THE COMPUTATIONAL OF STOCK MARKET VOLATILITY FROM THE PERSPECTIVE OF HETEROGENEOUS MARKET HYPOTHESIS

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1 Chin Wen CHEONG, PhD Research Cluser of Compuaional Sciences Faculy of Compuing and Informaics Mulimedia Universiy 6300 Cyberjaya Selangor, Malaysia THE COMPUTATIONAL OF STOCK MARKET VOLATILITY FROM THE PERSPECTIVE OF HETEROGENEOUS MARKET HYPOTHESIS Absrac: This sudy invesigaes inerday and inraday ime-varying volailiy modelling and forecasing based on he heerogeneous marke hypohesis. The rading aciviies of heerogeneous marke paricipans can be caegorized ino several ime duraions. These characerisics can be modelled by he auoregressive condiional heeroscedasiciy and heerogeneous auoregressive models using he Sandard and Poor (S&P500) index as he empirical sudy. Besides he common sum-of-square inraday realized volailiy, we also advocae wo power variaion realized volailiies o overcome he possible abrup jumps during he credi crisis wih various frequencies. The empirical forecas evaluaions consisenly show ha he realized volailiy models are ouperformed he inerday daa models for differen frequency daa. These empirical findings have implicaions for financial economerics modelling, porfolio sraegies and risk managemens. Keywords: realized volailiy, fracionally inegraed, heerogeneous auoregressive, marke efficiency. JEL classificaion: C, C5, C58, G4. Inroducion The behaviour of high frequency volailiy (realized volailiy) financial ime series has closely linked o informaionally marke efficiency (Fama, 998) concep. Over decades, researchers and invesors have invesigaed possible new findings o improve he exisence efficien marke hypohesis (EMH) in order o Lo (004,005) proposes a new framework called he Adapive markes hypohesis (AMH) ha reconciles marke efficiency wih behavioural alernaives by applying he principles of evoluion such as compeiion, adapaion and naural selecion o financial ineracions. This hypohesis emphasises on he counerexamples o economics raionaliy such as loss aversion, overconfidence, overreacion, menal accouning and oher

2 Chin Wen Cheong undersand he acual informaion flow underlies financial markes. Heerogeneous marke hypohesis is among he new ideas ha recommended nonhomogeneous marke paricipans. This concep has been inroduced by Muller e.al. (997), Dacorogna e.al. (998) and Peers (994) in he sock and FX markes. Lux and Marchesi (999) relaes his concep using simulaion models ha include paricipans wih differen ineres and sraegies. Under he normaliy assumpion, heir models are able o capure he empirical sylized facs such as heavy-ailed, long-range dependence and scaling law properies. Anoher relaed ineresing approach inroduces by Andersen and Bollerslev (997) is developed heerogeneiy wih a mixure of normal disribuions. By aggregaing disribuions wih differen shape (variance) and locaion (mean) parameers, he model assembles he marke by a differen group of paricipans wih dissimilar ineres and sraegies. This approach is also exhibied heavier ails han a normal disribuion. The heerogeneiy of a paricular financial marke may arise from reacion of marke paricipans o new informaion enering marke. All he marke paricipans may differ from heir endowmens, ineress, risk profiles, degree of informaion, conracual consrains, moivaions of rading and reacions o news. invesmen Shor-erm Medium-erm Long-erm sec min hour day week monh year decade Reacion ime Figure. Heerogeneous paricipans reac over differen ime scales In his sudy, we concenrae on he differen ime scales of marke paricipans in heir invesmen profiles. The ime horizons of invesmens can be characerized by shor-erm, medium-erm and long-erm ranging from seconds o decades. Some lieraures group he invesors in differen invesmens syles (Muller e al., 997;Lynch and Zumbach,003). The shor-erm invesors may refer o marke makers (eg. NASDAQ, KLSE, among ohers, consiss over 500 firms ha quoe boh bid and offer prices for a paricular asse) and inraday speculaors who rade over very shor ime horizons (seconds o hours) in order o gain profis (or minimize losses). Nex group of invesors involve hedge funds and porfolio invesmens in medium ime horizon. The former invesors rade over a few days or based on daily closing prices whereas he laer may ake weeks or monhs o adjus he porfolio according o invesed companies condiions and prices in he benchmark indices. For long-erm invesmens such as cenral banks and pension funds may rade over few years and even decades. Cenral banks ofen refer o long-erm macroeconomics view on FX and money marke raes. Pension funds behavioural biases. However, his concep is sill in he early sage of developmen, herefore, we do no discuss deails o i.

3 The Compuaion of Sock Marke Volailiy from he Perspecive of of Heerogeneous Marke Hypohesis invesors on he oher hand provide a common asse pool o generae sable growh over a very long ime horizon which allows hem o inves in long erm invesmen such as real esae ha normally generae capial gains over ime. shorerm volailiy mediumerm volailiy long-erm volailiy heerogeneo us marke volailiy Figure. Sruure of heerogenous marke The non-homogeneous marke paricipans inerpre same informaion differenly according o heir rading opporuniies. Each ime horizon rading aciviies creaes a unique volailiy under he flucuaing price movemens. Thus, he financial markes which compose by paricipans wih differen reacion imes o news have creaed volailiy cascade ranging from low o high frequencies. The combinaions of hese dissimilar volailiies (due o reacion imes) are believed o produce hyperbolic auocorrelaion decays or long-range dependence propery in financial markes. In shor, he srucure of heerogeneous marke volailiy can be illusraed in Figure. Based on he aforemenioned srucure of heerogeneous marke, high frequency daa have been widely used o measure he marke volailiy. Accurae volailiy esimaions are imporan in assising invesmen porfolio managemen and marke risk managemen. Realized volailiy is one of he famous model-free measures of laen volailiy ha normally canno be observed direcly from financial ime series. The ineres of high frequency volailiy esimaion has seadily increased afer i has been proven (Andersen and Bollerslev, 998; Blair e al.,00) significanly improve he modelling and forecas performance in foreign exchange and sock markes. There are wo major research direcions in realized volailiy lieraure. The firs group of researchers incorporae inraday informaion wih he GARCH model as he condiional variance regressor (Maren & Djik, 007; Taylor & Xu,997) while he oher groups advocae he inraday informaion direcly for economeric modeling and forecasing (Corsi, 009; Engle and Gallo,006). This sudy aims o furher invesigae he clusering volailiy and long memory in realized volailiy. For former sylized fac, Corsi e al. (008) and Cheong e al. (007) have considered he GARCH model o cope ime-dependen condiional heeroskedasiciy in he realized volailiy. The laer long memory behaviour is commonly capured by ARFIMA (Andersen e al. 003; Baillie e al., 996) model. This sylized fac is well explained by he concep underlies heerogeneous marke hypohesis (Dacorogna e al., 00). In his research, we include boh he clusering volailiy and long memory behaviour in realized volailiy model which based on he framework of HAR (Corsi, 009) and ARFIMA (Andersen e al.,

4 Chin Wen Cheong 003). In addiion, we also consider wo realized volailiy esimaors which are immune o abrup jumps namely he realized power variaion and realized bi-power variaion (Barndorff-Nielsen and Shephand, 004) o avoid possible abrup jumps during he subprime morgage crisis in year 008. Boh he esimaors are examined using 5-minue and 5-minue inervals. Alhough he error disurbances are commonly assumed o be normally disribued, our empirical analysis has used a non-gaussian condiional disribuion namely he generalized error disribuion (Nelson, 99) which, o our knowledge, has no ye been used in he lieraure of realized volailiy modelling. Overall, here are realized volailiy models based on hree ypes of inraday daa. The bes in-sample forecass among he models are seleced based on hree informaion crieria. Nex, rolling one-day-ahead ou-of-sample forecass are conduced for he duraion of six monhs rading days in year 009. Three loss funcions are used for forecas evaluaions. For he purpose of comparison, he exponenial GARCH (Nelson, 99) and fracionally inegraed exponenial GARCH (Bollerslev and Mikkelsen, 996) are also considered in forecas evaluaions. The paper is organized as follows. Secion describes he daa source and reurn definiion. Secion 3 presens he ime-varying HAR and ARFIMA model, esimaion, diagnosics and forecas evaluaions. Secion 4 discusses he empirical resuls, and Secion 5 concludes his sudy..0 Daa source This sudy calculaes he daily volailiy from sample variance of inraday reurn using wo differen frequencies, namely 5- and 5-minue inervals o bale microsrucure problem and noisy esimaion issue of realized volailiy. The empirical daa consiss of he S&P500 sock exchange index from January 005 o June 009 (3 rading days wih observaions) wih rading hours from o In general, inraday reurns are calculaed as he difference beween successive close o close log prices and express in percenages as follows: close close r, i 00 ln P, i ln P, i () i =,, M and =,, T. Thus, a full rading day for 5-minue inerval consiss of M = 78 minues wih M equally-spaced subinervals of lengh. For 5-minue inerval, M denoes 6 inraday reurns. The inraday daily reurn wih M= becomes M close close r r, j ln P, M ln P, i 3.0 Mehodology The auoregressive condiional heeroscedasiciy model The auoregressive condiional heeroscedasiciy (ARCH) model is inroduced by Engle (98) in he sudy of Unied-Kingdom inflaion uncerainy. One of he well known exensions of ARCH is advocaed fracionally inegraed differencing parameer (Granger and Joyeux,980) in he ARCH framework or commonly known as he FIGARCH (Baillie, Bollerslev and Mikkelsen,996) model wih he following specificaion ()

5 The Compuaion of Sock Marke Volailiy from he Perspecive of of Heerogeneous Marke Hypohesis () where and are lag polynomials and d i d( d )...( d i! i ( d i d i B ) ( ) L wih i i ). To avoid covariance-nonsaionary and nonnegaiviy issues, Bollerslev and Mikkelsen (996) used logarihmic log nesed wih Nelson (99) specificaion in FIEGARCH model () wih leverage effec if <0. The esimaion resuls of inerday ARCH model are mainly for comparison purposes wih he inraday realized volailiy models. 3. Long memory ime-varying realized volailiy models This sudy considers he heerogeneous auoregressive (HAR) and auoregressive fracionally inegraed moving average (ARFIMA) models in he realized volailiy modelling. For ARFIMA-GARCH ime-dependen heeroskedasiciy model, he specificaion can be wrien as:, (3) where denoes he fracional differencing operaor, and are lag polynomials. In his sudy, he is assumed o be a generalized error disribuion (GED) and RV i represens he ype of logarihmic realized volailiy (Andersen e al., 003). Anoher model which is capable o capure his sylized fac is he heerogeneous auoregressive realized volailiy model (HAR-RV) wih linear cascading of differen ime horizon realized volailiy componens. This model provides a simple auoregressive srucure where he curren volailiy is dependence by previous daily, weekly and monhly realized volailiies. Laer, Cheong e al. (007) and Corsi,e al. (008) exended he HAR wih he inclusion of ime-varying volailiy in realized volailiy. The modified HAR-GARCH(,) model can be wrien as:, (4) Baillie e al. (996) in inflaion.

6 Chin Wen Cheong where follows a condiional densiy wih ime-varying variance. For HAR componens, and. For boh he models, he RV i is eiher sum squared of inraday realized volailiy (Andersen, e al.,999), realized power variaion (Barndorff-Nielsen and Shephand, 004) or realized bipower variaion (Barndorff-Nielsen and Shephand, 004) as follows: (7) (8) (5) where M is he oal inerval wihin a day and 0 < z <. All he realized volailiies are examined using frequency 5-minue and 5-minue inervals. When very high frequency is used, M approaches and converges in probabiliy o he coninuous par of he price process. Under his condiion, boh he volailiy esimaors are immune o abrup jumps. Boh he esimaors are based on absolue reurn where i is more persisence (Ding e al., 993) han oher counerpars such as squared reurn. In pracical applicaions, 5-minue daa are normally used o lessen he impac of marke noise (ABDL,003) whereas 5-minue daa are recommended by ABDL (000) in order o reduce he biasness issue in esimaion. 3.3 Esimaion, diagnosic and model selecion In his sudy, he error are assumed o be followed a generalized error disribuion (Nelson,99) under he maximum likelihood esimaion o capure he heavy-ail propery ha ofen exhibied in financial ime series wih he densiy funcion v exp f ( z ; v) v v λ v z λ v where [ ] is he gamma funcion and, (6) v [ ] [3vv ] 0.5 wih v< for heavier ail compared o normal disribuion v=. The log-likelihood for normal and GED are v T v z LT, GED ln ln ln ln (7) v v For faser and easier compuaion, we use he Marquard (963) mehod where only he ouer producs of he gradien vecors are compued in he numerical analysis esimaions. In he model diagnosic, he Ljung-Box serial correlaion and

7 The Compuaion of Sock Marke Volailiy from he Perspecive of of Heerogeneous Marke Hypohesis Engle ARCH ess are used o examine he sandardized and squared sandardized residuals under he null hypohesis ha he noise erms are serially uncorrelaed or random. Afer ha, he model selecions are based on he Akaike informaion crierion (AIC), Schwarz informaion crierion (SIC) and Hannan-Quinn informaion crierion (HIC) which evaluaed from he adjused (penaly funcion due o addiional number esimaed parameers) average log likelihood funcion (L T ) are seleced for he esimaion evaluaion. The informaion crieria can be expressed as:, (8) where k is he number of esimaed parameers. 3.4 Forecas evaluaions Each volailiy model is esimaed H imes based on fixed inerval of 007 observaions (Jan 005 unil Jan 009). The in-sample esimaion afer he srucural change conains observaions from = o =007. A rolling parameer esimaions is implemened, for example, he firs one-day ahead forecas a =008, is using he esimaion from = o 007 while he esimaion from = o 008 is used o forecas he volailiy a =009. Therefore, H (Feb 009 unil Jun 009) one-day ahead volailiy forecass can be obained by using he rolling esimaions procedures for ˆ, where h=007,, H. Three loss funcions are ( h ), used o evaluae he predicive accuracy: (9) L : RMSE = L : MAE = H H h H H h σ RV σ, h, h σ RV σ Forecas, h Forecas, h ; ; L 3 : TIC = H h H H σ H h RV, h σ RV, h σ H Forecas, h H h σ Forecas, h (0) where he acual and forecas represened heir respecive RV and forecased volailiy respecively. Roo mean square error (RMSE), mean absolue error (MAE), and Theil inequaliy coefficien (TIC) are he common loss funcions in forecasing evaluaions. The hree loss funcions repor he evaluaions direcly based on he deviaions among he forecass and realizaions. TIC on he oher

8 Chin Wen Cheong hand saes ha he value lies beween 0 and wih a perfec fi if he score value is zero. 4.0 Empirical resuls Table. Descripive saisics for naural logarihm non-parameric volailiy esimaor Saisic RV 05 RV 5 RBP 05 RBP 5 RPV 05 RPV 5 Mean Sd. Dev Skewness Kurosis JB es 38.44* * 5.608* 97.3* * * Hurs (0.976) (0.980) 0.708(0.973) (0.978) (0.983) (0.985) Noe: Jacque-Bera es, H 0 : normaliy; Hurs parameer: rescale-range mehod (long range dependence if 0.5<H<.0) wih coefficien of deerminaion in parenhesis. * Significan a 5% level. Table shows he descripive saisics of hree uncondiional realized volailiies wih 5-minue and 5-minue frequencies. All he esimaors are deviaed from normal disribuion according o he Jarque-Bera es. Nex he degree of persisence of he esimaors is measured by he rescaled-range mehod where he ime series is long memory if he Hurs value lies from 0.5 o.0. Overall, he inensiy of persisence are almos he same for all he esimaors (average 0.730), however, he 5-minue esimaors are consisenly greaer han 5- minue. In oher words, he esimaor based on 5-minue inerval consiss of more predicabiliy componen for fuure volailiy. As a conclusion, non-gaussian and long memory behaviours should be aken accoun in model specificaion. Table and Table 3 repors he maximum likelihood esimaions for boh he HAR-GARCH(,) and ARFIMA(,d,0)-GARCH(,) models wih GED disribued error. Firs, he ail parameer, v wih value less han wo convinces he inadequacy of normaliy assumpion for boh models. Second, boh ype of models indicae he ime-dependen heeroskedasiciy volailiy have been eliminaed by he GARCH(,) in realized volailiy. Third, he fracional difference parameer, d for ARFIMA models indicae he presence of long memory volailiy whereas he addiive volailiy cascade of differen ime horizons of HAR are all saisically significan differen from zero. For HAR models, he impac of volailiy componens are he sronges for pas daily volailiy, follows by weekly and monhly volailiy. This is a common fac where he neares hisorical informaion has he highes influence o he recen volailiy movemens. In Table, he in-sample forecas using ime-dependen condiional volailiy model, HAR-GARCH (GED) shows improvemen in goodness of fi, measured by AIC, BIC and HIC crieria over he HAR and HAR-GARCH (Normal). The HAR indicaes significan heeroskedasiciy effec under he Ljung-Box correlaion and ARCH-LM ess. Alhough he condiional heeroskedasic effec can be removed by HAR-GARCH (normal), he HAR-GARCH (GED) seems o provide beer

9 The Compuaion of Sock Marke Volailiy from he Perspecive of of Heerogeneous Marke Hypohesis goodness of fi ess. Similar resuls also have been observed 3 in he ARFIMA- GARCH (GED) esimaions. The in-sample forecas evaluaion can be analyzed in wo ways. Firs, overall he ARFIMA-ype models are slighly ouperformed he HAR-ype models by referring o heir AIC, BIC and HQC crieria. Alhough he HAR model is more preferable (in erms of srucurally inline wih heerogeneous marke hypohesis), he ARFIMA model has he advanage of parsimonious srucure wih less number of parameers o be esimaed (hree agains four). As a comparison, he realized power variaion indicaes he bes fiing resuls for boh he HAR and ARFIMA models. This follows by he sum of squared realized volailiy and finally he realized bipower variaion. Second, overall he 5-minue inerval volailiy esimaors perform beer han 5-minue. These resuls sugges ha he microsrucure noise problem (ABDL,003) is more severe han he biasness issue(abdl,000) in he S&P500 marke. Finally, Table 4 presens he ou-of-sample forecas evaluaions based on RMSE, MAE and TIC. For he purpose of comparison, he forecas evaluaions also conduced using EGARCH and FIEGARCH for logarihmic volailiy. As indicaes in Table 4, boh he HAR and ARFIMA models perform subsanially beer han he daily GARCH models. Comparing he hree evaluaion crieria across 4 models show ha he ARFIMA-ype models perform marginally beer han he HAR-ype models wih he same volailiy proxies (, and ). Among he models, boh he ARFIMA-GARCH (GED) and HAR-GARCH (GED) provides he bes forecass using 5-minues, follows by and lasly. As a summary, he overall ranking based on he forecas evaluaion crieria is presened in Table 4. 3 Due o space scarciy, only he ARFIMA-GARCH (GED) is presened in Table 3.

10 Chin Wen Cheong Table. The maximum likelihood esimaion for Heerogeneous Auoregressive GARCH Esimaion 0 day week monh HAR * ( ) * ( ) * ( ) * ( ) HAR-normal HAR-GARCH * (0.08) 0.40 * ( ) 0.49 * ( ) * ( ) * (0.0087) * (0.0038) * ( ) RV * ( ) * ( ) * ( ) * ( ) * (0.0044) * ( ) * (0.0796) * (0.0879) RBP * (0.0493) * ( ) * ( ) * ( ) * ( ) * ( ) * (0.0590) * ( ) HAR-GARCH GED RPV * ( ) * ( ) * ( ) * (0.0463) * ( ) * ( ) * (0.0469) * ( ) RV * (0.0455) * ( ) * ( ) * ( ) * ( ) * (0.056) * ( ) * ( ) RBP * ( ) * ( ) * ( ) * ( ) * (0.005) * (0.0563) * ( ) * (0.604) RPV * ( ) * ( ) * ( ) * ( ) * ( ) * (0.0539) * ( ) * ( ) v Model selecion L AIC SIC HIC Diagnosic a ~, LB () 5.777(0.0) 6.85(0.57) 6.768(0.59) 7.45(0.35) (0.94) 5.89 (0.6).087 (0.049) (0.5) ~ a, LB () * (0.000) 6.708(0.6) 5.440(0.8) 7.567(0.8) 4.98(0.4) (0.7) (0.67) 8.830(0.77) LM- ARCH().936 * (0.0005).338(0.909).6(0.593) 0.60(0.86).88(0.865) 0.778(0.679) 0.894(0.559) 0.774(0.677) Noes. a ~ represens he sandardized residual. Ljung Box Serial Correlaion Tes (Q-saisics) on a ~ and ~ a : Null hypohesis No serial correlaion; LM ARCH es: Null hypohesis - No ARCH effec. For esimaion, he parenheses values represen sandard error;3. For diagnosic, he parenheses values represen p-value 4. * denoes 5% level of significance.

11 The Compuaion of Sock Marke Volailiy from he Perspecive of of Heerogeneous Marke Hypohesis Esimaion d RV * (0.3667) * ( ) * ( ) * ( ) * (0.0659) * ( ) * ( ) Table 3. The maximum likelihood esimaion for ARFIMA-GARCH RBP * (0.438) * ( ) * ( ) * ( ) * (0.0086) * ( ) * (0.0748) ARFIMA-GARCH (GED) RPV * (0.336) * ( ) * ( ) * ( ) * (0.0669) * (0.0546).5547 * (0.0839) RV * (0.3803) * ( ) * ( ) ( ) * (0.065) * ( ) * (0.0687) RBP * ( ) * ( ) * ( ) (0.09) (0.0756) * ( ) * (0.508) RPV * (0.353) * ( ) * ( ) ( ) * ( ) * ( ).7500 * (0.0774) v Model Selecion L AIC SIC HIC Diagnosic a ~, LB ().7(0.354).3(0.353).70(0.385).76(0.3).30(0.44).87(0.34) ~ a, LB ().457(0.55) (0.78).3(0.77) 9.069(0.55) 9.96 (0.504) (0.537) LM-ARCH().04(0.433) 0.54(0.899) 0.983(0.46) 0.793(0.657) 0.839(0.60) 0.787(0.664) Noes:. a ~ represens he sandardized residual. Ljung Box Serial Correlaion Tes (Q-saisics) on a ~ and ~ a : Null hypohesis No serial correlaion; LM ARCH es: Null hypohesis - No ARCH effec;. For esimaion, he parenheses values represen sandard error;3. For diagnosic, he parenheses values represen p-value 4. * denoes 5% level of significance.

12 Chin Wen Cheong No. Table 4: Forecas evaluaion Model RMSE rank MAE rank TIC rank Overall ranking GARCH-ype model: EGARCH FIEGARCH ARFIMA- GARCH: 3 wih 4 wih 5 wih 6 wih 7 wih RV RV RBP RBP RPV wih RPV HAR-GARCH: 9 wih 0 wih wih wih 3 wih 4 wih RV RV RBP RBP RPV RPV In summary, here are wo facors ha influence he accuracy of forecasing evaluaions, namely he ype of model and also he frequency used in he volailiy esimaion. In his specific sudy, he ARFIMA allowing for ime-varying volailiy of realized volailiy is marginally provides beer forecas accuracy han he HAR-ype models. This may due o is simpliciy (parsimonious principal) as compares o is counerpar. For volailiy esimaor frequency, he 5-minue inerval shows significan improvemen in volailiy poin forecass over 5-minue inerval for S&P500 index. Due o high liquidiy of S&P500 index maure marke, he 5-minue informaion has also been used by oher researchers (Maheu and McCurdy,00; Marens e al.,004). 5.0 Conclusion This sudy has shown ha he ARFIMA and HAR realized volailiy models allowing of ime-dependen heeroskedasiciy are ouperformed in boh he insample and ou-of-sample forecass han he sandard ARFIMA, HAR, EGARCH

13 The Compuaion of Sock Marke Volailiy from he Perspecive of of Heerogeneous Marke Hypohesis and FIEGARCH models. Moreover, he heavy-ailed disribued error erms in he model specificaion has also gained beer fiing evaluaion based on he in-sample esimaion informaion crieria. Besides he model specificaion, he daa frequency for volailiy esimaor is also played an imporan role o ensure superioriy in forecas evaluaion, for his specific sudy, he 5-minue frequency daa. As a conclusion, his sudy is relevance o risk managemen and invesmen porfolio managemen where he marke risk (in erm of value-a-risk) and porfolio hedging for single or muli-asse invesmens can be deermined direcly from he forecas resuls. REFERENCES [] Andersen, T., & Bollerslev, T. (998), Answering he Skepics: Yes, Sandard Volailiy Models Do Provide Accurae Forecass. Inernaional Economic Review, 39, ; [] Andersen, T.G., T. Bollerslev, F.X. Diebold, and P. Labys (999), Undersanding, Opimizing Using and Forecasing Realized Volailiy and Correlaion. Working paper; [3] Andersen, T.G., T. Bollerslev, F.X. Diebold and P. Labys (000), Grea Realizaions. Risk Magazine 3, 05-08; [4] Andersen, T.G., T. Bollerslev, F.X. Diebold and P. Labys (003), Modeling and Forecasing Realized Volailiy. Economerica 7, ; [5] Baillie, R.T., T. Bollerslev and H.-O. Mikkelsen, (996), Fracionally Inegraed Generalized Auoregressive Condiional Heeroskedasiciy. Journal of Economerics,74, 3-30; [6] Baillie, R.T., Chung, C.F., Tieslau, M.A. (996), Analysing Inflaion by he Facionally Inegraed ARFIMA-GARCH Model. Journal of Applied Economerics, 44: ; [7] Barndorff-Nielsen, O. E., & Shephard, N. (004), Power and Bipower Variaion wih Sochasic Volailiy and Jumps. Journal of Financial Economerics,, 48; [8] Blair, J. B., Poon, S. H., & Taylor, S. J. (00), Forecasing S&P00 Volailiy: The Incremenal Informaion Conen of Implied Volailiies and High Frequency Index Reurns. Journal of Economerics, 05, 5 6; [9] Bollerslev, T., H.O. Mikkelsen (996), Modeling and Pricing Long-memory in Sock Marke Volailiy. Journal of Economerics, 73, 5-84; [0] Cheong, C.W., Isa, Z., Abu Hassan S.M.N. (007), Modelling Financial Observable-volailiy using Long Memory Models. Applied Financial Economics Leers, 3: 0-08; [] Corsi, F. (009), A Simple Approximae Long Memory Model of Realized Volailiy. Journal of Financial Economerics, 7, 74 96;

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