[R] Comparison between GARCH and ARMA

[Megh Dal]

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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