In the previous chapters the most popular measurements of risk,
volatility and Value-at-Risk, have been introduced. Both are most
often defined as conditional standard deviations or as conditional
quantiles respectively, based on a given historical information
set. As with other non-parametric methods the neural network can
also be used to estimate these measurements of risk. The advantage
of the neural network based volatility and VaR estimators lies in
the fact that the information used for estimating the risk can be
represented by a large dimensional data vector without hurting the
practicality of the method. It is possible, for example, to
estimate the conditional 5% quantile of the return process of a
stock from the DAX given the individual returns of all of the DAX
stocks and additional macroeconomic data such as interest rates,
exchange rates, etc. In the following section we briefly outline
the necessary procedure.
As in (13.1) we assume a model of the form
 |
(19.4) |
to estimate the volatility, where
are independent,
identically distributed random variables with
,
.
represents as in
the previous section the exogenous information available at date
which we will use in estimating the risk of the time series
. The time series
given by (18.4) is a non-linear AR(p) ARCH(p) process with exogenous components.
To simplify we use
Then it holds for
that
The conditional expectation function
is approximated as
in the previous section by a neural network function
of the form (18.1). With the non-linear least
squares estimator
we obtain for
an
estimator for
Analogously we could estimate the conditional mean
by approximating the function with a neural network with output
function
and estimate its parameter
with a least squares estimator
within a
sufficiently large compact subset
, such as
, chosen from a fundamental
range:
As an estimator for the conditional volatility we immediately
obtain:
This estimator is in general guaranteed to be positive only for
. In order to avoid this restriction one can follow the
procedure used by Fan and Yao (1998), who have studied a
similar problem for the kernel estimator of the conditional
variance in a heteroscedastic regression model. Using this
application the residuals
are approximated by the sample residuals
Since the
has mean 0 and variance 1,
We could approximate this function directly with a neural network
with
neurons and the output function
,
whose parameter are estimated by
The resulting estimators for the
conditional volatility, which through the
is
also dependent on
, is then
Figure 18.10 shows the conditional volatilities
estimated from the log returns of the exchange rate time series
BP/USD together with some financial indicators using the procedure
described above (
periods are considered as time dependency and
radial basis functions networks are used).
Fig.:
Log-returns of exchange rate BP/USD and the estimated conditional variances by RBF neutral network.
SFEnnarch.xpl
|
It is for arbitrary
automatically non-negative. Since the
number of neurons essentially determines the smoothness of the
network function, it can make sense when approximating
and
to choose different networks with
neurons when it is
believed that the smoothness of both functions are quite different
from each other.
When the distribution of the innovations
is additionally
specified in the model (18.4), we immediately obtain
together with the estimators of
and
an estimator of
the conditional Value-at-Risk. If the distribution of
is,
for example,
N
then the conditional distribution of
given the information
and
at date
is
also a normal distribution with mean
and variance
If
is the
quantile of the standard normal distribution, then the VaR process
, i.e., the conditional
quantile of
given
is:
An estimator for this conditional Value-at-Risk based on a neural
network can be obtained by replacing
and
with the
appropriate estimator:
 |
(19.5) |
In doing this we can replace the standard normal distribution with
another distribution, for example, with a standardized
-distribution with mean 0 and variance 1.
is then the corresponding
quantile of the innovation
distribution, i.e., the distribution of
.
The estimator (18.5) for the Value-at-Risk assumes that
is a non-linear ARX-ARCHX process of the form
(18.4). Above all, however, it has the disadvantage of
depending on the critical assumption of a specific distribution of
. Above all the above mentioned procedure, in assuming a
stochastic volatility model from the standard normal distribution,
has been recently criticized in financial statistics due to
certain empirical findings. The thickness of the tails of a
distribution of a financial time series appears at times to be so
pronounced that in order to adequately model it even the
distribution of the innovations must be assumed to be leptokurtic.
Due to the simplicity of the representation a
-distribution
with only a few degrees of freedom is often considered. In order
to avoid the arbitrariness in the choice of the distribution of
the innovations, it is possible to estimate the conditional
quantile directly without relying on a model of the form
(18.4). This application goes back to the regression
quantile from Koenker and Bassett and has been applied by
Abberger (1997) to time series in connection with kernel
estimation.
We assume that
is a stationary time series. As in Chapter
17
represents the forecast distribution,
i.e., the conditional distribution of
given
With
we depict the corresponding conditional
distribution function
for
is the conditional
quantile, i.e.,
the solution to the equation
The conditional quantile function
solves the minimization problem
 |
(19.6) |
where
and
represent the positive and negative parts of
.
In order to estimate the quantile function directly with a neural
network with
neurons, we approximate
with a
network function
of the form
(18.1), whose weight parameter
lies in a
fundamental range
.
is estimated, however, not with the least
squares method, but with the minimization of the corresponding
sample values from (18.6):
As an estimator for the quantile
function we obtain
and with this the estimator for the conditional Value-at-Risk
given
,
White has shown that under suitable assumptions the function
estimators
converge in probability to
when the sample observations
and
when at the same time the number of neurons
at a suitable rate.