Asymmetry and Long Memory in Volatility Modelling*

Transcription

1 Asymmery and Long Memory in Volailiy Modelling* Manabu Asai Faculy of Economics Soka Universiy, Japan Michael McAleer Economeric Insiue Erasmus School of Economics Erasmus Universiy Roerdam and inbergen Insiue he Neherlands and Insiue of Economic Research Kyoo Universiy, Japan Marcelo C. Medeiros Deparmen of Economics Ponifical Caholic Universiy of Rio de Janeiro, Brazil EI Revised: Ocober 00 * he auhors are mos graeful o a Co-Edior, Associae Edior and wo referees for very helpful commens and suggesions, and Marcel Scharh for efficien research assisance. For financial suppor, he firs auhor acknowledges he Japan Minisry of Educaion, Culure, Spors, Science and echnology, Japan Sociey for he Promoion of Science, and Ausralian Academy of Science, he second auhor is mos graeful o he Ausralian Research Council, Naional Science Council, aiwan, and Japan Sociey for he Promoion of Science, and he hird auhor wishes o acknowledge CNPq, Brazil.

2 Absrac A wide variey of condiional and sochasic variance models has been used o esimae laen volailiy (or risk). In his paper, we propose a new long memory asymmeric volailiy model which capures more flexible asymmeric paerns as compared wih exising models. We exend he new specificaion o realized volailiy by aking accoun of measuremen errors, and use he Efficien Imporance Sampling echnique o esimae he model. As an empirical example, we apply he new model o he realized volailiy of Sandard and Poor s 500 Composie Index o show ha he new specificaion of asymmery significanly improves he goodness of fi, and ha he ou-of-sample forecass and Value-a-Risk (VaR) hresholds are saisfacory. Overall, he resuls of he ou-of-sample forecass show he adequacy of he new asymmeric and long memory volailiy model for he period including he global financial crisis. Keywords: Asymmeric volailiy, long memory, realized volailiy, measuremen errors, efficien imporance sampling.

3 Inroducion he accurae specificaion and modelling of risk are inegral o opimal porfolio selecion and risk managemen using high frequency and ulra high frequency daa. In his conex, a wide variey of condiional and sochasic variance models has been used o esimae laen volailiy (or risk) using high frequency daa, while he availabiliy of ick daa has led o alernaive models of realized volailiy o esimae inegraed volailiy in analysing ulra high frequency daa (see McAleer (005) for a comprehensive review of univariae and mulivariae, and symmeric and asymmeric, condiional and sochasic volailiy models, and Asai, McAleer and Yu (006) for a deailed review of alernaive specificaions and esimaion algorihms for mulivariae sochasic volailiy models). In he framework of diffusion processes, he daily variance of sock reurn is expressed as an inegral of he inraday variance, which is called he inegraed variance. If he microsrucure noise is ignored, we may esimae he inegraed variance by he sum of squared reurns of ulra high frequency daa. Such an esimaor is called he realized variance, which corresponds o an esimae of he inegraed variance, namely he rue daily variance. In his paper, we refer o he square roo of he inegraed variance and of he realized variance as he Inegraed Volailiy (IV) and Realized Volailiy (RV), respecively. For a recen exensive review of he RV lieraure, see McAleer and Medeiros (008), and Bandi and Renò (008), odorov (009) and Shephard and Sheppard (00), among ohers, for more recen developmens regarding he modelling and esimaion of sochasic volailiy using high frequency daa. Recen empirical resuls from he RV lieraure show wo ypical feaures in volailiy, namely he asymmeric effec on volailiy caused by previous reurns, and he long-range dependence in volailiy. he former issue has been invesigaed by Bollerslev and Zhou (006), Bollerslev, Liovinova and auchen (006), Bollerslev, Sizova and auchen (00), Chen and Ghysels (008), Marens, van Dijk and de Pooer (009), and Paon and Sheppard (00), among ohers. Wih respec o he laer poin, he auoregressive fracionally inegraed model has been used by Andersen, Bollerslev, Diebold and Labys (00), Koopman, Jungbacker and Hol (005) and Pong, Shackelon, aylor and Xu (004), among ohers, while oher sudies have used he heerogeneous auoregressive model of Corsi (009) o approximae he hyperbolic decay raes associaed wih long memory models. 3

4 he purpose of he paper is o propose a new specificaion of he asymmeric and long memory volailiy model, which allows flexible paerns in order o capure empirical regulariies. Based on he general specificaion, we examine alernaive sochasic volailiy models ha have recenly been developed and esimaed. Some of he corresponding SV models are in Harvey and Shephard (996), Danielsson (994), and Asai and McAleer (005, 00), wih similar aemps having been considered by Bollerslev, Sizova and auchen (00), Marens, van Dijk and de Pooer (009), and Corsi and Renò (00). Bollerslev, Sizova and auchen (00) develop an equilibrium model wih a coninuous ime long memory process, while our paper akes a discree-ime approach. Compared wih Marens, van Dijk and de Pooer (009) and Corsi and Renò (00), our model incorporaes a more general specificaion of he asymmeric effec and exac long memory process. Upon esimaing RV by using ulra high frequency daa, one of he major problems ha has arisen is ha of microsrucure noise. Several auhors have proposed alernaive mehods for removing he microsrucure noise (see, for example, Bandi and Russell (006), Barndorff-Nielsen, Hansen, Lunde and Shephard (008), Zhang, Mykland and Aï-Sahalia (005), and Hansen, Large and Lunde (008)). Some mehods have provided bias-correced and consisen esimaors of he inegraed variance, while oher mehods have no. Recenly, Asai, McAleer and Medeiros (009) have shown ha, even when a bias-correced and consisen esimaor is used, non-negligible measuremen errors remain in esimaing and forecasing IV. Barndorff-Nielsen and Shephard (00) considered he decomposiion of RV as he sum of IV and measuremen error, which hey call he RV error. In oher words, RV is considered o be a proxy for IV. Wih respec o he hird of our aims, we propose a new asymmeric model for RV by exending he general asymmeric volailiy model, wih an addiional erm o capure RV errors. I should be noed ha inroducing a correcion for measuremen error in he RV process renders he rue volailiy process unobservable. In order o esimae he proposed model, we employ he efficien imporance sampling (EIS) ML mehod proposed by Liesenfeld and Richard (003, 006). he EIS evaluaes he log-likelihood funcion of he model, including he laen process, by using simulaions, such as he Mone Carlo Likelihood (MCL) echnique of Durbin and Koopman (997). Compared wih he MCL mehod, he EIS mehod is applicable o various kinds of laen models (see he discussion in Liesenfeld and Richard (003)). 4

5 he remainder of he paper is organized as follows. Secion develops a general long-memory asymmeric volailiy model, and examines five kinds of asymmeric SV models. By using he srucure of asymmeric effecs, Secion 3 proposes a new model for RV based on correcing for RV errors. Secion 4 discusses he EIS-ML mehod, while Secion 5 presens he empirical resuls for he RV model using Sandard and Poor s 500 Composie Index, and evaluaes he new specificaion of asymmery wih respec o goodness of fi, ou-of-sample forecass, and Value-a-Risk (VaR) hresholds. Secion 6 gives some concluding remarks. Srucure of Asymmeric Volailiy Models wih Long Memory In his secion, we propose a new asymmeric volailiy model, and compare i wih sochasic volailiy (SV) models ha have recenly been developed and esimaed. he reurn process is given by r m Vz, z ~ i.i.d. 0,, where m and V are he ime-varying mean and volailiy processes, respecively, and z is he sandardized disurbance. We assume ha he log-volailiy follows an ARFIMA(p,d,q) process, d lnv L L L, () where L is he lag-operaor, L and L are he lag polynomials for he AR and d MA coefficiens, and L is he fracional difference operaor. As suggesed by Nelson (990, 99) for condiional volailiy models, he innovaion erm in he volailiy equaion plays an imporan role in considering asymmery and leverage effecs. We sugges a generalized error, such ha 5

6 N E, ~ 0,, z z z I 0 z I z, 3 3 () where, and 3 are parameers, and I0 z is he indicaor funcion, which akes he value of one if 0 z, and zero oherwise. he firs wo erms in play similar roles as in he EGARCH model. As shown in Harvey and Shephard (996), he negaive sign of he coefficien of z produces he dynamic relaionship beween curren reurn and fuure volailiy, which is called he leverage effec. Generally, a sufficien condiion for univariae SV models o have a leverage effec is ha is negaively correlaed wih z. For our new model, a negaive sign for is expeced. Hence, z conrols he leverage effec in he new model. On he oher hand, z governs he size effec. When 3 0, he erm makes he log-volailiy increase according o he size of he sandardized error. urning o he las wo erms in, hey conribue o capuring asymmeric effecs wih greaer flexibiliy. Figure shows he relaionship beween and z, and implies ha negaive shocks and large posiive shocks increase fuure volailiy via, bu small posiive shocks decrease volailiy. Such a phenomenon has recenly been observed in Chen and Ghysels (008) wih a semi-parameric mehod for realized volailiy. Recenly, Paon and Sheppard (00) also aemp o explain i by considering he sign of jumps on he realized volailiy measure. We consider five special cases as follows: 6

7 Model (i) Equaions (3) and (4), wih resricions 3 0. Model (ii) Equaions (5) and (6), wih resricions 0 and 3 0. Model (iii) Equaions (7) and (8), wih 3 0 and z replaced by r m. Model (iv) Equaions (9) and (0), wih 3 0 and z replaced by r m. Model (v) Equaions () and (), wih 3 0. In order o undersand hese conceps, i is convenien o consider a simple AR() model of log-volailiy. Seing 0 d, L L, and L in (3), we have lnv lnv. (4) aking V as he laen process for he sochasic volailiy, we may find he following correspondence. Model (i) is he basic SV model of aylor (98), which is symmeric as posiive and negaive shocks o reurns have idenical effecs on fuure volailiy. Model (ii) corresponds o he SV model suggesed in Harvey and Shephard (996), and re-examined by Yu (005) (see also Asai and McAleer (009) for a correcion of Yu s (005) news impac funcion). Model (iii) was proposed in Danielsson (994), and was esimaed in Asai and McAleer (005). Model (iv) was suggesed in Asai and McAleer (005) o capure boh leverage and asymmeric effecs. Model (v) adaps he EGARCH model of Nelson (99) o he SV lieraure, and was suggesed and esimaed in Asai and McAleer (00). In conras o Model (iii), Model (v) uses he sandardized reurns in forecasing fuure volailiy, and can capure various ypes of asymmeric and leverage effecs. As compared wih exising models, he new model in (5) and (6) allows log-volailiy o 7

8 follow he ARFIMA process, and incorporaes more flexible asymmeric effecs. 3 Model Specificaion for Realized Volailiy Le ( ) p be he logarihmic price of a given asse a ime 0 on day,,. We assume ha p ( ) follows a coninuous ime diffusion process, dp( ) ( ) d ( ) dw ( ), (7) where is he drif componen, ( ) is he insananeous volailiy (or sandard deviaion), and W ( ) is a sandard Brownian moion. Le r be he daily reurn, defined as p( ) p( ). Condiionally on r ( ), ( ) 0, which is he -algebra (informaion se) generaed by he sample pahs of and ( ) 0, we have ~ ( ), ( ) 0 0. r N d d he erm ( ) d 0 is known as he inegraed variance, which is a measure of he day- ex pos volailiy. he inegraed variance is ypically he objec of ineres as a measure of he rue daily volailiy. Wih respec o he model of he insananeous volailiy, here are several specificaions, which are called coninuous-ime Sochasic Volailiy (SV) models (see Ghysels, Harvey and Renaul (996), for example). Hull and Whie (987) allow he squared 8

9 volailiy o follow a diffusion process: d d db, (8) where B is a second Brownian moion, and and are parameers. Here, we have omied ( ) in order o simplify he noaion. Hull and Whie (987) assume a negaive correlaion beween W and B, hereby incorporaing leverage effecs. he model in (9) is closely relaed o he GARCH diffusion, which is derived as he diffusion limi of a sequence of GARCH(,) models (see Nelson (990)). Wiggins (987) assumes ha he log-volailiy follows a Gaussian Ornsein-Uhlenbeck (OU) process: dlog log d db. (0) In he specificaion, we may inroduce leverage effecs by assuming a negaive correlaion beween W and B. he asymmeric SV model of Harvey and Shephard (996) is considered o be an Euler-Maruyama approximaion of he coninuous-ime model (), wih negaive correlaion. hree major exensions of such diffusion-based SV models incorporae jumps o volailiy process (Eraker, Johannes and Polson (003)), model volailiy as a funcion of a number of facors (Chernov e al. (003)), and allow he log-volailiy o follow a long memory process (Come and Renaul (998)). If he underlying process of he insananeous volailiy is a coninuous-ime SV model, he resuling inegraed variance is sill a sochasic process. A his sage, i may be useful o disinguish he differences and similariies among he condiional variance, sochasic variance, and inegraed variance. As shown in Nelson (990), i is possible o consider he diffusion limis of ypical condiional variance models, such as he GARCH model and he exponenial GARCH model of Nelson (99). Hence, condiional variance models are considered o be approximaions of coninuous-ime SV models. Alernaive approximaions are he (discree-ime) SV models of aylor (98) and Harvey and Shephard (996), which are obained by he Euler-Maruyama discreizaion of he coninuous-ime SV models. Compared wih he class of GARCH models, discree-ime SV models give beer approximaions in he sense ha he laer can be derived sraighforwardly from coninuous-ime SV models. herefore, he condiional and 9

10 (discree-ime) sochasic variance can be considered as approximaions of he inegraed variance obained by coninuous-ime SV models. In he lieraure, here have been numerous exensions of GARCH models, while exensions of SV models are sill being developed. here are many cases where i is no sraighforward o consider a coninuous-ime SV model which corresponds o such an exension. For his reason, in he previous secion we considered asymmeric long-memory models of he inegraed volailiy direcly. Alhough he inegraed variance is unobservable, i is possible o esimae i using high frequency daa. Such esimaes are called Realized Volailiy (RV). Zhang, Mykland and Aï-Sahalia (005) and Barndorff-Nielsen, Hansen, Lunde and Shephard (008) have proposed consisen esimaor of he inegraed variance, under he exisence of microsrucure noise (for exensive reviews of he RV lieraure, see Bandi and Russell (006) and McAleer and Medeiros (008)). As observed in Barndorff-Nielsen and Shephard (00), we can always decompose RV as he sum of IV and a measuremen error, which hey call he RV error. According o heir analysis, even if we have a consisen esimaor of IV, he RV conains a measuremen error, which is no negligible. A his sage we should consider he possible confusion regarding condiional volailiy. he RV is an esimaor of IV, which is he ex-pos daily variance of he price process condiional on he sigma algebra, defined afer equaion (). However, his is quie differen from he condiional volailiy in he ARCH class, as he laer is condiional on he sigma algebra defined by pas observed informaion, such as he reurn series (see he deailed discussion in Andersen, Bollerslev, Diebold and Labys (00) and Andersen, Bollerslev and Diebold (00)). herefore, he condiional volailiy based on he exensions of he ARCH models conains less informaion as compared wih he IV. Now, we specify he new asymmeric model for realized volailiy (RV), noing he correspondence ha ( ), 0 m d 0 and ~ 0, V ( ) d z N. Assume ha he RV is a consisen esimaor of inegraed volailiy (IV). Barndorff-Nielsen and Shephard (00) refer o he measuremen error, defined by he difference beween RV and IV, as he RV error. Barndorff-Nielsen and Shephard (00), Bollerslev and Zhou (00) and Asai, McAleer and Medeiros (009) showed i is useful o employ an ad-hoc approach which accommodaes an error erm wih consan variance. 0

11 Le y be he daily log RV, in which RV is a consisen esimae of IV. he new asymmeric model for RV o be analysed in he paper is given by y V U E U V U ln, 0, u, d lnv L L L N E, ~ 0,, (3) z z z I 0 z I z, 3 3 z r V, where z is he sandardized reurn and follows he sandard normal disribuion. his specificaion enables U o capure he measuremen errors in RV. We will refer o his model as he RV-ARFIMA(p,d,q)-AS,, 3 -noise model. he model allows various ypes of symmeric and/or asymmeric effecs, long-memory propery, and akes accoun of he realized volailiy errors. If he measuremen errors are negleced, we will have a special case wih u 0. I should be noed ha we consider he mean subraced reurn, r, insead of reurn. 4 EIS-ML Esimaion he likelihood funcion for he asymmeric model in equaion (4) includes high-dimensional inegraion, which is difficul o calculae numerically. We employ he Efficien Imporance Sampling (EIS) mehod developed by Liesenfeld and Richard (003, 006) for evaluaing he log-likelihood. he pilo mehod for he EIS is he Acceleraed Gaussian Imporance Sampling (AGIS) approach, as developed in Danielsson and Richard (993). he AGIS approach is designed o esimae dynamic laen variable models, where he laen variable follows a linear Gaussian process. While he AGIS echnique has limied applicabiliy, he EIS is

12 applicable o models wih more flexible classes of disribuions and specificaions for he laen variables. As in he case of AGIS, EIS is a Mone Carlo echnique for he evaluaion of high-dimensional inegrals. he EIS relies on a sequence of simple low-dimensional leas squares regressions o obain a very accurae global approximaion of he inegrand. his approximaion leads o a Mone Carlo sampler, which produces highly accurae Mone Carlo esimaes of he likelihood. 4. Likelihood Evaluaion via EIS Le y be an observable variable and densiy of Y y and H h as, ; h lnv be a laen variable. We denoe he join f Y H, indexed by he unknown parameer vecor. In dynamic laen variable models, he join densiy is ypically formulaed as:, ;,,,,,,,, f Y H f y h Y H g y h Y p h H Y where g denoes he condiional densiy of y given h, y, and p he condiional densiy of h given H, Y. For ease of noaion, i is assumed ha he iniial condiions are known consans, bu EIS can easily accommodae alernaive (sochasic) assumpions. I should be noed ha we excluded he densiy of reurn series. his approach is no efficien, bu he loss in efficiency is minor, by consrucion. he likelihood funcion is given by he -dimensional inegral: ;, ; L Y f Y H dh, and a naural MC esimae of L Y is given by ; N ˆ i L; Y g y h, Y, N i, (5)

13 where i i h denoes a rajecory drawn from he sequence of densiies. Each h is drawn from he condiional densiy p h H, Y,. i In order o undersand he EIS, we firs noe ha EIS searches for a sequence of samplers ha explois he sample informaion on h conveyed by y. Le mh H, x denoe a sequence of auxiliary samplers, indexed by he auxiliary parameers Xn x. Regardless of he values of he auxiliary parameers, he likelihood funcion, L Y rewrien as ;, is f y,,, h Y H L; Y mh H, xdh mh H, x, and he corresponding imporance sampling MC esimae of he likelihood is given by L ; Y, X i i N f y, h xy, H x, N, (6) i i i mh x H x, x where samplers, m. i h x denoes a rajecory drawn from he sequence of auxiliary imporance he EIS chooses a sequence of m densiies by selecing values of he auxiliary parameers, X, which provide a good mach beween he produc in he numeraor and ha in he denominaor in equaion (6) o minimize he MC sampling variance of L ; Y, X order o implemen EIS, i requires consrucing a posiive funcional approximaion, kh; x, for he densiy f y, h Y, H,. In, wih he requiremen ha i be 3

14 analyically inegrable wih respec o role of a densiy kernel for mh, H x h. In Bayesian erminology, ;, which is hen given by k H x plays a m h H, x k H; x, (7) H, x H, ; x k H x dh. hen, he EIS requires solving a back-recursive where sequence of low-dimensional leas squares problems of he form: N i i i xˆ arg min ln f y, h Y, H, H, x x i c ln k H ; x, i (8) for :, wih H x,. As in equaion (5), i h denoes a rajecory drawn from he p densiies, and he c are unknown consans o be esimaed joinly wih x. If he densiy kernel ; k H x is chosen wihin he exponenial family of disribuions, he EIS leas squares problems become linear in x under he canonical represenaion of exponenial kernels. he EIS esimae of he likelihood funcion for a given value of is obained by subsiuing xˆ for x in equaion (6). In order o obain maximally efficien imporance samplers, a small number of ieraions of he EIS algorihm is required, where he naural samplers p are replaced by he previous sage imporance samplers. For such ieraions o converge o fixed values of he auxiliary parameers, x ˆ, which are expeced o produce opimal imporance samplers, i is necessary o apply he echnique of Common Random Numbers (CRNs). 4. Implemenaion Issues 4

15 As we consider he nonlinear ARFIMA(p,d,q) process, i is no sraighforward o incorporae i in he likelihood funcion. Hence, we sugges using an AR(J) approximaion of he AR represenaion of he ARFIMA par, which is similar o he MA(J) approximaion of he FIEGARCH model by Bollerslev and Mikkelsen (996), in he sense ha he coefficien of he J-h lagged erm is almos zero and is negligible for large J, such as J = 000. Based on he above runcaion, we have he disribuions of y and h. he RV-ARFIMA(p,d,q)-AS,, 3 -noise model in equaion (9) assumes ha RVs, y, given he laen log-volailiy, h, follow he normal disribuion: g y h y h, exp. Condiional on H, r, he log-volailiy, h lnv, follows he normal disribuion: p h H, r, exp h, l where 0 for l ihi E for,, J i J ihi E for J,, i and 5

16 C for J, J J J, J, J J, J C for,, J for J,, h where h h h h re r e re I re I re, is he variance of deermined by uncondiional covariance marix of h,, hj, and C j is he,, 3,,, J,,,, J. Noe ha i is assumed ha p h follows he normal disribuion wih mean zero and he uncondiional variance of h. Regarding he iniial disribuions for,, J, we used he decomposiion: J,, J,,,,, p h h p h p h h h r r. which produces he combinaion of he condiional and uncondiional mean and variance given above. We chose m as he parameric exension of he naural samplers, p. Hence, he parameerizaion for k is given by ;,,,, k H x r p h H r h x, where he auxiliary funcion h, x is iself a Gaussian densiy kernel. Under his parameerizaion, he naural sampler, p, cancels ou in he leas squares problem in equaion (8), o he effec ha ln h, x ln H, x, r serves o approximae ln g y h, Y, r,. In paricular, he appropriae auxiliary funcion for he asymmeric model is given by ln h, x exp x h x h, wih x x, x kernels of he imporance samplers have he form, and he densiy 6

17 l kh ; x, r exp x h x h. Accordingly, he condiional mean and variance of h on m are given by l m, m, x, m,, x (30) respecively. Inegraing ;, k H x r wih respec o h, and omiing irrelevan muliplicaive facors, leads o he following expression for he inegraing consan: H l m,, x, r exp m,. (3) Based on hese funcional forms, he compuaion of an EIS esimae of he likelihood for he asymmeric model requires he following seps: Sep (0): Use he naural samplers, p, o draw N rajecories of he laen variable, i h. Sep (): : : Use hese random draws o solve he back-recursive sequence of leas squares problems, as defined in equaion (8). he sep leas squares problem is characerized by he following linear auxiliary regression: i y ln, ˆ h h x consan x h x h u, i: N, i i i 7

18 where i u denoes he regression error erm. he iniial condiion for he inegraing consan (in equaion (3)) is given by h x r,,. Sep ( + ): he EIS samplers, m h H, xˆ, which are characerized by he condiional mean and variance given in equaion (30), are used o draw N rajecories i h aˆ calculaed according o equaion (6)., from which he EIS esimae of he likelihood is We se N 50, as Liesenfeld and Richard (003) repored ha 50 is sufficien for univariae and nonlinear laen variable models, such as SV. Afer 7-0 ieraions, L ; Y, X, R converged for each. he nex secion gives he EIS-ML esimaes for he asymmeric model of RV. For he case of neglecing measuremen errors (ha is, u 0 ), h is observable, so i is possible o perform maximum likelihood esimaion wihou simulaions. By comparing he log-likelihood wih he EIS log-likelihood above, we have he convenional likelihood raio es saisics, which follows he 0. u disribuion under he null hypohesis ha 4.3 Mone Carlo Experimens In his subsecion we presen he resuls of a Mone Carlo sudy o invesigae he small sample performance of he esimaion procedure presened in subsecion 4.. We generae R simulaed ime series for RV-ARFIMA(,d,0)-AS,, 3 -noise model in equaion (3) and for some given rue parameer vecor. Subsequenly, we rea as unknown and esimae i for each series using he EIS maximum likelihood mehod described in subsecions 4. and 4.. We compue he sample mean, sandard deviaion and roo mean squared error (RMSE) and compare i wih he rue parameer value. 8

19 he rue parameer values for generaing Mone Carlo samples are given in he firs column of able, which is obained by our empirical analysis in Secion 5. he resuls given in able are for he ypical sample size = 500 wih he number of ieraions se o R = 300. able shows ha he mos of he values of he sandard deviaion are close o hose of he RMSE, indicaing ha he bias in finie samples is negligible. 5 Empirical Resuls 5. Daa and Preliminary Resuls he empirical analysis focuses on he RV of Sandard and Poor s 500 Composie Index. In order o esimae he daily realized volailiy, we use he wo ime scales esimaor (SE) of Zhang, Mykland and Aï-Sahalia (005) wih five-minue grids, which is a consisen esimaor of he daily realized volailiy. he sample period is Jan/3/996 o March/9/007, giving = 796 observaions of RV. As a preliminary analysis, we consider he new Fracional Inegraed EGARCH- models given in Secion 3 as r z, z ~ S, d L L L ln, E, (33) z z z I 0 z I z, 3 3 where S denoes he sandardized disribuion, wih degrees of freedom given by v. Noe ha his model implicily specifies ha 0, so ha is deermined by he pas informaion. We denoe his as he FIEGARCH(p,d,q)--AS,, 3 model and, for he case d=0, as he EGARCH(p,q)--AS,, 3 model. 9

20 We esimaed wo kinds of models, namely EGARCH(,)--AS,, 3 and FIEGARCH(,d,)--AS,,0. able shows he ML esimaes of hese models, wih iniial values of 000. For he former model, all he esimaed parameers, excep for and 3, are significan a he five percen level. he esimae of is close o 0.99, showing high persisence in volailiy. he esimae of is negaive, while ha of is posiive. he esimae of is 0.08, indicaing ha he esimae of is close o 3. he resuls are ypical for he EGARCH- specificaion. For he long memory model, all he esimaed parameers, excep for and, are significan. his specificaion shows he lack of imporance of asymmeric effecs and heavy-ailed condiional disribuions. he AIC and BIC favour he FIEGARCH(,d,)--AS,,0 model. Similar resuls are also found in he lieraure wih he FIEGARCH- specificaion. 5. Esimaes for RV Models In he following, we will show ha he empirical resuls for RV models are subsanially differen from hose associaed wih EGARCH models. I should be noed ha i is inadequae o compare he log-likelihood of EGARCH models wih ha of RV models as he former is based on r while he laer is based on he RV, y. Furhermore, he fa ails of he condiional disribuion of r are irrelevan for he esimaion of he RV model. able 3 shows he EIS-ML resuls of he RV-AR()-AS,, 3 -noise model. Regarding asymmery, we consider four specificaions, namely AS0,0,0, AS,0,0, AS,,0, and AS,,. All he esimaed parameers are significan a he 5% 3 level. As he AS,, 3 model has he smalles AIC and BIC, we repor he empirical resuls only for his specificaion. he esimae of u is close o 0.4, showing ha he RV errors are no negligible. he esimae of is 0.986, while ha of is 0., which are ypical of SV models. he 0

21 esimae of is negaive, while ha of is posiive. Unlike he esimaes of he EGARCH model, he esimae of 3 is negaive and significan. Figure gives he news impac from z o lnv, showing ha negaive shocks and large posiive shocks increase fuure volailiy, bu small posiive shocks decrease volailiy. able 4 presens he EIS-ML resuls for he RV-ARFIMA(,d,0)-AS,, 3 -noise model. As before, we consider four kinds of asymmeric effecs. he AIC and BIC seleced he AS,, 3 model, so we will concenrae he empirical analysis on his model. All he esimaed parameers are significan a he five percen level. he esimae of u is close o 0.4, indicaing ha he RV errors are no negligible. he esimae of d is 0.47, showing ha he log-volailiy has long memory and is a saionary process. he esimae of is posiive and close o 0.4, which is agains he ypical value of -0. in he RV lieraure. he difference can be explained by he exisence of RV errors, U y lnv. As shown in he Mone Carlo experimens of Asai, McAleer and Medeiros (009), even minor RV errors can cause bias in he esimaes if he RV error is negleced in esimaion. he signs of, and 3 are he same as in he case of able 3. Figure 3 shows he news impac from z o lnv. From ables and 3, we find ha he RV-ARFIMA(,d,0)-AS,, 3 -noise model has he smalles AIC, while BIC chooses he RV-AR()-AS,, 3 -noise model. hese ables indicae ha having he addiional erm, 3, significanly improves he goodness of fi of he model. 5.3 Forecasing Analysis Regarding he RV-ARFIMA(,d,0)-AS,, 3 -noise model, we examine he performance of he ou-of-sample forecass using he following four approaches: (i) es for equal forecas accuracy; (ii) es model specificaion; (iii) es he forecass of he VaR hresholds; (iv) model selecion. he benchmark model is he Leverage Heerogeneous Auoregressive (LHAR) model, suggesed in Corsi and Renò (00). he LHAR model is based on he Heerogeneous Auoregressive (HAR) model of Corsi (009), which

22 approximaes a long memory process, wih an exension regarding he leverage effec. Hence, he LHAR model accommodaes boh long range dependence and he leverage effec. he LHAR model is given by r I r r I r y y y y r I r error, 0 0 where y h denoes he h-horizon normalized realized volailiy, defined by r h and y h y y y h h is defined by he same manner. 0, I r is he indicaor funcion which ake one if r is negaive, and zero oherwise. A similar model is suggesed by Marens, van Dijk and de Pooer (009). Noe ha i is possible o include he posiive par of heerogeneous reurns, bu hey are usually insignifican. Fixing he sample size a,500, we re-esimaed he model and compued one-sep-ahead forecass of log-volailiy for he las 50 days. Firs, we repor he resul for he Harvey, Leybourne, and Newbold (997) modificaion of he Diebold and Mariano (995) es of equal predicive accuracy. he new asymmeric and long-memory volailiy model is compared agains he LHAR model. he es saisic follows he sandard normal disribuion asympoically under he null hypohesis of equal accuracy. able 5 shows he es resuls, indicaing he difference beween he wo forecass. Second, we es he model specificaion, based on he Mincer-Zarnowiz regression, namely x abxˆ e,,,,50 where x can be he observed RV or log-rv on day, and xˆ is he one-sep-ahead

23 forecas of x on day. If he model is correcly specified, hen a 0 and b. able 6 show he esimaes of he coefficiens and he heeroskedasiciy-consisen F es saisics for he join null hypohesis, regarding he LHAR model and he RV-ARFIMA(,d,0)-AS,, 3 -noise model, respecively. Wih respec o he LHAR model, he F ess in boh cases rejeced he null hypohesis ha he model is correcly specified. However, for he new asymmeric and long-memory model, he F es did no rejec he null hypohesis. As he new model is based on log-rv, he esimaes for log-rv are very close o he values expeced under he null hypohesis. hird, we calculaed he VaR hresholds, accommodaing he filered hisorical simulaion (FHS) approach, which is an effecive mehod for predicing VaR hresholds (see Kueser e al. (006) for some recen sudies regarding he FHS approach). In shor, he FHS approach esimaes he empirical disribuion of he sandardized reurns, hen obains he 00p perceniles o compue he 00p percen VaR hresholds. In our analysis, each ime we esimaed he model wih,500 observaions, we compued he 00p perceniles of he empirical disribuion based on he las 500 observaions, discarding he firs,000 observaions. Combined wih he one-day-ahead forecass of log-volailiy, we compued he 00p percen VaR hresholds. In order o assess he esimaed VaR hresholds, he uncondiional coverage and independence ess developed by Chrisoffersen (998) are widely used. A drawback of he Chrisoffersen (998) es for independence is ha i ess agains a paricular alernaive of a firs-order dependence. he duraion-based approach in Chrisoffersen and Pelleier (004) allows for esing agains more general forms of dependence bu sill requires a specific alernaive. Recenly, Candelon e al. (00) have developed a more robus procedure which does no need a specific disribuional assumpion for he duraions under he alernaive. Consider he hi sequence of VaR violaions, which akes a value of one if he loss is greaer han he VaR hreshold, and akes he value zero if he VaR is no violaed. If we could predic he VaR violaions, hen ha informaion may help o consruc a beer model. Hence, he hi sequence of violaions should be unpredicable, and should follow an independen Bernoulli disribuion wih parameer p, indicaing ha he duraion of he hi sequence should follow a geomeric disribuion he GMM duraion-based es developed by Candelon e al. (00) works wih he J-saisic based on he momens defined by he orhonormal polynomials associaed wih he geomeric disribuion. he condiional coverage es and independence es based on q 3

24 orhnormal polynomials have asympoic q and disribuions under heir q respecive null disribuions. he uncondiional coverage es is given as a special case of he condiional coverage es wih q =. able 6 shows he percenage of VaR violaions and es resuls for he LHAR model and new asymmeric and long-memory volailiy model, respecively. For boh he LHAR model and RV-ARFIMA(,d,0)-AS,, 3 -noise model, he ess did no rejec he null hypohesis for he 5% and % VaR hresholds, indicaing ha he esimaed VaR hresholds are saisfacory. We also conduced he uncondiional coverage and independence ess developed in Chrisoffersen (998), and he resuls are unchanged. Finally, we selec he forecass using he following MZ equaion: x abxˆ b xˆ e,,,,50 AS LHAR i where xˆ i AS, LHAR is he one-sep-ahead forecas of x on day, based on he RV-ARFIMA(,d,0)-AS,, 3 -noise model (AS) and he LHAR model. We selec he forecass by he convenional es. As before, we consider wo dependen variables, namely volailiy and log-volailiy. able 8 gives he resuls. In boh cases, he ˆ LHAR coefficiens of x are insignifican, indicaing ha he daa prefer he forecass of he RV-ARFIMA(,d,0)-AS,, 3 -noise model. Overall, he resuls of he ou-of-sample forecass favour our new asymmeric and long memory volailiy model. 5.4 Global Financial Crisis In addiion o he previous analysis, we examine he adequacy of he new RV-ARFIMA(,d,0)-AS,, 3 -noise model for he period including he global financial crisis, saring from he bankrupcy of Lehman Brohers, ha is, Sep/5/008. 4

25 For he analysis, we chose IBM as he individual sock for he period Jan/03/000 o April/7/009, giving = 334 observaions for RV. We obained one-sep-ahead forecass as before for he las 50 observaions corresponding o he period saring from he bankrupcy of Lehman Brohers. We use he LHAR model as a benchmark. able 9 gives he esimaes for he MZ equaions. he new model does no rejec he null hypohesis, a 0 and b, showing he adequacy of he new model, while he LHAR model does rejec he null hypohesis. he resuls wih wo forecass show he significance of he forecas of he new model and insignificance for he LHAR model. We also conduced he HLN ess for volailiy and log-volailiy, producing values of he es saisics of 4.97 and 9.4, respecively. he resuls indicae he superioriy of he RV-ARFIMA(,d,0)-AS,, 3 -noise model, rejecing equal predicive accuracy. For he forecasing period, he sock price of IBM is so volaile ha here is no day in which a negaive reurn exceeds he boundary of -.98 imes RV. Hence, we canno conduc ess of he VaR hresholds. hus, we repor ha he number of violaions for he % hreshold is zero for he new model, while i is 4 imes (0.07%) for he LHAR model. Overall, he resuls of he ou-of-sample forecass show he adequacy of he new asymmeric and long memory volailiy model for he period including he global financial crisis. 6 Concluding Remarks We proposed a new asymmeric and long-memory volailiy model. Regarding he leverage effec, he new model sensiively capures he effecs of boh large and small, and posiive and negaive, shocks. Based on he new specificaion, his paper examined alernaive univariae volailiy models ha have recenly been developed and esimaed. We exended he specificaion of asymmeric and long memory volailiy in order o model RV by aking accoun of he RV errors. his is a general model which includes no only various kinds of asymmeric effecs, bu also shor and long memory specificaions. We applied he EIS-ML mehod o esimae he model of RV, and repored he resuls for a Mone Carlo experimen. he empirical resuls for he RV of Sandard and Poor s 500 Composie Index showed he exisence of RV errors. he esimaes of he shor and long memory models suppored he 5

26 new specificaion of asymmeric effec, which saisfies he following hree condiions: (i) negaive shocks o reurns increase fuure volailiy; (ii) large posiive shocks o reurns increase fuure volailiy, bu a negaive shock has a larger effec on volailiy han does a posiive shock of equal magniude; and (iii) small posiive shocks o reurns decrease fuure volailiy. Overall, he new specificaion of asymmery significanly improved he goodness of fi, and he ou-of-sample forecass and VaR hresholds were saisfacory. 6

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32 able : Mone Carlo Resuls for EIS-ML Esimaor for RV-AR()-AS,, 3 -noise Model Parameers rue Mean Sandard deviaion RMSE d (0.0366) [0.039] (0.0655) [0.0660] (0.056) [0.065] (0.009) [0.0067] (0.056) [0.093] (0.055) [0.0686] (0.054) [0.0754] (0.6) [0.33] u (0.0089) [0.0090] able : ML Esimaes of he New EGARCH Class Parameers New EGARCH- FIEGARCH(,d,0)- d (0.056) (0.0038) (0.0450) (0.4560) 0.30 (0.0777) (0.038) 0.05 (0.044) (0.004).4407 (0.0776) (0.0967) (0.4408) (0.09) (0.040) Log-Like AIC BIC Noe: Sandard errors are in parenheses. he firs,000 observaions are used for he iniial values for he FIEGARCH- model. 3

33 able 3: EIS Esimaes of RV-AR()-AS,, 3 -noise Parameers AS 3 0,0,0 AS (0.005) (0.009) (0.09),0,0 AS,,0 AS,, (0.0040) 0. (0.0075) (0.0788) (0.0046) (0.0044) 0.0 (0.0074) (0.37) (0.0043) (0.0074) (0.0044) 0.03 (0.0070).46 (0.57) (0.006) (0.0079) (0.047) (0.0605) u (0.0073) (0.0067) (0.0067) (0.0067) Log-Like AIC BIC Noe: Sandard errors are in parenheses. 33

34 able 4: EIS Esimaes of RV-ARFIMA(,d,0)-AS,, 3 -noise Parameers AS d 3 0,0,0 AS (0.0039) 0.36 (0.0603) 0.46 (0.05) (0.394),0,0 AS,,0 AS,, ( ) (0.0438) (0.047) 0.00 (0.006) (0.006) (0.0090) (0.0538) 0.85 (0.057) (0.007) (0.0063) 0.06 (0.0077) (0.0076) (0.09) (0.0080) (0.000) (0.0075) 0.05 (0.0076) (0.087) (0.096) u (0.008) (0.008) (0.0085) (0.0067) Log-Like AIC BIC Noe: Sandard errors are in parenheses. 34

35 able 5: HLN ess for Equal Forecas Accuracy HLN es Sa. P-value Volailiy Log-Volailiy Noe: HLN is he es for equal forecas accuracy of Harvey, Leybourne, and Newbold (997), where he new asymmeric volailiy model is compared wih LHAR. he es saisic follows he sandard normal disribuion asympoically under he null hypohesis of equal accuracy. able 6: ess for Model Specificaion by MZ Equaion x abxˆ e Model LHAR RV-ARFIMA(,d,0)-AS,, -noise 3 Dependen variable Consan Forecas F es Volailiy Log-Volailiy Volailiy Log-Volailiy (0.049) (0.0740) (0.38) (0.305) (0.0798) (0.0753) (0.408) (0.09) [0.0078] [0.0000] [0.56] [0.8698] Noe: Heeroskedasiciy-consisen sandard errors are in parenheses, and p-values are in brackes. F es denoes he value of he heeroskedasiciy-robus F es for he null hypohesis H : a0, b. 0 35

36 able 7: Backesing VaR hresholds Model LHAR RV-ARFIMA(,d,0)-AS,, -noise 3 VaR % Violaion 5% % UC ID CC [0.860] [0.9630] [0.9880] [0.433] [0.878] [0.948] % Violaion UC ID CC [0.8669] [0.8793] [0.9456] [0.383] [0.8408] [0.9] Noe: % Violaion is he percenage of days when reurns are less han he VaR hreshold. UC, IND CC are he GMM duraion-base ess for uncondiional coverage, independence and condiional coverage, developed by Candelon e al. (00). he number of orhonormal polynomials is se o 5. P-values are in brackes. able 8: Model Selecion by MZ Equaion x abxˆ b xˆ e,,,,50 AS LHAR Dependen Variable Cons xˆ AS xˆ LHAR Volailiy Log-Volailiy (0.83) (0.346).334 (0.830) (0.93) -0.3 (0.896) 0.09 (0.6) Noe: Heeroskedasiciy-consisen sandard errors are in parenheses. 36

37 able 9: Model Selecion by he MZ equaion for IBM daa x abxˆ b xˆ e,,,,50 AS LHAR Dependen Variable Cons xˆ AS xˆ LHAR Volailiy Volailiy Volailiy Log-Volailiy Log-Volailiy Log-Volailiy.367 (.3667) (.7943) (.650) (0.049).69 (0.080) (0.050).64 (0.85) (0.7) (0.0603) (0.078) (0.7576).3978 (0.764) (0.0903) (0.0870) Noe: Heeroskedasiciy-consisen sandard errors are in parenheses. 37

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