Modelling Volatility of Returns on the Egyptian Stock Exchange

Transcription

1 Modelling Volailiy of Reurns on he Egypian Sock Exchange Submied by Amr Saber Ibrahim Algarhi o he Universiy of Exeer as a disseraion owards he degree of Masers of Science in Economics and Economerics in Sepember 007 I cerify ha all he maerial in his disseraion which is no my own work has been idenified and ha no maerial is included for which a degree has previously been conferred upon me. Amr S Algarhi

2 Acknowledgemens I would like o hank my supervisor, Dr Andreea Halunga, for being an excellen influence from he beginning of his academic year and for her coninual help and suppor. I would also like o hank all oher eaching saff in he Economic Deparmen for he knowledge hey provided. The usual disclaimer applies.

3 3 To my family

4 Absrac This disseraion invesigaes condiional variance paerns in daily reurn series. For his purpose, wo symmeric and hree asymmeric GARCH models are employed. The analysis is conduced for CASE 30 Price Index which is considered over he period 998 o 007. Furhermore, Engle and Ng (993) diagnosic ess for news impac were applied. The esimaion resuls reveal ha he sock reurns can be reasonably well modelled using linear specificaions and GARCH- capures mos of he volailiy characerisics including volailiy persisence and fa-ailed disribuion of sock reurns. The AR(3)-GARCH-(,) and AR(3)-EGARCH(,), respecively, can be regarded as he mos appropriae models. JEL classificaion: C; C5; C53; G0. Keywords: volailiy modeling; GARCH models; asymmeric GARCH. 4

5 Liss of Conens I-Inroducion.. 7 II- Lieraure Review...8 III- Daa and Mehodology.... III-- Symmeric Volailiy models.4 III-- Asymmeric Volailiy models....6 III-3- Esimaion of he ARCH/GARCH models. 8 III-4- Diagnosic ess based on he news impac curve... 9 V- Empirical Resuls. 0 VI- Conclusion.. 8 References...30 Appendix

6 Lis of Tables Table : Descripive Saisics and Normaliy Tess for he Sock Reurns.0 Table : ADF and PP uni roo ess. Table 3: The Calendar (day-of-he-week) Effec.. Table 4: Parameer esimaes of differen GARCH models.4 Table 5: Tesing for ARCH effecs in differen GARCH. 6 Table 6: Skewness, Excess Kurosis and Normaliy Tess for he normalized errors..7 Table 7: Engle and Ng (993) diagnosic es..7 6

7 I- Inroducion: In convenional economeric models, he variance of he disurbance erm is assumed o be consan. This assumpion is called homoskedasiciy. However, many financial ime series exhibi periods of unusually large volailiy, followed by periods of relaive ranquilliydescribed as volailiy clusering. As a resul, he assumpion of a consan variance is inappropriae. The auoregressive condiional heeroskedasiciy (ARCH) and he generalized auoregressive condiional heeroskedasiciy (GARCH) models have become widespread ools for dealing wih such changing variances (ime series heeroskedasic models). The seminal Auoregressive Condiionally Heeroskedasic (ARCH) process proposed by Engle (98) o model his phenomenon has given a huge push o boh economeric model building and applied research. Engle s process proposed o model imevarying condiional volailiy using pas innovaions o esimae he variance of he series. Based on he ARCH model, Bollerslev (986) suggesed he more widely used Generalized Auoregressive Condiional Heeroskedasiciy (GARCH) model, which is an imporan ype of ime series model for heeroscedasic daa. I models a ime-varying condiional variance as a linear funcion of pas squared residuals and of is pas values. A variey of new exensions were produced since hen, some of which were moivaed by pure heory, while ohers were simply empirical rial-and-error suggesions. However, he GARCH model ofen does no fully capure he fa-ail propery of high frequency financial ime series, as financial asse reurns end o have disribuions ha exhibi fa ails and excess peakedness a he mean a characerisic known as lepokurosis. This has naurally led o he use of non-normal disribuions o improve he excess kurosis. Furhermore, he financial lieraure has documened asymmeric behaviour volailiy o shock- negaive shocks have higher impac on volailiy han posiive shocks. This asymmeric behaviour is referred o as he leverage effec and leads o he developmen of asymmeric volailiy models. Alhough here is a huge amoun of empirical research on sock marke behaviour, mos of he sudies have concenraed on he major developed sock markes. There has been no research devoed o invesigae he Egypian sock marke and o model he volailiy of reurns on he Egypian sock exchange. Egyp s Sock Exchange is comprised 7

8 of wo exchanges, Cairo and Alexandria. Boh are governed by he same board of direcors and share he same rading, clearing and selemen sysems. Boh exchanges were acive in he 940s and combined ogeher ranked he fifh in he world. However, he cenral planning and socialis policies adoped in he mid 950s led o he Sock Exchange dormancy beween 96 and 99. In 99, he Egypian governmen s resrucuring and economic reform program resuled in he revival of he Egypian sock marke which leaded o a major change in he organisaion of he Cairo and Alexandria Sock Exchange in January 997. This disseraion would, firs, give a brief survey on he main conribuions o he lieraure of he ARCH/GARCH models. There are a collecion of excellen survey aricles on ARCH/GARCH models ha are available. See, for example, Bollerslev, Chou and Kroner (99), Bollerslev, Engle and Nelson (994), Bera and Higgins (993), Engle (00) and Li, Ling and McAleer (00). Tesing for he presence of he day-of-he-week effec will be invesigaed for he Cairo & Alexandria Sock Exchage (CASE) 30 price index. Then, a specificaion of he condiional mean is provided. Moreover, his disseraion will esimae differen symmeric and asymmeric volailiy models and asymmery ess by Engle and Ng (993) are also employed. The plan of his disseraion is as follows. Secion II reviews he lieraure. Secion III provides a descripion of he daa and mehodology used. In secion IV, he resuls are represened and a discussion of he empirical evidence is provided. Finally, Secion IV gives some concluding remarks. II- Lieraure Review: Mandelbro (963) and Fama (965) observed ha he uncerainy of sock prices as measured by variances and covariances are changing hrough ime- financial reurns displayed volailiy clusering. Fama (965) also noed ha sock reurns end o exhibi nonnormal uncondiional sampling disribuions, in he form of skewness and excess kurosis. In oher words, he daa exhibi lepokurosis where he marginal disribuion of he daa ends o be heavy-ailed. In addiion o he lepokuric disribuion of he sock reurn daa, Black (976) has noed ha curren reurns and fuure volailiy are negaively The Alexandria Sock Exchange was esablished in 888, while Cairo Sock Exchange was esablished in 903. See hp:// 8

9 correlaed, inroducing he leverage effec which was furher invesigaed by Chrisie (98). According o he leverage effec, a reducion in he equiy value would raise he deb-o-equiy raio, hence raising he riskiness of he firm as manifesed by an increase in fuure volailiy (Bollerslev e al, 99, pp.3-4). As he resul of observing volailiy clusering, he assumpion of homoskedasiciy becomes inappropriae leading researchers o find an answer o how o model he volailiy clusering. The sandard approach of heeroscedasiciy is explicily o inroduce an independen variable x which predics volailiy. Consider he model y = ε x, where ε is a whie noise disurbance erm wih variance σ. Wih he realizaions of he { x } sequence are no all equal, he condiional variance of y is simply x σ and hence depends on he realized value of x. This soluion seems inappropriae, because i requires a specificaion of wha causes he changes in variance, raher han recognizing he condiional means and variances over ime. Then Granger and Anderson (978) developed he bilinear model which allows he condiional variance o depend on he pas realizaion of he series where he model is y = ε y and hen he condiional variance isσ. However, he uncondiional variance is eiher zero or infiniy which makes his an unaracive formulaion (Engle, 98, pp ). I was no unil he inroducion of The Auoregressive Condiional Heeroskedasiciy (ARCH) 3 model by Rober Engle (98) ha researchers sared modelling ime variaion in second or higher-order momens. Since hen he ARCH model has become a prominen ool o characerize such changing variances. Engle (98) sough a model o asses he validiy of a deducion of Milon Friedman (977) ha unpredicabiliy of inflaion was a main cause of business cycle. Friedman assumed ha he uncerainy abou fuure coss and prices no he level of inflaion- would preven enrepreneurs from invesing and lead o recession. This assumpion could be plausible if he uncerainy were changing over ime. Insead of he sandard approach of heeroscedasiciy and using he ad hoc variable choices of x, Engle (98) shows ha i is possible o simulaneously model he mean and he variance of a series. His original conribuion considered he enire class of higher-order ARCH(q) processes. Engle (98) considered he residuals of a simple model of he wage/price spiral for he UK over he 958:II-977:II period. In esing for he y 3 The erminology iself was invened by David Hendry. 9

10 ARCH errors, he Lagrange muliplier es for ARCH(4) errors was significan; hence Engle concludes ha here are ARCH errors. Engle shows ha he esimaion of he parameers of he mean and he variance equaions can be considered separaely wihou loss of asympoic efficiency. Firs, he mean equaion was esimaed using OLS and save he residuals. From hese residuals, an esimae of he variance equaion can be obained. The very imporan developmen o he ARCH model by Bollerslev (986), Engle s ousanding suden, called he Generalized Auoregressive Condiional Heeroskedasiciy (GARCH) is oday he mos widely used model. Bollerslev (986) exended Engle s work by developing a echnique ha allows he condiional variance o be an ARMA process. This essenially generalizes he purely auoregressive ARCH model o an auoregressive moving average model as he GARCH(p,q) allows for boh auoregressive and moving average componens in he heeroskedasic variance. Bollerslev (986) provided a comparison of a sandard auoregressive ime series model assuming a consan variance, a model wih ARCH errors and a model wih GARCH errors. Mos empirical implemenaions of GARCH(p,q)models adop low orders for he lag lenghs p and q, for example: GARCH(,), GARCH(,), or GARCH(,l) models. However, he applicaion ha appeared in Engle (98) and Bollerslev (986) was o inflaion in he UK and he USA respecively; he ARCH and GARCH lieraure has grown enormously and is applicaions have expanded o include: sock reurns, ineres raes, foreign exchange, ec. In finance, he risk/reurn effecs are of primary imporance and daa on daily frequencies are readily available o form accurae volailiy forecass. Thus finance is he field in which he grea richness and variey of ARCH models developed. There are numerous empirical applicaions of he ARCH mehodology in characerizing sock reurn variances and covariances. In equiy markes, ARCH effecs have been found o be highly significan. For example, highly significan es saisics for ARCH have been repored for individual sock reurns by Engle and Musafa (99), for index reurns by Akgiray (989), and for fuures markes by Schwer (990). High-order ARCH models could be found due o he weekendeffec in which he variance of reurns ends o be higher on days following closures of he marke. This effec is documened by French and Roll (986) using daily uncondiional variances. However, i remains significan in he low-order ARCH models for daily index reurns presened in French, Schwer, and Sambaugh (987), Nelson (989, 99), and Connolly (989), and a failure o ake proper accoun of such deerminisic influences 0

11 migh lead o a spurious seasonal ARCH effec. The imporance of adjusing for ARCH effecs in he residuals has been analyzed in Morgan and Morgan (987), Bera, Bubnys, and Park (988), Connolly (989), and Schwer and Seguin (990), where i is argued ha inferences can be affeced by ignoring he ARCH error srucure (Bollerslev, 99, pp.- ). As menioned before, financial daa exhibi lepokurosis. The condiional normaliy assumpion in ARCH generaes some degree of uncondiional excess kurosis, bu ypically less han adequae o fully accoun for he fa-ailed properies of he daa. One soluion o he kurosis problem is he adopion of condiional disribuions wih faer ails han he normal disribuion. Bollerslev (987) proposed ha η ha migh be drawn from a - disribuion wih ν degrees of freedom. In Baillie and DeGennaro (990), he assumpion of condiionally -disribued errors ogeher wih a GARCH(, ) model for he condiional variance is adoped, and i is found ha failure o model he fa-ailed propery can lead o spurious resuls in erms of he esimaed risk-reurn radeoff. As GARCH models can forecas condiional volailiy, he risk of an asse can be hen measured over he holding period. Hence, a number of exensions of he basic GARCH model have been developed ha are especially suied o esimaing he condiional volailiy of financial insrumens. A couple of mos imporan exensions will be menioned here. Firs, he non-negaively condiions in GARCH model may be violaed by he esimaed model and he only way o avoid his would be o replace arificial consrains on he model coefficiens in order o force hem o be non-negaive. Tha leads o he enormously imporan generalizaion Exponenial GARCH or EGARCH model of Daniel Nelson 4 (99). He recognized ha volailiy could respond asymmerically o pas forecas errors. In a financial conex, negaive reurns seemed o be more imporan predicors of volailiy han posiive reurns. Large price declines forecas greaer volailiy han similarly large price increases. Second, GARCH models can no accoun for leverage effecs, which lead o anoher exension o GARCH Model. The Threshold GARCH (TGARCH) model is developed by Glosen, Jagannahan and Runkle, which is a modified GARCH-M model originally allowing (Glosen e al, 993): () seasonal paerns in volailiy; i was made o capure he widely known January effec in sock marke. This sock marke anomaly has risen because socks in general and small socks in paricular have hisorically generaed 4 Dan Nelson premaurely passed away in 995. In his shor academic career, his conribuions were exremely influenial.

12 abnormally high reurns during he monh of January () Asymmeries in he condiional volailiy equaion - posiive and negaive innovaions o reurns o have differen impacs on condiional volailiy which was moivaed by early sudy of Black (976); Thus, he TGARCH model allows posiive and negaive innovaions o reurns o have differen impacs on condiional variance. This is accomplished by inroducing a dummy variable or indicaor funcion ino condiional variance equaion. (3) Nominal ineres rae o predic condiional variance. Furher generalizaions have been proposed by many researchers. There is an alphabe soup of ARCH models including: AARCH, APARCH, FIGARCH, FIEGARCH, STARCH, SWARCH, GJR-GARCH, TARCH, MARCH, NARCH, SNPARCH, SPARCH, QARCH, SQGARCH, CESGARCH, Componen ARCH, Asymmeric Componen ARCH, Taylor-Schwer, Suden--ARCH, GEDARCH, and many ohers. These models recognize ha here may be imporan non-lineariy, asymmery and long memory properies of volailiy and ha reurns can be non-normal wih a variey of parameric and nonparameric disribuions. III- Daa and Mehodology: The daily daa on CASE 30 Price Index raded on Cairo & Alexandria Sock Exchange (CASE) has been used in his disseraion. The daa used for CASE 30 Price Index were obained from CASE websie 5. The sample period spans from he firs ransacion day, January 998, o 9 March 007. The CASE 30 Price Index includes he op 30 companies in erms of liquidiy and aciviy in Egyp. I is weighed by marke capializaion adjused by he free floa. The daily reurn series { r } used for esimaions are obained by aking he naural logarihmic difference of he daily daa on CASE 30 Price Index P, r = ln( P P ) = ln P ln P () The resuls of OLS regressions will be spurious, as saed by Granger and Newbold (974), if he dependen variable is non-saionary. Firs sep is o deermine wheher he reurn series is saionary. One way of deermining wheher he reurn series is saionary is o use a formal es of saionariy, ha is, he Augmened Dickey-Fuller (ADF) es and Phillips-Perron (PP) es. 5 See hp://

13 In he financial lieraure, i is ofen documened ha sock reurns daa exhibis calendar effecs. The mos imporan calendar effecs sudied are he day-of-he-week effec where reurn of socks varies by he day of he week (significanly differen reurns on some day of he week; usually higher Friday reurns and lower Monday reurns) 6. In he Egypian Sock Exchange, he weekend covers Friday and Saurday which leads o he expecaion of higher Thursday reurns and lower Sunday reurns. To es for he calendar effec in CASE 30 reurns, a day-of-he-week dummy variable is used. The dummy variable akes a value of uniy for a given day and a value of zero for all oher days. Dummy variables for all days (Sunday, Monday, Tuesday, Wednesday and Thursday) are specified and an inercep is omied o avoid he dummy rap. The coefficien of each dummy variable measures he mean reurn of each day. The exisence of calendar effec will be confirmed from he F- saisics and deermine he day-of-he-week when he coefficien of a leas one dummy variable is saisically significan using -es. Thus, he calendar effec is esed as follows: 5 r = α D i= i i + u () where he coefficiens α i represens he mean reurn for each day and he u is he whie noise error erm. However, regressing he reurns on appropriaely defined dummy variables o capure he day-of-he week effec using he sandard OLS mehod has wo downsides. Firs, errors in he model may be auocorrelaed, as a resul cause misleading inferences. Second, he error variances may be ime dependen (heerosckedasiciy problem) ha affecs he esimaes of regression parameers making hem biased and inefficien. By including he lagged values of he reurn variables ino he equaion, he auocorrelaion could be ackled. To overcome he second downside, variances of errors are allowed o be ime dependen o include a condiional heeroscedasiciy ha capures ime variaion of variance in sock reurns. The heeroskedasiciy of sock reurns has been a subjec of a variey of sudies. There are differen ypes of condiional heeroskedasiciy models suggesed in he lieraure. 6 See Cross (973), French (980) and Cous e al. (000) for he lieraure of he day-of-he-week effec and see Aggarwall and Rivoli (993) and Dubois (986) for more applicaions on differen equiy markes. 3

14 III. Symmeric Volailiy Models ARCH(q) As menioned before, ime varying volailiy model was iniially expressed by Engle (98) as an auoregressive condiional heeroskedasiciy (ARCH) model. I is defined by is firs and second momen which can be referred o as he mean and variance equaion. In such models, he mean equaion is given by an AR process: r, he sock reurns series, is regressed on is pas values. r p = a0 + a r i= i i + ε (3) where he unpredicable shock ε = η h and η ~ iid(0,). η is he normalized error process. The errors ε ~ (0, h ) are assumed o be condiionally normally disribued F wih a zero mean and h variance, based on he informaion se, F, available a ime -. By definiionε is serially uncorrelaed, however ε is no independen asε are serially correlaed. ε will correspond o he innovaion in he mean for he sochasic process r. The variance equaion is formulaed o depic volailiy as he clusering of large shocks o he dependen variable and i can be represened by an ARCH(q) process. h q 0 + α jε j = α 0 α( L) ε, j= = α + (4) where α f 0 0 and α 0 j and L denoes he lag operaor. ARCH(q) process is saionary q if α j j= p. Errors are assumed o be condiionally normally disribued wih a zero mean and h variance, based on he informaion se, F, available a ime. This model is able o capure he volailiy clusering, i.e., for large price changes ( ε ε,..., ε, q large) o be followed by oher large price changes (he variance ofε is also large)., however ε is serially uncorrelaed. are 4

15 GARCH(p,q) The ARCH model was exended by Bollersev (986) leading o he class of generalised ARCH models, GARCH(p,q) 7, in which he condiional variance depends no only on he squared errors, bu also on is own pas values. The inenion of GARCH is ha i can parsimoniously represen a higher order ARCH process. For simpliciy, only he GARCH(,) model is shown here, as he GARCH(,) wih normally disribued errors is a widely used form for modelling he sock marke volailiy. Almos all of he oher GARCH models family members are developed from i, including symmeric and asymmeric ones. Parameers α f 0 0, 0 h α andβ 0 = + + α 0 αε βh so as o ensure ha he condiional variance h be nonnegaive. The parameer β measures he combined marginal impacs of he lagged innovaions, while α capures he marginal impac of he mos recen innovaion in he condiional variance. Empirical sudies of financial reurns show ha β esimaes are markedly higher han αesimaes. In addiion, i is necessary haα + β p. This condiion secures covariance (weakly) saionary case of he condiional variance. Ifα +β =, hen he process is no weakly saionary (uni roo case) and he condiional variance would no exis. A sraighforward inerpreaion of he esimaed coefficiens in (6) is ha he consan erm α 0 is he long-erm average volailiy. (5) GARCH- In he simple GARCH(,) mode, i is assumed ha he errors are normally disribued. However, The lieraure shows ha his assumpion is no saisfied as he disribuion funcion for he rae of reurn is fa-ailed. A fa-ailed disribuion has more weigh in he ails han a normal disribuion. Bollerslev (987) found ha GARCH models wih normal disribued shocks failed o model fa-ailed propery of sock reurns. He proposed he GARCH model wih -disribued errors, ε (0, h ) where n is he number of degrees of ~ n freedom for he condiional Suden s -disribuion. For finie n, his process will exhibi 7 ARCH model is a special case of a GARCH model, in which here are no lagged forecas variances in condiional variance equaion, p=0. 5

16 lepokurosis no only in he original daa series, bu also in he series of sandardized residuals. III. Asymmeric Volailiy Models Changes in sock prices end o be negaively relaed o he changes in sock volailiy. In oher words, i is ofen observed ha downward movemens in he marke are followed by higher volailiies han upward movemens of he same magniude. This phenomenon is known as a leverage effec. Symmeric GARCH models fail o accoun for his asymmery in he condiional variance. Accordingly, his problem has given rise o an array of asymmeric models. Among asymmeric models o capure volailiy clusering and asymmeric effecs of pas shocks of volailiy are: EGARCH, TGARCH and AGARCH. EGARCH The exponenial GARCH model, EGARCH, was proposed by Nelson (99) and one of is advanages is ha i explicily akes skewed disribuion ino accoun. The specificaion for he condiional variance is ln h α α η ψ η = ln h Noe ha he lef-hand side is he log of he condiional variance, so he condiional variance equaion is in log-linear form. This implies ha he leverage effec is exponenial and condiional variance is guaraneed o be nonnegaive. This overcomes he problem wih a sandard GARCH model where i is necessary o ensure ha all of he esimaed coefficiens are posiive. The EGARCH model uses η (he normalized value ofε ). Nelson (99) argues ha his sandardisaion allows for a more naural inerpreaion of he size and persisence of shocks. The presence of leverage effecs can be esed by he hypohesis ha ψ p 0. The EGARCH model capures boh he size (depending on α ) and he sign (depending onψ ) of pas shocks o volailiy. β (6) TGARCH The Threshold ARCH (TGARCH) was inroduced by Glosen, Jaganahan, and Runkle (993). The specificaion for he condiional variance is where I = if ε p 0, and I = 0 h α α ε = I + h 6 ψ oherwise. ε β (7)

17 In his model, good news and bad news have differenial effecs on he condiional variance- good news has an impac ofα, while bad news has an impac of α + ψ. If ψ f 0, we say ha he leverage effec exiss. If he coefficien ψ is saisically differen from zero, i can be concluded ha he daa conain a hreshold effec. AGARCH The Asymmeric GARCH model is used in he following specificaion: h α α ε ψ β = 0 + ( + ) + h The asymmeric effec in he variance equaion is rendered by he coefficienψ. If i is posiive, he unexpeced posiive innovaions will resul in higher volailiy, compared o he effec of negaive innovaions. The AGARCH capures he asymmery by allowing is new impac curve o be cenred a a posiiveε. (8) Tes of Misspecificaion A general es of misspecificaion for he GARCH models is o check if he fied model is able o capure all volailiy clusering and hus all dependence inε. Sinceε = η h, where ε ~ (0, h ) and η ~ iid(0,). Then if he condiional variance is correcly F specified hen η should be uncorrelaed wih any pas informaion. If GARCH model is capuring all dependence in he errorsε, hen he normalized error η will be independen. And if here are remaining ARCH effecs, he squared normalized residuals will be correlaed. ˆ η m = π 0 + π ˆ sη s + s= e (9) 7

18 III.3 Esimaion of he ARCH/GARCH models Suppose ha values of { ε } are drawn from a normal disribuion have a mean zero and a consan variance. From sandard disribuion heory, he likelihood of any realizaion of ε is ε L = exp. Since ε are independen, he likelihood of he join πσ σ realizaions is L T = πσ ε exp σ = and by aking he naural log of each side hen, T T ln L= ln(π ) lnσ σ T = ( ε ) (0) / Now suppose he model ε = η h where η ~ iid(0,) where h = α 0 + αε α qε q for ARCH(q) and h = α 0 + αε α ε β... β h for GARCH(q). Since q q + h + + he model is no longer of he usual linear form, OLS can no be used for GARCH model esimaion and anoher echnique, maximum likelihood (ML), is adoped. In general, maximum likelihood esimaion (MLE) is used o esimae he parameers of he condiional variance α α, α,..., α ) joinly for ARCH(q) and α α, α,..., α, β,..., β ) for = ( 0 q p q = ( 0 q p GARCH(q). A log-likelihood funcion is formed and he values of he parameers ha maximize i are sough. MLE can be employed o find parameer values for boh linear and non-linear models. The log-likelihood funcion relevan for a GARCH model can be consruced in he same way as for he homoskedasic case, as in equaion (0), by replacing σ wih h, T T ln L= ln(π ) ln h T = ε h () Inuiively, maximizing he log-likelihood funcion involves joinly minimizing T ln h T ε and h = T, since hese erms appear preceded wih a negaive sign and ln(π ) is jus a consan wih respec o he parameers. Equaion () of he likelihood funcion assumed ha 8 ε η = has a normal h (Gaussian) disribuion. However, he uncondiional disribuion of mos financial ime series has fa-ails. Bollerslev (987) proposed ha η ha migh be drawn from a -

19 disribuion wih ν degrees of freedom, where ν is regarded as a parameer o be esimaed by maximum likelihood. The log-likelihood funcion becomes hen, Γ ln L= T ln π where Γ( ) is a gamma funcion. [( ν + ) / ] ν + T ln h ln(+ ) = ν Γ( ν / ) T Bollerslev and Wooldridge (99) offers a consisen esimaes of boh he condiional mean and variance equaions by using a condiional Gaussian quasi-maximum likelihood (QML) funcion. Even if he assumpion ha η ~ (0,) is no valid and he rue disribuion is no normal, assuming a normal disribuion for he esimaion crierion will provide consisen esimaes 8. η () III.4 Diagnosic ess based on he news impac curve Engle and Ng (993) diagnosic ess based on he news impac curve were adoped in his assignmen o guaranee a mehodical analysis of he naure of condiional volailiy. Sign Bias Tes, Negaive Size Bias Tes, Posiive Size Bias Tes, and Join Tes provide a ool making i possible o esablish wheher one can predic he squared normalized residual by means of some variables observed in he pas which are no included in he volailiy model used. If hese variables can predic he squared normalized residual, hen he variance model is misspecified. The Sign Bias Tes examines he asymmeric impac of posiive and negaive innovaions on he volailiy which canno be prediced by he discussed model. The es has he following specificaion: where η is he sandardized residuals, η = a + bs + e S is a dummy variable ha akes he value of ifε is negaive and 0 oherwise. If he volailiy model being used is correc, hen b = 0. The egaive Size Bias Tes focuses on he differen impac of large and small negaive innovaions on he volailiy no prediced by he discussed model. The es has he following form: η = + bs ε a + e (3) (4) 8 See Hamilon (994, pp ) for more deailed discussion of Maximum Likelihood Esimaion. 9

20 On he oher hand, he Posiive Size Bias Tes examines he differen impac of large and small posiive innovaions. The es has he following specificaion: η a ( ε + e (5) = + b S ) Then, he Join Tes combines all above menioned ess in he following form: η = + b S + bs ε + b3 ( S ) ε a + e (6) If b = b = b 0, hen he volailiy model being used is correc. The -saisics of b 3 = parameer in he firs hree ess is used for possible biases, while in he Join Tes he F- saisics is used. IV- Empirical Resuls: Figure (), in he appendix, shows he ime series of he index reurns. The empirical phenomenon of volailiy clusering can be easily observed, i.e. periods of large changes will be followed by periods of similar large changes, while periods of small changes follow small changes. Table () provides he descripive saisics for he reurn series. As can be seen from he able, he series exhibi lepokuric and he skewness is differen from zero, mosly o he righ. This suggess he presence of asymmery oward posiive values. Besides, he disribuion of he sock reurns has heavy (faer) ails and hey are more peaked around he mean han he normal disribuion. As a resul, he Jarque-Bera es rejecs he null of he normaliy, as shown in Table (). Hence, he assumpion of a normal disribuion seems no o be saisfied. Table : Descripive Saisics and Normaliy Tess for he Sock Reurns # of Obs. Mean SD Max Min Skew Kur. Reurns oe: ** denoes significance a % risk levels. Jarque- Bera (p-value) (0.000)** ormaliy es (p-value) 84. (0.000)** The resuls of he OLS regressions will be spurious in he presence of non-saionary variables. To check for saionariy in he reurn series, uni roo ess 9 are applied o he CASE 30 price index series in log levels (no reurn series) in accordance wih he Dickey 9 Only he model including a drif and no rend is esed. 0

21 and Panula (987): he ADF (Augmened Dickey- Fuller) and PP (Phillips-Perron) ess are applied o he series in level and o firs difference. Table () gives he resuls and suggess ha, he CASE 30 series conain a uni roo in levels bu are saionary in firs difference; in oher words, he CASE 30 series is inegraed of order.therefore, by ransforming i in reurns, he series will be saionary. Table : ADF and PP uni roo ess Level Firs Difference ADF PP ADF PP -saisic ** ** oe: ** denoes he rejecion of he null of uni roo in he series a % risk levels. The lag lengh of 3 and is used for he ADF respecively and he runcaion lag 8 for he PP 0. Equaion () was esimaed which includes he day-of-he-week dummy variables on he righ hand side of he equaion. The resuls are presened in Table (3). From he resuls of equaion (), here are wide variaions of reurns across he days. Reurns for he days of Thursday, Wednesday and Sunday are higher han reurns of oher days; however, Wednesday and Sunday are no saisically significan. The maximum average reurn of occurs in Thursday, while reurns in Monday and Tuesday are negaive, and respecively [as shown in Figure ()]. The significance of F-saisics suggess he presence of he calendar effec. However, he Ljung-Box Q-saisic rejecs he null of no auocorrelaion indicaing ha he residuals of he model are auocorrelaed. As a resul, Newey and Wes (987) heeroskedasiciy and auocorrelaion consisen covariance marices, which provide consisen esimaor in he presence of auocorrlaion and heeroskedasiciy, are employed in calculaing he -saisics. Table (3) reveals ha he calendar effec (he day-of-he-week effec) can be presened in Thursday as he las day in he week as menioned before. As he coefficien of he dummy variables for Thursday δ 5 is saisically significan, i will be included in he mean equaion for he res of he disseraion. 0 For he PP es, he runcaion lag 5 for he Newey-Wes correcion is specified, ha is, he number of periods of serial correlaion o include. I is based solely on he number of observaions and is calculaed as, 9 q = floor 4( T ) 00. See Appendix for all figures.

22 Table 3: The Calendar (day-of-he-week) Effec Coefficien Sd. Error -value p-value Sunday (0.0009).05 (.66) 0.04 (0.097) Monday (0.0008) Tuesday (0.0007) Wednesday (0.0007) Thursday ** (0.0008) -0. (-0.3) (-0.53). (.9).89 (3.0) 0.83 (0.80) 0.69 (0.594) 0.77 (0.98) (0.006) F( ) Durban Wason.63 oe: ** Significan a he % level. * Significan a he 5% level. The ewey-wes s sandard errors, -values and p-values are repored in parenheses. R Equaion (3) was esimaed. The mean equaion of he reurn series is assumed o be modelled as an auoregressive process (AR). An AR(3) process is specified for which he lag lengh is obained using he Akaike informaion crierion (AIC). In using his crierion o compare alernaive models, he models mus be esimaed over he same sample period so ha hey will be comparable. AR(3) was compared wih alernaive models, where he number of observaions were kep fixed. AR(3) was chosen as i had he smalles AIC value ( ). The fi of he model improves, when he AIC approaches. r = r 0.048r r 3 + ε (0.0003) (0.0) (0.0) (0.0) he sandard error Durban-Wason=.00 F-es=30.9** The same AR(3) is hen used when esimaing differen GARCH models for he condiional variance in he res of he assignmen. To es he misspecificaion of he condiional mean equaion, many useful diagnosic checks were employed. Firs, he correlogram of he residuals of he fied model is shown in fig (5). To es for he error auocorrelaion, he Godfrey-Breusch Tes which was applied. The AIC usually chooses more lags han Schwarz Bayesian Crierion (SBC) does, however AIC will be preferred here o assure he absence of auocorrelaion. According o SBC (more parsimonious), lag one would be chosen.

23 The F-form for lag, 9, 6, and 3 were.93, 0.97,.07 and 0.89 respecively, where he null hypohesis was no rejeced, indicaing ha he residuals are no correlaed. This is srong evidence ha he AR(3) model fis he daa well. On he oher hand, he normaliy and asympoic ess were rejeced a % significance, and respecively. Tesing for heeroskedasiiciy based on Whie (980), where F-form is.83, rejecs he null a % significance implying he presence of heerskedasiciy. Engle (98) developed he Lagrange Muliplier (LM) es ha invesigaes he presence of ARCH effecs. However, he es for ARCH effec requires prior specificaion for he lag q, which was found o be 5. Tesing for he presence of ARCH(5) by applying F-es, he rejecion of he null hypohesis, F(5,63)= a % significance, suggess he presence of condiional heeroskedasiciy in he reurn series. Therefore, he presence of condiional heeroskedasiciy has been proved in he daa, and hen he following sep is o proceed wih esimaing differen symmeric and asymmeric GARCH models. Table (4) shows he resuls for esimaing: GARCH(,), GARCH-(,), EGARCH(,), TGARCH(,) and AGARCH(,). Noe ha quasilikelihood (QML) robus sandard errors described by Bollerslev and Wooldridge (99) were used since he residuals are lepokuric. Firs, GARCH(,) wih normally disribued error erm was esimaed which will serve as he benchmark model. The esimaed coefficiens are presened in Table (4). I is no surprise o find he sums of αand βcoefficiens are less han, which is he indicaion of he saisfacion of he saionary condiion and i also indicaes persisence of volailiy and ha long-memory ype ARCH process could be appropriaed o model daily reurn s volailiy. In addiion, hese wo coefficiens are saisically significan. Besides all hese coefficiens are posiive, which fulfils he sign resricion. Thus boh indicae ha his model is plausible. Moreover, he GARCH (,) model wih -disribued residuals was esimaed. The erm ψ is corresponding o suden- in he PCgive, which was found o be saisically significan. In he above symmeric GARCH models, he sings of he residuals have no impac in he variance equaion. However, in equiy markes, as menioned before, volailiy is no symmeric. To capure such asymmeric feaures knows as leverage effec. Three asymmeric GARCH models were esimaed. The EGARCH model was esimaed under he assumpion of normal errors. The leverage effec ermψ, denoed is posiive and 3

24 saisically differen from zero, indicaing he exisence of he leverage effec during he sample period. The esimaed coefficiens are presened in Table (4). Since exponeniaion always ensures posiiviy, EGARCH does no impose sign resricions on is coefficiens, also he saionary condiion becomes meaningless for EGARCH model. The TARCH(,) model fied o he daily CASE 30 reurns were esimaed. The hreshold effec erm ψ is negaive and significanly differen from zero, so here appears o be asymmeric effec. The esimaed AGARCH(,) shows ha he asymmeric parameer ψ is also significan. Table 4: Parameer esimaes of differen GARCH models GARCH GARCH- EGARCH TGARCH AGARCH Mean Equaion a (0.33) (0.0) (.69) (0.8) (.0) a (0.)** 0.38 (9.77)** 0.58 (9.98)** 0.49 (0.6)** 0.50 (0.7)** a (-3.35)** (-.8)** (-3.8)** (-3.7)** (-3.7)** a (3.49)** (3.55)** (.36)* (3.63)** 0.08 (3.57)** δ (.5)** Variance Equaion α (3.7)** (.5)* (.64)** (.37)* (-3.8)** (.53)* (3.3)** (.39)* (.6)* α 0.4 (5.8)** (5.8)** 0.35 (6.5)** 0.33 (5.8)** 0.8 (6.49)** β (35.)** (30)** (97.5)* (46.6)* (46.9)** ψ (9.33)** (.89)** (-3.9)** 46.9 (-.)* Log. Likelihood oe: ** Significan a he % level. * Significan a he 5% level. The --value is recorded in parenheses 4

25 Generally, almos all coefficiens, in variance equaion, are significan a he % level. The parameer measuring he leverage effec ψ in he variance equaion is saisically significan in all asymmeric models (EGARCH, TGARCH and AGARCH). I means ha news does have an asymmeric impac and ha posiive and negaive innovaions affec asymmerically he sock marke volailiy. This proves ha he daa have asymmeric feaures which are quie reasonable for high frequency/daily daa. Then, he asymmeric volailiy models EGARCH, TGARCH and AGARCH respecively are a beer fi for his daily daa o he GARCH model. I is clear ha he dummy variable for Thursday is saisically significan in he mean equaion. Furhermore, from Table (4), based on he performance of he goodness-of-fi saisics, he log-likelihood funcion of GARCH- is much higher han he oher models. Consequenly, GARCH- model is prevailing compared o ohers where i capures he fa-ailed propery of sock reurns. Therefore, i should be preferred over he oher models as i is a beer fi for he daa. In GARCH models, he errors should be uncorrelaed; however, he squared errors are correlaed. The degree of auoregressive decay of he squared residuals isα + β. From Table (4), i is obvious ha GARCH- and EGARCH have more persisence in volailiy compared o he oher models. Large values of α and β ac o increase he condiional volailiy, bu hey do so in differen ways. The largerα, he larger is he response of h o new informaion. Hence, ifαis large, a shock η has a sizable effec on ε and h. From figure (6), as α for EGARCH and GARCH- are 0.35 and 0.65 indicaing he effec of a shock η is more pronounced in he succeeding periods han he oher models. Furhermore from figure (6), he peaks in he series generaed using EGARCH are more persisen han he oher models, due o he larger value of β =0.96 and hence heir condiional variance displays more auoregressive persisence. Table (5) shows he es of misspecificaion of he differen GARCH models. The Lagrange Muliplier (LM) es invesigaes he presence of he remaining ARCH effecs. As daily daa are available, ARCH es for lags, 9, 6 and 3 are calculaed. All resuls for GARCH- model found o be insignifican indicaing he absence of any remaining ARCH effecs, which is a reasonable resul for daily daa. On he oher hand, he resuls for he oher GARCH models found o be significan. Hence, he GARCH- model capure he dependence in errors and herefore i is well specified. + 5

26 Table 5: Tesing for ARCH effecs in differen GARCH ARCH(3) ARCH(6) ARCH(9) ARCH() GARCH (0.058)* (0.0535) (0.0308)* (0.0000)** GARCH-.46 (0.39).3597 (0.73).7709 (0.0689).0900 (0.05) EGARCH (0.00)**.977 (0.0068)**.64 (0.005)** 7.83 (0.0000)** TGARCH (0.008)*.43 (0.0487)*.03 (0.035)* (0.000)** AGARCH 3.4 (0.8)*.9 (0.04)*.50 (0.08)* (0.000)** oe: ** Significan a he % level. * Significan a he 5% level. The p-values are repored in parenheses. To check he performance of he differen GARCH models, anoher specificaion ess were conduced. These various diagnosic checks for model adequacy are performed on he normalized residuals from he series ˆ η = ˆ ε ĥ. Firs he ACF of squared sandardized residuals for he various GARCH models were drawn. Table (6) shows us he esimaed sandardized skewness, excess kurosis and es for normaliy of he series. Normaliy es were applied o he sandardized residuals of GARCH models. The resuls presened in Table (6) lead o he conclusion ha GARCH models assuming normal disribuion canno ake ino reflec fa-ailed disribuion of reurns. In addiion, he resuls of skewness and excess kurosis are now saisfied in he asymmeric volailiy models. The values of skewness and excess kurosis of normalized residuals in he symmeric models are leaving furher from he sandard even han ha of he asymmeric models. This measuremen shows ha he symmeric volailiy model GARCH(,) fail o capure he asymmeric feaure of he condiional variance of he reurns. 6

27 Table 6: Skewness, Excess Kurosis and Normaliy Tess for he normalized errors Skewness Excess Kurosis ormaliy es GARCH (0.000)** GARCH (0.000)** EGARCH (0.000)** TGARCH (0.000)** AGARCH (0.000)** oe: ** Significan a he % level. Diagnosic ess by Engle and Ng (993) are o es wheher he esimaed sandard GARCH model has capured he leverage effec- wheher posiive and negaive shocks have differen effecs on he condiional variance. Usually, his es is conduced only on he sandardized residuals of GARCH (,) model wih normally disribued error erm, since i is he benchmark model for furher asymmeric analysis. However, his Disseraion employed diagnosic ess for all he volailiy models. The resuls are presened in Table(7). Table 7: Engle and Ng (993) diagnosic es Sign bias es egaive-size bias es Posiive size bias es Join es GARCH(,) -4.98* -3.47*.08* 3.8* GARCH-(,) EGARCH(,) -7.4* * TGARCH(,) -3.47* * AGARCH(,) oe: * denoes significance a 5%. For he join es, F-saisics is used insead of -value. The sign bias es uses he regression of equaion (3). The -es for he GARCH shows ha b is saisically differen from zero which means ha he sign of he curren period shock will make a difference in predicing he condiional volailiy, and GARCH(,) does no capure he asymmeric news impac, concluding ha here is leverage effec and he GARCH model is no correc model. However, on he oher hand, he -es for GARCH- indicaes ha b is no saisically differen from zero, which means 7

28 ha he sign of he curren period shock will no make any difference in predicing he condiional volailiy, Thus, GARCH-(,) capure he asymmeric news impac. Esimaing equaion (6) for he GARCH-(,) is o deermine wheher he effecs of posiive and negaive shocks also depend on heir size. Using he F-saisic, he null hypohesis b = b = b 0 was no rejeced. Table (7) also shows he esimae for 3 = asymmeric volailiy model: EGARCH, TGARCH or AGARCH. However, he form used in esing procedure omis he vecor of addiional explanaory variables and according o Engle and Ng in his case he es is conservaive. V- Conclusion: This disseraion aemped o model sock reurn volailiy of he CASE 30 price index, using high frequency daily daa covering a period from January 998 hrough 9 March 007. Volailiy clusering was observed in he Egypian Sock Exchange. This disseraion was also o analyze feaures of condiional variance. For his purpose, wo groups of GARCH models are esimaed: he symmeric GARCH (,) wih differen disribuions: normal and suden-; in addiion, hree asymmeric GARCH models: EGARCH(,), TGARCH(,) and AGARCH(,), in order o capure he leverage effec, which was also observed. Firs, he normaliy of reurns was esed and he resul was ha he reurns on CASE 30 price index are lepokuric, which is consisen wih general empirical findings. Furhermore, here is evidence ha he day-of-he-week effec is presen for reurns of CASE 30 price index, represened in Thursday wih he highes and saisically significan reurns. Tesing of he AR order of he mean equaion was hen performed and i is shown ha he reurns follow AR(3) process. Tess for heeroskedasiciy resuled in srong rejecion of he null hypohesis of homoskedasiciy for he series. The esimaion resuls of he differen volailiy models reveal ha he CASE 30 reurns can be reasonably well modeled using GARCH-(,) model and by using asymmeric models, mainly EGARCH(,). All leverage parameers are saisically significan, indicaing he asymmeric feaure of he daa because negaive shocks hi much harder he Egypian marke han posiive news. In oher words, reurns exhibi much more asymmery which corroboraes he usual observaion ha emerging sock markes like he Egypian marke may collapse much more suddenly and recover more slowly han 8

29 developed sock markes. The GARCH-(,) and EGARCH(,) over-performed ohers in erms of values of log-likelihood; he former also ou-weighed ohers wih respec o he improvemen of lepokurosis. The Engle and Ng (993) es was adoped which assures he asymmeric feaure of he daily daa. The resuls indicaes ha GARCH-(,) and asymmeric GARCH models capure asymmeric news impac, volailiy persisence and fa-ailed disribuion of he CASE 30 reurns. However, he resuls are ambiguous as he form used in he esing procedure omis he vecor of addiional explanaory variables and according o Engle and Ng in his case he es is conservaive. Finally, several exensions could emerge from his disseraion. One of hem is o improve he knowledge abou he volailiy forecasing models ha drive volailiy in he Egypian Sock Exchange. A comparison of he efficiency of ML, QML, and GMM esimaes using differen insrumen ses would also be recommended. The GARCH parameers of almos all he models ha were esimaed for he CASE 30 when added were very close o uniy, which indicaes ha inroducing models wih persisen shocks incorporaed ino hem e.g., IGARCH, FIGARCH and FIAGARCH- migh provide superior resuls. 9

30 References: Aggrawal. R. and P. Rivoli (989), Seasonal and day of he week effec in four emerging sock markes, Financial Review, 4, pp Akgiray, Veda (989), Condiional heeroskedasiciy in ime series of sock reurns: Evidence and forecass, Journal of Business, 6, pp Baillie, Richard T. and Ramon P. DeGennaro (990), Sock reurns and volailiy, Journal of Financial and Quaniaive Analysis, 5, pp Bera, A and Higgins M (993), ARCH models: properies, esimaion and esing, Journal of Economic Surveys, 7, pp Bera, Anil K., Edward Bubnys, and Hun Park (988), Condiional heeroskedasiciy in he marke model and efficien esimaes of beas, Financial Review, 3, pp.0-4. Black, Fischer (976), Sudies in sock price volailiy changes, Business Meeing of he Business and Economic Saisics Secion, American Saisical Associaion, pp Bollerslev, Tim (986), Generalized Auoregressive Condiional Heeroskedasicy, Journal of Economerics, April, 3:3, pp Bollerslev, T., (987), A condiional heeroskedasic ime series model for speculaive prices and raes of reurns, Review of Economics and Saisics, 69, pp Bollerslev T, Chou R and Kroner K (99), ARCH modelling in finance: a review of he heory and empirical evidence, Journal of Economerics, 5, pp Bollerslev, Tim and Jeffrey M. Wooldridge, (99), Quasi-Maximum Likelihood Esimaion and Inference in Dynamic Models wih Time Varying Covariances, Economeric Reviews,, pp Bollerslev T, Engle R and Nelson D (994), ARCH model, In Handbook of Economeics Volume IV, Engle R and McFadden D (eds), Elsevier Science: New York, pp Chrisie, Andrew A., (98), The sochasic behavior of common sock variances: Value, Leverage and ineres rae effecs, Journal of Financial Economics, 0, pp Connolly, Rober A., 989, An examinaion of he robusness of he weekend effec, Journal of Financial and Quaniaive Analysis, 4, pp Cous, J.A., C. Kaplanidis, C., and J. Robers (000), Securiy price anomalies in an emerging marke: The case of he Ahens Sock Exchange, Applied Financial Economics, 0, pp

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