Forecasting bank stock market prices with a Hybrid method: the case of Alpha Bank/Vertybiniu popieriu kainu prognozavimas Hibridiniu metodu: Alpha Bank pavyzdys.
Koutroumanidis, Theodoros ; Ioannou, Konstantinos ; Zafeiriou, Eleni 等
1. Introduction
Forecasting in Finance and Economics has been a subject of extended
study within the last decade (McAdam, McNellis 2005). In most cases the
forecasts are based on time series modeling (Borovkova et al. 2003).
Forecasting can also be based on the application of different methods
like, bootstrapping (De Peretti 2003; Hatemi, Roca 2006) and Artificial
Neural Networks (Kiani, Kastens 2008). The combined use of Bootstrap
methods and Artificial Neural Networks (ANNs) has also been used in
Forecasting in the past (Focarelli 2005). In our study we present a
different method which relies on the application of ANNs for the
estimation of the [(1-[alpha]).sup.*]100%(C.I) of the predicted values
of the estimated time series. The estimation of this time series is
based on the application of the Bootstrap method on the residuals.
In detail we apply a hybrid method that includes the combined
application of ANNs on the Upper Confidence Limit (UCL) and Lower
Confidence Limit (LCL) of the [(1-[alpha]).sup.*]100% (C.I), that have
been a result of the Bootstrap method on the residuals, aiming at the
prediction of the confidence intervals.
The paper is organised as follows, section 1 introduces the subject
of the study, section 2 is a review of the literature, and section 3
describes the methodology used to generate predictions. The empirical
results are given in section 4, and finally section 5 provides some
concluding remarks.
2. Literature review
Claeskens and Keilgom (2003), construct bootstrap confidence bands
for regression curves. Kolsrud (2007) proposes principles and methods
for the construction of a time-simultaneous prediction band for a
univariate time series. The methods are entirely based on a learning
sample of time trajectories, and make no parametric assumption about its
distribution. The expected coverage probability of a band can be
estimated with a bootstrap procedure. In Kim (2002), the construction of
bootstrap prediction intervals is based on the percentile and
percentile-t methods, employing the standard bootstrap as well as the
bootstrap-after-bootstrap method.
The use of GARCH models in modelling the stock prices; behavior is
used greatly within the last years (Teresiene 2009; Aktan et al. 2010).
Tambakis and Van Royen (2002), on the other hand, used the bootstrap
methodology to estimate the data's conditional predictability using
GARCH models. This result is then compared to predictability under a
random walk and a model using the prediction bias in uncovered interest
parity (UIP). Mark (1995) suggested bootstrapping in testing the null
hypothesis of no predictability. Based on the bootstrap tests, the
author found strong evidence favouring the forecast accuracy of the
monetary model relative to the random walk. Thombs and Schuchany (1990)
have developed a method of calculating bootstrap conditional prediction
intervals for autoregressive models.
McCullough, (1994), applies the bootstrap method in estimating
forecast intervals for an AR(p) model. Ankenbrand and Tomassini (1996),
present an integrated approach for modelling the behaviour of financial
markets with ANNs. Fernando Fernandez-Rodriguez et al. (2000),
investigate the profitability of a simple technical trading rule based
on ANNs. Ioannou et al. (2009), used ANNs in order to predict the future
prices of fuelwood. It is obvious that within the last decade, there has
been an increasing interest in surveying the predictable components in
stock prices (Fama 1991). Patterns in asset prices improved stock-market
forecast ability with different techniques (Fernandez-Rondriquez et al.
1997). One of the approaches that improved the ability of forecasting
security markets is the ANNs (Van Eyden 1995; Gencay, Stengos 1998a).
Brock et al. (1992) used bootstrap simulations of various null asset
pricing models and found that simple technical trading rule profits
cannot be explained by popular statistical models of stock index
returns. Dogan (2007) proposed the bootstrapping method for confidence
interval estimation and hypothesis testing in system dynamics models and
provided an overview of the issues related to the proper application of
bootstrapping in dynamic models.
Gencay and Stengos (1998b, 1999), confirm predictive power of
simple technical trading rules in forecasting the current returns using
feed forward network and NN regressions, while Gencay and Stengos
(1997), and Gencay and Stengos (1998c), find evidence of nonlinear
predictability in stock market returns by using the past buy and sell
signals of the moving average rules. Regarding foreign exchange markets,
Le Baron (1992) and Le Baron (1998), use the bootstrap methodology to
demonstrate the statistical significance of the technical trading rules
against several parametric null models of exchange rates. Furthermore,
Le Baron (1999) and Sosvilla-Rivero et al. (1999), discover that excess
returns from extrapolative technical trading rules in foreign exchange
markets are high during periods of central bank intervention. Gencay
(1999), by using feed forward network and NN regressions, finds
statistically significant forecast improvements for the current returns
over the random walk model of foreign exchange returns.
Skabar and Cloete (2002), describe a methodology in which neural
networks can be used indirectly, through a genetic algorithm based on
weight optimisation procedure, in order to determine buy and sell points
for financial commodities traded on a stock exchange. A number of
studies applied the simulation of trading agents based on ANNs (White
1988; Kimoto et al. 1990; Weigend and Gershenfeld 1994). The traditional
approach to supervise neural network weight optimisation is the
well-known back propagation algorithm (Rumelhart, McClelland 1986),
while Beltratti and Terna (1996), suggest the use of genetic search for
neural network weight optimisation in this field.
Ruiz and Pascual (2002), review the application of bootstrap
procedures in inference and prediction of financial time series.
However, bootstrap methods are not adequate in this context. Korajczyk
(1985) presents one of the earliest applications of bootstrap methods to
analyze financial problems. Given that the basic bootstrap techniques
were originally developed for independent observations, the bootstrap
inference has not the desired properties when applied to raw returns;
Bookstaber and McDonald (1987), Chatterjee and Pari (1990), Hsie and
Miller (1990) and Levich and Thomas (1993) present the problems of
surveys where returns are directly bootstrapped. Maddala and Li (1996)
pointed out the shortcomings in the application of bootstrap methods in
finance. On the other hand, Thombs and Schucany (1990), as well as Kim
(2002), argue that bootstrap-based methods can also be used to obtain
prediction densities and intervals for future values of a given variable
without making distributional assumptions on the innovations and, at the
same time, allowing the introduction, into the estimated prediction
densities, of the variability due to parameter estimation. Mizuno et al.
(1998), employ ANN to Tokyo stock exchange to predict buying and selling
signals with an overall prediction rate of 63%. Sexton et al. (1998)
concluded that the use of momentum and start of learning at random
points may solve the problems that may occur in training processes. Phua
et al. (2000), applied neural networks with genetic algorithms to the
stock exchange market of Singapore and predicted the market direction
with an accuracy of 81%. In Turkey, ANNs are mostly used in predicting
financial failures (Yildiz 2001). There is no empirical survey
concerning the prediction of Turkish stock market values with exception
that of Birgul Egeli et al. (2003), who use artificial neural networks
to predict Istanbul Stock Exchange (ISE) market index value. To be more
specific, the aim of their study was to use ANNs in order to forecast
Istanbul Stock Exchange (ISE) market index values.
What must also be mentioned is the greater predictability
performance of ANN compared to that of other conventional models like
autoregressive models, as provided by the current literature. Evidently,
according to Al Saba and El Amin (1998), F-M Tseng et al. (2002),
Gutierrez-Estrada et al. (2003), Koutroumanis et al. (2009), Prybutok et
al. (2000), Artificial Neural Networks tend to perform better and
predict better results when compared to Auto Regressive Moving Average
(ARIMA) models.
Le Baron and Weigend (1997) by using a bootstrap or resampling
method, compare the uncertainty in the solution stemming from the data
splitting with neural network specific uncertainties (parameter
initialization, choice of number of hidden units, etc.).
Parisi et al. (2008), analyze recursive and rolling neural network
models to forecast one-step-ahead sign variations in gold price.
Different combinations of techniques and sample sizes are studied for
feed forward and ward neural networks. White and Racine (2001) suggest
tests for individual and joint irrelevance of network inputs. Tests of
this type can be used to determine whether an input or group of inputs
belong and quote in a particular model permitting valid statistical
inference to be based on estimated feed forward neural network models.
The approaches employ well known statistical resampling techniques.
Lento and Gradojevic (2007) determine the profitability of technical
trading rules by evaluating their ability to outperform the naive
buy-and-hold trading strategy. The bootstrap methodology is used to
determine the statistical significance of the results.
3. Artificial neural networks
A neural network consists of a number of elements called neurons.
Each neuron receives a number of signals which come as an input to it.
The neuron has some possible states on which his internal structure may
be, that receives the input signals and finally has only one output
which is a function of input signals. Each signal transmitted from one
neuron to another through the neural network is coupled with a weight
value, w, which indicates how closely these two neurons are connected
with this weight. This value fluctuates on a specific interval, for
example, the interval between -1 and 1, although this interval is an
arbitrary choice and is dependent on the problem we want to solve. The
meaning of the weight value is to show us the importance of the
contribution of the specified signal to the configuration of the
structure of the network for the two neurons that connects. When w is
big then the contribution of the signal is also big.
The primary aim of an Artificial Neural Network is to solve
specific problems that we present to it, or to perform certain tasks
(like image recognition). In order to solve or perform these tasks the
network must be trained. This is exactly the main characteristic of
neural networks meaning that they learn by training. By using the word
"training" in neural networks we mean that we provide some
input and we get some outputs. Inputs are in essence, the presentation
to the network of some signals taking arithmetic values, i.e. a binary
number consisting of 0 and 1. These numbers given at the input of the
network constitute a prototype. There is a possibility that for a given
problem many prototypes are required. To each prototype corresponds a
correct answer, which is the signal we must receive at the output or
else the objective.
When the network stops changing weight values, we assume that
training is complete. This happens because the error at the input is
nearly or equal to zero. Typically the architecture of an ANN consists
of the Input Layer where we provide the data, the Hidden Layer where
data are being processed and may consist of various levels and finally
the Output layer from where we read the results of the network
(Argyrakis 2001).
4. Bootstrap method
Bootstrapping is the practice of estimating properties of an
estimator (such as its variance) by measuring those properties when
sampling from an approximating distribution. One standard choice for an
approximating distribution is the empirical distribution of the observed
data. In the case where a set of observations can be assumed to be from
an independent and identically distributed population, this can be
implemented by constructing a number of resamples of the observed
dataset (and of equal size to the observed dataset), each of which is
obtained by random sampling with replacement from the original dataset.
It may also be used for constructing hypothesis tests. It is often
used as an alternative to inference based on parametric assumptions when
those assumptions are in doubt, or where parametric inference is
impossible or requires very complicated formulas for the calculation of
standard errors (Fig. 1).
[FIGURE 1 OMITTED]
The idea of bootstrap is depicted in the diagram above. Suppose
that the researcher wants to assess statistical accuracy of the sample
data (statistics of sample), he can take N bootstrap samplings and
compute the statistics from each bootstrap sampling. The values of
bootstrap statistics are used to evaluate the statistical accuracy of
the original sample statistics (Teknomo 2006).
The advantage of bootstrapping against other analytical methods is
its great simplicity it is straightforward to apply the bootstrap to
derive estimates of standard errors and confidence intervals for complex
estimators of complex parameters of the distribution, such as percentile
points, proportions, odds ratio, and correlation coefficients (Efron
1982).
5. Hybrid method
In this study we propose a hybrid methodology that may allow us to
make forecasts on the confidence intervals of the predicted values of a
time series. This method includes ANNs and Bootstrap methods. Those two
methodologies can be combined by the use of Excel and Visual Basic for
Applications. This application is implemented in the following steps:
1st Step: ANN is applied in order to estimate the real values of
the time series and to make forecasts on its future values.
2nd Step: Initially, the residuals are estimated. The Bootstrap
method is applied on the new residuals. The application of Breusch
Godfrey LM test as well as ARCH and Breusch Pagan test did not trace any
problem of autocorrelation and heteroskedasticity. This result implies
that the bootstrap sample of the residuals [e.sup.*] of size N, can be
considered as random independent and identically distributed sample
drawn with replacement from the empirical distribution function (EDF) of
the residuals. Furthermore, Bootstrap based methods can also be used to
obtain prediction densities and intervals for future values of a given
variable without making distributional assumptions on the innovations
and, at the same time, allowing the introduction, into the estimated
prediction densities, of the variability due to parameter estimation
(Kim 2001; Thombs, Schucany 1990).
The sample [e.sup.*] can be considered a randomly resampled version
of e residuals: its elements are the same as those of the original data
set but some may appear once, some two or more times and some others may
not appear at all. Supposing B is independent bootstrap time series of
the residuals [e.sup.*1], [e.sup.*2], ........., [e.sup.*B]. Each time
series consists of N data values generated with replacement from e
(Mooney, Duval 1993). This means that for every real value of the time
series we take B+1 residuals randomly distributed.
3rd step: For every residual B new bootstraping residuals are
calculated. Within this process the [(1-[alpha]).sup.*]100% (Bootstrap
C.I.) of each residual is estimated, a process that is repeated for all
B+1 residuals. Based on the B.C.I, we estimate the C.I. of the
predictions. The technique of bootstrapping applied on the residuals has
been extensively used in the past (Shao, Tu 1955; Efron, Tibshirani
1986; Hall 1986, 1988; Beran 1988; Franklin, Wasserman 1992; Simar,
Wilson 1998; Glaz, Sison 1999; Bjornstad, Falck 2001; Tribouley 2004;
Chou 2006; Pesavento, Rossi 2006; Kapetanios 2008; Xiong, Li 2008;
Charitos et al. 2009; Jun Li et al. 2009; Annaert et al. 2009; Kascha,
Mertens 2009; Barnes et al. 2009).
4th step: Based on the process mentioned above two new time series
of the upper and lower limits of the B.C.I. are generated. In the fourth
step with the application of ANN on the upper and lower limit of BCI, we
can make forecasts about the upper and lower limits of the C.I. of the
predicted value respectively. Consequently, we can estimate the C.I. of
the predicted values regarding the initial time series.
6. Application of the hybrid method--results
This methodology is applied to a time series of the stock prices of
Alpha Bank for the time period from 28/01/2004 till 30/11/2005 (daily
prices) that the initial ANN is used. In order to implement an accuracy
test for the forecasts we used the last twenty observations of our time
series, with the assistance of statistical tests. For the evaluation of
forecasting accuracy the following statistical tests were used; RMSE,
MAPE, NOF and Theil'--U Statistic. An A.N.N is created using as
input the real values of the stock price of Alpha bank for the same time
period (28/01/2004 till 30/11/2005) through which we estimate the
forecasted values of the Alpha Bank stock prices. Then the residuals are
estimated with the application of ANN. In order to train the neural
network we used the Kalman filter which consists of one input neurons,
24 hidden neurons and 1 output neuron. During the ANN development, we
created several other ANN's with different numbers of neurons. The
network described here, provided the best results. The application we
used (Neural Ware Predict) automatically uses a part of the time series
for training, testing and validation, thus protecting the network from
overfitting. Kalman filters are based on linear dynamical systems
discretised in the time domain. They are modelled on a Markov chain
built on linear operators perturbed by Gaussian noise. The state of the
system is represented as a vector of real numbers. At each discrete time
increment, a linear operator is applied to the state to generate the new
state, with some noise mixed in, and optionally some information from
the controls on the system if they are known. Then, another linear
operator mixed with more noise generates the visible outputs from the
hidden state. The Kalman filter may be regarded as analogous to the
hidden Markov model, with the key difference that the hidden state
variables are continuous (as opposed to being discrete in the hidden
Markov model). Additionally, the hidden Markov model can represent an
arbitrary distribution for the next value of the state variables, in
contrast to the Gaussian noise model that is used for the Kalman filter
(Haikin 2001). In our case, we used the Kalman filter in order to train
the network which consists of two input neurons, 24 hidden neurons and 1
output neuron.
Kalman filters are based on linear dynamic systems. They are
modelled on a Markov chain built on linear operators perturbed by
Gaussian noise. The state of the system is represented as a vector of
real numbers. At each discrete time increment, a linear operator is
applied to the state to generate the new state, with some noise mixed
in, and optionally, some information from the controls on the system, if
they are known. Then, another linear operator mixed with more noise
generates the visible outputs from the hidden state. The Kalman filter
may be regarded similar to the hidden Markov model, with the key
difference that the hidden state variables are continuous (as opposed to
being discrete in the hidden Markov model). Additionally, the hidden
Markov model can represent an arbitrary distribution for the next value
of the state variables, in contrast to the Gaussian noise model that is
used for the Kalman filter. There is a strong duality between the
equations of the Kalman Filter and those of the hidden Markov model. The
Kalman filter model assumes the true state at time k has evolved from
the state at (k - 1) according to:
[x.sub.k] = [F.sub.k][x.sub.k-1] + [B.sub.k][u.sub.k] + [w.sub.k],
(1)
where: [F.sub.k] is the state transition model which is applied to
the previous state [x.sub.k-1], [B.sub.k] is the control-input model
which is applied to the control vector uk; [w.sub.k] is the process
noise which is assumed to be drawn from a zero mean multivariate normal
distribution with covariance [Q.sub.k].
[w.sub.k] ~ N(0, [Q.sub.k]). (2)
At time k an observation (or measurement) [z.sub.k] of the true
state [x.sub.k] is made according to:
[z.sub.k] = [H.sub.k][x.sub.k] + [v.sub.k], (3)
where: [H.sub.k] is the observation model which maps the true state
space into the observed space; [v.sub.k] is the observation noise which
is assumed to be zero mean Gaussian white noise with covariance
[R.sub.k].
[v.sub.k] ~ N(0, [R.sub.k]). (4)
The initial state, and the noise vectors at each step {[x.sub.0],
[w.sub.1], ..., [w.sub.k], [v.sub.1] ... [v.sub.k]} are all assumed to
be mutually independent. In order to use the Kalman filter to estimate
the internal state of a process, given only a sequence of noisy
observations, one must model the process in accordance to the framework
of the Kalman filter. This means specifying the matrices [F.sub.k],
[H.sub.k], [Q.sub.k], [R.sub.k], and sometimes Bk for each time-step k,
as described below. The Kalman filter is an efficient recursive filter
that estimates the state of a dynamic system from a series of incomplete
and noisy measurements. This means that only the estimated state from
the previous time step and the current measurement are needed to compute
the estimate for the current state. In contrast to batch estimation
techniques, no history of observations and/or estimates is required. It
is unusual in being purely a time domain filter; most filters (for
example, a low-pass filter) are formulated in the frequency domain and
then transformed back to the time domain for implementation. The state
of the filter is represented by two variables:
[[??].sub.k/k] - the estimate of the state at time k;
[P.sub.k/k] - the error covariance matrix (a measure of the
estimated accuracy of the state estimate).
The Kalman filter has two distinct phases: Predict and Update. The
predict phase uses the estimate from the previous time step to produce
an estimate of the current state. In the update phase, measurement
information from the current time step is used to refine the prediction
in order to arrive at a new, (hopefully) more accurate, estimate.
Predict Phase:
[x.sub.k] = [F.sub.k][x.sub.k-1] + [B.sub.k][u.sub.k] (Predicted
state), (5)
[P.sub.k/k-1] = [F.sub.k][P.sub.k-1]/k-1][F.sup.T.sub.k] +
[Q.sub.k] (Predicted estimate covariance). (6)
Update Phase:
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (Innovation or
measurement residual), (7)
[S.sub.k] = [H.sub.k][P.sub.k/k-1][H.sup.T.sub.k][R.sub.k]
(Innovation (or residual) covariance), (8)
[K.sub.k] = [P.sub.k/k-1][H.sup.T.sub.k][S.sup.-1.sub.k] (Optimal
Kalman gain), (9)
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (Updated state
estimate), (10)
[P.sub.k/k] = (I - [K.sub.k][H.sub.k])[P.sub.k/k-1] (Updated
estimate covariance). (11)
In the case of the ANN studied here, the sigmoid function was used,
as the activation function of each neuron. Because of this, the values
of the data variables in the model must be normalized onto range [0.1]
before applying the ANN methodology. This problem was solved through the
following scaling:
[V.sup.*.sub.b] = [V.sub.b] - [V.sub.min,b]/[V.sub.max,b] -
[V.sub.min,b], (12)
where: [V.sub.b] are the values of the data variables;
[V.sup.*.sub.b] is the scaled value of the variable; [V.sub.min,b] is
the minimum value of variable [V.sub.b] minus 15%; [V.sub.max,b] is the
maximum value of variable [V.sub.b] plus 15%.
Hence the scaled series are in the range [0.1]. This scale has the
advantage of mapping the desired range of a variable to the full working
range of the network input and moreover, the scaled series lies in the
central zone of the sigmoid function, where the function is
approximately linear. Therefore, during the validation model which is
described next, the problem of the output signal saturation that can
sometimes be encountered in ANN applications is avoided.
With the use of the Bootstrap method 120 time series of residuals
are generated, and consequently to every real value correspond 120
residuals which are randomly distributed around it. The application of
the Bootstrap technique is through a special menu called Bootstrap menu.
Thus, the Bootstrap method is activated by using as input data of a time
series determined by the user. The application of this technique is
being realized through sampling with reset, and a number of times series
is generated whose number is also determined by the user. The menu is
connected internally to a Visual Basic for Applications (VBA) code that
implements the method by using the RAND function and by calculating for
every real value of the time series of the residuals. This application
may determine the number of the bootstrap time series generated by the
initial time series (Fig. 2).
[FIGURE 2 OMITTED]
Then by using the Insertion sort (Knuth 1998) we estimate the 95%
(C.I.) of the 120 generated residuals, and thus we estimated the Upper
and Lower Confidence Limits of the 95% (C.I.) of the predicted values.
The insertion technique sorts each series by repeatedly taking the next
item and inserting it into the final data structure in its proper order
with respect to items already inserted.
An example of the code used in order to implement the insertion
technique is shown in the following example:
Module InsertionSort
Sub InsertionSort(ByRef a() As Integer)
Dim i As Integer
For i = 0 To a.Length - 1
insert a(i) into sorted suhlist
Next
End Sub
test main
End Module
Two new time series are created, one of the Upper Confidence Limit
(UCL) and the other of the Lower Confidence Limit (LCL) of the
[(1-[alpha]).sup.*]100% (C.I) of the predicted prices. Consequently, we
use the initial ANN for the calculation of twenty new values of the
stock price of Alpha Bank, by using as an input the closing prices of
EFG, and also by using an ANN having as input the upper and lower values
of the residuals that were calculated by using Bootstrap, we calculate
and the new expected upper and lower limits of the forecasted prices,
given by the initial ANN.
The results of the methodology are presented in Fig. 3. In this
figure we present the observed and the predicted prices of the stock
prices of Alpha Bank, the Upper Confidence Limit (UCL) and the Lower
Confidence Limit (LCL) of the [(1-[alpha]).sub.*]100% (C.I) of the
predicted prices.
[FIGURE 3 OMITTED]
Table 1 presents the forecasted prices, the Upper Confidence Limit
(UCL) and the Lower Confidence Limit (LCL) of the
[(1-[alpha]).sup.*]100% (C.I) of the forecasted prices for the last 20
observations of the sample.
Figure 4 depicts the forecasted prices and the Upper Confidence
Limit (UCL) and the Lower Confidence Limit (LCL) of the (1-a)*100% (C.I)
of the forecasted prices for the last twenty observations of the sample.
[FIGURE 4 OMITTED]
As it becomes evident by the figure the observed and the forecasted
prices are bounded by the confidence limits, giving us an indication for
the accuracy of the methodology. The following section presents the
quantitative evaluation of the forecasting accuracy of the particular
methodology.
7. Evaluation of forecasting accuracy of the forecasted prices
The validity of output from ANNs model was tested with the
application of different criteria: Root Mean Square Error (RMSE), Mean
Absolute Percentage Error (MAPE), Normalized Objective Function (NOF)
and Theil's U--Statistic. The parameters RMSE, MAPE and NOF have to
be as close to 0.0 as possible for the forecast to be considered
satisfactory. However, when the parameter NOF is less than 1.0, then the
theoretical method is reliable and can be used with sufficient accuracy
(Hession et al. 1994; Kornecki, Sabbagh 1999; Tsihrintzis et al. 1998).
The Theil's U--Statistic must be less than one.
According to the results we calculate the following evaluation
criteria of accuracy of forecasting; RMSE = 0.8192, MAPE = 4.4819%, NOF
= 0.048087 and Theil's U--Statistic = 0.048082.
The NOF is the ratio of the RMSE to the overall mean
<[[??].sub.t]> of the forecasted by the model data (Tsihrintzis et
al. 1998), defined as:
NOF = RMSE/<[[??].sub.t]>, (13)
where: <[[??].sub.t]> = 1/m [M.summation over (1)]
[[??].sub.t] (14)
is the average value of the model output data.
The Theil's U--Statistic defined as:
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (15)
where: [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] the
forecasted prices and M the number of the forecasted prices.
8. Evaluation of forecasting accuracy of the [(1-a).sup.*]100%
Confidence Intervals of the Forecasted prices
For the Evaluation of forecasting accuracy of the [(1-a).sup.*]100%
Confidence Intervals of the Forecasted prices, a statistical test as
follows is introduced;
We define the following distances:
[absolute value of Observed price - Forecasted price], [absolute
value of Forecasted price - UCL], [absolute value of Forecasted price -
LCL] for all the Forecasted prices.
We also define the min {[absolute value of Forecasted price - UCL],
[absolute value of Forecasted price - LCL]} for all the Forecasted
prices.
If the Probability,
P (|Observed price - Forecasted price [absolute value of [less than
or equal to] min { Forecasted price - UCL], [absolute value of
Forecasted price - LCL]} [greater than or equal to] 1-a, for all the
forecasted prices then we can agree that [(1-a).sup.*]100% Confidence
Intervals of the forecasted prices give a satisfactory forecast.
The absolute value of the differences (Observed - Forecasted),
(Forecasted - UCL), (Forecasted - LCL) for all the Forecasted prices are
given in Table 2.
Consequently, given that the probability;
P ([absolute value of Observed price - Forecasted price] [less than
or equal to] min { [absolute value of Forecasted price - UCL], [absolute
value of Forecasted price - LCL] } [greater than or equal to] 0.95 for
all the forecasted prices, then we may argue that the
[(1-a).sup.*]100%Confidence Intervals of the forecasted prices may give
us a satisfactory forecast.
9. Conclusion--Discussion
This is the first time that the hybrid method described above is
used in Finance. The particular methodology is combinational, since the
Bootstrap method and ANNs are used for the determination of the
[(1-[alpha]).sup.*]100 % (C. I.). The method was used to estimate the
C.I. involving the predicted values of stock prices. We used the
aforementioned Hybrid method for first time in the field of Finance. The
method is completed in four steps. The main objective of this survey was
to estimate the C.I. of the predicted values regarding the initial time
series and its main accomplishment was to amplify the validity of the
[(1-[alpha]).sup.*]100% Confidence Interval of the forecasted prices. We
used different forecasting criteria like RMSE, MAPE, NOF and
Theil's U--Statistic. In order to test the forecasting ability of
the particular methodology, based on the results we calculated the
following evaluation criteria of accuracy of forecasting RMSE = 0.8192,
MAPE = 4.4819%, NOF = 0.048087 and Theil's U--Statistic = 0.048082,
that all confirm a satisfactory forecast.
In the future, this method could be further developed with the use
of another programming language in order to create a stand alone
software toolbox for the prediction of stock prices.
doi: 10.3846/16111699.2011.555388
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Theodoros Koutroumanidis (1), Konstantinos Ioannou (2), Eleni
Zafeiriou (3)
(1, 3) Democritus University of Thrace, Department of Agricultural
Development, Pantazidou 193, Orestiada 68200, Greece
(2) Laboratory of Forest Informatics, School of Forestry and
Natural Environment, Aristotle University of Thessaloniki, Box 247, 54
124 Thessaloniki, Greece
E-mails: (1) tkoutrou@agro.duth.gr; (2)
ioannou.konstantinos@gmail.com;
(3) ezafir@agro.duth.gr (corresponding author)
Received 12 May 2010; accepted 17 December 2010
Theodoros KOUTROUMANIDIS was born in Komotini. He is a Professor in
Democritus University of Thrace with PhD in the Department of Civil
Engineering in the Polytechnic School of Thrace. His scientific
interests are related to time series Analysis (mainly ARIMA models),
Fuzzy Logic analysis. He has written in different cited Journals like
Forest Policy and Economics, Journal of Hydrology, Energy Policy and
others, while he has been a reviewer in different journals.
Konstantinos IOANNOU is a PhD, MSc Forester, specialized in the
development of Information Systems using modern computer languages and
statistical tools. In detail he works in the field of Artificial
Intelligence, creating and studying Decision support Systems, Expert
Systems, Artificial Neural Networks. He has great teaching experience in
Universities, where he teaches Information Technologies, Geographical
Information Systems and Computer Aided Design. His published work
includes 13 papers in international science journals, most of them
recorded by Web of Science. Additionally he has published 14 papers in
Greek and International Conferences, on Planning and Development of
Natural Resources, with the use of information systems and statistical
tools.
Eleni ZAFEIRIOU was born in Thessaloniki, Greece in 1973. She is a
Lecturer in Democritus University of Thrace, with PhD in Applied
Econometrics, and her scientific interests are related to the time
series Analysis, cointegration, chaotic behavior and others. She has
written in cited Journals like Forest Policy and Economics and Journal
of Hydrology, while she has been a reviewer in different journals like
Computational Economics and Operational Research. She has been employed
as a scientific partner in different projects elaborated by the
Aristotle University of Thessaloniki.
Table 1. Forecasted prices Alpha Bank stock, Upper Confidence Limit
(UCL) and the Lower Confidence Limit (LCL) of the (1-[alpha]) * 100%
(C.I) of the forecasted prices for the last twenty observations of our
sample
Observed Forecasted Upper Confidence Lower Confidence
prices prices Limit of the 95% Limit of the 95%
(C.I.) of the (C.I.) of the
forecasted prices forecasted prices
17.93 17.315485 18.42955947 16.09441543
17.89 17.50035477 18.60822296 16.27662981
17.96 17.57427025 18.67965317 16.34950793
18.11 17.18260002 18.30110979 15.96350431
17.86 17.12118912 18.24173307 15.9030273
17.87 16.95606422 18.08198738 15.74048936
17.79 17.0229435 18.14670765 15.80630672
17.86 17.0229435 18.14670765 15.80630672
17.63 16.94869041 18.07484949 15.73323393
17.61 16.91200066 18.03932714 15.69713652
18.03 16.95606422 18.08198738 15.74048936
18.14 17.3791275 18.49106801 16.15713298
17.99 17.18260002 18.30110979 15.96350431
17.84 16.76817131 17.89993477 15.55569184
17.74 16.80368614 17.93437803 15.59060836
17.53 16.5951004 17.73177969 15.38562953
17.66 16.73997498 17.872576 15.52797508
17.66 16.73997498 17.872576 15.52797508
17.91 17.12118912 18.24173307 15.9030273
17.66 16.84669304 17.97606468 15.63289917
Table 2. The absolute value of the differences (Observed--
Forecasted), (Forecasted--UCL), (Forecasted--LCL) for all the
Forecasted prices
Absolute Absolute Absolute
(Observed--Forecasted) (For--UCL) (For--LCL)
0.614515 1.1140745 1.22106957
0.389645 1.1078682 1.22372496
0.38573 1.1053829 1.22476232
0.9274 1.1185098 1.21909571
0.738811 1.120544 1.21816182
0.913936 1.1259232 1.21557486
0.767056 1.1237642 1.21663678
0.837056 1.1237642 1.21663678
0.68131 1.1261591 1.21545648
0.697999 1.1273265 1.21486414
1.073936 1.1259232 1.21557486
0.760873 1.1119405 1.22199452
0.8074 1.1185098 1.21909571
1.071829 1.1317635 1.21247947
0.936314 1.1306919 1.21307778
0.9349 1.1366793 1.20947087
0.920025 1.132601 1.2119999
0.920025 1.132601 1.2119999
0.788811 1.120544 1.21816182
0.813307 1.1293716 1.21379387
