Modeling of stock indices with HMM-SV models

Transcription

1 Thorcal ad Appld Ecoomcs Volum XXIV 7 No. 6 Summr pp Modlg of sock dcs wh HMM-SV modls E.B. NKEMNOLE Urs of Lagos Ngra kmol@ulag.du.g J.T. WULU Urs of Marlad Urs Collg USA joh.wulu@facul.umuc.du Absrac. Th us of olal modls o coduc olal forcasg s gag momum mprcal lraur. Th prformac of olal prssc as dcad b h smad paramr Sochasc Volal SV modl s pcall hgh. Sc fuur alus SV modls ar basd o h smao of h paramrs hs ma lad o poor olal forcass. Furhrmor hs hgh prssc accordg o som rsarch scss s du o h srucur chags.g. shf of olal lls h olal procsss whch SV modl cao capur. Hdd Marko Modls HMMs allow for prods wh dffr olal lls characrzd b h hdd sas. Ths work dals wh h problm b brgg h SV modl basd o Hdd Marko Modls HMMs calld HMM-SV modl. Va hdd sas HMMs allow for prods wh dffr olal lls characrzd b h hdd sas. Wh ach sa SV modl s appld o modl codoal olal. Emprcal aalss usg h proposd HMM-SV modls dos o ol addrss h srucur chags bu also prods br olal forcass ad sablshs a ffc forcasg srucur for olal modlg. Kwords: forcasg hdd Marko modl sochasc olal sock chag. JEL Classfcao: C3 C5 C8.

2 46 E.B. Nkmol J.T. Wulu. Iroduco A gra dal of ao has b pad fac as wll as mprcal lraur for praccall masurg rsk o modlg ad forcasg h olal of sock mark dcs a sochasc olal SV modl. No doub forcasg h olal of sock dcs s a mpora aspc of ma facal dcsos. For sac sm maagrs opo radrs ad h facal maagral bods ar all rsd olal forcass ordr o hr cosruc lss rsk porfolos or oba hghr profs Paa ad Slăscu. Varous olal modls ha b rcommdd o dscrb h sascal faurs of facal m srs. Sourcs of h mark olal ha b prsd b Shllr 993. Th mos commo modls amog hs ar h SV modls. Ohr olal modls clud bu o lmd o h auorgrss codoal hroskdasc ARCH modls b Egl 98 ad dd o gralzd ARCH GARCH b Bollrsl al Thr succss ls hr abl o capur som slzd facs of facal m srs such as m-arg olal ad olal clusrg. SV modlg has b appld o m arg olal Talor For ampl SV modls ca b usd o modl h arac as a uobsrd compo ha follows a parcular sochasc procss. I SV modls m-arg arac s o rsrcd o a fd procss. I SV modls s usual o modl olal as a logarhmc frs ordr auorgrss procss. SV modls rprs a dscr m approach o h dffuso procss usd h opo prcg lraur Hull ad Wh 987. Ths modl hough horcall arac s mprcall challgg as h uobsrd olal procss rs h modl a o-lar fasho whch lads o h lklhood fuco dpdg upo hgh-dmsoal grals. I hs papr w propos a soluo o h problm of of hgh olal prssc SV modl b brgg HMM o allow for dffr olal sas prods wh dffr olal lls m srs. Also wh ach sa w allow SV modl o modl h codoal arac. Th sug HMM-SV modl dd lds br olal forcas compard o SV modls for arfcal daa ad ral facal daa -sampl as wll as ou-of-sampl.. Rlad lraur A umbr of ard approachs ca b usd o sma h SV modl. Two rc suds ha compar h usfulss of h SV modl wh GARCH modls appld forcasg suaos ca b s So al. 999 ad Yu. So al. 999 ascrad ha modlg ad forcasg forg chag ras h SV modl smad as a sa spac modl dos o ouprform GARCH modl. Yu o hs ow par usd h SV modl o forcas dal sock mark olal for Nw Zalad. B mas of forcas accurac ss h dscord ha h SV modl surpasss prformac of GARCH modls. Th md rsuls from hs wo paprs suggs h d for furhr rsarch o h rla mrs of SV modls appld forcasg suaos. Alhough sadard SV

3 Modlg of sock dcs wh HMM-SV modls 47 modls mpro h -sampl f a lo compard wh cosa arac modls umrous suds fd ha SV modls g usasfacor forcasg prformacs Fglwsk 997. Xog-F ad La-Wa 4 argud ha h usuall orsad olal prssc SV modls ma b h caus of poor forcasg prformacs. For Lamouru 99 hs wll-kow hgh prssc ma orga from h srucur chags h olal procsss whch SV modls cao capur. Lamouru dmosrad ha a shf h srucur of facal m srs.g. h shf of ucodoal arac s lkl o lad o mssmag of h SV paramrs such a wa ha h al oo hgh a olal prssc. Nlso 99 ad Glos al. 993 ha usd h GARCH modl o compu h dffrc whch has ffcs of ga ad pos shocks o olal. Km al. 998 rc ms appld Hdd Marko modl HMM sad of ARCH o hadl h ffcs coomc daa. Th dffrc bw h HMM ad ARCH s h ucodoal arac. If hr ar squal chags rgm som rsarchrs ads ha som mor u approachs d o b cosdrd ad usg dffr rgms ma corbu o h rur-grag procss h mark. Hamlo ad Susml 994 apprhdd ha h log ru arac could ob rgm shf; h suggsd a ARCH procss. Th ffc wll ash f h us wkl daa bcaus spars m po maks h dpdc wakr. I s raoal o am h prc of sock mark b usg HMM. I usg HMM Chu al. 996 chos a wo-sag procss o rprs h rur bhaor h sock mark. Th frs cosdrd h rur bhaor sock mark as a Marko procss. Th h dffr rur rgms drd from h frs sag wr ulzd o sma h olal. Lasl h foud ha h ga daos rurs ca ha largr cras olal ha h pos o. Accordgl h hk h rur ad olal ar o larl bu asmmrcall. HMMs ha b appld for a las hr dcads sgal-procssg applcaos spcall auomac spch rcogo. Now hs hor ad applcao has padd o ohr flds. A HMM s Rabr 989 cluds wo sochasc procsss of whch o s a udrlg sochasc procss ha s o obsrabl ad h ohr procss s h obsrao squc.. HMM-SV modl.. Hdd Marko modl Alhough orgall roducd ad sudd 957 ad arl 97 s h comporar rpuao of sascal mhods of HMM s o quso. A HMM s a bara dscr-m procss X k Y k whr X s a homogous Marko cha k k k whch s o drcl obsrd bu ca ol b obsrd hrough Y k k ha produc h squc of obsrao. s a squc of dpd radom arabls such Y k k

4 48 E.B. Nkmol J.T. Wulu ha h codoal dsrbuo of Y k ol dpds o X k k s calld h sa. X k.th udrlg Marko cha HMM ar dfd hrough a fucoal rprsao kow as sa spac modl. Th sa spac modl Douc ad Johas 9 of a HMM s rprsd b h followg wo quaos: Sa quao f w Obsrao quao g whr f ad g ar hr lar or olar fucos whl w ad ar rror rm. Modls rprsd b - ar rfrrd o as sa spac modl ad hs comprss a class of HMMs wh o-lar Gaussa sa-spac modl for sac sochasc olal SV modl... Sochasc olal modl Sochasc olal modls Shphard 996 ar a ara of h gral sa spac approach prsd hr. SV modl blog o class of Hdd Marko modl wh olar Gaussa sa-spac modl ad h ak h olal of h daa o accou. Th SV modl du o Talor 98 ca b prssd as a auorgrss AR procss: r w 3 p 4 whr w ~ N ~ N ~ N { r } s h log-rurs o da w call h cosa scalg facor so ha { } rprss h log of olal of h daa log whr ar r.i ordr o sur saoar of r s assum ha. B akg h logarhm of h squard rurs of quao 4 rsuls a lar quao 5 z 5 Equaos 3 ad 5 form h rso of h SV modl whch ca b modfd ma was; oghr h form a lar o-gaussa sa-spac modl for whch 5 s h obsrao quao ad 9 s h sa quao.... Sochasc olal wh ha ald dsrbuo Th sadard form of h SV modl s g quaos 3 ad 4. I quao 4 follows a ormal dsrbuo. Varous auhors ha argud ha ral daa ma ha har als ha ca b capurd b h sadard SV modl.

5 Modlg of sock dcs wh HMM-SV modls 49 A so of h larzd rso of h SV modl s quao 3 ad 5 whr s assumd ha h obsraoal os procss z s a sud- dsrbuo s cosdrd. Th modl frs prsd Shumwa ad Soffr 6 ras h sa quao for h olal as: w bu h proposd sud- dsrbuo wh dgrs of frdom for h obsrao rror rm z ffcs a chag h obsrao quao: z z ~ 6 For h paramr smas of h proposd SV modl wh sud- h lklhood fucos ha b mamzd b usg h Squal Mo Carlo Epcao Mamzao algorhm Nkmol al. 5 h MATLAB opmzao rous..3. HMM wh sochasc olal modl Our modl s a bld of h orgal SV modl ad HMMs. To sar wh w us HMMs o dd h r m srs o rgms wh dffr olal lls. Th rur of h m srs s assumd o b modld b a mur of probabl dss ad ach ds fuco corrspods o a hdd sa wh s ma ad arac. I h HMMs SMCEM algorhm s mplod fdg h sa squc h m srs. Subsqul w g h subss of orgal m srs corrspodg o dffr sas olal lls. Afrwards wh ach rgms w allow SV modl wh dffr paramr ss o modl h codoal arac as: z w w ~ N whr dos h sa of h m srs a m. ad ar h paramr ss of h SV modl rlad o sa. Th for h olal forcas { } rprss h log of olal of h daa log whr ar r of h global modl hr s d for us o prdc h sa of m srs a m sa. Afr h sa a m has b drmd w choos h corrspodg SV modl wh paramr ss ad o mak olal forcas. Crra for assssg h accurac of h modls o prdc whch cluds MAE MSE MAPE ar lsd o pags 3 ad 4. SPSS ad MATLAB wr usd o aals h daa o produc fgurs ad rsuls of h modls.

6 5 E.B. Nkmol J.T. Wulu.4. Squal Mo Carlo Epcao Mamzao SMCEM Algorhm Aalss Esmao procdurs Th r smao procdur cosss of hr ma sps: flrg smoohg ad smao. Wh h oupu of flrg ad smoohg sp a approma pcd lklhood s calculad. { } ar smad o modl h chagg olal..4.. Flrg sp Th algorhm for h flrg ad smoohg sps shows a so of Godsll al. 4 ad Km ad Soffr 8. From hr M sampls from f Y for ach wr obad. Gra f ~ N For Gra a radom umbr ~ N j M w Compu p f w a. Compu w p p b. Gra f b rsamplg wh wghs j w..4.. Smoohg sp I h smoohg sp parcl smoohrs ha ar dd o g h pcd lklhood h pcao sp of h EM algorhm wr go: Suppos ha quall wghd parcls { for from h flrg sp. j Choos [ s ] [ f ] wh probabl. M For o } M from f Y ar aalabl f Calcula w f s for ach j f j s p f j p ~ s ~ j s

7 Modlg of sock dcs wh HMM-SV modls 5 Choos ] [ ] [ j f s wh probabl j w. } { : s s s s h radom sampl from Y f Rpa - for M ad calcula M M M E M s s p M s p M s.4.3. Esmao sp Ths sp cosss of obag paramr smas b sg h dra of h pcd lklhood of h compl daa } { g } { wh rspc o ach paramr o zro ad solg for ad. Th compl lklhood of } { s Y X f log p log log p log log 7 B h abo mhod w go h followg smas S S S S S 8 log 9

8 5 E.B. Nkmol J.T. Wulu whr S S p p S p 3. Volal forcas aluao ad comparso 3.. Daa ad mhodolog Boh smulad daa ss ad ral facal daa ss wr ulzd h olal forcas prms. Also h -sampl ad h ou-of-sampl forcasg prformacs wr cosdrd. To sar wh w usd smulad daa s o rf f h proposd modl sols h problms of css prssc SV modl; w grad mor ha obsraos ad dscardd h al sampls. Th w mplod h us of ral facal daa ss our prms o sablsh h abl of h proposd modl. Th ral facal daa ss coss of h dal chag ra srs of h Ngra Nara Ghaa Cd Brsh Poud ad Euro all agas h US Dollars from Jauar o Dcmbr Jarqu-Bra sascs Jarqu-Bra sascs s appld o am h o-ormal of h chag ra srs. Fgur. Nara/dollar chag ra d summar sascs 58 NIGERIA - USD Echag Ra Sascs Nara/Dollar ra Ma Sd. D Skwss.8 Kuross Jarqu-Bra Probabl.

9 Modlg of sock dcs wh HMM-SV modls 53 Fgur shows a pos skwss.8 ad a hgh pos kuross Wh rfrc o h Jarqu-Bra sascs Nara/dollar chag ra d s oormal a h cofdc ral of 99% sc probabl s. whch s lss ha.. Cosqul hr s d o cor h Nara/dollar chag ra d srs o h rur srs. Fgur. Cd/dollar chag ra d summar sascs.6 4 Ghaa Cds - USD.55 Echag Ra Sascs Cd/Dollar Ma.3455 Sd. D Skwss.4 Kuross Jarqu-Bra Probabl. Fgur shows a pos skwss.4 as wll as a pos kuross As dcad b Jarqu-Bra sascs h Cd/dollar chag ra d s o-ormal a h cofdc ral of 99% sc probabl s. whch s lss ha.. So h d also arss o cor h Cd/dollar chag ra d srs o h rur srs. Fgur 3. Euro/dollar chag ra d summar sascs.3 Euro - USD.. Rurs Sascs Euro/Dollar Ma.8556 Sd. D..3 Skwss.4487 Kuross Jarqu-Bra 95.8 Probabl. Fgur 3 shows a pos skwss.4487 ad a pos kuross As dcad b h Jarqu-Bra sascs Euro/dollar chag ra d s o-ormal a h cofdc ral of 99% sc probabl s. whch s lss ha.; hc h d o cor h Euro/dollar chag ra d srs o h rur srs.

10 54 E.B. Nkmol J.T. Wulu Fgur 4. Poud/dollar chag ra d summar sascs.74 Brsh Poud - USD Echag Ra Sascs Poud/Dollar Ma.598 Sd. D Skwss.855 Kuross.4943 Jarqu-Bra Probabl. Fgur 4 shows a pos skwss.855 ad a pos kuross As dcad b h Jarqu-Bra sascs Euro/dollar chag ra d s o-ormal a h cofdc ral of 99% sc probabl s. whch s lss ha.; hc h d o cor h Euro/dollar chag ra d srs o h rur srs Trasformao of h chag ra d srs of h Ngra Nara Ghaa Cd Brsh Poud ad Euro O h whol h moms of h sock dcs srs ar o-saoar ad hrfor o suabl for h sud purpos. Th sock dcs srs ar rasformd o hr rurs so ha w g saoar srs. Th rasformao s: p r l p whr r h m. p s h chag ra a m d p h chag ra jus pror o Augmd Dck-Fullr ADF Ts ad Phllps-Prro PP Ts o Nara/Dollar Cd/Dollar Poud/Dollar ad Euro/Dollar chag ras d Rurs Srs Boh h ADF ad PP ss ar usd o oba rfcao rgardg whhr Nara/Dollar Cd/Dollar Poud/Dollar ad Euro/Dollar chag ras rur srs s saoar or o. Tabl. ADF s o Nara/Dollar Cd/Dollar Poud/Dollar ad Euro/Dollar chag ras rurs -Sasc Nara/Dollar d Cd/Dollar d Poud/Dollar d Euro/Dollar d ADF s sasc % ll % ll Ts crcal alus % ll Prob.....

11 Modlg of sock dcs wh HMM-SV modls 55 Tabl shows ha h alus of ADF s sasc s lss ha s s crcal alu a 5% ll of sgfcac whch mpls ha h Nara/Dollar chag ras rur srs s saoar. Th rsul of ADF s also dmosras ha h Cd/Dollar Poud/Dollar ad Euro/Dollar rur srs ar saoar as h alus of ADF s sasc s lss ha s s crcal alu. Tabl. PP s o Nara/Dollar Cd/Dollar Poud/Dollar ad Euro/Dollar chag ras rurs -Sasc Nara/Dollar d Cd/Dollar d Poud/Dollar d Euro/Dollar d PP s sasc % ll % ll Ts crcal alus % ll Prob..... Tabl llusras h rsuls of h PP s ad pros ha h Nara/Dollar d rurs srs s saoar as h alus of PP s sasc s lss ha s s crcal alu a h ll of sgfcac of 5%. Th oucom of h PP s quall shows ha h Cd/Dollar Poud/Dollar ad Euro/Dollar chag ras rurs srs ar saoar sc h alus of PP s sasc s lss ha s s crcal alu. 3.. Summar Sascs of h Nara/Dollar Cd/Dollar Poud/Dollar ad Euro/Dollar chag ras rurs Fgur 5. Nara/dollar chag ra d rurs summar sascs.5 Ngra Nara - US.4.3. Rurs Sascs Nara/Dollar ra Ma Sd. D Skwss Kuross Jarqu-Bra Probabl. Fgur 5 rals a ga skwss ad a pos kuross As dcad b h Jarqu-Bra sascs h Nara/dollar chag ra d rurs srs s o-ormal a 95% cofdc ll sc probabl s. whch s lss ha.5.

12 56 E.B. Nkmol J.T. Wulu Fgur 6. Cd/dollar chag ra d rurs summar sascs. Euro - USD. Echag Ra Sascs Cd/Dollar Ma.658 Sd. D Skwss Kuross Jarqu-Bra 3.55 Probabl. Fgur 6 also rals a ga skwss ad a pos kuross Basd o h Jarqu-Bra sascs h Cd/dollar chag ra d rurs srs s o-ormal a 5% ll of sgfcac bcaus h probabl. s lss ha.5. Fgur 7. Euro/dollar chag ra d rurs summar sascs. Euro - USD. Echag Ra Sascs Euro/Dollar Ma.676 Sd. D Skwss Kuross Jarqu-Bra Probabl. Fgur 7 also rals a ga skwss ad a pos kuross Basd o h Jarqu-Bra sascs h Euro/dollar chag ra d rurs srs s o-ormal a 5% ll of sgfcac bcaus h probabl. s lss ha.5.

13 Modlg of sock dcs wh HMM-SV modls 57 Fgur 8. Poud/dollar chag ra d summar sascs. Brsh Poud - USD.5..5 Rurs Sascs Poud/Dollar Ma.83 Sd. D Skwss Kuross 4.69 Jarqu-Bra Probabl. Fgur 8 also rals a ga skwss ad a pos kuross Basd o h Jarqu-Bra sascs h Poud/dollar chag ra d rurs srs s o-ormal a 5% ll of sgfcac bcaus h probabl. s lss ha Emprcal rsuls ad aluao As h acual olal a m s o obsrabl hr s d for som masurs of olal o assss h forcasg prformac. I hs papr w appl h sadard approach suggsd b Paga ad Schwr 99. A pro for h acual olal s g b r r whr r s h ma of h m srs or h sampl prod. Th sascal prformac masurs Ma Squard Error MSE Ma Absolu Error MAE ad Ma Absolu Prcag Error MAPE ar appld o slc h bs prformg modl boh h -sampl ad h ou-of-sampl daa s dpdl hs sud: MSE = MAE= 3-4 MAPE = - / whr s h forcasd arac ad h umbr of forcass. 5 h acual arac m prod ad s

14 58 E.B. Nkmol J.T. Wulu 4.. Sascal prformac Th aluao rsuls ar show Tabls 3 ad 4 blow. A wo-sa HMM-SV modl was usd our prms. I boh abls - rprss ru alu HSV sads for HMM-SV modl ad SV sads for SV modl. s ad s dsga h wo sas wh low ad hgh olal lls rspcl. MSE MAE ad MAPE ar h -sampl MSE MAE ad MAPE whl MSE MAE ad MAPE ar h ou-of-sampl MSE MAE ad MAPE. Tabl 3. Sascal prformac rsuls for h smulad daa s ad h ru paramr ss compard wh hos obad from HMM-SV ad SV modls Modls MSE MAE MAPE MSE MAE MAPE - S S SV HMMSV S S Tabl 4. Sascal prformac rsuls for h sock rur daa ss ad h paramr ss obad from HMM-SV ad SV modls Sock Modls MSE MAE MAPE MSE MAE MAPE Echag Nara/Dollar SV HMMSV S S Cd/Dollar SV HMMSV S S Poud/Dollar SV HMMSV S S Euro/Dollar SV HMMSV S S Th abo rsuls ar dca ha ha HMM-SV modl capur h olal srucur chags procsss bw wo dffr olal rgms wh dffr olal prssc. Nohlss h SV modl cao capur such olal srucur chags ad alwas show r hgh olal prssc. Cosqul HMM-SV modl offrs br olal forcass as h MSE MAE of HMM-SV modl s cosdrabl smallr ha h SV modls mos of h m. 5. Cocluso Th olal prssc of wdl-usd SV modl s usuall oo hgh ladg o poor olal forcass. Th roo for hs css prssc sms o b h srucur chags.g. shf of olal lls h olal procsss whch h SV modl cao capur. As w dlopd our HMM-SV modl o allow for boh dffr olal sas m srs ad sa spcfc SV modl wh ach sa h mprcal rsuls for boh arfcal daa ad ral facal daa o ol aks car of h srucur chags hc

15 Modlg of sock dcs wh HMM-SV modls 59 gg br olal forcass bu also hlps o sablsh a profc forcasg srucur for olal modls. Accordgl h rsuls for boh -sampl ad ou-of-sampl aluao forcasg prformac cofrm ha our modl ouprforms wdl-usd SV modl Hc h rsuls suggs ha s promsg o dp h sud of olal prssc h hdd rgm-swchg mchasms clus. O log ru hs wll mpro olal forcass fuur rsarch. Rfrcs Bhar R. ad Hamor S. 4. Hdd Marko Modls: Applcao o Facal Ecoomcs. Dordrch. Kluwr Acadmc Publshrs. Bollrsl T Gralzd Auorgrss Codoal Hroskdasc. Ecoomrcs 3 pp Bollrsl T. Egl R.F. ad Nlso D.B ARCH modls. I: R.F. Egl ad D.L. Mc-Fadd Eds.. Hadbook of Ecoomrcs Vol. 4 Norh-Hollad Nw York chapr 49. Chu C.J. Sao G.J. ad Lu T Sock mark olal ad rgm shfs rurs. Iformao Sccs 77 pp Douc A. ad Johas A.M. 9. A uoral o parcl flrg ad smoohg: Ff ars lar. I: Crsa D. ad Rozosk B. Eds.. Oford Hadbook of Nolar Flrg Oford Urs Prss. Egl R. 98. Auorrgrss Codoal Hroskdasc wh smas of Ud Kgdom Iflao Ecoomrca 54 pp Glos L.R. Jagaaha R. ad Rukl D.E O h rlao bw h pcd alu of ad h olal of omal css rur o socks. Joural of Fac 53 pp Hamlo J.D. ad Susml R Auorgrss codoal hroscdasc ad chags rgm. Joural of Ecoomrcs 64 pp Jarqu C. ad Bra A A Ts for Normal of Obsraos ad Rgrsso Rsduals Iraoal Sascal Rw 55 pp Km C.J. Nlso C.R. ad Sarz R Tsg for ma rrso hroscdasc daa basd o Gbbs samplg augmd radomsao. Joural of Emprcal Fac 5 pp Fglwsk S Forcasg Volal. Facal Marks Isuos ad Isrums 6 pp Lamouru C. 99. Prssc Varac Srucural Chag ad h GARCH Modl. J. Bus Eco. Sas 8: pp MacKa R.J. 3. Hdd Marko Modls: Mulpl Procsss ad Modl Slco. Ph.D. Thss - Dp. of Sascs Th Urs of Brsh Columba.

16 6 E.B. Nkmol J.T. Wulu Nkmol E.B. Abass O. ad Kasumu R.K. 3. Paramr Esmao of a Class of Hdd Marko Modl wh Dagoscs Joural of Modr Appld Sascal Mhods pp Nkmol E.B. Abass O. ad Kasumu R.K. 5. A -dsrbuo Basd Parcl Flr for Uara ad Mulara sochasc olal modls Joural of h Ngra Mahmacal Soc 34 pp Nlso D.B. 99. Codoal hroskdasc ass rurs: a w approach. Ecoomrca 85 pp Paa I. ad Slăscu F.O.. Usg GARCH--ma modl o sga olal ad prssc a dffr frqucs for Buchars Sock Echag durg Thorcal ad Appld Ecoomcs 95 pp Rabr L.R A Tuoral o Hdd Marko Modls ad Slcd Applcaos Spch Rcogo I Procdgs of h IEEE 77 pp Shllr R.J OMark Volal. Th MIT Prss Cambrdg Massachuss. Shumwa R.H. ad Soffr D.S. 6. Tm Srs Aalss ad s Applcaos. Sprgr Nw York. Prro P Th gra crash h ol prc shock ad h u roo hpohss Ecoomrca 57 pp Paga A.R. ad Schwr G.W. 99. Alra Modls for Codoal Sock Volal. J. Ecoomrcs 45 pp Shphard N Sascal aspcs of ARCH ad sochasc olal. I Tm Srs Modls: I: Ecoomrcs Fac ad Ohr Flds Eds.. D.R. Co D.V. Hkl ad O.E. Bardorff-Nls. Lodo: Chapma ad Hall pp So M.K.P. Lam K. ad L W.K Forcasg chag ra olal usg auorgrss radom arac modl. Appld Facal Ecoomcs 9 pp Talor S.J Modlg Facal Tm Srs. Wl Chchsr. Talor S.J. 98. Facal rurs modlld b h produc of wo sochasc Procsss A sud of dal sugar prcs I: Tm Srs Aalss: hor ad Pracc Ed.. O.D. Adrso. Nw York: Elsr Scc Publshg Co. Amsrdam: Norh-Hollad Publshg Co. pp Xog-F Zhuag ad La-Wa Cha 4.Volal Forcass Facal Tm Srs wh HMM-GARCH Modls. IDEAL ol. 377 of Lcur Nos Compur Scc pp Yu J.. Forcasg olal h Nw Zalad sock mark Appld Facal Ecoomcs pp Zhag D. Ng X. Lu X. ad Hogw M. 7. Prdco Hdd Marko Modls Usg Squal Mo Carlo Mhods I: Procdgs of IEEE Iraoal Cofrc o Gr Ssms ad Illg Srcs pp