Nonlinear Extension of Asymmetric Garch Model within Neural Network Framework

Jan Zizka et al. (Eds) : CCSEIT, AIAP, DMDB, MoWiN, CoSIT, CRIS, SIGL, ICBB, CNSA-2016
pp. 101–111, 2016. © CS & IT-CSCP 2016 DOI : 10.5121/csit.2016.60609
NONLINEAR EXTENSION OF
ASYMMETRIC GARCH MODEL WITHIN
NEURAL NETWORK FRAMEWORK
Josip Arnerić1
and Tea Poklepović2
1
Faculty of Economics and Business Zagreb,
Department of Statistics, Trg J. F. Kennedyja 6, 1000 Zagreb, Croatia
jarneric@efzg.hr
2
Faculty of Economics Split, Department of Quantitative Methods,
Cvite Fiskovića 5, 21000 Split, Croatia
tpoklepo@efst.hr
ABSTRACT
The importance of volatility for all market participants has led to the development and
application of various econometric models. The most popular models in modelling volatility are
GARCH type models because they can account excess kurtosis and asymmetric effects of
financial time series. Since standard GARCH(1,1) model usually indicate high persistence in the
conditional variance, the empirical researches turned to GJR-GARCH model and reveal its
superiority in fitting the asymmetric heteroscedasticity in the data. In order to capture both
asymmetry and nonlinearity in data, the goal of this paper is to develop a parsimonious NN
model as an extension to GJR-GARCH model and to determine if GJR-GARCH-NN outperforms
the GJR-GARCH model.
KEYWORDS
conditional volatility, GARCH model, GJR model, Neural Networks, emerging markets
1. INTRODUCTION
Modelling volatility, i.e. returns fluctuations, has been a topic of interest to economic and
financial researchers. Portfolio managers, option traders and market makers are all interested in
volatility forecasting in order to get higher profits or less risky positions.
The most popular models in modelling volatility are generalized autoregressive conditional
heteroskedasticity (GARCH) type models which can account excess kurtosis and asymmetric
effects of high frequency data, time varying volatility and volatility clustering. The first
autoregressive conditional heteroscedasticity model (ARCH) was proposed by Engle [1] who
won a Nobel Prize in 2003 for his contribution to modelling volatility. The model was extended
by Bollerslev [2] by its generalized version (GARCH). However, standard GARCH(1,1) model
usually indicates high persistence in the conditional variance, which may originate from
structural changes in the variance process. Hence the estimates of a GARCH model suffer from a

102 Computer Science & Information Technology (CS & IT)
substantial upward bias in the persistence parameters. Also, it is often difficult to predict
volatility using traditional GARCH models because the series is affected by different
characteristics: non-stationary behaviour, high persistence in the conditional variance,
asymmetric behaviour and nonlinearity. Due to practical limitations of these models different
approaches have been proposed in the literature. Some of them are developed for resolving the
asymmetric behaviour problem and some of them the nonlinearity in variance. Diebold [3] found
that volatility models that fail to adequately incorporate nonlinearity are subject to an upward
bias in the parameter estimates which results in strong forms of persistence that occurs especially
in high volatility periods in financial time series and this influences the out-of-sample forecasts of
single regime type GARCH models. The empirical researches reveal that among asymmetric
models, Glosten, Jaganntahn and Runkle’s [4] sign-ARCH model, i.e. GJR-GARCH model
outperforms all the other GARCH-type models. Moreover, to account for nonlinearity, in recent
researches much attention is given to neural network models (NN) in forecasting volatility.
The NNs are a valuable tool for modelling and prediction of time series in general ([5]; [6]; [7];
[8]; [9]; [10]; [11]; [12]; [13]). Most financial time series indicate existence of nonlinear
dependence, i.e. current values of a time series are nonlinearly conditioned on information set
consisting of all relevant information up to and including period 1t − ([14]; [15]; [16]; [17]). The
feed-forward neural networks (FNN), i.e. multilayer perceptron, are most popular and commonly
used. They are criticized in the literature for the high number of parameters to estimate and they
are sensitive to overfitting ([18]; [19]).
The objective of this paper is to develop a parsimonious NN model as an extension to GJR-
GARCH model which will capture the nonlinear relationship between past return innovations and
conditional variance. The second objective of this paper is to determine if GJR-GARCH-NN
model outperforms GJR-GARCH models when there is high persistence of the conditional
variance. This paper contributes to existing literature in several ways. Firstly, this paper
introduces NN as semiparametric approach, which combines flexibility of nonparametric
methods and the interpretability of parameters of parametric methods, and attractive econometric
tool for conditional volatility forecasting. NN models have continuously been observed as a
nonparametric method relying on automatically chosen NN provided by various software tools,
which is unjustified from the econometric perspective. Therefore, in this paper the “black box”
will be opened. Secondly, in this paper new NN model is defined, as an extension to GJR-
GARCH models, and estimated. Although this paper relies on paper from Donaldson and
Kamstra [20] it contributes to the literature by estimating an additional parameter λ which was
previously set in advance. Finally, this paper contributes to the literature of emerging market
economies with the newest data.
The remainder of this paper is organized as follows. Section 2 presents the literature review.
Section 3 describes the data and methodology. Section 4 presents the obtained empirical results
and discussion. Finally, some conclusions and directions for future research are provided in
Section 5.
2. LITERATURE REVIEW
Donaldson and Kamstra [20] in their paper construct a seminonparametric nonlinear GARCH
model, based on NN approach, and evaluate its ability to forecast stock return volatility on stock
exchanges in London, New York, Tokyo and Toronto using daily stock returns from 1970 to

Computer Science & Information Technology (CS & IT) 103
1990. They compared this constructed NN model with performances of other most commonly
used volatility models, i.e. GARCH, EGARCH and GJR-GARCH model, in in- and out-of-
sample comparison and within different markets. The results reveal that GJR-GARCH model fits
the asymmetric heteroscedasticity in the data better than GARCH and EGARCH models,
however the best performing model of all seems to be the newly introduced NN model. The
authors present the new methodology which is applied in advancing markets, however, the
properties of selected methodology are not yet tested in emerging markets. Moreover, the number
of hidden neurons is obtained by selecting the best alternative model in the grid [0,5] parameter
space using Schwarz information criterion. In this paper the number of hidden units in three-layer
NN is specified in advance for the more suitable comparison between models. This paper
contributes in estimating an additional parameter λ which was in presented paper set in advance.
Teräsvirta et al [13] present similar methodology as [20] based on Medeiros et al [11] approach
and use it as AR-NN type model showing the potential of their proposed modelling approach in
two applications: sunspot series and US unemployment series. Moreover, they clearly combine
the NN model so as to be able to explain it as econometric model.
Bildirici and Ersin [21] analyse the volatility of stock returns on Istanbul Stock Exchange (ISE)
in period from 1987 to 2008 using daily closing prices of ISE 100 index. They compare and
combine GARCH, EGARCH, GJR-GARCH, TGARCH, NGARCH, SAGARCH, PGARCH,
APGARCH, NPGARCH with NN models in their forecasting abilities. The NN models are
retrained with conjugate gradient descent algorithm after the training with backpropagation. They
conclude that NN models improved the generalization and forecasting ability of GARCH models.
In their paper the models are not properly explained from an econometric perspective, nor are the
findings explained from the perspective to the real time data. Moreover, the parameters of the
models are nor presented or explained.
Their later paper, Bildirici and Ersin [22], relies on paper from [20] and [21] to analyse the
nonlinearity and leptokurtic distribution of stock returns on ISE in period from 1986 to 2010 and
benefits from both LSTAR and NN type of nonlinearity, i.e. this paper proposes several LSTAR-
GARCH-NN family models. GARCH, FI-GARCH, APGARCH and FIAPGARCH models are
augmented with a NN model. They conclude that extended GARCH models forecast better than
GARCH models; LSTAR-LST-GARCH show significant improvement in out-of-sample
forecasting; MLP-GARCH models provide similar results to LSTAR-LST-GARCH models;
LSTAR-LST-APGARCH-MLP model provided the best overall performance. To estimate NN
models, the number of hidden neurons ranges from 3 to 10 and the best model is selected based
on MSE or RMSE. Moreover, each of the selected model architecture is estimated 20 times for 8
different NN models and to obtain parsimony the appropriate model is selected based on AIC.
Although there is a vast number of econometric models for modelling conditional volatility
presented and estimated in this paper, along with an econometric presentation of NN models,
estimation of 100 different NN models with hidden neurons ranging from 3 to 10 and comparing
models with different specifications is econometrically unjustified.
Bildirici and Ersin [23] propose a family of nonlinear GARCH models that incorporate fractional
integration (FI) and asymmetric power (AP) properties to MS-GARCH processes. Moreover,
they augment the MS-GARCH type models with NN to improve forecasting accuracy. Therefore,
the proposed MS-ARMA-FIGARCH, APGARCH, and FIAPGARCH processes are further
augmented with MLP, RBF, Recurrent NN, and Hybrid NN type NNs. The MS-ARMA-GARCH

family and MS-ARMA-GARCH-NN family are utilized for modelling the daily stock returns of
the ISE Index. Forecast accuracy is evaluated with MAE, MSE, and RMSE error criteria and
Diebold-Mariano test for predictive accuracy. They conclude that the FI and AP counterparts of
MS-GARCH model provided promising results, while the best results are obtained for their NN
based models. Moreover, among the models analysed, the models MS-ARMA-FIAPGARCH-
HNN and MS-ARMA-FIAPGARCH-RNN provided the best forecast performances over the
single regime GARCH models and over the MS-GARCH model. Parameters of NN models are
not explained econometrically, although NN are regarded as econometric model instead of
nonparametric model.
Mantri et al [24] apply different methods, i.e. GARCH, EGARCH, GJR-GARCH, IGARCH and
NN models for calculating the volatilities of Indian stock markets. Two networks are presented:
single input (low index) single output (high index level) and multiple inputs (open, high and low
index level) single output (close index level). The data from 1995 to 2008 of BSE Sensex and
NSE Nifty indices are used to calculate the volatilities. The authors conclude that the MISO-NN
model should be used instead of SISO-NN model and that there is no difference in the volatilities
of Sensex and Nifty estimated under the GARCH, EGARCH, GJR-GARCH, IGARCH and NN
models.
In their later paper Mantri et al. [25] focused on the problem of estimation of volatility of Indian
Stock market. The paper begins with volatility calculation by ARCH and GARCH models of
financial computation up to lag 3. The results are compared to NN model using R2. It can be
concluded that NN can be used as a best choice for measuring the volatility of stock market.
These papers provide no information about the particular NN model used, NN is not explained as
econometric model, and therefore the papers are not suitable for deciding on suitability of the
models.
Bektipratiwi and Irawan [26] propose an alternative forecasting model based on the combinations
between RBF and EGARCH model to model stock returns of Bank Rakyat Indonesia Tbk for the
period from 2003 to 2011. They use RBF to model the conditional mean and EGARCH to model
the conditional volatility and propose a regression approach to estimate the weights and the
parameters of EGARCH using maximum likelihood estimator. The relevant explanatory variables
are chosen based on its contribution of giving greater reduction in the in-sample forecast errors.
They selected 11 inputs for RBF model, and 5 hidden neurons based on trial and error procedure.
Based on SIGN test, the best forecast is obtained by RBF-EGARCH model for 100 steps ahead.
All of the above researches combine GARCH-type and NN models by adding the NN structure to
existing GARCH-type models in search of the suitable model for forecasting conditional variance
of stock returns. The proposed methodology is empirically tested on developed markets, however
not on developing capital markets of Central and Eastern Europe. Because of the uniqueness of
these emerging capital markets, it is important to test features of proposed methodology in this
particular segment. Moreover, some of the papers use NN as nonparametric estimation technique,
neglecting the interpretability of parameters which could be obtained by using NN as
econometric tool. In this paper NN will be observed as semiparametric approach combining the
flexibility of nonparametric methods and the interpretability of parameters of parametric
methods. Another contribution of the paper is in a priori specified structure of NN models in
order to be comparable to the GARCH-type models and the estimation of additional parameter
which was not estimated before.

3. DATA AND METHODOLOGY
The most widespread approach to volatility modelling consists of the GARCH model of
Bollerslev [2] and its numerous extensions that can account for the volatility clustering and
excess kurtosis found in financial time series. The accumulated evidences from empirical
researches suggest that the volatility of financial markets can be appropriately captured by
standard GARCH(1,1) model ([27]) since it gives satisfactory results with small number of
parameters to estimate. According to Bollerslev [2] GARCH (1,1) can be defined as:
( )
2
2 2 2
1 1
. . . 0,1
t t t
t t t
t
t t t
r
u
u i i d
µ ε
ε σ
σ α β ε γ σ− −
= +
= ⋅
= + ⋅ + ⋅
:
(1)
where tµ is the conditional mean of return process { }tr , while { }tε is the innovation process
with its multiplicative structure of identically and independently distributed random variables tu .
The last equation in (1) is conditional variance equation with GARCH(1,1) specification which
means that variance of returns is conditioned on the information set 1tI − consisting of all relevant
previous information up to and including period 1t − . GARCH(1,1) model is covariance-
stationary if and only if 1β γ+ < ([2]). In particular, GARCH(1,1) model usually indicates high
persistence in the conditional variance, i.e. integrated behavior of the conditional variance when
1β γ+ = (IGARCH). The reason for the excessive GARCH forecasts in volatile periods may be
the well-known high persistence of individual shocks in those forecasts. Relevant researches
([28]; [29]) show that this persistence may originate from structural changes in the variance
process. High volatility persistence means that a long time period is needed for shocks in
volatility to die out (mean reversion period).
Although GARCH models are the most popular and widely used in empirical researches and
among practitioners due to their ability of describing the volatility clustering, excess kurtosis and
fat-tailedness of the data, they cannot capture the asymmetric behavior of volatility. This means
that negative shocks affect volatility quite differently than positive shocks. Therefore, different
asymmetric models have been developed and used in empirical researches such as EGARCH,
GJR-GARCH, TARCH, PGARCH, APGARCH among the others. However, the results from
Engle and Ng [30] of Japanese stock returns suggest that Glosten, Jaganntahn and Runkle’s [4]
sign–ARCH model, usually called GJR model, shows the most potential in outperforming the
traditional GARCH model. Moreover, in recent literature it also proved to capture the asymmetric
behavior in data better that the other models ([20]). Therefore, GJR-GARCH model is
considered, i.e. GJR-GARCH(1,1,1) is given by:
2 2 2 2
1 1 1 1
1
1
1
1 0,
0 0.
t t t t t
t
t
t
D
if
D
if
σ α β ε γ σ φ ε
ε
ε
− − − −
−
−
−
= + ⋅ + ⋅ + ⋅ ⋅
 <
= 
≥
(2)

As can be seen from (2), GJR-GARCH model is just an augmentation of GARCH model that
allows past negative unexpected returns to affect volatility differently than positive unexpected
returns. When 0φ > negative shocks will have a larger impact on conditional variance. For GJR-
GARCH stationarity condition is satisfied if 2 1β γ φ+ + < .
An alternative solution to overcome the problems found for standard GARCH (1,1) model is to
define appropriate neural network (NN), i.e. by extending the GJR-GARCH (1,1,1) model with
NN model, significant improvements can be found.
The NN is an artificial intelligence method, which has recently received a great deal of attention
in many fields of study. Usually NN can be seen as a nonparametric statistical procedure that uses
the observed data to estimate the unknown function. A wide range of statistical and econometric
models can be specified modifying the structure of the network, however NN often give better
results. Empirical researches show that NN are successful in forecasting extremely volatile
financial variables that are hard to predict with standard econometric methods such as: exchange
rates ([7]), interest rates ([6]) and stocks ([8]). The most commonly used type of NN in empirical
researches is multi-layer feed-forward neural networks (FNN).
The FNN forwards information from input layer to output layer through a number of hidden
layers. Neurons in a current layer connect to neuron of the subsequent layer by weights and an
activation function. In order to obtain weights backpropagation (BP) learning algorithm, which
works by feeding the error back through the network, is mostly used. The weights are iteratively
updated until there is no improvement in the error function. This process requires the derivative
of the error function with respect to the network weights. The mean of squared error (MSE) is the
conventional least square objective function in a NN, defined as mean of squared differences
between the observed and fitted values of time series. The FNN with linear component can be
written as:
















+++= ∑∑∑ ===
p
i
itihch
q
h
hoit
p
i
iocot xgxfy
1
,
1
,
1
ˆ φφφφφ (3)
where t is a time index, ˆty is the output vector, ,t ix is the input matrix with i variables, f(·) and
g(·) are activation functions (usually linear and logistic respectively). Index c is the constant, i is
the input, h is the hidden, and o is the output neuron. coφ denotes the weight of the direct
connection between the constant and output, ioφ denote the weights of direct connection from
inputs to output, chφ denote the weights for the connections between the constant and hidden
neurons. The weights ihφ and hoφ denote the weights for the connections between the inputs and
hidden neurons and between the hidden neurons and output. NN with p inputs and q outputs has
the abbreviation FNN(p,q).
However, the disadvantage of FNN is the problem of overfitting. It occurs due to the inclusion of
multiple hidden layers or multiple neurons in hidden layer which, with existing theoretically
based number of inputs (independent variables) and lagged outputs (dependent variables),
increases the number of parameters to estimate. Therefore, in this paper NN are observed only as
an extension to GJR-GARCH type model with the structure defined in advance to benefit from

parsimonious model in order to avoid the problem of overfitting. The GJR-GARCH-NN(1,1,1,1)
as a nonlinear extension to GJR-GARCH model is defined as:
( )
( )
( )
( )
1
2 2 2 2
1 1 1 1
1
1
1
1
1
2
1 0,
0 0.
1
1
.
t
t t t t t t
t
t
t
t z
t
t
D z
if
D
if
z
e
E
z
E
λ
σ α β ε γ σ φ ε ξψ λ
ε
ε
ψ λ
ε ε
ε
−
− − − −
−
−
−
−
−
= + ⋅ + ⋅ + ⋅ ⋅ +
 <
= 
≥
=
+
−
=
(4)
where ( )tzψ λ specifies the logistic function in hidden unit of neural network with 1 hidden
neuron, 1tz − provides a normalization of ε necessary to prepare the lagged unexpected returns as
inputs into the nodes. All the data are transformed using the in-sample mean and variance.
Donaldson and Kamstra [20] chose λ in advance from a uniform random number generator so
they lie between -2 and 2 in order to achieve the identification of parameters ξ , and then
parameters , , , ,α β γ φ ξ are estimated with maximum likelihood. In this paper λ are defined
between -2 and 2, however they are estimated with maximum likelihood just as other parameters.
The data set consists of returns of the daily closing prices obtained from stock exchanges in
period from January 2011 until September 2014 for selected European emerging markets, i.e.
Bulgaria, Croatia, Czech, Romania, Slovakia and Slovenia. Data is obtained from Thomson
Reuters database.
4. EMPIRICAL RESEARCH
In order to investigate GARCH-type models it is important to give an overview of the sample, i.e.
descriptive statistics. From Table 1 can be seen that in observed period European emerging
markets have negative expected returns. The lowest risk is observed in Slovakia and Slovenia and
the highest risk in Czech Republic and Romania. Each distribution shows asymmetric behavior
and leptokurtosis. Moreover, time series is not stationary since the variance of returns is time
varying. Detailed results are omitted due to a lack of space. They are available from authors upon
request.
Parameters for GJR-GARCH(1,1,1) model are estimated in SAS software using the maximum
likelihood method and the results for selected markets with estimated parameters and the value of
Log-Likelihood (LL) is given in Table 2. The results reveal that asymmetric behavior is
statistically significant in all markets and since 0φ < the positive shocks will have a larger
impact on conditional variance. In developed markets this parameter is usually positive indicating
the opposite conclusions.

Table 1. Descriptive statistics of daily returns for selected markets
N Min Max µ σ α3 α4
BULGARIA 2177 -0,1136 0,0729 -0,00044 0,01307 -1,05 10,39
CROATIA 2177 -0,1076 0,1477 -0,00024 0,01273 0,14 18,43
CZECH 2177 -0,1618 0,1109 -0,00032 0,01521 -0,82 15,32
ROMANIA 2177 -0,1311 0,1056 -0,00005 0,01639 -0,50 8,64
SLOVAKIA 2177 -0,1481 0,1188 -0,00022 0,01153 -1,52 29,90
SLOVENIA 2177 -0,0843 0,0835 -0,00032 0,01189 -0,46 7,47
Table 1. Parameter estimates of GJR-GARCH(1,1,1) model with values of Log-Likelihood (LL)
BULGARIA CROATIA CZECH ROMANIA SLOVAKIA SLOVENIA
ߤ -0.00007 -0.00021 -0.00008 0.000367* 5.36E-06 -0.00014
ߙ 7.14E-06*** 4.56E-07*** 4.42E-03*** 4.27E-06*** 0.000028*** 0.000012***
ߚ 0.307146*** 0.116087*** 0.171614*** 0.190021*** 0.078577*** 0.279281***
ߛ 0.703246*** 0.912962*** 0.846355*** 0.824793*** 0.797184*** 0.697366***
߶ -0.08503** -0.05222*** -0.08193*** -0.04265* -0.09016*** -0.14047***
LL 6970.803 7227.855 6611.161 6470.448 6649.694 6945.997
Note: Parameter estimates are significant at 1% (***), 5% (**) and 10% (*) significance level
Parameters for GJR-GARCH-NN(1,1,1,1) model are estimated in SAS software using the
maximum likelihood method and the results for selected markets with estimated parameters and
the value of Log-Likelihood (LL) is given in Table 3. This model has two additional parameters
to estimate: ξ and λ . Parameter ξ is in each market positive and statistically significant.
Moreover, the Log-Likelihood is in GJR-GARCH-NN model larger than in simpler model. All
these findings lead to a conclusion that extending the GJR-GARCH with the NN model, i.e.
adding the nonlinearity in the model, is statistically significant and improves the models’ fit
Table 3. Parameter estimates of GJR-GARCH-NN(1,1,1,1) model with values of Log-Likelihood (LL)
BULGARIA CROATIA CZECH ROMANIA SLOVAKIA SLOVENIA
ߤ -0.00007 -0.00018 -0.00011 0.000391 -0.0002 -0.00015
ߙ -0.03772*** -0.03771*** -0.03772*** -0.03761*** -0.03787*** -0.03776***
ߚ 0.311529*** 0.12463*** 0.151603*** 0.19557*** 0.364962*** 0.349813***
ߛ 0.703012*** 0.911607*** 0.848221*** 0.830741*** 0.6018284*** 0.697389***
߶ -0.09305*** -0.06524*** -0.04615 -0.06322*** -0.27835*** -0.27628***
ߦ 0.075462*** 0.075429*** 0.075448*** 0.075225*** 0.075971*** 0.075546***
ߣ 0.000028 0.000039 -0.00021 0.000137 0.000798* 0.000519***
LL 6970.826 7228.358 6612.326 6470.998 6657.298 6950.291
Note: Parameter estimates are significant at 1% (***), 5% (**) and 10% (*) significance level

5. CONCLUSIONS
Modelling volatility, i.e. returns fluctuations, is in the main focus of the paper. This research
begins with the most widespread approach to volatility modelling, i.e. GARCH(1,1) model. Due
to its disadvantage in capturing the asymmetric behavior GJR-GARCH(1,1,1) model is
introduced. However, both models fail to model nonlinearity in data and, therefore NN model as
an extension to GJR-GARCH model is defined, i.e. parsimonious GJR-GARCH-NN model. This
paper estimates the parameters of both simple and extended GJR-GARCH model and compares
these models using data for selected European emerging markets. Moreover, NN are presented as
an econometric tool. Results of this paper confirm conclusions of previous researches about
superiority of NN versus other linear and nonlinear models. However, they are still a challenge
for the researchers in order to improve their performances in forecasting conditional variance of
stock returns and time series in general. The out-of-sample predictive performance, inclusion of
more hidden neurons or other architectures, the use of different algorithms in the network
training, open space for future work and further studies.
ACKNOWLEDGEMENTS
This work has been fully supported by Croatian Science Foundation under the project “Volatility
measurement, modeling and forecasting” (5199).
REFERENCES
[1] Engle, R.F. (1982) “Autoregressive conditional heteroscedasticity with estimates of the variance of
UK inflation”, Econometrica, Vol. 41, pp. 135-155.
[2] Bollerslev, T. (1986) “Generalized autoregressive conditional heteroscedasticity”, Journal of
Econometrics, Vol. 31, pp. 307-327.
[3] Diebold, F. (1986) “Comment on modelling the persistence of conditional variances,” Econometric
Reviews, Vol. 5, pp. 51–56.
[4] Glosten, L.R., Jagannathan, R. & Runkle, D.E. (1993) “On the Relation between the Expected Value
and the Volatility of the Nominal Excess Return on Stocks”, The Journal of Finance, Vol. 48, No. 5
(Dec., 1993), pp. 1779-1801.
[5] Balkin, S.D. (1997) “Using Recurrent Neural Networks for Time Series Forecasting”, Working Paper
Series number 97-11, International Symposium on Forecasting, Barbados.
[6] Täppinen, J. (1998) “Interest rate forecasting with neural networks”, Government Institute for
Economic Research, Vatt-Discussion Papers, 170.
[7] Dunis, C.L. & Williams, M. (2002) “Modelling and trading the euro/US dollar exchange rate: Do
neural networks perform better?”, Journal of Derivatives & Hedge Funds, Vol. 8, No. 3, pp. 211-239.
[8] Zekić-Sušac, M. and Kliček, B. (2002) “A Nonlinear Strategy of Selecting NN Architectures for
Stock Return Predictions”, Finance, Proceedings from the 50th Anniversary Financial Conference
Svishtov, Bulgaria, 11-12 April, Svishtov, Veliko Tarnovo, Bulgaria: ABAGAR, pp. 325-355.

[9] Tal, B. (2003) “Background Information on our Neural Network-Based System of Leading
Indicators”, CBIC World Markets, Economics & Strategy
[10] Ghiassi, M., Saidane, H. & Zimbra, D.K. (2005) “A dynamic artificial neural network model for
forecasting time series events”, International Journal of Forecasting, Vol. 21, pp. 341-362.
[11] Medeiros, M.C., Teräsvirta, T. & Rech, G. (2006) “Building Neural Network Models for Time
Series: A Statistical Approach”, Journal od Forecasting, No. 25, pp. 49-75.
[12] Kuan, C.-M. & White, H. (2007) “Artificial neural networks: an econometric perspective”,
Econometric Reviews, Vol. 13, pp. 1-92.
[13] Teräsvirta, T., Tjøstheim, D. & Granger, C.W.J. (2008) Modelling nonlinear economic time series,
Advanced texts in econometrics, Oxford, New York, Oxford University Press
[14] Gonzales, S. (2000) “Neural Networks for Macroeconomic Forecasting: A Complementary Approach
to Linear Regression Models”, Working Paper 2000-07.
[15] Hwarng, H.B. (2001) “Insights into neural-network forecasting of time series corresponding to
ARMA(p,q) structures”, Omega, Vol. 29, pp. 273-289.
[16] Zhang, G.P. (2003) “Time series forecasting using hybrid ARIMA and neural network model”,
Neurocomputing, Vol. 50, pp. 159-175.
[17] Aminian, F., Suarez, E.D., Aminian, M. & Walz, D.T. (2006) “Forecasting Economic Data with
Neural Networks”, Computational Economics, Vol. 28, pp. 71-88.
[18] Lawrence, S., Giles, C.L. & Tsoi, A.C. (1997) “Lessons in Neural Network Training: Overfitting
May be Harder than Expected”, Proceedings of the Fourteenth National Conference on Artificial
Intelligence, AAAI-97, pp. 540-545.
[19] Franses, P.H. & van Dijk, D. (2003) Nonlinear Time Series Models in Empirical Finance, Cambridge
University Press.
[20] Donaldson, R.G. & Kamstra, M. (1997) “An Artificial Neural Network - GARCH Model for
International Stock Return Volatility”, Journal of Empirical Finance, Vol. 4, No. 1, pp. 17-46.
[21] Bildirici, M. & Ersin, Ö.Ö. (2009) “Improving forecasts of GARCH family models with the artificial
neural networks: An application to the daily returns in Istanbul Stock Exchange”, Expert Systems
with Applications, No. 36, pp. 7355-7362
[22] Bildirici, M. & Ersin, Ö.Ö. (2012) “Nonlinear volatility models in economics: smooth transition and
neural network augmented GARCH, APGARCH, FIGARCH and FIAPGARCH models”, MPRA
Paper No. 40330
[23] Bildirici, M. & Ersin, Ö.Ö. (2014) “Modelling Markov Switching ARMA-GARCH Neural Networks
Models and an Application to Forecasting Stock Returns”, Hindawi Publishing Corporation: The
Scientific World Journal, Vol. 2014, Article ID 497941, 21 pages
[24] Mantri, J.K., Gahan, P. & Nayak, B.B. (2010) “Artificial Neural Networks – An Application to Stock
Market Volatility”, International Journal of Engineering Science and Technology, Vol. 2, No. 5, pp.
1451-1460.

[25] Mantri, J.K., Mohanty, D. &Nayak, B.B. (2012) “Design Neural Network for Stock Market
Volatility: Accuracy Measurement”, International Journal on Computer Technology & Applications,
Vol. 3, No. 1, pp. 242-250.
[26] Bektipratiwi, A. & Irawan, M.I. (2011) “A RBF-EGARCH neural network model for time series
forecasting”, Proceedings of “The IceMATH 2011, Topic, pp. 1-8.
[27] Visković, J., Arnerić, J. &Rozga, A. (2014) “Volatility Switching between Two Regimes”, World
Academy of Science, Engineering and Technology, International Science Index 87, International
Journal of Social, Management, Economics and Business Engineering, Vol. 8, No. 3, pp. 682 - 686.
[28] Lamoureux, C. & Lastrapes, W. (1990) “Persistence in variance, structural change, and the GARCH
model”, Journal of Business and Economic Statistics, Vol. 8, pp. 225–234.
[29] Wong, C.S. & Li, W.K. (2001) “On a mixture autoregressive conditional heteroscedastic model”,
Journal of American Statistical Association, Vol. 96, No. 455, pp. 982–995.
[30] Engle, R.F. & Ng, V.K. (1993) “Measuring and Testing the Impact of News on Volatility”, The
Journal of Finance, Vol. 48, No. 5 (Dec., 1993), pp. 1749-1778.
AUTHORS
Josip Arnerić
Assistant Professor, PhD. University of Zagreb, Faculty of Economics and Business
Zagreb, Croatia. Scientific affiliation: econometric methods and models, financial
time series and volatility, VAR models and cointegration, GARCH and MGARCH
models, stochastic processes and risk management. Phone: 0038512383361. Fax:
0038512332618. E-mail: jarneric@efzg.hr
Tea Poklepović
Teaching Assistant, Doctoral student. University of Split, Faculty of Economics,
Split, Croatia. Scientific affiliation: statistics and econometrics in business, finance
and macroeconomics, especially econometric methods and models, time series and
neural networks. Phone: 0038521430761. Fax: 0038521430701. E-mail:
tpoklepo@efst.hr.

Nonlinear Extension of Asymmetric Garch Model within Neural Network Framework

More Related Content

What's hot (16)

Similar to Nonlinear Extension of Asymmetric Garch Model within Neural Network Framework (20)

Recently uploaded (20)

Nonlinear Extension of Asymmetric Garch Model within Neural Network Framework