17.2 Nonparametric ARMA Estimates
GARCH processes are closely related to ARMA processes. If we square a
GARCH
[3](1,1) process
given by (17.1) then we get an ARMA(1,1) process
where
is white noise, i.e. a sequence
of pairwise uncorrelated random variables, with mean 0. Therefore, we study as
an intermediate step towards GARCH processes the nonparametric estimation for
ARMA models which is more closely related to the errors-in-variables
regression of Fan and Truong (1993). A linear ARMA(1,1) model with
non-vanishing mean
is given by
with zero-mean white noise
. We consider the nonparametric generalization
of this model
 |
(17.3) |
for some unknown function
which is monotone in the second argument
.
Assume we have a sample
observed
from (17.3). If
does not depend on the second argument,
(17.3) reduces to a nonparametric autoregression of order 1
and the autoregression function
may be estimated by common kernel
estimates or local polynomials. There exists extensive literature about
that type of estimation problem, and we refer to the review paper of Härdle, Lütkepohl and Chen (1997).
In the general case of (17.3) we again have the problem
of estimating a function of (partially) non-observable variables. As
depends also on the observable time series
, the basic idea of
constructing a nonparametric estimate of
is to combine a common kernel
smoothing in the first variable
with a deconvolution kernel smoothing in
the second variable
To define the estimate we have to introduce some
notation and assumptions.
We assume that the innovations
have a
known probability density
with
distribution function
and with Fourier transform
for all
and

for
for some constants
The nonlinear ARMA process (17.3) has to be stationary and strongly
mixing with exponentially decaying mixing coefficients. Let
denote the
density of the stationary marginal density of
The smoothing kernel
in
-direction is a common kernel function with
compact support
satisfying
for all
.
The kernel
which is used in the deconvolution part has a Fourier transform
which is symmetric around 0, has compact support
and satisfies some smoothness conditions (Holzberger; 2001).
We have chosen a kernel with the following Fourier transform:
For convenience, we use the smoothing kernel
to be
proportional to that function:
. The kernel
is hence an Epanechnikov
kernel with modified boundaries.
Let
be the bandwidth for smoothing in
-direction, and let
be the smoothing parameter for deconvolution in
-direction
where
and
are some constants. Then,
is a common Rosenblatt-Parzen density estimate for the stationary density
Let
denote the stationary density of the random variable
and let
be its conditional density given
An
estimate of the latter is given by
 |
(17.4) |
where the deconvolution kernel
is
In (17.4) we use a deconvolution smoothing in the direction of the
second argument of
using only pairs of observations
for which
i.e.
By integration, we get
the conditional distribution function of
given
and its estimate
for some
for
Due to technical
reasons we have to cut off the density estimate in regions where it is still
unreliable for given
. The particular choice of denominator guarantees
that
in practice, since
is a cumulative distribution function.
To estimate the unconditional density
of
we use a standard deconvolution density
estimate with smoothing parameter
Let
be the conditional density of
given
and let
be the corresponding conditional distribution
function. An estimate of it is given as
where again we truncate at
To obtain the ARMA function
, we can now compare
and
.
In practice this means to relate
and
. The
nonparametric estimate for the ARMA function
depending on
smoothing parameters
and
is hence given by
if
is increasing in the second argument, and
if
is a decreasing function of
for any
.
denotes
the inverse of the function
for fixed
. Holzberger (2001) has shown that
is a consistent estimate for
under suitable
assumptions and has given upper bounds on the rates of bias and variance of the estimate.
We remark that the
assumption of monotonicity on
is not a strong restriction.
In the application to GARCH processes which we have in mind it seems to be intuitively reasonable that the volatility
of today is an increasing function of the volatility of yesterday which
translates into an ARMA function
which is
decreasing in the second argument.
Let us illustrate the steps for estimating a nonparametric ARMA
process. First we generate time series data and
plot
versus
.
library("times")
n=1000
x=genarma(0.7,0.7,normal(n))
The result is shown in Figure 17.1. The scatterplot in the right panel
of Figure 17.1 defines the region where we can estimate the function
.
Figure 17.1:
ARMA(1,1) process.
|
To compare the deconvolution density estimate with the density of
we use now our own routine (myarma
) for generating
ARMA(1,1) data from a known function (f
):
proc(f)=f(x,e,c)
f=c[1]+c[2]*x+c[3]*e
endp
proc(x,f)=myarma(n,c)
x=matrix(n+1)-1
f=x
e=normal(n+1)
t=1
while (t<n+1)
t=t+1
f[t]=f(x[t-1],e[t-1],c)
x[t]=f[t]+e[t]
endo
x=x[2:(n+1)]
f=f[2:(n+1)]
endp
n=1000
{x,f}=myarma(n,0|0.7|0.7)
h=0.4
library("smoother")
dh=dcdenest(x,h) // deconvolution estimate
fh=denest(f,3*h) // kernel estimate
Figure 17.2:
Deconvolution density estimate (solid) and kernel density estimate (dashed)
of the known mean function of an ARMA(1,1) process.
|
Figure 17.2 shows both density estimates. Note that the smoothing
parameter (bandwidth
) is different for both estimates since different
kernel functions are used.
- f =
nparmaest
(x {,h {,g {,N {,R } } } } )
- estimates a nonparametric ARMA process
|
The function
nparmaest
computes the function
for an
ARMA process according to the algorithm described above. Let us first consider
an ARMA(1,1) with
, i.e.
Hence, we use myarma with c=0.3|0.6|1.6 and call the estimation routine by
f=nparmaest(x)
The optional parameters N and R are set to 50 and 250,
respectively. N contains the grid sizes used for
and
.
R is an additional grid size for internal computations.
The resulting function is therefore computed on a grid of size N
N. For comparison, we also calculate the true function on the same grid.
Figure 17.3 shows the resulting graphs. The bandwidths h
(corresponding to
) for the one-dimensional deconvolution kernel
estimator
and g for the two-dimensional (corresponding to
and
) are chosen according to the rates derived in Holzberger (2001).
Figure 17.3:
Nonparametric estimation of a (linear) ARMA process. True vs. estimated
function and data.
|
As a second example consider an ARMA(1,1) with a truly nonlinear function
,
i.e.
where
denotes the sigmoid function
In contrast to the previous example, this function is obviously not
dependent on the first argument. The code above has to be modified by
using
proc(f)=f(x,e,c)
f=c[2]/(1+exp(-c[3]*e))+c[1]
endp
c=-2.8|8|6
The resulting graphs for this nonlinear function are shown in
Figure 17.4. The estimated surface varies obviously
only in the second dimension and follows the
-shaped underlying
true function. However, the used sample size and the internal
grid sizes of the estimation procedure do only allow for a
rather imprecise reconstruction of the tails of the surface.
Figure 17.4:
Nonparametric estimation of a (nonlinear) ARMA process. True vs. estimated
function and data.
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