To forecast the future development of financial time series an
autoregressive model is particularly suitable. The value of the
time series at date is a function of infinite many
observations from the past in addition to an innovation
independent of the past:
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The result of this procedure is illustrated in Figure
18.9: it shows the forecasting of the exchange rate
time series JPY/USD using neural networks considering periods
of time dependency.
The asymptotic normality of the parameters and of the function
estimators and the consistency of
as an estimator of
for an increasing
remain robust even in the case where the stochastic process
is
-mixing with exponentially
decreasing mixing coefficients, White (1989b) and
White (1990). Franke, Kreiss, Mammen and Neumann (2003) have formulated
conditions for the case where
for the autoregression
function
and for the distribution of the innovations
, which guarantee for the NLAR(1) process the
strongest
-mix properties with exponentially decreasing
coefficients. Next to technical details it is essential that
The conditions on the autoregression function is comparatively
weak and obvious when one considers the stationarity conditions
for linear AR(1) processes
, where
. Accordingly also
for NLAR(
) process of large order
it is sufficient to
use weaker conditions on
, which above all guarantees
stationarity in order to make the neural network a useful tool as
a non-parametric estimator of
.
For the practical forecast one not only wants to use the last
values in the time series, but also economic data available at
time such as exchange rates, index values, oil prices or the
non-linear transformation of prices. To do this the non-linear
autoregressive process with exogenous components of order
(NLARX(
)) process is suitable:
The practical application of the forecast on financial time series with neural networks is illustrated with a pilot study that was done in cooperation with the Commerzbank AG, Franke (1999). The goal was to develop a trading strategy for a portfolio made up of 28 of the most important stocks from the Dutch CBS-Index. We will restrict ourselves here to the buy-and-hold strategy with a time horizon of a quarter of a year (60 trading days), i.e., the portfolio is created at the beginning of a quarter and then held for three months with no alterations. At the end of the three months the value of the portfolio should be as large as possible.
As a basis for the trading strategy a three month forecast of the
stocks is used. represents the price of one of the 28
stocks. To model the time series
we use a NLARX process of
the form (18.3); the system function
is approximated
with a network function
. Here
is a vector made up of constant non-linear transformations of
that were taken from the technical market
analysis, for example, a moving average, momentum or
Bollinger-intervals, Müller and Nietzer (1993),
Welcker (1994). The random vector
represents the
chosen market data such as index prices, exchange rates,
international interest rates, etc. As is expected with a forecast
horizon of 60 units of time into the future, the actual forecasts
of the stock prices in 60 days,
In choosing a suitable network and in estimating the network
weight vector the data from 1993 to 1995 is used. In
choosing the network structure a statistical model selection
technique and the experience of the experts was used. The
resulting network is a multiple layered perceptron with one hidden
layer made up of
neurons. The input vector
has the dimension 25, so that a parameter vector
needed to be estimated.
To check the quality of the network based trading strategy, it is
applied to the data from 1996. At the beginning of every quarter a
portfolio made up of 28 stocks is created based on the network
based forecast. At the end of the quarter the percentage increase
in value is considered. As a comparison the increase in value of a
portfolio replicating the CBS Index exactly is considered. Since
in the years considered the market was of the most part in an
increasing phase, it is known from experience that it is hard to
beat an index. As Table 18.1 shows, the network portfolio
achieved a higher percentage increase in value in every quarter
than the index portfolio, that is in the quarters, such as the
first and fourth, where the index has substantially increased, as
well as in the quarters, such as the second, where the index has
minimally decreased. Nevertheless the results need to be
interpreted with a bit of caution. Even in the training phase
(1993-1995) the CBS Index tended to increase, so that the network
was able to specialize in a trend forecast in a generally
increasing market. Presumably one would need to use a different
network as a basis for the trading strategy, when the market
fluctuates within a long-term lateral motion or when the index
dramatically decreases.
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