Volatility Modeling for Forecasting Stock Index with Fixed Parameter Distributional Assumption

Transcription

1 Journal of Appled Fnance & Banng, vol. 3, no. 1, 13, 19-1 ISSN: (prn verson), (onlne) Scenpress Ld, 13 Volaly Modelng for Forecasng Soc Index wh Fxed Parameer Dsrbuonal Assumpon Md. Mosafzur Rahman 1, Md. Azzur Rahman and Md. Alamgr Hossan 3 Absrac The am of hs paper s o emprcally nvesgae he n sample and ou of sample forecasng performance of several GARCH-ype models such as GARCH, EGARCH and APARCH model wh Gaussan, suden-, Generalzed error dsrbuon (GED), suden- wh fxed DOF 1 and GED wh fxed parameer 1.5 dsrbuonal assumpon n case of Colombo Soc Exchange (CSE), Sr Lana. The daly All Share Prce Index (ASPI) of CSE from January, 199 o December 9, for a oal number of 15 observaons s used for emprcal analyss. We consder frs 195 observaons for n sample esmaon and las observaons for ou of sample forecasng evaluaon. Our emprcal sudy showed ha fxed DOF 1 of suden- densy and fxed parameer 1.5 of GED densy fal o mprove he n sample esmaon performance compared o suden- and GED dsrbuonal assumpon. Among all of hese models, APARCH model wh suden- densy gve beer n sample esmaon resuls. In case of ou-of-sample forecasng performance we found ha APARCH model wh all dsrbuonal assumpon gve lower value of Mean Squared Error (MSE) and Mean Absolue Error (MAE). Accordng o he denses suden- dsrbuon wh fxed DOF 1, suden- and Gaussan dsrbuonal assumpons gve beer resuls n case of GARCH, EGARCH and APARCH model respecvely. The esmaon resuls of SPA es sugges ha APARCH model wh Gaussan dsrbuonal assumpon gve beer forecasng performance n case of all share prce ndex of CSE, Sr Lana. 1 Asssan Professor, Deparmen of Sascs, Sascs and Mahemacs School, Yunnan Unversy of Fnance and Economcs, Kunmng-51, P.R. Chna e-mal: mosafz_bd1@yahoo.com Lecurer n Sascs, Braldah College, Rashah Educaon Board, Rashah, Bangladesh 3 Lecurer n Sascs, Kafura Degree College, Bangladesh Naonal Unversy, Bangladesh Arcle Info: Receved : Ocober, 1. Revsed : Ocober 3, 1. Publshed onlne : January 15, 13

2 11 Md. Mosafzur Rahman e al. JEL classfcaon numbers: C5, C53, H5, G15. Keywords: GARCH model, suden- wh fxed DOF 1, GED wh fxed parameer 1.5, MSE, MAE, SPA es. 1 Inroducon Generally fnancal me seres daa conans he propery of volaly cluserng, serally correlaon n he squared log reurn and long aled ness.e. he reurns show normally longer al han Gaussan dsrbuon where volaly cluserng ndcae large changes end o be followed by large changes and vce versa. To capure he volaly cluserng and long al propery of fnancal me seres daa Engle [11] nroduced Auoregressve Condonal Heerosedasc(ARCH) model. Earler emprcal research shows ha n order o accoun he dynamc of condonal varance hgh ARCH order s ulzed.e. we need o esmae many parameers. To solve hs hgh ARCH order problem Bollerslev [] proposed Generalzed Auoregressve Condonal Heerosedasc (GARCH) model. Bu mos of me GARCH model fal o capure he hc al propery compleely. Ths excess uross naurally leads o use non normal dsrbuon for errors. To solve hs problem many auhors esmae GARCH models by usng suden- dsrbuon for errors (for example, Bollerslev [5], Balle and Bollerslev [] and Bene e al [3]) whle oher auhors such as Nelson [5] and Kaser [17] suggesed Generalzed Error Dsrbuon (GED). Peers [9] examned he forecasng performance of GARCH, EGARCH, GJR and APARCH models under fa al and sewed dsrbuonal assumpon for FTSE 1 and DAX 3 ndces and found ha asymmerc GARCH models wh fa al densy gve beer resuls n case of n-sample esmaon bu usng non normal error dsrbuon does no clearly shows forecasng effcency. Alhough researchers showed ha someme GARCH models gve beer n-sample esmaon bu very poor forecasng performance, Anderson and Bollerslev [1] argued ha GARCH models provde good volaly forecass. There are several reasons such as usng nadequae measure for volaly or choosng wrong sascal loss funcon lead o provde worse forecasng performance n case of GARCH models. A large number of earler sudes fnd ou he approprae GARCH model and her forecasng performance bu hey dd no fnd any unque model for GARCH esmaon whch always gves beer resul for every soc mare. Kang e al. [1] showed ha componen GARCH and fraconally negraed GARCH models whch capure long-memory volaly provde beer forecasng performance compare o smple GARCH and negraed GARCH models. On he oher hand Cheong [] showed ha smple GARCH model characerze he Bren crude ol daa gve beer resuls han Asymmerc GARCH models. Ramon [3] used he specfcaons for he mean, varance and error usng ARMA, SARMA and GARCH models o predc he volaly of Phlppne nflaon rae. He esmae GARCH model wh Gaussan, suden- wh 1 DOF and GED dsrbuon wh 1.5 fxed parameer and found suden- dsrbuon wh fxed DOF 1 s he mos adequae choce for he varance of he error dsrbuon. We e al [3] compared he performance of dfferen number of lnear and non-lnear GARCH models o capure he volaly feaures of wo crude ol mares-bren and Wes Texas Inermedae (WTI) and found ha no model can ouperform all of he oher models for eher he Bren or he WTI mare across dfferen loss funcons. Lu e al [1] nvesgaed he specfcaon of reurn dsrbuon nfluences he performance of volaly forecasng for wo Chnese soc ndexes usng wo GARCH models and found

3 Volaly Modelng for Forecasng Soc Index 111 ha GARCH model wh sewed generalzed error dsrbuon gve beer resuls han GARCH model wh normal dsrbuon. The Colombo Soc Exchange (CSE) s he man soc exchange n Sr Lana whch was founded a 195. Recenly CSE maes remarable developmen as presen annual growh rae 1.% n whch was 3% n -. The world Federaon of Exchanges raed he CSE as he bes performng soc radng place for he fscal year of 9. There are wo ndces avalable for CSE and hese are: All Share Prce Index (ASPI) and Mlana Prce Index (MPI). Kumar and Mal [] nvesgaed wheher he common fndng regardng he asymmerc mpac of news on he volaly of reurns and found no sgnfcan asymmery n he volaly facors. Jaleel and Samaraoon [1] examned he mpac of lberalzaon of he Sr Lanan Soc mare on reurn volaly usng GARCH and TGARCH models for he perod from 195 o and found ha lberalzaon of he mare o foregn nvesors sgnfcanly ncreased he reurn volaly n he Colombo soc Exchange. Hanffa [13] examned soc reurn volaly of he All Share Prce Index (ASPI) usng Auo Regressve Condonal Heerosedasc (ARCH) and Generalsed ARCH models ha capure mos common sylsed facs on asse reurns. Addonally, he also examned he effec of exchange rae flucuaons have any mpac on he volaly of ndex reurn or no. The prevous sudy ndcae ha only few researchs have been conduced based on Sr Lanan Soc Mare and none of hem compare he forecasng performance of dfferen symmerc and asymmerc GARCH model wh fxed and flexble parameer ogeher for he error dsrbuon by robus es. Snce dfferen models f well for dfferen soc mares and dfferen error dsrbuon mprove he forecasng performance, so o fgure ou accurae forecasng models for any parcular soc amre s always neresng. For forecasng evaluaon we used mos commonly used wo loss funcon and Superor Predcve Ably(SPA) es of Hansen [1]. Therefore, he am of hs paper s o emprcally re-examne he n sample and ou of sample forecasng performance of several GARCH-ype models such as GARCH, EGARCH and APARCH model wh Gaussan, suden-, Generalzed error, suden- wh 1 DOF and GED wh fxed parameer 1.5 denses by usng wo loss funcons and SPA es of Hansen [1] n case of CSE. Ths sudy s mporan because hs s he frs me o compare he performance of dfferen GARCH ype models wh fxed parameer dsrbuonal assumpon by SPA es for CSE. Res of he paper s organzed as follows: secon presen he mehodology, secon 3 presen he characersc of daa and emprcal analyss and fnally secon presen he conclusons. Mehodology.1 Model and Error Dsrbuon Engle s [11] ARCH model has been wdely used n fnancal me seres analyss. In general fnancal reurn seres can be formalzed as: y / y1 h where logy / log (1) y 1 s he log asse prce, s he condonal mean of he reurns, h s he volaly process and ~ N(,1 ). In he case of auoregressve condonal

4 11 Md. Mosafzur Rahman e al. heerosedasc (ARCH) model we need long lag propery o mprove he goodness of f. For hs reason Bollerslev [] proposed generalzed ARCH (GARCH) model. The GARCH model consss of squared resduals and lag of condonal varance. Olowe [] used sx GARCH se models o nvesgae he volaly of Ngeran exchange rae. The Generalzed ARCH (GARCH) model of Bollerslev [] can be expressed as h w q 1 p h where w,,, 1,,..., q and 1,,..., p confrm posve condonal varance and he nnovaon can be expressed as he produc of an..d process wh mean and varance 1. If he sum of he parameers 1 hen he equaon () wll be saonary and f hey are close o 1 hen volaly parameer wll be more perssen. Snce fnancal me seres daa normally show hgh uross value whch ndcae he asymmery of he daa and GARCH model wh Gaussan dsrbuon fal o accoun such asymmery. The weaness of GARCH models encourages he researchers o develop dfferen GARCH se models whch can nclude sewness and asymmery. The popular models of asymmerc volaly ncludes Exponenal GARCH (EGARCH) model. Ths model was proposed by Nelson [5]. The specfcaon for condonal varance s: log( h ) w q p r log( h ) (3) 1 1 h 1 h The lef hand sde of equaon (3) s he log of he condonal varance. Ths mples ha he leverage effec s exponenal raher han quadrac. The presence of he leverage effecs can be esed by he hypohess. The mpac asymmerc f. In he EGARCH model no resrcons are requred o ensure he posve ness of he condonal varance. Anoher exenson of Asymmerc model was APARCH model. Taylor [3] and Schwer [31] nroduced he sandard devaon of GARCH model, where he sandard devaon s modeled raher han he varance. Dng e al [1] nroduced he Asymmerc Power ARCH (APARCH) model. In hs model he power parameer of he sandard devaon can be esmaed raher han mposed and oher oponal parameers are added o capure asymmery. The APARCH (p, q) model can be expressed as: h w p 1 q 1 h () where w,, ( 1,,..., p), and 1 1,( 1,,..., q). Snce APARCH model can consder he leverage effec no accoun so, s neresng o use for fnancal me seres daa. Furhermore, hs model ncludes seven oher ARCH exensons as specal cases (See Dng e al [1], Peers [9], Karlsson [19]). To complee he ARCH model esmaon we need he assumpon of condonal dsrbuon for he error erms. Fnancal me seres daa normally presen he fa-aled propery. In order o capure hs fa al propery we need o use some fa aled dsrbuon. EVews 5 sofware presen he esmaon of GARCH model under Gaussan, suden-, GED, suden- wh fxed degrees of freedom (DOF) and GED wh fxed ()

5 Volaly Modelng for Forecasng Soc Index 113 parameer (De Swar e al [9]). GARCH models are normally esmae by consderng he condonal dsrbuon of he error erms are Gaussan. The normal or Gaussan dsrbuon s a symmerc dsrbuon wh densy funcon: 1 ( x) / f ( x) e (5) Where s he mean value and s he varance of he sochasc varable. The sandard Gaussan dsrbuon consders he mean value and varance 1. Alhough Gaussan dsrbuon hold he lepourc propery bu hs lepouross s no enough o explan he lepouross propery whch s found n mos of he fnancal daa (Bollerslev, [5]). Therefore, one should ae hs no accoun and use condonally lepourc dsrbuon for he error. One alernave possbly s Suden- dsrbuon. The densy funcon of suden- s gven by f ( x) [( v 1) / ] v [ v / ](1 x / v) ( v1) / Where v s he degree of freedom (df) v. If v ends o he suden- dsrbuon converges o normal dsrbuon. Anoher mos commonly used fa al dsrbuon s he Generalzed Error dsrbuon (GED). The GED s a symmerc dsrbuon and playurc. The GED has he followng densy funcon. () 1 ve f ( x) ( v1) x / v 1/ v (7) Where 1/ v 3/ v / v 1/ I ncludes he normal dsrbuon f he parameer v has he value. For DOF v < ndcae fa al dsrbuon. Las wo error dsrbuons gve flexble condon for he user o used DOF and dfferen parameer. If he DOF of suden- dsrbuon has value abou 3 or above hen can be argued ha he suden- dsrbuon s close o he normal dsrbuon (Soyanov e al [33]). Ramon [3] esmae GARCH se model wh under Gaussan, suden- wh 1 DOF and GED dsrbuon wh fxed parameer 1.5. So, a our sudy we consder wo fx parameer dsrbuon and chec eher hese fx parameer can mprove esmaon effcency for boh n sample or ou of sample or no.. Forecasng Technque The forecasng performance requres he mnmzaon of he loss funcon propery. There s no unque creron whch s always conssen for provdng bes forecasng performance (see Bollerslev e al [] and Lopez []). Many auhors argued for usng real loss funcons o evaluae he volaly forecasng. As for example, Engle e al [1] and

6 11 Md. Mosafzur Rahman e al. Wes e al [35] suggesed prof based and uly based crera for evaluang he volaly forecass. Marcucc [3] used seven dfferen loss funcons such as Mean Squared Error, Mean Absolue Devaon, RLOG loss funcon of Pagan and Schwer [7], QLIKE loss funcon of Bollerslev e al [] and HMSE of Bollerslev and Ghysels [7] for evaluang complee forecasng performance of dfferen volaly models. Laer We e al [3] used sx dfferen loss funcons o measure he dfference beween he realzed and he esmaed condonal varances. These loss funcons are Mean squared Error and Mean Absolue Error, Heerosedascy-adused MSE and MAE, Logarhmc Loss (LL) funcon and QLIKE loss funcon. Therefore, we use sandard loss funcon such as Mean Squared Error (MSE) and Mean Absolue Error (MAE) for our forecasng evaluaon. These are esmaed by he followng equaon: ˆ 1 n MSE h h () n 1 n (9) 1 MAE h hˆ n 1 Besdes hese wo forecasng crera we also used he SPA es of Hansen [1] o es he accuracy of forecasng models. Several auhors used hs SPA es wh dfferen loss funcon (see Hou and Suasd [15], Mederos and Vega [] ec.). Hansen [1] appled a supremum over he sandardzed performances and ess he null hypohess H max : 1,,... m where ( ) E f, And he es sascs 1/ S max n f T max, (1) hˆ where ĥ s he sandard devaon of n / f 1 hen f 1/ n f, and f, l, l,. The earler denfed loss funcon a me s defned by for he benchmar model where l, ndcae he value of correspondng loss funcon for anoher compeng model, n s he number of ou of sample daa. To reduce he nfluence of poor performng models whle preservng he nfluence of he alernaves wh, Hansen [1] proposes he followng conssen esmaor for : where c c ˆ f 1 1 / n fˆ / ˆ ln ln n (11), 1,..., m 1. s an ndcaor funcon. Hansen argued ha he hreshold rae lnlnn ensures ˆ s conssen esmaor ha effecvely capures all alernave wh, and hs leads o a conssen esmae of he null dsrbuon, whch mproves he power of he es. The dsrbuon of he es sasc under null hypohess can be approxmaed by he emprcal dsrbuon derved from he boosrap resample based on he saonary boosrap of Pols and Romano []. n 1 l,

7 Volaly Modelng for Forecasng Soc Index 115 U * f h( f ) for b 1,..., B and 1,..., n (1) *, b,, b, where h( fˆ ) f 1 1 / n fˆ / ˆ ln ln n. The p-value of he SPA es can be obaned by he frs calculaon of 1/ S* maxn f T b max, (13) hˆ for each b 1,..., B and hen he comparng T S o he quanles of S* T b. P S B b1 1 B S* S Tb T (1) 3 Daa, Resuls and Dscussons 3.1 Characerscs of Daa For our emprcal sudy we use he daly All Share Prce Index (ASPI) of Colombo Soc Exchange, Sr Lana. The daa of he range from January, 199 o December 9, for a oal 15 observaons. The frs 195 observaons are aen for n sample esmaon and las observaons consder for ou of sample forecasng performance. For analyss we used EVews5. and MATLAB 7.. In order o oban he saonary seres we ransformed hese daa no her reurns. Daly reurns of ASPI are ploed a Fgure 1 whch ndcaes ha he daa s more volale n he perod of July, 3 o March. The descrpve sascs of ASPI reurns are dsplayed n Table-1. From Table-1 we found ha mean reurns of he CSE s.. Volaly whch s measured by sandard devaon s The reurns hold he propery of lepouross and posve sewness. The normaly of he reurns s reeced based on he Jarque-Berra sascs. The ARCH es confrms he presence of ARCH effec. Overall hese resuls clearly suppor for he reecon of he hypohess ha CSE me seres of daly ASPI reurns are me seres wh ndependen daly values Daly reurns of CSE Fgure 1: Daly reurns of Colombo Soc Exchange

8 11 Md. Mosafzur Rahman e al. Table 1: Descrpve Sascs for ASPI daly reurns Sample sze Mean Mn. Max. Sandard Devaon Sewness Kuross Jarque-Bera es (.) ARCH es.1 (.95) 3. In Sample Esmaon In our sudy we consder wo ypes of GARCH model such as symmerc and asymmerc GARCH models (.e. EGARCH and APARCH) wh dfferen dsrbuonal assumpons wh fxed and flexble parameer dsrbuon of suden- and generalzed error dsrbuon. For fxed parameer we use suden- dsrbuon wh fxed DOF 1 and GED dsrbuon wh fxed parameer 1.5. Parameer esmaon resuls for GARCH model wh Gaussan, suden-, GED, suden- wh fxed degrees of freedom (DOF) 1 and GED wh fxed parameer 1.5 dsrbuonal assumpons are gven a Table-. From Table- we found ha all of he parameers of GARCH model under all dsrbuonal assumpon are sgnfcan a 5% level of sgnfcance. Table-: GARCH model esmaon wh dfferen dsrbuonal assumpon w Gaussan. (3.31) Suden-.531 (3.5) GED.7 (3.) Suden-.55 Fxed (3.7) DOF GED Fxed Parameer Table-3: EGARCH model esmaon wh dfferen dsrbuonal assumpon w (or ) Gaussan.391 (.751) (-.1).9 (.).971 (.7).79 (77.) Suden-.9 (3.159) -.11 (-13.3).595 (1.3) -.51 (-1.).75 (9.3) 3. (11.99) GED. (3.) -.31 (-1.9). (15.) -.73 (-.3).7 (9.71) 1.7 (39.75) Suden- Fxed DOF.79 (.1) -.73 (-.17).5793 (19.3).53 (.39) -.11 (-.593) GED Fxed Parameer.51 (3.15).77 (.97) -.51 (-5.3).1593 (1.5).15 (.1).11 (.5).17 (.773).179 (1.35). (7.1).1 (.7).397 (7.95).91 (1.). (1.).539 (3.53).1 (.7).37 (33.17).55 (15.).53 (15.93).5353 (.).1 (3.7).3 (1.9) 3.5 (11.757) 1. (37.)

9 Volaly Modelng for Forecasng Soc Index 117 Table-: APARCH model esmaon wh dfferen dsrbuonal assumpon w (or Gaussan.35 (3.59) Suden-.9 (3.19) GED.71 (3.1) Suden-.511 Fxed (.95) DOF GED Fxed Parameer.5 (3.1).19 (13.393).133 (.99).1 (.577).1177 (.5).1319 (9.5).71 (.1).17 (.95).5 (1.5).31 (13.931).39 (19.71).37 (1.9).5 (17.5).51 (1.7).5797 (.9).59 (1.7) ) (-3.9).15 (3.5).171 (.397).3 (.7) -.19 (-.9).3 (11.79) 1.99 (.) 1.75 (.5) 1. (.39) 1.1 (9.) 3.37 (11.91) 1.1 (3.37) Parameer esmaon resuls of EGARCH and APARCH model wh dfferen dsrbuonal assumpons are gven a Table-3 and Table- respecvely. From Table-3 we found ha mos of he parameers of EGARCH model wh dfferen dsrbuonal assumpons are sgnfcan a 5% level of sgnfcance excep parameer β n case of suden- and GED dsrbuon and parameer γ or γ n case of suden- wh fxed DOF 1 and GED wh fxed parameer 1.5. Table- showed ha mos of he parameers of APARCH model wh all dsrbuonal assumpons are sgnfcan a 5% level excep he asymmerc parameer γ or γ n case of suden- wh fxed DOF and GED wh fxed parameer. The sum of GARCH parameers under all dsrbuonal assumpons are less han one whch sugges ha he volaly are lmed and he daa are saonary whch explan ha he models are fed well. In order o fnd ou he bes performng model n case of n sample esmaon we used some model comparson crera such as Box-Perce sascs for boh resduals and squared resduals, Aae Informaon Crera (AIC), Log Lelhood value. These esmaon resuls are gven a Table-5. From hs able we found ha all he models seem o do a good ob n descrbng he dynamc of he frs wo momens of he seres based on Box-Perce sascs for boh of he resduals, whch are all non-sgnfcan a 5% level. The Aae Informaon Crera (AIC) and he log-lelhood values sugges ha APARCH model gve beer resuls han GARCH and EGARCH models under all dsrbuonal assumpon. Among hese models EGARCH model showed he wors n sample esmaon resul n case of CSE. Regardng he denses, Suden- and Generalzed error dsrbuons clearly ouperform han suden wh fxed DOF 1 and GED wh fxed parameer 1.5 and all of hese densy show beer performance han Gaussan densy. In he case of suden- dsrbuon he AIC value for he model GARCH, EGARCH and APARCH are less han oher denses. The log lelhood value s srcly ncreasng n case of suden- dsrbuonal assumpons where fxed parameer of suden- densy can no mprove he log lelhood value. Smlar resuls also found n case of GED and GED wh fxed parameer. So, fnally from Table-5 we found ha n case of n sample performance APARCH model wh suden- dsrbuon gve beer resuls han oher models for CSE, Sr Lana. Table-5: Model comparson based on n sample

10 11 Md. Mosafzur Rahman e al. Error Model Q () Q () AIC Log-lelhood Dsrbuon Gaussan GARCH EGARCH APARCH Suden- GARCH EGARCH APARCH GED GARCH EGARCH APARCH Suden- Fxed DOF GED Fxed parameer GARCH EGARCH APARCH GARCH EGARCH APARCH Ou of Sample Forecasng Performance Snce he ou of sample es can conrol he possble over fng or over parameerzaon problems, herefore many emprcal researchers became neresed o have good volaly forecas based on ou of sample esmaon nsead of good n sample esmaon. In our paper we use ou of sample evaluaon of one sep ahead volaly forecas based on he loss funcon MSE and MAE. To beer assess he forecasng performance of he varous models we use he Superor Predcve Ably (SPA) es of Hansen [1]. The esmaon resuls are gven a Table-. Table-: Forecasng performance comparson based on ou of sample Error Model MSE MAE SPA(p-value) Dsrbuon MSE MAE Gaussan GARCH EGARCH APARCH Suden- GARCH EGARCH APARCH GED GARCH EGARCH APARCH Suden- GARCH Fxed EGARCH DOF APARCH GED Fxed parameer GARCH EGARCH APARCH From Table- we found ha APARCH model wh all dsrbuonal assumpon gve he

11 Volaly Modelng for Forecasng Soc Index 119 lowes value of MSE and MAE where EGARCH model wh all dsrbuonal assumpons provde poores forecasng performance. The comparson among denses sugges ha GARCH model, suden- dsrbuon wh fxed DOF 1 gve beer resuls han oher denses and for EGARCH model, suden- densy gve beer resuls and for APARCH model, Gaussan densy gve lowes value of MSE and MAE. We also found ha models wh suden- densy wh fxed DOF 1 and GED wh fxed parameer 1.5 gve beer forecasng performance han suden- and GED wh flexble parameer denses respecvely. Among hese models APARCH model wh Gaussan densy gve beer resuls. Snce we found ha varous dsrbuonal assumpons gve beer resuls for dfferen models. So, n order o fnd ou unque forecasng model for ASPI ndex of CSE we use SPA es of Hansen [1]. A hs able we only repored he p-value of he SPA es under MSE and MAE loss funcon. Under he null hypohess he base model s no ouperformed by all of he oher models, he hgher p-value ndcae he superory of he forecasng performance. The P-value for he APARCH model wh Gaussan dsrbuon s.9 and.9 for MSE and MAE loss funcon respecvely whch s vrually close o 1 suggesng ha APARCH model wh Gaussan densy presens he hghes forecasng accuracy han oher models n case of CSE Forecas of Varance for GARCH-N Forecas of Varance for EGARCH-N Forecas of Varance for APARCH-N Forecas of Varance for GARCH- Forecas of Varance for EGARCH- Forecas of Varance for APARCH Forecas of Varance for GARCH- fxed Forecas of Varance for EGARCH- fxed Forecas of Varance for APARCH- fxed Forecas of Varance for GARCH-GED Forecas of Varance for EGARCH-GED Forecas of Varance for APARCH-GED Forecas of Varance for GARCH-GED fxed Forecas of Varance for EGARCH-GED fxed Fgure : Forecas of varance Forecas of Varance for APARCH-GED fxed

12 1 Md. Mosafzur Rahman e al. Conclusons In hs paper we compared n sample and ou of sample forecasng performance of several GARCH-ype models such as GARCH, EGARCH and APARCH model wh Gaussan, suden- and generalzed error dsrbuon (GED), suden- wh fxed DOF 1 and GED wh fxed parameer 1.5 n case of Colombo Soc Exchange. Our emprcal resuls show ha noceable mprovemens can be made when usng asymmerc GARCH model wh non normal dsrbuonal assumpons n case of n-sample esmaon. The log lelhood value s srcly ncreasng n case of suden- dsrbuonal assumpons where fxed parameer of suden- and GED densy fal o mprove he log lelhood value compared o boh suden- and GED dsrbuon. Among hese models APARCH model wh suden- dsrbuonal assumpon gve beer n sample esmaon resuls of ASPI ndex of CSE. In he case of ou-of-sample forecasng performance we found ha APARCH model wh all dsrbuonal assumpon gve he lowes value of MSE and MAE. Accordng o he denses suden- dsrbuon wh fxed DOF 1, suden- and Gaussan dsrbuonal assumpons gve beer resuls n case of GARCH, EGARCH and APARCH model respecvely. I s also observed ha fxed parameer of suden- and GED mproves he forecasng performance han suden- and GED densy. The esmaon resuls of SPA es sugges APARCH model wh Gaussan dsrbuonal assumpon gve beer forecasng performance n case of CSE for hs sudy perod. Therefore, he resul suggess ha APARCH model wh suden- dsrbuonal assumpon gve beer n sample esmaon resuls and APARCH model wh Gaussan dsrbuonal assumpon s he bes forecasng model n case of ASPI ndex of Colombo soc mare, Sr Lana. ACKNOWLEDGEMENTS: Ths research was suppored by Yunnan Unversy of Fnance and Economcs (YCT117). The auhors are graeful o Dr. A. B. Zahufer for provdng daa. References [1] T. Andersen and T. Bollerslev, Answerng he Crcs: yes, ARCH Models Do Provde Good Volaly Forecass, NBER Worng Paper Seres, 3, [] R. Balle, and T. Bollerslev, The Message n Daly Exchange Raes: A Condonal- Varance Tale, Journal of Busness and Economc Sascs, 7, (199), [3] K. Bene, S. Lauren and C. Lecour, Accounng for Condonal Lepouross and Closng Days Effecs n FIGARCH Models of Daly Exchange Raes, Appled Fnancal Economcs, Vol. 1, (), 59-. [] T. Bollerslev, Generalzed auoregressve condonal heerosedascy, Journal of Economercs, 31, (19), [5] T. Bollerslev, A Condonally Heerosedasc Tme Seres Model for Speculave Prces and Raes of Reurn, Revew of Economcs and Sascs, 9, (197), [] T. Bollerslev, R.F. Engle and D.B. Nelson, ARCH Models, Hand Boo of Economercs, Vol., (199), [7] T. Bollerslev and E. Ghysels, Perodc Auoregressve condonal heeroscedascy,

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