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
>
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>
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