[R-SIG-Finance] Non-gaussian (L-stable) Garch innovations
Spencer Graves
spencer.graves at pdf.com
Tue Dec 25 21:50:56 CET 2007
Hi, Christopher:
Thanks very much. Now that I know the answer, I see it should
have been obvious from the web site I referenced.
Thanks again.
Happy holidays to all -- and a very big "THANK YOU" to all who
have contributed to my education and others through this listserve.
Best Wishes,
Spencer Graves
Christopher G. Green (L) wrote:
> 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]]
>>>> >
>>>> >
>>>> >
>>>>
>>>>
>>> -------------------------------------------------------------
>>>
>> -----------
>>
>>>> >
>>>> >_______________________________________________
>>>> > R-SIG-Finance at stat.math.ethz.ch
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>>>> >
>>>>
>>>>
>>>>
>>>>
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