Financial Volatility Forecasting by Least Square Support Vector Machine Based on GARCH, EGARCH and GJR Models: Evidence from ASEAN Stock Markets

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1 Inernaonal Journal of Economcs and Fnance February, Fnancal Volaly Forecasng by Leas Square Suppor Vecor Machne Based on GARCH, EGARCH and GJR Models: Evdence from ASEAN Sock Markes Phchhang Ou (correspondng auhor) School of Busness, Unversy of Shangha for Scence and echnology Rm, Inernaonal Exchange Cener, No. 6, Jun Gong Road, Shangha 93, Chna Fax: E-mal: Hengshan Wang School of Busness, Unversy of Shangha for Scence and echnology Box 46, No. 6, Jun Gong Road, Shangha 93, Chna el: E-mal: hs work s suppored by Shangha Leadng Academc Dscplne Projec, Projec Number: S34. Absrac In hs paper, we am a comparng sem-paramerc mehod, (Leas square suppor vecor machne), wh he classcal GARCH(,), EGARCH(,) and GJR(,) models o forecas fnancal volales of hree major ASEAN sock markes. More precsely, he expermenal resuls sugges ha usng hybrd models, GARCH-, EGARCH- and GJR- provdes mproved performances n forecasng he leverage effec volales, especally durng he recenly global fnancal marke crashes n 8. Keywords: Leas squared suppor vecor machne, Forecasng Volaly, GARCH, EGARCH, GJR.. Inroducon me seres mehod plays a val role n fnancal areas, parcularly volaly modelng and forecasng. Mos of he fnancal researchers and praconers are manly concerned wh modelng volaly n asse reurns. In hs conex, volaly s he varably n he asse prces over a parcular perod of me. I refers o he sandard devaon of he connuously compounded reurns of a fnancal nsrumen wh a specfc me horzon. I s ofen used o quanfy he rsk of he nsrumen over ha me perod. Invesors wan a premum for nvesng n rsky asses. A rsk manager mus know oday he lkelhood ha hs porfolo wll declne n he fuure and he may wan o sell before becomes oo volale. herefore, he ably o forecas fnancal marke volaly s mporan for porfolo selecon and asse managemen as well as he prcng of prmary and dervave asses. Researches on me varyng volaly usng he me seres models have been acve ever snce Engle nroduced he ARCH (auoregressve condonal heeroscedascy) model n 98. Snce s nroducon, he GARCH model generalzed by Bollerslev (986) has been exended n varous drecons. Several exensons of he GARCH model amed a capurng he asymmery n he response of he varance o a shock. hese exensons recognze ha here may be mporan nonlneary, asymmery, and long memory properes n he volaly process as suggesed by varous researchers based on emprcal evdences. he popular approaches can be referred o Exponenal GARCH model by Nelson (99) as well as he GJR model by Glosen, Jaganahan, and Runkle (993) whch boh accoun for he asymmerc relaon beween sock reurns and changes n varance; see Black (976) he begnnng sudy of he asymmerc effec; Engle and Ng (993) for furher dscusson. Oher models such as APARCH, AGARCH, GARCH and QGARCH models have also been developed (by Dng, Granger and Engle (993); Engle (99); Zakoan (994) and Senana (99)) for he flexbly of he models. However, all of he models do requre specfed dsrbuon of nnovaons n order o esmae he model specfcaon and o appropraely forecas fuure values. One of he mos classc one s Gaussan process and s wdely used n mos of leraure; bu oher dsrbuons of nnovaons are also araced afer he emprcal sudes of modelng reurns have shown he volaon of normaly condons. For example, Suden s dsrbuon by Bollerslev (987), GED n Nelson (99), Granger and Dng (99) for he Laplace dsrbuon and Hseh (989) for boh Suden s and GED as dsrbuonal alernave models for nnovaons. he researches have found ha reurns usually exhb emprcal regulares ncludng hck als, volaly cluserng, leverage effecs (Bollerslev e al,994).

2 Vol., No. Inernaonal Journal of Economcs and Fnance Sem-paramerc approaches do no requre any assumpons on daa propery (.e. reurn dsrbuon). hese models have been successfully shown for modelng and forecasng me seres, ncludng volaly. One of hem s NN (neural nework) and s a powerful ool for predcon problems due o her bes ably o esmae any funcon arbrary wh no pror assumpon on daa propery (Haykn, 999). Donaldson and Kamsra (997) proposed neural nework o model volaly based GJR-GARCH; her hybrd approach capured asymmerc effecs of new mpac well lke paramerc model and also generaed beer forecasng accuracy. Bldrc & Ersn(9) fed neural nework based on nne dfferen models of GARCH famly such as NN-GARCH, NN-EGARCH, NN-GARCH, NN-GJR, NN-SAGARCH, NN-PGARCH, NN-NGARCH, NN-APGARCH, and NN-NPGARCH o forecas Isanbul sock volaly and mos of he hybrd models mproved forecasng performance. hs ndcaes ha he hybrd model s also able o capure he sylzed characerscs of reurn. Anoher effcen (sem-paramerc) model s SVM (suppor vecor machne) orgnally nroduced by Vapnk (99). he SVM, a novel neural nework algorhm, guaranees o oban globally opmal soluon (Crsann & Shawe-aylor, ), and hence solves he problems of mulple local opma n whch he neural nework usually ge rapped no. Perez-Cruz e al (3) predced GARCH(,) based volaly by SVM and he proposed model yelded beer predcve capably han he paramerc GARCH(,) model for all suaon. Chen e al (8) developed recurren SVM as a dynamc process o model GARCH(,) based volaly. he expermenal resuls wh smulaed and real daa also showed he model generaed beer performance han MLE (maxmum lkelhood esmaon) based GARCH model. More applcaons of SVM n GARCH predcon based on dfferen kernels, wavele and splne wavele can be referred o ang e al (8, 9). Anoher verson of SVM s (Leas squares suppor vecor machne), modfed by Suykens e al (999). he SVM algorhm requres Epslon nsensve loss funcon o oban convex quadrac programmng n feaure space, whle jus uses leas square loss funcon o oban a se of lnear equaons (Suykens, ) n dual space so ha learnng rae s faser and he complexy of calculaon n convex programmng n SVM s also relaxed. In addon, he avods he drawback faced by SVM such as rade-off parameers ( C,, ) selecon, nsead requres only wo hyper-parameers (, ) whle ranng he model. Accordng o Suykens e al (), he equaly consrans of can ac as recurren neural nework and nonlnear opmal conrol. Due o hese nce properes, has been successfully appled for classfcaon and regresson problems, ncludng me seres forecasng. See Van Gesel e al (4) for dealed dscusson on classfcaon performance of and Ye e al (4) for predcve capably of n chaoc me seres predcon. Van Gesel e al () proposed o predc me varyng volaly of DAX 3 ndex by applyng Bayesan evdence framework o. he volaly model s consruced based on nferred hyperparameers of formulaon whn he evdence framework. he proposed model provded a beer predcve performance han GARCH(,) and oher AR() models n erm of MSE and MAE. In hs paper, we am a comparng he mehod wh he classcal GARCH(,), EGARCH(,) and GJR(,) models o forecas fnancal volales of ASEAN sock markes as a new concep o be nvesgaed. he hybrd models denoed as GARCH-, EGARCH-, and GJR- are consruced by usng lagged erms as npu and presen erm as oupu whch corresponds o he paramerc models. he hybrd models are no he same as he volaly model proposed by Van Gesel e al () bu hey are smlarly bul accordng wh he resuls by Donaldson & Kamsra (997) and Bldrc & Ersn(9) wh neural nework approach, and Perez-Cruz e al (3) wh SVM mehod. In our expermen, we consder wo sage forecass for he whole year 7 as frs perod and 8 as he second sage whch cover global fnancal crss perod. Several mercs MAD, NMSE, HR, and lnear regresson R squared are employed o measure he model performances. he paper s organzed as follows. Nex secon brefly revews formulaon. Secon 3 dscusses volaly modelng of hybrd models based on GARCH, EGARCH and GJR. Secon 4 llusraes he expermenal resuls and he fnal secon s abou he concluson.. Leas squared suppor vecor machnes In formulaon, he daa are generaed by nonlnear funcon y f ( x ) e for,, N whch may be approxmaed by anoher nonlnear funcon y w ( x ) b e. () he model parameer w s called wegh and e s random nose. Oupu y R can be referred as reurn of an asse or n n n volaly, whle he npu vecor x R may conss of lagged reurns or lagged volaly. Mappng ( ) : R R s nonlnear funcon ha maps he npu vecor x no a hgher dmensonal feaure space. he wegh vecor w and funcon () are never calculaed explcly; nsead, Mercer condoned kernel K( x, x) ( x ) ( x) whch s symmerc and posve defne s appled. One defnes he opmzaon problem as objecve funcon N mn J ( w, e) w w e () w, b, e

3 Inernaonal Journal of Economcs and Fnance February, subjec o he consrans e y ( w ( x ) b),, N. (3) Here he equaly consran s used n nsead of he nequaly consran n SVM. Lagrangan can be defned o solve he above mnmzaon problem as N ( L w, b, e ; ) J ( w, e) ( w ( x ) b e y ) where denoes Lagrange mulplers (also called suppor values). From he Karush-Kuhn-ucker (KK) heory, a sysem of equaons s obaned as he followng L N w ( x ) w L N b (4) L e,, N e L b y w ( x ) e,,, N. Noe ha sparseness s los from he condon e. By elmnang w and e, he followng lnear sysem s wren as follow v b () v D y where y [ y,, y N ], v [,, ], e [ e,, e N ], [,, N ], D dag([,, N ]). Marx j ( x ) ( x j ) K( x, x j ) for, j,, N sasfes Mercer s condon and he LS-SVM model for esmang funcon s obaned as N h( x) w ( x) b K( x, x ) b. (6) K(.,.) s he Mercer s kernel funcon represenng he hgh-dmensonal feaure space ha nonlnearly mapped from he npu space. In hs work, Gaussan kernel or RBF(radal bass funcon) s used as ends o gve a good performance under general smoohng assumpons. he kernel s defned as K( x, x ) exp( x ). he kernel and x regularzed parameers (, ) are uned by grdsearch echnque o avod overfng problem. Malab oolbox s used n he whole expermen. 3. Predcve model of volaly 3. Model buldng Le P be he sock prce a me. hen y. log ( P / P ) (7) denoes he connuously compounded daly reurns of he parcular sock a me. Le F be he pas nformaon se avalable up o me ; hs nformaon se conans he realzed values of all prevous relevan varables. he expeced reurn a me afer observng he pas nformaon up o defned as E y / F ] f ( F ). (8) [ he volaly o nvesors nvesng n he parcular sock a me s denoed as follow where f (.) and h (.) are well defned funcons wh h (.). hen he reurn of sock y can be modelled Var y / F ] h ( F ) (9) y [ () 3

4 Vol., No. Inernaonal Journal of Economcs and Fnance and. z where z s (d) ndependen dencally dsrbued random varables wh mean and varance. I s common o assume so ha square reurn n () s obaned o be he shock squared, y z. () Here, we am a esmang volaly (or condonal varance of reurn) n (9) by kernel regresson (called sem-paramerc mehod) based on paramerc models of GARCH, EGARCH and GJR. One parcular approach of he kernel regresson s (leas square suppor vecor machne) presened n he prevous secon. approxmaes GARCH(,).. () by nonlnear funcon obaned from he algorhm, h (, ). (3) Smlarly, he hybrd model esmaes EGARCH(,) log.[ / E( / ) ] / log (4) by log h (log, [ / E( / ) ], / ) () where npus log, [ / E( / ) ], / mus be obaned before ranng. he expecaon E( / ) s esmaed by s correspondng average (or mean) value so ha specfed dsrbuon on nnovaon (or reurn n hs case) s no requred. Fnally, he GJR- obans GJR(,)... S (6) as h (,, S ) (7) where S f and oherwse. So S s he squared value of negave shock a. 4 Here ˆ k yk suggesed by Perez-Cruz (3), and he funcon h (.) s obaned by algorhm n (6). 3. Forecasng Ou of sample forecass by he hybrd models are obaned as follow: h (, ) for GARCH- from (3), exp h (log, [ / E( / ) ], / ) for EGARCH- from () and h (,, S ) for GJR- from (7). 4. Expermenal Analyss We examne hree sock prce ndexes from hree major ASEAN sock markes ncludng Sras me ndex of Sngapore sock marke, of he Phlppnes and of Kula Lumpur sock marke. Each sock ndex prce s colleced from Yahoo Fnance webse and s ransformed no log reurn as n (7) before makng analyss. able repors he n-sample and ou-of-sample perods of each marke for wo sages, basc sascs of he daa and dagnoscs. From he able, we see ha mean of all reurns s close o zero. wo ndexes and have posve skewed reurns whle SI produce negave skewed coeffcen. he excess kuross appears n all seres and he larges s from (4.). he Jarque Bera sascs srongly sugges ha all reurns are non normal. Ljung Box es for squared reurn a lag and Engle LM es sgnfcanly ndcae all reurn seres exhb ARCH effecs; ha means he homoscedascy hypohess s srongly rejeced. hs shows he presence of volaly cluserng and he leverage effecs ha could be caused by he excess kuross. Fgure plos prce and log reurn of each ndex seres for he enre sample. hough movemen of he ndex prces of he hree markes s almos n smlar drecon, he reurns behave dfferenly. From he plos, we can see some hgh volaly on log reurn seres afer fnancal crss n ASEAN n 997 and durng he recen crss of global marke crashes; hs s obvously seen ha he plos of each sock prce fall down sharply durng 8. 4

5 Inernaonal Journal of Economcs and Fnance February, 4. Esmaon resuls hree paramerc models GARCH, EGARCH and GJR are fed o all reurn seres by (), (4) and (6) respecvely. Each model s esmaed wce for each marke reurn as frs sage and second sage esmaons wh updang n-sample. able.a,.b, and.c presen he model parameers and her correspondng sandard errors n brackes. he saonary condons of he models hold for all seres. Furhermore, sgnfcance of negave leverage coeffcens n EGARCH and posve leverage coeffcens of he correspondng GJR ndcae he presence of asymmerc effecs o he reurns for boh sages whch may be caused by global fnancal crss. By log lkelhood, AIC and BIC crera n able.a and.c, GJR model s more adequae o he boh sage esmaons of SI and reurns. For he reurn n able.b, he GJR model fs well o he n-sample daa a he frs sage bu he second sage esmaon daa s favour o EGARCH model accordng o he log lkelhood, AIC and BIC. Now we proceed o esmaon resuls obaned from ranng he leas square suppor vecor machne. Frs, reurn seres from all ndexes are ransformed no npu and oupu forma and hen ge hem raned by algorhm n (6) so as o ge he esmaed nonlnear funcon n (3) for GARCH hybrd, () for EGARCH case and (7) for GJR model. he ranng resuls are summarzed n able 3.A and 3.B for frs and second sages respecvely. Each second column of he able 3.A and 3.B shows he coss of ranng measured by he mean square errors. he hrd and fourh columns dsplay he opmal regularzed parameers and opmal kernel parameers obaned by grdsearch echnque whle ranng. he las column ells he bas erm of resuled funcon obaned by he. From he frs sage of he cos column n SI ndex, smalles cos falls o GARCH- and he larges value goes o EGARCH-. For seres, GJR- generaes he leas cos and he larges cos s from GARCH-, bu hese errors are no far from one anoher. Fnally, seres produces he smalles error o EGARCH- and he error s a b far from he errors drven by GARCH- and GJR-. In he second sage, ranng mean square error for SI and are analogue o he mean square errors for SI and n he frs sage respecvely; ha s he SI s n favour wh GARCH- and produces he smalles cos whle geng raned by EGARCH-. For Kula Lumpur sock marke, GARCH- gves he smalles cos, bu EGARCH- sll produces he hghes value of cos lke before. In he nex secon, hese hybrd models wll be performed o forecas volaly of he hree markes and also be compared wh he paramerc approaches esmaed n he prevous secon. 4. Forecasng resuls he followng Evaluaon mercs are used o measure he performance of proposed models n forecasng of he hree dfferen sock markes volales. hey are Mean Absolue Devaon (MAD), Normalzed Mean Square Error(NMSE) and H Rae (HR) whch defned as he followng: MAD n n n d n n a p, NMSE ( a p ) n s where s n n ( a a ). HR ( a a )( p p ) where d oherwse Here a y acual values and p ˆ forecased volaly. Here n n. We also use lnear regresson o evaluae he forecasng performance of he volaly model. We smply regress square reurns on a consan and he forecased volaly for ou-of-sample me pon,,,..., n, y c c ˆ e. he -sasc of he coeffcens s a measure of he bas and he square correlaon R s a measure of forecasng performance. In hs regresson, he consan erm c should be close o zero and he slope c should close o. able 4.A and 4.B llusrae forecasng performances by dfferen models for each marke. he MAD, NMSE, HR and R squared wh c and c are shown n he second o sevenh columns. Frs sage for 7: Begnnng wh SI seres, he hybrd approaches perform beer han paramerc models for almos all mercs: MAD, NMSE, HR, and R squared. Only R squared creron s n favour o EGARCH model ha generaes he hghes value. Among all he models, EGARCH- s bes a a predcve performance because provdes hghes HR (.873), smalles values of MAD and NMSE and also sasfes o (c and c) values whch are no far from ( and ) respecvely. Now by consderng Kula Lumpur marke, based on MAD and NMSE, he hybrd models are much beer bu n erm of HR and R squared some sem-paramerc models especally EGARCH- s unable o defea s counerpar, EGARCH. he reurn s well modelled by EGARCH lke SI case snce generaes leas NMSE,

6 Vol., No. Inernaonal Journal of Economcs and Fnance hghes R squared and HR among he ohers. Lookng a c and c crera, he hybrd models are more sasfed han he paramerc approaches. For, he sem-paramerc models are superor o he paramerc models for all cases. Second sage for 8: he SI reurn seres s well forecased by EGARCH model lke s prevous performance n he frs sage forecas due o he hghes values of HR and R squared, whch can be seen from he able 4.B. For he oher formed GARCH and GJR, s beer han he paramerc models. From he able 4.B, he values of c and c of EGARCH model (wh values -3.7,.), hough generaed bes performance, devae far from he approprae norm (, ) respecvely due o he global fnancal marke crashes. However, EGARCH- and oher s are more ressan n forecasng performance o he crashes snce her c and c are no much far from and respecvely. For and, hybrd approaches bea all paramerc models for all crera and EGARCH- s superor among he ohers. hese evdences can argue ha s more robus han he paramerc models n forecasng volaly n spe of he hgh volale suaon durng he global fnancal marke crashes. Fgures, 3, & 4 plo he ou of sample forecass by paramerc models of GARCH, EGARCH and GJR and he correspondng hybrd models for SI, and respecvely. From he plos, he forecas lnes by hybrd models capure more exreme pons han he paramerc models do and herefore hey mprove forecasng performance. Noceably, he algorhm here has no been mposed he sparsy and robusness condons proposed by Suyken e al ().. Concluson In hs paper, we combne Leas square suppor vecor machne () wh GARCH(,), EGARCH(,) and GJR(,) models as a hybrd approach o forecas leverage effec volaly of ASEAN sock markes. o check he performance of he proposed models, we employ he correspondng paramerc models o compare wh he hybrd models. he forecass are conduced wce n whch he whole year 7 s reaed as he frs sage and he second sage s for 8 ncludng he recen global fnancal crss perod. From he expermenal resuls, s found ha he hybrd models are ressan and robus o he hgh volale suaon of he fnancal marke crashes and hence hey generae mproved forecasng performance. hs suppors he general dea ha s he promsng machne learnng sysem whch s good a esmang nonlnear funcon whou assumpons on daa propery n me seres applcaons. References Bldrc, M. & Ersn O. O. (9). Improvng forecass of GARCH famly models wh he arfcal neural neworks: An applcaon o he daly reurns n Isanbul Sock Exchange, Exper Sysems wh Applcaons, Black, F. (976). Sudes of sock prce volaly changes, Proceedngs of he 976 Meengs of he Busness and Economcs Sascs Secon, Amercan Sascal Assocaon, Bollerslev,., Engle R.F. & Nelson, D.B. (994). ARCH models, Handbook of Economercs, Volume IV, Elsever Scence B.V. Bollerslev,. (987) A condonal heeroskedasc me seres model for speculave prces and raes of reurn, Rev Econ Sa 69:4-47 Bollerslev. (986). Generalzed Auo Regressve Condonal Heeroskedascy, Journal of Economercs, pp Chen, S., Jeong, K. & Hardle, W. (8). Suppor Vecor Regresson Based GARCH Model wh Applcaon o Forecasng Volaly of Fnancal Reurns, SFB 649 Dscusson Paper of Economc Rsk, Berln. Crsann, N., & Showe-aylor, J. (). An Inroducon o Suppor Vecor Machnes and Oher Kernel-based Learnng Mehods, London: Cambrdge Unversy Press. Dng, Z., C. W.J. Granger & R. F. Engle. (993). A long memory propery of sock marke reurns and a new model, Journal of Emprcal Fnance,, 83-6 Norh-Holland. Donaldson, R. G. & Kamsra, M. (997). An arfcal neural nework-garch model for nernaonal sock reurn volaly, Journal of Emprcal Fnance, pp Engle, R.F. (98). Auoregressve condonal heeroskedascy wh esmaes of varance of UK nflaon, Economerca, pp Engle, R. F. (99), Dscusson: Sock marke volaly and he crash of 87, Revew of Fnancal Sudes, 3, 3-6. Engle, R. F. & Ng, V. K. (993). Measurng and esng he Impac of News on Volaly, Journal of Fnance, pp Glosen, L., Jagannahan, R. & Runkle, D. (993). On he relaonshp beween he expeced value and he volaly of he nomnal excess reurn on socks, Journal of Fnance 46, pp

7 Inernaonal Journal of Economcs and Fnance February, Granger, C. & Dng, Z. (99) Some properes of absolue reurn an alernave measure of rsk, Ann Econ Sa 4:67-9. Hseh, D.A. (989) he sascal properes of daly foregn exchange raes: , J. Inernaonal Economcs 4:9-4. Haykn, S. (999). Neural neworks: a comprehensve foundaon, Englewood clffs, Prence Hall. Nelson, D.B. (99). Condonal Heeroskedascy n Asse Reurns: A New Approach, Economerca 9, pp Perez-Cruz, F., Afonso-Rodrguez, J.A. & Gner J. (3). Esmang GARCH models usng suppor vecor machnes, Journal of Quanave Fnance, pp Senana, E. (99), Quadrac ARCH models, Revew of Economc Sudes, 6(4), Suykens, J.A.K, Vandewalle, J. (999). Leas squares suppor vecor machne classfers. Neural Processng Leers (9) Suykens, J.A.K. (). Leas squares suppor vecor machnes for classfcaon and nonlnear modelng, Neural Nework World, Vol., pp Suykens, J.A.K, Vandewalle, J., De Moor, B. (). Opmal conrol by leas squares suppor vecor machnes. Neural Neworks (4) 3-3. Suykens, J.A.K, & De Brabaner, J., Lukas, L., & Vandewalle, J. (). Weghed leas squares suppor vecor machnes: Robusness sparse approxmaon. Neurocompung, 48: -4, 8-. ang, L.B., Sheng, H.Y. & ang, L.X., (8). Forecasng volaly based on wavele suppor vecor machne, Exper Sysems wh Applcaons. ang, L.B., Sheng, H.Y. & ang, L.X. (9). GARCH predcon usng splne wavele suppor vecor machne. Journal of Neural Compung and Applcaon, Sprnger-Verlag London. Van Gesel,.V., Suykens, J.A.K, Baesaens, D.E., Lambrechs, A., Lanckre G., Vandaele B., De Moor B. & Vandewalle, J.(). Fnancal me Seres Predcon usng Leas Squares Suppor Vecor Machnes whn he Evdence Framework, IEEE ransacons on Neural Neworks, () Van Gesel,., Suykens, J.A.K, Baesens, B., Vaene, S., Vanhenen, J., Dedene, G., Moor B.D., & Vandewalle, G. (4). Benchmarkng Leas Squares Suppor Vecor Machne Classfers, Machne Learnng, (4) -3. Vapnk, V.N. (99). he naure of sascal learnng heory, Sprnger-Verlag, New York. Ye, M.Y. & Wang, X.D. (4). Chaoc me seres predcon usng leas squares suppor vecor machne, J. Chnese Physcs, IOP Publshng Ld. IP address: Zakoan, J.M. (994). hreshold Heeroscedasc Models, Journal of Economc Dynamcs and conrol, 8, Acknowledgemen We would lke o hank Dr. Nahale Vlla-Valanex from Insu de Mahémaques de oulouse, oulouse, France for echncal commens on hs research. 7

8 Vol., No. Inernaonal Journal of Economcs and Fnance able. Descrpve sascs of each reurn seres SI In-sample ( s sage) //998 - /9/6 //998 - /9/6 //998 - /9/6 Ou of sample( s sage) /3/7 - /3/7 /3/7 - /3/7 //7 - /8/7 In-sample ( nd sage) /4/999 - /3/7 /4/999 - /3/7 /4/999 - /8/7 Ou of sample( nd sage) //8 - /3/8 //8 - /3/8 //8 - /4/8 oal sample sze Mnmum Maxmum Mean Medan Varance Sdev Skewness Kuross JB a Q () b ARCH-LM c Noe: a JB s he Jarque Bera es for normaly b Q () s he Ljung-Box es for squared reurns c ARCH-LM s he Engle s Lagrange Mulpler es for condonal heeroskedascy wh lags able.a. MLE esmaon of he Paramerc models for Sras mes ndex SI Frs sage Second sage Sascs GARCH EGARCH GJR GARCH EGARCH GJR.66[.]*.79[.]*.44[.]**.7[.]*.[.]**.[.]*.6[.]*.64[.]*.6[.]*.6[.]*.48[.]*.7[.]*.3[.]*.46[.]**.76[.]*.9[.]*.349[.3]*.73[.]*.876[.]*.9[.]*.88[.]*.886[.]*.98[.]*.886[.]* -.67[.]*.8[.]* -.[.]*.7[.]* LL AIC BIC Noe: Values n bracke [ ] ndcaes sandard error of esmaes; LL denoes Log lkelhood values. * sgnfcan a he % level, ** sgnfcan a % level. able.b. MLE esmaon of he Paramerc models Frs sage Second sage Sascs GARCH EGARCH GJR GARCH EGARCH GJR.43[.]*.[.].3[.].[.]*.4[.]*.4[.]*.[.]*.74[.]*.[.]*.8[.]*.6[.]*.9[.]*.4[.]*.3[.]**.9[.]*.[.]*.49[.]*.8[.]*.87[.]*.9[.]*.876[.]*.893[.]*.977[.]*.888[.]* -.3[.]*.6[.]* -.4[.]*.9[.]* LL AIC BIC Noe: Values n bracke [ ] ndcaes sandard error of esmaes; LL denoes Log lkelhood values. * sgnfcan a he % level, ** sgnfcan a % level. 8

9 Inernaonal Journal of Economcs and Fnance February, able.c. MLE esmaon of he Paramerc models Frs sage Second sage Sascs GARCH EGARCH GJR GARCH EGARCH GJR.7[.].8[.].3[.].3[.].9[.].8[.].[.]*.38[.]*.94[.]*.4[.3]*.[.]*.[.3]*.3[.]*.7[.]*.74[.]*.[.]*.4[.]**.67[.]*.8[.]*.964[.]*.83[.]*.769[.]*.936[.]*.78[.]* -.36[.]*.73[.]* -.4[.]*.86[.]* LL e AIC BIC Q () f Noe: Values n bracke [ ] ndcaes sandard error of esmaes; LL denoes Log lkelhood values. * sgnfcan a he % level, ** sgnfcan a % level. able 3.A. ranng resuls by for s sage SI Cos (MSE)* Opmal Gamma** Opmal Sgma** b*** GARCH EGARCH GJR GARCH EGARCH GJR GARCH EGARCH GJR *Cos of esmaon by MSE measure. ** Opmal parameers (Gamma and Sgma) seleced by grdsearch echnque. *** b s he nercep value of he funcon esmaed by. able 3.B. ranng resuls by for s sage SI Cos (MSE)* Opmal Gamma** Opmal Sgma** b*** GARCH EGARCH GJR GARCH EGARCH GJR GARCH EGARCH GJR *Cos of esmaon by MSE measure. ** Opmal parameers (Gamma and Sgma) seleced by grdsearch echnque. *** b s he nercep value of he funcon esmaed by. 9

10 Vol., No. Inernaonal Journal of Economcs and Fnance able 4.A. Forecas performances of ASEAN sock volales by dfferen models for 7 SI MAD NMSE c c R HR GARCH [-.4].3[9.6] GARCH [-.].[9.] EGARCH [-6.].8[.66] EGARCH [-.3].44[3.8] GJR [-.].3[9.94] GJR [-.66].[.44] MAD NMSE c c R HR GARCH [-.3].6[.4].3.76 GARCH [-.].6[.36] EGARCH [-6.34].36[9.3] EGARCH [-.3].34[7.] GJR [-.87].9[.34] GJR [-.37].[3.7] MAD NMSE c c R HR GARCH [-4.6].97[.4] GARCH [-.6].7[3.69] EGARCH [-4.39].33[9.7] EGARCH [-.].4[3.4] GJR [-3.8].9[.73].3.77 GJR [-.].[.7] Noe: hgher R squared and HR s preferred, whle smaller values of MAD and NMSE ndcae he forecased volaly s closer o he acual values. he coeffcens of c and c should be close o (, ) respecvely showng small forecasng errors. able 4.B. Forecas performances of ASEAN sock volales by dfferen models for 8 SI MAD NMSE c c R HR GARCH [-.34].[.] GARCH [-.34].4[6.7] EGARCH [-.7].[8.38] EGARCH [-.8].6[.47] GJR [-.3].4[.6] GJR [-.].43[6.] MAD NMSE c c R HR GARCH [-.4].6[8.] GARCH [-.77].8[8.9] EGARCH [9.89] -.98[-3.69] EGARCH [-.89].64[.6] GJR [8.6] -.[-.3] GJR [-.7].39[9.6] MAD NMSE c c R HR GARCH [-4.66].[3.9] GARCH [-.66].37[3.89] EGARCH [-6.] 3.3[3.4] EGARCH [-3.].84[8.64].87.7 GJR [-4.69].3[4.84] GJR [-.].4[9.9] Noe: hgher R squared and HR s preferred, whle smaller values of MAD and NMSE ndcae he forecased volaly s closer o he acual values. he coeffcens of c and c should be close o (, ) respecvely showng small forecasng errors. 6

11 Inernaonal Journal of Economcs and Fnance February, 4 Prce of Sras mes Index SI Log reurn of Sras mes Index SI Jan 998 Jan Jan Jan 4 Jan 6 Dec 8 - Jan 998 Jan Jan Jan 4 Jan 6 Dec 8 6 Prce of FSE Bursa Malaysa Log reurn of FSE Bursa Malaysa Jan 998 Jan Jan Jan 4 Jan 6 Dec 8 - Jan 998 Jan Jan Jan 4 Jan 6 Dec 8 4 Prce of he PSE Compose Index Log reurn of he PSE Compose Index Jan 998 Jan Jan Jan 4 Jan 6 Dec 8 - Jan 998 Jan Jan Jan 4 Jan 6 Dec 8 Fgure. Plos of Prces and log reurns of each marke ndex Plos of each ndex prce (lef) and log reurn (rgh) for he whole sample. From he lef sdes, we can see ha all ndex prces movemen are almos smlar drecon bu he reurns behave dfferenly. he prce seres of each marke falls down sharply a he las perod due o global fnancal crss. he log reurn plos exhb hgh breaks a 998 (afer ASEAN fnancal crss 997) and n 8 (he recen fnancal marke crashes). 6

12 Vol., No. Inernaonal Journal of Economcs and Fnance 8 6 Volaly forecas 7 GARCH GARCH- 3 3 Volaly forecas 8 GARCH 4 SI SI Volaly forecas 7 EGARCH 3 3 Volaly forecas 8 EGARCH 4 SI SI Volaly forecas 7 GJR 3 3 Volaly forecas 8 GJR 4 SI SI Fgure. Volaly Forecass of Sngapore Sock Marke (SI). Noe: Plos n lef par are referred o he Frs sage forecas n 7 (before crss) and plos n he rgh sde are referred o he second sage forecas for whole 8 (durng fnancal crss). Small do lne s forecased by paramerc models (GARCH, EGARCH and GJR) whle dash lne s obaned by hybrd approaches. 6

13 Inernaonal Journal of Economcs and Fnance February, 8 6 Volaly forecas 7 GARCH 3 Volaly forecas 8 GARCH Volaly forecas 7 EGARCH 3 Volaly forecas 8 EGARCH Volaly forecas 7 GJR 3 Volaly forecas 8 GJR Fgure 3. Volaly Forecass of Kula Lumpur Sock Marke (). Noe: Plos n lef par are referred o he Frs sage forecas n 7 (before crss) and plos n he rgh sde are referred o he second sage forecas for whole 8 (durng fnancal crss). Small do lne s forecased by paramerc models (GARCH, EGARCH and GJR) whle dash lne s obaned by hybrd approaches. 63

14 Vol., No. Inernaonal Journal of Economcs and Fnance 4 4 Volaly forecas 7 GARCH 4 4 Volaly forecas 8 GARCH Volaly forecas 7 EGARCH 4 4 Volaly forecas 8 EGARCH Volaly forecas 7 GJR 4 4 Volaly forecas 8 GJR Fgure 4. Volaly Forecass of he Phlppnes sock marke (). Noe: Plos n lef par are referred o he Frs sage forecas n 7 (before crss) and plos n he rgh sde are referred o he second sage forecas for whole 8 (durng fnancal crss). Small do lne s forecased by paramerc models (GARCH, EGARCH and GJR) whle dash lne s obaned by hybrd approaches. 64