Description
The prediction of complex stock market data requires nonlinear techniques 0. Artificial neural networks (ANNs) are a significant tool for solving classification and prediction problems and therefore attract a great attention from the field of financial markets.
No. 3/2011
19
THE IMPACT OF FINANCIAL CRISIS ON THE
PREDICTABILITY OF THE STOCK MARKETS OF
PIGS COUNTRIES – COMPARATIVE STUDY
OF PREDICTION ACCURACY OF TECHNICAL
ANALYSIS AND NEURAL NETWORKS
Katarína Hi?ovská – Martina Lu?kani?ová – Ján Šterba
Katarína Hi?ovská: Technical University of Košice, Faculty of Economics
N?mcovej 32, Košice, [email protected]
Martina Lu?kani?ová: Technical University of Košice, Faculty of Economics
N?mcovej 32, Košice, [email protected]
Ján Šterba: University of Economics in Bratislava, Faculty of Economic Informatics
Dolnozemská cesta 1/b, Bratislava, [email protected]
Abstract: To a degree the financial crisis influenced all European countries but the
most affected are the PIGS (Portugal, Ireland, Greece and Spain). We investigated the
effect of the financial crisis on the prediction accuracy of artificial neural networks on
the Portuguese, Irish, Athens and Madrid Stock Exchange. We applied three-layered
feed-forward neural networks with backpropagation algorithm to forecast the next day
prices and we compared the paper returns achieved before and after the recent financial
crisis. This method failed in forecasting the direction of the next day price movement
but performed well in absolute price changes. However, it achieved better results than
the strategy based on technical analysis in the period before the crisis. On the other
hand, technical analysis performed better during the crisis.
Key words: Stock return, Prediction, Feed-Forward Neural Network, Technical
Analysis, Financial Crisis
JEL Classification: G15, G17
Financial Assets and Investing
20
Introduction
Lane and Milesi-Ferretti 0 probed possible differences across European economies in
their vulnerability to shift in global imbalances. There is a bi-modal distribution of
account balances within the European economy. While one group runs sizeable
surpluses, Portugal, Ireland, Greece, Spain and Central and Eastern European countries
have deficits of a magnitude similar to the US deficit. The correction of global
imbalances involves an increase in global interest rates with a positive impact on the
financial terms of trade of countries with a positive net debt and position and a
corresponding negative impact on countries with a negative net debt position. This
analysis was confirmed in the recent financial crisis where Portugal, Ireland, Greece,
and Spain (hereinafter PIGS) became the most affected European countries. According
to the study executed by Rose and Spiegel 0, there is little evidence that the intensity of
the crisis across countries can be easily modelled using quantitative techniques. Hence
we tried to undertake research focused primarily on the prediction of stock price
indexes using artificial neural networks.
The prediction of complex stock market data requires nonlinear techniques 0. Artificial
neural networks (ANNs) are a significant tool for solving classification and prediction
problems and therefore attract a great attention from the field of financial markets.
ANNs mimic the human brain in two aspects: information is collected in ANNs during
learning and connections between neurons (synaptic weights) are used to store
knowledge. ANNs are able to supplement or substitute statistical estimations and
techniques of Moving Averages 0 used in technical analysis.
Halbert White 0 was the first to apply ANNs to the prediction of stock prices in 1988.
He employed the feed-forward network to analyze the daily stock returns of IBM. He
did not find any predictive rules but his research pointed out the prediction potential of
ANNs on stock markets. Recently there is an abundance of studies attempting to
forecast the price levels of international stock market indices 0000. Researchers
suggest utilizing ANNs for a trading strategy to encourage higher returns than
alternative strategies 00. Our goal was to set up an ANN model with a good prediction
performance in price changes to create a profitable trading strategy on analyzed
markets.
Aim and methodology
The objective of our research was to build a prediction system with a capability to
forecast closing prices of Portugal PSI General Price Index, Ireland SE Overall Price
No. 3/2011
21
Index, ATHEX Composite Price Index, and Madrid SE General Price Index. As an
alternative strategy for paper return comparison a trading strategy based on technical
analysis was built.
Analysis of price indexes
The descriptive statistic for daily returns of price indexes (Fig. 1) shows differences in
logarithmic return during the analysed periods. The variance in returns is higher during
the crisis for each index, which confirms the higher volatility on stock markets during
unstable phases of economic cycles. The analysed returns have negative relative
skewness which implies frequent small gains and few extreme losses except Portugal
PSI General and Madrid SE General during the crisis period. The excess kurtosis is
leptokurtic for each of analysed stock return although for the period during the crisis
the frequency of extremely large deviations from mean is higher than a normal
distribution. The lowest kurtosis during the crisis in comparison to period before the
crisis is observable only in returns of ATHEX Composite Price Index.
Fig. 1 The descriptive statistic of logarithmic returns on price indexes of PIGS
countries before and during the financial crisis
PORTUGAL
PSI GENERAL
IRELAND SE
OVERALL
ATHEX
COMPOSITE
MADRID SE
GENERAL
Mean 1E-05 (8E-05) 2E-04 (-4E-05) -5E-04 (7E-05) -9E-05 (10E-05)
Mean Error 2E-04 (8E-05) 3E-04 (1E-04) 4E-04 (1E-04) 3E-04 (1E-04)
Median 2E-04 (2E-04) 0.000 (9E-05) 0.000 (3E-05) 1E-04 (2E-04)
St. Dev. 0.007 (0.004) 0.009 (0.005) 0.010 (0.005) 0.009 (0.005)
Variance 4E-05 (1E-05) 8E-05 (3E-05) 9E-05 (2E-05) 7E-05 (2E-05)
Kurtosis 10.458 (5.641) 5.601 (4.626) 1.792 (3.367) 6.657 (3.703)
Skewness 0.155 (-0.866) -0.646 (-0.224) -0.121 (-0.124) 0.420 (-0.170)
Minimum -0.046 (-0.026) -0.061 (-0.029) -0.044 (-0.028) -0.042 (-0.032)
Maximum 0.044 (0.015) 0.042 (0.033) 0.040 (0.033) 0.060 (0.027)
Note: Entries in brackets correspond to the values before the crisis.
Source: Authors
We tested the null hypothesis of equal mean returns within two periods (before and
during the crisis) to test the weak form efficiency on selected markets (Fig. 2). The
rejection of null hypothesis indicated a specific observable pattern in the Greece stock
market returns and a possibility to build a profitable trading strategy.
Financial Assets and Investing
22
Fig. 2 T-statistic for distribution comparison of logarithmic returns on price indexes
of PIGS countries before and during the financial crisis
Logarithmic returns before crisis ? Logarithmic returns during crisis
p-value t-statistic
PORTUGAL PSI GENERAL 0.554 0.592
IRELAND SE OVERALL 0.134 1.498
ATHEX COMPOSITE 0.018 2.364*
MADRID SE GENERAL 0.326 0.983
* rejection at significance level ? = 0.05
Note: The table represents t-statistic for differences in forecasting errors measured as absolute
differences between the real and the forecasted values. Only ANNs with the best prediction
accuracy for each price index and different groups of inputs were used for final comparison.
Source: Authors
Artificial Neural Network for stock market forecasting
Artificial neural networks are composed of simple elements operating in parallel. Like
in a biological nervous system, the function of ANNs is determined mostly by the
connections between elements. With the explicit knowledge about target values the
network is able to “learn” by adjusting the values between connections (weights
between elements).
The feed-forward neural network (FFNN) is one of the most applied ANNs for one-
step ahead stock return forecasting. The topology of FFNN is designed as a network
with one input layer, one or more hidden layers and one output layer (a three-layered
network). Each neuron from the input layer is connected with each neuron in the
hidden layers and each neuron from hidden layers is linked with each neuron in the
output layer.
FFNN is used to process information from one layer to the next by an activation
function. The jth node in the hidden layer is defined as
g
]
= ¡
]
(o
0]
+? w
ì]
x
ì ì?]
), (1)
where x
i
is the value of the ith input node, f
j
(.) is a logistic activation function
No. 3/2011
23
¡
]
(z) =
cx p(z)
1+cx p(z)
, (2)
?
0j
is called the bias, the summation i?j means summing over all input nodes feeding
to j, and w
ij
are the weights.
For the output layer the node is defined as
o = ¡
o
(o
0o
+? w
]o
g
] ]?o
), (3)
where the activation function f
o
is linear.
o = o
0o
+ ? w
]o
g
]
k
]=1
, (4)
where k is the number of nodes in the hidden layer. Combining the layers in one also
allows for a direct connection from the input layer to the output layer, the output of
FFNN can be written as
o = ¡
o
+ |o
0o
? w
]o
g
] ]?1
(o
0]
+ ? w
ì] ì?]
x
ì
)], (5)
where the first summation is summing over the input nodes 0. There is no
transformation in the output units (f
0
is an identity function).
The application of ANN involves two steps. The first step is to train the network (in-
sample analysis) and the second step is to execute the forecasting (out-of-sample
analysis). The available data for network training is defined as
{r
t
. x
t
|t = 1. .I], (6)
r
t
= ln
P
t
P
t+1
, (7)
where x
t
denotes the vector of inputs, r
t
is the logarithmic stock return in a given period,
P
t
is a stock price in time t, and o
t
is the output of the network.
Fitting criterion of the least squares can be used during the network training. To ensure
the smoothness of the fitted function the Levenberg-Marquardt back propagation
learning algorithm can be applied
S
2
= ? (r
t
- o
t
)
2 1
t=1
. (8)
Financial Assets and Investing
24
The data used for the analysis employing ANNs were 5,901 daily closing prices for the
period of 9/30/1988 to 5/13/2011. Daily normalized closing prices were used as inputs
into ANNs. Normalization was used to reduce the range of the dataset to values
appropriate for inputs (the range between 0 and 1) defined as
x =
¡ - mìn
mux-mìn
, (9)
where x is the normalized variable, y is the variable before normalization min is the
minimum value and max is the maximum value of variable during the reference period.
The 90% of the entire dataset was used to train the network (in-sample analysis). The
rest of the data were used in the second step to execute the forecast (out-of-sample
analysis) and to calculate the profitability of the strategy. The data were divided into
two non-overlapping subsamples in the training stage. The first 90% subsample of the
training dataset was used to estimate the parameters of a given FFNN. To prevent
overtraining the remaining 10% of the training subsample was used as a validation set.
These data were used to test the generalisation ability of the network.
FFNN has been used with a different number of neurons in the hidden layers. The
number of neurons in the input layer which corresponded with the length of the input
vector was stable. A 5-day delay of each input variable was used to set up the input
matrix. The number of neurons in the hidden layers varied from 1 to 7. The aim of the
testing was to predict one variable (next day closing price); therefore, the number of
neurons in the output layer was set to 1. ANNs with different topologies were created
and ANN of every topology was trained. The criterion of stopping training was set to
99% prediction accuracy and 250 cycles if no progress was attained. After attaining
this goal, the training was stopped. The investigated set up is depicted in Fig. 3. The
forecast accuracy of networks with the best performance for each index were analyzed
before
3
the financial crisis (03/22/2002 – 09/12/2008) and during
4
the crisis
(09/15/2008 – 05/13/2011). This required dividing the testing (out-of-sample) data into
two samples. The paper returns achieved with the trading strategy based on the forecast
of the ANN for both periods were compared to the profitability of the strategy based on
technical analysis.
3
We considered the bankruptcy of the investment bank Lehman Brothers as the critical point for
distinguishing between the periods. The period before crisis excluded the time period used for in-sample
analysis of Neural Network during training.
4
We assumed that the slump still remains in the analyzed countries.
No. 3/2011
25
Fig. 3 Scheme of the prediction system using feed-forward neural networks with
stock prices and indicators of technical analysis used as inputs into network
Source: Authors
Only ANNs with the best prediction accuracy for each stock and input groups were
used for the final performance comparison. The prediction accuracy of FFNN models
was compared using Square Error (RMSE), Mean Absolute Percentage Error (MAPE),
and Mean Absolute Error (MAE) defined as:
RHSE = _
1
n
?
t
- p
t
)
2
n
t=1
, (10)
HAPE =
100
n
¡
¡
t
-p
t
¡
t
¡
n
t=1
, (11)
HAE =
1
n
? |:
t
- p
t
|
n
t=1
, (12)
where n is number of forecasting period, v
t
is actual time series value at time t and p
t
is the predicted value of time series.
Daily closing prices
Data
transformatio
n to matrix
Input
layer
Output layer
Hidden
layer
Forecasted
closing
prices
Financial Assets and Investing
26
Technical Analysis in stock market forecasting
The technical analysis is an approach of financial market prediction based on an
analysis of historical prices to predict their probable future values. Technical analysis
uses two elementary tools – the analysis of charts (trend lines) and the analyses based
on technical indicators 0. An indicator is a mathematical calculation that can be applied
to a security price and/or volume fields. The result is a value that is used to anticipate
future changes in prices.
The Moving Average Convergence Divergence (MACD) used in our analysis is a trend
following momentum indicator. The MACD shows the relationship between two
moving averages of prices (26-day and 12-day exponential moving averages). As the
“signal” line to show buy/sell opportunities a 9-day exponential moving average line
was utilised. The signals for long positions were generated when MACD line crossed
above the signal line (9-day EMA of MACD) and sell crossover occurred when
MACD crossed down the signal line. We applied a hold strategy in the situation when
no signal was created.
Trading strategies
To compare the profitability of trading strategies based on MACD and neural network
the paper return was considered. The initial amount of money for trading was set to
10,000 EUR for both trading strategies and for both reference periods, before and
during the crisis. The transaction costs were excluded from the calculations and during
the observed periods, the maximum available amount of money was invested all the
time (depending on the price of shares). All generated buy/sell/hold signals were
considered (in case of neural networks the repeating buy or sell signals were
considered as a hold signal). When NN forecasts gave a signal for selling at the
beginning of the period, the allowed shortage amount was set to 10.000 EUR and was
covered by the next buy signal.
The trading strategy based on neural network forecast followed rules:
p
t+1
> :
t
? buy (13)
p
t+1
< :
t
? sell (14)
No. 3/2011
27
If the forecasted value for the following day was higher than the actual closing price,
we bought the stocks. Reversely, if the forecasted value for the following day was
lower than the actual closing price, we sold the amount of stock which was available
according to our disposable amount of money remaining from the previous trade. If the
buy/sell signal continued, we kept the hold position.
Results
Only the topologies of networks with the best performance for each stock were used
for final comparison. The performance of ANNs with the best prediction accuracy for
each stock achieved in the two analysed periods is depicted in Fig. 4. Entries in
brackets represent the forecasting error for the period before the crisis. Apart from
ATHEX Composite Price Index, the forecasting error according to all error measures
was lower in period during the crisis. This result suggests that the strategy based on
neural network forecast should be more profitable in the period during the crisis at
least for Portugal, Ireland, and Madrid Price Index.
Fig. 4 Performance comparison of feed-forward artificial network on price indexes
of PIGS countries
MAE RMSE MAPE
PORTUGAL PSI GENERAL 0.067 (0.249) 0.087 (0.437) 1.820 (4.664)
IRELAND SE OVERALL 0.107 (0.492) 0.119 (0.786) 6.961 (8.028)
ATHEX COMPOSITE 0.209 (0.206) 0.268 (0.245) 2.006 (3.617)
MADRID SE GENERAL 0.143 (0.778) 0.188 (1.019) 2.804 (10.508)
Notes: This table reports results from variance decomposition for the frontier emerging markets
and the developed markets in the period before and during the crisis. Entries in brackets
correspond to the values before the crisis period.
Source: Authors
We confronted the absolute difference between the real and the forecasted values in the
period before and during the crisis using T-statistic (Fig. 5). We confirmed that the
difference in the prediction accuracy before and during the crisis is significant across
analysed indexes. The forecasting error in the period before and during the crisis was
not proved as significant only for ATHEX.
Financial Assets and Investing
28
Fig. 5 T-statistic for distribution comparison of feed-forward artificial networks
applied on PIGS stock markets before and during the financial crisis
Forecasting error before crisis ? Forecasting error during crisis
p-value t-statistic
PORTUGAL PSI GENERAL 8.1E-39 13.268*
IRELAND SE OVERALL 2.2E-58 16.558*
ATHEX COMPOSITE 0.62300 0.492
MADRID SE GENERAL 1.7E-46 14.632*
* rejection at significance level ? = 0,05
Note: The table represents t-statistic for differences in forecasting errors measured as absolute
differences between real and forecasted values. Only ANNs with the best prediction accuracy for
each price index and different groups of inputs were used for final comparison.
Source: Authors
The profitability of trading strategies based on neural networks and technical analysis
are shown in Fig. 6. Trading according to MACD indicator showed more optimistic
results when predicting the price during the financial crisis. All observed stocks
showed positive returns from which the Athens Stock Exchange obtained the highest
percentage return of 251.01%. Portuguese Stock Exchange earned 89.36%, Irish
48.35%, and Madrid Stock Exchange 12.72%. When looking at the period before the
crisis, all positions ended with losses out of which the Portuguese was the highest (-
40.76%). The trading strategy based on neural networks achieved generally lower
returns in comparison to the technical analysis. However ANNs performed better in the
period before the crisis except the Irish Stock Market where the return was -18.24% in
comparison to -4.15% achieved by MACD. The average return before the crisis gained
with the neural network was 5.61% in comparison to -25.92% with the Technical
analysis. The strategy based on neural networks was profitable both in the period
before and during the crisis for the Portuguese and Madrid Stock Exchange. On the
other hand, for the Irish and Athens Stock Exchange the returns were negative both in
the period before and during the crisis. The comparable results within the analysed
period were expected in ATHEX Composite Price Index because the difference
between forecasting error means was not rejected.
No. 3/2011
29
Fig. 6 The comparison of paper returns achieved by trading strategy based on neural
network and technical analysis
5
applied on PIGS stock markets
Neural Network MACD
PORTUGAL PSI GENERAL 36.83 (21.28) 89.36 (- 40.76)
IRELAND SE OVERALL - 31.83 (- 18.24) 48.35 (- 4.15)
ATHEX COMPOSITE - 31.28 (- 6.29) 251.01 (- 20.30)
MADRID SE GENERAL 87.91 (25.69) 12.72 (- 38.45)
Note: The table represents the paper return (in %) of trading strategies with initial amount of
10,000 EUR excluding transaction costs. Entries in brackets correspond to the values before the
crisis period and only neural networks with the best performance for each Price Index were
considered.
Source: Authors
We can conclude that during the crisis the strategy based on MACD seems to be more
appropriate while before the crisis the neural networks performed better. Better
performance in the period before the crisis was anticipated according to a lower
absolute error between the real and the forecasted values in comparison to the period
during the crisis.
Conclusion
This article focused on the analysis of the predictability of PIGS Stock Markets before
and during the financial crisis. The neural networks failed in forecasting the direction
of next day price movement but performed well in absolute price changes. When
comparing the profitability of the trading strategies, the strategy based on the technical
analysis was more profitable during the crisis while the strategy based on ANN forecast
performed better in the period before the crisis. Although the ANN strategy seemed to
be more stable because of a lower variance in returns, the MACD strategy always
achieved a positive return in the period during the crisis. ANN did not perform well in
forecasting extreme changes in stock prices in the period of huge bubbles appearing in
Portuguese, Irish, and Spanish stock markets.
5
The table reports the percentage of paper returns while only the Neural Networks with the highest
achieved prediction accuracy was used.
Financial Assets and Investing
30
Hence, a different trading strategy of the neural network should be tested and more
evidence is needed to prove the results achieved in this study. The proposed analysis
undertaken on other European national stocks would be helpful to provide an
exhaustive comparison of differences in returns before and during the crisis within
Europe. Another interesting issue is also the situation in the post crisis period.
References
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between Fama and French´s model and artificial neural network in predicting the Chinese stock
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0548.
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SAFER, M. A comparison of two data mining techniques to predict abnormal stock market
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Tilakaratne, C. D.; Mammadov, M. A.; Morris S. A. Modified Neural Network Algorithms for
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Attachment
Fig. 7 The prediction accuracy of neural network models for price indexes
PORTUGAL PSI GENERAL - PRICE INDEX
before during crisis
IRELAND SE OVERALL - PRICE INDEX
Financial Assets and Investing
32
Source: Authors
ATHEX COMPOSITE - PRICE INDEX
MADRID SE GENERAL - PRICE INDEX
closing prices forecasted closing prices
doc_696330470.pdf
The prediction of complex stock market data requires nonlinear techniques 0. Artificial neural networks (ANNs) are a significant tool for solving classification and prediction problems and therefore attract a great attention from the field of financial markets.
No. 3/2011
19
THE IMPACT OF FINANCIAL CRISIS ON THE
PREDICTABILITY OF THE STOCK MARKETS OF
PIGS COUNTRIES – COMPARATIVE STUDY
OF PREDICTION ACCURACY OF TECHNICAL
ANALYSIS AND NEURAL NETWORKS
Katarína Hi?ovská – Martina Lu?kani?ová – Ján Šterba
Katarína Hi?ovská: Technical University of Košice, Faculty of Economics
N?mcovej 32, Košice, [email protected]
Martina Lu?kani?ová: Technical University of Košice, Faculty of Economics
N?mcovej 32, Košice, [email protected]
Ján Šterba: University of Economics in Bratislava, Faculty of Economic Informatics
Dolnozemská cesta 1/b, Bratislava, [email protected]
Abstract: To a degree the financial crisis influenced all European countries but the
most affected are the PIGS (Portugal, Ireland, Greece and Spain). We investigated the
effect of the financial crisis on the prediction accuracy of artificial neural networks on
the Portuguese, Irish, Athens and Madrid Stock Exchange. We applied three-layered
feed-forward neural networks with backpropagation algorithm to forecast the next day
prices and we compared the paper returns achieved before and after the recent financial
crisis. This method failed in forecasting the direction of the next day price movement
but performed well in absolute price changes. However, it achieved better results than
the strategy based on technical analysis in the period before the crisis. On the other
hand, technical analysis performed better during the crisis.
Key words: Stock return, Prediction, Feed-Forward Neural Network, Technical
Analysis, Financial Crisis
JEL Classification: G15, G17
Financial Assets and Investing
20
Introduction
Lane and Milesi-Ferretti 0 probed possible differences across European economies in
their vulnerability to shift in global imbalances. There is a bi-modal distribution of
account balances within the European economy. While one group runs sizeable
surpluses, Portugal, Ireland, Greece, Spain and Central and Eastern European countries
have deficits of a magnitude similar to the US deficit. The correction of global
imbalances involves an increase in global interest rates with a positive impact on the
financial terms of trade of countries with a positive net debt and position and a
corresponding negative impact on countries with a negative net debt position. This
analysis was confirmed in the recent financial crisis where Portugal, Ireland, Greece,
and Spain (hereinafter PIGS) became the most affected European countries. According
to the study executed by Rose and Spiegel 0, there is little evidence that the intensity of
the crisis across countries can be easily modelled using quantitative techniques. Hence
we tried to undertake research focused primarily on the prediction of stock price
indexes using artificial neural networks.
The prediction of complex stock market data requires nonlinear techniques 0. Artificial
neural networks (ANNs) are a significant tool for solving classification and prediction
problems and therefore attract a great attention from the field of financial markets.
ANNs mimic the human brain in two aspects: information is collected in ANNs during
learning and connections between neurons (synaptic weights) are used to store
knowledge. ANNs are able to supplement or substitute statistical estimations and
techniques of Moving Averages 0 used in technical analysis.
Halbert White 0 was the first to apply ANNs to the prediction of stock prices in 1988.
He employed the feed-forward network to analyze the daily stock returns of IBM. He
did not find any predictive rules but his research pointed out the prediction potential of
ANNs on stock markets. Recently there is an abundance of studies attempting to
forecast the price levels of international stock market indices 0000. Researchers
suggest utilizing ANNs for a trading strategy to encourage higher returns than
alternative strategies 00. Our goal was to set up an ANN model with a good prediction
performance in price changes to create a profitable trading strategy on analyzed
markets.
Aim and methodology
The objective of our research was to build a prediction system with a capability to
forecast closing prices of Portugal PSI General Price Index, Ireland SE Overall Price
No. 3/2011
21
Index, ATHEX Composite Price Index, and Madrid SE General Price Index. As an
alternative strategy for paper return comparison a trading strategy based on technical
analysis was built.
Analysis of price indexes
The descriptive statistic for daily returns of price indexes (Fig. 1) shows differences in
logarithmic return during the analysed periods. The variance in returns is higher during
the crisis for each index, which confirms the higher volatility on stock markets during
unstable phases of economic cycles. The analysed returns have negative relative
skewness which implies frequent small gains and few extreme losses except Portugal
PSI General and Madrid SE General during the crisis period. The excess kurtosis is
leptokurtic for each of analysed stock return although for the period during the crisis
the frequency of extremely large deviations from mean is higher than a normal
distribution. The lowest kurtosis during the crisis in comparison to period before the
crisis is observable only in returns of ATHEX Composite Price Index.
Fig. 1 The descriptive statistic of logarithmic returns on price indexes of PIGS
countries before and during the financial crisis
PORTUGAL
PSI GENERAL
IRELAND SE
OVERALL
ATHEX
COMPOSITE
MADRID SE
GENERAL
Mean 1E-05 (8E-05) 2E-04 (-4E-05) -5E-04 (7E-05) -9E-05 (10E-05)
Mean Error 2E-04 (8E-05) 3E-04 (1E-04) 4E-04 (1E-04) 3E-04 (1E-04)
Median 2E-04 (2E-04) 0.000 (9E-05) 0.000 (3E-05) 1E-04 (2E-04)
St. Dev. 0.007 (0.004) 0.009 (0.005) 0.010 (0.005) 0.009 (0.005)
Variance 4E-05 (1E-05) 8E-05 (3E-05) 9E-05 (2E-05) 7E-05 (2E-05)
Kurtosis 10.458 (5.641) 5.601 (4.626) 1.792 (3.367) 6.657 (3.703)
Skewness 0.155 (-0.866) -0.646 (-0.224) -0.121 (-0.124) 0.420 (-0.170)
Minimum -0.046 (-0.026) -0.061 (-0.029) -0.044 (-0.028) -0.042 (-0.032)
Maximum 0.044 (0.015) 0.042 (0.033) 0.040 (0.033) 0.060 (0.027)
Note: Entries in brackets correspond to the values before the crisis.
Source: Authors
We tested the null hypothesis of equal mean returns within two periods (before and
during the crisis) to test the weak form efficiency on selected markets (Fig. 2). The
rejection of null hypothesis indicated a specific observable pattern in the Greece stock
market returns and a possibility to build a profitable trading strategy.
Financial Assets and Investing
22
Fig. 2 T-statistic for distribution comparison of logarithmic returns on price indexes
of PIGS countries before and during the financial crisis
Logarithmic returns before crisis ? Logarithmic returns during crisis
p-value t-statistic
PORTUGAL PSI GENERAL 0.554 0.592
IRELAND SE OVERALL 0.134 1.498
ATHEX COMPOSITE 0.018 2.364*
MADRID SE GENERAL 0.326 0.983
* rejection at significance level ? = 0.05
Note: The table represents t-statistic for differences in forecasting errors measured as absolute
differences between the real and the forecasted values. Only ANNs with the best prediction
accuracy for each price index and different groups of inputs were used for final comparison.
Source: Authors
Artificial Neural Network for stock market forecasting
Artificial neural networks are composed of simple elements operating in parallel. Like
in a biological nervous system, the function of ANNs is determined mostly by the
connections between elements. With the explicit knowledge about target values the
network is able to “learn” by adjusting the values between connections (weights
between elements).
The feed-forward neural network (FFNN) is one of the most applied ANNs for one-
step ahead stock return forecasting. The topology of FFNN is designed as a network
with one input layer, one or more hidden layers and one output layer (a three-layered
network). Each neuron from the input layer is connected with each neuron in the
hidden layers and each neuron from hidden layers is linked with each neuron in the
output layer.
FFNN is used to process information from one layer to the next by an activation
function. The jth node in the hidden layer is defined as
g
]
= ¡
]
(o
0]
+? w
ì]
x
ì ì?]
), (1)
where x
i
is the value of the ith input node, f
j
(.) is a logistic activation function
No. 3/2011
23
¡
]
(z) =
cx p(z)
1+cx p(z)
, (2)
?
0j
is called the bias, the summation i?j means summing over all input nodes feeding
to j, and w
ij
are the weights.
For the output layer the node is defined as
o = ¡
o
(o
0o
+? w
]o
g
] ]?o
), (3)
where the activation function f
o
is linear.
o = o
0o
+ ? w
]o
g
]
k
]=1
, (4)
where k is the number of nodes in the hidden layer. Combining the layers in one also
allows for a direct connection from the input layer to the output layer, the output of
FFNN can be written as
o = ¡
o
+ |o
0o
? w
]o
g
] ]?1
(o
0]
+ ? w
ì] ì?]
x
ì
)], (5)
where the first summation is summing over the input nodes 0. There is no
transformation in the output units (f
0
is an identity function).
The application of ANN involves two steps. The first step is to train the network (in-
sample analysis) and the second step is to execute the forecasting (out-of-sample
analysis). The available data for network training is defined as
{r
t
. x
t
|t = 1. .I], (6)
r
t
= ln
P
t
P
t+1
, (7)
where x
t
denotes the vector of inputs, r
t
is the logarithmic stock return in a given period,
P
t
is a stock price in time t, and o
t
is the output of the network.
Fitting criterion of the least squares can be used during the network training. To ensure
the smoothness of the fitted function the Levenberg-Marquardt back propagation
learning algorithm can be applied
S
2
= ? (r
t
- o
t
)
2 1
t=1
. (8)
Financial Assets and Investing
24
The data used for the analysis employing ANNs were 5,901 daily closing prices for the
period of 9/30/1988 to 5/13/2011. Daily normalized closing prices were used as inputs
into ANNs. Normalization was used to reduce the range of the dataset to values
appropriate for inputs (the range between 0 and 1) defined as
x =
¡ - mìn
mux-mìn
, (9)
where x is the normalized variable, y is the variable before normalization min is the
minimum value and max is the maximum value of variable during the reference period.
The 90% of the entire dataset was used to train the network (in-sample analysis). The
rest of the data were used in the second step to execute the forecast (out-of-sample
analysis) and to calculate the profitability of the strategy. The data were divided into
two non-overlapping subsamples in the training stage. The first 90% subsample of the
training dataset was used to estimate the parameters of a given FFNN. To prevent
overtraining the remaining 10% of the training subsample was used as a validation set.
These data were used to test the generalisation ability of the network.
FFNN has been used with a different number of neurons in the hidden layers. The
number of neurons in the input layer which corresponded with the length of the input
vector was stable. A 5-day delay of each input variable was used to set up the input
matrix. The number of neurons in the hidden layers varied from 1 to 7. The aim of the
testing was to predict one variable (next day closing price); therefore, the number of
neurons in the output layer was set to 1. ANNs with different topologies were created
and ANN of every topology was trained. The criterion of stopping training was set to
99% prediction accuracy and 250 cycles if no progress was attained. After attaining
this goal, the training was stopped. The investigated set up is depicted in Fig. 3. The
forecast accuracy of networks with the best performance for each index were analyzed
before
3
the financial crisis (03/22/2002 – 09/12/2008) and during
4
the crisis
(09/15/2008 – 05/13/2011). This required dividing the testing (out-of-sample) data into
two samples. The paper returns achieved with the trading strategy based on the forecast
of the ANN for both periods were compared to the profitability of the strategy based on
technical analysis.
3
We considered the bankruptcy of the investment bank Lehman Brothers as the critical point for
distinguishing between the periods. The period before crisis excluded the time period used for in-sample
analysis of Neural Network during training.
4
We assumed that the slump still remains in the analyzed countries.
No. 3/2011
25
Fig. 3 Scheme of the prediction system using feed-forward neural networks with
stock prices and indicators of technical analysis used as inputs into network
Source: Authors
Only ANNs with the best prediction accuracy for each stock and input groups were
used for the final performance comparison. The prediction accuracy of FFNN models
was compared using Square Error (RMSE), Mean Absolute Percentage Error (MAPE),
and Mean Absolute Error (MAE) defined as:
RHSE = _
1
n
?
t
- p
t
)
2
n
t=1
, (10)
HAPE =
100
n
¡
¡
t
-p
t
¡
t
¡
n
t=1
, (11)
HAE =
1
n
? |:
t
- p
t
|
n
t=1
, (12)
where n is number of forecasting period, v
t
is actual time series value at time t and p
t
is the predicted value of time series.
Daily closing prices
Data
transformatio
n to matrix
Input
layer
Output layer
Hidden
layer
Forecasted
closing
prices
Financial Assets and Investing
26
Technical Analysis in stock market forecasting
The technical analysis is an approach of financial market prediction based on an
analysis of historical prices to predict their probable future values. Technical analysis
uses two elementary tools – the analysis of charts (trend lines) and the analyses based
on technical indicators 0. An indicator is a mathematical calculation that can be applied
to a security price and/or volume fields. The result is a value that is used to anticipate
future changes in prices.
The Moving Average Convergence Divergence (MACD) used in our analysis is a trend
following momentum indicator. The MACD shows the relationship between two
moving averages of prices (26-day and 12-day exponential moving averages). As the
“signal” line to show buy/sell opportunities a 9-day exponential moving average line
was utilised. The signals for long positions were generated when MACD line crossed
above the signal line (9-day EMA of MACD) and sell crossover occurred when
MACD crossed down the signal line. We applied a hold strategy in the situation when
no signal was created.
Trading strategies
To compare the profitability of trading strategies based on MACD and neural network
the paper return was considered. The initial amount of money for trading was set to
10,000 EUR for both trading strategies and for both reference periods, before and
during the crisis. The transaction costs were excluded from the calculations and during
the observed periods, the maximum available amount of money was invested all the
time (depending on the price of shares). All generated buy/sell/hold signals were
considered (in case of neural networks the repeating buy or sell signals were
considered as a hold signal). When NN forecasts gave a signal for selling at the
beginning of the period, the allowed shortage amount was set to 10.000 EUR and was
covered by the next buy signal.
The trading strategy based on neural network forecast followed rules:
p
t+1
> :
t
? buy (13)
p
t+1
< :
t
? sell (14)
No. 3/2011
27
If the forecasted value for the following day was higher than the actual closing price,
we bought the stocks. Reversely, if the forecasted value for the following day was
lower than the actual closing price, we sold the amount of stock which was available
according to our disposable amount of money remaining from the previous trade. If the
buy/sell signal continued, we kept the hold position.
Results
Only the topologies of networks with the best performance for each stock were used
for final comparison. The performance of ANNs with the best prediction accuracy for
each stock achieved in the two analysed periods is depicted in Fig. 4. Entries in
brackets represent the forecasting error for the period before the crisis. Apart from
ATHEX Composite Price Index, the forecasting error according to all error measures
was lower in period during the crisis. This result suggests that the strategy based on
neural network forecast should be more profitable in the period during the crisis at
least for Portugal, Ireland, and Madrid Price Index.
Fig. 4 Performance comparison of feed-forward artificial network on price indexes
of PIGS countries
MAE RMSE MAPE
PORTUGAL PSI GENERAL 0.067 (0.249) 0.087 (0.437) 1.820 (4.664)
IRELAND SE OVERALL 0.107 (0.492) 0.119 (0.786) 6.961 (8.028)
ATHEX COMPOSITE 0.209 (0.206) 0.268 (0.245) 2.006 (3.617)
MADRID SE GENERAL 0.143 (0.778) 0.188 (1.019) 2.804 (10.508)
Notes: This table reports results from variance decomposition for the frontier emerging markets
and the developed markets in the period before and during the crisis. Entries in brackets
correspond to the values before the crisis period.
Source: Authors
We confronted the absolute difference between the real and the forecasted values in the
period before and during the crisis using T-statistic (Fig. 5). We confirmed that the
difference in the prediction accuracy before and during the crisis is significant across
analysed indexes. The forecasting error in the period before and during the crisis was
not proved as significant only for ATHEX.
Financial Assets and Investing
28
Fig. 5 T-statistic for distribution comparison of feed-forward artificial networks
applied on PIGS stock markets before and during the financial crisis
Forecasting error before crisis ? Forecasting error during crisis
p-value t-statistic
PORTUGAL PSI GENERAL 8.1E-39 13.268*
IRELAND SE OVERALL 2.2E-58 16.558*
ATHEX COMPOSITE 0.62300 0.492
MADRID SE GENERAL 1.7E-46 14.632*
* rejection at significance level ? = 0,05
Note: The table represents t-statistic for differences in forecasting errors measured as absolute
differences between real and forecasted values. Only ANNs with the best prediction accuracy for
each price index and different groups of inputs were used for final comparison.
Source: Authors
The profitability of trading strategies based on neural networks and technical analysis
are shown in Fig. 6. Trading according to MACD indicator showed more optimistic
results when predicting the price during the financial crisis. All observed stocks
showed positive returns from which the Athens Stock Exchange obtained the highest
percentage return of 251.01%. Portuguese Stock Exchange earned 89.36%, Irish
48.35%, and Madrid Stock Exchange 12.72%. When looking at the period before the
crisis, all positions ended with losses out of which the Portuguese was the highest (-
40.76%). The trading strategy based on neural networks achieved generally lower
returns in comparison to the technical analysis. However ANNs performed better in the
period before the crisis except the Irish Stock Market where the return was -18.24% in
comparison to -4.15% achieved by MACD. The average return before the crisis gained
with the neural network was 5.61% in comparison to -25.92% with the Technical
analysis. The strategy based on neural networks was profitable both in the period
before and during the crisis for the Portuguese and Madrid Stock Exchange. On the
other hand, for the Irish and Athens Stock Exchange the returns were negative both in
the period before and during the crisis. The comparable results within the analysed
period were expected in ATHEX Composite Price Index because the difference
between forecasting error means was not rejected.
No. 3/2011
29
Fig. 6 The comparison of paper returns achieved by trading strategy based on neural
network and technical analysis
5
applied on PIGS stock markets
Neural Network MACD
PORTUGAL PSI GENERAL 36.83 (21.28) 89.36 (- 40.76)
IRELAND SE OVERALL - 31.83 (- 18.24) 48.35 (- 4.15)
ATHEX COMPOSITE - 31.28 (- 6.29) 251.01 (- 20.30)
MADRID SE GENERAL 87.91 (25.69) 12.72 (- 38.45)
Note: The table represents the paper return (in %) of trading strategies with initial amount of
10,000 EUR excluding transaction costs. Entries in brackets correspond to the values before the
crisis period and only neural networks with the best performance for each Price Index were
considered.
Source: Authors
We can conclude that during the crisis the strategy based on MACD seems to be more
appropriate while before the crisis the neural networks performed better. Better
performance in the period before the crisis was anticipated according to a lower
absolute error between the real and the forecasted values in comparison to the period
during the crisis.
Conclusion
This article focused on the analysis of the predictability of PIGS Stock Markets before
and during the financial crisis. The neural networks failed in forecasting the direction
of next day price movement but performed well in absolute price changes. When
comparing the profitability of the trading strategies, the strategy based on the technical
analysis was more profitable during the crisis while the strategy based on ANN forecast
performed better in the period before the crisis. Although the ANN strategy seemed to
be more stable because of a lower variance in returns, the MACD strategy always
achieved a positive return in the period during the crisis. ANN did not perform well in
forecasting extreme changes in stock prices in the period of huge bubbles appearing in
Portuguese, Irish, and Spanish stock markets.
5
The table reports the percentage of paper returns while only the Neural Networks with the highest
achieved prediction accuracy was used.
Financial Assets and Investing
30
Hence, a different trading strategy of the neural network should be tested and more
evidence is needed to prove the results achieved in this study. The proposed analysis
undertaken on other European national stocks would be helpful to provide an
exhaustive comparison of differences in returns before and during the crisis within
Europe. Another interesting issue is also the situation in the post crisis period.
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Attachment
Fig. 7 The prediction accuracy of neural network models for price indexes
PORTUGAL PSI GENERAL - PRICE INDEX
before during crisis
IRELAND SE OVERALL - PRICE INDEX
Financial Assets and Investing
32
Source: Authors
ATHEX COMPOSITE - PRICE INDEX
MADRID SE GENERAL - PRICE INDEX
closing prices forecasted closing prices
doc_696330470.pdf