17.3 Nonparametric GARCH Estimates
In the following, we consider nonparametric GARCH(1,1) models which depend
symmetrically on the last observation:
Here,
denotes a smooth unknown function and the innovations
are chosen as in as in
Section 17.2. This
model covers the usual parametric GARCH(1,1) process (17.1)
but does not allow
for representing a leverage effect like the TGARCH(1,1) process. We show
now how to transform
(17.5) into an ARMA model. First, we define
By (17.5), we have now
with
Now, we can estimate the ARMA function
from the logarithmic squared
data
as in Section 17.3
using the nonparametric
ARMA estimate
of (17.5). Reverting the
transformations, we get
or, combining both equations,
as an estimate of the symmetric GARCH function
We have to be aware, of course, that the density
used in the
deconvolution part of estimating
is the probability density of the
, i.e. if
denotes the density of
If
is a common parametric GARCH(1,1) process of form
(17.1), then
and the corresponding ARMA function is
This is a
decreasing function in
which seems to be a reasonable assumption in the
general case too corresponding to the assumption that the present
volatility is an increasing function of past volatilities.
As an example, we simulate a GARCH process from
proc(f)=gf(x,e,c)
f=c[1]+c[2]*x+c[3]*e
endp
proc(e,s2)=mygarch(n,c)
e=zeros(n+1)
f=e
s2=e
z=normal(n+1)
t=1
while (t<n+1)
t=t+1
s2[t]=gf(e[t-1]^2,s2[t-1]^2,c)
e[t]=sqrt(s2[t]).*z[t]
endo
e=e[2:(n+1)]
s2=s2[2:(n+1)]
endp
- f =
npgarchest
(x {,h {,g {,N {,R } } } } )
- estimates a nonparametric GARCH process
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The function
npgarchest
computes the functions
and
for a
GARCH process using the techniques described above. Consider
a GARCH(1,1) with
Hence, we use
n=1000
c=0.01|0.6|0.2
{e,s2}=mygarch(n,c)
and call the estimation routine by
g=npgarchest(e)
Figure 17.5 shows the resulting graph for the estimator of
together with the true function (decreasing in
) and the data (
versus
).
As in the ARMA case, the estimated function shows the underlying structure only
for a part of the range of the true function.
Figure 17.5:
Nonparametric estimation of
for
a (linear) GARCH process. True vs. estimated
function, data
.
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Finally, we remark how the the general case of nonparametric GARCH models
could be estimated. Consider
where
may depend asymmetrically on
We write
As
depend only on the squared arguments we can estimate them as
before. Again, consider
Let
be the number of all
with
and
Then, we set
are defined as in
Section 17.2
with
replacing
and
, and, using both estimates of conditional distribution functions
we get an ARMA function estimate
Reversing the
transformation from GARCH to ARMA, we get as the estimate of
The estimate for
is analogously defined
where, in the derivation of
and
are replaced by
and