Forecasting Volatility Evidence from Indian Stock and Forex Markets

Transcription

1 IIMK/WPS/08/FIN/006/06 Forecasng Volaly Evdence from Indan Sock and Forex Markes Dr. S S S Kumar Assocae Professor, Indan Insue of Managemen Kozhkode, Kozhkode (emal: ssskumar@mk.ac.n)

2 Forecasng Volaly Evdence from Indan Sock and Forex Markes Volaly forecasng s an mporan area of research n fnancal markes and lo of effor has been expended n mprovng volaly models snce beer forecass ranslae n o beer prcng of opons and beer rsk managemen. In hs drecon hs paper aemps o evaluae he ably of en dfferen sascal and economerc volaly forecasng models n he conex of Indan sock and forex markes. These compeng models are evaluaed on he bass of wo caegores of evaluaon measures symmerc and asymmerc error sascs. Based on an ou of he sample forecass and a majory of evaluaon measures we fnd ha GARCH (4, ) and EWMA mehods wll lead o beer volaly forecass n he Indan sock marke and he GARCH (5, ) wll acheve he same n he forex marke. The same models perform beer on he bass of asymmerc error sascs also. Volaly s he varably of he asse prce changes over a parcular perod of me and s very hard o predc correcly and conssenly. In fnancal markes volaly presens a srange paradox o he marke parcpans, academcans and polcy makers whou volaly superor reurns are can no be earned, snce a rsk free secury offers meager reurns, on he oher hand f s hgh wll lead o losses for he marke parcpans and represen coss o he over all economy. Therefore here s no gansayng wh he saemen ha volaly esmaon s an essenal par n mos fnance decsons be asse allocaon, dervave prcng or rsk managemen. However he queson as o wha model should be used o calculae volaly, here s no unque answer as dfferen volaly models were proposed n he leraure and were beng used by praconers and hese varyng models lead o dfferen volaly esmaes. In he pas wo decades hs has been a ferle area for research n fnancal economcs for boh academcans as well as praconers. Unforunaely mos of he work was done n he conex of developed markes n he conex of sock and forex markes. Ths paper s an aemp o examne he effcacy of he compeng volaly forecasng models n he Indan marke. The works of Mandelbro (963) and Fama (965) were he frs few works ha examned he sascal properes of sock reurns; n he same srand Akgray s (989) See Poon and Granger (003)

3 work proceeds furher whch no only nvesgaes he sascal properes bu also presens evdence on he forecasng ably of ARCH and GARCH models vs-à-vs EWMA (exponenally weghed movng average) and he Hsorc smple average mehod. Whle Pagan and Schwer (990) repor ha GARCH and EGARCH models enhanced wh erms suggesed by nonparamerc mehods yelds sgnfcan ncreases n explanaory power. In he same year Dmson and Marsh (990) came up wh raher neresng fndng ha smple models perform beer han he exponenal smoohng or regresson based mehods. Of course has o be noed ha her sudy does no nclude he popular ARCH famly of models n conras o hs Tse (99), Tse and Tung (99) fnd ha EWMA models provde beer forecass han he GARCH models. These sudes were conduced n dfferen markes he former was carred n U K sock marke whle he laer examned n Japanese and Sngapore markes respecvely. Franses and van Djk (996) examned he forecasng ably of he GARCH famly of models agans random walk model n fve European sock markes and found ha random walk model fares beer even when he perod of 987 crash was ncluded. Bralsford and Faff (996) nvesgaed he forecasng models n he Ausralan marke and found ha ARCH class of models and smple regresson provde beer forecass bu he rankngs were sensve o he error sasc used o assess he accuracy of he forecas. In he conex of foregn exchange markes Wes and Cho (995) fnd evdence n favour of GARCH model over shorer nervals and n he longer horzon no model fare beer. Some of he more recen works were by Loudon e al (000), Mcmllan e al (000), Yu (00), Klaassen (0), Vlasuso (00) and Balaban (004). In he Indan conex Varma (999) nvesgaed he volaly esmaon models comparng GARCH and he EWMA models n he rsk managemen seng. Pandey (00) explored he exreme value esmaors and found ha hey perform beer han he radonal close o close esmaors alhough hs sudy does no consder he performance of exreme value esmaors versus me varyng volaly models. Kaur (004) examned he naure and characerscs of sock marke volaly n Inda. From he leraure revew he followng pons emerge:. There s no conclusve evdence as o he supremacy of any volaly forecasng model n he leraure on developed markes 3

4 . In he Indan conex research on hs mporan opc s fragmened as here s no work ha compares he ably of all he mporan compeng models. Though o some exen he work by Varma (999) s an mporan conrbuon, he work consders only he sock marke and ha oo wh a dfferen objecve. 3. In he conex of foregn exchange marke hs opc s no addressed a all 4. Evdence n he form of ou of sample forecasng and how he smple models fare agans sophscaed models s sll unanswered n Indan leraure. In hs conex he presen work ses ou o nvesgae he relave ably of varous forecasng models rangng from naïve models o relavely advanced models n boh sock and forex markes of Inda. Daa descrpon In hs sudy we consdered he Nfy ndex as a proxy for he sock marke and accordngly he closng ndex values were colleced from Jun ll Dec The exchange rae daa was peranng o he Indan rupee/us dollar exchange rae over he perod Jan ll Dec and he same was colleced from he Pacfc Exchange Rae Servce (hp://fx.sauder.ubc.ca/). Ou of he oal observaons he daa peranng o July 990 ll Dec 000 oalng 6 monhly observaons of NIFTY were used for esmaon of he model parameers and he remanng observaons wll be used for ou of sample forecasng also known as hold ou sample. In he case of foregn exchange marke he daa peranng o Jan 994 ll Dec 000 oalng 85 monhly observaons were used for esmaon of he model parameers and he remanng observaons wll be used for ou of sample forecasng. Therefore he frs monh for whch ou of sample forecass are obaned s Jan 00 and he ou of sample forecass were consruced for 60 monhs ll Dec 005. The daly observaons were convered no connuous compounded reurns n he sandard mehod as he log dfferences: r = ln where I sands for he closng ndex value/exchange rae on day ; followng Meron (980) he monhly volaly s obaned as he sum of he squared daly reurns n ha monh whch s shown below: I I 4

5 = N r = Where r s he daly reurn on day and N s he number of radng days n he monh under queson. The descrpve sascs of he daa were presened n Table and fgures and plos he reurn seres of Nfy and exchange rae respecvely. The mean daly reurn for nfy was 0.063% whle for exchange rae was 0.09% and he annualzed volaly for nfy s around 7.88% and for he exchange rae was 4.78%, boh he seres exhb excess kuross ndcang ha he uncondonal reurn dsrbuons are no normally. The Jarque-Bera (JB) sasc confrms ha normaly s rejeced a a p-value of almos. From fgures 3 and 4 we can noe ha he reurns exhb fa als whch s more promnen for he exchange rae seres. The plo of reurn seres n Fgures and shows ha here s perssence and volaly cluserng s a feaure of boh he markes whch suggess ha he volaly s predcable. The Ljung- Box Q sascs for he reurn and squared reurn seres show ha he null hypohess of no seral correlaon canno be rejeced a 36 h lag for boh seres. To es for possble un roos he augmened Dckey-Fuller (ADF) sasc s calculaed and he resuls are presened n he las row of Table. The null hypohess of un roo can be rejeced n boh he cases a % level of sgnfcance. See Table & Fgures,, 3 and 4 The absence of un roo means he seres s saonary, combned wh he phenomenon of volaly cluserng mples ha volaly can be predced and he forecasng ably of he dfferen models can be generalzed o oher me perods also. Compeng Models: Takng cues from he earler sudes n he nernaonal conex menoned n he leraure revew hs work examnes he forecasng capables of he followng models: 5

6 . Random walk 5. Exponenal weghed movng. Hsorcal mean average 3. Movng average 6. Smple and hgher order GARCH 4. Smple regresson models The models ha were consdered n hs parcular sudy are no exhausve bu cover a very large varey of models rangng from naïve models o he advanced models lke GARCH and he preferred model by praconers vz.,. We red o nclude hose models whose effcacy s no examned prorly n he Indan conex and s also guded by he researcher s assessmen of he models used by praconers lke exponenal smoohng (RskMercs s beng used by more han 65 nsuons world wde Source: Web se of Rsk mercs). In he followng paragraphs we aemp o gve a bref descrpon of all he canddae models: Random Walk Model As per hs model, he bes forecas for hs perod s volaly s he las perod s realzed volaly = where = for Nfy and = for he exchange rae seres. Hsorcal Mean Model Assumng he condonal expecaon of he volaly consan, hs model forecass volaly as he hsorcal average of he pas observed volales = = where = for Nfy and = for he exchange rae seres. Movng Average Model In he hsorc mean model he forecas s based on all he avalable observaons and each observaon wheher s very old or mmedae s gven equal wegh hs may lead o sale prces effecng he forecass. Ths s adjused n a movng averages mehod whch s a radonal me seres echnque n whch he volaly s defned as he equally weghed average of realzed volales n he pas m monhs. 6

7 m = m = The choce of m s raher arbrary and n hs paper we nvesgae fve models 3, 6,, 4 and 60 monhs. Smple Regresson In hs mehod he famlar regresson of acual volales on lagged values s run, n oher words s he frs auoregresson s performed on he frs par of daa whch s mean for esmang he parameers and he esmaes hus obaned were used for forecasng he volaly for he nex monh. Accordngly he frs par nvolves runnng he followng regresson: = α + β α and β are esmaed over he year perod from July 990 ll Dec 000 for Nfy and for he exchange rae was done for 7 year perod from Jan 994 o Dec 000. Now for he nex forecas he volaly for Feb 00 he parameers α and β are reesmaed by omng he mos dsan pas observaon.e., July 990 and ncludng he Jan 00 acual volaly observaon. Ths process s repeaed and hus he esmaon wndow moves forward, he same process s carred ou for exchange rae seres also. By followng hs mehodology we acually ulze he me-varyng parameers for each monh. Exponenal Weghed Movng Average 3 Exponenal smoohng s an adapve forecasng mehod ha gves greaer wegh o more recen observaons so ha he fne memory of he marke s represened. Ths mehod adjuss he forecass based on pas forecas errors and he forecas s calculaed as a weghed average of he mmedae pas observed volaly and he forecased value for ha same perod.e., ˆ = α + ( α) here α s known as smoohng facor and s consraned o 0 < α <. The smoohng facor deermnes he wegh ha s gven o acual volaly observed n he mmedae pas 3 Srcly speakng hs mehod s exponenal smoohng however some praconers and n parcular RskMercs call as exponenal weghed movng average hence o avod confuson we are also callng hs approach by he same name 7

8 monh and as α means more recen observaons ge more wegh and α can be chosen based on he analys s nuve judgemen or can be objecvely deermned so as o produce he bes f by mnmzng he sum of he squared devaon beween acual and forecased volales n he esmaon perod.e., usng nsample daa. Pas sudes vz., Dmson and Marsh (990) esmae opmal α for each year bu n hs sudy we reesmae on a monhly bass. ARCH and GARCH ARCH sands for auoregressve condonally heeroskedascy and hese models are a sophscaed group of me seres models nally nroduced by Engle (98) and ARCH models capure he volaly cluserng phenomenon usually observed n fnancal me seres daa. In he lnear ARCH (q) model he me varyng condonal varance s posulaed o be a lnear funcon of he pas q squared nnovaons. In oher words varance s modeled as a consan plus a dsrbued lag on he squared resdual erms from earler perods r = γ + ε and = ϖ + α ε q = Where ε ~ d N(0,) For sably Σ α I <.0 and heorecally q may assume any number bu generally s deermned based on some nformaon crera lke AIC or BIC. In fnancal markes he ARCH() model s mos ofenly used and hs s a very smple model ha exhbs consan uncondonal varance bu non-consan condonal varance. Accordngly he condonal varance s modeled as = α 0 + α ε As wh smple regresson he parameers n ARCH and GARCH models (dscussed nex) are esmaed a monhly nervals usng a rollng wndow of monhly year wndow. The problem wh he ARCH models s nvolves esmaon of a large number of parameers and f some of he parameers become negave hey lead o dffcules n forecasng. Bollerslev (986) proposed a Generalzed ARCH or GARCH (p, q) model where volaly a me depends on he observed daa a -, -, q as well as on volales a -, -, p. The advanage of GARCH formulaon s ha hough recen nnovaons ener he model nvolves only esmaon of a few parameers 8

9 hence here wll be lle chance ha hey wll ll-behaved. In GARCH here wll be wo equaons condonal mean equaon gven below r = γ + and he condonal varance equaon shown below, = ω + q = α ε ε + p = β he parameers n boh he equaons are esmaed smulaneously usng maxmum lkelhood mehods once a dsrbuon for he nnovaons ε has been specfed generally s assumed ha hey are Gaussan. The smples and mos commonly used member of he GARCH famly s he GARCH (, ) model shown below + = ω + α ε β GARCH forecas for he nex day s compued gven as + = ω + α ε + β If he forecas s requred for more han one day for nsance n day forecas s gven as ˆ n n ˆ ( ˆ ˆ) ( ˆ ˆ) ˆ + n = ω α + β + α + β = 0 Followng Schwarz Informaon Crera and Akake Informaon Crera 4 we found ha he bes model n he GARCH (p, q) class for p [, 5] and q [, ] was a GARCH (4, ) n he sock marke and GARCH (5, ) n he forex marke. We also esed for wheher he GARCH (4,) adequaely capured all he perssence n he varance of reurns by usng Ljung-Box Q- sasc a he 36 h lag of he sandardzed squared resduals was (p = 0.357) ndcang ha he resduals are no serally correlaed. However n hs sudy we evaluaed he performance of he GARCH (, ) model also ha s ofen used n fnancal economcs leraure. In our forecasng exercse frs we esmaed he GARCH parameers usng he esmaon perod.e., Jul990 o Dec 000 for Nfy and Jan 994 o Dec 000 for exchange rae seres and hen used hese parameers o oban he forecass for he radng days n Jan 00 and hese daly forecass were aggregaed o oban he forecas for he monh of January 00. Then he begnnng and end observaons for parameer 4 For conservng space and o manan he flow he values are no presened and are avalable up on reques 9

10 esmaon were adjused by ncludng he daa for Jan 00 and omng he daa peranng o Jul 990. The procedure s repeaed for every monh usng a rollng wndow of years for Nfy daa and 7years rollng wndow for he exchange rae daa. Emprcal Resuls We compare he forecas performance of each model usng he followng error sascs used n pas sudes vz. Yu(00) and Bralsford and Faff (996) Mean absolue error (MAE), Roo Mean Square Error (RMSE), Thel s U (TU) and MAPE. These are defned as follows: MAE = n ˆ n = RMSE = Thel U = n n = n = n = ( ˆ ) ( ˆ ) ( ˆ ) MAPE = ( ˆ ) n / n = In all he above sascs n sand for number of ou of sample forecass whch s equal o 60 as he same was done over a fve year perod. Thel's U s a sasc ha uses he random walk as a benchmark for comparng he qualy of forecas models behnd hs noon s he belef ha f a forecasng model canno do beer han a naïve forecas, hen he model s no dong an adequae job. These sascs are generally ermed as symmerc forecas error sascs as hey penalze boh over forecas and under forecas equally. Snce over forecas and under forecas may lead o dfferen prof/cos consequences o buyers and sellers dfferenly hey have o be reaed dfferenly.e., a dfferenal weghng s requred snce under forecas of volaly may be desrable for an opon buyer snce he resulng prce wll be less bu wll be undesrable for he seller as he buyer s gan wll be he seller s loss, followng Bralsford and Faff (996) 0

11 he below menoned asymmerc error sascs mean mxed error sascs MME(U) and MME(O) were consruced MME(U) = 60 O = ˆ + U = ˆ O U MME(O) = ˆ + ˆ 60 = = Where O s he number of over predcons and U s he number of under predcons. MME(U) penalze more he under predcons and MME(O) penalzes more he over predcons. Resuls and Dscussons Table presens he resuls of he error sascs explaned n he earler secon. In hs able we presen he acual sasc along wh he relave rankng of ha parcular mehod among he compeng models from he resuls we can make he followng observaons. Frsly, based on Thel s-u and MAE he GARCH models ouperform oher models n boh he markes vz., sock and forex markes. Whle on he bass of MAPE here s unanmy of he superory of EWMA mehod n boh he markes. On he bass of RMSE, EWMA mehod fares well n he sock marke and n he forex marke agan GARCH (5, ) model ranks he bes. In he sock marke EWMA s found o perform beer on he bass of wo measures RMSE and MAPE whle on he bass of MAE hough s ranked second he dfference n forecas error as per GARCH (4, ) and EWMA s very nomnal. On he oher hand he GARCH models perform clearly ahead of EWMA n he forex marke on he bass of hree measures. Second, all he measures ndcae hsorcal mean model as he wors performng model n he forex marke and n he sock marke hsorcal mean model s ranked worse by wo measures MAE and MAPE whle random walk model s caegorzed as he worse by RMSE and Thel s U. I s cusomary for Thel s U o compare he performance of compeng models agans he smples of he forecas mehods ermed he "naïve" model - he random walk - whch usually consss of a forecas repeang he mos recen value of he varable bu n forex marke hree oher models hsorc mean model, 5-yr movng average, and smple regresson were found o produce worse resuls han he random walk model. The

12 performance of he popular smple regresson mehod s raher no encouragng n eher of he markes - n forex marke was ranked as nnh by wo measures and an equal number ranked as eghh. Thrd, n he sock marke he forecas accuracy ncreases on an average of 70% by usng he GARCH models vs-à-vs he worse performng models and n he forex marke hs mprovemen s o he exen of 80%. In oher words f we consder he forecas error of las ranked model on he bass of one of he measures (for nsance) RMSE as he base hen by usng he op ranked model EWMA he forecas error reduces by around 49% whch s a sgnfcan reducon n forecas error. Fourh, he hgher order GARCH models lke he GARCH (4, ) and GARCH (5, ) perform beer vs-à-vs he smple GARCH (, ) and oher compeng models and he mprovemen n forecas accuracy as ndcaed by Thel s-u, MAE or MAPE s que sgnfcan. Ffh, on he bass of asymmerc loss error sascs, one can noe ha only random walk model provdes unbased forecass meanng he probably of over predcons s equal o he probably of under predcons whch s equal o 50% and he null hypohess of an equal number of under and over forecass canno be acceped for any oher model bu for random walk model a convenonal levels of sgnfcance. In boh he markes he superor ranked models vz., EWMA or GARCH models have a endency o produce over forecass bu wh relavely small errors. On he bass of MME (U) he bes f GARCH (5, ) emerged beer model n he forex marke whle he smple GRACH (, ) model ranked hgher and on he bass of MME (O) he exponenal smoohng mehod comes ou as he bes model. See Table followed by Table 3 Concluson Volaly forecasng s an mporan area of research n fnancal markes and n hs paper we evaluae he comparave ably of dfferen sascal and economerc volaly forecasng models n he conex of Indan sock and forex markes. A oal of en dfferen models were consdered n hs sudy and hese compeng models are evaluaed on he bass of wo classes of evaluaon measures symmerc and asymmerc error sascs. Based on he ou of sample forecass and he number of evaluaon measures ha rank a parcular mehod as superor we can nfer ha EWMA wll lead o

13 mprovemens n volaly forecass n he sock marke and he GARCH (5, ) wll acheve he same n he forex marke. These fndngs are conrary o he fndngs of Bralsford and Faff (996) who found no sngle mehod as superor. Bu he resuls n sock marke are smlar o he fndngs of Akgray (989) and McMllan e al (000) and Anderson and Bollerslev (998) and Anderson e al (999) n he forex marke. The nferences reman same even on he bass of asymmerc error sascs.e., GARCH (4, ) and GARCH (5, ) models when under forecass are penalzed heavly n he sock marke and forex marke respecvely and EWMA when over forecass are penalzed heavly. 3

14 Table Nfy reurns Exchange rae reurns Mean Sandard Error Sandard Devaon Kuross Skewness Mnmum Maxmum JB (0.000) 095.9(0.000) Q(36) 45.5(0.000) (0.004) Q (36) 40.(0.000) 57.78(0.000) ADF sasc** **The McKnnon crcal value a % level of sgnfcance s and he es s conduced wh 4 four lags. The nferences reman he same for he Phllp-Perron es also. For an explanaon of Augmened Dckey Fuller es see Hamlon (994) Table Forecas Error Sascs: Symmerc Loss Funcon Forex marke TU Rank MAE Rank RMSE Rank MAPE Rank EWMA GARCH (,) GARCH(5,) Hs mean MA E MA MA MA Random Walk Smple reg Sock Marke EWMA GARCH (,) GARCH (4,) Hs mean MA MA MA MA Random Walk Smple reg

15 Table 3 Forecas Error Sascs: Asymmerc Loss Funcon Forex Marke MME (U) Rank MME(O) Rank *UF *OF Bnomal Probably EWMA GARCH (,) GARCH(5,) Hs mean MA MA MA MA Random Walk Smple reg Sock Marke EWMA GARCH (,) GARCH (4,) Hs mean MA MA MA MA Random Walk Smple reg *UF and OF sand for number of under forecass and number of over forecass 5

16 Fgure Daly reurns on Nfy Fgure Exchange rae Fgure 3 30 Densy Plo of Nfy Reurn seres Fgure 4 Densy Plo of Exchange Rae seres

17 References Akgray, V. (989) Condonal Heeroscedascy n Tme Seres of Sock Reurns: Evdence and Forecass, Journal of Busness, 6: Andersen, T. G. and T. Bollerslev (998) Answerng he Skepcs: Yes, Sandard Volaly Models Do Provde Accurae Forecass, Inernaonal Economc Revew, 39: Andersen, T. G., T. Bollerslev and S. Lange (999) Forecasng Fnancal Marke Volaly: Sample Frequency vs-à-vs Forecas Horzon, Journal of Emprcal Fnance, 6: Balaban, E. (004) Comparave Forecasng Performance of Symmerc and asymmerc Condonal Volaly Models of an Exchange Rae, Economcs Leers, 83: Bollerslev, T. (986) Generalzed Auoregressve Condonal Heeroscedascy, Journal of Economercs, 3: Bollerslev, T. R., Y. Chou and K. F. Kroner (99) ARCH Modelng n Fnance. A Revew of he Theory and Emprcal Evdence, Journal of Economercs, 5: 5-59 Bollerslev, T., R. F. Engle and D. B. Nelson (994) ARCH Models, n R.F. Engle and D. McFadden, Eds.. Handbook of Economercs, Vol. 4 Norh-Holland: Amserdam. Bralsford, T. J. and R. W. Faff (996) An Evaluaon of Volaly Forecasng Technques, Journal of Bankng and Fnance, 0: Brooks, C. and S. P. Burke (998) Forecasng Exchange Rae Volaly Usng Condonal Varance Models Seleced by Informaon Crera, Economcs Leers, 6: Dmson, E. and P. Marsh (990) Volaly Forecasng Whou Daa-Snoopng, Journal of Bankng and Fnance, 4: Engle, R. F (98) Auoregressve Condonal Heeroscedascy wh Esmaes of he Varance of U.K. Inflaon, Economerca, 50: Fama Eugene F. (965) The Behavor of Sock-Marke Prces, Journal of Busness, 38: Franses, P. H. and Dck Van Djk (996) Forecasng sock marke volaly usng (nonlnear) GARCH Models, Journal of Forecasng 3:

18 Franses, P. H. and H. Ghjsels (999) Addve Oulers, GARCH and Forecasng Volaly, Inernaonal Journal of Forecasng, 5: -9 Hamlon, James D (994) Tme Seres Analyss, Prnceon Unversy Press Kaur H (004) Tme varyng volaly n he Indan Sock Marke, Vkalpa 94.: 5-4 Klassen, F. (00) Improvng GARCH Volaly Forecass, Emprcal Economcs, 7: Lee, K. Y. (99) Are he GARCH Models Bes n Ou-of-Sample performance? Economcs Leers, 37: Loudon, G. W. Wa and P. Yadav (000) An Emprcal Analyss of Alernave Paramerc ARCH Models, Journal of Appled Economercs, 5: Mandelbro, B. (963) The varaon of ceran speculave prces, Journal of Busness, 36: McMllan, D., A. Spegh and O. Gwlym (000) Forecasng UK Sock Marke Volaly, Appled Fnancal Economcs, 0: Meron, Rober (980) On esmang he expeced reurn on he marke: An explanaory nvesgaon, Journal of Fnancal Economcs, 8: 33-6 Pagan, A. R. and G. W. Schwer (990) Alernave Models for Condonal Sock Volaly, Journal of Economercs, 45: Pandey A (00) The Exreme Value Volaly Esmaors and Ther Emprcal Performance n Indan Capal Markes, NSE Workng Paper No. 5 downloadable from Poon, S. and C. Granger (003) Forecasng Volaly n Fnancal Markes: A Revew, Journal of Economc Leraure, 4: Taylor, S. J. (987) Forecasng he Volaly of Currency Exchange Raes, Inernaonal Journal of Forecasng, 3: Tse, S. H. and K. S. Tung (99) Forecasng Volaly n he Sngapore Sock Marke, Asa Pacfc Journal of Managemen, 9: -3. Tse, Y. K. (99) Sock Reurns Volaly n he Tokyo Sock Exchange, Japan and he World Economy, 3: Varma, J. R. (999) Value a rsk models n he Indan sock marke IIMA Workng paper no downloadable from 8

19 Vlasuso, J., (00) Forecasng Exchange Rae Volaly," Economc Leers, 76: Wes, K. D. and D. Cho (995) The Predcve Ably of Several Models of Exchange Rae Volaly, Journal of Economercs, 69: Yu, J., (00) Forecasng Volaly n he New Zealand Sock Marke, Appled Fnancal Economcs, :