Modelling Exchange Rate Volatility Using Asymmetric GARCH Models (Evidence from Sierra Leone)

Transcription

1 Inernaonal Journal of Scence and Research (IJSR) ISSN (Onlne): Imac Facor (01): Modellng Exchange Rae Volaly Usng Asymmerc GARCH Models (Evdence from Serra Leone) Mlon Abdul Thorle 1, Lxn Song, Xaoguang Wang 3, Muhammad Amn 4 School of Mahemacal Scences, Dalan Unversy of Technology, Dalan, 11603, P.R. Chna Absrac: Ths arcle examnes he accuracy and forecasng erformance of volaly models for he Leones/USA dollars exchange rae reurn, ncludng he ARMA, Generalzed Auoregressve Condonal Heeroscedascy (GARCH), and Asymmerc GARCH models wh normal and non-normal (suden s and skewed Suden ) dsrbuons. In fng hese models o he monhly exchange rae reurns daa over he erod January 004 o December 013, we found ha, he Asymmerc (GARCH) and GARCH model beer fs under he non-normal dsrbuon han he normal dsrbuon and mrove he overall esmaon for measurng condonal varance. The GJR-GARCH model usng he skewed Suden - dsrbuon s mos successful and beer forecas he Serra Leone exchange rae volaly. Fnally, he sudy suggess ha he gven models are suable for modelng he exchange rae volaly of Serra Leone and he Asymmerc GARCH models shows asymmerc n exchange rae reurns, resulng o he resence of leverage effec. Gven he mlcaon of exchange rae volaly, he sudy would be of grea value o olcy makers, nvesors and researchers a home and abroad n romong develomen of he caal marke and foregn exchange marke sably n emergng economes. Keywords: Asymmerc GARCH Model, Exchange raes volaly, Leverage effec, Fnancal Marke, Forecasng 1. Inroducon Modelng and forecasng of exchange rae volaly has become a relevan asec and ask n fnancal markes and he economy. The s because he volaly of exchange rae reurn can be seen as a measuremen of he rsk for nvesmen, asse allocaon and rovdes essenal nformaon for nvesors o make he correc decsons. I has ganed consderable aenon o marke arcans, nvesors, olcy makers n order o fully undersand he changes and he fnancal sably of an economy. Exensve research reflecs s moran n he volaly of nvesmen analyss, secury valuaon, and rsk managemen, radng and hedgng sraegy, sock marke and moneary olcy decson makng. In fnance, researchers always u a lo of neress n modelng and forecasng volaly of exchange rae reurns, falng whch could lead o crss and ossble falure n he fnancal marke. A crucal ar of rsk managemen s measurng he oenal fuure losses of a orfolo of asses, and n order o measure hese oenal losses, esmaes mus be made for fuure volales and correlaons. Ths research does no undermne he mlcaons on fnancal markes exchange rae volaly. However, sudes on exchange rae volaly are rear n emergng fnancal markes lke Serra Leone. Every economy uses boh moneary and fscal olces for sablzaon. In Afrca economes, banks as well as oher fnancal nsuons lays moran role as deosores and rovdes fnancal nsrumen for household wealh, mananng aymen sysem and used as vehcles for mlemenng moneary and fscal olces for mananng confdence n he fnancal secor and hence economc growh. They usually nves n foregn exchange nsrumens hus he need for accurae modelng and forecasng of volaly. Ths s he case for Serra Leone, he Mnsry of Volume 3 Issue 11, November Lcensed Under Creave Commons Arbuon CC BY Fnance and he Bank of Serra Leone are he auhores ha manage fscal and moneary olces resecvely. The fscal olcy am a enhancng domesc revenue moblzaon, reduces he overall budge defc as well as reduce domesc deb. Moneary olcy focuses on mananng rce sably, conssen wh hgh and susanable economc growh. The Bank of Serra Leone srenghens s mlemenaon n he asec of moneary olcy hrough mrovng s lqudy forecasng, develo and deeen he nerbank marke, hrough roer lqudy managemen sraegy and nroduce a reserve requremen on foregn currency deoss o conrol he growh n he growng sze of oal commercal banks deoss. The Serra Leone economy s very sensble o flucuaons n he Leones/USD exchange rae gven he fac ha he counry generally mor n US dollars. In general, sudes have aemed o examne exchange rae and macroeconomc managemen (moneary and fscal olces) n Serra Leone, bu here s no secfc sudy on he modelng and forecasng of exchange rae volaly n Serra Leone, hs herefore movae he sudy. Ths sudy s he frs laboraory es case, whch ams a fng a volaly models o he Serra Leone Leones and USA dollar exchange rae and o assess he accuracy of he forecass roduced. Fnancal me seres exhb ceran characersc feaures such as volaly cluserng, leokurc and he Generalzed Auoregressve Condonal Heeroscedascy (GARCH) model of Bollerslev (1986) has ganed n oulary of s ably o address hese ssues bu, somemes fals o caure he fa-al roery of fnancal daa. Ths has lead o he use of non-normal dsrbuons (Suden-,Generalzed Error Dsrbuon and Suden-), whn many non-lnear exensons of he GARCH model whch have been roosed by Exonenal GARCH (EGARCH) of Nelson (1991), he Glosen, Jagannahan, and Runkle, he so-called GJR (1993) and he Asymmerc Power ARCH (APARCH) of Dng, Granger, and Engle (1993), o beer model he fa-aled (he excess kuross), skewness and leverage effec characerscs. Paer ID: OCT

2 Inernaonal Journal of Scence and Research (IJSR) ISSN (Onlne): Imac Facor (01): These models assume Gaussan normal dsrbuon n modelng and forecasng he reurns of fnancal me seres (Le Baron, 1999). Modelng her volaly s moran o asses hese reserve asses n banks and n currency orfolo managemen. Exorers and morers encounered ransacon losses f no managed roerly, and hus accurae forecasng models are needed o avod hese losses hrough hedgng and o reduce he cos of foregn exchange ransacon. The romnen asecs abou me seres models s abou volaly modelng dscussed by (Engle, 1995) and laer exensons covered no only volaly (Andersen, 1997) bu also excess kuross by (Balle, 1989; Hseh, 1989) and volaly cluserng (Lux, 000). Anoher moran area s deermnng he dsrbuon for reurns generaed by fnancal me seres (Barndorff e.al 001). The reurns (shocks) whch are creaed by he changes n rces of sock marke or n currency marke are o be modeled n order o enhance cos effcen managemen. Dfferen deas exs among researchers regardng he shae of he dsrbuons of hese reurns. The Gaussan normal dsrbuon s he mos oular among hem s symmerc and hus fal o caures he fa als (Jensen, 001), kuross and skewness roeres (Arfovc and Gencay, 000) whch are wdely revalen n he reurns generaed by he fnancal asse rce changes. Researchers and raconers wdely used non-normal dsrbuons o draw random effec whle analyzng he fuure exchange raes. The mlcaon s ha hese varous mehods n modelng and forecasng volaly are raher n conclusve n naure. Therefore, n hs aer, we use normal dsrbuon, suden -dsrbuon and skewed Suden - dsrbuon for modelng Serra Leone exchange rae volaly. In vew of he leraure survey and relevan dscussons clearly reveals ha no dealed sudes were conduced on modelng of he exchange rae volaly n Serra Leone. The am of hs arcle s o conrbue o he exsng leraure on he grounds of modelng exchange raes volaly and fnd he model ha bes esmaes he volaly of exchange rae, and o evaluae he forecasng erformance of he Asymmerc GARCH models wh he normal, suden s and skewed- dsrbuons for he Leones/Uned Saes dollars exchange raes of Serra Leone. To hs end, we ask crucal and moran emrcal quesons () how bes he model fs exchange rae volaly n Serra Leone? () why do we need o forecas exchange rae volaly? () how bes he sudy wll nform olcy marker and nernaonal develomen organzaons who are asssng n he develomen and romoon of moneary and fscal olces? These herefore also movae he sudy, as s execed ha he oucomes of he sudy would be relevan o academcs, olcy makers, nvesors, nernaonal organzaons such as he World Bank, he Inernaonal Moneary Fund (IMF) and foregn governmens ha are neresed n faclang rade and enhance develomen of he caal marke and foregn exchange marke n emergng economes. Snce sable exchange rae volaly s convenonally requred o enhance susanable economc growh and develomen of emergng markes n an economy. The res of he arcle s srucured as follows: In secon, s Leraure revew, secon 3 se ou he mehodology and Volume 3 Issue 11, November Lcensed Under Creave Commons Arbuon CC BY model secfcaons, secon 4 resens he emrcal resuls and dscussons. Fnally, concluson s resened n secon 5.. Leraure Revew Ths secon revews he body of exsng knowledge on he connecon beween exchange rae volaly and forecasng n an economy. There are varous research and onons on modelng he volaly of exchange rae n a gven economy. Condonal heeroscedascy models for me seres have lay a val role n fnancal forecasng, rsk managemen and s desgns o make fnancal decsons on he bass of he observed rce of asse. Volaly s an moran arameer n rsk assessmen and managemen and changes as he marke rces of fnancal roducs change. To caure he morance of negave reurns GJR-GARCH and DGE -GARCH models nroduces he leverage arameer. Ths aer ncororaes he GARCH, GJR-GARCH and DGE -GARCH models and negraes wh ARMA o comue he reurn and forecasng exchange raes. Though hese models have been horoughly researched n he las wo decades sll a large ga s uncovered n he raccal alcaon as hey all model volaly ndvdually and hey come ou wh her fndngs. The volaly and ARMA models ulmaely ends u n forecasng he fnancal me seres lke share rces and exchange raes (Gullaume e. al, 1997) whch are acvely ursued no only for buyng and sellng decsons bu also for roecng he asse orfolos ha s carred ou for sasfyng he nvesors, regulaors, governmens and oher develomen arners who nves n hese fnancal nsrumens. Ryan & Worhngon (004) used he GARCH n Mean o assess he mac of marke, neres rae and foregn exchange rae rsks on he sensvy of Ausralan bank Sock Reurns. Olowe (009) examned he volaly of Ngeran Nara / Dollar Exchange rae by fng sx unvarae GARCH models wh suden s nnovaons usng monhly daa and found ha he bes erformng models are he Asymmerc Power ARCH and TS - GARCH. Accordng o Hung-Chung e al. (009) shown ha he GARCH model wh an underlyng leokurc asymmerc dsrbuon ouerforms one wh an underlyng normal dsrbuon for modelng volaly of he Chnese Sock Marke and smlar sudes underaken by Wlhelmsson (006) uses nne ossble error dsrbuons o model he volaly of he Sandard & Poor s 500 sock ndex wh he leokurc dsrbuons workng ou bes and also esablshed ha he use of fa aled error dsrbuons whn a GARCH (1,1) framework leads o mroved volaly forecass. In Ghana, Adjas (008) examned he mac of exchange rae volaly on he local Sock Exchange usng an Exonenal GARCH model for her urose and observe ha here s a negave relaonsh beween he exchange rae volaly and sock marke reurns. Balaban (004) comared he forecasng erformance of symmerc and asymmerc GARCH models wh he US Dollar/Deusche Mark reurns seres was flered usng an AR (1) rocess and he GARCH (1, 1), GJR-GARCH(1,1) and EGARCH(1,1) volaly equaons are used. The auhor found ha he EGARCH model erforms beer n roducng ou of samle forecass wh he GARCH (1, 1) closely followng whereas he GJR-GARCH fares wors. Also Aggarwal (1981) exlore he Paer ID: OCT

3 Inernaonal Journal of Scence and Research (IJSR) ISSN (Onlne): Imac Facor (01): relaonsh beween changes n he dollar exchange raes and change n ndces of sock rces usng monhly daa from 1974 o 1978 on sock rces and effecve exchange raes for he USA and found ha here s a osve correlaon and sronger relaonsh exs n he shor run han n he long run among he varables examned. Govannn and Joron (1987) also arrved wh smlar resuls of Aggarwal (1981) n case of he USA. Rahman and Uddn (009) examned he relaonsh beween exchange raes and sock rces for hree Souh Asan counres (Bangladesh, Inda and Paksan) and found negave relaonsh among he varables. In accordance o (Ken Johnson, 000) esablshed ha exchange raes s more moran, bu less suded varable when comared o shares, bonds and eques. Fnancal me seres end o be non-saonary (Hamlon, 1994) meanng ha addonal daa wll no only change he mean bu also he varance, whch s an medmen n forecasng. The argumen of non saonary naure s aken care of by naural logarhm dfferencng. Few sudes rove ha he reurn dsrbuons are no-erfecly normal and hey are eher skewed or wh leokurc roery wh fa als (Lux, 1998) and show - dsrbuon aern. Fnancal me seres rsk managemen s concerned abou he negave reurns a he lef al of a dsrbuon (Belra, 1999) and hey are o be quanfed for effecve hedgng decsons. Our aer s alcaon orened and rovdes esmaon and comares he forecasng accuracy of he hree models ha forecas he exchange rae volaly (Leones/USA dollar) of he Serra Leone economy. A vas number of emrcal sudes whch examne he exchange rae volaly n boh develoed and develong counres hs ncludes, Maren (001), McKenze (1997), McKenze and Mchell (00), Sanchez-Fung (003), Tse (1998), Andersen and Bollerslev (1998), Vlasuso (00), Balle and Bollerslev (1989), Bollerslev, Engle, and Nelson (1994), (Taylor, 1986; Andersen, 1994), and Bene, Lauren and Lecour (000). The economc heory suggess ha neres raes, nflaon, money suly, rce level and oher macro elemens are moran varables n undersandng he oeraons of he economy and also for redcng he volaly and rends n exchange raes. Accordng o (Kuy, 010) suggesed ha a lower sock rce may lead o currency derecaon. There has been consderable aenon on wheher he exchange raes and sock rces have any emrcal relaonsh. Parcularly, hs ssue has become more aracve among researchers, nvesors and olcy makers afer he Asan Fnancal Crses (1997) and Global Fnancal Crses (009). Emrcally, s argued ha f he exchange raes and sock rces are ner-relaed hen s ossble o reven such crses by lookng a he drecon of causaly, gven ha he causaly runs from sock rces o exchange raes hen he olcy makers should kee an eye on sablzng he sock markes by enforcng he desrable economc olces. Based on he overall leraure survey ndcaes ha here are sudes whch have been aemed o modeled exchange raes volaly n boh develoed and develong economes as well as emergng fnancal markes. The emrcal resuls of hese sudes are mosly nconclusve n naure. In vew of he above leraure survey, he urose of resen sudy s o conrbue Volume 3 Issue 11, November Lcensed Under Creave Commons Arbuon CC BY o he exsng leraure on he grounds of modelng exchange raes volaly usng asymmerc GARCH models. To he bes of our knowledge, n he Serra Leone conex here are some sudes have aemed o sudy he deermnans of he real exchange rae n general, and here s no secfc sudy on modelng he volaly of exchange raes usng asymmerc GARCH models, n arcular. Hence, hs movaes us o rovde a modelng analyss of exchange raes volaly n he Serra Leone ersecve. The man lmaon s he avalably of daa coverage n erms of n-deh analyss wh resec o me. Dese hs lmaon he daa oban for hs sudy from (January 004 o December 013) s arorae o carry ou an emrcal sudy n modelng exchange rae volaly n he conex of Serra Leone. 3. Mehodology In hs secon, we brefly resen he models secfcaon, condonal dsrbuons and forecasng creras as well as daa se we use o model he Leones/US Dollars exchange rae reurns volaly n he Serra Leone economy. Ths arcle analyses he volaly of he Serra Leone exchange rae usng varous volaly models such as Auoregressve Movng Average (ARMA), GARCH, he Glosen,Jagannahan and Runkle(GJR) GARCH, Asymmerc Power Auoregressve condonal Heeroskedascy APARCH model of Dng e.al (1993) as well as he condonal dsrbuons such as normal, Suden- and skewed suden s- dsrbuons. In hs sudy hree dfferen crera s, Mean Squared Error (MSE), Mean Absolue Error (MAE) and Adjused Mean Absolue Percenage Error (AMAPE) are used o evaluae he forecasng erformance for he condonal heeroscedascy models. 3.1 ARMA Model The ARMA model was esablshed by Box and Jenkns (1970) n order o f daa so as o remove he lnear deendence n he seres and o oban he resduals whch are uncorrelaed. The seres y sasfy ARMA (, q), hen y can be descrbed as y y j j 1 j1 q, where and,for 1,, and j 1,, q are j and var. arameers. The 0 3. GARCH Model Emrcal evdence has shown ha a hgh ARCH order has o be obaned o cach he dynamcs of he condonal varance. The Generalzed ARCH (GARCH) model nroduced by Bollerslev (1986) ends o address hs ssue. I s based on an nfne Auoregressve Condonal Heeroscedascy (ARCH) secfcaon and allows reducng he number of esmaed arameers by mosng non-lnear resrcons on hem. The sandard GARCH (,q) model secfcaon can be exressed as follows: y x, 1, T, N0, (3.1) q (3.) j j 1 j1 Paer ID: OCT

4 Inernaonal Journal of Scence and Research (IJSR) ISSN (Onlne): Imac Facor (01): , 0, 0, s wde saonary f and only f 1. The mean equaon (3.1) can be exressed as a funcon of exogenous varables wh error erms. The s uncorrelaed bu s condonal varance s changng over me as he funcon of he as errors defned n Engle (198), and hen generalzed by Bollerslev (1986) who exended he ARCH model o have longer memory and more flexble lag Srucure. From (3.) s a consan erm, s he ARCH erm and s he GARCH erm. Ths model s wdely used n j forecasng and modelng he condonal heeroscedascy n fnancal me seres analyss. 3.3 The APARCH Model The Asymmerc Power Auoregressve Condonal Heeroskedascy (APARCH) model was nroduced by Dng e al.,(1993). I changes he second order of he error erm no a more flexble varyng exonen wh an asymmerc coeffcen akes he leverage effec no accoun. A me seres y, x : 1,, sasfes a lnear model wh APARCH errors, gven by y x, 1, T (3.3) The mean equaon of (3.3) can be wren as y E y F 1, where E y F 1 x such ha x F and 1 F : ys, xs1, s s where F mles he nformaon se a me and E y F 1 s he condonal mean of y gven F he nformaon se ll 1 me 1. The erm n (1) reresen he resdual reurns whch are he nnovaon of he me seres rocess.. The condonal varance equaon of APARCH (, q) can be wren as q j j 1 j1, (3.4) Where 0, 0, 0, 1 1, 1,,, 0, j 1,, q. Here j and q denoes he number of lagged erms and erms. The arameers,,, and j are o be esmaed where, reflecs he leverage effec or he asymmerc resonse arameer. Posve and negave nformaon wh regards wll resul n dfferen levels of effec on he rce volaly of he exchange rae. As a resul, when s osve mles negave nformaon ha ndcaes sronger mac on he rce volaly of he fnancal asse han osve nformaon. The condonal varance s secfed by a consan erm, he arameers 0 and 0 are requred o ensure ha s srcly osve. and j are weghs assgned o he lagged squared reurns and lagged varances resecvely, whch have an mac on he condonal varance. They rovde nformaon abou volaly of rces of socks from revous erod s n order o have exlanaory owers on curren volaly of marke rces n he fnancal marke. The arameer measures he volaly sllover effec based on Volume 3 Issue 11, November Lcensed Under Creave Commons Arbuon CC BY he nformaon n fnancal marke. The volaly s drecly lnked wh he rae of nformaon flow beween he fnancal markes whch wll cause changes of rces n he marke. The APARCH (, q) rocess s saonary and enals a general class of models whch ncludes secal cases as ARCH by Engle (198), GARCH by Bollerslev (1986), TS-GARCH by Taylor and Schwer (1986 ced n Dng e al., 1993), GJR-GARCH by Glosen e al. (1993 ced n Dng e al., 1993), and TARCH by Zakoan (1994 ced n Dng e al., 1993). The GJR-GARCH and DGE-GARCH models are dscussed. a) GJR-GARCH When we have q j j 1 j1 For 0, 0, 1 1, 1,,,, 0, j1,,, q j b) DGE-GARCH q j j 1 j1, where 0, 0, 0, 1 1, 1,,,, 0, j1,,, q j 3.4 Dsrbuon Assumons Emrcal evdence shows ha fnancal me-seres ofen exhbs non-normaly aerns such as excess kuross and skewness. I may be execed ha excess kuross and skewness dslayed by he resduals of condonal heeroscedascy models wll be reduced when a more arorae dsrbuon s used. In hs aer: he normal dsrbuon, he Suden- dsrbuon and he skewed Suden- dsrbuon are consdered n order o ake no accoun he skewness, excess kuross and heavy-als of reurn dsrbuons. I s clear ha Suden- and skewed Suden- dsrbuons exhb heavy-als Normal Dsrbuon The normal dsrbuon s wdely used n esmang and forecasng GARCH models. If he error erm follows a Gaussan, he log-lkelhood funcon of he sandard normal dsrbuon s gven by T 1 1 T ln ln ln 1 z s ndeendenly and dencally dsrbued L e z, Where (..d) and he 0, z var 1 and T s he number of observaon wh regards he reurn seres. z 3.4. Suden s -Dsrbuon The suden s -dsrbuon s used for fng GARCH model as suggesed by Bollerslev (1987) for he sandardzed error o beer caure he observed fa als n he reurn seres. I s symmerc around mean zero. The log-lkelhood funcon s of he form Paer ID: OCT

5 Inernaonal Journal of Scence and Research (IJSR) ISSN (Onlne): Imac Facor (01): T v1 v 1 1 z LT Tln ln ln v ln 1 vln 1, 1 v Where v s he shae arameer, v and (.) s he gamma funcon. The lower v s, he faer he als Suden s -dsrbuon Fernandez and Seel (1998) as ced n Aberg e al. (008) rovdes analyss of he suden- dsrbuon by addng a skew arameer whch was aled by Lamber and Lauren (000,001) o he GARCH model. The log-lkelhood funcon for he skew -dsrbuon s gven by v 1 v 1 LT T v s ln ln ln ln ln 1 sz m v T 1 ln 1 ln 1, 1 v here s he skew arameer or asymmery arameer, v s he degree of freedom of he dsrbuon and m 1 f z s, v1 v and 1 I 1 m m s 1 m. 1 f z v s 3.5 Daa Ths sudy used he monhly daa on exchange raes of he Leones agans he USA dollars (Le/USD) of he Serra Leone economy obaned from Cenral Bank of Serra Leone from January 004 o December 013 and hen ransformed no logarhmc reurn seres. The corresondng ransform rce seres no monhly logarhmc reurn are calculaed by usng he formula: r log E log E 1. Where E s he exchange rae and r denoes he reurns. 4. Emrcal Resul and Dscussons In hs secon, we resen he emrcal resuls as well as dscussons of esmaon resuls we obaned o accoun for he Leones/US Dollars exchange rae reurns volaly n Serra Leone. The analyss was done usng he R ackage o rovde emrcal resuls of he Serra Leone monhly exchange rae rces daa agans Uned Saes dollars. The arameer esmaon mehod ha we choose s he Maxmum Lkelhood Esmaon (MLE), and esmaes he models wh he gven dsrbuonal assumon o deermne he bes erformance of forecasng model of exchange rae volaly of Serra Leone examned. Some summary sascs for he monhly exchange rae reurns (Le/USD) are dslayed n Table 1 Table 1: Summary Sascs of Serra Leone monhly Exchange rae Reurns (r) (Leones/ USA ($)) Samle sze Mean Medan Var. S.Dev. Mn. Max. Skew. Kur. Jarque Bera Tes P-value e -16 The summary sascs of hs sudy s resened n able 1. Ths ndcaes ha he reurns seres have a daly osve mean of (0.0043) whle he daly volaly s (0.0101), whou loss of generaly he mean grows a a lnear rae whle he volaly grows aroxmaely a a square roo rae. The lowes monhly reurns corresond o ( ) and he bes monhly exchange rae reurns s (0.0603). The reurns seres of he exchange rae shows osve skewness. Ths mles ha he seres s flaer o he rgh. The kuross value s hgher han he normal value of erfecly normal dsrbuon n whch value for skewness s zero and kuross s hree and hs sugges ha he kuross curve of he exchange rae reurn seres s leokurc. The resuls of hs sudy reveal ha, he seres s no normally dsrbued. Our emrcal resul s conssen wh he Jarque-Bera (JB) ess oban above whch s used o assess wheher he gven seres s normally dsrbued or no. Here, he null hyohess s ha he seres s normally dsrbued. Resuls of JB es fnd ha he null hyohess s rejeced for he reurn seres and sugges ha he observed seres are no normally dsrbued. Table resens he arameer esmaon resuls of ARMA (1, 1) - GARCH (1, 1), GJR-GARCH (1, 1) and DGE-GARCH (1, 1) models wh he normal, suden s- and skewed Suden- dsrbuons and her corresondng -values. The resuls show ha he arameers esmaed n hese hree models are all sgnfcan under he gven condonal dsrbuons exce for he coeffcens of Mu under he hree condonal dsrbuons for he GARCH and Volume 3 Issue 11, November Lcensed Under Creave Commons Arbuon CC BY GJR-GARCH model whch are no sgnfcan. Under he normal dsrbuon, he sum of he GARCH arameer esmaes j s greaer han 1, mlyng ha he volaly rae model s srcly saonary whle ha of DGE-GARCH model s less han 1, whch ndcaes ha he model s well fed. For he suden s- dsrbuon, he sum of he GARCH arameer for GJR-GARCH model s less han 1, whch ndcae ha he volaly s lmed and he daa s saonary and he model s well fed, whle n he case of he GARCH and DGE-GARCH models he sum of he GARCH arameers s greaer han 1. Smlarly he sum of he GARCH arameer s less han 1 for he GJR-GARCH and DGE-GARCH models wh he skewed Suden -dsrbuon whch also show ha he shocks n volaly s lmed and saonary and he model s well fed, he sum s greaer han 1 n case of he GARCH model. The leverage effec erm (gamma) n boh he GJR model and he DGE s sascally sgnfcan bu s negave, mlyng ha negave shocks resuls o a hgher nex erod condonal varance han osve shocks of he same sgn, ndcaes ha he bad news (negave shocks) effec he volaly more han he good news. The able shows ha he esmaed of he DGE-GARCH model under he normal dsrbuon s whch s sgnfcanly dfferen from (GJR-GARCH) and decreases from o as he condonal dsrbuon changes o suden s- dsrbuon and skewed Suden- dsrbuon resecvely. Paer ID: OCT

6 Inernaonal Journal of Scence and Research (IJSR) ISSN (Onlne): Imac Facor (01): Table : Parameer Esmaon of he ARMA (1, 1)-GARCH (1, 1), GJR (1, 1) and DGE (1, 1) Models wh he Condonal Dsrbuons GARCH GJR-GARCH DGE-GARCH Condonal Normal Suden Normal Suden Normal Suden Dsrbuon Suden Suden Suden 6.39e -05.8e e e e e e e e -04 Mu(μ) (0.4066) (0.7471) (0.9646) (0.3490) (0.477) (0.4754) (0.0008) ( ) ( ) ar1 1 (< e-16) (< e-16 ) (< e-16) (< e-16) (< e-16) (< e-16) (< e-16) (< e-16) (< e-16) Ma1 1 (7.40e-08) (7.00e-08) (.51e-08) (1.69e-11) (5.39e-13) (5.35e-13) (< e-16 ) (< e-16) (< e-16 ) 9.36e e e e e e e e-03.90e-03 Omega (ω) (0.088) (0.01) (0.004) (0.0049) (0.0018) (0.0158) (0.0111) (0.0161) (0.0015) Alha (α1) (0.0060) ( ) ( ) (0.0803) ( ) (0.0678) ( ) ( ) (0.0001) Bea(β1) ( ) (7.9e -05) (5.81e -05) (0.011) (4.97e-14) (9.08e -13) (3.3e -06) (< e-16 ) (< e-16) Gamma(γ1) ( ) ( ) (0.0037) ( ) (< e-16) (< e-16) Dela (δ) ( ) ( ) ( ) Shae v (0.005) (0.076) ( ) (0.0444) (0.0134) (0.0101) Skew (1.4e-14) (5.1e-09) (.6e-13) Noe: The coeffcens reored as shown n he able are he maxmum lkelhood esmaes of he arameers and he -values are n arenheses for he ARMA (1, 1) - GARCH (1, 1), GJR-GARCH (1, 1) and DGE-GARCH (1, 1), models. The esmaon resuls of he models wh he condonal dsrbuons, ncludng log-lkelhood value, he Box-Perce sascs of lags 10, 15 and 0 of he sandardzed and squared sandardzed resduals, he Akake Informaon Creron (AIC) and Bayesan Informaon Creron (BIC), he ARCH es and her resecve -values are lsed n Table 3. Comarng he log-lkelhood, he Akake Informaon Creron (AIC) and Bayesan Informaon Creron (BIC) values among hese models GARCH and GJR-GARCH models beer esmae he exchange rae reurn seres han he DGE-GARCH model wh he skewed Suden -dsrbuon assumon gves beer resuls. The resuls also show ha, he skewed suden -dsrbuon and he suden s -dsrbuon ouerforms he normal dsrbuon, wh he skewed Suden -dsrbuon ouerforms he oher wo condonal dsrbuonal assumons dscussed n hs aer. Among hese models, GJR-GARCH wh skewed Suden -dsrbuon gves he hghes log-lkelhood value of The AIC and BIC values of he GARCH and GJR-GARCH models under he hree condonal dsrbuon gves he lowes values when comared o he DGE-GARCH model and ha he GJR-GARCH model wh he suden s -dsrbuon rovdes he smalles values of AIC (-7.599) and BIC ( ) resecvely, hs mles ha GJR-GARCH model under he suden s -dsrbuon rovdes a beer f for he monhly exchange rae reurns accordng o hs crerons. Table 3: Analyss of sandardzed resduals and fed arameers GARCH GJR-GARCH DGE-GARCH Condonal Normal -dsrbuon Dsrbuon -dsrbuon Normal -dsrbuon Normal -dsrbuon -dsrbuon -dsrbuon Log lkelhood Jarque-Bera Tes (0.0043) (0.0000) (0.0000) (0.0069) (0.0000) (0.0000) (0.0000) (0.0000) (0.0000) Ljung-Box Tes R (Q10) (0.1560) (0.65) (0.998) (0.310) (0.4510) (0.433) (0.57) (0.0647) (0.107) Ljung-Box Tes R (Q15) (0.0644) (0.0617) (0.0654) (0.4939) (0.0903) (0.0867) (0.4807) (0.0630) (0.0789) Ljung-Box Tes R (Q0) (0.0973) (0.1671) (0.176) (0.0963) (0.1865) (0.1756) (0.7096) (0.1197) (0.1669) Ljung-Box TesR^(Q10) (0.6610) (0.684) (0.7411) (0.7181) (0.8405) (0.807) (0.13) (0.139) (0.3517) Ljung-Box TesR^(Q15) (0.498) (0.09) (0.10) (0.5439) (0.3730) (0.3596) (0.0553) (0.15) (0.461) Ljung-Box TesR^(Q0) (0.5735) (0.377) (0.3359) (0.6670) (0.5038) (0.4801) (0.0637) (0.377) (0.697) LM Arch Tes (0.5734) (0.301) (0.948) (0.7698) (0.4467) (0.4511) (0.0814) (0.171) (0.4585) AIC BIC Volume 3 Issue 11, November Lcensed Under Creave Commons Arbuon CC BY Paer ID: OCT

7 Inernaonal Journal of Scence and Research (IJSR) ISSN (Onlne): Imac Facor (01): Noe: The able shown he -sascs and -values are n arenheses for ARMA (1, 1)- GARCH(1,1),GJR(1,1) and DGE(1, 1) models.(aic) reresen Akake Informaon Creron, (BIC) s Bayesan Informaon Creron (BIC), Ljung-Box Tes R (Sandardzed Resduals and Ljung-Box TesR^ (Square Sandardzed Resdual) R- ackage wh he es funcon o evaluae a en se ahead forecas usng 119 observaons for he monhly exchange rae reurns. The forecass are evaluaed usng hree dfferen measures whch rovde robusness n choosng he omal redcs models for he reurn seres. We consder he followng measures. The Jarque-Bera sasc o es he null hyohess of wheher he sandardzed resduals are normally dsrbued. The resuls resened n able 3 show ha he sandardzed resduals are leokurc and he Jarque-Bera sasc srongly rejecs he hyohess of normal dsrbuon whch means ha he fa-aled asymmerc condonal dsrbuons ouerform he normal for modelng and forecasng he Serra Leone exchange raes volaly reurns. The Ljung Box ess for he resduals have -values ha are sascally no sgnfcan ndcang ha no seral correlaon exss. The Ljung-Box sascs for u o weneh-order seral correlaon of squared resduals are no sgnfcan suggesng ha no sgnfcance correlaon exs. As for he LM-ARCH es he resuls reveals ha he condonal heeroskedascy ha exsed n he exchange rae reurns me seres have successfully removed, ndcang ha no sgnfcan aearance of he ARCH effec. 4.1 Forecasng The forecasng ably of he GARCH models has been dscussed recsely by Poon and Granger (003). We use he 1). Mean Squared Error (MSE): I quanfes he dfference beween values wh regards he esmaor and he rue values of he quany beng esmaed whn a gven samle. I s defned as follows Sh 1 MSE ˆ h 1 s ). Mean absolue error (MAE): I akes no consderaon he average of he absolue value of he resduals. I s smlar o he MSE bu s less sensve o large errors. S h 1 MAE ˆ h 1 S 3). Adjused mean absolue ercenage error: Adjused Mean Absolue Percenage Error (AMAPE) s a measure based on ercenage errors. 1 S h ˆ AMAPE h S 1 ˆ Here h s he number of lead ses, S he samle sze, ˆ s he forecased varance and s he acual varance. Table 4: Forecasng Analyss for he Exchange rae reurns wh he Condonal dsrbuons Exchange rae reurns GARCH GJR-GARCH DGE-GARCH (Le /USA($) Normal Suden- Skew- Normal Suden- Skew- Normal Suden- Skew- MSE MAE AMAPE Noe: MSE denoes he Mean Squared Errors, MAE s Mean Absolue Error, and AMAPE s Adjused Mean Absolue Percenage Error The resuls, as shown n he able 4 above, ndcae ha he forecasng erformance of he GJR-GARCH and DGE-GARCH models, esecally when fa-aled asymmerc condonal dsrbuons are aken no accoun n he condonal volaly, s beer han he GARCH model. However, he comarson beween he models wh normal, suden- and skewed Suden- dsrbuons shows ha, accordng o he dfferen measures used for evaluang he erformance of volaly forecass, he GJR GARCH model rovdes he bes forecass and clearly ouerforms DGE-GARCH and GARCH models and he DGE-GARCH model rovdes less sasfacory forecas resuls whle he oores forecas resuls was regsered for he GARCH model. Moreover, s found ha he skewed Suden- dsrbuon s more arorae for modelng and forecasng he exchange rae reurns volaly. 5. Concluson Modelng exchange rae volaly has receved consderable aenon from academes, marke arcan, olcy makers, nvesors and raconers n recen years as rovde a measure of rsk n he fnancal marke. I s moran o noe ha, orfolo selecon, asse valuaons, rsk managemen, oon rcng and hedgng sraeges rovdes he morance of modelng and forecasng he condonal volaly of exchange rae reurns. Ths arcle conrbues o he exsng leraure of volaly modelng and forecasng by he followng asecs by esmang and evaluae he forecasng erformance of volaly models for exchange rae reurns ncludng he ARMA, GARCH, GJR-GARCH, and DGE-GARCH wh normal, suden- and skewed Suden- dsrbuons for modellng he Serra Leones/USA dollar exchange rae reurns volaly. The resuls show ha he forecasng erformance of asymmerc GARCH Models (GJR and DGE), esecally when fa-aled asymmerc condonal dsrbuons are aken no consderaon n he condonal volaly, s beer han GARCH model. The esmaed arameers of he Models are sascally sgnfcan exce, he coeffcens of Mu for he GARCH and GJR- GARCH models under he hree condonal dsrbuons. Also, he coeffcens on he sandardzed resduals and squared resduals of 10, 15 and 0 and he ARCH effec are no sascally sgnfcance whch mles ha no seral correlaon exss n he exchange rae reurn seres and ha no sgnfcan aearance of he ARCH effec n he reurns seres and he varance equaon s correcly secfed. Volume 3 Issue 11, November Lcensed Under Creave Commons Arbuon CC BY Paer ID: OCT

8 Inernaonal Journal of Scence and Research (IJSR) ISSN (Onlne): Imac Facor (01): In relaon o he resuls found wh regards he exchange rae markes, he leverage effec erm (gamma) n boh he GJR-GARCH model and he DGE-GARCH model s sascally sgnfcan bu negave, ndcang ha he exsence of leverage effec s observed n reurns of he exchange rae marke. However, accordng o he dfferen measures used for evaluang he erformance of volaly forecass he GJR-GARCH model under he skewed Suden -dsrbuon rovdes he bes forecass and clearly ouerforms he GARCH and DGE-GARCH models. Moreover, s found ha he skewed Suden- dsrbuon s more arorae for modelng and forecasng he Serra Leone/USA dollars exchange rae volaly n arcular. Among all of hese models, he GJR-GARCH wh skewed Suden -dsrbuon gves he hghes log-lkelhood value and he smalles AIC and BIC values under he suden s -dsrbuon. The overall resuls, clearly suggess ha GJR-GARCH model couled wh a skewed Suden -dsrbuon erforms very well wh he daase, and wll be more suable and relevan n faclang rsk managemen sraegy for he Serra Leone/USA dollars exchange rae reurns volaly, whch wll rovde useful nformaon and benchmarks o nvesors and olcy makers for arorae decson makng n undersandng nvesmen sraeges and enhancng exchange rae sably n he economy. Fnally, fuure researches drecons could be nvesgaed o mrove he modelng Serra Leone/USA dollars exchange rae volaly whch could be beer esmaed by selecng shorer me nervals as well as nroducng long run erssence of shocks n he volaly wh fraconally negraed models and asymmerc models (FIGARCH, FIAPARCH, TS-GARCH and EGARCH) ha allow o beer caure he dynamcs of he reurn seres 6. Acknowledgemens We would lke o hank he Edor and referees for her commen and valuable suggesons. Ths work s suored by he Naonal Naural Scences Foundaon of Chna under he Grans [ ], [ ] and he Naonal Naural Scence Foundaon Research Funds for he Cenral Unverses n Chna (DUT1LK9). References [1] Adjas, C., H. S. A. D. (008). Effec of exchange rae volaly on he Ghana sock exchange, Afrcan Journal of Accounng, Economcs, Fnance and Bankng Research. 3, [] Aggarwal, R., (1981). Exchange Raes and Sock Prces: A Sudy of he US Caal Markes under Floang Exchange Raes Akron Busness and Economc Revew, 1, [3] Alberg, D., Shal, H. and Yosef, R. (008). Esmang sock marke volaly usng asymmerc GARCH models. Aled Fnancal Economcs, 18, [4] Andersen, T.G., (1994). Sochasc Auoregressve volaly: a framework for volaly modelng. Mahemacal Fnance 4, [5] Andersen T. and Bollerslev T., (1997). Heerogeneous nformaon arrvals and reurns volaly dynamcs, Journal of fnance, Volume 3 Issue 11, November Lcensed Under Creave Commons Arbuon CC BY [6] Andersen, T.G., & Bollerslev,T. (1998). 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