Analysis of Volatility and Forecasting General Index of Dhaka Stock Exchange

Transcription

1 American Journal of Economics 03, 3(5): 9- DOI: 0.593/j.economics Analysis of Volailiy and Forecasing General Index of Dhaka Sock Exchange M Monimul Huq, M Mosafizur Rahman,*, M Sayedur Rahman, Md. Abu Shahin, Md. Ayub Ali Deparmen of Saisics, Universiy of Rajshahi, Rajshahi, 605, Bangladesh Saisics and Mahemaics School, Yunnan Universiy of Finance and Economics, Kunming, 650, China Absrac The purpose of he presen sudy is o empirically examine he performance of differen kinds of volailiy modeling and heir forecasing performance for he general index of an emerging sock marke, namely Dhaka Sock Exchange from he period December 06, 00 o March, 03. We mainly used Box-Jenkins modeling sraegy hereafer he volailiy model. The descripive saisics, correlogram, uni roo es, ARMA, ARCH, GARCH, TARCH, EGARCH and several model selecion crieria are used in his sudy. The Buerworh filer is used for removing he noise of he re urn series of general index. All he parameers in his sudy are esimaed hrough Maximum Likelihood mehod. The descripive saisics show general index decrease slighly overime wih posiively skewed and lepokuric. The reurn series follows ARMA(,) model wih volailiy provide evidence of he superioriy of GARCH(,) and GARCH(,) over he all order of oher GARCH models. Finally, we found ha he fied model on filered general index of Dhaka Sock Exchange are ARMA(,) wih GARCH(, ) and GARCH(,) model. This model can be used for fuure policy implicaion hrough is accurae forecas. Keywords Buerworh filer, Reurn series, Uni roo, ARMA, GARCH, TARCH, EGARCH, Model selecion crieria. Inroducion Tradiional regression ools have shown heir limiaion in he modeling of high-frequency daa, assuming ha only he mean response could be changing wih covariaes while he variance remains consan over ime ofen revealed o be an unrealisic assumpion in pracice. This fac is paricularly obvious in series of financial daa where cluser of volailiy can be deeced visually. Indeed, i is now widely acceped ha high frequency financial reurns are heeroskedasic. Volailiy is undersood as he spread of asse reurns and is closely relaed o risk. Volailiy has pronounced role in modern finance as i is used in muliple risk managemen soluions. Volailiy is he mos imporan variable in valuaing derivaive insrumens. I has cenral role in risk managemen, asse valuaion and invesmen in general. Mos socks are raded on exchange, which are places where buyers and sellers mee and decide on a price. Banglades h has wo Sock Exchanges, Dhaka Sock Exchange (DSE) and Chiagong Sock Exchange (CSE), and hese exchanges are self-conrolled, privae secor bodies which are obliged o ge heir funcioning policies acceped by he SEC. Dhaka Sock Exchange (DSE) is he major sock exchange of Bangladesh and founded a Moijheel a he * Corresponding auhor: mosafiz_bd@yahoo.com (M Mosafizur Rahman) Published online a hp://journal.sapub.org/economics Copyrigh 03 Scienific & Academic Publishing. All Righs Reserved cener of he Dhaka own in 95 where buying and selling is carried ou by Compuerized Auomaed Trading Sysem. Sock marke volailiy has been he subjec of many sudies over he pas few decades. The main impeus for his ineres began afer he 987 sock marke crash where, for example, he Sandard & Poor s (S&P) composie porfolio dropped from 8.70 o.8 (0. %) and he Dow-Jones Average fell by 508 poins in one day. The erm sock marke volailiy refers o he characerisic of he sock marke o rise or fall sharply in price wihin a shor-erm period (from day o day or week o week). Generally, increased volailiy has been viewed as an undesirable consequence of desabilizing marke forces such as speculaive aciviy, noise rading or feedback rading. Increased volailiy could come as a resul of an innovaion, by reflecing he acual variabiliy of informaion regarding fundamenal values. So an increased volailiy may no necessarily be undesirable, Bollerslev e al[]. On he oher hand policymakers may pursue regulaory reforms by eiher rying o reduce volailiy direcly or by assising financial markes and insiuions o adap o increased volailiy. In pracice policy makers have focused on he laer, improving he abiliy of financial markes and insiuions o weaher increased volailiy. For financial insiuions direcly exposed o increased volailiy, such as deposiory insiuions and marke makers, policymakers have encouraged greaer capializaion. Increased capial allows hese insiuions o weaher greaer financial volailiy wihou incurring he liquidiy and solvency

2 30 M Monimul Huq e al.: Analysis of Volailiy and Forecasing General Index of Dhaka Sock Exchange problems ha migh disrup he funcioning of financial markes. The opic of volailiy is of significan imporance o anyone involved in he financial markes. In general volailiy has been associaed wih risk, and high volailiy is hough of as a sympom of marke disrupion, wih securiies unfairly priced and he malfuncioning of he marke as whole. Especially, wihin he derivaive securiy marke volailiy and volailiy forecasing is vial as managing he exposure of invesmen porfolios is crucial, Figlewski[7]. More recenly he lieraure has focused on he abiliy o forecas volailiy of asse reurns. There are many reasons why forecasing volailiy is imporan according o Walsh and Yu-Gen Tsou[8], for example, opion pricing has radiionally suffered wihou accurae volailiy forecass. Conrolling for esimaion error in porfolios consruced o minimize ex ane risk, wih accurae forecass we have he abiliy o ake advanage of he correlaion srucure beween asses. Finally when building and undersanding asse pricing models we mus ake ino accoun he naure of volailiy and is abiliy o be forecased, since risk preferences will be based on marke assessmen of volailiy. There are many researchers analysis he share price index and hey have noed ha ARIMA wih GARCH model is adequae. The average daily share price indices of he daa series of Square Pharmaceuicals Limied (SPL) have been used and found ha i is non-saionary a level, even afer log-ransformaion, finally found ha he ARIMA (,,) is he bes model (Paul e al.,[]). The relaionship beween he sock marke volailiy and volailiy in macroeconomic variables such as he real GDP, inflaion, and ineres rae are examine and found ha here was a bi-direcional causal relaionship beween sock marke volailiy and real GDP, and he sock prices were no significan in explaining he inflaion rae and ineres rae, and vice versa (Oseni and Nwosa,[]). The analysis of daily reurn of Dhaka Sock Marke indices include he daily price indices of all securiies lised on he DSE general, DSI (All Share), DSE op 0 indices, and Daily indices lised in he marke during he pas years and showed ha DSE does no follow he random walk model and so he Dhaka sock exchange (DSE) is no efficien even in weak form (Khandoker e al.[9]). The daa se of monhly DSE General Index (DSE-GEN) which covers he weny hree year long period commencing from January, 987 o March, 00 showed ha i is follows random walk model bu reurn series is no and he monhly DSE reurns follow Generalized Auoregressive condiional Heeroskedasiciy (GARCH) properies (Rayhan e al.[7]). Rahman e al[5] examined a wide variey of popular volailiy models wih normal, Suden- and GED disribuional assumpion for Chiagong Sock Exchange (CSE) and found ha Random Walk GARCH (RW-GARCH), Random Walk Threshold GARCH (RW-TGARCH), Random Walk Exponenial GARCH (RW-EGARCH) and Random Walk Asymmeric Power ARCH (RW-APA RCH) models under Suden- disribuional assumpions are suiable for CSE. Rahman e al[6] invesigaed he in-sample and ou-of-sample forecasing performance of GARCH, EGARCH and APARCH models under fa ail and skewed disribuions in case of Dhaka Sock Exchange (DSE) from he period January 0, 999 o December 9, 005. From he above discussion i seems ha volailiy is imporan, since i direcly and indirecly affecs he financial sysem and he economy as a whole. The main aim of his research is o find ou he appropriae model for he general index of Dhaka Sock Exchange (DSE). A furher insigh ino volailiy forecasing will be given bu firs i is necessary o explore he differen models used when esimaing volailiy. Therefore, he Objecive of his Sudy is o compare he performance of differen kind of volailiy modeling and o esablish he forecasing efficiency of he fied model in case of DSE. Res of he paper is organized as follows: Secion presens he mehod and maerials, Secion 3 discuss he empirical resuls and finally Secion presens he conclusion.. Mehods and Maerials We consider daily sock exchange daa (general index) for building ime series modeling and forecas. To fulfill his purpose we collec daa from he daabank of he Dhaka Sock Exchange websie addres s rke_informaion.php. The daa series is from December 06, 00 o March, 03 and he lengh is 5. In analyzing general index daa, he general index mus be convering as reurn series because i is seen as he whie noise process which indicaes ha index price is a random walk, he series should be idenically and independenly disribued wih zero mean and consan variance (Akgiray,[]). A whie noise suggess ha he fuure values of he expeced mean and variance of he series canno be prediced by using pas values of he series. The reurn series are consruced using DSE general index o allow a marke wide measure of volailiy o be examined. By convenion, he daily reurns were calculaed as he coninuously compounded reurns which are he firs difference in logarihm of closing prices of DSE general index of successive days: r ln( GI / GI ) ln( GI ) ln( GI ) Generally, he reurn series (or general index) is random bu i is widely affeced by he noise. Since he main purpose of modeling for general index is o find ou acual naure hence we can apply differen filering for removing noise. There are several filer mehod are available bu we only consider Buerworh filer in his sudy. The Buerworh filer proposed by Briish engineer Sephen Buerworh in his paper eniled "On he Theory of Filer Amplifiers a 930 is a ype of signal processing filer designed o have as fla a frequency response as possible in he pass band. Le us consider a ime series x T for considered as filering. We are ineresed in isolaion componen of x,

3 American Journal of Economics 03, 3(5): 9-3 denoed y wih period of oscilla ions beween p, where p l p. u u p and l Consider he following decomposiion of he ime series The componen x y x y is assumed o have power only in he,, frequencies in he inerval b a, b a and b are relaed o p l and a. p by a, b, p u If infinie amoun of daa is available, hen we can use he ideal band pass filer y B( L) x where he filer, B (L), is given in erms of he lag operaor L and defined as u p l j k B( L) B L, L x x j j k The ideal band pass filer weighs are given by sin( jb) sin( ja) B j j b a B0 The digial version of he Buerworh high pass filer is described by he raional polynomial expression (he filer s z-ransform) ( z) n n ( z) ( z ) n ( z ) ( z) n n ( z The ime domain version can be obained by subsiuing z for he lag operaor L. Pollock derives a specialized finie-sample version of he Buerworh filer on he basis of signal exracion heory. Le s be he rend and c cyclical co mponen of y, hen hese componens are exraced as y n ( L) s c v n ( L) ( L) where ~ N(0, ) v and ~ N(0, ) v ) nd If we consider drif is rue, hen he drif adjused series is obained as ~ xt x x, T 0,,..., T where x~ is he undrafed series. The sofware Microsof Excel, Eviews, R and Microsof Word are used whole analysis and wrien in his paper... Auoregressive Moving Average (ARMA) Process n Of course, i is quie likely ha a ime series has characerisic of boh AR and MA and is herefore ARMA. An ARMA process of order p and q is denoed by ARMA(p, q) and is wrien in he form: Y C Y... Y... p p q q () Where C is consan coefficiens, respecively, and, i' s and ' i s are AR and MA ' i s whie noise disurbance erms. In general he ARMA (p, q) process can be wrien as a saionary ARMA process has he following form of heoreical ACF and PACF; The heoreical ACF decays, The heoreical PACF are also decays... Auoregressive Condiional Heeroscedasic (ARCH) Model Le us consider a univariae ime series y. If is he informa ion se (i.e. all he info rmaion available) a ime, we can define is funcional form as: y E[ ] () y where E [..] denoes he condiional expecaion operaor and is he disurbance erm (or unpredicable par), wih. E[ ] 0 and E[ ] 0, s The s erm in equaion () is he innovaion of he process. The condiional expecaion is he expecaion condiional o all pas informaion availab le a ime. The Auoregressive Condiional Heeroscedasic (ARCH) process of Engle[5] is any } of he form where { z (3) z is an independenly and idenically disribued (i.i.d.) process, [ ] 0 E, [ ] z Var z and where is a ime-varying, posiive and measurable funcion of he informa ion se a ime. By defin iion, is serially uncorrelaed wih mean zero, bu is condiional variance equals o and, herefore, may change over ime, conrary o wha is assumed in OLS esimaions. Specifically, he ARCH (q) model is given by () 0 i The models considered in his paper are all ARCH-ype. They differ on he funcional form of bu he basic logic is he same..3. Generalized ARCH (GARCH) Model Engle used he condiional variance o be auoregressive. Bollerslev[] exended Engle s ARCH o include he lagged variance according o ARMA process. Bollerslev s generalized auoregressive condiional heeroskedasic

4 3 M Monimul Huq e al.: Analysis of Volailiy and Forecasing General Index of Dhaka Sock Exchange process ake he following form. The error process is given by q p 0 i i j j (5) i ji Using he lag or backshif operaor L, he GARCH (p, q) model is L) ( L) (6) 0 ( wih ( L) L L... ql p ( L) L L... pl and based on equaion (6), i is sraighforward o show ha he GARCH model is based on an infinie ARCH specificaion. If all he roos of he polynomial ( L) 0 of equaion (6) lie ouside he uni circle, we have ( L)] ( L)[ ( L)] or equivalenly 0[ 0 i j... p i which may be seen as an ARCH( ) process since he condiional variance linearly depends on all previous squared residuals. GARCH(,) model can be obain from equaion (5) by using he value p q... Threshold GARCH (TARCH) Model TARCH or Threshold ARCH and Threshold GARCH were inroduced independenly by Zakoïan[9] and Glosen, e al[6]. The generalized specificaion for he condiional variance is given by: q p r 0 j j ii kk k (7) j i k where, if 0 = 0 oherwise. In his model 0 and 0 i have differenial effecs on he condiional variance; 0 has an impac of, while 0 i 0, 0 i i i i i has an impac of i i q. If increases volailiy, and we say ha here is a leverage effec for he news impac is asymmeric. i h order. If 0.5. Exponenial GARCH(EGARCH) Model he The EGARCH or Exponenial GARCH model was proposed by Nelson[0]. The specificaion for condiional variance is: q p r i k log( ) 0 j log( j ) i k j i (8) i k k The lef hand side of equaion (8) is he log of he i condiional variance. This implies ha he leverage effec is exponenial raher han quadraic and ha forecass of he condiional variance are guaraneed o be non-negaive. The presence of he leverage effecs can be esed by he hypohesis 0. The impac asymmeric if 0. i 3. Resuls and Discussions We collec daa of general index from Dhaka Sock Exchange, Bangladesh from December 06, 00 o March, 03 wih 5 realizaions. We divided sample ino wo pars such as firs 99 and las samples for raining and es sample, respecively o permi more efficien model. The mos popular programming sofware R and economerical sofware Eviews used o whole analysis according o objecive of presen research. Throughou his paper, he general index is convered ino sock reurns are defined coninuously compued or log reurns (hereafer reurns) a ime because i is more sable hen original general index. I has been observed ha he reurn series as well as general index is very much affeced by noise. So he acual behavior of he series is hidden by noise and hence we canno mee o he main objecive of analysis. Therefore, we can remove he noise from reurn series using filered mehods. There are many mehods are available for filering of a ime series, in his case we consider Buerworh high pass filer bu his filer is very much affeced by differen values of order of filer and cu-off frequency. So, we consider differen values of order of filer and cu-off frequency for filering reurn series and finally chose he filered series if i gives smalles mean square error. The differen values of order of filer and cu-off frequency wih mean square error are presened in Table. From Table, i is clear ha he filered series for 3 rd order filer wih cu-off frequency is (Sl. No. ) gives smalles MSE (mean square error) as considered de-noise acual sudy series. From he seleced filered reurn series (Sl. No. in Table ), he rend indicaed by red color (op of Figure ) is used as sudy series; cyclical componen is ignored and indicaes noise of original reurn series. We observed ha he deviaions (cyclical componen) are very small. In he nex, we will be used his filered reurn series (rend) for fulfill according o our objecives. The Table repors summary saisics for he filered reurn series of general index. The mean and median of daily reurns are negaive and hey are no significanly differen from zero. I suggess ha sock prices in general decrease slighly overime. The skewness and kurosis parameers respecively measure he asymmery and peakness of he probabiliy disribuion of filered daily reurns compared wih normal curve usually hese wo parameers should be zero and hree in normal disribuion. The Jarque-Bera saisic wih skewness and kurosis are used o signify he disribuion characerisics of reurn series. In he daa se, he evidence for reurn series indicaes posiively skewness of daily reurn disribuion. In oher words, reurn series disribuion has significanly faer ails i

5 American Journal of Economics 03, 3(5): 9-33 han he normal disribuion. The coefficien of skewness indicaes ha he series ypically have asymmeric disribuions skewed o he righ. The kurosis saisic which is equal o 5.5 indicaes he lepokuric characerisic of he daily reurn disribuion. The reurn disribuion has a more acue peak around he mean han he normal disribuion. The implicaion of non-normaliy is suppored by he Jarque-Bera es saisic which poins ou ha he null hypohesis of normal disribuion is rejeced. In accordance wih he skewness and kurosis, he Jarque-Bera saisic is anoher evidence o conclude ha he daily reurn series is no normally disribued. Also from he filered reurn series indicaes ha he series variance is no consan over ime suggess ha models employed need o ake ino accoun heeroskedaiciy issues, indicaing he reasonable use of GARCH ype models (Bollerslev,[]). Researchers are ineresed in he firs sep of he analysis of any ime series is o plo he daa o see he visual srucure. Time series plo gives an iniial clue abou he naure of he series or shows an upward or downward rend, seasonal or cyclical flucuaion ec. Graphical represenaion suggess ha he ime series is saionary or no. The filered reurn series (red color named rend locaed op of Figure ) shows ha he mean reurns is consan bu he variances change over ime around some normal level, wih large (s mall) changes ending o be followed by large (small) changes of eiher sign, i.e. volailiy ends o cluser. Periods of high volailiy can be disinguished from low volailiy periods. Now ACF and PACF of he filered series are ploed in Figure and 3, and see almos all of he coefficiens of ACF and PACF inside he confidence limis. Again he ACF converge o zero very quickly, which indicaed ha he series is saionary. Finally, uni roo es is used o confirm he series is saionary or no. The uni roo es resuls for filered reurn series are shown in Table 3. The Table 3 showed differen uni roo es wih p-values and criical values for filered reurn series of general index of DSE. The resuls of Dickey-Fuller (DF), Augmened Dickey-Fuller (ADF) and Phillips-Perron (PP) ess showed ha he filered series daa is saionary. So we can say form graphical represenaion, correlogram and uni roo es he filered daa series is saionary. We now proceed o build models. Now we can build ARMA Model using he filered daily reurn series because i is saionary a level. Using he Box-Jenkins modeling sraegy he model ARMA(,) is beer among he various combinaions of model parameers p and q of ARMA(p,q). Now we wan o es wheher he heeroskedaiciy problem is exis or no and ha is permi using more formal Lagrange muliplier es for ARCH disurbances proposed by Engle[5]. The calculae value of es saisics (F-value) of Lagrange muliplier es up o lag is wih probabiliy indicaes he presence of ARCH effec in he residual series of ARMA(,) model. Therefore, we have o search he family of GARCH ype Models. We simulaneously model he mean and he variance of Dhaka Sock Exchange, considering he ARCH, GARCH, TARCH and EGARCH models for condiional variance. Table presen he resuls from seven alernaive models including ARCH(), ARCH(), ARCH(3), GARCH (,), GARCH(,), TARCH(,) and EGARCH(,) employed in his disseraion o esimae volailiy for he period from December 06, 00 o January 0, 03 in Dhaka sock Exchange. We already saw ha in our daa he heeroskedasic problem is presen. Thus he ARCH class models may give he beer resul. Now we apply he various ARCH models under he assumpion of normal disribuion for reurn daa which gives he resuls shown in Table. A firs we examined symmeric ARCH () model. This specificaion gives ha he coefficien of ARCH() model is significan in parameers. Also mean equaion wih ARMA(,) model parameers are also significan. We also observed ARCH(), ARCH(3) models where all he parameers also significan. Bu ARCH() and higher order model parameers are insignifican. We also perform a higher order of GARCH esima ion allowing he ARCH and GARCH erm o ener he variance equaion wih more han one lags. Therefore, we hen carry ou he GARCH(,q) model allowing more lags of GARCH erms involved in volailiy equaion. The coefficiens in variance equaion esimaed from GARCH(,) and GARCH(,) models are posiive and highly saisically significan. However, when he higher order of GARCH erms are added ino he variance equaion, he saisics of he new GARCH erm is less significan or i makes he oher erms saisic less significan. Thus resul suggess ha GARCH(,) and GARCH(,) models are sufficien enough o capure volailiy characerisics of Dhaka Sock marke wihou he need of higher-order GARCH models. The skewness saisics of he reurn series which are generaed in Table and Table imply he asymmery in he daa se. Therefore, he asymmeric specificaions of GARCH model, TARCH and EGARCH, are also employed in his disseraion. However, regarding he resuls of hose asymmeric models, he so-called asymmeric effec parameer γ is no saisically significan a 0% and 5% level for TARCH and EGARCH, respecively. Though, he coefficien of mean equaion in boh models is significan a % level. This implies he weak evidence of asymmery in he reurn series. In oher words, he condiional variance is no higher in he presence of negaive innovaion or he marke seems no nervous when bad news akes place. The finding is ineresing because i is likely a phenomenon in many maure markes ha he invesors reac more dramaically o negaive shocks han posiive shocks. However, he oucome is consisen wih he findings in some oher emerging markes such as ransiion markes of Cenral Europe (Harouounian and Price,[8]). According o Chou[3], he sum of ARCH and GARCH erms coefficiens α + β presen he change in he response funcion of shocks o condiional variance per period. Poerba and Summber[] showed ha he impac of he volailiy on sock prices crucia lly depends on he persisence of shocks o variance over a long ime. The parameers from

6 3 M Monimul Huq e al.: Analysis of Volailiy and Forecasing General Index of Dhaka Sock Exchange Table poin ou ha he sum of α and β is close o uniy across he wo models including GARCH(,), GARCH(,) models. The high persisence of shock o variance is suppored by previous sudies in he case of emerging capial markes. The finding implies ha available informaion is relevan in forecasing condiional variance a long ime horizon. Generally, coefficiens in variance equaion obained from wo esimaion models including GARCH(,), GARCH(,) are highly saisically significan a % level. Moreover, he higher order GARCH (p,q) models generae he insignifican parameers, indicaing he sufficiency of GARCH(,) and GARCH(,) in modelling volailiy of Dhaka sock index reurns. Meanwhile, he findings from TARCH and EGARCH provide weak evidence of asymmery in reurn series, which is consisen wih empirical sudies in some emerging markes. In Table 5 we examined R-square value, Akaike Informaion Crierion (AIC) and Schwarz -Crierion (SC) from differen models. From Table 5 we found ha he value of R-square, AIC and BIC of GARCH(,) and GARCH(,) are almos equal and s malles of all oher models. So, we can proceeds hese wo models for furher analysis of diagnosic check and forecasing. Firsly, we can consider ARMA(,) wih GARCH(,) model for diagnosic checking and forecasing. The graphical presenaion of his fied model is helpful for inspecion he performance of fied model and presened as Figure. We observed from Figure which is he plo of residual, acual and fied model daa and conclude ha he GARCH(,) model is fi well. Model diagnosic checking is an imporan sep for selecing an appropriae model. Usually residual analysis is he bes way for model checking. Now we shall observe he correlogram, uni roo es of residual o check i is saionary or no. In order o check he whie noise propery of he esimaed ARMA(,) wih GARCH(,) model residual, we consruced ACF and PACF of he residual series presened by Figure 5 and 6. From Figure 5 and 6, he ACF and PACF apper of very quickly, his implies ha he residual series is saionary. Then we can say ha he model is appropriae. The Table 6 showed differen uni roo es wih p-values and criical values for residual of ARMA(,) wih GARCH(,) model. The resuls of Dickey-Fuller (DF), Augmened Dickey-Fuller (ADF) and Phillips-Perron (PP) ess showed ha he residual is saionary. So we can say ha he assumpion of chosen model is appropriae. To es of ARCH effec in he residual series of fied he ARMA(,) wih GARCH(,) model, we can consider Lagrange muliplier (LM) es of he residuals. We observed ha he calculaed value of F saisic for Lagrange muliplier es is wih associaed probabiliy is indicaing no ARCH effec in residuals of ARMA(,) wih GARCH(,) model. Therefore, he chosen model for volailiy may be adequae. The properies of sandardized residuals are employed o define he bes fi daa models. Sandardized residual plo is a very useful ool o check he presence of unusual observaions. Sandardized residual from he model in presened in Figure 7 has some posiive and some negaive values ha falls in hree sandard deviaion confidence inerval excep one. Bu i falls in four sandard deviaion confidence inerval. This is he srong indicaion ha he ARMA(,) wih GARCH(,) model is appropriae. The es of normaliy in his sudy we used Jarque-Bera es. We observed ha he calculae value of Jarque-Bera es is wih Probabiliy indicae he null hypohesis The residual series of fied model is normal is acceped. Therefore, he esimaed residuals of he fied ARMA(,) wih GARCH(,) model is normally disribued. A forecas is generally defined as a saemen concerning fuure evens. Forecasing is one of he mos common uses of economeric mehods. According o Akgiray[], here are wo reasons why forecasing volailiy aracs ineress of invesors. Firsly, good forecas capabiliy of volailiy models provides a pracical ool for sock marke analysis. Secondly, as proxy for risk, volailiy is relaed o expeced reurns, hence good forecas models enable invesors give more appropriae securiies pricing sraegies. However, in-sample and ou-of-sample forecas evaluaion poenially provides more useful comparison. Now we will cheek he adequacy of he fied model using he in-sample and ou-of-sample, roo mean square error, mean absolue error, mean absolue percen error, Theil inequaliy coefficien, bias proporion, variance proporion and covariance proporion of forecas properies. If he in-sample roo mean square error, mean absolue error, mean absolue percen error, Theil inequaliy coefficien, bias proporion, variance proporion and covariance proporion are greaer han he ou-sample wih same crieria hen his fied model is adequae. We have used all daa from /05/00 o 0/0/03 (sample size 99) as he in-sample period and he ou sample daa from 0/3/03 o 03//03 (las sample size ) as he ou-of-sample period. Now he calculaed values of forecas properies for in sample and ou sample are presened in Table 7. From Table 7, he values of roo mean square error, mean absolue error, mean absolue percen error, Theil inequaliy coefficien, variance proporion and covariance proporion for in sample is greaer han for ou sample implies ha he seleced model is adequae. The bias proporion for in sample is less han ou sample ha is unexpeced bu ha is occurs because he ou sample size is very smaller compared wih in sample. Therefore we can say on he basis of forecas properies he seleced model ARMA(,) wih GARCH(,) is adequae. Thus, he forecasing performance of seleced model can be represened by graphically as Figure 8. The Figure 8 shows he acual filered series, forecased series and heir average for he period 0/3/03 o 03//03. I is eviden from Figure 8, forecas values, observed values and average values are very close o each oher. So he model ARMA(,) wih GARCH(, ) forecasing performance is reasonable. Secondly, we can consider ARMA(,) wih GARCH(,) model for diagnosic checking and forecasing. We can explain he ARMA(,) wih GARCH(,) model similar

7 American Journal of Economics 03, 3(5): 9-35 way of he ARMA(,) wih GARCH(,) model. The graphical presenaion of his fied model represened as Figure 9. The fied model ARMA(,) wih GARCH(,) (Figure 9) shows he acual and fied values of reurn series is very close and he error is very small indicaing model is fi well. Now, we shall need o check he esimaed residual is saionary or no. We can check saionariy of he residual series using auocorrelaion and parial auocorrelaion funcion and uni roo es. The auocorrelaion and parial auocorrelaion funcion of residual series of ARMA(,) wih GARCH(,) model presened in Figure 0 and. We observed he Figure 0 and, he spikes of auocorrelaion and parial auocorrelaion funcion are converse o zero very quickly imply he residual series is saionary. To conform he residual series of ARMA(,) wih GARCH(,) model is saionary, we used he formal es of saionary Dickey Fuller (DF), Augmened Dickey Fuller (ADF) and Phillips Perron (PP) ess are presened in Table 8. The null hypohesis The residual series conain uni roo is rejeced for all es (DF, ADF and PP es) implies he residual series is saionary. We observed he reurn series of general index is conains volailiy problems so we used he GARCH ype of model for conrolling volailiy effec. So afer such volailiy modelling we mus check wheher heir volailiy effec exis or no. The Lagrange muliplier es is used for esing ARCH effec. The calculae value of es saisics (F-value) of Lagrange muliplier es wih lag is wih probabiliy imply he null hypohesis is acceped and i is evidence of he absen of ARCH effec in he ARMA(,) wih GARCH(,) model of filered reurn series. The sandardized residual plo is used wheher he daa series conain he some specific values (usually ±3 sandard deviaion) oherwise consider as ouliers. The sandardized residual plo for ARMA(,) wih GARCH(,) model is presened in Figure. From Figure, i is clear ha he all sandardized residuals are lies wihin (-3 o +3) excep one. Hence, model is fied well. The normaliy of residual series can be es using Jarque Bera es saisics. The null hypohesis of Jarque Bera es saisics is he residual series is normally disribued. The calculaed vale of Jarque Bera es saisics is.335 wih probabiliy indicaing he null hypohesis is acceped ha means he residual series of ARMA(,) wih GARCH(,) model is normally disribued. Finally, we can check he model adequacy using various model selecion crieria for in-sample and ou-sample forecas of chosen model. The forecasing evaluaion based on various forecasing error crieria of ARMA(,) wih GARCH(,) model is summarized he in Table 9. We know ha if he Roo mean square error, mean absolue error, mean absolue percen error, Theil inequaliy coefficien, bias proporion, variance proporion and covariance proporion are smaller for ou-sample forecas han in-sample forecas considered as bes model. We observed he Table 9, he all value for Roo mean square error, mean absolue error, mean absolue percen error, Theil inequaliy coefficien, variance proporion and covariance proporion are smaller for ou-sample forecas han in-sample forecas bu bias proporion is reverse. The bias proporion for in sample is less han ou sample ha is unexpeced bu ha is occurs because he ou sample size is very smaller compared wih in sample. Hence we can say he chosen ARMA(,) wih GARCH(,) model is adequae. The acual, forecas and average values for ou-sample are presened as Figure 3 and indicaes he difference beween acual and forecas values are very small. Therefore, he chosen ARMA(,) wih GARCH(,) model is accepable.. Conclusions The general index of sock marke is very imporan index for he analysis and predicing invesmen. The general index of Dhaka Sock Marke over he ime 5 December 00 o 0 January 03 wih sample size is 99 were used in his sudy. The preliminary analysis of daa se suggess he non-normal disribuion and srong evidence of GARCH effecs in filered general index reurn series. A he beginning of he sudy general index daa of Dhaka Sock Exchange has been explored and analyzed using visual inspecion of original daa, reurn series ha is also filered by using Buerworh filer. The filered reurn series has consan mean bu he variance is no consan. To check he saionariy, we perform es of correlogram and uni roo es which indicae ha he reurn series is saionary. Then we consruc several mean models as well as variance models. Boh symmeric and asymmeric models used in his sudy and finally found ha he model ARMA(,) wih GARCH(,) and GARCH (,) are more appropriae model for he general index of Dhaka Sock Exchange (DSE) for his sudy period. This resul will provide informaive guide line for he researchers and policymaker. ACKNOWLEDGEMENTS This research was suppored by Yunnan Universiy of Finance and Economics.

8 36 M Monimul Huq e al.: Analysis of Volailiy and Forecasing General Index of Dhaka Sock Exchange Appendix A Table. Buerworh filer wih differen order, cu-off, drif and mean square error Sl. No. Filer order of filer cu-off frequency drif Mean square error.5 08 True True True True True True True Table. Descripive Saisics for daily reurn series of DSE General Index Mean Median Maximum Minimum Sandard Deviaion Skewness Kurosis Jarque-Bera (Probabiliy) ( ) Table 3. Uni roo es of filered reurn series Tes Saisics Type Value of Tes of Saisics (p-value) 5% Criical Value DF Inercep Trend and Inercep ADF Inercep (0.0080) Trend and Inercep (0.000) PP Inercep (0.0000) Trend and Inercep (0.0000) Variance Equaion ARCH() ARCH() ARCH(3) GARCH(,) GARCH(,) TARCH(,) EGARCH(,) Table. Maximum Likelihood esimaes of several ARCH, GARCH, TARCH and EGARCH models Mean Equaion α0 α β β γ μ δ 3.5E-05* * * (.5E-06) ( ) (0.039) ( ).0E-05* * * 0.939* * (.79E-06) (0.0879) (0.099) (0.007) (0.0096) 9.9E-06* 0.759* * 0.395** * * (.88E-06) (0.0776) ( ) ( ) (0.0388) (0.0063).79E-06* 0.339* 0.673* * * (.0E-06) (0.0678) (0.066) (0.0608) (0.0050).70E-06* 0.390* 0.307* 0.985* * * (.6E-06) ( ) (0.8) ( ) (0.099) (0.0038).85E-06* * * * * (.0E-06) (0.67) (0.0689) (0.5) (0.076) ( ) * * 0.903* * * (0.338) ( ) ( ) (0.059) (0.0630) ( ) Table 5. The following able gives R-square, AIC and SC of some differen models Mean Model: ARMA(,) wih R-square AIC SC ARCH() ARCH() ARCH(3) GARCH(,) GARCH(,) Table 6. ADF and PP es of he residual series of ARMA(,) wih GARCH(,) model Tes Saisics Value of Tes of Saisics (p-value) 5% Criical Value DF ADF (0.0000) PP (0.0000)

9 American Journal of Economics 03, 3(5): 9-37 Table 7. Resuls of in-sample and he ou-of-sample forecas properies of fied model Model In-sample Ou-of-sample Roo mean square error Mean absolue error Mean absolue percen error Theil inequaliy coefficien Bias proporion Variance proporion Covariance proporion Table 8. Uni roo es of residual series for ARMA(,) wih GARCH(,) model Tes Saisics Value of Tes of Saisics (p-value) 5% Criical Value DF ADF (0.0000) PP (0.0000) Table 9. Resuls of in-sample and he ou-of-sample forecas properies of fied model Model In-sample Ou-of-sample Roo mean square error Mean absolue error Mean absolue percen error Theil inequaliy coefficien Bias proporion Variance proporion Covariance proporion Appendix-B Figure. Plo of Buerworh filer of reurn series of GI series and deviaion from rend

10 Parial ACF ACF 38 M Monimul Huq e al.: Analysis of Volailiy and Forecasing General Index of Dhaka Sock Exchange Lag Number Figure. ACF of filered reurn daa of DSE Lag Number Figure 3. PACF of filered reurn daa of DSE ARMA(,) wih GARCH(,) Model Residual Acual Fied Figure. Acual, fied and residual of ARMA(,) wih GARCH(,)

11 Parial ACF ACF American Journal of Economics 03, 3(5): Lag Number Figure 5. ACF of residual of ARMA(,) wih GARCH(,) Lag Number Figure 6. PACF of residual of ARMA(,) wih GARCH(,) ARMA(,) wih GARCH(,) Model Sandardized Residual Figure 7. Sandardized residual obained from ARMA(,) wih GARCH(,) model

12 ACF 0 M Monimul Huq e al.: Analysis of Volailiy and Forecasing General Index of Dhaka Sock Exchange Figure 8. Forecas, observed and average values of ou-of-sample forecas for ARMA(,) wih GARCH(,) model ARMA(,) wih GARCH(,) Model Residual Acual Fied Figure 9. Acual, fied and residual graph for ARMA(,) wih GARCH (,) model Lag Number Figure 0. ACF of residual of ARMA(,) wih GARCH(,)

13 Parial ACF American Journal of Economics 03, 3(5): Lag Number Figure. PACF of residual of ARMA(,) wih GARCH(,) ARMA(,) wih GARCH(,) Model Sandardized Residual Figure. Sandardized residual plo of ARMA(,) wih GARCH(,) Figure 3. Forecas of ou of sample of ARMA(,) wih GARCH(,)

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