Inernaional Journal of Probabiliy and Saisics 04, 3: -7 DOI: 0593/is040300 Condiional Variance Parameers in Symmeric Models Onyeka-Ubaka J N,*, Abass O, Okafor R O Dearmen of Mahemaics, Universiy of Lagos, Akoka, Lagos, +34, Nigeria Dearmen of Comuer Science, Universiy of Lagos, Akoka, Lagos, +34, Nigeria Absrac he arameers α and β are resriced o be non-negaive in GARCH model, which have some conseuences for he saionariy condiion, and alhough he disurbances have mean 0, hey are clearly no whie noise because of heir ime-varying asymmeric robabiliy densiy funcions hus, his saionary rocess is caable of cauring well known henomena resen in financial markes such as volailiy clusering, marginal disribuions having heavy ails and hin cenres Leokurosis; reurn series aearing o be almos uncorrelaed over ime bu o be deenden hrough higher momens he ossibiliy of having deendence beween higher condiional momens, mos noably variances, involves examining nonlinear sochasic rocesses from a more realisic ersecive in ime series daa his moivaes he consideraion of nonlinear models he resuls obained hrough Mone Carlo simulaions esablished he racicabiliy and small samle erformance of symmeric models under Gaussian disribuions he resuls also showed ha he corresonding sandard errors are very small indicaing ha esimaors are asymoically unbiased, efficien and consisen a leas wihin he samle Keywords GARCH, Symmeric, Simulaion, Condiional Variance, Saionariy Inroducion Financial ime series resen various forms of non linear dynamics, he crucial one being he srong deendence of he variabiliy of he series on is own as and furhermore, wih he fiing of he sandard linear models being oor in hese series he auoregressive condiional heeroskedasiciy ARCH model has become one of he mos imoran models in financial alicaions hese models are non-consan variances condiioned on he as, which are a linear funcion on recen as disurbances Onyeka-Ubaka and Abass[9] and Dallah, Okafor and Abass[5] his means ha he more recen news will be he fundamenal informaion ha is relevan for modelling he resen volailiy Moreover, he accuracy of he forecas over ime imroves when some addiional informaion from he as is considered Secifically, he condiional variance of he innovaions will be used o calculae he ercenile of he forecasing inervals, insead of he homoskedasic formula used in sandard ime series models Recen develomens in economic and financial economerics sugges he use of nonlinear ime series srucures o model he aiude of invesors oward risk and execed reurns his is because he condiional variance is no consan over ime Hence, his aer, herefore, is se ou * Corresonding auhor: onyeka-ubaka@unilagedung Onyeka-Ubaka J N Published online a h://ournalsauborg/is Coyrigh 04 Scienific & Academic Publishing All Righs Reserved o esablish he racicabiliy and small samle erformance of Generalied Auoregressive CondiionalHeeroskedasici y GARCH model wihin he Gaussian framework Lieraure Review Emirical sudies show ha models which resen some nonlineariy can be modelled by condiional secificaions, in boh condiional mean and variance A sochasic rocess { y } is a model ha describes he robabiliy srucure of a seuence of observaions over ime Zhu[3] A ime series y is a samle realiaion of a sochasic rocess ha is observed only for a finie number of eriods, indexed by,,,τ Onyeka-Ubaka[8] Any sochasic rocess can be arially characeried by he firs and second momens of he oin robabiliy disribuion: he se of means, µ Ε y and he se of variances and covariances cov y, yh Ε y µ yh µ h,, h In order o ge consisen forecas mehods, we need ha he underlying robabilisic srucure would be sable over ime So a sochasic rocess is called weak saionary or covariance saionary when he mean, he variance and he covariance srucure of he rocess is sable over ime, ha is: Ε y µ < Ε µ y <
Onyeka-Ubaka J N e al: Condiional Variance Parameers in Symmeric Models Le { } y refer o he univariae discree ime-valued sochasic rocess o be rediced eg he rae of reurn of a aricular sock or marke orfolio from ime - o where is a vecor of unknown arameers and Ε y Ε y µ denoes he condiional mean given he informaion se available in ime - he innovaion rocess for he condiional mean, { }, is hen given by y µ wih corresonding uncondiional variance V Ε, ero uncondiional mean and Ε h 0, h he condiional variance of he rocess given by is defined by V y V y Ε Since invesors would know he informaion se when hey make heir invesmen decisions a ime -, he relevan execed reurn o he invesors and volailiy are µ and, resecively An ARCH rocess, { }, can be resened as: i i d ~ Ν0, g,, ;,, ; v, v, where Ε 0, V, is a ime-varying osiive and measurable funcion of he informaion se a ime -, v is a vecor of redeermined variables included in, g is a linear or nonlinear funcional form By definiion, is serially uncorrelaed wih mean ero, bu a ime varying condiional variance eual o he condiional variance is a linear or nonlinear funcion of lagged values of and, and redeermined variables v, v, included in In he seuel, for noaional convenience, no exlici indicaion of he deendence on he vecor of arameers,, is given when obvious from he conex Since very few financial ime series have a consan condiional mean ero, an ARCH model can be resened in a regression form by leing be he innovaion rocess in a linear regression: y x b + 3 ~ Ν0, where x is a k vecor of endogenous exlanaory variables included in he informaion se, b is a k vecor of unknown arameers, is he random disurbance erm, and he subscri indicaes ha X and Y are series of eually saced observaions hrough ime he use of he ime series regression model underlies a number of assumions concerning he form of he model, he indeenden variable and he disurbance erms Provided ha hese assumions hold: i Zero mean: E[ e ] 0 ii Consan Variance: E[ e ] iii Non-auoregression: E[ e e h ] 0 h 0 I is ossible o esimae oimally he regression arameers and heir variances wih he following formulas: where bˆ X X Y Y aˆ X X Var bˆ Y bx ˆ 4 5 Var aˆ se + s s e X X X X X Y Yˆ e Yˆ aˆ + bx ˆ 6 7 8 he esimaors are oimal in he sense ha hey are unbiased, efficien and consisen Unbiased esimaors are hose in which he execed value of he esimaor, say bˆ, is eual o he rue value, b An esimaor is relaively efficien if i has a smaller variance han any oher esimaor of b, and i will be consisen if boh is bias and variance aroach ero as he samle sie aroaches infiniy ogeher hese roeries mean ha an esimaor will be cenered around he rue value as he samle sie increases Of aricular ineres
Inernaional Journal of Probabiliy and Saisics 04, 3: -7 3 are assumions i and iii, which, ogeher, imly ha he covariance of any wo disurbance erms i e, cov[ ee h ] is eual o ero his can be seen as follows: Cov[ e e h ] Ε [ e Ε e ] Ε[ e h Ε e h ] Ε[ e 0] Ε[ e h 0] E[ e e h ] 0 his means ha one assumes ha disurbances a one oin in ime are no correlaed wih any oher disurbances he basic indicaor of wheher he non-auoregression assumion is violaed is wheher here is a samle correlaion beween he various random disurbance erms o obain a visual indicaion of he naure of he correlaion, i is helful o consruc a correlogram ha rovides a grahical reresenaion of he esimaed auocorrelaion funcion wih ime lags and auocorrelaion coefficiens forming he axes he aer observes ha he main roblem wih an ARCH model of Engle[6] in emirical alicaion is ha i reuires a large number of lags o cach he naure of he volailiy; his can be roblemaic as i is difficul o decide how many lags o include and roduces a non-arsimonious model where he non-negaiviy consrain could be failed o faciliae he comuaional roblems of ARCH model, Bollerslev[] roosed a generaliaion of he ARCH rocess o allow for as condiional variances in he curren euaion E[ ] 0, ω+ α i i+ β 9 i where ω > 0, αi 0, i,,, β 0,,,, hese condiions on arameers ensure srong osiiviy of he condiional variance 9 If he sudy wries he euaion 9 in erms of lag-oeraor B, i ges ω + α Β + β Β 0 where and β α Β α Β + α Β + + α Β Β β Β + β Β + + β Β he model is covariance saionary if all he roos of α Β + β Β lie ouside he uni circle, or euivalenly if αi + β < ha is, if i ω + αi + β i α β ω i i α β i ω i hen, is long-run average variance uncondiional variance is eual o ω α i i β his model differs o he ARCH model in ha i incororaes suared condiional variance erms as addiional exlanaory variables his allows he condiional variance o follow an ARMA rocess in he suared innovaions of orders max, and,[armamax,, ], resecively: + i i + v + v i ω α β β 3 he GARCH model is usually much more arsimonious and ofen a GARCH, model is sufficien, his is because he GARCH model incororaes much of he informaion ha a much larger ARCH model wih large number of lags would conain 3 Mehodology In his aer, he modelling is roceeded by secifying and esimaing a model and hen checking is adeuacy If model defecs are deeced a he laer sage, model revisions are made unil a saisfacory model has been found hen he model may be used for forecasing Figure deics he main ses of a GARCH analysis and i is on hese ses ha his research is organied accordingly
4 Onyeka-Ubaka J N e al: Condiional Variance Parameers in Symmeric Models Figure GARCH analysis figure adaed from Box-Jenkins 976 se by se aroach 4 Resuls and Discussion he esimaion of GARCH model involves he maximiaion of a likelihood funcion consruced under he auxiliary assumion of an indeenden idenically disribued iid disribuion for he sandardied innovaion Le ; η f denoe he densiy funcion for /, wih mean ero and variance one, where η is he nuisance arameer, k H R η is he vecor of he arameer of f o be esimaed o imlemen he maximum likelihood rocedure, we use normal, he mos commonly disribuion in he lieraure 4 Since he normal disribuion is uniuely deermined by is firs wo momens, only he condiional mean and variance arameers ener he log-likelihood funcion in 9 ie he log-likelihood is I follows ha he score vecor ; y l s akes he form: µ / y µ + 3 / / } ex{ } ; [ / π η f ln ln l π s /
Inernaional Journal of Probabiliy and Saisics 04, 3: -7 5 When µ + µ + α, β where α are he condiional mean arameers and β are he condiional variance arameers, he score akes he form: l α s l β µ α α β α, α α β β β β l l where β 5 Even in he case of he symmeric GARCH, wih normally disribued innovaions, we have o solve a se of k + + + nonlinear euaions in 9 Numerical echniues are used in order o esimae he vecor of arameers he roblem faced in nonlinear esimaion, as in he case of he GARCH models, is ha here are no closed form soluions So, an ieraive mehod has o be alied o obain a soluion Ieraive oimiaion algorihms work by aking an iniial se of values of he 0 arameers, say, hen erforming calculaions based on hese values o obain a beer se of arameers values he rocess is reeaed unil he likelihood funcion y, y, y; y ; S s 6 0 no longer imroves beween ieraions If is a rial value of he esimae, hen exanding L { y }; / 0 and reaining only he firs ower of -, we obain L L L 0 + 0 0 O L / should eual ero A he maximum, 0 Rearranging erms, he correcion for he iniial value,, obained is Le 0 L L 0 0 0 7 i denoe he arameer esimaes afer he i h ieraion Based on 7 he Newon-Rahson algorihm i+ comues as: i+ i i L i L 8 he scoring algorihm is a mehod closely relaed o he Newon-Rahson algorihm and was alied by Engle[6] o esimae he arameers of he ARCH model he difference beween he Newon-Rahson mehod and he mehod of scoring is ha he former deends on observed second derivaives, while he laer deends on he execed values of he second derivaives So, he scoring algorihm i+ comues as: i+ i i i L L + E 9 he assumion of normally disribued sandardied innovaions is ofen violaed by he daa If he rue disribuion is insead leokuric, hen he maximum is sill consisen, bu no longer efficien In his case he maximum likelihood mehod is inerreed as he Quasi-Maximum Likelihood QML mehod Bollerslev and Wooldridge[3], based on Wesis[] and Pagan and Sabau[0], showed ha he maximiaion of he normal log-likelihood funcion can rovide consisen esimaes of he arameer vecor even when he disribuion of in non-normal, rovided ha E 0 E hese esimaes are, however, inefficien wih he degree
6 Onyeka-Ubaka J N e al: Condiional Variance Parameers in Symmeric Models of inefficiency increasing wih he degree of dearure from normaliy Pagan and Schwer[] So, he sandard errors of he arameers have o be adused Le ˆ be he esimae ha maximies he normal log-likelihood funcion, in euaion 6, based on he normal densiy funcion in 5 and le 0 be he rue value hen, even when is non-normal, under cerain regulariy condiions: ˆ 0 d Ν0, Α ΒΑ hus,ˆ is a maximum likelihood esimaor based on a mis-secified model Under regulariy condiions, he QML esimaor converges almos surely o he seudo rue value 0 For symmeric dearures from normaliy, he uasi-maximum likelihood esimaion is generally close o he exac maximum likelihood esimaion MLE Bu, for non-symmeric disribuion, Engle and Kroner[7] showed ha he loss in efficiency may be uie high he racical alicabiliy and small samle erformance of he MLE rocedure for he GARCH rocess is sudied by Mone Carlo simulaions 4 Mone Carlo Exerimens A hybrid Mone Carlo exerimen was erformed using he normal disribuion as daa generaing rocesses by inroducing an auxiliary vecor o avoid random walk behaviour he momenum samles are discarded afer samling he end resul of hybrid Mone Carlo exerimen is ha he roosals move across he samle sace in large ses and are herefore less correlaed and converge o he arge disribuion more raidly hrough he Mone Carlo exerimen, he model considered for GARCH, given by Υ µ + α + α 0 for +, β y µ is a, n, where is a sandard Normal random variable and n 50, 60, 70, 80, 50, 700, 000 and 3000 he condiional mean, µ, is assumed o follow an AR model in he simulaion exerimen able liss he Mone Carlo mean, sandard error Sd error and -saisic for he arameer vecor ω across M 000 Mone Carlo simulaions he simulaion algorihm generaes n + 000 observaions for each series, saving only he las n his oeraion is erformed in order o avoid deendence on iniial values he calculaions were carried ou in MALAB R008b For any se of arameer ω, α, β, he saring esimae for he variance of he firs observaion is ofen aken o be he observed variance of he residuals I is easy o calculae he variance forecas for he second he GARCH udaing formula akes he weighed average of he uncondiional variance, he suared residuals for he firs observaion and he saring variance and esimaes of he second his is inu ino he forecas of he hird variance, and so forh Ideally, his series is large when he residuals are large and small when hey are small he likelihood funcion rovides a sysemaic way o adus he arameers ω, α, β o give he bes fi able Esimaed Parameers and he Esimaed Sandard Errors for he Cenered Normal GARCH, Model ˆα N 0 Mean Sd error -saisic 50-00450 0737-006 048 60-0535 0653-09890 0590 70-0497 8838-0455 0090 80-0005 0545-00946 0045 50-00067 00768-0087 0047 700-00046 00590-00780 0084 000-000 00388-00567 00486 3000-0006 0096-0054 0093 ˆα Mean Sd error -saisic 030 0368 065 00376 0988 0499 0 0045 0709 0575 0534 00843-0048 0306-06768 0065 064 0075 835 00039 05 00555 38757 0047 0999 0040 74788 000879 050 0003 3645 00057 ˆβ Mean Sd error -saisic 0000 00896 607 0000 0885 0886 43399 0076 0363 0779 09 075 0357 0780 845 00569 09447 00499 8939 0005 098 0003 956 0007 09809 00076 90657 00009 09895 00030 398333 00004
Inernaional Journal of Probabiliy and Saisics 04, 3: -7 7 Insecion of able reveals ha for all samle sies, he averages obained from he exac MLE are close o he rue arameer values he corresonding sandard errors are very small indicaing ha esimaors are asymoically unbiased, efficien and consisen he sandard GARCH model has news imac curve, which is symmeric and cenred a 0 ha is, osiive and negaive reurn shocks of he same magniude roduce he same amoun of volailiy Also, larger reurn shocks forecas more volailiy a a rae roorional o he suare of he sie of he reurn shock If a negaive reurn shock causes more volailiy han a osiive reurn shock of he same sie, he GARCH model under redics he amoun of volailiy following bad news and over redics he amoun of volailiy following good news Furhermore, if large reurn shocks cause more volailiy han a uadraic funcion allows, hen he sandard GARCH model under redics volailiy afer a large reurn shock and over redics volailiy afer a small reurn shock able summaries he esimaes coefficiens from he GARCH, model wih he rue value se of arameers ˆ α, ˆ α, ˆ β, ˆ } {-0006, 050, 09895}, ha is, he { 0 c simulaion model: 0006 + 050 + 09895 he P-values of he esimaed arameers shown in he brackes are saisically significan wih values almos less han 005 Where Sd error is he Sandard errors and -saisic is Saisics obained by dividing he mean by he sandard error Mone Carlo simulaions are comued 000 relicaions Each relicaion gives a samle sie n 50, 60, 70, 80, 50, 700, 000 and 3000 observaions 5 Conclusions he GARCH model is characeried by a symmeric resonse of curren volailiy o osiive and negaive lagged errors, since is uncorrelaed wih is hisory his could be inerreed convenienly as a measure of news enering a financial marke a ime he aer esablished ha he GARCH, model caures volailiy clusering and leokurosis resen in high freuency financial ime series daa he Mone Carlo simulaions showed he racicabiliy and samle erformance analysis of he GARCH, model under normal disribuion he simulaed resuls also showed ha he corresonding sandard errors are very small indicaing ha esimaors are asymoically unbiased, efficien and consisen From he discussions, we observe ha he simle srucure of he model imoses imoran limiaions on GARCH models hus: i GARCH models, however, assume ha only he magniude and no he osiiviy or negaiviy of unaniciaed excess reurns deermines condiional variance If he disribuion of is symmeric, he change in variance omorrow is condiionally uncorrelaed wih excess reurns oday ii he GARCH models are no able o exlain he and observed covariance beween his is ossible only if he condiional variance is exressed as an asymmeric funcion of iii GARCH models essenially secify he behaviour of he suare of he daa In his case a few large observaions can dominae he samle iv he negaive correlaion beween sock reurns and changes in reurns volailiy, ha is, volailiy ends o rise in resonse o bad news, excess reurns lower han execed and o fall in resonse o good news excess reurns higher han execed are also observed his limiaion is called leverage effec and can be ackled wih asymmeric models REFERENCES [] Bollerslev, 986 Generalied Auoregressive Condiional Heeroskedasiciy, Journal of Economerics, 3, 307-37 [] Bollerslev, Engle and Wooldridge 988 A Caial Asse Pricing Model wih ime-varying Covariances Journal of Poliical Economy 96, 6-3 [3] Bollerslev, and Wooldridge, J M 99 Quasi- Maximum Likelihood Esimaion and Inference in Dynamic Models wih ime-varying Covariances Economeric Reviews,, 43-73 [4] Box, GEP and Jenkins, G W 976 ime Series Analysis: Forecasing and Conrol, Holden-Day, San Francisco [5] Dallah, H, Okafor, R O and Abass, O 004 A Model-Based Boosra Mehod for Heeroskedasiciy Regression Models Journal of Scienific Research and Develomen, 9, 9 - [6] Engle, R F 98 Auoregressive Condiional Heeroskedasiciy wih Esimaes of Unied Kingdom Inflaion Economerica, 50, 987-007 [7] Engle, R F and Kroner, K F 995 Mulivariae Simulaneous Generalied ARCH Economeric heory,, -50 [8] Onyeka-Ubaka, J N 03 A Modified BL-GARCH Model for Disribuions wih Heavy ails A PhD hesis, Universiy of Lagos, Akoka, Nigeria [9] Onyeka-Ubaka, J N and Abass, O 03 Cenral Bank of Nigeria CBN Inervenion and he Fuure of Socks in he Banking Secor American Journal of Mahemaics and Saisics,36: 407-46 [0] Pagan, A R and Sabau, H 987 On he Inconsisency of he MLE in Cerain Heeroskedasic Regression Models, Universiy of Rocheser, Dearmen of Economics, Mimeo [] Pagan, A R and Schwer, G W 990 Alernaive Models for Condiional Sock Volailiy, Journal of Economics, 45, 67-90 [] Weiss, A 986 Asymoic heory for ARCH Models: Esimaion and esing Economeric heory,, 07-3 [3] Zhu, F 0 A negaive binomial ineger-valued GARCH model Journal of ime Series Analysis, 3, 54-67