GARCH MODEL INDENTIFICATION USING NEURAL NETWORK
DSc. André Machado
Caldeira
Fuzzy Consultoria,
Brazil
Email: mmachado@ibmecrj.br
DSc. Maria Augusta
Soares Machado
IBMEC/RJ, Brazil
Email:
mmachado@ibmecrj.br
Ph.D. Reinaldo
Castro Souza
Pontificia Universidade
Católica (PUC/RJ), Brazil
Email:
reinaldo@ele.pucrio.br
Ph.D. Ricardo
Tanscheit
Pontificia
Universidade Católica (PUC/RJ), Brazil
Email:
ricardo@ele.pucrio.br
Submission:
04/11/2013
Accept:
20/11/2013
ABSTRACT
GARCH models are being largely used to estimate the volatility of
financial assets, and GARCH (1,1) is the one most used. However, identification
of GARCH models is not fully explored. Some specialist systems technology have
been used in some applications of time series models such as time series
classification problems, ARMA models identification, as well as SARIMA. The aim
of this paper is to develop an intelligent system that can accurately identify
the specification of GARCH models providing the right choice of the model to be
used, thus avoiding the indiscriminate usage of GARCH (1,1) model.
Keywords: GARCH,
Volatility, Identification.
1
INTRODUCTION
ARCH
and GARCH models have being largely explored technically and empirically since
their creation in 1982 and 1986, respectively. However, the focus is always on
stylized facts of financial time series or volatility forecast, where GARCH (1,1)
is commonly used. Hardly ever do we find a study concerning the identification
of GARCH models. Some studies have been developed using specialized systems
applied to time series models (REYNOLDS, et al., 1995) and identification of
both ARMA (MACHADO, 2000) and SARIMA (SILVA, 2005) models. In this context,
this paper has as its aim the development of an intelligent system which could
improve the specification identification, thus avoiding the indiscriminate
usage of GARCH (1,1) model. In order to validate the accuracy and efficacy of
the system proposed, simulated time series will be used. The results derived
from such system will then be compared to chosen model derived from AIC (Akaike Information Criterion) and BIC (Bayesian
Information Criterion) criteria.
This
paper is developed in five chapters. The following chapter presents theoretical
concepts relevant to this paper such as the foundation to the development of
the system. The third and fourth chapters present the identification results
using AIC and BIC criteria and using the specified system proposed by this
paper. The concluding chapter focuses on further discussion on the subject
meter and proposes new development.
1.1
GARCH
Models
Currently,
financial markets suffer significant influence of daily news. On analyzing a
series of financial asset returns which present a shift between periods of high
and low volatility forming clusters, volatility can be defined “as a
conditional variance of a time series” (VEIGA, et al., 1993).
During
the highlevel volatility period, the investor may feel reluctant to invest,
and as a consequence many assets values are penalized because of their
liquidity. However, when volatility is
not so high, it is good for the financial market.
The
excess of volatility can bring many consequences into the financial market such
as:
·
in asset prices: the volume of investment reduces and
investors are induced to change from a highrisk to a lowrisk asset in other
markets;
·
in interest rate: the cost of credit increases, and as
a consequence there may be an impact on the economy level;
·
in currency exchange rate: whenever there is a
significant decrease in the total amount of importation, the price of important
and exported goods may increase due to exchange rate risks. In addition there
may be a decrease in consumption levels of imported goods. .
Volatility
is extremely important for the economy and financial markets, and by taking
this into account, studies concerning financial time series are being developed
using models different then the classic ARMA time series models (BOX; JENKINS,
1976). Such classic models cannot reproduce financial time series with
essentials characteristics known as stylized facts.
Many
kinds of models have being developed to estimate volatility, for example, the
Exponential Weight Moving Average (Known as Risk Metrics), stochastic
volatility models and GARCH models. This study focuses on GARCH models; for
further details about other models see Clark (1973), Taylor (1980, 1986 e
1994), Tauchen and Pitts (1983), Hull and White (1987) and Harvey et al.
(1994).
The
concepts of stylized facts of financial time series are really necessary to
understand the inspirations of GARCH models. For further information on
stylized facts of financial time series, see Bernardo and Fernandes (1999).
The
main stylized facts of financial time series could be ranked as such:
·
stylized
fact 1: Stationary Series – Statistical proprieties are
static over time.
·
stylized
fact2: Weak or no linear dependence and nonlinear
dependence (GARCH effect). Series are not or are little autocorrelated, but
the quadratic series are autocorrelated.
·
stylized
fact3: NonGaussian – Financial time series commonly
presents skewness and higher kurtosis.
·
stylized
fact4: Existence of volatility clusters – Financial time
series commonly present alternate periods of high volatility and low
volatility. The conditional variance is time dependent.
A
central hypothesis of the option valuation model proposed by Black e Scholes
(1973) is that the financial time series performs as a Brownian movement, or
the distribution of the returns is lognormal with the same mean and variance
over the time. However, Mandelbrot (1963) and Fama (1963 and 1965) proposed
that those series have higher kurtosis and they discussed the existence of
volatility clusters. Those characteristics were interpreted as an evidence of
stochastic volatility of financial assets.
For
the purpose of representing those characteristics, since approximately two
decades ago, GARCH models are being largely used in financial studies,
especially in financial derivatives studies. The initial success of ARCH models
to represent the nonlinear dependence made possible many extensions.
1.1.1
GARCH
models representation
The
first model from the GARCH family was introduced by Engle (1982). This model
can represent some stylized facts of a financial time series. Engle proposed to
model the quadratic of the return time series using an autoregressive model
with q parameters (AR(q)). This model was called Autoregressive Conditional
Heteroskedastic or ARCH(q), which can be written by the expression:
Where
is a white noise:
_{}
Sometimes
it is convenient to rewrite this expression like this:
Suppose:
Where:
If _{} is written as:
This implies:
So, if is generated by e , then follows an ARCH(q)
process, and if and are used in , it becomes:
Using specification,
the innovation in AR(q)
representation for in can be expressed by:
Notice that even if the
unconditional variance of is assumed to be a
constant in , the
conditional variance of changes over time.
Thus, the ARCH model can describe volatility clusters.
In 1986, Bollerslev observed, by
empirical evidence, that it would be necessary to estimate ARCH models with
high orders to reproduce the conditional variance dynamics. In order to solve
this problem, he proposed a more general and parsimonious form of ARCH model,
which he called Generalized Autoregressive Conditional Heteroskedastic (GARCH) (BOLLERSLEV,
1986).
The same idea of parsimony used in
ARMA models was then applied to GARCH models. So, it can be demonstrated that a
Moving Average model (MA) with order one is equivalent to an Autorregressive
model (AR) with infinite order. In order to reduce the number of parameters to
be used, the AR is merged with MA, thus creating the ARMA model. GARCH model is
based on ARCH model with infinite order and can be expressed as:
_{}
For the same reason that ARCH models
depend on some restrictions concerning to be positive for
every t, GARCH models depend on restrictions of , and . Nelson and Cao (1992) observed that the conditions and were sufficient, but not necessary. So, they argued that by
imposing such conditions could be excess of precaution and could become a
limitation considering some empirical works, and in practical applications,
even if there is some negative coefficients, the conditional variance becomes
positive. Such restrictions could be relaxed and in practical works it is used
to estimate the coefficients with none of those restrictions.
In many applications using high
frequency time series, the estimated conditional variance
by a GARCH (p,q) model demonstrates a strong persistence, that is:
If , the process () is second order stationary and the noise on the conditional
variance of has a decrease impact
on , when h increases, and is asymptotically insignificant. This
feature is called persistence.
Other variations of GARCH models
were proposed having in mind many
objectives, as for example the Exponential GARCH (EGARCH) (NELSON, 1991; ENGLE;
NG, 1993) and the TGARCH (ZAKOIAN, 1991;
GLOSTEN, et al., 1993; RABEMANANJARA;
ZAKOIAN, 1993), that were proposed to capture the asymmetric effect
on the volatility clusters
GARCH and ARCH models will be
applied in this paper.
1.1.2 Modeling Strategy
Franses and Djik (2000) proposed a
modeling sequence which uses the following steps:
·
calculate some time series statistics (ACF,
Autocorrelation Function, and PACF, Partial Autocorrelation Function);
·
compare those values with theoretical values to
specify the right model (Identification);
·
estimate parameters of the specified model
(Estimation);
·
evaluate the specified model using adequacy metrics
(Validation);
·
respecify the model if necessary;
·
use the model to make the forecast (Forecasting).
The specification of the appropriate
structure (identification) for the equation of the conditional variance of a
time series which follows a GARCH process is the main concern of this paper.
Autocorrelation function (ACF) and Partial Autocorrelation function (PACF) are
commonly used in the identification and validation of the ARMA model
specification (BOX; JENKINS, 1976). On the other hand, Bollerslev et al. (1988)
showed that those functions, when applied on the square of the time series,
could be used to the specification and validation of the GARCH model.
Suppose that is the nth
autocorrelation and is the kth partial
autocorrelation of obtained through the
solution to the equations for the GARCH models, analogues of YuleWalker
equations. Thus, the usual interpretation for ARMA models can be used for GARCH
models. For an ARCH(q) process, has an abrupt cut
after the qlag, which behavior is identical to the partial autocorrelation
function of an AR(q) process. On the other hand, the autocorrelation function
of for a GARCH(q,p)
process is different from zero and has an exponential decay. By using these
patterns, such functions can help identify the right specification of the GARCH
model.
Another way to identify the
specification of GARCH models is to use the AIC (Akaike Information Criterion)
and BIC (Bayesian Information Criterion) statistics. The model which shows the
lowest statistic is the one selected to be the identified model. Some results
using this means of identification are presented in chapter 3.
2
PROPOSED
IDENTIFICATION METHODOLOGY
The proposed identification
methodology blend the procedure of autocorrelation and partial autocorrelation
functions described by Bollerslev, et al. (1988) with the identification using
AIC and BIC, and further test overspecification of Box and Jenkins (1974). Figure
1 represents such
methodology.
First
step is to train a neural network to represent the pattern configuration of
each model using autocorrelation, partial autocorrelation function and
statistics AIC and BIC. Therefore, the next step is to test models with high
orders then the selected one. Using both steps the final identification is
done.
Figure
1: Proposed Identification Methodology
3
Applied
Study on Simulated Data
In
order to compare the identification performance of the statistics AIC and BIC
with the proposed neural network, the first step is to simulate a time series
sample generated by GARCH processes using MatLab software for this purpose.
The
models to be compared are ARCH(1), ARCH(2), GARCH(1,1), GARCH(2,1) and
GARCH(1,2). The simulated data total showed 8,000 series, of which 1,600 were
series generated for each model, divided in four lengths of series, in which
one moth was represented by 22 observations, one quarterly period was
represented by 66 observations, one semester was represented by 132
observations and one year was represented by 264 observations. Each length had
400 series for each model.
Random
numbers between zero and one were used to represent the coefficients of the
specified model, taking into account two restrictions: lower lags have higher
coefficient than higher ones, and the sum of all coefficients is lower than
one, which is a condition for GARCH models.
3.1
Model
Identification using AIC e BIC
By
using those simulated data, the model selected as the best was the one which
has the lower AIC and BIC. Table 1 show this identification criteria results.
Because the data are simulated, the generated model is known. So, it is
possible to know whether AIC or BIC classified them with accuracy or not.
Table 1:
Results of identification using AIC and BIC
Series Length observations 
Correctly classified series by AIC
identification 
Correctly classified series by BIC
identification 

Series 
Percentage 
Series 
Percentage 

22 
488 
24.4% 
465 
23.3% 
66 
809 
40.5% 
734 
36.7% 
132 
1,070 
53.5% 
947 
47.4% 
264 
1,371 
68.6% 
1,200 
60.0% 
Total 
3,738 
46.7% 
3,346 
41.8% 
Identification with AIC and BIC
present high level of misclassified percentage, higher than 50% considering the
total data classification. Considering just the annual series, (264
observations) that identification reached almost 70% of correctly classified
series, but taking a look on the smaller series, the results presented a lower
level of correctly classified series. For example, considering data from recent
Initial Public Offering (IPO), those data should probably show high probability
of misclassification.
Tables
2 and 3 show the percentage of correctly classified series of each model using
AIC and BIC criteria. It can be observed that when the number of parameters
increases, misclassification also increases, as it is already expected by the
time that AIC and BIC penalize the model when a new parameter is introduced
with the aim of looking for parsimony. Therefore, those criteria tend to bias
the classification due to parsimony.
As
long as the AIC presents a higher percentage of correctly classified series,
such criteria will be used from now on as a benchmark in this study.
Table 2:
Percentage of correctly classified series using AIC
Series Length observations 
ARCH(1) 
ARCH(2) 
GARCH(1,1) 
GARCH(2,1) 
GARCH(1,2) 
Total 
22 
94.5% 
22.5% 
3.5% 
1.5% 
0.0% 
24.4% 
66 
92.8% 
64.8% 
29.5% 
8.0% 
7.3% 
40.5% 
132 
92.3% 
82.5% 
54.0% 
23.3% 
15.5% 
53.5% 
264 
93.0% 
89.8% 
74.5% 
53.0% 
32.5% 
68.6% 
Total 
93.1% 
64.9% 
40.4% 
21.4% 
13.8% 
46.7% 
Table 3: Percentage of correctly
classified series using BIC
Series Length observations 
ARCH(1) 
ARCH(2) 
GARCH(1,1) 
GARCH(2,1) 
GARCH(1,2) 
Total 
22 
96.8% 
16.0% 
3.0% 
0.5% 
0.0% 
23.3% 
66 
98.8% 
56.5% 
25.0% 
2.0% 
1.3% 
36.7% 
132 
99.3% 
79.0% 
53.8% 
3.5% 
1.3% 
47.4% 
264 
99.5% 
91.5% 
83.0% 
19.8% 
6.3% 
60.0% 
Total 
98.6% 
60.8% 
41.2% 
6.4% 
2.2% 
41.8% 
3.2
Intelligent
System Identification
As
presented in section 2, the first step of intelligent system identification is
to specify the neural network to be trained. Figure 2 represents proposed
neural network specification.
Figure 2: Neural Network Specification to identify GARCH structure
Neurons
of the hidden layers and the neuron of the output layer are represented by
sigmoid functions. Once more MatLab software was used.
The
same 8,000 series of section 3.1 were used to test the neural network. For the
training data, were generated other 8,000 series with the same characteristics of
the simulated data as seen in section 3.1.
Before
specifying the best structure for the neural network to be trained, it is
needed to select which input features will be used, therefore it was applied
the Fischer Score feature selection method (BISHOP, 1995). Such method selected
the following variables: ACF (lag1), ACF (lag2), ACF (lag3), PACF (lag1), PACF
(lag2), Difference between ACF (lag2) and ACF (lag1), and Difference between
ACF (lag3) and ACF (lag2).
After
the feature selection, neural network topology needs to be specified, so a
sensitive analysis was done, varying the number of neurons and the number of
layers, the results can be seen on Figure 3.
Figure 3: Accuracy by varying number of layer and number of
neurons
By
analyzing previews chart it can be seen that the best result was reached by the
topology with two hidden layers using twenty neurons in each hidden layer. It
can also be observed that the misclassification of the neural network with
lower layers increases as the number on neurons increases. Such results might
indicate overfitting.
Table
3 and Table
4 present
classification results using neural network methodology.
Table 3:
Percentage of correctly classified series using Neural Network
Series Length observations 
ARCH(1) 
ARCH(2) 
GARCH(1,1) 
GARCH(2,1) 
GARCH(1,2) 
Total 
22 
66.3% 
23.5% 
27.3% 
9.0% 
38.8% 
33.0% 
66 
86.0% 
61.8% 
25.5% 
8.5% 
54.5% 
47.3% 
132 
89.3% 
81.5% 
42.5% 
23.5% 
62.8% 
59.9% 
264 
92.0% 
89.5% 
57.0% 
53.0% 
69.3% 
72.2% 
Total 
83.4% 
64.0% 
38.1% 
23.5% 
56.3% 
53.1% 
Table 4:
Crossclassification percentage using Neural Network
Real / Classified 
ARCH(1) 
ARCH(2) 
GARCH(1,1) 
GARCH(2,1) 
GARCH(1,2) 
ARCH(1) 
84.1% 
2.9% 
3.1% 
0.8% 
9.1% 
ARCH(2) 
15.8% 
64.0% 
5.9% 
6.8% 
7.6% 
GARCH(1,1) 
21.1% 
15.4% 
33.3% 
6.6% 
23.6% 
GARCH(2,1) 
12.8% 
26.7% 
17.3% 
29.0% 
14.2% 
GARCH(1,2) 
17.5% 
4.7% 
16.0% 
4.9% 
56.9% 
The
experiment suggests that AIC and BIC can be improved by using computational
intelligence. The specified neural network presented 53.1% of correctly
classified series, representing an improvement of 640 bps considering the
results of AIC, and if they are compared to the BIC results, there is an
improvement of 1130 bps, especially considering GARH(1,2).
Those
results suggest that the neural network methodology increases the percentage of correctly classified series.
However, Table
4 shows many GARCH(2,1)
misclassified as GARCH(1,1) or ARCH(2), and GARCH(1,1) misclassified as
ARCH(1), for example. Even though, it is notorious that ARCH(1) and ARCH(2)
have better classifications results.
Those
results can be improved by over specifying such models. In other words, the
number of parameters of the identified model can be increased and its
significance tested. Thus, for this purpose, Table
5 describes which
models are tested using the Ttest for the significance of the new parameter.
Table 5:
Overspecify procedure
Identified
Model 
Overspecified
model 1 
Overspecified
model 2 
ARCH(1) 
ARCH(2) 
GARCH(1,1) 
ARCH(2) 
GARCH(2,1) 
None 
GARCH(1,1) 
GARCH(2,1) 
GARCH(1,2) 
GARCH(2,1) 
None 
None 
GARCH(1,2) 
None 
None 
Figure
4 shows the classification performance as the significance of Ttest is applied
on the over specifying procedure.
Figure 4: Accuracy by varying Ttest significance
By
applying best results for each length of series, the performance improved by
almost 5%. The results can be shown on Table
6 and Table
7.
Table 6:
Right classification percentage using Neural Network after over specifying
N 
ARCH(1) 
ARCH(2) 
GARCH(1,1) 
GARCH(2,1) 
GARCH(1,2) 
Total 
22 
55.8% 
24.5% 
26.0% 
23.3% 
48.5% 
35.6% 
66 
72.0% 
57.5% 
35.3% 
41.8% 
47.8% 
50.9% 
132 
86.3% 
79.8% 
44.8% 
50.5% 
56.8% 
63.6% 
264 
92.5% 
91.0% 
56.8% 
54.3% 
77.3% 
74.4% 
Total 
76.6% 
63.2% 
40.7% 
42.4% 
57.6% 
56.1% 
Table 7:
Crossclassification percentage using Neural Network after over specifying
Real / Classified 
ARCH(1) 
ARCH(2) 
GARCH(1,1) 
GARCH(2,1) 
GARCH(1,2) 
ARCH(1) 
76.6% 
5.9% 
7.4% 
0.9% 
9.1% 
ARCH(2) 
13.8% 
63.2% 
6.4% 
8.6% 
8.0% 
GARCH(1,1) 
8.2% 
16.3% 
40.7% 
10.3% 
24.5% 
GARCH(2,1) 
1.4% 
18.5% 
21.7% 
42.4% 
16.0% 
GARCH(1,2) 
0.4% 
7.3% 
23.1% 
11.7% 
57.6% 
Table
6 shows that the over
specification improved the results by almost 5%, so if those results are
compared to the AIC results they present an increase of 20% of right
identification on overall results. They improved the overall performance
classification from 47% of the AIC identification to 56% of neural network.
4
FINAL
CONSIDERATIONS
As
presented in section 3.1, the statistics AIC and BIC were able to classify
correctly just 46.7% and 41.8% respectively. However, if the annual series are
excluded, the performance reaches 39.5% and 35.8% respectively. Therefore, the
performance of the proposed neural network improves considerably, from 46.7% of
overall correctly identified series by the AIC to 56.1% of overall correctly
identified by neural network, demonstrating that there are opportunities to gain
performance in the identification of the GARH model.
As a
follow up to this study, an application using real time series can be done to
test predicted performance of each model selected by the criteria tested. This
application can be of great importance especially to emerging capital markets
as they can be good resources capitalization option to middlesized and
largesized companies.
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