Application of logit model and self organizing maps SOMs for the prediction of financial

Description
The purpose of this paper is to examine two different approaches in the prediction of the
economic recession periods in the US economy

Journal of Financial Economic Policy
Application of logit model and self-organizing maps (SOMs) for the prediction of
financial crisis periods in US economy
Eleftherios Giovanis
Article information:
To cite this document:
Eleftherios Giovanis, (2010),"Application of logit model and self-organizing maps (SOMs) for the prediction
of financial crisis periods in US economy", J ournal of Financial Economic Policy, Vol. 2 Iss 2 pp. 98 - 125
Permanent link to this document:http://dx.doi.org/10.1108/17576381011070184
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Application of logit model
and self-organizing maps (SOMs)
for the prediction of ?nancial
crisis periods in US economy
Eleftherios Giovanis
Analysis Center, Serres, Greece
Abstract
Purpose – The purpose of this paper is to examine two different approaches in the prediction of the
economic recession periods in the US economy.
Design/methodology/approach – A logit regression was applied and the prediction performance
in two out-of-sample periods, 2007-2009 and 2010 was examined. On the other hand, feed-forwards
neural networks with Levenberg-Marquardt error backpropagation algorithm were applied and then
neural networks self-organizing map (SOM) on the training outputs was estimated.
Findings – The paper presents the cluster results from SOM training in order to ?nd the patterns of
economic recessions and expansions. It is concluded that logit model forecasts the current ?nancial
crisis period at 75 percent accuracy, but logit model is useful as it provides a warning signal three
quarters before the current ?nancial crisis started of?cially. Also, it is estimated that the ?nancial
crisis, even if it reached its peak in 2009, the economic recession will be continued in 2010 too.
Furthermore, the patterns generated by SOM neural networks show various possible versions with
one common characteristic, that ?nancial crisis is not over in 2009 and the economic recession will be
continued in the USA even up to 2011-2012, if government does not apply direct drastic measures.
Originality/value – Both logistic regression (logit) and SOMs procedures are useful. The ?rst one is
useful to examine the signi?cance and the magnitude of each variable, while the second one is useful
for clustering and identifying patterns in economic recessions and expansions.
Keywords Binary logic, Pattern recognition, Neural net devices, Error analysis, National economy,
United States of America
Paper type Research paper
1. Introduction
One major challenge of macroeconomists and ?nancial managers is the prediction of
?nancial crisis and economic recessions and expansions periods. There are various
approaches which have been developed, applied, and examined. The ?rst approach
includes from logit and probit models for currency and banking crises prediction
(Eichengreen and Rose, 1998; Demirguc-Kunt and Detragiache, 1998) to signal to noise
ratio for identifying for various variables as potential indicators of crisis (Kaminsky
and Reinhart, 1996; Kaminsky et al., 1998) using binary variables of indicating crisis or
no crisis period. From another perspective the Markov switching autoregressive
regime model (Hamilton, 1989) and vector autoregressive models (Sims, 1980) have
been developed. Since 1990 new approaches like neural networks, fuzzy logic, and
genetic algorithm started to be examined and to gain signi?cant territory of research
and present superior results and higher forecasting and estimating better performance
than the traditional logit model and multivariate discriminant analysis (Zhang et al.,
1999; O’Leary, 1998; Cheng et al., 2006; Serrano Cinca, 1996).
The current issue and full text archive of this journal is available at
www.emeraldinsight.com/1757-6385.htm
JFEP
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98
Journal of Financial Economic Policy
Vol. 2 No. 2, 2010
pp. 98-125
qEmerald Group Publishing Limited
1757-6385
DOI 10.1108/17576381011070184
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In this paper, we showhowself-organizing map (SOM) neural networks can provide a
reliable prediction of ?nancial and economic expansion and recession periods and how
macroeconomists and governments can use it in combination with the results derived by
econometric models. We propose the scienti?c ?ndings and methods of arti?cial
intelligence (AI) because most studies have found superior results, especially in stock
prediction, economic data prediction, than the common logit models, multiple
discriminant analysis, autoregressive conditional heteroskedasticity (ARCH), moving
average method among others (Coats and Fant, 1993; Brockett et al., 2006; Zhang et al.,
1999; Fan and Palaniswami, 2000; Salchenberger et al., 1992; Ni and Yin, 2009).
So economists and ?nancial managers should adopt in their portfolio of research tools
with the AI methods and approaches. Furthermore, ?rst SOMs and other neural
networks approaches are non-parametric models and these are not based on statistical
methodology, so the speci?c methodology procedures do not suffer fromautocorrelation
and other econometric problems. Second, other widely used method, the fuzzy logic, is
based on possibilities rather than on probabilities as statistical science does.
Additionally fuzzy logic contains the imprecision philosophy where the statistics are
unable to do it, while the categorization of dummy variables is based on crisp number.
Therefore, these models are able to capture both imprecision and non-linearities by
introducing fuzzy rules.
The structure of the paper is follows. In Section 2, we present a brie?y synopsis of
previous researches of ?nancial and economic crisis predictions. In Section 3, we
describe the methodology of binary logit model, a speci?cation score test proposed by
Silva (2001) for choosing between probit and logit models, the process of feed-forward
neural networks (FFNNs) with Levenberg-Marquardt error backpropagation algorithm
training. In the remained part of Section 3, we discuss the training by SOMs neural
networks and the formulation of patterns of economic recessions. In Section 4, we
provide the data and the research sample, which will be used for estimation and
prediction. In Section 5, the empirical estimated and forecasting results are presented.
In Section 6, we discuss the general conclusions of our ?ndings.
2. Literature review
Many studies have been written in an effort to provide reliable approaches for
prediction of ?nancial crisis periods. One approach is the ?rst generation models
(Krugman, 1979; Flood and Garber, 1984), which describe that ?nancial crisis, are
results of speculative attacks because of rational arbitrage. Then the second generation
models have been developed (Krugman, 1998; Obstfeld, 1994), which showmechanisms
that even sustainable currency pegs may be attacked and broken.
There are other models beyond the second generation like the Moral Hazard models
(Allen and Gale, 2000) which try to explain cycle of investment boom and bust as a
result of important and rapid withdrawal from ?nancial assets because the asset prices
are declined sharply and suddenly, rather than a simple currency crisis. Speci?cally,
these models indicate that a possible index of ?nancial system’s fragility can be used as
a reliable pre-warning indicator.
Two related approaches have been estimated in the literature for the designing of an
early or pre-warning system for ?nancial crisis, the probit/logit models and the signal
to noise ratio. In the ?rst approach the construction of a crisis dummy variable is
required. There are various examples of application in currency crisis of this approach
Prediction of
?nancial crisis
periods
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(Frankel and Rose, 1996; Glick and Rose, 1998; Glick and Moreno, 1999). The great
advantage of this approach is that allows for testing and examining the signi?cance
and the magnitude of each explanatory variable to the possibility of ?nancial
crisis or no crisis period occurrence. But the disadvantage of this approach is the
possibility of serial correlation existence in the explanatory variables. Furthermore,
heteroskedasticity and ARCH effects in high frequency data are two additional
problems. The second approach optimizes the signal to noise ratio for the potential
crisis indicators (Kaminsky and Reinhart, 1996; Kaminsky et al., 1998). The positive
results with this approach is that it is possible to rank directly the variables as crisis
indicators, but weaknesses of this approach is that it does not allow for statistical
testing and also ignores about the possible correlations between the variables.
A different approach is followed by Mariano et al. (2001), which is a Markov
switching model with time varying transition probabilities. This model allows for sharp
movements between the regimes, so it is able to describe sudden shifts and changes in
behavior can be determined. Furthermore, the model avoids the misclassi?cation errors
and serial correlation as in probit and logit models.
The last approaches concern neural networks applications, where the most research
papers present results of supporting them. Cheng et al. (2006) propose a fuzzy
regression for a pre-warning ?nancial distress model with triangular membership
function and they found better results with fuzzy regression than with logit model.
Nachev and Stoyanov (2007) propose that the prediction of corporate bankruptcy can be
viewed as a pattern recognition problem and they estimate a predictive adaptive
resonance theory network model for ?nancial diagnosis. Other researches compare the
neural networks with traditional statistical approaches and their results show that
neural networks outperformsigni?cant the other statistical methods (Zhang et al., 1999;
Coats and Fant, 1993; Fernandez and Olmeda, 1995). Sookhanaphibarn et al. (2007)
examine various neural network models for the bankruptcy prediction in Thailand.
Speci?cally, the authors apply experiments with the learning vector quantization,
probabilistic neural network and FFNN with backpropagation learning, where the ?rst
one outperforms the other two. Chen and Du (2009) compare arti?cial neural networks
(ANN) and data mining techniques developing a ?nancial distress model and they
found that ANN presents better prediction accuracies than data mining techniques,
suggesting that AI can be a more suitably methodology than traditional statistical
approaches.
There were views and opinions that the economic recession was completed in the
beginning of the third quarter of 2009. Omarova (2009) claimed that it is very possible
that the economic crisis reached in bottom and ended its downturn on July 2009 based
on the Internationale Nederlanden Groep recession index and therefore we should not
see surprised negative economic data. The prediction was actually wrong while the
unemployment rate in the USA rose at 10.2 percent in October 2009 and US trade de?cit
widened to 36.5 billion dollars in September 2009 from 30.8 billion dollars in August
2009 (Veneziani, 2009). Additionally, the calculation of unemployment index does not
include millions of individuals who have just stopped or given up looking for work.
Also the Reuters/University Michigan index of consumer sentiment fell to 66.0 in
November, down from 70.6 in October, while analysts erroneous had been hoping to see
the measure rise as high as 72 in the latest report (Reese, 2009). This shows the
weakness of the traditional statistical process against AI. We should notice that AI is
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not panacea, but in the cases where the traditional statistics and econometrics fail or are
unable to give clear answers, AI can be ef?ciently used. Furthermore, the approaches to
the AI might be mathematically simpler, but smarter than the complicate econometric
models, indicating that AI because of its mathematical simplicity can lead to an
ef?ciently targeted formulation of the problems.
3. Methodology
In the ?rst part of this section we describe the methodology of logit model, arti?cial
feed-forward neural networks (AFNNs) and SOMs procedures. More speci?cally in
Section 3.1, we present the methodology of logit regressions. In Sections 3.2 and 3.3, we
describe the AFNN and SOMs procedures. Finally, in Section 3.4, we present the score
test for selection between probit and logit models developed by Silva (2001), wherein
the empirical results section we conclude that we prefer logit model.
3.1. Binary logit regression
The logistic distribution is de?ned as (Greene, 2008; Wooldridge, 2006):
Prob(Y = 1}x) =
e
x
/
b
1 - e
x
/
b
= F(x
/
b) (1)
The marginal partial effects of explanatory variables are given by:
›E[ y}x]
›x
= F(x
/
b)[1 2F(x
/
b)]b (2)
The logistic regression analyzes the binomial distributed data and it is:
Y
i
,B(n
i
; P
i
); i = 1; 2; . . . ; n (3)
where n
i
denotes the number of Bernoulli trials and are known, while p
i
denotes the
probabilities of success, which are unknown. The model proposes for each i a set of
explanatory variables and the model can take the following form:
P
i
= E
Y
i
n
i
}x
i
_ _
(4)
Next the unknown probabilities are modeled as linear function of variables x
i
:
y
i
= ln
p
i
1 2p
i
_ _
= b
0
-b
1
x
1;i
-b
2
x
2;i
- · · · -b
k
x
k;i
(5)
An alternative formulation of the model is:
p
i
=
1
1 - e
2(b
0
-b
1
x
1;i
-b
2
x
2;i
-· · ·-b
k
x
k;i
)
(6)
Logit model can be written a general form regression as:
y = a -

n
i=1
b
i
x
i
-1 (7)
Prediction of
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periods
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where variable y is a binary dummy variable taking value 1 if the economy is on crisis
period and value zero otherwise (no crisis period), x indicates the explanatory variables,
a is the constant, b
i
are the regression estimators, and x
i
are the explanatory variables.
The classi?cation of dummy variable is based on the de?nition by National Bureau of
Economic Research (NBER), where a recession begins when the economy reaches a
peak of activity. Based on the de?nitions of NBER the most important and conceptual
measures of the economic activity is employment rate and the domestic production.
Furthermore, according to NBER the last economic recession began in December of
2007 so we include this sample in the fourth quarter of 2007 as we use in our analysis
quarterly data. The prediction or the classi?cation percentage is done based on the
estimated coef?cients from the in-sample period each time using as the cut-off point the
value of 0.5. For the forecasting and the classi?cation performance of the binary logistic
model is:
.
If y
*
. 0.5, then the economy is on the ?nancial or economic crisis period.
.
If y
*
# 0.5, then the economy is not on crisis period.
Variable y
*
denotes the predicted values.
3.2. Feed-forward neural networks
The FFNNs model is a widely used approach known for its speed and accuracy.
A FFNN shown in Figure 1. To be speci?c, in Figure 1 we present a FFNN with an
input layer of m
0
nodes for n = 1, . . . , m
0
, one hidden layer and a single output layer.
Figure 1.
A FFNNs with one hidden
layer and one output layer
Input layer Hidden layer Output layer
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The input layer includes the input variables. The hidden layer consists of hidden
neurons or units placed in parallel. Each neuron in the hidden layer performs a
weighted summation of the weights, which then passes a non-linear activation
function. The output layer of the neural network is formed by another weighted
summation of the outputs of the neurons in the hidden layer (Haykin, 1999).
The FFNN model is estimated based on the error backpropagation algorithm (Rojas,
1996; Krose and Smagt, 1996; Haykin, 1999; Graupe, 2007). This algorithm adopts a
learning process referred as error correction learning. Speci?cally, the learning process
has as the main target the minimization of the cost function leading to a learning rule
known as the delta rule or Widrow-Hoff rule (Widrow and Hoff, 1960). The cost
function which is minimized is de?ned as:
e
k
= d
k
2y
k
(8)
where e
k
is the error signal, y
k
is the neural network output signal, and d
k
is the
desired target. So in the case we study d
k
is the binary dummy variable indicating
crisis or no crisis periods y
k
is the neural network output signal, which is trained
based on the independent variables. The purpose of the neural network learning
process is to apply corrective adjustments to the synaptic weight of neuron k in order to
make the output y
k
to come closer to the desired response d
k
in a step-by-step
manner. The minimization of the cost function is:
f =
1
2
e
k
(9)
We denote the w
kj
as the value of the synaptic weight w
kj
of neuron k excited by
element x
j
on the signal input vector x
j
at time step n, where input vector contains
the independent variables we examined. Based on the delta rule the adjustment Dw
kj

applied to the synaptic weight w
kj
at time step n is given by the following relation:
Dw
kj
= he
k
x
j
(10)
where the Greek letter h denotes the learning rate. After the computation of the
synaptic adjustment Dw
kj
the synaptic weight w
kj
is updated in the following
ways:
w
kj
(n - 1) = w
kj
-Dw
kj
(11)
In other words w
kj
and w
kj
(n - 1) can be viewed as the old and new values,
respectively, (Haykin, 1999). We used the sigmoid transfer function from input to
hidden layer while linear transfer function was used from the hidden to output layer.
Five hidden neurons are used from input to hidden layers. The sigmoid transfer
function is a real function where sig
c
: R!(0; 1) and is de?ned by the expression:
f
c
(x) =
1
1 - e
2cx
(12)
The process from the input to output layer is the forward pass, where the inputs x are
fed into the network. The transfer functions at the nodes and their derivatives
are evaluated in each node and then derivatives are stored. The purpose of the
backpropagation algorithm, which is the backward pass from output to input layer,
Prediction of
?nancial crisis
periods
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is the derivation of relation (12). This can be written as:
d
dx
f (x) =
e
2x
(1 - e
2x
)
2
= f (x) · (1 2f (x)) (13)
More speci?cally, the ?rst step is the forward pass. The second step is the
backpropagation to the output layer. This can be written as:
e
(B)
j
= y
j
(1 2y
j
)( y
j
2d
j
) (14)
where e
(B)
j
is de?ned as the backpropagation error, y
j
is the signal or output of the
output layer and d
j
is the desired output, where in our case is the dummy variable. The
partial derivative is:
›E
›w
(B)
kj
= y
j
(1 2y
j
)( y
j
2d
j
) ho
j
= e
(B)
j
· ho
j
(15)
where E is the error-cost function (9), ho
j
denotes the output values from the hidden
layer and ›w
(B)
kj
is the synaptic weight matrix from output to hidden layer. The third
step and the next process of backpropagation algorithm is the backpropagation to the
hidden layer. This is:
e
(A)
j
= ho
j
(1 2ho
j
)

n
k=1
w
(B)
kj
e
(B)
j
(16)
where e
(A)
j
is de?ned as the backpropagation error to hidden layer, ho
j,
e
(B)
j
and ›w
(B)
kj
are de?ned as previously. The partial derivative is:
›E
›w
(A)
kj
= 2e
(B)
j
· I
k
(17)
I
k
denotes the inputs and ›w
(A)
kj
is the synaptic weight matrix fromhidden to input layer.
The ?nal step is the updating of weights as we described above in relations (10)-(11). We
use Levenberg-Marquardt error backpropagation algorithm and is de?ned as:
w
i-1
= w
i
2(J
T
i
J
i
-l I )
21
· J
T
i
· 1
i
(18)
where w
i
is the vector containing all the weights of the networks, 1
i
is a vector of network
errors and J
i
is the Jacobian matrix which contains the ?rst derivatives of the network
errors with respect to weights and biases. Jacobian matrix is a N-by-Tmatrix, where Nis
the number of entries or inputs in the network and Tis the total number of inputs, which
is the sum of weights and biases. More speci?cally Jacobian matrix is presented in
equation (19). Matrix (J
T
J) must be positive de?nitive. Relation (18) includes lI, where
I is the identity matrix and l is the learning rate or know as the damping term. Actually
the signi?cant difference between Newton and Levenberg-Marquardt algorithms is that
the last one introduces the term lI (Rojas, 1996):
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T
)
J =
›F(x
1
; w)
›w
1
. . . . . .
›F(x
1
; w)
›w
T
›F(x
2
; w)
›w
1
. . . . . .
›F(x
2
; w)
›w
T
. . . . . . . . . . . .
. . . . . . . . . . . .
›F(x
N
; w)
›w
1
. . . . . .
›F(x
N
; w)
›w
T
_
¸
¸
¸
¸
¸
¸
¸
¸
¸
¸
¸
¸
_
_
¸
¸
¸
¸
¸
¸
¸
¸
¸
¸
¸
¸
_
(19)
where wis de?ned as in equation (18), x denotes the inputs and F is the neural network
function which can be viewed as a non-linear function of the following general form:
F(x; w) = y (20)
where x and w are de?ned as previously and y is the output of the neural network.
3.3. Self-organizing maps neural networks
SOMs are arti?cial neural networks based on unsupervised training and have been
proposed by Kohonen (1989). SOMs are non-linear non-parametric techniques which
are applied in order to ?nd relationships between inputs and outputs and to organize
the data and to disclose unknown patterns. In a SOM the neurons are placed at the
nodes of lattice that is usually one or two dimensional. The neurons then are placed in
classes of input patterns through a course of competitive learning process (Haykin,
1999). The SOM as we mentioned above is a non-linear process so it can be viewed as a
non-linear generalization of principal components analysis. The algorithm of SOM
assigns small values picked from a random number generator for the initialization of
the synaptic weights of the network. As the network is properly initialized then three
processes are involved in the formation of the SOM. The ?rst process is the competition,
where for each input pattern the neurons in the network compute their respective values
of a discriminant function and then this function provides the basis for the competition
among the neurons. The neuron with the largest discriminant function is the winner of
the competition. The second process is the cooperation, where the winning neuron
determines the spatial location of a topological neighborhood of neurons providing the
basis for the cooperation among the neighborhood neurons. The last process is the
synaptic adaptation, where in this process the neurons increase their individual values
of the discriminant function in relation to the input pattern through suitable adjustment
procedures applied to their synaptic weights. Below we provide a brief description of
the mathematical formulation of SOM (Haykin, 1999). For the competitive process we
consider a space of input variables with dimension m:
x = [x
1
; x
2
; x
3
; . . . ; x
m
]
T
(21)
The synaptic weight vector of neuron j can be written as:
w
j
= [w
j1
; w
j2
; w
j3
; . . . ; w
jm
]
T
; j = 1; 2; . . . ; k (22)
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where k is the total number of neurons in the network. In order to ?nd the best match of
the input vector x with the synaptic weight vector w
j
we compare the inner products
w
j
T
x for j = 1, 2, . . . , k and we choose the largest. Actually the maximization of the inner
product w
j
T
x is mathematically equivalent to the minimization of the Euclidean distance
between vectors x and w
j
. We have:
i(x) = arg min
j
|x 2w
j
|; j = 1; 2; . . . ; k (23)
where i(x) is the best matching or the winning neuron for the input vector x.
For the next process which is the cooperative process the winning neuron locate the
center of a topological neighborhood of the cooperating neurons. We de?ne c
j,i
as the
topological neighborhood centered on winning neuron i and encompassing a set of
cooperating neurons which each one is denoted as j. We express d
j,i
as the lateral
distance between winning neuron i and cooperating neuron j. Then we assume that c
j,i
has the Gaussian function:
c
j;i
(x) = exp 2
d
2
j;i
2s
2
_ _
(24)
The parameter s is the effective width of the topological neighborhood, which
measures the degree to which cooperating neurons in the vicinity of the winning
neuron participate in the learning process. Variable d
2
j,i
is de?ned as:
d
2
j;i
= |r
j
2r
i
| (25)
where the vectors r
j
and r
i
are discrete vectors denoting the position of cooperating
neuron j and wining neuron i, respectively, measured in discrete output space. For
cooperation among neighboring neurons to hold is necessary that c
j,i
be dependent on
d
2
j,i
between winning neuron i and exciting neuron j. Furthermore, c
j,i
shrinks with time
as s decreases. The dependence of s on discrete time h is the exponential decay
described by Ritter et al. (1992), as:
s = s
0
exp 2
n
t
1
_ _
(26)
where s
0
denotes the value of s in the initiation of SOM algorithm and t
1
is a time
constant. Because, the topological neighborhood c
j,i
assumes a time-varying from it is:
c
j;i(x)
= exp 2
d
2
j;i
2s
2

_ _
(27)
where s is de?ned as in equation (26) and n is the time or the number of iterations,
which as is increased the width s decreases at an exponential rate and topological
neighborhood c
j,i,(x)
shrinks. The purpose of a wide c
j,i,(x)
is to correlate the
directions of the weight updates of a large number of excited neurons in lattice, so as
the width of c
j,i,(x)
is decreased so the number of neurons, decreased, whose update
directions are correlated.
The last step is the adaptive process where the synaptic weight vector w
j
of neuron j
in the network is necessary to change in relation with the inputs x in order for the
network to be self-organized. The adaptive learning process is de?ned as:
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Dw
j
= hc
j;i(x)
· (x 2w
j
) (28)
Parameter h denotes again the learning rate, x denotes the inputs, w
j
is the synaptic
weights vector and c
j,i
is de?ned as previously. The updating of weights becomes
similarly as in the case of FFNNs and it is:
w
j
(n - 1) = w
j
-hc
j;i(x)
· (x 2w
j
) (29)
where w
j
(n - 1) and w
j
denote the new and old values as in the case of FFNN and n
denotes the time of iterations. We observe that the learning rate h should be
time-varying so it should be decreased gradually as the time n increases. Similarly, the
exponential decay will be:
h = h
0
exp 2
n
t
2
_ _
(30)
where h
0
is the initial value of h at the initiation of SOM algorithm and t
2
is another
time constant. According to Kohonen (1982) the adaptive process is divided by two
phases. The ?rst phase is the self-organizing phase where the initial value of the
learning rate h should be 0.1 and should decrease gradually until to remain above 0.01.
Then the topological neighborhood c
j, i, (x)
initially includes all neurons in the
network centered at the winning neuron i and then c
j, i, (x)
shrinks through time.
Finally, c
j, i, (x)
is permitted to reduce to a small value of a couple of neighboring
neurons around a winning neuron.
In the second phase, which is the convergence phase and it is needed for the
accurate statistical quanti?cation of the input space. This requires that the learning
rate should be maintained at 0.01 and it must not be allowed to reach 0. Additionally,
the neighborhood function c
j, i
should contain only the nearest neighbors of the winning
neuron, which may be reduced to one or even to zero neighboring neurons.
3.4. Speci?cation test for selection between probit and logit models
In this section, we describe brie?y the test to decide whether to estimate with probit or
logit model proposed by Silva (2001). This test is particularly and especially convenient
for binary models as in the case we examine. We de?ne logit regression as the Model 1
and probit regression as Model 2. We compute the new variable z
i
(0) as:
z
i
(0) =
^
P
1
i
1 2
^
P
1
i
_ _
^
f
1
i
ln
^
P
1
i
1 2
^
P
2
i
_ _
^
P
2
i
1 2
^
P
1
i
_ _
_
¸
_
_
¸
_ (31)
where P
1
i
and P
2
i
are the predicted probabilities of Models 1 and 2 or logit and
probit models, respectively, i denotes the observations and f
i
1
denotes the individual
observations on the density for the null model, which is logit model. The next step is to
re-estimate Model 1 or logit regression added z
i
(0) as additional dependent variable.
The H0 is that the coef?cient of the new variable z
i
(0) is zero. Accepting the H0 favors
Model 1 or logit, while rejecting the H0 then we choose Model 2 or probit regression.
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4. Data and research sample design
We use quarterly data of seven indexes as independent variables. The data source is
the Federal Reserve Bank of St Louis and the NBER. These are the National Income,
the balance of accounts, which is de?ned as the sum of trade balance (exports minus
imports) plus the net factor income from the abroad plus the net transfer payments.
The remaining variables are the industrial production, the bank prime loan rate, the
unemployment rate, the total investments at all commercial banks and the total loans
at all commercial banks. We could use in?ation rate and public debt, but they were
found statistically signi?cant or presented the wrong sign. The above variables were
used in logit regression.
In SOMs neural networks we also include additionally six more variables, the
in?ation rate, the oil prices, the S&P 500 index prices, the interest rates of three-monthly
Treasury Bills, the total borrowings of depository institutions from Federal Reserve
System and the US public debt in order to achieve statistical signi?cance of the major
variables. The choice of variables is based on various research papers and studies
(Demirguc-Kunt and Detragiache, 1998; Eichengreen and Rose, 1998; Glick and Moreno,
1999; Fioramanti, 2006), as also based on NBER, which de?nes real gross domestic
product (GDP), real income unemployment rate, industrial production and retail sales
as the most important factors de?ning the economic activity in US economy.
In Figures 2-3 the variables included in our analysis are presented. We observe that
it is possible that the variables are not stationary in the levels, but probably are in the
?rst or second differences. To be speci?c we con?rm this assumption by applying
Augmented Dickey-Fuller (ADF) (Dickey and Fuller, 1979), Phillips-Perron (PP) unit
root tests (Phillips and Perron, 1988) and Kwiatkowski-Phillips-Schmidt-Shin (KPSS)
stationary test (Kwiatkowski et al., 1992). The ADF test is de?ned from the following
relation:
Dy
t
= m -gy
t21
-f
1
Dy
t21
-f
2
Dy
t22
- · · · -f
p
Dy
t2p
-bt -1
t
(32)
where y
t
is the variable we examine each time. In the right hand of regression (32) the
lags of the dependent variable are added with order of lags equal with p. Additionally
regression (26) includes the constant or drift m and trend parameter b. The disturbance
term is de?ned as 1
t
. In the next step, we test the hypotheses:
H0. w = 1, b = 0 = .y
t
, I(0) with drift.
against the alternative:
H1. }w} , 1=.y
t
, I(1) with deterministic time trend.
Similarly, PP test estimates the equation (32), but it modi?es the t-ratio of the g
coef?cient and it is based on the following statistic:
~
t
g
=
^
d
0
^
l
_ _
1=2
2
1
2
T ^ s
^
d
0
2
^
l
_ _
s
^
l
1=2
(33)
where t
g
is the t-ratio of g, s is the coef?cient standard error, s is the standard error of
the test regression, and d
0
is a consistent estimation of the error variance in equation
(32) and is calculated as s
2
(T 2 k)/T, where k denotes the number of regressors.
Finally, l is an estimator de?ned as:
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l =

T
t=j-1
f (
^
j ) · K
j
l
_ _
(34)
where l is a bandwidth parameter, K is a kernel function and f( j) is the jth sample
autocovariance of the residuals 1
t
and is de?ned as:
f (
^
j) =

T
t=j-1
^ 1
t
^ 1
t21
T
(35)
Figure 2.
Line graphs during period
1947-2009
0
2,000
4,000
6,000
8,000
10,000
12,000
14,000
50 55 60 65 70 75 80 85 90 95 00 05 50 55 60 65 70 75 80 85 90 95 00 05
National income
–1,000
–800
–600
–400
–200
0
200
(a)
50 55 60 65 70 75 80 85 90 95 00 05
(c)
50 55 60 65 70 75 80 85
Notes: (a) National income; (b) balance of accounts; (c) industrial production; (d) bank prime loan rate;
(e) unemployment rate; (f) total investments at all commercial banks
90 95 00 05
(e)
(b)
50 55 60 65 70 75 80 85 90 95 00 05
(d)
50 55 60 65 70 75 80 85 90 95 00 05
(f)
0
20
40
60
80
100
120
0
4
8
12
16
20
24
2
3
4
5
6
7
8
9
10
11
0
400
800
1,200
1,600
2,000
2,400
Balance of accounts
Industrial production
Bank prime loan rate
Unemployment rate
Total investments
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Figure 3.
Line graphs during period
1947-2009
0
2,000
4,000
6,000
8,000
10,000
50 55 60 65 70 75 80 85 90 95 00 05
–1.5
–1.0
–0.5
0.0
0.5
1.0
1.5
(a)
50 55 60 65 70 75 80 85 90 95 00 05
(c)
50 55 60 65 70 75 80 85 90 95 00 05
(e)
50 55 60 65 70 75 80 85 90 95 00 05
(b)
50 55 60 65 70 75 80 85 90 95 00 05
(d)
50 55 60 65 70 75 80 85 90 95 00 05
(f)
50 55
Notes: (a) Total loans at all commercial banks; (b) inflation rate; (c) oil prices; (d) S&P 500 index prices;
(e) interest rates of three-monthly Treasury Bills; (f) total borrowings of depository institutions from
Federal Reserve System; (g) US public debt
60 65 70 75 80 85 90 95 00 05
(g)
0
20
40
60
80
100
120
140
0
400
800
1,200
1,600
0
4
8
12
16
0
100
200
300
400
500
600
700
0
2,000
4,000
6,000
8,000
10,000
12,000
Total loans at all commercial banks
Inflation rate
Oil prices
S&P 500 index
3-monthly
treasury bills
interest rates
Total borrowings
Public debt
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In both ADF and PP tests, we accept that a variable is stationary if we reject the H0 of
unit root test. On the other hand in KPSS test series is assumed to be stationary under
the H0. The series is detrended by regressing y on a random walk process x
t
, i.e.
x
t
= x
t21
- u
t
and a deterministic term bt:
y
t
= x
t
-bt -1
t
(36)
KPSS statistic is based on the residuals for the ordinary least squares regression (36).
Let the partial sum series of 1
t
be s
t
. It is:
s
t
=

t
j=1
e
j
; (37)
The KPSS statistic is then de?ned as:
KPSS = T
22

T
t=1
s
2
t
^ s
2
( p)
(38)
where T is the number of sample and ^ s
2
( p) is the long-run variance of 1
t
and can be
constructed from the residuals 1
t
as:
^ s
2
( p) =
1
T

T
t=1
1
2
t
-
2
T

p
t=1
w
j
( p) -

T
t=j-1
1
t
1
t2j
; (39)
where p is the truncation lag, w
j
( p) is an optional weighting function that corresponds
to the choice of a special window (Bartlett, 1950). Under the H0 of level stationary:
KPSS !
_
1
0
V
1
(r)
2
dx (40)
where V
1
(x) is a standard Brownian bridge: V
1
(r) = B(r) – rB(1) and B(r) is a Brownian
motion (Wiener process) on r [ [0, 1]. Because relation (40) is refereed in testing only
on the intercept and not in the trend and as we are testing with both intercept and trend
we have the second-level Brownian bridge V
2
(x) and it is:
KPSS !
_
1
0
V
2
(r)
2
dx
where V
2
(x) is given by:
V
2
(r) = W(r) - (2r 23r
2
)W
1
- (26r - 6r
2
)
_
1
0
W
s
(s) ds (41)
In Table I ADF and PP unit root and KPSS stationary test are provided. We observe
that in the most cases variables are stationary in the ?rst differences so they are I(1),
with the exception of total investments, total loans, and public debt, which are
stationary in the second differences, I(2), based on KPSS test. Also, some variables are
stationary in their levels or we have I(0), based on ADF and PP tests for a = 0.05 and
a = 0.10. We prefer to reject the H0 of ADF and PP tests as also to accept the H0 of
KPSS test for all three 1, 5, and 10 percent signi?cance levels. Finally, we observe that
variable total borrowings is I(0) based on ADF test.
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The ?rst estimating period used for logit procedure is the period 1947-2007, based on
the availability of data. The ?rst out-of-sample period is de?ned as the period of
2007-2009 in order to predict the current crisis. Then we estimate the period 1947-2009
and using the expected or estimated values for the independent variables in 2010 we
estimate the forecasts for 2010.
In the case of SOM neural networks we use period 1913-2009 and we obtain different
variables in relation with the previous cases and these are the in?ation rate, the
industrial production, the Dow Jones Industrial Index prices, the total borrowings of
depository institutions from Federal Reserve System, the exchange rate circulation and
ADF t-stat. PP t-stat. KPSS LM-stat.
Variables in levels
National income 20.4323 20.5415 0.4820
Balance of accounts 23.6671 21.6275 0.3479
Industrial production 23.0027 22.2827 0.3222
Bank prime loan rate 22.1678 22.2072 0.3696
Unemployment rate 24.5627 22.9859 0.2146
Total investments 22.8280 23.3405 0.4546
Total loans 22.8698 23.9385 0.4718
In?ation rate 23.8613 28.6985 0.3230
Oil price 20.7535 22.8299 0.1527
S&P 500 22.0631 22.0449 0.4197
Three-monthly treasury bill interest rates 21.6602 22.0603 0.4186
Total borrowings 236.341 23.1415 0.1518
Public debt 22.5482 23.5526 0.5193
Variables in ?rst differences
National income 29.0561 29.0739 0.0598
Balance of accounts 213.340 213.328 0.0723
Industrial production 27.2208 27.4079 0.0871
Bank prime loan rate 212.555 212.515 0.0266
Unemployment rate 25.7803 0.0501
Total investments 26.0458 211.402 0.2356
Total loans 23.7757 210.243 0.2467
In?ation rate 219.093 236.647 0.0712
Oil price 24.6874 212.796 0.0698
S&P 500 210.728 210.838 0.0561
Three-monthly treasury bill interest rates 27.3398 212.127 0.0261
Total borrowings 213.527 0.0914
Public debt 25.1645 27.6341 0.2003
Variables in second differences
Total investments 0.0500
Total loans 0.0777
Critical values for ADF and PP tests
a
a = 0.01 23.996
a = 0.05 23.428
a = 0.10 23.137
Public debt 0.0778
Critical values for KPSS test
b
a = 0.01 0.216
a = 0.05 0.146
a = 0.10 0.119
Sources:
a
MacKinnon (1996);
b
Kwiatkowski et al. (1992)
Table I.
ADF, PP unit root and
KPSS stationary tests
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the producer price index, in order to predict the current ?nancial crisis. More
speci?cally, we set up the period 1913-1923 as the in-sample period and 1924-2009 as
the out-of-sample period.
5. Empirical results
5.1. Logit regression
In this sub-section, we present the estimation results by logit model and the correctly
classi?cation percentage rates in both in-sample and out-of-sample periods. First, we
present the results of Silva (2001) test, described in previous section, in Table II, in an
effort to decide whether we should obtain logit or probit model. It should be noticed
that the estimated results of probit and logit regressions are not presented, but only the
estimated coef?cient of variable z
i
(0) is presented because in this phase we are
interesting only on deciding which model is more appropriate. From Table II, we
observe that we accept the H0 which favors Model 1 and this is logit regression. For
this reason we present only the estimated results generated by logit model.
In Table III, we present the estimation results and we observe that all the coef?cients
have the expected sign and are statistically signi?cant. As it was expected an increase of
1 percent in national income will lead to a decrease of 0.9937 – e
20.00627
– of ?nancial
crisis to take place. The balance of accounts has positive effect to lead in a ?nancial crisis
period and this possibility is increased at 1.013 if the balance of accounts increased by 1
percent. Similar magnitude of effect presents the total loans which is 1.022. The
strongest positive effects leading to ?nancial crisis periods is the unemployment rate
and the industrial production, which are 6.827 and 1.491, respectively, followed by bank
prime loan rate with possibility of 1.244. It should be noticed that someone would
probably expect a negative sign for the industrial production. The positive sign can be
explained as that the international oil prices and the money supply impose a positive
impact on the industrial production, as also the drop of real exchange rate, where the US
dollar dropped signi?cant during 2008 against euro, has as a result the increase of the
industrial production ( Jiranyakul, 2006). Additionally the positive sign of industrial
production is similar with that of Japan’s during 1930s which was doubled. This
phenomenon was a result of two policies. First, was the large ?scal stimulus involving a
de?cit spending and second, was the devaluation of the currency (Cha, 2003). The
situation is very similar with US economy with large ?scal stimulus packages which
increased the de?cit and the devaluation of the US dollar, especially during the period
2008-2009. On the other hand the positive sign of the total loans can be explained by the
fact of over-indebtedness, where fuelled speculation and asset bubbles (Fisher, 1933;
Fortune, 2000) were some of the possible factors, which had led to the Great depression
as also happened before the current crisis. Furthermore, when the market falls, brokers
call in these loans, these cannot be paid back. Finally, if total investments increased by 1
percent then the possibility of ?nancial crisis occurrence is decreased at 0.945. So we
conclude that unemployment rate, the industrial production, the bank prime loan rate,
the total loans and the balance of accounts affect signi?cant the possibility of ?nancial
Coef?cient a Standard error z-statistic p-value
0.4845 0.4102 1.18 0.238
Table II.
Estimated coef?cient a
of variable z
i
(0) with logit
regression
Prediction of
?nancial crisis
periods
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s
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o
n
s
t
a
n
t
2
2
7
.
1
9
7
(
2
3
.
2
1
)
*
B
a
n
k
p
r
i
m
e
l
o
a
n
r
a
t
e
0
.
2
1
8
(
2
.
3
2
)
*
*
L
R
(
7
)
x
2
5
2
.
9
1
[
0
.
0
0
0
]
N
a
t
i
o
n
a
l
i
n
c
o
m
e
2
0
.
0
0
6
2
7
(
2
2
.
6
0
)
*
U
n
e
m
p
l
o
y
m
e
n
t
r
a
t
e
1
.
9
2
0
(
3
.
5
2
)
*
P
s
e
u
d
o
R
2
0
.
4
2
9
7
B
a
l
a
n
c
e
o
f
a
c
c
o
u
n
t
s
0
.
0
1
3
6
(
1
.
7
8
)
*
*
*
T
o
t
a
l
i
n
v
e
s
t
m
e
n
t
s
2
0
.
0
5
6
(
2
3
.
2
4
)
*
L
o
g
-
l
i
k
e
l
i
h
o
o
d
2
3
5
.
1
0
4
I
n
d
u
s
t
r
i
a
l
p
r
o
d
u
c
t
i
o
n
0
.
3
9
9
(
2
.
4
7
)
*
*
T
o
t
a
l
l
o
a
n
s
0
.
0
2
1
9
(
3
.
5
0
)
*
N
o
t
e
s
:
S
t
a
t
i
s
t
i
c
a
l
l
y
s
i
g
n
i
?
c
a
n
t
a
t
*
0
.
0
1
,
*
*
0
.
0
5
,
*
*
*
0
.
1
0
l
e
v
e
l
s
,
r
e
s
p
e
c
t
i
v
e
l
y
;
z
-
s
t
a
t
i
s
t
i
c
s
i
n
p
a
r
e
n
t
h
e
s
e
s
;
p
-
v
a
l
u
e
s
i
n
b
r
a
c
k
e
t
Table III.
Results of binary logistic
regression for period
1947-2006
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crisis occurrence, so an appropriate controlling of these economic variables can lead in
an effort of avoiding crisis. In Table III, we present the likelihood-ratio (LR) x
2
. More
speci?cally the LR x
2
statistic is de?ned as (Agresti, 2002):
LR
k
(x
2
) = 2

f · ln
f
f
i
_ _
(42)
where f and f
i
indicate, respectively, the observed and the expected frequency of crisis
and no-crisis periods and k denotes the number of independent variables in the logistic
regression. Using the LR statistic of (42) we test the following hypothesis:
H0. b
0
= b
1
= · · · = b
k
= 0, indicating that logit regression is statistically
insigni?cant, against the alternative hypothesis.
H1. b
0
– b
1
– · · · – b
k
– 0, indicating that logit regression is statistically
signi?cant.
Likelihood-ratio x
2
statistic follows x
2
distribution with k degrees of freedom. In the
case, we examine there are seven independent variables and so it is k = 7. In Table III,
we observe that LR
(7)
x
2
is 52.91 and it is higher than the x
2
(7)
at a = 0.10, a = 0.05,
and a = 0.01 which are 12.017, 14.067, and 16.013, respectively,. Moreover, the p-value
of LR
(7)
x
2
, in Table III, is zero and we reject the H0 indicating that the logit regression
is statistically signi?cant.
In Table IV we present the correctly classi?cation performance of logit regression in
the period 1947-2007, which is the in-sample period, and we observe that it is not able
to forecast with success the economic recessions periods.
In Table V, we present the forecasts of logit model for the period 2007-2009 and
because we have quarterly data, so there are 12 periods, we observe that logit model
predicts with accuracy 75 percent the economic recession periods. We should mention
that the predicted probabilities calculated by logit regression give a signal of economic
recession in whole period 2007-2009, while the of?cial beginning of the economic
crisis is set up at the fourth quarter of 2007 based on NBER. This fact is not necessary
Prediction
Actual Crisis No crisis Correctly percentage rate
Crisis 14 6 70.00
No crisis 8 116 93.54
Overall percentage 90.27
MAE 0.1548 RMSE 0.2698
Table IV.
Prediction results of
binary logistic regression
for in-sample period
1947-2006
Prediction
Actual Crisis No crisis Correctly percentage rate
Crisis 9 3 75.00
No crisis 0 0 NAN
Overall percentage 75.00
MAE 0.2500 RMSE 0.5000
Table V.
Prediction results of
binary logistic regression
for out-of-sample period
2007-2009
Prediction of
?nancial crisis
periods
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a disadvantage because logit regression is able to give a warning signal three quarters
before the current crisis took place. Furthermore, logit estimation provide us also with
a warning signal two quarters before the sub prime crisis, which began in June of 2007.
Then we present the estimation results of logit regression in Table VI for the period
1947-2009. We observe that the estimated coef?cients are almost the same with the
results of Table I. The percentage of correct classi?cation rates and forecasts for the
in-sample period 1947-2009 are presented in Table VII, where the forecasting
performance for the ?nancial crisis periods is increased. Next we provide the forecasts
for period 2010 in Table VIII and we conclude that the economic recession will be
continued. The forecasting performance of logit regression in both in-sample and
out-of-sample periods is counted based on the percentage of correct classi?cation, the
mean absolute error (MAE) and root mean squared error (RMSE).
5.2. Feed-forward neural networks and self-organizing maps
For the FFNNs and the Levenberg-Marquardt error backpropagation algorithm we
used 0.1 as the learning rate and we set up the value of l at 0.01. The maximum
number of epochs (replications) was set up at 50 and the training process stopped after
16 epochs, where the training error, which is the sum squared error of network errors,
was found equal with 0.01436 (Fioramanti, 2006). For the SOMs the initial learning rate
was set up at 0.9 and the number of epochs was equal with 200. The literature review
includes various studies in choosing the size of SOMs, but the research of these studies
is restricted in the examination of the forecasting performance in bankruptcy crisis and
failure of enterprises and not in predicting ?nancial crises of national economies. Our
criteria in choosing the size in our analysis is based on the possible clusters of crises in
the period 1913-2009, we examine and their speci?c and unique characteristics (Burns
and Wesley, 1946; Moore, 1983; Gordon, 1986; Allen, 2000). The ?rst cluster includes
the crises of 1918-1921, 1945, 1948-1949, and 1953-1954 where the common cause of all
these crises was the war. Speci?cally, 1918-1921 is the post-First World War crisis,
1945 and 1948-1949 are the post-Second World War crises and ?nally 1953-1954 was
the post-Korean war crisis. The second cluster includes the mild recessions of
1910-1913, 1926-1927, 1960-1961, 1969-1970, the second half of 1980 and 1990-1991. The
third cluster includes the sharp recession of 1923-1924, while the fourth cluster is
consisted by the Great Depression of 1929-1933 and the ?fth cluster is consisted by the
crisis of 1937-1938. The sixth cluster is consisted of crisis 1957-1958 which was mostly
due to the tightened monetary policy of the Federal Reserve, the seventh cluster
includes the crises of 1973-1975 and 1981-1982, where the main reasons were caused by
The Organization of the Petroleum Exporting Countries and Iranian oil embargo. The
eighth cluster includes the crisis of 2000-2002 which was caused by an economic boom
in computer and software sales caused by the Y2K scare and created a boom and
subsequent bust in Internet businesses. Finally, the ninth and last cluster is the current
crisis of 2007. Because we categorize the crises in nine cluster for this reason we set up
the size of SOM neural networks at [3 3].
In the ?gures which are followed the line graphs represent the dummy variable
indicating crisis or no crisis. So for values of one the economy is on crisis period, while
for values of zero the economy characterized by no crisis periods. In Figure 4, we
present the SOM hexagonal topology 3 £ 3 as also the neighbor weight distances. In
Figures 5 and 6, the predicted versus the actual values for in-sample 1913-1923 and
JFEP
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s
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i
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o
n
s
t
a
n
t
2
2
7
.
8
4
4
(
2
3
.
3
8
)
*
B
a
n
k
p
r
i
m
e
l
o
a
n
r
a
t
e
0
.
1
9
6
(
2
.
1
0
)
*
*
L
R
(
7
)
x
2
8
5
.
0
5
[
0
.
0
0
0
]
N
a
t
i
o
n
a
l
i
n
c
o
m
e
2
0
.
0
0
5
2
9
(
2
2
.
6
3
)
*
U
n
e
m
p
l
o
y
m
e
n
t
r
a
t
e
1
.
9
6
6
(
3
.
6
4
)
*
P
s
e
u
d
o
R
2
0
.
5
3
7
2
B
a
l
a
n
c
e
o
f
a
c
c
o
u
n
t
s
0
.
0
1
3
7
(
2
.
0
6
)
*
*
T
o
t
a
l
i
n
v
e
s
t
m
e
n
t
s
2
0
.
0
5
6
(
2
3
.
2
7
)
*
L
o
g
-
L
i
k
e
l
i
h
o
o
d
2
3
6
.
6
3
6
I
n
d
u
s
t
r
i
a
l
p
r
o
d
u
c
t
i
o
n
0
.
4
1
0
(
2
.
6
0
)
*
T
o
t
a
l
l
o
a
n
s
0
.
0
2
0
1
(
3
.
5
5
)
*
N
o
t
e
s
:
S
t
a
t
i
s
t
i
c
a
l
l
y
s
i
g
n
i
?
c
a
n
t
a
t
*
0
.
0
1
,
*
*
0
.
0
5
,
*
*
*
0
.
1
0
l
e
v
e
l
s
,
r
e
s
p
e
c
t
i
v
e
l
y
;
z
-
s
t
a
t
i
s
t
i
c
s
i
n
p
a
r
e
n
t
h
e
s
e
s
;
p
-
v
a
l
u
e
s
i
n
b
r
a
c
k
e
t
Table VI.
Results of binary logistic
regression for period
1947-2009
Prediction of
?nancial crisis
periods
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out-of-sample 1924-2009, respectively, are presented. We observe that SOM is able to
predict with accuracy the depressions and the sharp economic recessions, with the
exception the post-Korean war crisis in 1953-1954 and the recession of 1981-1982. On
the other hand the predicting power of SOM for some mild recessions, as the recessions
of 1960-1961, 1969-1970, and 1990-1991 is rather poor. In Tables IX and X the
forecasting performance of SOM for 1913-1923 and 1924-2009 periods, respectively, are
reported. The cutoff point for classi?cation of crisis or no crisis period is the same with
that we have used in logit, where for values higher than 0.5 the economy is in recession
and for values lower than 0.5 the economy is in expansion.
Finally, in Table XI we present the prediction results of SOMfor the current ?nancial
crisis of 2007. We observe that SOM predicts at 100 percent correct the crisis periods,
while the correct classi?cation for no crisis periods is 75 percent. The overall correct
classi?cation percentage is 91.67. Based on RMSE, MAE, and the correct classi?cation
percentage we conclude that SOM outperforms the forecasting performance of logit
model for the current ?nancial crisis.
Generally, we observe that the forecasting performance of SOM is rather poor in the
in-sample period 1913-1923, while is considerably improved in the out-of-sample period
1924-2009 and especially in current crisis 2007-2009 period. Additionally, we observe
that the forecasting performance of SOM outperforms the respective of logit regression
Prediction
Actual Crisis No crisis Correctly percentage rate
Crisis 23 5 82.14
No crisis 9 119 92.97
Overall percentage 91.03
MAE 0.1470 RMSE 0.2709
Table VII.
Prediction results of
binary logistic regression
for in-sample period
1947-2009
Prediction
Crisis No crisis
4 0
Table VIII.
Prediction results of
binary logistic regression
for 2010
Figure 4.
Self-organizing maps
(a) (b)
Notes: (a) Hexagonal topology with size [3 3]; (b) neighbor weight distances
JFEP
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Figure 5.
Actual and predicted
values of SOM for period
1913-1923
0
0.2
0.4
0.6
0.8
1
1.2
J
a
n
-
1
9
1
3
J
u
l
-
1
9
1
3
J
a
n
-
1
9
1
4
J
u
l
-
1
9
1
4
J
a
n
-
1
9
1
5
J
u
l
-
1
9
1
5
J
a
n
-
1
9
1
6
J
u
l
-
1
9
1
6
J
a
n
-
1
9
1
7
J
u
l
-
1
9
1
7
J
a
n
-
1
9
1
8
J
u
l
-
1
9
1
8
J
a
n
-
1
9
1
9
J
u
l
-
1
9
1
9
J
a
n
-
1
9
2
0
J
u
l
-
1
9
2
0
J
a
n
-
1
9
2
1
J
u
l
-
1
9
2
1
J
a
n
-
1
9
2
2
J
u
l
-
1
9
2
2
J
a
n
-
1
9
2
3
J
u
l
-
1
9
2
3
Predicted values
Actual values
Figure 6.
Actual and predicted
values of SOM for period
1924-2009
0
0.2
0.4
0.6
0.8
1
1.2
J
a
n
-
1
9
2
4
J
a
n
-
1
9
2
6
J
a
n
-
1
9
2
8
J
a
n
-
1
9
3
0
J
a
n
-
1
9
3
2
J
a
n
-
1
9
3
4
J
a
n
-
1
9
3
6
J
a
n
-
1
9
3
8
J
a
n
-
1
9
4
0
J
a
n
-
1
9
4
2
J
a
n
-
1
9
4
4
J
a
n
-
1
9
4
6
J
a
n
-
1
9
4
8
J
a
n
-
1
9
5
0
J
a
n
-
1
9
5
2
J
a
n
-
1
9
5
4
J
a
n
-
1
9
5
6
J
a
n
-
1
9
5
8
J
a
n
-
1
9
6
0
J
a
n
-
1
9
6
2
J
a
n
-
1
9
6
4
J
a
n
-
1
9
6
6
J
a
n
-
1
9
6
8
J
a
n
-
1
9
7
0
J
a
n
-
1
9
7
2
J
a
n
-
1
9
7
4
J
a
n
-
1
9
7
6
J
a
n
-
1
9
7
8
J
a
n
-
1
9
8
0
J
a
n
-
1
9
8
2
J
a
n
-
1
9
8
4
J
a
n
-
1
9
8
6
J
a
n
-
1
9
8
8
J
a
n
-
1
9
9
0
J
a
n
-
1
9
9
2
J
a
n
-
1
9
9
4
J
a
n
-
1
9
9
6
J
a
n
-
1
9
9
8
J
a
n
-
2
0
0
0
J
a
n
-
2
0
0
2
J
a
n
-
2
0
0
4
J
a
n
-
2
0
0
6
J
a
n
-
2
0
0
8
Predicted values
Actual values
Prediction
Actual Crisis No crisis Correctly percentage rate
Crisis 14 4 77.47
No crisis 8 18 69.23
Overall percentage 72.72
MAE 0.4304 RMSE 0.5314
Table IX.
Prediction results of SOM
for in-sample period
1913-1923
Prediction of
?nancial crisis
periods
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concerning the crisis periods, while logit model presents signi?cant higher predicting
power concerning the no crisis periods. But the main purpose of the paper is not only
to rely on the comparison of the forecasting power between these two different
procedures as each approach presents advantages which both approaches can be used
in combination in order to predict future possible recessions.
In Figure 7, the situation is not very clear as in 1997 and later only Pattern 4 is
presented. But we observe that in the whole sample after three years of no crisis with
Pattern 4, a crisis period is followed. For example, in years 1964-1967 the Pattern 4 is
presented and the crisis took place in 1969, two years later, so from 2002 to 2006 where
Pattern 4 is presented, we expect a crisis in 2007, very close to the actual time ?nancial
crisis period. In Figure 8, the SOMneural networks patterns for the period 1913-2009 are
presented. The patterns indicate that ?nancial crisis is not over. Even if currency crisis
is over we must estimate what happens and what will happen in the real economy, as the
GDP growth, the unemployment rate, the wages, the economic, and consumer sediment
among others. The patterns show that economy is very fragile and even if economy
starts its growth, this will be very small and slow with the danger to be followed from
a further recession. More speci?cally we observe that in years 2008 and 2009 Pattern 6
Prediction
Actual Crisis No crisis Correctly percentage rate
Crisis 32 5 86.48
No crisis 55 252 82.08
Overall percentage 82.55
MAE 0.1809 RMSE 0.2848
Table X.
Prediction results of SOM
for out-of-sample period
1924-2006
Prediction
Actual Crisis No crisis Correctly percentage rate
Crisis 8 0 100.00
No crisis 1 3 75.00
Overall percentage 91.67
MAE 0.0993 RMSE 0.2669
Table XI.
Prediction results of SOM
for out-of-sample period
2007-2009
Figure 7.
Patterns based on SOMs
neural networks for the
period 1913-2006
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1314 15 1617 18 19 20 212223 242526 27 2829 30 3132 33 3435363738 39 4041 42 4344 454647 484950 51 5253 54 5556 57 5859 60 616263 6465 66676869 7071 72 737475 7677 7879 80 8182 83 848586 87 8889 9091 92 9394 959697 98 9900 01 0203 04 05 06
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is presented. Speci?cally, Pattern 6 is associated with small period crisis but these
recessions were taking place and repeated in an interval of one year. So the crisis will
probably last for two years, and speci?cally for the period 2008-2010 as in the case of
period 1910-1912, followed by one year of economic growth in 2011 and then a new
economic recession in 2012. The second and very possible scenario is that ?nancial
crisis is on its downturn at the end of 2009, then an economic growth might take place in
2010, but the economic recession will be followed in the years of 2011-2012. The third
and worst scenario is that the economic recession reached its bottom in the end of 2009,
economic growth might be followed during 2010, but new crisis of four year duration
might take place in the years 2011-2014, similar to the pattern of the third quarter of
1926 through the fourth quarter of 1927 and then the Great depression of 1929-1933. The
last scenario is very probably not applicable as the ?nancial crisis of 2007 is the greatest
after the Great Depression and so the pattern is not similar with the crisis of 1926-1927,
which the last one has not the great impact of the crisis of 2007. On the other hand we
could have the following scenario, which is the most potential. More speci?cally the
crisis of zero down sub-prime lending in 2006-2007 was followed from the crisis of
2008 so in a similar manner the crisis of 1926-1927 was followed by the Great
Depression of 1929 which lasted for three years and seven months, so the current crisis
can last up to 2011.
We conclude that in both methodology approaches, the logit regression and the
SOM neural networks, the economic recession will be continued if drastic measures
will not obtained directly, leading the USA and the global economy into new ?nancial
crisis. Furthermore, the methodology of SOM neural networks can be applied for the
purpose of per one or two years ahead prediction in order to obtain the appropriate
measures to avoid a situation like that. Also we can examine the behavior for each
variable in each crisis or no crisis period with the combination of the patterns
recognition. For example, we could present in a graph how the US public debt behaves
in each period and each pattern. Because the variables are too many we do not provide
a description and analysis like that but we present the main ideas.
Furthermore, based on GDP, which was turned positive in the third quarter of 2009,
US recession probably ended. But, it should be noticed that this rule de?nition is not
consistent with that we follow in this paper, which is based on NBER de?nition as we
described in previous section. Additionally the increase of GDP at 2.2 percent was not
enough to create jobs and if a recovery has already been started then this is very ?at.
Also, the increase of GDP is owed at the economic stimulus package, where the
estimated increase would have been 0.77 percent. So the real consequences will be
Figure 8.
Patterns based on SOM
neural networks for the
period 1913-2009
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
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1314 15 1617 18 19 20 2122 23 242526 27 2829 30 3132 33 3435 363738 39 40 4142 43 44 454647 4849 50 5152 53 545556 57 5859 60 61 62 6364 65666768 69 7071 72 737475 7677 7879 80 8182 83 848586 87 8889 90 91 92 9394 959697 98 9900 01 0203 04 05 06
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clearer after the effects of economic stimulus package would fade and probably a new
economic stimulus package by the government might be necessary. Finally, even if
recession will end soon the post-crisis ?nancial situation of persons and enterprises
will be much lower than that of pre-crisis and it may take the economy a few years to
get back to pre-crisis levels (Amadeo, 2009; Miller, 2009). On the other hand a similar
policy was followed in Japan by large ?scal stimulus and de?cit spending, where
econometric studies have shown that the speci?c policy was identi?ed as especially
effective (Cha, 2003).
Even if SOM neural networks are able to predict the crisis periods this is not
enough, because the main responsibility of the governments, the ?nancial managers,
and economists is to guarantee that a crisis like that will not be repeated in the future.
Even if technology and the availability of data capable to observe and examine directly
the effects of the economic policies and the development of computers, software and
algorithms, in comparison with the unavailability in 1929, have not led to optimal
controls and actually economists failed to predict successfully the crisis and its strong
magnitude and impact and to obtain the appropriate measures. The inef?cient policy of
US economy led to a new ?nancial crisis, which led the global economy in crisis too. An
effective change in regulations of ?nancial markets and the more carefully economic
policy by the US Government and the president of the Federal Reserve System must be
the priorities in the future. Also, almost all researches have proved that AI is more
capable and powerful tool on the formulation of ?nancial crisis models, but also for
other economic models, from monetary and ?scal policy measures to risk control, so a
further research study and exploring must be done on this ?eld.
6. Conclusions
We examined and applied logit binary regression in order to predict the ?nancial crisis
in period 2007-2009 and then in period 2010. The model gave poor forecast results in
the in-sample period, speci?cally for the prediction of ?nancial crisis periods, rather no
crisis periods. Logit model predicts at 75 the crisis periods in 2007-2009 where ?nancial
crisis took place and we estimate based on this model that the economic recession will
be continued in 2010. The advantage of logit model is that it gives a warning signal
two quarters before the sub prime crisis and three quarters before the current crisis to
started of?cially in December of 2007 based on NBER. The problem with logit
regression as with all econometrical models which are used in ?nancial crisis
prediction models is that we need information and some estimated future values of the
independent variables, which is some times impossible and even estimations from
international institutions present signi?cant deviations from the actual values. For this
reason we estimated simultaneously SOMs neural networks, in order to ?nd the
patterns of economic business cycles. The problem with this methodology approach is
that is a non-parametric approach so we are unable to examine the sign, the effects and
the magnitude of each explanatory variable. But on the other hand with this approach
we do not need to know the future values of the explanatory variables, so we can
forecast the crisis or no crisis periods in a longer period than one year. So SOMs can be
applied simultaneously with logit or other parametric models, in order to measure the
magnitude and the statistically signi?cance of each possible candidate which can
explain the business cycles and to examine the patterns of them. The main conclusion
is that with both approaches and mainly with SOMs neural networks we estimate and
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predict that the economic recession is not over and we claim that will last at least one
with two years yet. This is just a proposed well-known method and this paper
examines only one case, the US economy. The purpose of this study is to give the
motives for further research using also other possible neural networks models, as also
fuzzy logic, genetic algorithms, and advanced hybrid and heuristic methods.
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Corresponding author
Eleftherios Giovanis can be contacted at: [email protected]
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This article has been cited by:
1. Eleftherios Giovanis. 2012. Study of Discrete Choice Models and Adaptive Neuro-Fuzzy Inference System
in the Prediction of Economic Crisis Periods in USA. Economic Analysis and Policy 42, 79-95. [CrossRef]
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