Modeling and Estimation of Volatility in the Indian Stock Market

Transcription

1 Inernaional Journal of Business and Managemen February, Modeling and Esimaion of Volailiy in he Indian Sock Marke Hojaallah Goudarzi (Corresponding auhor) Deparmen of Sudies in Commerce, Universiy of Mysore Mysore, India C. S. Ramanarayanan Deparmen of Commerce, Maharaja s Collage, Universiy of Mysore Mysore 57 5, Karnaaka, India prof.csramanarayanan@yahoo.com Absrac The inernaional financial markes urmoil, which sared around mid-7, has depreciaed subsanially since Augus 8. The financial marke crisis has led o he collapse of major financial insiuions. Neverheless, crashes and/or crises are no devoed o only developed markes and developing counries including India, are no excluded from his rule and i may face such a condiion. The sharp decline of Sensex price index from is closing peak of 873 on January 8, 8, o less han by Ocober 7, 8, in line wih similar large declines in oher major sock markes is good reminders of his fac. Volailiy as a measure of risk plays an imporan role in many financial decisions in such a siuaions. The main purpose of his sudy is o examine he volailiy of he Indian sock markes and is relaed sylized facs using ARCH models. The BSE5 sock index was used o sudy he volailiy in he Indian sock marke over a years period. Two commonly used symmeric volailiy models, ARCH and GARCH were esimaed and he fied model of he daa, seleced using he model selecion crierion such as SBIC and AIC. The adequacy of seleced model esed using ARCH-LM es and LB saisics. The sudy concludes ha GARCH (, ) model explains volailiy of he Indian sock markes and is sylized facs including volailiy clusering, fa ails and mean revering saisfacorily Keywords: India sock exchange, Volailiy, Sylize facs, ARCH models. Inroducion Flucuaion of sock prices is no desrucive per se and is a sign of marke efficiency in sock markes. In an efficien marke, sock price fully reflecs all available informaion. Thus, sock price flucuaes in response o new informaion. The main problem wih price flucuaion ha affecs he financial marke efficiency is desrucive excess volailiy ha ends up in crashes and/or crises in financial markes. In such a siuaion, difference beween sock inrinsic value and is relaed marke value is significan and has several consequences. The urmoil in he inernaional financial markes of advanced economies, ha sared around mid-7, has exacerbaed subsanially since Augus 8. The financial marke crisis has led o he collapse of major financial insiuions. Neverheless, crashes and/or crises are no devoed o only developed markes and developing counries including India, are no excluded from his rule and i may face such a condiion. Top- Indian sock marke crashes include Apr 99, May 4, May 6, April 7, July 7, Aug 7 Oc 7, Nov 7,Dec 7,Aug, 7and paricularly, Jan 8 are good reminders of his fac. Wih he volailiy in porfolio flows having been large during 7 and 8, he impac of global financial urmoil has been fel paricularly in he Indian equiy marke. The BSE Sensex increased significanly from a level of 3 7 as a end-march 7 o is peak of 873 on January 8, 8 in he presence of heavy porfolio flows responding o he high growh performance of he Indian corporae secor. Wih porfolio flows reversing in 8, parly because of he inernaional marke urmoil (Mohan,8) he Sensex fell from is closing peak of 873 on January 8, 8, o less han by Ocober 7, 8, in line wih similar large declines in oher major sock markes. In addiion, Beween January and Ocober 6 8, he rupee fell by nearly 5 per cen, even relaive o a weak currency like he dollar, from Rs 39. o he dollar o Rs (Chandrasekhar and Ghosh, 8).Hence, he sudy of financial asse volailiy is imporan o academics, policymakers, and financial markes paricipans for several reasons. Firs, predicion of financial marke volailiy is imporan o economic agens 85

2 Vol. 5, No. Inernaional Journal of Business and Managemen because i represens a measure of risk exposure in heir invesmens. Second, a volaile sock marke is a serious concern for policymakers because insabiliy of he sock creaes uncerainy and hus adversely affecs growh prospecs. Evidence shows ha when markes are perceived as highly volaile i may ac as a poenial barrier o invesing. Third, he sock marke volailiy causes reducion in consumer spending. Fourh, pricing of derivaive securiies and pricing of call opion is a funcion of volailiy. Finally, sock reurn forecasing is in a sense volailiy forecasing and his has creaed new job opporuniies for he professionals hose who are expers in volailiy forecasing (Onyeaso and Rogers, 4). Consequenly, i can be seen ha he sudy of sock marke volailiy and is characerisics is very imporan and can be helpful for formulaion of economic policies and forming rules and regulaions relaed o sock marke. While he volailiy and is relaionship wih sock price in developed financial markes has been well sudied, lile concenraion has been paid owards an exensive sudy of he volailiy of he emerging sock marke of India. I is now well known ha equiies from emerging capial markes have vasly differen characerisics han equiies from developed capial markes. There are a leas four disinguishing feaures of emerging marke reurns: higher sample average reurns, low correlaions wih developed marke reurns, more predicable reurns, and higher volailiy (Bekaer and Wu, ).These differences may have imporan implicaions for decision making by invesors and policy makers and pu emphasis on developed markes finding may mislead policy makers in making proper decisions. Therefore, in line wih developed markes sudies, he main objecive of his sudy is o invesigae volailiy and is relaed sylized facs in he Indian sock markes using ARCH models. The res of his paper is organized as follows. Secion deals wih he volailiy models considered for his paper. The review of lieraure is presened in secion 3. The descripion of he BSE5 daa and he mehodology is presened in secion4.the resuls and discussions are presened in secion 5 and finally secion 6 concludes he paper.. Models of Volailiy ARCH models are capable of modeling and capuring many of he sylized facs of he volailiy behavior usually observed in financial ime series including ime varying volailiy or volailiy clusering (Zivo and Wang, 6). The serial correlaion in squared reurns, or condiional heeroskedasiciy (volailiy clusering), can be modeled using a simple auoregressive (AR) process for squared residuals. For example, le y denoe a saionary ime series such as financial reurns, hen y can be expressed as is mean plus a whie noise if here is no significan auocorrelaion in y iself: y = c + () where c is he mean of y, and is iid wih mean zero. To allow for volailiy clusering or condiional heeroskedasiciy, assume ha Var -( ) = σ wih Var - (.) denoing he variance condiional on informaion a ime -, and p p σ = α + α + + α () Since has a zero mean, Var ( ) = σ, he above equaion can be rewrien as: = α + α + + α + u (3) p p u ( ) E Where = is a zero mean whie noise process. The above equaion represens an AR (p) process for, and he model in () and () is known as he auoregressive condiional heeroskedasiciy (ARCH) model of Engle (98), which is usually referred o as he ARCH(p ) model. Before esimaing a full ARCH model for a financial ime series, i is necessary o es for he presence of ARCH effecs in he residuals. If here are no ARCH effecs in he residuals, hen he ARCH model is unnecessary and misspecified. Since an ARCH model can be wrien as an AR model in erms of squared residuals as in equaion 3, A simple Lagrange Muliplier (LM) es for ARCH effecs can be consruced based on he auxiliary regression as in equaion 3. Under he null hypohesis ha here is no ARCH effecs: he es saisic is H α α α = = = = P = LM = T R χ ~ ( P ) 86

3 Inernaional Journal of Business and Managemen February, where T is he sample size and R is compued from he regression (3) using esimaed residuals. If P-value is smaller han he convenional 5% level, he null hypohesis ha here are no ARCH effecs will rejeced. In oher word, he series under invesigaion shows volailiy clusering or persisence. If he LM es for ARCH effecs is significan for a ime series, one could proceed o esimae an ARCH model and obain esimaes of he ime varying σ based on pas hisory. However, in pracice i is ofen found ha a large number of lags P, and hus a volailiy large number of parameers, is required o obain a good model fi. A more parsimonious model proposed by Bollerslev (986) replaces he AR model in (equaion ) wih he following formulaion: p q = + i i + b j j i = j = (4) σ α α σ where he coefficiens α i ( i =..., p) and bj ( j =..., q) are all assumed o be posiive o ensure ha he condiional variance σ is always posiive. The model in (equaion 4) ogeher wih (equaion ) is known as he generalized ARCH or GARCH (p, q) model. When q =, he GARCH model reduces o he ARCH model. Under he GARCH (p, q) model, he condiional variance of, σ, depends on he squared residuals in he previous p periods, and he condiional variance in he previous q periods. Usually a GARCH (, ) model wih only hree parameers in he condiional variance equaion is adequae o obain a good model fi for financial ime series (Zivo and Wang, 6). Arch Models Specificaion for BSE5 Before esimaing ARCH models for a financial ime series, aking wo seps is necessory.firs check for uni roos in he residuals and second es for ARCH effecs. The inpu series for ARMA needs o be saionary before we can apply Box-Jenkins mehodology.the series firs needs o be differenced unil is saionary. This needs log ransforming he daa o sabilize he variance. Since he raw daa are likely o be non-saionary, an applicaion of ARCH es is no valid. For his reason, i is usual pracice o work wih he logs of he changes of he series raher han he series iself. The presence of uni roo in a ime series is esed using Augmened Dickey- Fuller es. I ess for a uni roo in he univariae represenaion of ime series. For a reurn series R, he ADF es consiss of a regression of he firs difference of he series agains he series lagged k imes as follows: Or The null and alernaive hypoheses are as follows: r = α + δ r + β r + i i i = p r ; ln ( ) = r r r = R H : he series conains uni roo H : he series is saionary The accepance of null hypohesis implies non-saionary. If he ADF es rejecs he null hypohesis of a uni roo in he reurn series, ha is if he absolue value of ADF saisics exceeds he McKinnon criical value he series is saionary and we can coninue o analyze he series. Before esimaing a full ARCH model for a financial ime series, i is necessary o check for he presence of ARCH effecs in he residuals. If here are no ARCH effecs in he residuals, hen he ARCH model is unnecessary and misspecified (Zivo and Wang, 6)... Arch effec es process Consider he k-variable linear regression model. 87

4 Vol. 5, No. Inernaional Journal of Business and Managemen y = β + β x + + β x + u k k In addiion, assume ha condiional on he informaion available a ime (-), he disurbance erm disribued as Tha is, u ~, ( α + α u ) u is normally disribued wih zero mean and V ar ( u ) = ( α + α u ) u Tha is he variance of follows an ARCH () process. The variance of u a ime is dependen on he squared disurbance a ime (-), hus giving he appearance of serial correlaion. The error variance may depend no only on one lagged erm of he squared error erm bu also on several lagged squared erms as follows: Var( u ) = σ = α + α u + α u + + α u p p If here is no auocorrelaion in he error variance, we have H : α = α = = α p = In such a case, Var( u ) = α, and we do no have he ARCH effec. σ Since we do no direcly observe Engle has shown ha running he following regression can easily es he preceding null hypohesis: u垐 = α垐 + α u + α垐 u垐 + + α u p p u Where, as usual, denoe he OLS variance obained from he original regression model. The null hypohesis can be esed by he usual F es bu he ARCH-LM es of Engle 98 is a common es in his regard. Under ARCH-LM es he null and alernaive hypohesis for BSE5 sock index are as follows: H : α = and α = and α 3 = and... α q = H : α and α and α and... α 3 Null hypohesis in his case is homoskedasiciy or equaliy in he variance. Accepance of his hypohesis imply ha, here is no ARCH effecs in he under process series. In oher word, he daa do no show volailiy clusering i.e. here is no heeroskadasiciy or ime varying variance in he daa. Since an ARCH model can be wrien as an AR model in erms of squared residuals as in = α + α + + α + p p a simple Lagrange Muliplier (LM) es for ARCH effecs can be consruced based on he auxiliary regression. = α + α + + α + Under he null hypohesis ha here are no ARCH effecs: The es saisic is as follows: p p LM = T R χ Where T is he sample size R is compued from he regression ~ ( P ) = α + α + + α + p p u u u q using esimaed residuals. Tha is in large sample TR follows he Chi-square disribuion wih df equal o he number of auoregressive erms in he auxiliary regression. 88

5 Inernaional Journal of Business and Managemen February, The es saisic is defined as TR (he number of observaions muliplied by he coefficien of muliple correlaion) from he las regression, and i is disribued as a χ (q) (Gujarai,7). Thus, he es is one of a join null hypohesis ha all q lags of he squared residuals have coefficien values ha are no significanly differen from zero. If he value of he es saisic is greaer han he criical value from he χ disribuion, hen one can rejec he null hypohesis. The es can also be hough of as a es for auocorrelaion in he squared residuals. Alernaively, if P-value is smaller han he convenional α % level, he null hypohesis ha here are no ARCH effecs will rejeced. In oher word, he series under invesigaion shows volailiy clusering or volailiy persisence (Brooks, ). If an ARCH effec is found o be significan, hen he specificaion of an appropriae ARCH model is necessary. In order o idenify he ARCH characerisics in BSE5, he condiional reurn should be modeled firs; he general form of he reurn can be expressed as a process of auoregressive AR (p), up o (p) lags, as follows: R p = α + α R + i = R This general form implies ha he curren reurn depends no only on ( ) bu also on he previous (p) reurn value R p ( ). The nex sep is o consruc a series of squared residuals ( ) based on condiional reurn o drive he condiional variance. Unlike he OLS assumpion of a consan variance of (, s ), ARCH models assumes ha(, s ) have a non h consan variance or heeroscedasiciy, denoed by( ).Afer consrucing ime series residuals, we modeled he condiional variance in a way ha incorporaes he ARCH process of ( ) in he condiional variance wih (q) lags. The general forms of he condiional variance, including (q) lag of he residuals is as follows: q = + i = h β β The above equaion is wha Engle (98) referred o as he linear ARCH (q) model because of he inclusion of he (q) lags of he ( ) in he variance equaion. This model suggess ha volailiy in he curren period is relaed o volailiy in he pas periods, β For example in he case of AR() model, If is posiive,i suggess ha if volailiy was high in he previous β period, i will coninue o be high in he curren period, indicaing volailiy clusering.if is zero, hen here is no volailiy clusering. To deermine he value of q or he ARCH model order, we use he model selecion crierion such as AIC (Akaike Informaion Crierion) and SBIC (Schwarz Bayesian Informaion Crierion). The decision rule is o selec he model wih he minimum value of informaion crierion. This condiion is necessary bu no enough because he esimae mees he general requiremens of an ARCH model. The model o be adequae should have coefficien ha all are significan. If his requiremen mees hen he specified model is adequae and fi he daa well. 89

6 Vol. 5, No. Inernaional Journal of Business and Managemen. Garch model The problem wih applying he original ARCH model is he non-negaiviy consrain on he coefficien parameers of (βi's) o ensure he posiiviy of he condiional variance. However, when a model requires many lags o model he process correcly he non-negaiviy may be violaed. To avoid he long lag srucure of he ARCH (q) developed by Engle (98), Bollerslev (986), generalized he ARCH model, he so-called (GARCH), by including he lagged values of he condiional variance. Thus, GARCH(p,q) specifies he condiional variance o be a linear combinaion of (q) lags of he squared residuals from he condiional reurn equaion and (p) lags from he condiional variance follows: h σ j q p = β + β i + βh j i = j =.Then, he GARCH(p,q) specificaion can be wrien as j=,...p and i=,...q Where β, β > and ( β+ β) < is o avoid he possibiliy of negaive condiional variance. The above equaion saes ha he curren value of he condiional variance is a funcion of a consan and values of he squared residual from he condiional reurn equaion plus values of he previous condiional variance. To show he significance of he explanaion of condiional variance of one lag of boh h and, e.g. and h,he GARCH process should be employed by esimaing he condiional reurn o drive,and hen he esimaion of he condiional variance by using equaion below h = β + β + α h The adequacy of he GARCH model can be examined by sandardized residuals, ( ) σ, where ( σ ) is he condiional sandard deviaion as calculaed by he GARCH model, and( ) is he residuals of he condiional reurn equaion. R p = α + α R + i = If he GARCH model is well specified, hen he sandardized residuals will be Independen and Idenically Disribued (IID).To shows his, wo-sep es is needed. The firs sep is o calculae he Ljung-Box Q-Saisics (LB) on he squared observaion of he raw daa. This es can be used o es for remaining serial correlaion in he mean equaion and o check he specificaion of he mean equaion. If he mean equaion is correcly specified, all Q-saisics should no be significan. The nex sep is o calculae he Q-saisics of he squared sandardized residuals. This es can be used o es for remaining ARCH in he variance equaion and o check he specificaion of he variance equaion. If he variance equaion is correcly specified, all Q-saisics should no be significan. Pu anoher way, if he GARCH is well specified, hen he LB saisic of he sandardized residuals will be less han he criical value of he Chi-square saisic χm p q (Alsalman.A.E.). The es for mean equaion specificaion can be hough of as a es for auocorrelaion in he sandardized residuals. The es is one of a join null hypohesis ha here is no auocorrelaion up o order k of he residuals. If he value of he es saisic is greaer han he criical value from he Q-saisics, hen he null hypohesis can be rejeced. Alernaively, if p-value is smaller han he convenional significance level, he null hypohesis ha here are 9

7 Inernaional Journal of Business and Managemen February, no auocorrelaion will be rejeced. In oher words, he series under invesigaion shows volailiy clusering or volailiy persisence. The same is rue for variance equaion.the only difference is ha in his case he es will be done on squared sandardized residuals. In addiion o Ljang-Box Q-saisics he ARCH-LM es also can be used o es he adequacy of Arch model. The procedure is same as ARCH model. To model selecion, model selecion crieria such as SBC crieria and AIC is used.. 3 Mean reversion The high or low persisence in volailiy is generally capured in he GARCH coefficien(s)of a saionary GARCH model.for a sainary GARCH model he volailiy mean revers o is long run level,a rae given by he sum of ARCH and GARCH coefficiens,which is generally close o one for a financial ime series. The average number of ime periods for he volailiy o rever o is long run level is measured by he half life of he volailiy shock and i is used o forecas he BSE5 series volailiy on a moving average basis (Banerjee and sarkar, 6). A covariance saionary ime series { y } has an infinie order moving avarage represenaion of he form y = µ + ψ, i i i = ψ, ψ i < i = = The plo of he ψ i agains i is called he Impulse Response Funcion (IRF).The decay rae of IRF is someimes repored as a half-life, denoed by L half, which is he lag a which he IRF reaches.. 3. Calculaion of half-life of volailiy shock for a saionary GARCH (, ) process The mean revering form of he basic GARCH ( ) model is: ( σ ) = ( α + β ) ( σ ) + u β u where = /( ) σ α α β is he uncondiional long run level of volailiy and u σ = ( ). The mean α + β revering rae α + β implied by mos fied models is usually very close o.the magniude of conrols he speed of mean reversion.the half life of a volailiy shock is given by he formula L half = ln ( ) / ln ( α + β ) Measures he average ime i akes for σ α o decrease by one half.the closer + β is o one he longer is α he half life of a volailiy shock. If + β >, he GARCH model is nonsaionary and he volailiy will evenually explode o infiniy (Banerjee and sarkar,6). 3. Review of Lieraure Sock prices volailiy is an exremely imporan concep in finance for numerous reasons. The lieraure on sock price volailiy agrees on one key phenomenon. There is evidence of sever movemens in sock prices. In oher words, dynamic naure of sock prices behavior is an acceped phenomenon and all paricipans in sock markes include regulaors, professionals and academics have consensus abou i. Bu, wha causes sock prices volailiy is a quesion ha remains unseled in finance field. Answer o his quesion, because of he grea number of involved variables is no an easy ask and up o now here is no consensus abou i. However researchers in ques of answer his quesion has invesigaed he sock prices volailiy from differen angels. In his regards, from lae wenieh cenury and paricularly afer inroducing ARCH model by Engle (98), as said by Bollerslev (999) and Granger and Poon () several hundred research ha mainly accomplished in developed counry and o some exen in developing counries has been 9

8 Vol. 5, No. Inernaional Journal of Business and Managemen done by researchers in his area using differen mehodology. Our objecive in his secion is o give he reader jus a glimpse of hese sudies as follows: Engle (98) published a paper ha measured he ime-varying volailiy. His model, ARCH, is based on he idea ha a naural way o updae a variance forecas is o average i wih he mos recen squired "surprise"(i.e. he squired deviaion of he rae of reurn from is mean).while convenional ime series and economeric models operae under an assumpion of consan variance, he ARCH process allows he condiional variance o change over ime as a funcion of pas errors leaving he uncondiional variance consan. In he empirical applicaion of he ARCH model a relaively long lag in he condiional variance equaion is ofen called for, and o avoid problems wih negaive variance parameers a fixed lag srucure is ypically imposed. Bollerslev (986) o overcome he ARCH limiaions inroduced his model, GARCH, ha generalized he ARCH model o allow for boh a longer memory and a more flexible lag srucure. As noed above, in he empirical applicaion of he ARCH model, a relaively long lag in he condiional variance equaion is ofen called for, and o avoid problems wih negaive variance parameers a fixed lag srucure is ypically imposed. In he ARCH process he condiional variance is specified as a linear funcion of pas sample variance only, whereas he GARCH process allows lagged condiional variances o ener in he model as well. Engle, Lilien, and Robins (987) inroduced he ARCH-M model by exending he ARCH model o allow he condiional variance o be deerminan of he mean. Whereas in is sandard form, ARCH model expresses he condiional variance as a linear funcion of pas squired innovaions in his new model hey hypohesize ha, changing condiional variance direcly affec he expeced reurn on a porfolio. Their resuls from applying his model o hree differen daa ses of bond yields are quie promising. Consequenly, hey conclude ha risk premia are no ime invarian; raher hey vary sysemaically wih agen's percepions of underlying uncerainy. Nelson (99) exended he ARCH framework in order o beer describe he behavior of reurn volailiies. Nelson's sudy is imporan because of he fac ha i exended he ARCH mehodology in a new direcion, breaking he rigidness of he G/ARCH specificaion. The mos imporan conribuion was o propose a model (EARCH) o es he hypohesis ha he variance of reurn was influenced differenly by posiive and negaive excess reurns. His sudy found ha no only was he saemen rue, bu also ha excess reurns were negaively relaed o sock marke variance. Glosen, Jagannahan and Runkle (993),o modify he primary resricions of GARCH-M model based upon he ruh ha GARCH model enforce a symmeric response of volailiy o posiive and negaive shocks, inroduced GJR's (TGARCH) models. They conclude ha here is a posiive bu significan relaion beween he condiional mean and condiional volailiy of he excess reurn on socks when he sandard GARCH-M framework is used o model he sochasic volailiy of sock reurns. On he oher hand, Campbell's Insrumenal Variable Model esimaes a negaive relaion beween condiional mean and condiional volailiy. They empirically show ha he sandard GARCH-M model is misspecified and alernaive specificaions provide reconciliaion beween hese wo resuls. When he model is modified o allow posiive and negaive unanicipaed reurns o have differen impacs on condiional variance, hey find ha a negaive relaion beween he condiional mean and he condiional variance of he excess reurn on socks. Finally, hey also find ha posiive and negaive unexpeced reurns have vasly differen effecs on fuure condiional variance and he expeced impac of a posiive unexpeced reurn is negaive. Engle and Ng (993) measure he impac of bad and good news on volailiy and repor an asymmery in sock marke volailiy owards good news as compared o bad news. More specifically, marke volailiy is assumed o be associaed wih he arrival of news. A sudden drop in price is associaed wih bad news on he oher hand, a sudden increase in price is said o be due o good news. Engle and Ng find ha bad news creae more volailiy han good news of equal imporance. This asymmeric characerisic of marke volailiy has come o be known as he "leverage effec". The sudies of Black (976), Chrisie (98), FSS (987), Schwer (99) and Pagan and Schwer (989) also explain his volailiy asymmery wih he" leverage effec". However, heir models do no capure his asymmery. Engle and Ng (993) provide new diagnosic ess and models, which incorporae he asymmery beween he ype of news and volailiy, hey advise researchers o use such enhanced models when sudying volailiy. Bara [4] in an aricle eniled" sock reurn volailiy paerns in India examined he ime varying paern of sock reurn volailiy and asymmeric Garch mehodology. He also examined sudden shifs in volailiy and he possibiliy of coincidence of hese sudden shifs wih significan economic and poliical evens boh of domesic and global origin. Also, he examined sock marke cycles for variaion in ampliude, duraion and volailiy of he bull and bear phases over he reference period. His analysis revealed ha liberalizaion of he sock marke or he FII enry in paricular does no have any direc implicaions for he sock reurns volailiy. No srucural changes in he sock price volailiy around any liberalizaion even or more imporanly around he daes of breaks for volailiy in FII sales and purchases in India were observed. The apparen link generally drawn beween sock price volailiy and he sudden wihdrawal or heavy purchase by he FIIs i.e. he volaile FII invesmen in he sock marke did no seem o hold rue for India. In all he phases, as delineaed by heir srucural break analysis, he period beween 99:5 and 993: was he mos volaile 9

9 Inernaional Journal of Business and Managemen February, period wih he sandard deviaion of sock reurns exceeding ha in he oher periods. The sudy also showed ha in general over he references period he bull phases are longer, he ampliude of he bull is higher and he volailiy in he phases is also higher. He also concluded ha he gains during expansions are larger han he losses during he bear phases of sock marke cycles. The bull phase, in comparison wih is pre liberalizaion characer was more sable in he pos liberalizaion phase. The resuls of heir analysis also, showed ha he sock marke cycles have dampened in he recen pas. Finally, he sudy showed ha volailiy has declined in he pos liberalizaion phase for boh he bull and bear phase of he sock marke cycles. Kumar [6] in an aricle eniled comparaive performance of volailiy forecasing models in Indian markes" evaluaed he comparaive abiliy of differen saisical and economic volailiy forecasing models in he conex of Indian sock and forex markes. Based on he ou of sample forecass and he number of evaluaed measures ha rank a paricular mehod as superior he concluded ha i is possible o infer ha EWMA will lead o improvemens in volailiy forecass in he sock markes and he GARCH (5,) will achieve he same in he forex marke. As he concluded, his findings were conrary o he findings of Brailsford and Paff [996] who found no single mehod as superior, bu he resuls in sock marke were similar o he findings of Akigray [989], McNillian [], Anderson and Bollerslev[998] and Anderson e al [999] in he Forex marke. Banerjee and Sarkar [6] in an aricle eniled long memory propery of sock reurns; evidence from India examined he presence of long memory in asse reurns in he Indian sock marke. They found ha alhough daily reurns are largely uncorrelaed, here is srong evidence of long memory in is condiional variance. They concluded ha FIGARCH is he bes-fi volailiy model and i ouperforms oher Garch ype models. They also observed ha he leverage effec is insignifican in SenSex reurns and hence symmeric volailiy models urn ou o be superior as hey expeced. 4. Mehodology The required daa including 8 daily closing observaion for BSE5 price index covering he period 6/7/ hrough //9 were obained from he Bangalore Sock Exchange, and were based on daily closing prices. The BSE5 reurns ( r ) a ime are defined in he logarihm of BSE5 indices (p), ha is, r = log( p / p( ) ).Visual inspecion of he plo of daily reurns series of BSE5 proved very useful. I can be seen ha from figure ha reurn flucuaes around mean value ha is close o zero. Volailiy is high for cerain ime periods and low for oher periods. The movemens are in he posiive and negaive erriory and larger flucuaions end o cluser ogeher separaed by periods of relaive calm. The volailiy was highes in 4 and 8.Thus figure shows volailiy clusering where large reurns end o be followed by small reurns leading o coninuous periods of volailiy and sabiliy.volailiy clusering implies a srong auocorrelaion in squared reurn. The number of observaion is 8.The mean daily reurn is.53e-8.the volailiy (measured as a sandard deviaion) is.74.there is indicaion of negaive skewness (Skw= -.96) which indicaes ha he lower ail of he disribuion is icker han he upper ail, ha is,he index declines occur more ofen han is increases. The kurosis coefficien is posiive, having high value for he reurn series (Kur = 8.93) ha is he poiner of lepokurosis or fa aildness in he underlying disribuion. In fac, under he null hypohesis of normaliy he Jarque-Bera saisic asympoically follows a Qi-squire disribuion wih degree of freedom. The compued value of 75 wih P-value of zero rejecs he normaliy assumpion due o he high kurosis indicaing fa ail.q-q plo in figure also confirm he non-normaliy of he reurns series. As able. shows ARCH-LM es is saisically significan which indicaes he presence of ARCH effec in he residuals of mean equaion of BSE 5.The ADF es saisics rejecs he hypohesis of uni roo in he reurns series a % level of significance. A formal applicaion of ADF es on log reurns, rejecs he null hypohesis of a uni roo in he reurn series.there is rejecion a. level of significance because absolue value of ADF saisics exceeds McKinnon criical value These properies of he BSE5 reurns series are consisen wih oher financial imes series. The ARCH and GARCH models are esimaed for BSE5 reurns series using he robus mehod of Bollerslev-Wooldridge s quasi-maximum likelihood esimaor (QMLE) assuming he Gaussian sandard normal disribuion. Nex, we use informaion crieria such as AIC, SBIC values, and a se of model diagnosic ess (ARCH-LM es and Q-Saisics) o choose he volailiy models which represen he condiional variance of he BSE5 reurns series appropriaely. We esimaed he model using Eviews 4, Eviews 5. and S-plus

10 Vol. 5, No. Inernaional Journal of Business and Managemen 5. Findings To deec he presence of ARCH effec in he mean equaion of BSE5 we use he ARCH-LM (Lagrange muliplier) es. We esed for ARCH-effec for higher order and found ha coefficien of 3, 5, 6 and 8 found o be saisically insignifican. ARCH-LM es is saisically significan which indicaes he presence of ARCH effec in he residuals of mean equaion of BSE 5[able]. To deermine which ARCH model is adequae for describing he condiional heeroscedasiciy of he daa a 5% significance level we apply sample ACF and PACF of he squared residuals which showed he exisence of ARCH effecs. The sample PACF indicaed ha an ARCH (4) model migh be appropriae. Consequenly, we specify he ARCH (4) model as follows: r = µ + α r + σ = α + α + α + α + α The resuls for he ARCH (4) for daily log reurns of BSE5 are repored in able. As able shows he esimaes of α, α, α3 and α4 are all saisically significan a he 5% level of significance. Therefore, he model need no o be simplified. Therefore, we choose ARCH (4) for our daa se of BSE5. Using he AIC, SBIC and Loglikelihood model selecion crieria we achieved same resuls. To es he adequacy of he model we applied he ARCH-LM es up o four lag. The resul has repored in he able 3. As able 3 indicaes, boh es saisics are saisically insignificans. I means no ARCH effecs lef in he model. Thus, we found ha ARCH (4) can be possible represenaive of he condiional volailiy process for daily reurn series of BSE5.Hence we obain he following fied model for mean and variance equaions. r = r + σ = 8.E Garch 3 4 model Alhough he ARCH model is simple, i ofen requires many parameers o adequaely describe he volailiy process of an asse reurns. Bollerslev (986) proposes a useful exension known as he generalized ARCH (GARCH) model. The modeling process of ARCH models can also be used o build a GARCH model. However, specifying he order of GARCH model is difficul. For his reason only lower order of GARCH, models are used in mos applicaion. We fi he GARCH models wih differen orders (up o 5) o he daily reurns. To selec he order of GARCH model, we used SBC crieria. The model wih lower value of SBC fis he daa bes. The resuls are presened in able 4. As able 4 shows, The SBIC value is lowes for p= and q=. Therefore, we choose GARCH (,) for our daa se of BSE5.Thus,we found ha GARCH(,) can be possible represenaive of he condiional volailiy process for daily reurn series of BSE5.Table5 repors he saisics regarding GARCH(,).To es he adequacy of GARCH (,) model we apply ARCH-LM es up o lag. The resuls of ARCH LM es are repored in able 6. As resuls show he F-saisic and Obs*R-squared saisic boh are insignificance and indicaing no arch effecs lef in he series. Thus we employed GARCH (,) o model volailiy.the model of volailiy for BSE5 index using GARCH (,) are as follows: r = r + σ =.3E o.78674σ + As above model indicaes he value of α is and he value of β is The sum of parameers is.97.the saionary condiion ( α + β < ) is saisfied. The mean revering rae ( α + β ) =.97, implied by our fied 94

11 Inernaional Journal of Business and Managemen February, model is close o one. Therefore, he fied GARCH model implies ha condiional volailiy is very persisen. A large value of GARCH lag coefficien β (.78674) indicaes ha shocks o condiional variance akes a long ime o die ou, so he volailiy is persisen. Low value of error coefficien α i.e. (.79646) suggess ha large marke surprises induce relaively small revision in fuure volailiy. ( α + β ) =.97 is close o uniy and implies ha a shock a ime persiss for many fuure periods. A high value of his kind implies a long memory in he sock marke. Any shock will h lead o a permanen change in all he fuure values of, hence shocks o condiional variance are persisen. 5. Mean reversion To es he null of non saionary series or no mean reversion in he BSE5 reurns we applied wo ess. Firs we used he uni roo es. As i saed in he beginning of he chaper, he resuls of he ADF es showed ha he series is saionary. In oher words here was no evidence in favor of uni roo in he daa and we concluded ha he daa series is saionary. When he daa series is saionary, i is mean revering and volailiy finally revers o is long run average. Anoher way of esing mean reversion is using GARCH model. For a sainary GARCH model he volailiy mean revers o is long run level,a rae given by he sum of ARCH and GARCH coefficiens,which is generally close o one for a financial ime series. The average number of ime periods for he volailiy o rever o is long run level is measured by he half life of he volailiy shock and i is used o forecas he BSE5 series volailiy.here he sum of arch and garch erm is nearly.97 which is close o. The mean revering α + β rae α implied by our fied model is very close o.the magniude of + βconrols he speed of mean reversion. The half life of a volailiy shock Measures he average ime i akes for σ o decrease by one half α + β he closer α is o one he longer is he half life of a volailiy shock.if β + >,he GARCH model is nonsaionary and he volailiy will evenually explode o infiniy.in our case i is almos or approximaly one calendar monh.therefore, he null hypohesis of uni roo or no mean reversion is rejeced and we accep he alernaive hypohesis of saionary or mean revering in he underlying series. 6. Conclusions This sudy aemped o sudy he volailiy and is sylized facs in he Indian sock marke. The BSE5 index of Mumbai sock exchange is used as a proxy for he Indian marke. The daa used for analysis were 8 daily observaions for he period of 7/6/ o //9. Empirical resuls showed ha GARCH (,) model can adequaely describe he BSE5 sylized facs. The resuls sugges ha he volailiy in he Indian sock marke exhibis he persisence of volailiy and mean revering behavior. The condiional volailiy of he BSE5 was found o be quie persisence. Wihin he ARCH family ha used in his sudy, our resuls revealed ha he GARCH (,) model saisfacorily explains volailiy and is he mos appropriae model for explaining volailiy clusering, fa ails and mean revering in he series under analysis. The resuls of he sudy have useful implicaions for regulaor and policy makers in he Indian sock marke. Given he inefficiency of radiional mehods of calculaing volailiy such as Moving Average and EWMA in capuring sylized facs of sock marke i.e. volailiy clusering and mean reversion, using hese mehods in evaluaing risk needs o be reviewed and using GARCH-ype model should be considered in risk managemen decisions. References Alsalman. A.E. (). Empirical issues of financial marke volailiy in Kuwai sock exchange. submied hesis,howard universiy. Banerjee.A., and Sarkar.S. (6). Modeling daily volailiy of he Indian sock marke using inr-day daa. IIM CALCUTTA,Working paper. Bara.A. (4). Sock reurn volailiy paerns in India. Working Paper No.4. Bekaer. G., and Wu.G,, Asymmeric volailiy and risk in equiy markes. The review of financial sudies, vol, 3,. 95

12 Vol. 5, No. Inernaional Journal of Business and Managemen Bollerslev, Tim. (986). Generalized auoregressive condiional heeroskedasiciy. journal of Economerics, Vol, 3, Brooks Chris. (). Inroducory economerics for finance. (s ed) Cambridge Universiy Press. Campbell Y. John., Andrew w. Lo., and A. Craig McKinley. (6) The economerics of financial markes ( s Indian sub-coninen ed). New age inernaional(p) limied publicaion. Engle F. Rober. (98). Auoregressive condiional heeroskedasiciy wih esimaes of he variance of Unied Kingdom inflaion, Economerica. vol.5, 4, Engle, Rober F., and Vicor K. Ng (993) Measuring and esing he impac of news on volailiy. Journal of Finance, 48, Glosen, Lawrence R., Ravi Jagannahan., and David E. Runkle (993).On he relaion beween he expeced value and he volailiy of he nominal excess reurn on socks. Journal of Finance, 48, Gujarai Damodar n., and Sangeea. ( 7). Basic economerics. The McGraw Hill publishing company. Kummar S.S.S. (6). Comparaive performance of volailiy forecasing models in indian markes. Decisions, Vol Nelson, Daniel B. (99). Condiional heeroskedasiciy in asse reurns: a new approach. Economerica, 59, Onyeaso G., And Rogers. (4). An economeric invesigaion of he volailiy and marke efficiency of he u.s. small cap 6 sock index. Journal of Business and Economics, Vol. 43. Poon Ser Haung. (5). A pracical guide o forecasing financial marke volailiy. John Wiley and Sons. Poon., and Granger. (3). Forecasing volailiy in financial markes: a review. Journal of Economic Lieraure, Vol.4., 478. Tsuji Chickasha. (3). Is volailiy he bes predicor of marke crashes? Asia pacific Financial Marke, Vol., PP Wu.G. (). The deerminans of asymmeric volailiy. The review of financial sudies, vol. 4, 3. Zivo Eric., and Wang. (6). Modeling financial ime series wih s-plus, (nd ed). Springer. Web References Chandrasekhar C. P. Ghosh. (8).India and he global financial crisis, [Online] Available: hp:// Mohan R. (8). Global financial crisis and key risks: impac on india and asia. [Online] Available: hp://rbidocs.rbi.org.in/rdocs/speeches/pdfs/87784.pdf Table. ARCH-LM es of BSE5 log reurns series up o lags ARCH-LM TEST F-saisics Probabiliy. Obs*R-Squared Probabiliy. Table. ARCH (4) model parameers ARCH (4) model parameers Mean equaion Coefficien Sd. Error z-saisic Prob. C AR() Variance Equaion C 8.E-5 8.4E ARCH() ARCH() ARCH(3) ARCH(4)

13 Inernaional Journal of Business and Managemen February, Table 3. Arch-Lm Tes for Arch (4) Model Up o 4 Lag ARCH(4) Tes F-saisics.9846 Probabiliy.549 Obs*R-Squared Probabiliy.5393 Table 4. SBIC for differen Garch model Comparisons of he SBC for he GARCH(p,q) model wih differen combinaions of p and q for BSE5 p q Table 5. GARCH (, ) parameers GARCH(,) Parameers Mean equaion Coefficien Sd. Error z-saisic Prob. C AR() Variance Equaion C.3E-5.95E ARCH() GARCH() Table 6. ARCH-LM es for Garch (,) model up o lag ARCH() es F-saisics Probabiliy Obs*R-Squared Probabiliy

14 Vol. 5, No. Inernaional Journal of Business and Managemen LOGRT Residuals Figure. The Residuals of Bse5 Reurns 4 3 Normal Quanile LOGRT Figure. Q-Q Plo of BSE5 Daily Reurns Series 98