[R] Comparison between GARCH and ARMA
Spencer Graves
spencer.graves at pdf.com
Sat Nov 11 00:04:02 CET 2006
Have you tried to Monte Carlo ARMA and GARCH? If you plot the
resulting series in various ways, I suspect the differences will be
apparent to you. If you'd like more from this list, I suggest you
illustrate your question with commented, minimal, self-contained,
reproducible code, as suggested in the posting guide
"www.R-project.org/posting-guide.html".
Hope this helps.
Spencer Graves
gyadav at ccilindia.co.in wrote:
> Hi
>
> i was just going by this thread, i thought of igniting my mind and got
> something wierd so i thought of making it wired.
>
> i think whether you take ARMA or GARCH. In computer science these are
> feedback systems or put it simply new values are function of past values.
> In ARMA case it is the return series and the error series. In case of
> GARCH it is the errors and stdev or returns and shock with propotionality
> of coeficient. In any case we are trying to find the returns only. What if
> i put stdev in ARMA and returns in GARCH ? I want to ask what it would end
> up showing me. For me both are having similar structure having two parts :
>
> 1. regression depending on past values
>
> 2. trying to reduce errors by averaging them
>
> i hope i am correct. please correct me where i am wrong.
>
> thanks and regards
> Email(Office) :- gyadav at ccilindia.co.in , Email(Personal) :-
> emailtogauravyadav at gmail.com
>
>
>
>
> "Wensui Liu" <liuwensui at gmail.com>
> Sent by: r-help-bounces at stat.math.ethz.ch
> 08-11-06 12:24 AM
>
> To
> "Leeds, Mark (IED)" <Mark.Leeds at morganstanley.com>
> cc
> r-help at stat.math.ethz.ch, Megh Dal <megh700004 at yahoo.com>
> Subject
> Re: [R] Comparison between GARCH and ARMA
>
>
>
>
>
>
> Mark,
>
> I totally agree that it doesn't make sense to compare arma with garch.
>
> but to some extent, garch can be considered arma for conditional
> variance. similarly, arch can be considered ma for conditional
> variance.
>
> the above is just my understanding, which might not be correct.
>
> thanks.
>
> On 11/7/06, Leeds, Mark (IED) <Mark.Leeds at morganstanley.com> wrote:
>
>> Hi : I'm a R novice but I consider myself reasonably versed in time
>> series related material and
>> I have never heard of an equivalence between Garch(1,1) for volatility
>> and an ARMA(1,1) in the squared returns
>> and I'm almost sure there isn't.
>>
>> There are various problems with what you wrote.
>>
>> 1) r(t) = h(t)*z(t) not h(i) but that's not a big deal.
>>
>> 2) you can't write the equation in terms of r(t) because r(t) =
>> h(t)*z(t) and h(t) is UPDATED FIRST
>> And then the relation r(t) = h(t)*z(t) is true ( in the sense of the
>> model ). So, r(t) is
>> a function of z(t) , a random variable, so trying to use r(t) on the
>> left hand side of the volatility
>> equation doesn't make sense at all.
>>
>> 3) even if your equation was valid, what you wrote is not an ARMA(1,1).
>> The AR term is there but the MA term
>> ( the beta term ) Has an r_t-1 terms in it when r_t-1 is on the left
>> side. An MA term in an ARMA framework
>> multiples lagged noise terms not the lag of what's on the left side.
>> That's what the AR term does.
>>
>> 4) even if your equation was correct in terms of it being a true
>> ARMA(1,1) , you
>> Have common coefficients on The AR term and MA term ( beta ) so you
>> would need contraints to tell the
>> Model that this was the same term in both places.
>>
>> 5) basically, you can't do what you
>> Are trying to do so you shouldn't expect to any consistency in estimates
>> Of the intercept for the reasons stated above.
>> why are you trying to transform in such a way anyway ?
>>
>> Now back to your original garch(1,1) model
>>
>> 6) a garch(1,1) has a stationarity condition that alpha + beta is less
>> than 1
>> So this has to be satisfied when you estimate a garch(1,1).
>>
>> It looks like this condition is satisfied so you should be okay there.
>>
>> 7) also, if you are really assuming/believe that the returns have mean
>> zero to begin with, without subtraction,
>> Then you shouldn't be subtracting the mean before you estimate
>> Because eseentially you will be subtracting noise and throwing out
>> useful
>> Information that could used in estimating the garch(1,1) parameters.
>> Maybe you aren't assuming that the mean is zero and you are making the
>> mean zero which is fine.
>>
>> I hope this helps you. I don't mean to be rude but I am just trying to
>> get across that what you
>> Are doing is not valid. If you saw the equivalence somewhere in the
>> literature,
>> Let me know because I would be interested in looking at it.
>>
>>
>> mark
>>
>>
>>
>>
>>
>>
>> -----Original Message-----
>> From: r-help-bounces at stat.math.ethz.ch
>> [mailto:r-help-bounces at stat.math.ethz.ch] On Behalf Of Megh Dal
>> Sent: Tuesday, November 07, 2006 2:24 AM
>> To: r-help at stat.math.ethz.ch
>> Subject: [R] Comparison between GARCH and ARMA
>>
>> Dear all R user,
>>
>> Please forgive me if my problem is too simple.
>> Actually my problem is basically Statistical rather directly R related.
>> Suppose I have return series ret
>> with mean zero. And I want to fit a Garch(1,1)
>> on this.
>>
>> my is r[t] = h[i]*z[t]
>>
>> h[t] = w + alpha*r[t-1]^2 + beta*h[t-1]
>>
>> I want to estimate the three parameters here;
>>
>> the R syntax is as follows:
>>
>> # download data:
>> data(EuStockMarkets)
>> r <- diff(log(EuStockMarkets))[,"DAX"]
>> r = r - mean(r)
>>
>> # fit a garch(1,1) on this:
>> library(tseries)
>> garch(r)
>>
>> The estimated parameters are given below:
>>
>> ***** ESTIMATION WITH ANALYTICAL GRADIENT *****
>>
>>
>>
>> Call:
>> garch(x = r)
>>
>> Coefficient(s):
>> a0 a1 b1
>> 4.746e-06 6.837e-02 8.877e-01
>>
>> Now it is straightforward to transform Garch(1,1)
>> to a ARMA like this:
>>
>> r[t]^2 = w + (alpha+beta)*r[t-1]^2 + beta*(h[t-1] -
>> r[t-1]^2) - (h[t] - r[t]^2)
>> = w + (alpha+beta)*r[t-1]^2 + beta*theta[t-1] + theta[t]
>>
>> So if I fit a ARMA(1,1) on r[t]^2 I am getting following result;
>>
>> arma(r^2, order=c(1,1))
>>
>> Call:
>> arma(x = r^2, order = c(1, 1))
>>
>> Coefficient(s):
>> ar1 ma1 intercept
>> 9.157e-01 -8.398e-01 9.033e-06
>>
>> Therefore if the above derivation is correct then I should get a same
>> intercept term for both Garch and ARMA case. But here I am not getting
>> it. Can anyone explain why?
>>
>> Any input will be highly appreciated.
>>
>> Thanks and regards,
>> Megh
>>
>>
>>
>>
>>
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