Comparison of Forecasting Volatility in the Czech Republic Stock Market

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1 Appled Economcs and Fnance Vol., No. 1; February 15 ISSN E-ISSN Publsed by Redfame Publsng URL: p://aef.redfame.com Comparson of Forecasng Volaly n e Czec Republc Sock Marke Eleferos I. Talassnos 1, Ergnbay Ugurlu & Yusuf Muraoglu 3 1 Professor, Unversy of Praeus, Deparmen of Marme Sudes, Greece Asssan Professor, H Unversy FEAS, Deparmen of Economcs, Turkey 3 Researc Asssan, Gaz Unversy FEAS, Deparmen of Economcs, Turkey Correspondence: Eleferos I. Talassnos, Unversy of Praeus, Deparmen of Marme Sudes, Greece. Receved: Sepember, 14 Acceped: Ocober 13, 14 Avalable onlne: December 11, 14 do: /aef.v1.68 URL: p://dx.do.org/ /aef.v1.68 Absrac Te am of s paper s o examne dfferen GARCH models w ree dfferen dsrbuons n order o compare er forecasng power n erms of volaly exsng n e reurns of e Czec Sock Marke and more specfc n e PX ndex, for e perod We ave employed GARCH, GJR-GARCH and EGARCH models agans normal, suden- and generalzed error dsrbuons. Ten, we ave forecased sock marke volaly for e Czec Republc by s reurns usng e same models, GARCH, GJR-GARCH and EGARCH comparng er forecasng performance. Te resuls sow a reurn volaly can be caracerzed by sgnfcan perssence and asymmerc effecs. We ave esmaed e correspondng varances for all models for e full sample perod usng sac forecass. Afer comparng e forecasng performance of all nne models was found a e EGARCH model as e bes forecasng performance compared o oers. Keywords: GARCH models, sock marke volaly, forecasng performance JEL classfcaon: G15, G17 1. Inroducon Mos fnancal daa ave some feaures wc are lepokuross, volaly cluserng or poolng and leverage effecs. Lnear srucural models and me seres models are unable o explan some of e above mporan feaures. Lepokuross, volaly cluserng or poolng and leverage effecs are endences for fnancal asse reurns. Te endences of ese feaures are defned as: a) o ave dsrbuons a exb fa als and excess peakedness a e mean; b) o ave volaly n fnancal markes o appear n bunces wc means large reurns, of eer sgn and/or small reurns, of eer sgn; c) o ave volaly o rse more followng a large prce fall an followng a prce rse of e same magnude (Brooks, 8). Te mos popular non-lnear fnancal models are e Auoregressve Condonal Heeroscedasc (ARCH) or Generalzed Auoregressve Condonal Heeroscedasc (GARCH) wc ave been used for modelng and forecasng volaly proposed by Engle (198) and Bollerslev (1986). Te am of s paper s o forecas sock marke volaly for Czec Republc by s reurns usng GARCH, GJR-GARCH and EGARCH models and compare er forecasng performance. Te res of e paper s organzed as follows. Te second secon s a leraure revew of relaed sudes usng GARCH models and sock marke volaly. Te rd secon gves bref nformaon abou ARCH/GARCH models wle e esmaon resuls are presened n e four secon. Te ff secon summarzes and concludes e paper.. Leraure Revew Andersen, Bollerslev and Lange (1999) saed a e expeced fuure volaly of fnancal marke reurns s e man ngreden n assessng asse or porfolo rsk and plays a key role n dervave prcng models. Emerson, Hall and Zelweska-Mura (1997) ave nvesgaed e Bulgaran sock marke wle Scecer (1999) ave examned Pols sock reurns. Selds (1997) modeled e reurns of e Warsaw and Budapes sock excanges reurns. Scecer (1) ave caracerzed Hungary, Poland, and Czec markes as prncpal emergng sock markes n Europe. Te auor as esmaed a VEC model for eac marke modelng s volaly w a mulvarae GARCH (M-GARCH) srucure. Te fndngs sow a e nvesgaed counres ave lmed neracon and er volaly as a regonal 11

2 Appled Economcs and Fnance Vol., No. 1; 15 caracer. Vošvrda and Žıkeš (4) ave suded e beavor of volaly and e dsrbuonal properes of e Czec, Hungaran, and Pols sock markes usng weekly daa for e perod Tey ave used e PX-5 ndex for Czec Republc and ave found sascally sgnfcan resuls for GARCH (1,1) model concludng a e volaly of e reurns on PX-5 s very perssen. Syropoulos (7) as nvesgaed e relaonsps beween e emergng sock markes, wc are Czec Republc, Hungary, Poland, Slovaka and wo developed sock markes, Germany and e USA over e perod Harouounan and Prce (1) ave nvesgaed Czec Republc, Hungary, Poland, and Slovaka usng bo unvarae and mulvarae GARCH models, namely GARCH, NGARCH, EGARCH, GJR-GARCH, AGARCH, NAGARCH and VGARCH. Te researc as wo pars wc are e unvarae models and e mulvarae models. Relyng on e resuls of e unvarae models, ey ave concluded a srong GARCH effecs are apparen for all four markes w e excepon of e Czec Republc, were e coeffcen of e lagged squared reurns s no sgnfcan n ree ou of seven specfcaons of condonal volaly. Hajek (7) as examned e Effcen Marke Hypoess on e Czec capal marke for e perod usng monly, weekly and daly daa. In s researc, e auor as analyzed effcency and lnear dependency of several ndces closng values and sock closng prces on e Prague Sock Excange. He as concluded a bo daly sock reurns and daly ndex reurns are sgnfcanly lnearly dependen, so e eeroskedascy-conssen meodology mus be erefore appled o avod sgnfcan bases. Rocknger and Urga () ave nvesgaed wo groups of counres, wc are e ranson economes and e esablsed economes. Transon economes are Czec Republc, Poland, Hungary and Russa wle e esablsed economes are USA, Germany and UK. Aloug ey ave focused on a sample of Cenral and Easern European Fnancal Markes (CEEFM) 1 ey ave only used ese four counres. Te model resuls are very smlar for e Czec Republc, Hungary and Poland wle for all counres nvesgaed sgnfcan GARCH effecs ave been appeared. Talassnos and Pols (11) ave analyzed nernaonal sock markes for a number of counres usng conegraon analyss. Tey ave found nerrelaonsps beween e seleced varables a resul a s conssen w oer smlar sudes. Talassnos, Madnos and Pascalds (1) ave observed sgnfcan evdence regardng nsder radng n e Aens Sock Excange n e perod Meodology We ave used four dfferen ypes of GARCH models, ARCH, GARCH, GJR-GARCH and EGARCH srucures as follows: 3.1 ARCH Model ARCH model s based on e varance of e error erm a me and depends on e realzed values of e squared error erms n lagged perods. Te model s specfed as n equaons (1) o (3): y u (1) u ~ N, () q u (3) Ts model s referred as ARCH(q), were q refers o e order of e lagged squared reurns ncluded n e model. If we use ARCH(1) model equaon (3) becomes: Snce 1 1u (4) 1 s a condonal varance, s value mus always be srcly posve; a negave varance a any pon n me would be meanngless. To ave posve condonal varance esmaes, all of e coeffcens n e condonal varance are usually requred o be non-negave. Tus coeffcens mus sasfy e condon GARCH Model Bollerslev (1987) and Taylor (1986) developed e GARCH(p, q) model. Te model allows e condonal varance of j 1 Czec Republc, Poland, Hungary, Russa, Bulgara, Slovena, Romana, Croaa and Esona 1

3 Appled Economcs and Fnance Vol., No. 1; 15 e varable o be dependen upon prevous lags; frs lag of e squared resdual from e mean equaon presens news abou e volaly from e prevous perod wc s as follows: q 1 u In e leraure e mos used and e smples model s e GARCH(1, 1) process, for wc e condonal varance can be wren as n equaon (6): 1 1 p 1 u (6) 1 1 Under e ypoess of covarance saonary, e uncondonal varance uncondonal expecaon of equaon (6). Tus: Solvng equaon (6) we ave: 1 1 (5) can be found by akng e (7) For e exsence of s uncondonal varance s requred o ave 1 1 1and n order o be posve mus be. 3.3 GJR GARCH Model (8) Te GJR model s a smple exenson of GARCH w an addonal erm added o accoun for possble asymmeres (Brooks, 8). Glosen, Jagannaan and Runkle (1993) developed e GARCH model wc allows e condonal varance o ave a dfferen response o pas negave and posve nnovaons as sown n equaon (9): Were q 1 1 p u u d (9) s a dummy varable w values for bad and good news: d 1 1 f u f u 1 1 1, bad news j j, good news In s model, e effec of good news sows er mpac by, wle bad news sows er mpac by. In addon f news mpac s asymmerc and leverage effec exss. To sasfy non-negavy condon coeffcens would be,, and. Ta s e model s sll accepable, even f, provded a (Brooks, 8). 3.4 Exponenal GARCH MODEL Exponenal GARCH (EGARCH) proposed by Nelson (1991), as formed leverage effecs n e equaon. In EGARCH model e specfcaon for e condonal covarance s gven by e followng equaon (1): log q j log j j1 p 1 log even f e parameers are negave e equaon wll be posve; b) asymmeres are allowed under e EGARCH formulaon. In equaon (1), k represens leverage effec, wc accouns for e asymmery of e model. Wle e basc GARCH model requres ese resrcons, e EGARCH model allows unresrced esmaon of e varance (Tomas and Mcell, 5). If k, s s an ndcaon a leverage effec exs, were k ndcaes an asymmerc mpac n e model. Te meanng of e exsence of leverage effec because of bad news ncreases e volaly. Te wo advanages saed n Brooks (8) for e pure GARCH specfcaon are: a) by usng Wen applyng GARCH models o reurn seres, s ofen found a GARCH resduals sll end o be eavy aled. To accommodae s, raer an o use normal dsrbuon e Suden s and GED dsrbuon ave o be used o employ ARCH/GARCH ype models (Mnk, Paolella and Racev,, Nelson, 1991). u r k1 k u k k (1) 13

4 1/1/ 1/1/3 1/1/4 1/3/5 1//6 1/1/7 1/1/8 1/1/9 1/1/1 1/3/11 1//1 1/1/ 1/1/3 1/1/4 1/3/5 1//6 1/1/7 1/1/8 1/1/9 1/1/1 1/3/11 1//1 Appled Economcs and Fnance Vol., No. 1; Emprcal Applcaon We ave used daly daa n sock excanges of Czec Republc PX for e perod Sascal daa ave been colleced from Reuers. We ave used e reurn erm wc s defned n Yu () as follows: x r log (11) x 1 Were s e capal ndex. Grap 1 sows PX and s reurns wc s defned as RPX., 1,6 1, PX RPX Grap 1. Daly Prces of PX and RPX Source: Reuers Table 1 summarzes e descrpve sascs of PX and RPX, wc ave negave skewness and g posve kuross. Tese values sgnfy a e dsrbuons of e seres ave a long lef al and lepokurc form. Jarque-Bera (JB) sascs rejec e null ypoess of normal dsrbuon a 1% level of sgnfcance. Table 1. Descrpve Sascs RPX Mean.1 Medan.71 Maxmum Mnmum Sd. Dev Skewness Kuross Jarque-Bera Probably. Sum.694 Sum Sq. Dev..696 Observaons 899 See: p:// for descrpon of PX 14

5 Appled Economcs and Fnance Vol., No. 1; 15 Table. Sows Augmened Dckey-Fuller (ADF) Tes Resuls And Concludes Ta RPX Is Saonary. Wou Trend W Trend Varable ADF sa p ADF sa p RPX *** ***. Noe: *** denoes sgnfcan a 1% level. We ave also esed e mean model for an ARCH effec w e ARCH-LM Tes. Table 3 sows ARCH-LM es resuls. If e value of e es sasc s greaer an e crcal value from e dsrbuon, e null ypoess s rejeced ndcang a ere s no ARCH effec n e model. Table 3. ARCH (1) LM Tes Resuls Dependen Varable of Model ARCH(1)LM Sa P RPX 49,797***. Noe: *** denoes sgnfcan a 1% level. We ave employed ARCH, GARCH, GJR-GARCH and EGARCH processes usng bo Suden- and Generalzed Error dsrbuons n addon o Normal dsrbuon. Resuls sow a srong GARCH and GJR-GARCH effecs are apparen for reurns (Table 5, Table 6 and Table 7). Te sum of e coeffcens α and β are less an 1 for all dfferen dsrbuons usng GARCH and GJR-GARCH models. Nevereless, e esmaon of α s smaller an e esmaon of β n bo cases, wc reveals a negave socks aven a larger effec on condonal volaly an e posve socks of e same magnude. In GJR-GARCH model, wc means a e news mpac s asymmerc or n oer words, bad news ncrease volaly. In e E-GARCH model negave and sgnfcan leverage effec parameer ndcaes e exsence of e leverage effec n reurns. In EGARCH model, f GED parameer(r) value s reveals a normal dsrbuon or a lepokurc dsrbuon f s less an. In all models r s less an and sascally sgnfcan, wc ndcaes a RPX s lepokurc. Ts resul s conssen w e skewness values repored n Table 1. Afer all, ARCH effec s esed n all models. Te null ypoess a ere s no ARCH effec canno be rejeced n all models as sown n Table 5, 6 and 7. In e case of GJR-GARCH model w normal dsrbuon, e null ypoess s rejeced a 1% level of sgnfcance. Te n-sample evdence provdes e sory performance of e models. We ave esmaed varance for all models for full sample perod usng sac forecas. Ten, we compared e forecasng performance of models wc ave been used n s researc. We ave consdered four sascs for evaluang e forecasng accuracy, wc are employed n e researc of Wang and Wu (1). Tese four measures are e Mean Square Error (MSE) and e Mean Absolue Error (MAE) sown n equaons (1) o (15): n 1 MSE 1 n ˆ (1) 1 MSE n 1 1 n ˆ 1 (13) n 1 1 n ˆ 1 MAE (14) n 1 n ˆ 1 MAE (15) Were: n s e number of forecass s e acual volaly 15

6 Appled Economcs and Fnance Vol., No. 1; 15 ˆ s e volaly forecas a day. Table 4 sows e forecasng performance of GARCH, GJR-GARCH and EGARCH models. Generally, we ave found a EGARCH model as e greaes forecasng accuracy accordng o MSE, MAE1 and MAE, w only MSE1 sowng beer performance for GJR-GARCH. Table 4. Comparson Forecasng Performance of GARCH Models Normal Dsrbuon Suden Dsrbuon GED GARCH GJR-GARCH EGARCH GARCH GJR-GARCH EGARCH GARCH GJR-GARCH EGARCH MSE1 6.3E-7 6.5E-7 6.5E E-7 6.8E-7 6.9E E-7 6.6E-7 6.7E-7 MSE MAE MAE Noe: Te values n bold face refer o smalles loss. 4. Concluson Te arcle as examned ree GARCH models namely GARCH, GJR-GARCH and EGARCH usng ree dfferen dsrbuons, e Normal dsrbuon, Suden- and Generalzed Error dsrbuon, n order o compare er forecasng power for e volaly of e reurns of e PX ndex for e Czec Republc sock marke. Te resuls ave sown a sgnfcan ARCH and GARCH effecs are presen n e daa, wc ndcaes a e volaly n PX ndex reurns are caracerzed by sgnfcan perssence and asymmerc effecs. Tese resuls are conssen w e resuls of Rocknger and Urga (), Harouounan and Prce (1) and Vošvrda and Žkeš (4) wc ave used smlar GARCH ype models for e Czec sock marke. Fnally, we ave compared e n sample forecasng performance of e nne models for e perod Te evdence sows a EGARCH model as e bes forecasng performance based on real sascal daa. References Andersen, T. G., Bollerslev, T., & Lange, S. (1999). Forecasng Fnancal Marke Volaly: Sample Frequency vs-a-vs Forecas Horzon, Journal of Emprcal Fnance, 6(5), Bollerslev, T. (1986). Generalzed Auoregressve Condonal Heeroskedascy, Journal of Economercs, 31(3), Bollerslev, T. (1987). A Condonal Heeroskedasc Tme Seres Model for Speculave Prces and Raes of Reurn, Revew of Economcs and Sascs, 69(3), Brooks, C. (8). Inroducory Economercs For Fnance: Second Edon, Cambrdge Unversy Pres. Emerson, R., Hall, S., & Zelweska-Mura, A. (1997). Evolvng Marke Effcency w an Applcaon o Some Bulgaran Sares, Economcs of Plannng, 3(-3), Engle, R. F. (198). Auoregressve Condonal Heeroskedascy W Esmaes of e Varance of UK İnflaon, Economerca, 5(4), Glosen, L. R., Jagannaan, R., & Runkle, D. (1993). On e Relaon Beween e Expeced Values and e Volaly of e Nomnal Excess Reurn on Socks, Journal of Fnance, 48(5), Hájek, J. (7). Czec Capal Marke Weak-Form Effcency, Seleced Issues, Prague Economc Papers, 4, Harouounan, M. K., & Prce, S. (1). Volaly İn Te Transon Markes of Cenral Europe, Appled Fnancal Economcs, 11, Mnk, S., Paolella, M. S., & Racev, S. T. (). Saonary of Sable Power-GARCH Processes, Journal of Economercs, 16, Nelson, D. B. (1991). Condonal Heeroscedascy n Asse Reurns: A New Approac, Economerca, 59(), Rocknger, M., Urga, G. (). Tme Varyng Parameers Model o Tes for Predcably and Inegraon n Sock Markes of Transon Economes, CERP Dscusson Papers

7 Appled Economcs and Fnance Vol., No. 1; 15 Scecer, M. (1999). Modelng Pols Sock Reurns. n Helmensen, C. (ed.), Capal Markes n Transon Economes. Celenam, Edward Edgar, UK, Scecer, M. (1). Te Comovemens of Sock Markes n Hungary, Poland and e Czec Republc. Inernaıonal Journal of Fnance and Economcs, 6, Selds, K. (1997). Sock Reurn Volaly on Emergng Easern European Markes. Te Manceser Scool Supplemen, Syropoulos, T. (7). Dynamc Lnkages Beween Emergng European And Developed Sock Markes: Has Te Emu Any Impac, Inernaonal Revew of Fnancal Analyss 16, Taylor, S. J. (1986). Modellng Fnancal Tme Seres, Jon Wley & Sons, Cceser. Talassnos, I. E., & Pols, D. E., (11). Inernaonal Sock Markes: A Conegraon Analyss, European Researc Sudes Journal, XII (), Talassnos, I. E., Madnos, D., & Pascalds, A. (1). Observng Evdence of Insder Tradng n e Aens Sock Excange (ASE), Journal of Economc Srucures, 1(8), Dec. Tomas, S. & Mcell, H. (5). GARCH Modelng of Hg-Frequency Volaly n Ausrala s Naonal Elecrcy Marke, Dscusson Paper, Melbourne Cenre for Fnancal Sudes. Vošvrda, M., & Žıkeš, F. (4). An Applcaon of e GARCH- Model on Cenral European Sock Reurns, Prague Economc Papers, 1, Wang, Y., & Wu, C. (1). Forecasng Energy Marke Volaly Usng GARCH Models: Can Mulvarae Models Bea Unvarae Models, Energy Economcs. Yu, J. (). Forecasng Volaly n e New Zeal& Sock Marke, Appled Fnancal Economcs, 1, Appendx Table 5. Normal Dsrbuon Mean equaon GARCH(1,1) GJR-GARCH E GARCH Value p Value p Value p Varaon equaon 4.69E E AIC SIC DW-sa ARCH-LM (1) Tes p-value of C Sq Obs

8 Appled Economcs and Fnance Vol., No. 1; 15 Table 6. Suden s T Dsrbuon Mean equaon GARCH(1,1) GJR-GARCH E GARCH Value p Value p Value p Varaon equaon 4.34E E AIC SIC DW-sa ARCH-LM(1) Tes p-value of C Sq. Table 7. Generalzed Error Dsrbuon(GED) Obs. 899 Mean equaon GARCH(1,1) GJR-GARCH E GARCH Value p Value p Value p Varaon equaon 4.56E E r AIC SIC DW-sa ARCH-LM (1) Tes p-value of C Sq Obs. 899 Ts work s lcensed under a Creave Commons Arbuon 3. Lcense. 18