Volatility Modelling of the Nairobi Securities Exchange Weekly Returns Using the Arch-Type Models

Transcription

1 Inernaonal Journal of Appled Scence and Technology Vol. No. 3; March 1 Volaly Modellng of he Narob Secures Exchange Weekly Reurns Usng he Arch-Type Models ADOLPHUS WAGALA Chuka Unversy College Deparmen of Busness Admnsraon P.O BOX 19-CHUKA-64, KENYA DANKIT K. NASSIUMA Kabarak Unversy School of Engneerng P.O PRIVATE BAG KABARAK-157 ALI S. ISLAM Egeron Unversy Deparmen of Mahemacs P.O Box 536, Egeron 115 Phone: JESSE W. MWANGI Egeron Unversy Deparmen of Mahemacs P.O BOX 536,Egeron-115 Absrac In hs paper we denfy he mos effcen ARCH-ype model ha can be appled o he Narob sock exchange daa for forecasng and predcon of volaly whch n urn s mporan n prcng fnancal dervaves, selecng porfolos, measurng and managng rsks more accuraely. The esablshmen of an effcen sock marke s ndspensable for an economy ha s keen on ulzng scarce capal resources o acheve s economc growh. The purpose of hs sudy was o deermne he mos effcen model from he symmerc and he asymmerc GARCH models. The models were evaluaed by use of he Shwarz Bayesan Crera (SBC), Akake Informaon Crera (AIC) and he Mean Squared Error (MSE). The resuls show ha he AR-Inegraed GARCH (IGARCH) models wh suden s -dsrbuon are he bes models for modellng volaly n he Narob Sock Marke daa. Key words: ARCH, Model Effcency, MSE, Volaly 1. Inroducon Sock marke volaly s one of he mos mporan aspecs of fnancal marke developmens, provdng an mporan npu for porfolo managemen, opon prcng and marke regulaon (Poon and Granger, 3). An nvesor s choce of a porfolo s nended o maxmze he expeced reurn subjec o a rsk consran, or o mnmze hs rsk subjec o a reurn consran. An effcen model for forecasng of an asse s prce volaly provdes a sarng pon for he assessmen of nvesmen rsk. To prce an opon, one needs o know he volaly of he underlyng asse. Ths can only be acheved hrough modellng he volaly. Volaly also has a grea effec on he macro-economy. Hgh volaly beyond a ceran hreshold wll ncrease he rsk of nvesor loses and rase concerns abou he sably of he marke and he wder economy (Hongyu and Zhchao, 6). Fnancal me seres modellng has been a subjec of consderable research boh n heorecal and emprcal sascs and economercs. Numerous paramerc specfcaons of ARCH models have been consdered for he descrpon of he characerscs of fnancal markes. Engle (198) nroduced he Auoregressve Condonal Heeroscedascy (ARCH) for modellng fnancal me seres whle Bollerslev (1986) came up wh he Generalzed ARCH (GARCH) o parsmonously represen he hgher order ARCH model whle Nelson (1991) nroduced he Exponenal GARCH o capure he asymmerc effec. 165

2 Cenre for Promong Ideas, USA Oher specfcaons of he GARCH model ncludes: he TGARCH nroduced by Zakoan (1994), IGARCH by Engle and Bollerslev (1986), he Quadrac GARCH (QGARCH) model nroduced by Senana (1995), he GJR model by Glosen e al., (1993) jus o menon bu a few. The Sub-Saharan Afrca has been under-researched as far as volaly modellng s concerned. Sudes carred ou n he Afrcan sock markes nclude, Frmpong and Oeng-Abaye (6) who appled GARCH models o he Ghana Sock Exchange, Brooks e al., (1997) examned he effec of polcal change n he Souh Afrcan Sock marke, Appah-Kus and Pasceo (1998) nvesgaed he volaly and volaly spllovers n he emergng markes n Afrca. More recenly, Ogum e al., (6) appled he EGARCH model o he Kenyan and Ngeran Sock Marke reurns. From he avalable leraure, he NSE jus lke oher Sub Saharan Afrca Equy Markes has been under-researched as far as marke volaly s concerned and herefore hs sudy conrbues o he small leraure avalable on he Narob sock marke. There s a sgnfcan amoun of research on volaly of sock markes of developed counres. The focus of fnancal me seres modellng has been on he ARCH model and s varous exensons hereby gnorng he aspec of effcency whn he ARCH-ype of models. As a maer of fac, he subjec of he effcency of he models for fnancal modellng has receved lle aenon as far as economerc modellng s concerned. Ths sudy herefore ams a fndng he mos effcen model from amongs he auoregressve condonal heeroscedascy class of models. The remander of hs paper s arranged as follows; secon presens he properes of fnancal daa, secon 3 gves an overvew of he ARCH-ype models consdered n hs paper, n secon 4 we presen he resuls and dscussons, secon 5 hghlghs he summary and conclusons whle secon 6 conans he references.. Properes of Fnancal Daa Fnancal me seres daa ofen exhb some common characerscs. Fan and Yao (3) summarzes he mos mporan feaures of fnancal me seres as: The seres end o have lepokurc dsrbuon,.e hey have heavy aled dsrbuon wh hgh probably of exreme values. In addon, changes n sock prces end o be negavely correlaed wh changes n volaly, ha s; volaly s hgher afer negave shocks han afer posve shocks of he same magnude. Ths s referred o as he leverage effec. The sample auocorrelaons of he daa are small whereas he sample auocorrelaons of he absolue and squared values are sgnfcanly dfferen from zero even for large lags. Ths behavour suggess some knd of long range dependence n he daa. The dsrbuon of log reurns over large perods of me (such as a monh, a half a year, a year) s closer o a normal dsrbuon han for hourly or daly log-reurns. Fnally, he varances change over me and large (small) changes of eher sgn end o be followed by large (small) changes of eher sgn (Mandelbro, 1963). Ths characersc s known as volaly cluserng. These are facs characerzng many economc and fnancal varables. Researchers have appled dfferen models o he socks daa from me o me. Mandelbro (1963) ulzed he nfne varance dsrbuons when consderng he models for sock marke prce changes. Fama (1965) smlarly poned ou nally, her applcaon n cases of economcs parcularly n modellng sock marke prces. Fama e al., (1969) used a random walk o model he prce changes. Andrew and Whney (1986) esed he random walk hypohess for weekly sock marke reurns by comparng he varance esmaors. Here he random walk model was srongly rejeced. In recen sudes, varous specfcaons of ARCH models have been consdered for he descrpon of he characerscs of fnancal markes. Some sudes n whch ARCH-ype models were ulzed nclude; Gary and Mngyuon (4) who appled he GARCH model o Shangha Sock Exchange, Berram (4) modelled Ausralan Sock Exchange usng ARCH models and Baudouha (4) used he GARCH model n analyzng he Nordc fnancal marke negraon. In addon, Curo () employed he GARCH model o explan he volaly of he Poruguese equy marke, Waler (5) appled he srucural GARCH model o porfolo rsk managemen whle Frmpong and Oeng-Abaye (6) modelled he Ghana Sock Exchange volaly usng he GARCH models. More over, Ogum e al., (6) appled EGARCH model o he Kenyan and Ngera daly sock marke daa. 3. Auoregressve Condonal Heeroscedascy (ARCH) models An ARCH process s a mechansm ha ncludes pas varances n he explanaon of fuure varances (Engle, 4). Auoregressve descrbes a feedback mechansm ha ncorporaes pas observaons no he presen. ARCH models specfcally ake he dependence of he condonal second momens n modellng consderaon. 166

3 Inernaonal Journal of Appled Scence and Technology Vol. No. 3; March 1 Ths accommodaes he ncreasngly mporan demand o explan and o model rsk and uncerany n fnancal me seres (Degannaks and Xekalak, 4; Engle, 4; Fan and Yao, 3). An ARCH process can be defned n erms of he dsrbuon of he errors of a dynamc lnear regresson model. The dependen varable y s assumed o be generaed by y x =1,,T 1 where x s a kx1 vecor of exogenous varables, whch may nclude lagged values of he dependen varable and s a kx1 vecor of regresson parameers. The ARCH model characerzes he dsrbuon of he sochasc error condonal on he realzed values of he se of varables 1 { y 1, x 1, y, x,...}. In pracce, s assumed ha / 1 ~ N (, h ) where h qq wh > and, 1,..., q o ensure ha he condonal varance s posve (Engle s (198). An explc generang equaon for an ARCH process s h where ~..d N (,1). Snce h s a funcon of 1 and s herefore fxed when condonng on 1, s clear ha as wll be condonally normal wh E( / 1 ) h E( / 1) and Var( / 1) h, Var( / 1) h. The Generalzed Auoregressve Condonal Heeroscedascy (GARCH) model developed by Bollerslev (1986) s a generalzed ARCH (GARCH) where he condonal varance s h qq 1h 1... ph p wh he nequaly condons >, for =1,,q, for = 1,,p o ensure ha he condonal varance s srcly posve. When he parameer esmaes n GARCH (p,q) models are close o he un roo bu no less han un,.e p 1 q j 1, for he GARCH process, he mul-sep forecass of he condonal varance do no approach j1 he uncondonal varance. These processes exhb he perssence n varance/volaly whereby he curren nformaon remans mporan n forecasng he condonal varance. Engle and Bollerslev (1986) refer o hese processes as he Inegraed GARCH or IGARCH and hey do no possess a fne varance bu are saonary n he srong sense (Nelson, 199). The smples GARCH(1,1) s ofen found o be he benchmark of fnancal me seres modellng because such smplcy does no sgnfcanly affec he precseness of he oucome. Anoher exenson s he GARCH-M model developed by Engle e al., (1987) whose key posulae was ha me varyng prema on dfferen erm nsrumens can be modelled as rsk prema where he rsk s due o unancpaed neres raes and s measured by he condonal varance of he one perod holdng yeld. The GARCH (1,1)-M model s presened as x y 1 h where x and h are defned as before whle y 1 s a vecor of addonal explanaory varables. Jus lke he GARCH model, he GARCH-M s unable o capure asymmerc characerscs of fnancal daa. The Exponenal GARCH (EGARCH) models were nroduced by Nelson (1991) n an aemp o address he wo major lmaons of he GARCH models. Here he volaly depends no only on he magnude of he shock bu also on her correspondng sgns. The non-negavy resrcons are no mposed as n he case of GARCH snce he EGARCH model descrbes he logarhm of he condonal varance whch wll always be posve. The specfcaon for he condonal varance (Nelson, 1991) s gven as, q p p Log log Noe ha where ~..d (,1). 1 The parameer ( ) n equaon (1) measures he mpac of nnovaon on volaly a me whle parameer ( ) s he auo-regressve erm on lagged condonal volaly, reflecng he wegh gven o prevous perod s condonal volaly. I measures he perssence of shocks o he condonal varance. 167

4 Cenre for Promong Ideas, USA The saonary requremen s ha he roos of he auo-regressve polynomal le ousde he un crcle. For EGARCH (1,1) hs ranslaes no 1 <1 (Ogum e al., 6). Unlke he lnear GARCH, n he EGARCH model a negave shock can have a dfferen mpac compared o a posve shock f he asymmery parameer s nonzero. Threshold GARCH models (TGARCH) were nroduced by Zakoan (1994). The generalzed specfcaon of he condonal varance equaon s gven by, q p r jh j k k j1 1 k1 h I 1 k where h and I 1, f and zero oherwse. In hs model, good news,, and bad news, have dfferenal effecs on he condonal varance; good news has an mpac of, whle bad news has an mpac of. If, bad news ncreases volaly whle f, he news mpac s asymmerc. When he hreshold erm s se o zero, hen equaon (1) becomes a GARCH (p,q) model. 4. Resuls and Dscussons In hs sudy, four ses of daa conssng of he weekly average share prces for Bambur Cemen Ld, Naonal Bank of Kenya Lmed (NBK), Kenya Arways (KQ) Ld as well as he weekly average NSE share ndex were used. The daa was obaned from he Narob Sock Exchange (NSE) for he perod beween 3 rd June 1996 o 3 h Ocober 11 for he company share prces whle for he NSE -share ndex daa was for perod beween nd March 1998 o 3 h Ocober 11. The NSE -share ndex s a weghed mean wh 1966 as he base year a 1. I s based on companes calculaed on a daly bass. The ndex s useful n deermnng he performance of he NSE by measurng he general prce movemen n he lsed shares of he sock exchange. Bambur Cemen, Ld. was founded n 1951 and manufacures cemen n sub-saharan Afrca. The Kenya Arways prncpal acves nclude passengers and cargo carrage. I was ncorporaed n 1977 as he Eas Afrcan Arways Corporaon (EAA). The company was lsed n he NSE n 1996 and has been a major player n he Narob sock marke. The Naonal Bank of Kenya Lmed (NBK) was ncorporaed on 19 h, June 1968 and offcally opened on Thursday 14 h, November Is man objecve was o help Kenyans o ge access o cred and conrol her economy afer ndependence. The prelmnary analyss was done by use of me plos for he varous seres presened n Fgures 1 and NBK BAMB KQ NSEINDEX Fgure 1: Tme plos for he weekly average prces 168

5 Inernaonal Journal of Appled Scence and Technology Vol. No. 3; March 1 A vsual nspecon of he me plos clearly shows ha he mean and varance are no consan, mplyng nonsaonary of he daa. The non-consan mean and varance suggess he ulzaon of a nonlnear model and preferably a non-normal dsrbuon for modellng he daa. The seres were ransformed by akng he frs dfferences of he naural logarhms of he values n each he seres. The ransformaon was amed a aanng saonary n he frs momen. The equaon represenng he ransformaon s gven by X ln( P ) ln( P 1),where P represens he weekly average value for each seres. The sequence plos for he reurns are presened n Fgure Log_dff_NBK Log_dff_BAMB Log_dff_KQ Log_dff_NSEINDEX Fgure : Tme plos for log dfferenced seres The basc sascal properes of he daa show ha mean reurns are all posve and close o zero a characersc common n he fnancal reurn seres. All he four seres have very heavy als showng a srong deparure from he Gaussan assumpon. The Jarque-Bera es also clearly rejecs he null hypohess of normaly. Noable s he fac ha all he four seres exhb posve Skewness esmae. Ths means ha here are more observaons on he rgh hand sde. The seres havng exhbed heeroscedascy as shown by he me plos were esed for he ARCH dsurbances usng Engle s (198) Lagrange Mulpler (LM) whle he Pormaneau Q es (McLeod and L, 1983) based on he squared resduals was used o es for he ndependence of he seres. Snce boh he Q sasc and he LM are calculaed from he squared resduals, hey were used o denfy he order of he ARCH process. For all he reurn seres, he Q sascs and he Lagrange Mulpler (LM) ess ndcaed srong heeroscedascy for all he lags from 1 o 1.Ths suggesed an ARCH model of order q= Emprcal Resuls and Dscussons ARCH models The frs se of models mplemened n hs sudy was he orgnal Engle s (198) ARCH models. The suden s - dsrbuon and he General Error Dsrbuon (GED) were esed for all he seres. The suden s dsrbuon assumpon provded a beer model for NBK and KQ whle he GED performed well for NSE Index and Bambur. Ths could be due o he fac he fnancal daa s hghly heavy aled and s beer capured by he suden s -dsrbuon snce he GED dsrbuon has a hgher peak han he suden s -dsrbuon. Alhough he GED dsrbuon may be beer able o capure peaks, s far worse for capurng fa als. The Jarque-Bera (198) sasc also srongly rejeced he normaly assumpon n he sandardzed resduals for all he seres. The fed models were adequae snce her sandardzed resduals were no sgnfcanly correlaed n all he four seres basng on he Ljung-Box Q sascs. The squared resduals were also no sgnfcanly correlaed for lags up o 1 for all he four seres. 169

6 Cenre for Promong Ideas, USA GARCH models The nex class of models mplemened was he GARCH models. The auoregressve models were appled o capure he auocorrelaon presen n he seres. The GARCH models for dfferen values of p and q were fed o he daa, dagnosed and from he dagnoss and goodness of f sascs, he GARCH (1,1) was found o be he bes choce. Ths s conssen wh mos emprcal sudes nvolvng he applcaon of GARCH models n fnancal me seres daa. The Maxmum Lkelhood Esmaon (MLE) mehod was employed n he parameer esmaon. The GARCH parameer esmaes for he varance equaon was sgnfcan for all he seres excep for he NSE Index n whch α 1 was no sascally sgnfcan. In he GARCH model, he parameers and mus sasfy for saonary. However, he GARCH (1,1) esmaes volaed he resrcon mposed.e. n all cases, 1. Ths 1 1 mples ha he fed GARCH model s no weakly saonary and he condonal varance ( ) does no approach he uncondonal varance ( ) and hus he seres mgh no have fne uncondonal varance. Ths calls for he mplemenaon of Inegraed GARCH (1,1) model snce s capable of beng saonary n he srong sense even hough 1 (Nelson,199). 1 1 Two dsrbuons were esed (.e suden s and GED) for he specfc GARCH (p,q) model and he bes dsrbuon choce was deermned based on he SBC, AIC and he Log lkelhood Rao es n all he cases (see Table 4.8). For he NSE ndex, he dsrbuon of choce was he suden s -dsrbuon whle for NBK, Bambur, and KQ he Generalzed Error Dsrbuon was chosen. Ths shows ha he NSE ndex daa had faer als as compared o NBK, Bambur and KQ. The model adequacy was checked usng he Ljung-Box Q sascs for resduals and squared resduals n whch he null hypohess of no sgnfcan correlaons was no rejeced for all he seres mplyng ha he fed models were adequae. The JB sascs rejeced he null hypohess of normaly n he sandardzed resduals. Ths mples ha he models wh he respecve dsrbuons faled o normalze he resduals. The Goodness of f sascs and Dagnosc ess are presened Appendx Inegraed GARCH (IGARCH) Model Snce he parameer esmaes n GARCH (1,1) models were close o he un roo bu no less han un,.e p 1 q j 1,he IGARCH model was fed. The MLE mehod was ulzed for parameer esmaon for j1 he mean and varance equaons. The parameer esmaes for he varance equaon were sascally sgnfcan a.5 sgnfcance level n all he seres. In addon, for all he cases; mplyng ha mul-sep forecass of he condonal varance do no approach he uncondonal varance (.e he uncondonal varance s nfne). Despe he nfne uncondonal varance, one aracve feaure of he IGARCH model s ha s srongly saonary even hough s no weakly saonary. The resuls ndcae ha he daa ses used exhb he perssence n varance/volaly whereby he curren nformaon remans mporan n forecasng he condonal varance,.e. he curren nformaon n he NSE remans mporan n forecasng he condonal varance. Two dsrbuon assumpons namely, Generalzed error Dsrbuon and -dsrbuons were esed. Generalzed error Dsrbuon provded he bes f for he daa adequaely when modellng wh he IGARCH model n all he four seres. The models were fed and dagnosed usng he AIC, SBC and he Log lkelhood rao es. However, he fnal model was consdered adequae f s sandardzed resduals and squared resduals were no sgnfcanly correlaed a 5% sgnfcance level. The resdual correlaon was esed usng he Ljung-Box Q sascs. All he fed IGARCH models were adequae snce her resduals were no sgnfcanly correlaed. Furher, he sandardzed resduals were sll non-normal as shown by he JB sascs for normaly. The goodness of f sascs for he IGARCH(1,1) model and he dagnosc ess are presened n Appendx 1 and respecvely. In order o capure he leverage effecs, wo asymmerc ARCH-ype models; he Exponenal GARCH (EGARCH) and Threshold GARCH (TGARCH) were fed. 17

7 Inernaonal Journal of Appled Scence and Technology Vol. No. 3; March EGARCH models Despe he populary and apparen success of GARCH models n praccal applcaons, hey canno capure asymmerc response of volaly o news snce he sgn of he reurns play no role n he model specfcaon. Sascally, he asymmerc effec occurs when an unexpeced decrease n prce resulng from bad news ncreases volaly more han an unexpeced ncrease n prce of smlar magnude followng good news. Accordngly, Nelson s (1991) EGARCH model was fed. Unlke he GARCH (p,q) model, a negave shock can have a dfferen mpac on fuure volaly when compared o he posve shock f asymmery parameer γ 1 s no zero for he EGARCH model. I also does no need resrcons o be mposed on he parameers o ensure he nonnegavy. In he EGARCH model esmaon, he MLE creron was employed. Dfferen orders for p and q n he varance equaon were esed wh he bes resuls beng acheved for p=q=1. The Generalzed Error Dsrbuon emerged as he bes dsrbuon for all he seres (NSE, Bambur, KQ and NBK). Ths mples ha all he seres under nvesgaon have long als and are asymmerc. The EGARCH model parameer esmaes also reveal he perssence n volaly of he Narob equy marke. Ths s because he sum of α 1 and s approxmaely 1 n all he daa ses. The asymmerc parameer γ 1 1 s posve and sgnfcan for all he four seres namely NSE Index,NBK, Bambur and KQ. The posvy of γ 1 ndcaes ha posve shocks ncrease volaly more han he negave shocks of an equal magnude. Ths shows ha he concep of leverage effec (.e he negave shocks ncreasng volaly more han a posve shock of he same magnude) s no applcable o he ndvdual company socks. Ths s conssen wh he earler sudes on he Narob Sock Exchange for nsance Ogum e al., (5, 6) who found he asymmery parameer γ 1 o be posve when modellng he daly NSE Share Index usng he EGARCH models. Ths could arse from he fac ha he weekly reurn seres were used n hs sudy whle Ogum e al., (5, 6) modelled he daly reurns. Some nformaon could have been los when usng he weekly average for he NSE ndex and he share prces for he companes. In addon, he flow of nformaon n NSE mgh no be as effcen as n he developed equy markes. The model dagnoscs and goodness of f sascs are presened n Appendx 1 and respecvely. The dagnoscs ncluded he auocorrelaon of he sandardzed resduals and squared resduals respecvely. The Ljung-Box Q sascs represened by Q(1) and Q (1) for resduals and squared resduals respecvely were used whch were no sgnfcan n all cases confrmng he adequacy of he fed models. The models could hus explan he non-lnear dependence n he resduals.e he models capured he dependence n he varance shown by he orgnal seres of reurns. The EGARCH model, n all cases showed a smaller Kuross compared o he ARCH and GARCH models. In addon, he suden s -dsrbuon and Generalzed Error Dsrbuons also capured he al properes of he daa beer han he Gaussan dsrbuon n all he four cases. The JB sascs also srongly rejeced he null hypohess of normaly n he sandardzed resduals n all he seres under consderaon Threshold GARCH (1,1) The TGARCH (1,1) model whch falls n he asymmerc class of ARCH-ype models was also used. The model was fed, esmaed and dagnosed jus lke he prevous models. From he wo dsrbuons esed, he suden s dsrbuon emerged he bes for he NSE ndex whle GED was consdered he bes for he NBK, Bambur and KQ. Ths s because he GED and he sudens s -dsrbuons were able o capure he al properes of he daa. I s worh nong ha under he suden s dsrbuon, he convergence durng esmaon was a major problem. The algorhm converged very slowly and somemes weakly. Ths cass doubs on he sably of he parameer esmaes. In he varance equaon, he asymmery parameer γ 1 was less han zero for all he four seres. Ths mples ha good news ncreases volaly more han bad news. Ths s conssen wh he fndngs of Ogum e al., (5, 6) who appled EGARCH models o he daly NSE Share Index. Hence he leverage effec experenced n developed markes mgh no be a unversal phenomenon. The dagnosc ess and goodness of f sascs for he TGARCH models are presened n Appendx 1 and respecvely. Jus lke he prevous models, he bes dsrbuons were GED and he suden s -dsrbuon. 171

8 Cenre for Promong Ideas, USA Also, based on he Ljung-Box Q sascs, boh he resduals and he squared resduals were no sgnfcanly (5% level) correlaed mplyng ha he models were adequae. The JB sasc for normaly also rejeced he normaly assumpon n he sandardzed resduals. 4. Effcency Comparson beween he ARCH-ype Models Model effcences for each of he ARCH-ype models mplemened were evaluaed usng he varous MSE. The MSE for he chosen models are presened n Table 1. Table 1: MSE for he fed ARCH-ype models Seres ARCH(q) GARCH(1,1) IGARCH(1,1) EGARCH(1,1) TGARCH(1,1) NSE INDEX NBK BAMBURI KQ Consderng he MSE values n Table1, s clear ha ARCH, GARCH, EGARCH and IGARCH are all equally effcen n modellng volaly based on he MSEs only, snce he dfferen ARCH-ype models are almos equal for he respecve daa ses. The dsadvanage wh he ARCH model s ha so many parameers are o be esmaed. The GARCH, IGARCH, EGARCH and TGARCH models are able o parsmonously model he seres and hence are preferred o he orgnal ARCH model. Consderng he asymmerc properes of he daa and he respecve MSEs, he EGARCH (1,1) emerged as he bes model for he NSE Index and Bambur. For he NBK, boh he EGARCH and he TGARCH are equally good bu EGARCH s consdered he bes snce he parameer esmaes for he TGARCH are unsable due o weak convergence. The bes model for Kenya Arways was he GARCH model. The respecve models chosen are jusfed by her relavely lower values of resdual Kuross and MSE n addon o he oher dagnoscs consdered as well as he asymmerc parameer ha capures he leverage effec. However, n erms of saonary, he IGARCH model wh he Generalzed Error Dsrbuon (GED) emerged as he bes ARCH-ype model snce was srongly saonary hus beng more sable. Ths makes he IGARCH model o be he preferred model from he ARCH-ype models for modellng he Narob Sock Exchange daa for he perods beween nd March 1998 o 3 h Ocober 1 for NSE -Share ndex whle and 3 rd June 1996 o 3 h Ocober 1 for company share prces,.e NBK, Bambur and Kenya Arways. 5. Summary and Conclusons In hs sudy, he orgnal Engle s (198) ARCH (p) model and s hree exensons namely, sandard GARCH (p,q), IGARCH(p,q), EGARCH (p,q) and TGARCH (p,q) were appled o he daa. Dfferen orders for ARCH(p) were esed n all cases where p=8 was found o be he mos adequae for NSE ndex, Bambur and KQ whle for he NBK seres, p=9 provded he bes order for ARCH model. Four dfferen p and q values were esed for GARCH(p,q), EGARCH (p,q) and TGARCH (p,q): (1,1), (1,), (,1) and (,). The order p, q equal o (1,1) s by far he mos used values n GARCH research oday and resuls obaned s also conssen wh hs. In all he four seres, he order (1,1) s he bes choce. Comparng he dagnoscs and he goodness of f sascs, he IGARCH (1,1) ouperformed he ARCH, EGARCH and TGARCH models majorly due o s saonary n he srong sense. However, he IGARCH model s unable o capure he asymmery exhbed by he sock daa. The EGARCH (1,1) and he TGARCH (1,1) are he preferred models o descrbe he dependence n varance for all he four seres suded snce hey were able o model asymmery and parsmonously represen a hgher order ARCH(p). However, he sandardzed resduals sll dsplayed non-normaly n all cases. Judgng from he asymmerc parameer (γ 1 <) n he EGARCH model, he volaly ncreases more wh he bad news (negave shocks) han he good news (posve shocks) of he same magnude for he NSE Index. Ths s no conssen wh he fndngs of Ogum e al., (5, 6). However, for he ndvdual socks he asymmerc parameer (γ 1 >) meanng ha volaly ncreases more for good news more han bad news of he same magnude. Ths mples ha he leverage effec may no be a unversal phenomenon afer all. From he dfferen dsrbuons esed and esmaed, he suden s dsrbuon was he bes choce for NSE ndex whle GED was he bes for NBK, Bambur and Kenya Arways. The Gaussan assumpon provded he poores resuls and n some cases had convergence falures. 17

9 Inernaonal Journal of Appled Scence and Technology Vol. No. 3; March 1 6. References Akake, H. (1974). A new look a he sascal model denfcaon. IEE. Auom. conrol 19, Andrew,W and Whney, K. (1986). A large sample chow for he sngle lnear smulaneous equaon. Economc leers 19, Appah-Kus, J. and Pesceo, G. (1998). Volaly and Volaly Spll-Overs n Emergng Markes, CERF Dscusson Paper Seres No Baudouha, A.V. (4). Nordc Fnancal Marke Inegraon:An Analyss wh GARCH. Unpublshed Msc Thess, Göeborg Unversy Bera, A.K. and Hggns, M.L. (1993). On ARCH models: Properes, Esmaon and Tesng, Journal of Economc Surveys, 7, Bernd, E.K, Hall, B.H., Hall, R.E and Hausman, J. (1974). Esmaon and Inference n Nonlnear Srucural Models. Annals of Economc and Socal Measuremen, 3, Berram, W. (4). An emprcal nvesgaon of Ausralan Sock Exchange Daa. School of Mahemacs and Sascs, Unversy of Sydney. Bollerslev, T. (1986). Generalzed Auoregressve Condonal Heeroscedascy. Journal of economercs,31, Brooks, R. D., Davdson, S. and Faff, R. W. (1997). An Examnaon of he Effecs of Major Polcal Change on Sock Marke Volaly: The Souh Afrcan Experence, Journal of Inernaonal Fnancal Markes, Insuons & Money,7, Curo, J.J.D.(). Modellng he volaly n he Poruguese Sock Marke: A Comparave Sudy wh German and US markes. Unpublshed Phd Thess, Lsbon Unversy. Degannaks, S. and Xekalak, E. (4). Auoregressve Condonal Heeroscedascy Models: A Revew. Qualy Technology and Quanave Managemen, 1(), Engle, R.F. (198). Auoregressve Condonal Heeroskedascy wh Esmaes of he Varance of U.K nflaon, Economerca, 5, Engle, R.F. (1983). Esmaes of he Varance of US Inflaon Based on he ARCH Model. Journal of Money Cred and Bankng, 15, Engle, R.F.,Llen, D.and Robns,R. (1987). Esmang Tme Varyng Rsk Prema n he Term Srucure: The ARCH- M model, Economerca, 55 (), Engle, R.F. (4). Rsk and Volaly: Economerc Models and Fnancal Pracce. The Amercan Economc Revew, 94, Fama, E, Fsher, L., Jensen, M., Rolls, R. (1969). The adjusmen of sock prces o newnformaon. Inernaonal Economc Revew, 69, 1-1. Fama, E. (1965). The behavour of sock marke prces. J.Busness, 38, Fan, J. and Yao, Q.(3). Nonlnear Tme Seres: Nonparamerc mehods and Paramerc mehods. New York: Sprnger Seres n Sascs. Frmpong, J.M and Oeng-Abaye, E.F.(6). Modellng and Forecasng Volaly of Reurns on he Ghana Sock Exchange Usng Garch Models. Amercan Journal of Appled Scences, 3 (1), Gary, T and Mngyuan, G. (4). Inraday Daa and Volaly Models: Evdence from Chnese Socks. Unversy of Wesern Sydney. Glosen, L.,Jagannahan, R, and Runkle, D. (1993). On he Relaon beween he Expeced Value and he Volaly of he Nomnal Excess Reurn on Socks. Journal of Fnance, 48, Hongyu, P and Zhchao, Z.( 6). Forecasng Fnancal Volaly: Evdence from Chnese Sock Marke. Workng paper n economcs and fnance No. 6/. Unversy of Durham. Uned Kngdom. Mandelbro,B. (1963). The varaon of speculave prces. J. Bus, 36, McLeod, A.I. and L, W.K. (1983). Dagnosc checkng ARMA Tme Seres Models Usng Squared-Resdual Auocorrelaons, Journal of Tme Seres Analyss, 4, Mlls, T.C. (1999). The economerc Modellng of Fnancal Tme Seres, Cambrdge Unv.Press. Nelson, D. (199). Saonary and perssence n he GARCH(1,1) models. Economerc Theory 6, Nelson, D.B. (1991): Condonal Heeroskedascy n Asse Reurns: a New Approach. Economerca, 59., Ogum, G., Beer, F. and Nouyrga, G. (6). An Emprcal Analyss of he Kenyan and he Ngeran daly reurns usng EGARCH Models. In hp:// meda/nouyrga_-_isini.pdf Senana, E. (1995). Quadrac ARCH models. Revew of Economc Sudes. 6 (4), Poon, S.H. and C. Granger ( 3). Forecasng volaly n fnancal markes: a revew, Journal of Economc Leraure, XLI, Presly, M.B. (1988). Non-Lnear and Non-Saonary Tme Seres Analyss, Academc Press, New York Schwarz, G. (1978).Esmang he Dmenson of a Model. Ann. Ins.sasc. Mah.6, Waler,A.D. (5). A srucural GARCH model: An applcaon o porfolo rsk managemen Unpublshed PhD Thess, Unversy of Preora. Zakoan, J-M. (1994). Threshold Heeroskedasc Models. Journal of Economc Dynamcs and Conrol. 18,

10 Cenre for Promong Ideas, USA APPENDICES Appendx 1: The goodness of f sascs for ARCH models NSE INDEX NBK BAMBURI KQ ARCH (q) GARCH(1,1) IGARCH(1,1) EGARCH(1,1) TGARCH GED GED GED GED GED LR AIC SBC LR AIC SBC LR AIC SBC LR AIC SBC LR- Represens Log lkelhood Rao es JB- Represens Jarque-Bera sascs for normaly Q(1) - Represens Ljung-Box Q sascs for he sandardzed resduals Q (1) - Represens Ljung-Box Q sascs for squared sandardzed resduals P-Values are gven n he brackes Appendx : Dagnosc Tess for Sandardzed Resduals for ARCH-ype models Seres NSE INDEX NBK BAMBURI KQ Sascs ARCH(q) GARCH(1,1) IGARCH(1,1) EGARCH(1,1) TGARCH(1,1) Skewness Kuross JB (.) (.) (.) (.) (.) Q(1) (.345) (.74) (.59) (.79) 7.48 (.319) Q (1) (.34) (.45) (.56) (.511) (.38) Skewness Kuross JB 573 (.) (.) (.) (.) (.) Q(1) 1.75(.93) 1.64 (.31) (.53) (.55) 1.15 (.333) Q (1) (.659) (.943) (.).7684 (.973) 3.14 (.956) Skewness Kuross JB (.) 38.6(.) 56.8 (.) (.) (.) Q(1) (.17) (.6) (.67) (.18) (.6) Q (1).157 (1.) (.99) (.1) 1.91 (.965) (.984) Skewness Kuross JB (.) (.) 19.6 (.) (.) (.) Q(1) 14.4 (.7) 13.(.15) (.665) (.8) 1.37 (.135) Q (1) (.64) (.45) (.66) (.66) 174