Modeling Long Memory in The Indian Stock Market using Fractionally Integrated Egarch Model

Transcription

1 Inernaional Journal of Trade, Economics and Finance, Vol., No.3, Ocober, 00 ing Long Memory in The Indian Sock Marke using Fracionally Inegraed Egarch Hojaallah Goudarzi Absrac The weak form of marke efficiency assumes ha predicion of asse reurns based on hisorical informaion s is no possible. Neverheless, a grea number of sudies show ha asse reurns exhibi significan auocorrelaion beween observaions widely separaed in ime. This is one of he sylized facs of financial markes which is known as long memory. The presence of long memory can be defined in erm of persisence of auocorrelaion. This paper sudies he presence of long memory propery in he Indian sock marke. Using daa from BSE500 sock index, his sudy found evidence of long memory propery in he Indian sock marke as seen in developed sock markes and some oher emerging markes. I is found ha he FIE (, d, ) is he bes fi model and i ouperforms oher ARCH-ype models in modelling volailiy in he Indian sock marke. Index Terms Asse Reurns, Volailiy, Fracionally Inegraed E, Long Memory I. INTRODUCTION The saisical analysis of financial ime series provides evidence of various sylized facs, i.e. volailiy clusering, mean reversion, fa ails and leverage effecs, among which volailiy clusering has received considerable aenion. Many models have been added hroughou he years o he Auoregressive Condiional Heeroskedasiciy (ARCH) family, following he seminal paper by [], which capure he shor-run dependency of he condiional variances []. I is generally acceped ha many ime series of pracical ineres exhibi srong dependence, i.e., long memory. For such series, he sample auocorrelaions decay slowly. This necessiaes a class of models for describing such behavior. A popular class of such models is he auoregressive fracionally inegraed moving average (ARFIMA) of [3], [4], and [5] which is a linear process. However, here is also a need for nonlinear long memory models [6]. Among he empirical regulariies which volailiy models ry o capure, one is ha he decay exhibied by esimaed condiional variances seems o be decreasing hyperbolically raher han exponenially. Anoher way of expressing his feaure is ha he process possesses long memory properies in he condiional variance []. To capure long-range dependence in volailiy, [7] and [7] propose fracionally inegraed auoregressive condiional heeroskedasiciy (ARCH) models. In his class of model, fracional inegraion, suggesed by [3] and [5] for processes for he mean are applied o a framework [6]. The fracionally inegraed (FI) model of [7] can be viewed as a fracionally inegraed ARMA (ARFIMA) specificaion for he squared innovaions. Like he fracionally inegraed class of processes I(d), inroduced by [3], [4], and [5] he FI model avoids he sharp disincion beween I(0) and I() processes by allowing d o ake a value beween 0 and. Therefore, he ACF of he volailiy process can possess a rae of decay somewhere beween he exremes of an exponenial rae (I(0)) and infinie persisence (I())[9].Exensions o he FI specificaion include[7] who propose an exponenial version (FIE) while [0] exend he parameerizaion o include he asymmeric power ARCH srucure of []. The FI model has been successfully applied in several areas of empirical finance. Ref.[] invesigae he economic value of FIE forecass of volailiy, while [3] exend he model o a bivariae framework, and [4] and [5] sudy high frequency daa wih he model[9]. The advanage of modelling long memory applied o volailiy processes is ha he forecasing properies of he model so derived beer sui he needs of medium-o-long erm predicion which is crucial in derivaive pricing models. One class of models ha was suggesed in his direcion is he so-called Fracionally Inegraed (FI) process in which he ideas of fracional inegraion suggesed by [3] and [5] for processes for he mean are applied o a framework. Fracional inegraion serves he purpose of exending ARIMA processes o a more general class, ARFIMA, giving a coninuum of possibiliies beween he polar cases of uni roos processes and of inegraed processes of order 0. The order of inegraion in such a case becomes a real parameer d assuming values beween 0 and which can be esimaed in he ime or in he frequency domain []. The purpose of his sudy is o examine he long memory feaure of he volailiy of reurns for he BSE index which is one of he Indian Sock Marke indices using FIE model. The remainder of his paper is organized as follows. Review of he lieraure is presened in secion. We review he volailiy models in secion 3. The nex secion describes he esimaion and esing procedures for long memory, and empirical resuls are discussed in Secion 5. The final secion provides a brief conclusion. PhD Scholar, Universiy of Mysore, Mysore, India( hg50003@yahoo.com). 3 II. LITERATURE REVIEW Ref. [7] proposed Figarch model o capure long-range

2 Inernaional Journal of Trade, Economics and Finance, Vol., No.3, Ocober, 00 dependence in volailiy. Sandard models of he condiional variance implied an exponenial rae of decay for auocorrelaion funcion of squared innovaions. For example, he model possess his feaure and canno capure he degree of persisence in he sample ACF of absolue reurns. Typically auocorrelaions from a model sar oo high and decay much faser han he daa implies. This observaion has been used o conclude ha sandard volailiy models, such as he (p,q) class, are unable o accoun for he longrange dependence found in measures of volailiy. Ref.[] in an aricle eniled ling and pricing long memory in sock marke volailiy proposed a new class of more flexible fracionally inegraed E (FIE) models for characerizing he long-run dependencies in he US sock marke volailiy. They found ha he condiional variance for he S&P500 composie index is bes modeled as a mean revering fracionally inegraed process. Ref.[6] in heir paper eniled Long memory in he Greek sock marke, using he specral regression mehod, found significan evidence of fracional dynamics wih longmemory feaures in he sock reurns series of emerging capial markes, a he Ahens sock exchange in Greece. They found ha price movemen in he GSM appears o be influenced by realizaions from boh he recen pas and he remoe pas. The ou-of-sample long-memory forecasing resuled in significan improvemens in forecasing accuracy (especially over longer horizons). Compared wih RW forecas long-memory forecas also dominaed auoregressive forecass for horizons exceeding 6 monh. This evidence conradiced he maringle model, which saes ha condiioning on horizonal reurns is unpredicable. The pracical usefulness of developing longmemory models for he GSM was herefore esablished. They concluded ha he long-memory evidence obained for he Greek sock marke is in sharp conras o ha obained for major capial markes. This suggesed he possibiliy of differenial long-erm sochasic behavior beween major and emerging capial markes. Their findings suggesed ha long-memory dynamics may prove o be an imporan elemen of hose characerizaions. Ref.[7] in an aricle eniled Long memory in sock reurns invesigaed a se of monhly sock index reurns for long memory and applied a wide range of parameric and semi parameric esimaors in an effor o obain inference ha are robus o he non-normaliy in he reurns daa. The sudy considered markes which differ widely in erms of capializaion, sophisicaion and microsrucure o ensure robusness. The resuls showed ha semi-parameric mehods provide srong evidence of long memory in Souh Korean reurns and some evidence of long-range dependence in he German, Japanese and Taiwanese reurns series. The oher reurns series were consisen wih shor memory. The resuls also suggesed ha long horizon predicabiliy of sock reurns for UK, US, Hong Kong, Singapore and Ausralia is more likely o arise from ime variaion in expeced reurns or speculaive bubbles han long memory. Ref.[8] in an aricle eniled Long memory propery of sock reurns; evidence from India examined he presence 3 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 FI is he bes-fi volailiy model and i ouperforms oher 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. III. LONG MEMORY THORETICAL ISSUES Tradiional saionary ARMA processes have shor memory in he sense ha he auocorrelaion funcion decays exponenially. When he sample auocorrelaion decays very slowly, radiional saionary ARMA processes usually resul in an excessive number of parameers. A saionary process has long memory, or long range dependence, if is auocorrelaion funcion behaves like. α p k ρ ( k) C as k () where C P is a posiive consan, and α is a real number beween 0 and.thus he auocorrelaion funcion of a long memory process decays slowly a a hyperbolic rae. In fac, i decays so slowly ha he auocorrelaions are no summable [9]: k = ρ ( k ) = A. Long Memory s Following [], he process { ε } is said o follow an ARCH model if ε = σ z () where E ( ) 0 z = and Var ( ) z =, and σ is measurable wih respec o he ime - informaion se. Condiional variance of he ARCH(q) model can be wrien as a linear funcion of pas squared values of he process, σ = α + αε +Λ+ αε (3) 0 q q where α 0 > 0 and α,, 0 Λ α q. This model capures he endency for volailiy clusering, ha is, for large (small) price changes o be followed by oher large (small) price changes. [0], exended he ARCH class of models o he generalized ARCH, or which has an auoregressive moving average form for he condiional variance σ. Condiional variance of he (p,q) model is expressed as σ = α + αε +Λ+ αε + βσ +Λ+ β σ 0 q q p p = ω + α ( L) ε + β ( L) σ (4)

3 where 0 Inernaional Journal of Trade, Economics and Finance, Vol., No.3, Ocober, 00 ω > and α, Λ, αq, β, Λ, βq 0. ( L) ( L) α and β are he lag polynomials wih orders of q and p respecively. Tha is, by adding a parameer p o he ARCH model, longer lags can be considered in he model wih low orders. The (p,q) process in Eq.(4.38) can be rewrien as an ARMA process in ε. [ α( L) β( L) ε ω [ β( L)] v = + (5) v ε σ.the { where v } process is inerpreed as he innovaions for he condiional variance. To ensure condiional variance o be nonnegaive, i is assumed ha all he roos of he polynomial[ β ( L)] lie ouside he uni circle. To ake accoun of a uni roo in he auoregressive polynomial [ α( L) β ( L)], [], inroduced he I model. The I(p,q) process is given by φ( L)[ L] ε = ω + [ β( L)] v (6) where φ( L) = [ α( L) β( L)][ L). In he I process, curren informaion remains imporan for he forecas of he condiional variance for all horizons. To capure long-range dependence in volailiy, [7] exended he I mode in he Eq 6 o he FI model. A FI process of order (p,d,q) is defined by φ ε = ω+ β (7) d ( L)( L) [ ( L)] v where he parameer d is allowed o be any real number q beween 0 and and φ( L) = φl Λ φ L, and P β( L) = β L+Λ+ β L. All he roos of φ ( L) and P [ β ( L)] lie ouside he uni circle. In Eq.(7), persisence of shocks o he condiional variance, or he degree of long-erm dependencies is measured by he fracional differencing parameer d. From he fac ha Eq.(7) is idenical o he (p,q) model for d=0 and o he I(p,q) model for d=, we can see ha he FI process includes he and I processes as special cases []. The FI model direcly exends he ARMA represenaion of squared residuals, which resuls from he model, o a fracionally inegraed model. However, o guaranee ha a general FI model is saionary and he condiional variance σ is always posiive, usually complicaed and inracable resricions have o be imposed on he model coefficiens. Noing ha, an E model can be represened as an ARMA process in erms of he logarihm of condiional variance and hus always guaranees ha he condiional variance is posiive. [] proposed he following fracionally inegraed E (FIE) model: ln( σ ) ( )( ) ( ) d ω ψl φl = + + L g( ε) q 33 q d ( L)( L) ln = + ( bj x j + jx j) j= (8) φ σ α γ where φ( L) is defined as earlier for he FI model, γ j 0 allows he exisence of leverage effecs, and x is he sandardized residual: x ε = (9) σ [] showed ha he FIE model is saionary If 0 <d < [9]. IV. MODEL SPECIFICATION AND HYPOTHESIS TESTING To es he long memory properies of he Indian sock markes we se he following hypoheses: H 0 : d=0 or d= H : 0< d < Following [3] we choose he FIE(,d,) model o sudy he long memory propery of volailiy of he Indian sock reurns.also we seleced he AR() process for mean equaion of he series. r = ω + α r + ε g( ε ) = θε + γ ε E ε ω = ω + ln( + δ N ) Parameers of he above model are esimaed by he maximum likelihood esimaion. V. DATA AND EMPIRICAL RESULTS A. Daa The required daa including 08 daily closing observaion for BSE500 price index covering he period 6/7/000 hrough 0/0/009 were obained from he Bangalore Sock Exchange, and were based on daily closing prices. The BSE500 reurns ( r ) a ime are defined in he logarihm of BSE500 indices (p), ha is, r = log( p / p( ) ) B. Empirical Analysis Visual inspecion of he plos of residuals of daily reurns series of BSE500, shown below, proved very useful. I can be seen ha from Figs and 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.

4 Inernaional Journal of Trade, Economics and Finance, Vol., No.3, Ocober, 00 The volailiy was highes in 004 and 008.Thus Figs show 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. Alhough here is lile serial correlaion in he ime series BSE iself, i seems ha boh large changes and small changes are clusered ogeher, which is ypical of many high frequency macroeconomic and financial ime series To confirm his conjecure, we use he ACF o look a he auocorrelaion plos of BSE500 reurns and is squared reurns (Fig 3). Obviously, here is no auocorrelaion in he reurn series iself, while he squared reurns exhibi significan auocorrelaion a leas up o lag 6. Since he squared reurns measure he second order momen of he original ime series, his resul indicaes ha he variance of BSE500 condiional on is pas hisory may change over ime, or equivalenly, he ime series BSE500 may exhibi ime varying condiional heeroskedasiciy or volailiy clusering. Descripive saisics are repored in able. Generally he index has a large difference beween is maximum and minimum reurns..the mean daily reurn is The volailiy (measured as a sandard deviaion) which is indicaing a high level of flucuaions of he BSE500 reurns. There is indicaion of negaive skewness (Skw= ) 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. I means lef ail is paricularly exreme, an indicaion ha he BSE500 has non-symmeric reurns (see Fig 4). The kurosis coefficien is posiive, having high value for he reurn series (Kur = 8.94) which 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-square disribuion wih degree of freedom. The compued value of 750 wih P- value of zero, rejecs he normaliy assumpion due o he high kurosis. Figure,,3 presens he paern of he reurns series of BSE500 index series for he period under review. The index looks like a random walk. The Q-Q plo ha presened in Fig 5 also shows ha he reurn disribuion also exhibis fa ails confirming he resuls in able. TABLE : DESCRIPTIVE STATISTICS OF BSE500 DAILY RETURNS SERIES Mean skewness Max Kurosis 8.94 Figure : The Residuals of BSE500 Reurns Series Figure : Series of BSE500 ACF ACF Min Jarque-Bera Lag Lag Sd.Dev Probabiliy Figure 3 : ACF of BSE500 reurns series for sandard and squared residuals Sample: 6/07/000 Through 0/0/009 34

5 Inernaional Journal of Trade, Economics and Finance, Vol., No.3, Ocober, 00 Before modelling long memory, we make sure of ARCH effec in he underlying series. As able shows ARCH-LM es is saisically significan which indicaes he presence of ARCH effec in he residuals of mean equaion of BSE500.Therefore applying ARCH-ype models are appropriae. TABLE : ARCH-LM TEST ARCH Tes Saisics Value P-Value LM-Tes Figure 4 : Disribuion of BSE500 Daily Reurns Series 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, as shown in able 3 rejecs he null hypohesis of a uni roo in he reurn series. These properies of he BSE500 reurns series are consisen wih oher financial imes series. Figure 5: Q-Q Plo of BSE500 Daily Reurns Series Variables TABLE 3. UNIT ROOT TEST FOR BOTH LEVEL AND LOG OF BSE500 INDEX Period 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 In his case, he P-value is essenially zero, which is smaller han he convenional 5% level, so we rejec he null hypohesis ha here are no ARCH effecs. Long memory Since daily reurns usually have a mean very close o zero, he absolue reurn is someimes used as a measure of volailiy. The ploed sample auocorrelaion funcion of he daily absolue reurns are as Fig 6. ACF Lag Figure 6: Sample Auocorrelaion Funcion of he Daily Absolue Reurns 35

6 Inernaional Journal of Trade, Economics and Finance, Vol., No.3, Ocober, 00 As Fig 6 shows he auocorrelaion of absolue reurns is highly persisen and remains very significan a lag 00. Before esing for long memory, we modelled volailiy using symmeric and asymmeric ARCH-ype models. Using AIC, BIC and log likelihood informaion crierion and afer all pos hoc analysis we choose he (, ), T (, ) and E (, ). Since he resuls of esimaed asymmeric models confirmed he exisence of leverage effecs in he underlying series and FIE model simulaneously can capures long memory and leverage effecs, he sudy applies FIE model. To es he null hypohesis of no long memory we specified FIE (, d, ) model. The esimaed coefficiens of models are repored in he able 4. As i is shown all esimaed coefficiens are significan and we can concluded ha he esimaed models are correcly specified. In he FIE (, d, ) model when fracion erm is 0<d<0.5 he underlying series is saionary and has long memory; and when -/<d<0, he underlying series is saionary bu has shor memory. The summary of he informaion relaed o coefficiens of all models are repored in he Table 4. Inercep- Mean. TABLE 4: COEFFICIENTS OF ESTIMATED MODELS E T (,d,) 5.769e e AR ().33e e Inercep- Variance ARCH Term Term.06e e e e e e Leverage e Fracion FIE Inercep - Mean TABLE 5: ESTIMATED PARAMETERS OF FIE (, D, ) FIE Value Sd. Error T-Value e-003 AR() e-0 Inercep - Variance e+000 ARCH Term e+000 Term e+000 Leverage e+000 Fracion e+00 Adequacy Tes Afer making sure of he accuracy of he esimaed coefficien and selecing an appropriae order for seleced models he nex sep is o check he adequacy of he seleced models. To check he adequacy of he esimaed model we applied o commonly used es i.e. Ljung-Box and ARCH LM ess. 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 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. The resuls for Ljung-Box es are repored in he Table 6. The es is for check he auocorrelaion remain in he series. As resuls show he resuls for all model is insignifican and showing ha here is no auocorrelaion lef in he models. I means he models adequaely capure he auocorrelaion in he underlying series. Therefore according o he resuls, he null hypohesis of no auocorrelaion canno be rejeced. P-Value In our esimaed model, FIE (, d, ), he fracion erm is 0.43 which shows ha he series shows long memory. Therefore, we conclude ha he null hypohesis of no long memory canno be acceped. The esimaed parameers for FIE (, d, ) are repored in he Table 5. As resuls show, all he esimaed coefficiens of he model are significan and we can conclude ha he seleced model is correc. The fracion erm in he esimaed model is 0.43 which indicae ha he BSE500 reurns series show long memory. The sum of ARCH and erm also is less han indicaing ha he series is saionary. 36 E T FIE * Ljung-Box TABLE 6: AUTOCORRELATION TEST LB* P-value LB es P-value Tes for for SSR*** SR**

7 Inernaional Journal of Trade, Economics and Finance, Vol., No.3, Ocober, 00 **Sandardized Residuals *** Squared Sandardized Residuals As anoher es for checking he adequacy of he model we applied ARCH-LM es. The resuls of he es for all models are repored in he Table 7. TABLE 7: ARCH-LM TEST LM P-value F-sa P-value ARCH volailiy models urn ou o be superior, as expeced. The presence of long memory in asse reurns conradics he weak form marke efficiency hypohesis. These resuls have profound implicaions for he capial marke regulaor. Firs, asse prices are no random walk; second, volailiy of asse reurns can be modeled using reurns from he recen as sell as remoe pas and hence derivaive insrumens can now be more efficienly priced; and hird, a sock reurn generally follows a random walk wih long memory in condiional volailiy process [3] E T FIE(,d,) The resuls show ha he esimaed models adequaely capure he ARCH effecs in he underlying series. Boh LM and F-saisics are insignifican for all models indicaing ha here are no ARCH effecs lef in he model and he null hypohesis of no ARCH effecs canno be rejeced. Therefore we concluded ha all model are adequaely capure he ARCH effec in he underlying series. Finally we compared all esimaed models based on AIC, BIC and Log likelihood informaion crierion. As resuls in he Table 8 shows, according o all crierion he FIE (,d,) model seems o provide a slighly beer fi. TABLE 8.COMPARISON OF ESTIMATED MODELS BASED ON INFORMATION CRITERION E T (,D,) FIE AIC BIC LOGLIK VI. CONCLUSION This paper examined he presence of long memory in asse reurns in he Indian sock marke. Using BSE500 index in India, our sudy finds ha here is srong evidence of long memory in is condiional variance. Various saisical ess are conduced o invesigae he long memory propery. I is observed ha he FIE (, d, ) is he bes-fi volailiy model and i ouperforms oher -ype models. 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