NONLINEAR EXTENSION OF ASYMMETRIC GARCH MODEL WITHIN NEURAL NETWORK FRAMEWORK

NONLINEAR EXTENSION OF ASYMMETRIC GARCH MODEL WITHIN NEURAL NETWORK FRAMEWORK Josip Arnerić and Tea Poklepović 2 Faculy of Economics and Business Zagreb, Deparmen of Saisics, Trg J. F. Kennedyja 6, 000 Zagreb, Croaia jarneric@efzg.hr 2 Faculy of Economics Spli, Deparmen of Quaniaive Mehods, Cvie Fiskovića 5, 2000 Spli, Croaia poklepo@efs.hr ABSTRACT The imporance of volailiy for all marke paricipans has led o he developmen and applicaion of various economeric models. The mos popular models in modelling volailiy are GARCH ype models because hey can accoun excess kurosis and asymmeric effecs of financial ime series. Since sandard GARCH(,) model usually indicae high persisence in he condiional variance, he empirical researches urned o GJR-GARCH model and reveal is superioriy in fiing he asymmeric heeroscedasiciy in he daa. In order o capure boh asymmery and nonlineariy in daa, he goal of his paper is o develop a parsimonious NN model as an exension o GJR-GARCH model and o deermine if GJR-GARCH-NN ouperforms he GJR-GARCH model. KEYWORDS condiional volailiy, GARCH model, GJR model, Neural Neworks, emerging markes. INTRODUCTION Modelling volailiy, i.e. reurns flucuaions, has been a opic of ineres o economic and financial researchers. Porfolio managers, opion raders and marke makers are all ineresed in volailiy forecasing in order o ge higher profis or less risky posiions. The mos popular models in modelling volailiy are generalized auoregressive condiional heeroskedasiciy (GARCH) ype models which can accoun excess kurosis and asymmeric effecs of high frequency daa, ime varying volailiy and volailiy clusering. The firs auoregressive condiional heeroscedasiciy model (ARCH) was proposed by Engle [] who won a Nobel Prize in 2003 for his conribuion o modelling volailiy. The model was exended by Bollerslev [2] by is generalized version (GARCH). However, sandard GARCH(,) model usually indicaes high persisence in he condiional variance, which may originae from srucural changes in he variance process. Hence he esimaes of a GARCH model suffer from a Jan Zizka e al. (Eds) : CCSEIT, AIAP, DMDB, MoWiN, CoSIT, CRIS, SIGL, ICBB, CNSA-206 pp. 0, 206. CS & IT-CSCP 206 DOI : 0.52/csi.206.60609

02 Compuer Science & Informaion Technology (CS & IT) subsanial upward bias in he persisence parameers. Also, i is ofen difficul o predic volailiy using radiional GARCH models because he series is affeced by differen characerisics: non-saionary behaviour, high persisence in he condiional variance, asymmeric behaviour and nonlineariy. Due o pracical limiaions of hese models differen approaches have been proposed in he lieraure. Some of hem are developed for resolving he asymmeric behaviour problem and some of hem he nonlineariy in variance. Diebold [3] found ha volailiy models ha fail o adequaely incorporae nonlineariy are subjec o an upward bias in he parameer esimaes which resuls in srong forms of persisence ha occurs especially in high volailiy periods in financial ime series and his influences he ou-of-sample forecass of single regime ype GARCH models. The empirical researches reveal ha among asymmeric models, Glosen, Jagannahn and Runkle s [4] sign-arch model, i.e. GJR-GARCH model ouperforms all he oher GARCH-ype models. Moreover, o accoun for nonlineariy, in recen researches much aenion is given o neural nework models (NN) in forecasing volailiy. The NNs are a valuable ool for modelling and predicion of ime series in general ([5]; [6]; [7]; [8]; [9]; [0]; []; [2]; [3]). Mos financial ime series indicae exisence of nonlinear dependence, i.e. curren values of a ime series are nonlinearly condiioned on informaion se consising of all relevan informaion up o and including period ([4]; [5]; [6]; [7]). The feed-forward neural neworks (FNN), i.e. mulilayer percepron, are mos popular and commonly used. They are criicized in he lieraure for he high number of parameers o esimae and hey are sensiive o overfiing ([8]; [9]). The objecive of his paper is o develop a parsimonious NN model as an exension o GJR- GARCH model which will capure he nonlinear relaionship beween pas reurn innovaions and condiional variance. The second objecive of his paper is o deermine if GJR-GARCH-NN model ouperforms GJR-GARCH models when here is high persisence of he condiional variance. This paper conribues o exising lieraure in several ways. Firsly, his paper inroduces NN as semiparameric approach, which combines flexibiliy of nonparameric mehods and he inerpreabiliy of parameers of parameric mehods, and aracive economeric ool for condiional volailiy forecasing. NN models have coninuously been observed as a nonparameric mehod relying on auomaically chosen NN provided by various sofware ools, which is unjusified from he economeric perspecive. Therefore, in his paper he black box will be opened. Secondly, in his paper new NN model is defined, as an exension o GJR- GARCH models, and esimaed. Alhough his paper relies on paper from Donaldson and Kamsra [20] i conribues o he lieraure by esimaing an addiional parameer λ which was previously se in advance. Finally, his paper conribues o he lieraure of emerging marke economies wih he newes daa. The remainder of his paper is organized as follows. Secion 2 presens he lieraure review. Secion 3 describes he daa and mehodology. Secion 4 presens he obained empirical resuls and discussion. Finally, some conclusions and direcions for fuure research are provided in Secion 5. 2. LITERATURE REVIEW Donaldson and Kamsra [20] in heir paper consruc a seminonparameric nonlinear GARCH model, based on NN approach, and evaluae is abiliy o forecas sock reurn volailiy on sock exchanges in London, New York, Tokyo and Torono using daily sock reurns from 970 o

Compuer Science & Informaion Technology (CS & IT) 03 990. They compared his consruced NN model wih performances of oher mos commonly used volailiy models, i.e. GARCH, EGARCH and GJR-GARCH model, in in- and ou-ofsample comparison and wihin differen markes. The resuls reveal ha GJR-GARCH model fis he asymmeric heeroscedasiciy in he daa beer han GARCH and EGARCH models, however he bes performing model of all seems o be he newly inroduced NN model. The auhors presen he new mehodology which is applied in advancing markes, however, he properies of seleced mehodology are no ye esed in emerging markes. Moreover, he number of hidden neurons is obained by selecing he bes alernaive model in he grid [0,5] parameer space using Schwarz informaion crierion. In his paper he number of hidden unis in hree-layer NN is specified in advance for he more suiable comparison beween models. This paper conribues in esimaing an addiional parameer λ which was in presened paper se in advance. Teräsvira e al [3] presen similar mehodology as [20] based on Medeiros e al [] approach and use i as AR-NN ype model showing he poenial of heir proposed modelling approach in wo applicaions: sunspo series and US unemploymen series. Moreover, hey clearly combine he NN model so as o be able o explain i as economeric model. Bildirici and Ersin [2] analyse he volailiy of sock reurns on Isanbul Sock Exchange (ISE) in period from 987 o 2008 using daily closing prices of ISE 00 index. They compare and combine GARCH, EGARCH, GJR-GARCH, TGARCH, NGARCH, SAGARCH, PGARCH, APGARCH, NPGARCH wih NN models in heir forecasing abiliies. The NN models are rerained wih conjugae gradien descen algorihm afer he raining wih backpropagaion. They conclude ha NN models improved he generalizaion and forecasing abiliy of GARCH models. In heir paper he models are no properly explained from an economeric perspecive, nor are he findings explained from he perspecive o he real ime daa. Moreover, he parameers of he models are nor presened or explained. Their laer paper, Bildirici and Ersin [22], relies on paper from [20] and [2] o analyse he nonlineariy and lepokuric disribuion of sock reurns on ISE in period from 986 o 200 and benefis from boh LSTAR and NN ype of nonlineariy, i.e. his paper proposes several LSTAR- GARCH-NN family models. GARCH, FI-GARCH, APGARCH and FIAPGARCH models are augmened wih a NN model. They conclude ha exended GARCH models forecas beer han GARCH models; LSTAR-LST-GARCH show significan improvemen in ou-of-sample forecasing; MLP-GARCH models provide similar resuls o LSTAR-LST-GARCH models; LSTAR-LST-APGARCH-MLP model provided he bes overall performance. To esimae NN models, he number of hidden neurons ranges from 3 o 0 and he bes model is seleced based on MSE or RMSE. Moreover, each of he seleced model archiecure is esimaed 20 imes for 8 differen NN models and o obain parsimony he appropriae model is seleced based on AIC. Alhough here is a vas number of economeric models for modelling condiional volailiy presened and esimaed in his paper, along wih an economeric presenaion of NN models, esimaion of 00 differen NN models wih hidden neurons ranging from 3 o 0 and comparing models wih differen specificaions is economerically unjusified. Bildirici and Ersin [23] propose a family of nonlinear GARCH models ha incorporae fracional inegraion (FI) and asymmeric power (AP) properies o MS-GARCH processes. Moreover, hey augmen he MS-GARCH ype models wih NN o improve forecasing accuracy. Therefore, he proposed MS-ARMA-FIGARCH, APGARCH, and FIAPGARCH processes are furher augmened wih MLP, RBF, Recurren NN, and Hybrid NN ype NNs. The MS-ARMA-GARCH

04 Compuer Science & Informaion Technology (CS & IT) family and MS-ARMA-GARCH-NN family are uilized for modelling he daily sock reurns of he ISE Index. Forecas accuracy is evaluaed wih MAE, MSE, and RMSE error crieria and Diebold-Mariano es for predicive accuracy. They conclude ha he FI and AP counerpars of MS-GARCH model provided promising resuls, while he bes resuls are obained for heir NN based models. Moreover, among he models analysed, he models MS-ARMA-FIAPGARCH- HNN and MS-ARMA-FIAPGARCH-RNN provided he bes forecas performances over he single regime GARCH models and over he MS-GARCH model. Parameers of NN models are no explained economerically, alhough NN are regarded as economeric model insead of nonparameric model. Manri e al [24] apply differen mehods, i.e. GARCH, EGARCH, GJR-GARCH, IGARCH and NN models for calculaing he volailiies of Indian sock markes. Two neworks are presened: single inpu (low index) single oupu (high index level) and muliple inpus (open, high and low index level) single oupu (close index level). The daa from 995 o 2008 of BSE Sensex and NSE Nify indices are used o calculae he volailiies. The auhors conclude ha he MISO-NN model should be used insead of SISO-NN model and ha here is no difference in he volailiies of Sensex and Nify esimaed under he GARCH, EGARCH, GJR-GARCH, IGARCH and NN models. In heir laer paper Manri e al. [25] focused on he problem of esimaion of volailiy of Indian Sock marke. The paper begins wih volailiy calculaion by ARCH and GARCH models of financial compuaion up o lag 3. The resuls are compared o NN model using R2. I can be concluded ha NN can be used as a bes choice for measuring he volailiy of sock marke. These papers provide no informaion abou he paricular NN model used, NN is no explained as economeric model, and herefore he papers are no suiable for deciding on suiabiliy of he models. Bekipraiwi and Irawan [26] propose an alernaive forecasing model based on he combinaions beween RBF and EGARCH model o model sock reurns of Bank Rakya Indonesia Tbk for he period from 2003 o 20. They use RBF o model he condiional mean and EGARCH o model he condiional volailiy and propose a regression approach o esimae he weighs and he parameers of EGARCH using maximum likelihood esimaor. The relevan explanaory variables are chosen based on is conribuion of giving greaer reducion in he in-sample forecas errors. They seleced inpus for RBF model, and 5 hidden neurons based on rial and error procedure. Based on SIGN es, he bes forecas is obained by RBF-EGARCH model for 00 seps ahead. All of he above researches combine GARCH-ype and NN models by adding he NN srucure o exising GARCH-ype models in search of he suiable model for forecasing condiional variance of sock reurns. The proposed mehodology is empirically esed on developed markes, however no on developing capial markes of Cenral and Easern Europe. Because of he uniqueness of hese emerging capial markes, i is imporan o es feaures of proposed mehodology in his paricular segmen. Moreover, some of he papers use NN as nonparameric esimaion echnique, neglecing he inerpreabiliy of parameers which could be obained by using NN as economeric ool. In his paper NN will be observed as semiparameric approach combining he flexibiliy of nonparameric mehods and he inerpreabiliy of parameers of parameric mehods. Anoher conribuion of he paper is in a priori specified srucure of NN models in order o be comparable o he GARCH-ype models and he esimaion of addiional parameer which was no esimaed before.

3. DATA AND METHODOLOGY Compuer Science & Informaion Technology (CS & IT) 05 The mos widespread approach o volailiy modelling consiss of he GARCH model of Bollerslev [2] and is numerous exensions ha can accoun for he volailiy clusering and excess kurosis found in financial ime series. The accumulaed evidences from empirical researches sugges ha he volailiy of financial markes can be appropriaely capured by sandard GARCH(,) model ([27]) since i gives saisfacory resuls wih small number of parameers o esimae. According o Bollerslev [2] GARCH (,) can be defined as: r ε u 2 = u σ = µ + ε ( ) : i. i. d. 0, 2 2 2 = + + σ α β ε γ σ () where µ is he condiional mean of reurn process { r }, while { } ε is he innovaion process wih is muliplicaive srucure of idenically and independenly disribued random variables u. The las equaion in () is condiional variance equaion wih GARCH(,) specificaion which means ha variance of reurns is condiioned on he informaion se I consising of all relevan previous informaion up o and including period. GARCH(,) model is covariancesaionary if and only if β + γ < ([2]). In paricular, GARCH(,) model usually indicaes high persisence in he condiional variance, i.e. inegraed behavior of he condiional variance when β + γ = (IGARCH). The reason for he excessive GARCH forecass in volaile periods may be he well-known high persisence of individual shocks in hose forecass. Relevan researches ([28]; [29]) show ha his persisence may originae from srucural changes in he variance process. High volailiy persisence means ha a long ime period is needed for shocks in volailiy o die ou (mean reversion period). Alhough GARCH models are he mos popular and widely used in empirical researches and among praciioners due o heir abiliy of describing he volailiy clusering, excess kurosis and fa-ailedness of he daa, hey canno capure he asymmeric behavior of volailiy. This means ha negaive shocks affec volailiy quie differenly han posiive shocks. Therefore, differen asymmeric models have been developed and used in empirical researches such as EGARCH, GJR-GARCH, TARCH, PGARCH, APGARCH among he ohers. However, he resuls from Engle and Ng [30] of Japanese sock reurns sugges ha Glosen, Jagannahn and Runkle s [4] sign ARCH model, usually called GJR model, shows he mos poenial in ouperforming he radiional GARCH model. Moreover, in recen lieraure i also proved o capure he asymmeric behavior in daa beer ha he oher models ([20]). Therefore, GJR-GARCH model is considered, i.e. GJR-GARCH(,,) is given by: D 2 2 2 2 = + + + D σ α β ε γ σ φ ε if ε < 0, = 0 if ε 0. (2)

06 Compuer Science & Informaion Technology (CS & IT) As can be seen from (2), GJR-GARCH model is jus an augmenaion of GARCH model ha allows pas negaive unexpeced reurns o affec volailiy differenly han posiive unexpeced reurns. When φ > 0 negaive shocks will have a larger impac on condiional variance. For GJR- GARCH saionariy condiion is saisfied if β + γ + φ 2 <. An alernaive soluion o overcome he problems found for sandard GARCH (,) model is o define appropriae neural nework (NN), i.e. by exending he GJR-GARCH (,,) model wih NN model, significan improvemens can be found. The NN is an arificial inelligence mehod, which has recenly received a grea deal of aenion in many fields of sudy. Usually NN can be seen as a nonparameric saisical procedure ha uses he observed daa o esimae he unknown funcion. A wide range of saisical and economeric models can be specified modifying he srucure of he nework, however NN ofen give beer resuls. Empirical researches show ha NN are successful in forecasing exremely volaile financial variables ha are hard o predic wih sandard economeric mehods such as: exchange raes ([7]), ineres raes ([6]) and socks ([8]). The mos commonly used ype of NN in empirical researches is muli-layer feed-forward neural neworks (FNN). The FNN forwards informaion from inpu layer o oupu layer hrough a number of hidden layers. Neurons in a curren layer connec o neuron of he subsequen layer by weighs and an acivaion funcion. In order o obain weighs backpropagaion (BP) learning algorihm, which works by feeding he error back hrough he nework, is mosly used. The weighs are ieraively updaed unil here is no improvemen in he error funcion. This process requires he derivaive of he error funcion wih respec o he nework weighs. The mean of squared error (MSE) is he convenional leas square objecive funcion in a NN, defined as mean of squared differences beween he observed and fied values of ime series. The FNN wih linear componen can be wrien as: yˆ p q = + + + p f φ co φio x, i φho g φch φih x, i (3) i= h= i= where is a ime index, y ˆ is he oupu vecor, x, i is he inpu marix wih i variables, f( ) and g( ) are acivaion funcions (usually linear and logisic respecively). Index c is he consan, i is he inpu, h is he hidden, and o is he oupu neuron. φ co denoes he weigh of he direc connecion beween he consan and oupu, φ io denoe he weighs of direc connecion from inpus o oupu, φ ch denoe he weighs for he connecions beween he consan and hidden neurons. The weighs φ ih and φ ho denoe he weighs for he connecions beween he inpus and hidden neurons and beween he hidden neurons and oupu. NN wih p inpus and q oupus has he abbreviaion FNN(p,q). However, he disadvanage of FNN is he problem of overfiing. I occurs due o he inclusion of muliple hidden layers or muliple neurons in hidden layer which, wih exising heoreically based number of inpus (independen variables) and lagged oupus (dependen variables), increases he number of parameers o esimae. Therefore, in his paper NN are observed only as an exension o GJR-GARCH ype model wih he srucure defined in advance o benefi from

Compuer Science & Informaion Technology (CS & IT) 07 parsimonious model in order o avoid he problem of overfiing. The GJR-GARCH-NN(,,,) as a nonlinear exension o GJR-GARCH model is defined as: D z ( z λ ) 2 ( ε ) ( z ) 2 2 2 2 = + + + D + σ α β ε γ σ φ ε ξψ λ ψ if ε < 0, = 0 if ε 0. = λz + e ε E ( ε ) =. E (4) where ψ ( ) z λ specifies he logisic funcion in hidden uni of neural nework wih hidden neuron, z provides a normalizaion of ε necessary o prepare he lagged unexpeced reurns as inpus ino he nodes. All he daa are ransformed using he in-sample mean and variance. Donaldson and Kamsra [20] chose λ in advance from a uniform random number generaor so hey lie beween -2 and 2 in order o achieve he idenificaion of parameers ξ, and hen parameers α, β, γ, φ, ξ are esimaed wih maximum likelihood. In his paper λ are defined beween -2 and 2, however hey are esimaed wih maximum likelihood jus as oher parameers. The daa se consiss of reurns of he daily closing prices obained from sock exchanges in period from January 20 unil Sepember 204 for seleced European emerging markes, i.e. Bulgaria, Croaia, Czech, Romania, Slovakia and Slovenia. Daa is obained from Thomson Reuers daabase. 4. EMPIRICAL RESEARCH In order o invesigae GARCH-ype models i is imporan o give an overview of he sample, i.e. descripive saisics. From Table can be seen ha in observed period European emerging markes have negaive expeced reurns. The lowes risk is observed in Slovakia and Slovenia and he highes risk in Czech Republic and Romania. Each disribuion shows asymmeric behavior and lepokurosis. Moreover, ime series is no saionary since he variance of reurns is ime varying. Deailed resuls are omied due o a lack of space. They are available from auhors upon reques. Parameers for GJR-GARCH(,,) model are esimaed in SAS sofware using he maximum likelihood mehod and he resuls for seleced markes wih esimaed parameers and he value of Log-Likelihood (LL) is given in Table 2. The resuls reveal ha asymmeric behavior is saisically significan in all markes and since φ < 0 he posiive shocks will have a larger impac on condiional variance. In developed markes his parameer is usually posiive indicaing he opposie conclusions.

08 Compuer Science & Informaion Technology (CS & IT) Table. Descripive saisics of daily reurns for seleced markes N Min Max µ σ α 3 α 4 BULGARIA 277-0,36 0,0729-0,00044 0,0307 -,05 0,39 CROATIA 277-0,076 0,477-0,00024 0,0273 0,4 8,43 CZECH 277-0,68 0,09-0,00032 0,052-0,82 5,32 ROMANIA 277-0,3 0,056-0,00005 0,0639-0,50 8,64 SLOVAKIA 277-0,48 0,88-0,00022 0,053 -,52 29,90 SLOVENIA 277-0,0843 0,0835-0,00032 0,089-0,46 7,47 Table. Parameer esimaes of GJR-GARCH(,,) model wih values of Log-Likelihood (LL) BULGARIA CROATIA CZECH ROMANIA SLOVAKIA SLOVENIA -0.00007-0.0002-0.00008 0.000367* 5.36E-06-0.0004 7.4E-06*** 4.56E-07*** 4.42E-03*** 4.27E-06*** 0.000028*** 0.00002*** 0.30746*** 0.6087*** 0.764*** 0.9002*** 0.078577*** 0.27928*** 0.703246*** 0.92962*** 0.846355*** 0.824793*** 0.79784*** 0.697366*** -0.08503** -0.05222*** -0.0893*** -0.04265* -0.0906*** -0.4047*** LL 6970.803 7227.855 66.6 6470.448 6649.694 6945.997 Noe: Parameer esimaes are significan a % (***), 5% (**) and 0% (*) significance level Parameers for GJR-GARCH-NN(,,,) model are esimaed in SAS sofware using he maximum likelihood mehod and he resuls for seleced markes wih esimaed parameers and he value of Log-Likelihood (LL) is given in Table 3. This model has wo addiional parameers o esimae: ξ and λ. Parameer ξ is in each marke posiive and saisically significan. Moreover, he Log-Likelihood is in GJR-GARCH-NN model larger han in simpler model. All hese findings lead o a conclusion ha exending he GJR-GARCH wih he NN model, i.e. adding he nonlineariy in he model, is saisically significan and improves he models fi Table 3. Parameer esimaes of GJR-GARCH-NN(,,,) model wih values of Log-Likelihood (LL) BULGARIA CROATIA CZECH ROMANIA SLOVAKIA SLOVENIA -0.00007-0.0008-0.000 0.00039-0.0002-0.0005-0.03772*** -0.0377*** -0.03772*** -0.0376*** -0.03787*** -0.03776*** 0.3529*** 0.2463*** 0.5603*** 0.9557*** 0.364962*** 0.34983*** 0.70302*** 0.9607*** 0.84822*** 0.83074*** 0.608284*** 0.697389*** -0.09305*** -0.06524*** -0.0465-0.06322*** -0.27835*** -0.27628*** 0.075462*** 0.075429*** 0.075448*** 0.075225*** 0.07597*** 0.075546*** 0.000028 0.000039-0.0002 0.00037 0.000798* 0.00059*** LL 6970.826 7228.358 662.326 6470.998 6657.298 6950.29 Noe: Parameer esimaes are significan a % (***), 5% (**) and 0% (*) significance level

Compuer Science & Informaion Technology (CS & IT) 09 5. CONCLUSIONS Modelling volailiy, i.e. reurns flucuaions, is in he main focus of he paper. This research begins wih he mos widespread approach o volailiy modelling, i.e. GARCH(,) model. Due o is disadvanage in capuring he asymmeric behavior GJR-GARCH(,,) model is inroduced. However, boh models fail o model nonlineariy in daa and, herefore NN model as an exension o GJR-GARCH model is defined, i.e. parsimonious GJR-GARCH-NN model. This paper esimaes he parameers of boh simple and exended GJR-GARCH model and compares hese models using daa for seleced European emerging markes. Moreover, NN are presened as an economeric ool. Resuls of his paper confirm conclusions of previous researches abou superioriy of NN versus oher linear and nonlinear models. However, hey are sill a challenge for he researchers in order o improve heir performances in forecasing condiional variance of sock reurns and ime series in general. The ou-of-sample predicive performance, inclusion of more hidden neurons or oher archiecures, he use of differen algorihms in he nework raining, open space for fuure work and furher sudies. ACKNOWLEDGEMENTS This work has been fully suppored by Croaian Science Foundaion under he projec Volailiy measuremen, modeling and forecasing (599). REFERENCES [] Engle, R.F. (982) Auoregressive condiional heeroscedasiciy wih esimaes of he variance of UK inflaion, Economerica, Vol. 4, pp. 35-55. [2] Bollerslev, T. (986) Generalized auoregressive condiional heeroscedasiciy, Journal of Economerics, Vol. 3, pp. 307-327. [3] Diebold, F. (986) Commen on modelling he persisence of condiional variances, Economeric Reviews, Vol. 5, pp. 5 56. [4] Glosen, L.R., Jagannahan, R. & Runkle, D.E. (993) On he Relaion beween he Expeced Value and he Volailiy of he Nominal Excess Reurn on Socks, The Journal of Finance, Vol. 48, No. 5 (Dec., 993), pp. 779-80. [5] Balkin, S.D. (997) Using Recurren Neural Neworks for Time Series Forecasing, Working Paper Series number 97-, Inernaional Symposium on Forecasing, Barbados. [6] Täppinen, J. (998) Ineres rae forecasing wih neural neworks, Governmen Insiue for Economic Research, Va-Discussion Papers, 70. [7] Dunis, C.L. & Williams, M. (2002) Modelling and rading he euro/us dollar exchange rae: Do neural neworks perform beer?, Journal of Derivaives & Hedge Funds, Vol. 8, No. 3, pp. 2-239. [8] Zekić-Sušac, M. and Kliček, B. (2002) A Nonlinear Sraegy of Selecing NN Archiecures for Sock Reurn Predicions, Finance, Proceedings from he 50h Anniversary Financial Conference Svishov, Bulgaria, -2 April, Svishov, Veliko Tarnovo, Bulgaria: ABAGAR, pp. 325-355.

0 Compuer Science & Informaion Technology (CS & IT) [9] Tal, B. (2003) Background Informaion on our Neural Nework-Based Sysem of Leading Indicaors, CBIC World Markes, Economics & Sraegy [0] Ghiassi, M., Saidane, H. & Zimbra, D.K. (2005) A dynamic arificial neural nework model for forecasing ime series evens, Inernaional Journal of Forecasing, Vol. 2, pp. 34-362. [] Medeiros, M.C., Teräsvira, T. & Rech, G. (2006) Building Neural Nework Models for Time Series: A Saisical Approach, Journal od Forecasing, No. 25, pp. 49-75. [2] Kuan, C.-M. & Whie, H. (2007) Arificial neural neworks: an economeric perspecive, Economeric Reviews, Vol. 3, pp. -92. [3] Teräsvira, T., Tjøsheim, D. & Granger, C.W.J. (2008) Modelling nonlinear economic ime series, Advanced exs in economerics, Oxford, New York, Oxford Universiy Press [4] Gonzales, S. (2000) Neural Neworks for Macroeconomic Forecasing: A Complemenary Approach o Linear Regression Models, Working Paper 2000-07. [5] Hwarng, H.B. (200) Insighs ino neural-nework forecasing of ime series corresponding o ARMA(p,q) srucures, Omega, Vol. 29, pp. 273-289. [6] Zhang, G.P. (2003) Time series forecasing using hybrid ARIMA and neural nework model, Neurocompuing, Vol. 50, pp. 59-75. [7] Aminian, F., Suarez, E.D., Aminian, M. & Walz, D.T. (2006) Forecasing Economic Daa wih Neural Neworks, Compuaional Economics, Vol. 28, pp. 7-88. [8] Lawrence, S., Giles, C.L. & Tsoi, A.C. (997) Lessons in Neural Nework Training: Overfiing May be Harder han Expeced, Proceedings of he Foureenh Naional Conference on Arificial Inelligence, AAAI-97, pp. 540-545. [9] Franses, P.H. & van Dijk, D. (2003) Nonlinear Time Series Models in Empirical Finance, Cambridge Universiy Press. [20] Donaldson, R.G. & Kamsra, M. (997) An Arificial Neural Nework - GARCH Model for Inernaional Sock Reurn Volailiy, Journal of Empirical Finance, Vol. 4, No., pp. 7-46. [2] Bildirici, M. & Ersin, Ö.Ö. (2009) Improving forecass of GARCH family models wih he arificial neural neworks: An applicaion o he daily reurns in Isanbul Sock Exchange, Exper Sysems wih Applicaions, No. 36, pp. 7355-7362 [22] Bildirici, M. & Ersin, Ö.Ö. (202) Nonlinear volailiy models in economics: smooh ransiion and neural nework augmened GARCH, APGARCH, FIGARCH and FIAPGARCH models, MPRA Paper No. 40330 [23] Bildirici, M. & Ersin, Ö.Ö. (204) Modelling Markov Swiching ARMA-GARCH Neural Neworks Models and an Applicaion o Forecasing Sock Reurns, Hindawi Publishing Corporaion: The Scienific World Journal, Vol. 204, Aricle ID 49794, 2 pages [24] Manri, J.K., Gahan, P. & Nayak, B.B. (200) Arificial Neural Neworks An Applicaion o Sock Marke Volailiy, Inernaional Journal of Engineering Science and Technology, Vol. 2, No. 5, pp. 45-460.

Compuer Science & Informaion Technology (CS & IT) [25] Manri, J.K., Mohany, D. &Nayak, B.B. (202) Design Neural Nework for Sock Marke Volailiy: Accuracy Measuremen, Inernaional Journal on Compuer Technology & Applicaions, Vol. 3, No., pp. 242-250. [26] Bekipraiwi, A. & Irawan, M.I. (20) A RBF-EGARCH neural nework model for ime series forecasing, Proceedings of The IceMATH 20, Topic, pp. -8. [27] Visković, J., Arnerić, J. &Rozga, A. (204) Volailiy Swiching beween Two Regimes, World Academy of Science, Engineering and Technology, Inernaional Science Index 87, Inernaional Journal of Social, Managemen, Economics and Business Engineering, Vol. 8, No. 3, pp. 682-686. [28] Lamoureux, C. & Lasrapes, W. (990) Persisence in variance, srucural change, and he GARCH model, Journal of Business and Economic Saisics, Vol. 8, pp. 225 234. [29] Wong, C.S. & Li, W.K. (200) On a mixure auoregressive condiional heeroscedasic model, Journal of American Saisical Associaion, Vol. 96, No. 455, pp. 982 995. [30] Engle, R.F. & Ng, V.K. (993) Measuring and Tesing he Impac of News on Volailiy, The Journal of Finance, Vol. 48, No. 5 (Dec., 993), pp. 749-778. AUTHORS Josip Arnerić Assisan Professor, PhD. Universiy of Zagreb, Faculy of Economics and Business Zagreb, Croaia. Scienific affiliaion: economeric mehods and models, financial ime series and volailiy, VAR models and coinegraion, GARCH and MGARCH models, sochasic processes and risk managemen. Phone: 00385238336. Fax: 00385233268. E-mail: jarneric@efzg.hr Tea Poklepović Teaching Assisan, Docoral suden. Universiy of Spli, Faculy of Economics, Spli, Croaia. Scienific affiliaion: saisics and economerics in business, finance and macroeconomics, especially economeric mehods and models, ime series and neural neworks. Phone: 00385243076. Fax: 00385243070. E-mail: poklepo@efs.hr.