Except in the case of zero means, that is point (7) in your previous mail.
On 11/7/06, Leeds, Mark (IED) <[EMAIL PROTECTED]> wrote:
>
> but not the return squared squared which is what was written previously.
> .
>
>
> ------------------------------
> *From:* Hannu Kahra [mailto:[EMAIL PROTECTED]
> *Sent:* Tuesday, November 07, 2006 3:54 PM
> *To:* Wensui Liu
> *Cc:* Leeds, Mark (IED); r-help@stat.math.ethz.ch; Megh Dal
> *Subject:* Re: [R] Comparison between GARCH and ARMA
>
> A GARCH model can be regarded as an application of the ARMA idea to the
> squared innovation series. See, e.g. Ruey S. Tsay, Analysis of Financial
> Time Series, Wiley, 2nd edition, page 114.
>
> Hannu
>
> On 11/7/06, Wensui Liu <[EMAIL PROTECTED]> wrote:
> >
> > 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) <[EMAIL PROTECTED]> 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: [EMAIL PROTECTED]
> > > [mailto: [EMAIL PROTECTED] On Behalf Of Megh Dal
> > > Sent: Tuesday, November 07, 2006 2:24 AM
> > > To: r-help@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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> > > ______________________________________________
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> > > PLEASE do read the posting guide
> > > http://www.R-project.org/posting-guide.html
> > > and provide commented, minimal, self-contained, reproducible code.
> > > --------------------------------------------------------
> > >
> > > This is not an offer (or solicitation of an offer) to
> > buy/se...{{dropped}}
> > >
> > > ______________________________________________
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> > > PLEASE do read the posting guide
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> > >
> >
> >
> > --
> > WenSui Liu
> > (http://spaces.msn.com/statcompute/blog )
> > Senior Decision Support Analyst
> > Cincinnati Children Hospital Medical Center
> >
> > ______________________________________________
> > R-help@stat.math.ethz.ch mailing list
> > https://stat.ethz.ch/mailman/listinfo/r-help
> > PLEASE do read the posting guide http://www.R-project.org/posting-guide.html
> >
> > and provide commented, minimal, self-contained, reproducible code.
> >
>
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