Asian Economic and Financial Review VOLATILITY MEAN REVERSION AND STOCK MARKET EFFICIENCY. Hojatallah Goudarzi

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1 Asian Economic and Financial Review journal homepage: hp://aessweb.com/journal-deail.php?id=500 VOLATILITY MEAN REVERSION AND STOCK MARKET EFFICIENCY Hojaallah Goudarzi Deparmen of Finance and Insurance, Faculy of Managemen, Universiy of Tehran ABSTRACT Tradiional economeric models, such as he ordinary leas square mehod, are buil on he assumpion of consan variance. Financial ime series, unlike oher economic series, usually exhibi a se of peculiar characerisics i.e. mean reversion, volailiy clusering, fa ails and long memory. The main purpose of his sudy was o sudy marke efficiency hrough modeling one sylized facs of asse reurns series i.e. mean reversion in he Indian sock marke. To achieve his purpose, he sudy used ADF es and GARCH model. The sudy found ha he underlying series is saionary and herefore mean revering. Therefore, based on he resuls he sudy concluded ha, he Indian sock marke is informaionally weak-inefficien. Keywords: Sylized facs, Random Walk, Mean Reversion, Marke Efficiency, Uni Roo.. INTRODUCTION The efficien marke hypohesis (EMH) was ariculaed and developed by Fama during 960 s, and popularized hrough his highly influenial review of Efficien Capial Markes, published in 970 (Pesaran, 005). Efficien financial markes are hose ha do no allow invesors o earn above average reurns wihou acceping above average risks. In such a marke, neiher echnical analysis, which is he sudy of pas sock prices in an aemp o predic fuure prices, nor even fundamenal analysis, which is he analysis of financial informaion such as company earnings, asse values ec., o help invesors selec undervalued sock, would enable an invesors o achieve reurns greaer han hose ha could be obained by holding a randomly seleced porfolio of individual socks wih comparable risks (Mlkiel, 003). Sock marke efficiency implies ha prices respond quickly and accuraely o relevan informaion. An efficien sock exchange is characerized by a random walk process, which is a clue ha reurns of a sock marke are unpredicable from previous price changes (Narayan and Prasad, 007). A random walk process implies ha any shock o sock price is permanen and here is no endency of mean revering. In oher words, his suggess ha fuure reurns are unpredicable 68

2 based on pas observaions. Hence, i is imperaive o invesigae wheher he sock-price can be characerized as random walk (uni roo) or mean reversion process. (Mobarek, 009). Hence, esing for mean reversion could help o examine marke efficiency.tes for mean reversion also allows one o gauge wheher shocks o sock prices have a permanen or a ransiory effec. For insance, if i is esablished ha sock prices are mean revering, i.e. hey are saionary processes, hen his implies ha shocks o sock prices will have a ransiory effec, in ha prices will reurn o heir rend pah over ime. From an invesmen poin of view, his ensures ha one can forecas fuure movemens in sock prices based on pas behavior and rading sraegies can be developed so as o earn abnormal reurns. However, if i is found ha sock prices are nonsaionary hen shocks will have a permanen effec, implying ha sock prices will aain a new equilibrium and fuure reurns canno be prediced based on hisorical movemens in sock prices (Narayan and Prasad, 007). This paper conribues o he lieraure by esing marke efficiency by modeling mean reversion in daily sock prices for he Indian sock marke by employing ADF es and GARCH model on daa over a en years period from 000o 00. The res of his paper is organized as follows. Secion wo deals wih he heoreical issues considered for his paper. The review of lieraure is presened in secion hree. The resuls are provided in secion four and secion five concludes he paper.. THEORETICAL ISSUES.. ARCH Model To capure he serially correlaion of volailiy, Engle (98)proposed he class of Auoregressive Condiional Heeroscedasiciy (ARCH) model. This wries condiional variance as a disribued lag of pas squared innovaions. In order o idenify he ARCH characerisics in ime series, he condiional reurn mus be modeled firs, he general form of he reurn can be expressed as a process of auoregressive R AR ( p) p R 0 i, up o (p) lags, as follows: This general form implies ha he curren reurn depends no only on ( R ) bu also on he previous (p) reurn value ( R p ).The objecive of modeling he condiional reurn is o consruc a series of squared residuals ( ) from which o drive he condiional variance.unlike he OLS assumpion of a consan variance of (, s) ARCH assumes ha (ε s)have a no consan variance or heeroscedasiciy, denoed by ( h ).Afer consrucing ime series residuals, he 68

3 condiionalvariance can be modeled in a way ha incorporaes he ARCH process of ( ) in he condiional variance wih ( q) ( q) lag of he residuals is as follows: lagged. The general forms of he condiional variance, including 0 p p The above equaion is wha Engle (98) referred o as he linear ARCH ( q) model because of he inclusion of he ( ) lags of he p ( ) in he variance equaion (Brooks, 00)... GARCH Model To avoid he long lag srucure of he ARCH (q) developed by Engle (98),Bollerslev (986), generalized he ARCH model, he so-calledgeneralized Condiional Heeroscedasiciy (GARCH), by including he lagged values of he condiional variance. Thus, GARCH (, ) specifies he pq 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 ( j).then, he pq GARCH (, ) specificaion can be wrien as follows: p q 0 i i b j j i j 3 Where, b 0 i j and ( b ) i j 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 (Brooks, 00).... Modelling Mean Reversion Using GARCH Model Alhough financial markes may experience excessive volailiy from ime o ime, i appears ha volailiy will evenually sele down o a long run level. Given ha, he long run level of variance ε, for a saionary GARCH(,) model is 0 b 4 In his case, he volailiy is always pulled oward his long run level by rewriing he ARMA represenaion in ( b ) u b u

4 As follows ( ) ( b )( ) u b u. o o b b 6 If he above equaion is ieraed k imes, one can show ha ( ) ( b ) ( ) o k o k k b b 7 where is a moving average process. Since b < for a saionary GARCH(, ) model, ( b) k 0 as k. Alhough a ime here may be a large deviaion beween ε and he long run variance, 0 b 8 will approach zero on average as k ges large, i.e., he volailiy mean revers o is long run level o b. In conras, if b > and he GARCH model is non-saionary, he volailiy will evenually explode o infiniy as k. Similar argumens can be easily consruced for a GARCH (p,q) model (Zivo and Wang, 006). According o Banerjee and Sarkar (006), he 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 BSE500 series volailiy on a moving average basis. A covariance saionary ime series { y } has an infinie order moving avarage represenaion of he form i i i0 y, 0, i < i0 9 The plo of he agains i is called he Impulse Response Funcion(IRF).The decay rae of IRF is i someimes repored as a half-life,denoed by L half,which is he lag a which he IRF reaches. 684

5 Half-life of Volailiy Shock for a saionary GARCH(,) process 0 0 The mean revering form of he basic GARCH() model is: ( ) ( )( ) u u where 0 /( ) is he uncondiional long run level of volailiy and u ( ). The mean revering rae 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(zivo and wang,00): Lhalf ln( ) / ln( ) Measures he average ime i akes for o decrease by one half.he closer is o one he longer is he half life of a volailiy shock.if,he GARCH model is nonsaionary and he volailiy will evenually explode o infiniy and he volailiy will evenually explode o infiniy.in oher words,he series follow random walk To es he mean revering properies of BSE500 reurns series and assuming he GARCH(,), we se he following hypohesis: H H Or 0 H H : The BSE500 reurns series is non-saionary : The BSE500 reurns series is mean revering : ( ) 0 : ( ) 685

6 .3.Uni Roo Tes 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 follows: r r r Or i i i p r r r ; ln( ) r R The null and alernaive hypoheses are as follows: H H 0 : he series conains uni roo : he series is saionary k imes as 3 4 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 concluded ha he series do no follow random walk (Goudarzi and Ramanarayanan, 00). 3. LITERATURE REVIEW (Fama 965) reviews he exising lieraure on sock price behavior, examines he disribuion and serial dependence of sock marke reurns, and concludes ha "i seems safe o say ha his paper has presened srong and voluminous evidence in favor of he random walk hypohesis." Kendall and Hill (953)examined UK sock and commodiy price series. He concluded ha "in series of prices which are observed a fairly close inervals he random changes from one erm o he nex are so large as o swamp any sysemaic effec which may be presen. The daa behave almos like wandering series." The near-zero serial correlaion of price changes was an observaion ha appeared inconsisen wih he views of economiss. Neverheless, hese empirical observaions came o be labeled he "random walk model" or even he "random walk heory". (Osborne, 959) analyzed US sock price daa ou of pure academic ineres, presening his resuls o oher physiciss a he US Naval Research Laboraory. Osborne shows ha common sock prices have properies analogous o he movemen of molecules. He applies he mehods of saisical mechanics o he sock marke, wih a deailed analysis of sock price flucuaions from he poin of view of a physicis. (Robers, 959) demonsraed ha a ime series generaed from a sequence of random numbers was indisinguishable from a record of US sock prices - he raw maerial used by marke 686

7 echnicians o predic fuure price levels. "Indeed," he wroe, "he main reason for his paper is o call o he aenion of financial analyss empirical resuls ha seem o have been ignored in he pas, for whaever reason, and o poin ou some mehodological implicaions of hese resuls for he sudy of securiies." (Fama, 970) summarizes he early random walk lieraure, his own conribuions and oher sudies of he informaion conained in he hisorical sequence of prices, and concludes ha "he resuls are srongly in suppor" of he weak form of marke efficiency. He hen reviews a number of semi-srong and srong form ess, highlighing hose ha we cover in he nex wo secions, and concludes ha "in shor, he evidence in suppor of he efficien markes model is exensive, and (somewha uniquely in economics) conradicory evidence is sparse." He concedes, however, ha "much remains o be done", and indeed, Fama (99) subsequenly reurned o he fray wih a reinerpreaion of he efficien markes hypohesis in he ligh of subsequen research. Fama (99) noed in his second review, he es of he EMH involved a join hypohesis - marke efficiency and he underlying equilibrium asse pricing model. He concluded ha Thus, marke efficiency per se is no esable. This did no, however, mean ha marke efficiency was no a useful concep. Almos all areas of empirical economics are subjec o he join hypoheses problem. Sharma and Mahendru (009), invesigae he validiy of he Efficien Marke Hypohesis on he Indian securiies Marke. Alhough, he resuls lead hem ino believing ha he BSE is weak form efficien, ye hey choose o remain cauious in leing our belief ranscend ino a generalizaion. The findings of his sudy indicaed ha he BSE needs o srenghen is regulaory capaciy o boos invesors confidence. This would involve hem being more sringen in enforcing financial regulaions, performing regular marke. Vaidyanahan (994) esed he weak form efficiency of he Indian sock marke using serial correlaion, run es and filer ess. The evidence from all he hree ess suppors he weak form of Efficien Marke Hypohesis. Goudarzi and Ramanaraynan (00; 0), examined he volailiy of he Indian sock markes and is relaed sylized facs using ARCH models. The BSE500 sock index was used o sudy he volailiy in he Indian sock marke over a 0 years period. Several commonly used symmeric and asymmeric volailiy models, ARCH, GARCH,EGARCH,TGARCH,FIGARCH and FIEGARCH 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 was esed using ARCH-LM es and LB saisics. The sudy concluded ha GARCH (, ) model explains volailiy of he Indian sock markes and is sylized facs including volailiy clusering, fa ails, leverage effecs, mean reversion and long memory saisfacorily. 687

8 4. RESULTS The required daa including daily closing observaion for BSE500 price index covering a en years period were obained from he Bangalore Sock Exchange. The BSE500 reurns ( were defined in he logarihm of BSE500 indices (p), ha is, r log( p / p( ) ). r ) a ime The ARCH ype models were esimaed for BSE500 reurns series using he robus mehod of Bollerslev-Wooldridge s quasi-maximum likelihood esimaor (QMLE). The informaion crierion such as AIC, SBIC were used and a se of model diagnosic ess (ARCH-LM es and Q- Saisics) were applied o choose he volailiy models which represen he condiional variance of he BSE500 reurns series appropriaely. To es his hypohesis he ARCH ype models were used. Before esimaing ARCH models for a financial ime series, aking wo seps is necessary. Firs i is necessary o check for uni roos in he residuals and second is o es for ARCH effecs. A formal applicaion of ADF es presened in able, canno rejecs he null hypohesis of uni roo in he closing price series. I means he BSE500 sock index in he level form is nonsaionary. In oher words, i follows random walks and is no mean revering. Bu in he case of log reurns series, formal applicaion of ADF es on log reurn series rejecs he null hypohesis of uni roo in he series.there is rejecion a 0.0 level of significance because absolue values of ADF saisics -9.4 and exceeds McKinnon criical value and respecively. I is evidence of saionary ime series and means ha he log of series does no follow random walk and is mean revering. Second 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. To es he ARCH-effecs, he ARCH-LM es of Engle (98) was used. Under ARCH-LM es he null and alernaive hypohesis for BSE500 sock index are as follows: Or H 0: BSE reurn series is homascedasic H :BSE reurn series is heeroscedasic H : 0 and 0 and 0 and H : 0 and 0 and 0 and If he ARCH effecs exis in he daa hen using ARCH-ype models is appropriae. We es for ARCH effecs in he BSE500 reurns series. The resuls are presened in able. q q 688

9 The resuls confirmed he presence of ARCH effecs in he series for boh periods. Therefore, o es he null hypohesis, he ARCH model of Engle (98) and GARCH model of Bollerslev (986) were used. To selec he appropriae models of each class all possible models were examined and a las afer all pos hoc analysis using SBC informaion crierion he ARCH(4) and GARC(,) were seleced. The resuls are presened in able 3. To es he adequacy of he models, he ARCH-LM es was used o make sure no ARCH effecs lef in he series. The resuls are provided in ables 4 and5. Based on ARCH-LM es resuls presened in he able4 and 5, boh F saisics and LM saisics were insignifican for boh models. Therefore, we concluded ha here are no ARCH effecs lef in he series and ARCH (4) and GARCH (, ) models well represens he condiional heeroscedasiciy in he series. Given he significance of all esimaed coefficiens, he null of no ime varying variance in he daa was rejeced. I means he volailiy of asse reurns in he Indian sock marke is ime varying. This phenomenon is known as he volailiy clusering in he lieraure and is one of he common sylized facs or regulariies of volailiy in he sock markes. I implies a srong auocorrelaion in squared reurns. 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. In his sudy, The mean revering raes implied by our fied model is very close o. The sum of ARCH and GARCH erms presened in able 3 is nearly 0.97 which is close o.i suggesed ha he series does no follow random walk.in oher words,he series is mean revering. According o (Zivo, 009) 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.in our case, i is almos days 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. 5. CONCLUSIONS The informaion abou mean reversion is crucial for invesors, because if sock prices can be characerized as a uni roo process hen i implies ha shocks o prices have a permanen effec, in ha sock prices will aain a new equilibrium and fuure reurns canno be prediced based on hisorical movemens in sock prices. This also opens up he possibiliy ha volailiy in sock markes will increase in he long run wihou bound. On he oher hand, if sock prices are mean revering hen shocks o prices will have a emporary effec, ensuring ha one can forecas fuure movemens in sock prices based on pas behavior and rading sraegies can be developed so as o earn abnormal reurns. This paper considers mean reversion in he Indian sock marke by 689

10 employing ADF es and GARCH model on daily daa over he period 000 o 00. All ess indicae ha reurns series for Indian sock marke are characerized by mean reversion, inconsisen wih he efficien marke hypohesis. This evidence for he mean reversion shows he Indian sock marke o be in formaionally weak-inefficien relaive o he empirical invesigaion of he behavior of he BSE500 ha represen he sock marke benchmark. This inefficiency could be he resul of various facors. Therefore, o improve he efficiency of he marke and secure he flow of informaion o he marke paricipans, he policy makers mus ake his ino accoun o preven any speculaion which may affec he inrinsic value of he share and cause crashes and or crises. REFERNCES Banerjee, A. and S. Sarkar, 006. Modeling daily volailiy of he Indian sock marke using inr-day daa. IIM Calcua,Working paper. Bollerslev, T., 986. Generalized auoregressive condiional heeroskedasiciy. Journal of Economerics, 3(3): Brooks, C., 00. Inroducory economerics for finance. s Edn: Cambridge Universiy Press. Engle, F.R., 98. Auoregressive condiional heeroskedasiciy wih esimaes of he variance of Unied Kingdom inflaion, Economerica, 50(4): Fama, E., 965. The behavior of sock-marke prices. Journal of Business, 38(): Fama, E., 970. Random walks in sock marke prices,[online]. Available from hp:// Fama, E., 99. Efficien capial markes ii. The Journal of Finance, 46(5): Goudarzi, H. and C.S. Ramanarayanan, 00. Modeling and esimaion of volailiy in he Indian sock marke. Journal of Business and Managemen, 5(): Goudarzi, H. and C.S. Ramanaraynan, 0. Modeling asymmeric volailiy in he Indian sock marke. Inernaional Journal of Business and Managemen, 6(3): - 3. Kendall, M.G. and B. Hill, 953. The analysis of economic ime series-par i:prices. Journal of Royal Saisical Sociey. Series A(General), 6(): -34. Mlkiel, B.G., 003. The efficien marke hypohesis and is criics.[online]. Available from hp:// Mobarek, 009. Available from hp://papers.ssrn.com/sol3/papers.cfm?absrac_id= Narayan and Prasad, 007. Available from hp:// Osborne, M.F.M., 959. Brownian moion in he sock marke. Operaions Research, 7(): Pesaran, M.H., 005. Marke efficiency oday,[online]. Available from hp:// 690

11 Robers, H.V., 959. Sock-marke paerns and financial analysis: Mehodological suggesions. The Journal of Finance, 4(): -0. Sharma, G.D. and M. Mahendru, 009. Efficiency hypohesis of he sock markes:a case of Indian securiies [Online]. Available from hp://papers.ssrn.com/sol3/papers.cfm?absrac_id=875. Vaidyanahan, G., k., 994. Efficiency of he Indian capial markes,[online]. Available from hp:// Zivo, E., 009. Pracical issue in he analysis of univariae GARCH models.handbook of financial ime series. Available from hp:// Zivo, E. and Wang, 006. Modeling financial ime series wih s-plus. nd Edn: Springer. variables Table-. Uni Roo Tes for boh Level and Log of BSE500 Index 0 years daily daa Augmened Dickey- Fuller es saisic Tes criical values % 5% 0% PT LOGRT P is closing price of BSE500 sock index LOGRT is log reurns of BSE500 sock index Table-.ARCH-LM Tes Saisics 0 years daily daa F-saisics probabiliy 00 LM- saisics probabiliy 00 Table-3.Esimaed Coefficiens of All ARCH-Type Models ARCH-Type Models Coefficiens Models Coefficien Value Z-Value P-Value AIC BIC Log ARCH(4) GARCH(,) AR INTERCEPT ARCH ARCH ARCH3 ARCH4 AR INTERCEPT ARCH GARCH E E Likelihood

12 Table-4.ARCH-LM Tes for ARCH (4) Model Saisics 0 years daily daa F-saisics Probabiliy LM- saisics Probabiliy Table-5.ARCH-LM Tes for GARCH (, ) Model Saisics 0 years daily daa F-saisics Probabiliy LM- saisics Probabiliy