer Graves]

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
>
> 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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