Alternative Asymmetric Stochastic Volatility Models*

Transcription

1 Alernaive Asymmeric Sochasic Volailiy Models* Manabu Asai Faculy of Economics Soka Universiy, Japan Michael McAleer Economeric Insiue Erasmus School of Economics Erasmus Universiy Roerdam and Tinbergen Insiue The Neherlands and Insiue of Economic Research Kyoo Universiy Japan EI Updaed: November 00 * The auhors wish o acknowledge he insighful commens and suggesions of he Edior and a referee, and helpful discussions wih Felix Chan, Neil Shephard and Jun Yu, and seminar paricipans a Fondazione Eni Enrico Maei - Milan, Naional Universiy of Singapore, Universiy of Auckland, Universiy of Melbourne, Universiy of Milan-Bicocca, Universiy of Venice Ca Foscari, Ene Einaudi - Rome, and Universiy Pompeu Fabra. An earlier version of he paper was presened a he Symposium on Economeric Forecasing and High-Frequency Daa Analysis in Singapore, May 004. The firs auhor appreciaes he financial suppor of he Japan Sociey for he Promoion of Science, and he Ausralian Academy of Science. The second auhor is graeful for he financial suppor of he Ausralian Research Council, Naional Science Council, Taiwan, and he Japan Sociey for he Promoion of Science.

2 Absrac The sochasic volailiy model usually incorporaes asymmeric effecs by inroducing he negaive correlaion beween he innovaions in reurns and volailiy. In his paper, we propose a new asymmeric sochasic volailiy model, based on he leverage and size effecs. The model is a generalizaion of he exponenial GARCH (EGARCH) model of Nelson (99). We consider caegories for asymmeric effecs, which describes he difference among he asymmeric effec of he EGARCH model, he hreshold effecs indicaor funcion of Glosen, Jagannahan and Runkle (99), and he negaive correlaion beween he innovaions in reurns and volailiy. The new model is esimaed by he efficien imporance sampling mehod of Liesenfeld and Richard (003), and he finie sample properies of he esimaor are invesigaed using numerical simulaions. Four financial ime series are used o esimae he alernaive asymmeric SV models, wih empirical asymmeric effecs found o be saisically significan in each case. The empirical resuls for S&P 500 and Yen/USD reurns indicae ha he leverage and size effecs are significan, supporing he general model. For TOPIX and USD/AUD reurns, he size effec is insignifican, favoring he negaive correlaion beween he innovaions in reurns and volailiy. We also consider sandardized disribuion for capuring he ail behavior. The resuls for Yen/USD reurns show ha he model is correcly specified, while he resuls for hree oher daa ses sugges here is scope for improvemen. Key words: Sochasic volailiy, asymmeric effecs, leverage, hreshold, indicaor funcion, imporance sampling, numerical simulaions.

3 Inroducion I has long been recognized ha he reurns of financial asses are negaively correlaed wih changes in he volailiies of reurns (see Black (976) and Chrisie (98)) and, moreover, ha such volailiies end o change over ime. In he class of auoregressive condiional heeroskedasiciy (ARCH) models pioneered by Engle (98), several auhors have proposed exensions of he ARCH model and found evidence of such negaive correlaion. For insance, Nelson (99) proposed he exponenial generalized ARCH (EGARCH) model, while Glosen, Jagannahan and Runkle (99) developed GJR, a hreshold indicaor funcion GARCH model. The hreshold effec is ypically called asymmery when he hreshold is se o zero. A common idea used in such asymmeric models is he leverage effec, in which negaive shocks o reurns increase he predicable volailiy o a greaer exen han do posiive shocks of a similar magniude. On he oher hand, sochasic volailiy (SV) models are based on he direc correlaion beween he innovaions in boh reurns and volailiy. For a heoreical developmen in coninuous ime, Hull and Whie (987) generalized he Black-Scholes opion pricing formula o analyse sochasic volailiy and he negaive correlaion beween he innovaions. In empirical research, exensions of a simple discree ime model due o Taylor (986) have been analysed by Wiggins (987), Chesney and Sco (989), and Harvey and Shephard (996) in order o accommodae he direc correlaion. Alhough his exension has been called he asymmeric SV model, we will refer o he asymmeric behaviour based on he direc correlaion beween he innovaions as he leverage SV (SV-L) model o disinguish i from an alernaive model of asymmery. A comparison of a variey of univariae and mulivariae, condiional and sochasic, models is given in McAleer (005). In addiion o he leverage model, his paper considers a general asymmeric SV model based on he EGARCH specificaion of Nelson (99). As he EGARCH model incorporaes boh leverage and size effecs, we will refer o he asymmeric behaviour as he leverage and size effecs SV (SV-LS) model. The SV-LS model ness he SV-L model. As anoher special case of he SV-LS model, we will also consider he size effec SV (SV-S) model, which is a symmeric model. The SV-LS model will be esimaed and esed for an opimal and pracical represenaion of asymmery. The general model also permis he non-nesed SV-L and SV-S models o be esed agains 3

4 each oher. Recenly, So e al. (00) considered a hreshold effecs model in which he breaks in he consan and auoregressive parameer in he SV equaion depend on he signs of he previous reurns. Their model will no be discussed in deail here as he empirical resuls in Asai and McAleer (005) show ha heir model is generally inferior o he SV-L model given below in erms of he AIC and BIC crieria. The empirical analysis is concerned wih boh sock reurns and exchange rae reurns. Alhough Gallan, Hsieh and Tauchen (99) found ha he response of condiional volailiy o negaive and posiive shocks was essenially symmeric for he Briish pound/us dollar exchange rae by using he seminonparameric echnique of Gallan and Tauchen (989), we observed asymmeries in he exchange rae daa based on he SV-L and SV-S models, even hough such asymmeries may no be capured adequaely using he ARCH approach. In addiion o providing more accurae esimaes of volailiy, hese empirical resuls should assis in calculaing opimal Value-a-Risk (VaR) forecass and capial charges for purposes of porfolio and risk managemen. For esimaion of he SV model, recen developmens have been on he likelihood-oriened procedures (see Fridman and Harris (998), Sandmann and Koopman (998), Liesenfeld and Richard (003)), and on he Bayesian Markov Chain Mone Carlo (MCMC) echnique proposed by Jacquier, Polson and Rossi (994) (see, among ohers, Chib, Nardari and Shephard (00) and Shephard and Pi (998)). The Mone Carlo resuls conduced by Fridman and Harris (998) and Sandmann and Koopman (998) show ha he properies of hese mehods are very similar o hose of Jacquier, Polson and Rossi (994). While he procedures proposed by Fridman and Harris (998) are more compuaionally demanding han he MCMC echnique of Jacquier, Polson and Rossi (994), he simulaed maximum likelihood approaches proposed by Sandmann and Koopman (998) and Liesenfeld and Richard (003) are much easier o implemen compuaionally. Alhough he Mone Carlo likelihood mehod of Sandmann and Koopman (998) is compuaionally faser, he efficien imporance sampling (EIS) mehod of Liesenfeld and Richard (003) is flexible o various kinds of SV models. Wih regard o he Bayesian approach, Asai (005) compared several mehods wih regard o a numerical efficiency measure ha was proposed by Geweke (99). The remainder of he paper is organized as follows. Secion examines he SV-L, SV-S and SV-LS models, and invesigaes heir relaionships. A non-nesed esing 4

5 procedure o discriminae beween he SV-L and SV-S models is also discussed. Secion 3 compares a variey of leverage and asymmeric effecs in he framework of boh sochasic and condiional volailiy models. Secion 4 discusses some sandard esimaion echniques for SV models, and Secion 5 presens he resuls of Mone Carlo experimens regarding he finie sample performance of he esimaors of he alernaive SV models. In Secion 6, he wo asymmeric SV models and he SV-LS model are esimaed using S&P 500 Composie reurns, he Tokyo sock price index (TOPIX) reurns, and he exchange raes beween he USA and Ausralia and beween Japan and he USA. Secion 7 gives some concluding remarks. Leverage and Asymmery in SV Models In his paper we consider a new asymmeric SV model. Before going o he SV class, we review he ypical models for he ARCH class. There are wo sandard mehods of capuring asymmeric behaviour in ARCH-ype models, one of which is he exponenial generalized ARCH (EGARCH) model of Nelson (99). Alhough he EGARCH model has been used quie frequenly in empirical applicaions when asymmeric behaviour is observed, he presence of he absolue value of a sandardized shock in he model poses a problem regarding he saisical properies of he model. The oher frequenly used model of asymmeric behaviour in ARCH-ype models is he hreshold indicaor funcion ARCH model of Glosen, Jagannahan and Runkle (99) (GJR). This effec is ypically called asymmery when he hreshold is se o zero, in which case a disincion is made beween he effecs of posiive and negaive shocks on volailiy. Bu, as shown in he nex secion, he GJR model is very limied o express various kinds of asymmeries. The problem of EGARCH comes from he evaluaion of he derivaives based on he absolue value of he sandardized residuals. As shown in Henschel (995), we can avoid he problem by he wo ways; one is o esimae he mean and variance equaions separaely, and he oher is o approximae he absolue value funcion by using he recangular hyperbola roaed counerclockwise by 45 degrees. As we can handle he problem of absolue values, he new asymmeric SV model will be based on he EGARCH model. In his paper we consider he new asymmeric SV model, as follows: 5

6 y exp( h / ), ~ N(0,),,..., T, () h h E, ~ N(0, ), () where E( s) 0 for any and s, and y R is he mean-adjused reurn on an asse. Since many financial ime series exhibi lile or no dynamic behaviour in he mean bu pronounced serial dependence in he variance (see Bollerslev, Chou and Kroner (99), Bollerslev, Engle and Nelson (994), and McAleer (005) for useful surveys), he esimaion of is no he subjec of ineres in his paper. The model can be considered as a sochasic version of he EGARCH model of Nelson (99), and hus can express various kinds of asymmeries, which we will discuss in he nex secion. The model conains several special cases. The firs model is EGARCH model, which is obained by puing 0. Secondly, when 0, he model reduces o he basic SV model, in which he log-volailiy, h, follows a simple AR() process. The hird case is he asymmeric SV model of Harvey and Shephard (996). By seing 0, we have anoher represenaion of he model as follows: y exp( h / ), h h, (3) 0 ~ N,, 0 where and ; see also Harvey and Shephard (996). We will refer o his ype of asymmery as he SV wih Leverage (SV-L) model. Leverage capures asymmery by he negaive correlaion beween reurns and volailiy innovaions, which is described by puing 0. The fourh model is he case 0, yielding he symmeric size effec. In he model, posiive and negaive shocks increase fuure volailiy by he same amoun, if 0. We will refer o his model as he SV wih size effecs (SV-S) model. 6

7 The general model and, which we will refer o he SV wih leverage and size effecs (SV-LS) model, can be considered as an exension of Asai and McAleer (005). While Asai and McAleer (005) is based on he unsandardized error, y, he SV-LS model is based on he sandardized error,. Hence, he SV-LS model capure sandardized size effecs. The firs and second momens of he generalized error erm, E, is given by 0. The correlaion E and E coefficien beween and is. In he class of SV, when 0 and 0 in, his yields our asymmeric SV model, which may be inerpreed as eiher: (i) an asymmeric model which exhibis boh leverage and hresholds; or (ii) an arifac which is used solely for purposes of esing he non-nesed SV-L and SV-S models agains each oher. In he laer case, he four possible oucomes of he non-nesed ess of he SV-L and SV-S models agains each oher are as follows: (i) 0 and 0, which leads o rejecion of boh SV-L and SV-S; (ii) 0 and 0, which leads o rejecion of SV-S bu no SV-L; (iii) 0 and 0, which leads o rejecion of SV-L bu no SV-S; (iv) 0 and 0, which leads o rejecion of neiher SV-L nor SV-S. The reason why we consider he SV-S model in our analysis is o examine wheher or no he exchange rae reurns have symmeric effecs on fuure volailiies, in he framework of SV. Tess of non-nesed condiional volailiy models, specifically GARCH versus EGARCH, and GJR versus EGARCH, have been examined by Ling and McAleer (000) 7

8 and McAleer, Chan and Marinova (00), respecively. For furher deails regarding non-nesed esing procedures in he conex of economeric ime series and regression models, see McAleer (005). We may exend he SV-LS model and in order o incorporae he heavy-ailed reurn disribuions. We consider he sandardized disribuion for, for which he densiy funcion is given by f,, where represens he degrees-of-freedom and x denoes he gamma funcion. For convenience, we call i as he SV model wih disribuion, he leverage and size effecs (SV--LS). As long as 4, he kurosis of he disribuion is E, which is greaer han 3 if. Needless o say, he disribuion approaches he sandard normal disribuion when. I should be noed ha Asai (008) inroduced he disribuion in a differen way. Asai (008) decomposed he disribuion ino he sandard normal and chi-squared disribuions, and assumed he negaive correlaion using he sandard normal disribuion. 3 A Comparison of Sochasic and Condiional Volailiy Models Alhough he erms leverage and asymmery end o have similar meanings in he ARCH lieraure, i is insrucive o clarify any differences beween hem, as here is a greaer range of asymmeric effecs han is generally considered. Chrisie (98, p. 408) saes: Hisorically a variance/sock price relaion is par of marke folklore, he usual claim being ha he relaion is a negaive one; in oher words when he sock price increases he variance declines. Originally, Black (976) and Chrisie (98) invesigaed he negaive relaion beween 8

9 he ex-pos volailiy in he rae of reurns on equiy and he curren value of he equiy. We will refer o his phenomenon as he leverage effec. On he oher hand, we define asymmery as he differenial impacs of posiive and negaive shocks on volailiy. Given hese definiions, he leverage effec implies asymmery as a negaive shock increases volailiy, while a posiive shock decreases volailiy. On he oher hand, an effec ha is symmeric implies ha posiive and negaive shocks have he same effec on volailiy. Afer he developmen of he ARCH class of volailiy models in Engle (98), many auhors, including French, Schwer and Sambaugh (987), Pagan and Schwer (990), Nelson (99) and Glosen, Jagannahan and Runkle (99), analyzed he relaion beween reurns and volailiy using variaions of ARCH. Of he various ARCH models o capure asymmeric effecs, he EGARCH model proposed by Nelson (99) and he GJR model of Glosen, Jagannahan and Runkle (99), are he mos widely used. The GARCH, GJR and EGARCH models are defined as follows: y h z, z iid 0,, / GARCH: h y h, GJR: h y y I z h, EGARCH: h h exp z z, where h denoes condiional volailiy o disinguish i from sochasic volailiy, and I( z ) is an indicaor funcion such ha I( z) if z 0 and I( z) 0 oherwise. Alhough he GJR and EGARCH models are more flexible han he GARCH model, i is helpful o check heir flexibiliy wih respec o boh he leverage and asymmeric effecs. In order o disinguish several asymmeric effecs of a shock o reurns, we consider he following five cases, condiional on a negaive shock increasing volailiy: Case I (Symmery): Negaive and posiive shocks have idenical effecs in increasing volailiy. Case II (General Asymmery): A negaive shock increases volailiy, bu he effec of a posiive shock differs from ha of a negaive shock. Case III (Wide Asymmery): Negaive and posiive shocks increase volailiy 9

10 differenly (as a subse of Case II). Case IV (Sandard Asymmery): The impac of a negaive shock exceeds ha of a posiive shock (as a subse of Case III). Case V (Leverage): A negaive shock increases volailiy, whereas a posiive shock decreases volailiy. Case II provides he general asymmeric framework, and includes Cases III-V. Furhermore, Case III includes Case IV. The empirical resuls based on he GJR and EGARCH models ypically fall ino Case IV. Under he conceps of symmery, asymmery and leverage, Table provides he parameric resricions on he GJR, EGARCH, SV-L, SV-S and he general SV-LS models under Cases I-V. The resuls are summarized as follows: (i) (ii) (iii) In Case I, he GJR model reduces o GARCH. The GJR model canno accommodae leverage as he parameric resricions for leverage are 0 and 0, in which case volailiy is no guaraneed o be posiive. The EGARCH and our SV-LS models are enirely flexible wih respec o symmery, asymmery and leverage. As SV-L and SV-S are special cases of SV-LS model, hey canno capure he specific asymmeric effecs in Cases II, III and IV. Recenly, Bollerslev and Zhou (006) invesigaed no only asymmeric effecs, bu also he linear relaionship beween he conemporaneous reurns and volailiy, which hey ermed he volailiy feedback effec. Their specificaion is similar o he GARCH-M and SV-M models, alhough heir resuls are based on realized and implied volailiy. As hey repored ha he empirical resuls are sensiive o he insrumen choice of volailiy, an examinaion of his effec requires furher research. 4 Model Esimaion via EIS Mehod In his secion, we examine he finie sample properies of he ML esimaor for he SV-LS models, based on he efficien imporance sampling (EIS) mehod, in which he likelihood funcion is evaluaed hrough simulaion. Before considering he SV-LS model, i will be useful o inroduce several mehods 0

11 for esimaing he SV-L model. Wih respec o he SV-L model, i is sraighforward o apply he Mone Carlo likelihood (MCL) mehod of Sandmann and Koopman (998), as i is based on he sae space form derived by Harvey and Shephard (996) (see also Asai (008)). For he Bayesian MCMC mehod, we can use he approach of Omori e al. (007), which is based on he inegraion sampler of Chib, Nardari and Shephard (00). I should be noed ha Jacquier, Polson and Rossi (004) proposed a Bayesian MCMC echnique o esimae he SV-L model. However, his approach is based on Jacquier, Polson and Rossi (994), which is less efficien numerically han he mehod of Chib, Nardari and Shephard (00). Moreover, Yu (005) showed ha i was no clear how o ensure or inerpre he leverage effec in he model of Jacquier, Polson and Rossi (004). As here is no echnique o esimae he general SV-LS model, we will apply he EIS mehod, and invesigae he finie sample properies of he EIS esimaor in he reminder of his secion. The EIS is much easier o implemen compuaionally, and is flexible for he models discussed in his paper (see Liesenfeld and Richard (003) for a deailed explanaion of evaluaing SV models via he EIS mehod). Simulaion experimens were conduced in order o assess he performance of he EIS esimaor. The range of parameer values,,,, was seleced as follows. Firs of all, he auoregressive parameer and he mean of he log-volailiy are se o 0.95 and 0, respecively. Secondly,,, is seleced so ha he variances of he generalized error are he same in he hree models. Specifically, we se he parameer vecor o be,, 0., 0.08,0, 0.5,0,0.6, 0.00, 0.08,0.6, which represens he SV-L, SV-L and SV-LS models, respecively. Noe ha for each parameer se, he value of E is Wih respec o he SV-L model, he value of he correlaion coefficien in is given by For each, we generaed a sample of size T=000, and esimaed he SV-L, SV-S

12 and SV-LS models using he EIS mehod. The number of replicaions is 500. We consider a sample size of T 000 wih 500 replicaions. Table shows he sample means, sandard deviaions and roo mean squared errors of he MCL esimaors. The sample means are close o he rue values for all models, indicaing lile bias. Compared wih he Mone Carlo resuls in Table 3 of Sandmann and Koopman (998) for he MCL esimaor, which is limied o he parameer values given by 0, he sandard deviaions and roo mean squared errors presened in Table seem quie reasonable. Compared wih Table of Asai and McAleer (005), which is also limied o he SV-L model, we have similar resuls. 5 Empirical Resuls This secion examines he MCL esimaes of asymmeric behaviour in he SV, DL, AL, DAL and TSV models for four ses of empirical daa, namely Sandard and Poor's 500 Composie Index (S&P 500), Tokyo sock price index (TOPIX), US Dollar/Ausralian Dollar exchange rae (USD/AUD), and Japanese Yen/US dollar exchange rae (YEN/USD). The sample period is /4/999 o 8//007. The sample size for S&P is T=57, while hose for TOPIX, USD/AUD and Yen/USD are T=. The reurns, R, are defined as log P log P log P imes 00, where P is he closing price on day. The auocorrelaion srucure in he sock reurns, R m, was removed by using he following hreshold AR (TAR) model: m c R si si i i p, where s is zero if y 0, and one oherwise. As here was no evidence of serial correlaion for exchange rae reurns, we only subraced mean from he reurn series. Hereafer, for convenience we will refer o he sock and exchange rae reurns series as y R m and y R R, respecively. For sock reurns such as S&P 500 and TOPIX, a negaive correlaion would be expeced beween he innovaions in reurns and volailiy. Table 3 shows he EIS

13 esimaes for S&P 500 reurns. Four kinds of SV model were esimaed, namely he sandard SV model wih 0, SV-L wih 0, SV-S wih 0, and SV-LS wih no resricions. The significance of he esimaes of and/or in he various models leads o a srong rejecion of he basic SV model, which neglecs he leverage and size effecs. The esimaes of in he SV-L and SV-LS model are negaive and significan, indicaing he exisence of he leverage effecs. The esimaes of are negaive and significanin he SV-S and SV-LS models. AIC and BIC chose he SV-LS model. The likelihood raio ess favored he SV-LS model. A his sage, we need o discuss he sign of. While he esimaes of in he EGARCH models are always posiive in he lieraure, ha for he SV-LS was negaive. Our esimaes indicae ha a negaive shock increase volailiy, while a posiive shock decrease volailiy wih he differen magniude. This is Case V (Leverage). On he oher hand, ypical esimaes for he EGARCH model imply he Case IV (Sandard Asymmery). As in he EGARCH esimaes, i is reasonable ha large posiive and negaive shocks will increase volailiy. We need o pay aenion o he effec of small posiive shock. Recenly, Chen and Ghysels (007) found ha small posiive shock in S&P reurns decrease volailiy, by heir semi-parameric mehod. This is Case II (General Asymmery) in our caegory. However, as he direcion of he effec of a small posiive shock is differen form ha of a large posiive shock, he SV-LS and EGARCH models are no flexible o capure such effecs. If heir finding is applicable o our daa, he SV-LS model successfully describes he effec of small posiive shock, while i fail o capure he effec of large posiive shock, which may be absorbed by he innovaion erm,. On he oher hand, he EGARCH model may fail o capure he effecs of small posiive shocks, as i lacks he innovaion erm. Table 4 for TOPIX reurns shows ha he esimaes of are negaive and significan in he SV-L and SV-LS models, bu ha he esimae of is insignifican, indicaing he rejecion of he SV-S model. Boh AIC and BIC sugges ha SV-L is he bes model. The likelihood raio ess also seleced he SV-L. As in he case of S&P 500 reurns, he sandard SV model is clearly rejeced in favour of he wo SV models wih leverage effecs. The size effecs may be small so ha hey are absorbed by he innovaion erm of volailiy. Tables 5 and 6 presen he EIS esimaes for he USD/AUD and Yen/USD reurns, respecively. In Table 5, he resuls generally lead o similar implicaions as in he case of 3

14 he TOPIX reurns. AIC, BIC and he likelihood raio ess also prefer he SV-L model. The esimae of correlaion coefficien beween he innovaions of reurn and volailiy,, is for USD/AUD. The resuls for he YEN/USD reurns in Table 6 are similar o hose for S&P reurns in Table 3. In Tables 6, here is significan evidence of leverage and size effecs in he SV-L, SV-S and SV-LS models. The likelihood raio ess favor he SV-LS model. AIC suggess ha he SV-LS is he bes model, while BIC chose he SV-S. Overall, all four esimaes indicae ha he asymmeric effecs for he four daases are classified o Case V (Leverage). The slopes of negaive and posiive shocks are differen from each oher for he cases of he S&P and YEN/USD reurns. For he diagnosic checking, we calculaed filered esimaes of volailiy, Eexp h y, =,,,T. Table 7 shows he diagnosic saisics for he sandardized residuals. For all four series, we have he same resuls as follows. The Ljung-Box pormaneau ess based on weny lags suppor no serial correlaion. The Jarque-Bera ess rejec he null of normal disribuion. In order o consider he heavy-ailed condiional disribuion, we esimaed he SV--LS model. Table 8(a) presens he parameer esimaes of he SV--LS model for four series. For all series excep for YEN/USD, he esimaes of v are close o zero and insignifican, which may be caused by misspecificaion of he srucure of asymmery and/or ail-behavior. Wih respec o YEN/USD, he esimae of is negaive and significan while ha of is posiive and significan. Compared o he esimaes of SV-LS in Table 6, he sign of is changed. The esimae of v is significan, and he esimae of v is Table 8(b) shows he log-likelihood, AIC and BIC. For he case of YEN/USD, he likelihood raio es shows ha v is significan, while he model has he smalles AIC and BIC. For he oher hree series, AIC and BIC shows ha here is no improvemen form inroducing he heavy-ailed condiional disribuion. Table 8(c) shows he diagnosic saisics for he sandardized residuals. The Ljung-Box pormaneau ess based on weny lags suppor no serial correlaion for all series. We employed he es proposed by Godfrey and Orme (99) for asymmery under heavy-ailed disribuions. The es does no rejec he null of symmery excep for 4

15 TOPIX. The es for excess kurosis rejecs he null for YEN/USD, as expeced from he disribuion. 6 Conclusion In his paper, we suggesed a new asymmeric sochasic volailiy model, based on he leverage and size effecs, as an exension of he EGARCH model. We considered five caegories for asymmeric effecs, which describes he difference among he asymmeric effec of he EGARCH model, he hreshold effecs of GJR model, and he leverage effecs of SV model. We consider he SV wih leverage effecs (SV-L), he SV wih size effec (SV-S), and he general asymmeric SV model (SV-LS). These hree models are esimaed by he efficien imporance sampling mehod of Liesenfeld and Richard (003), and he finie sample properies of he esimaor are invesigaed using Mone Carlo experimens. We used four financial ime series are used o esimae he alernaive asymmeric SV models. The empirical resuls for S&P 500 and Yen/USD reurns preferred he SV-LS model, while TOPIX and USD/AUD reurns favored he SV-L model. For he case of sandardized disribuion, he resuls for Yen/USD reurns show ha he model is correcly specified, while he resuls for hree oher daa ses suggesed here was scope for improvemen. In addiion o providing more accurae esimaes of volailiy, hese empirical resuls should assis in calculaing opimal Value-a-Risk (VaR) forecass and capial charges for purposes of porfolio and risk managemen. This paper has made cerain conribuions, bu several exensions are sill possible. Firs, we may work wih muli-facors as in Chernov e al. (003) and Asai (008), in order o capure ail behavior. Secondly, we may consider more flexible specificaions of asymmery, which describe he resuls of Chen and Ghysels (007). Thirdly, he paper focuses on lepokuric disribuion, bu i is also worhwhile fiing skewed disribuions, including he skewed disribuion (Fernández and Seel (998)). 5

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20 Table : Parameric Resricions for Symmery, Asymmery and Leverage Model GJR EGARCH Case I Symmery 0, 0 0, 0 Case II General Asymmery NA Case III Wide Asymmery 0,, 0, 0 Case IV Sandard Asymmery 0, 0 0, Case V Leverage NA 0, SV-L NA NA NA NA 0 SV-S 0 NA NA NA NA SV-LS 0, 0 0, 0, 0, Noe: NA denoes no applicable. 0

21 Table : Finie Sample Performance of he EIS Esimaor for T=000 Parameer SV-L SV-S SV-LS (0.074) (0.58) (0.686) [0.076] [0.70] [0.697] (0.0) (0.06) (0.05) [0.05] [0.03] [0.09] (0.0334) (0.0330) (0.080) [0.0334] [0.033] [0.080] (0.0380) (0.049) [0.0380] [0.049] (0.068) (0.057) [0.0630] [0.0573] Noe: Sandard errors are in parenheses and roo mean squared errors are in brackes. True parameers are 0, 0.95, and,, 0., 0.08,0, 0.5,0,0.6, 0.00, 0.08,0.6, corresponding o he SV-L, SV-L and SV-LS models, respecively.

22 Table 3: EIS Esimaes for S&P 500 Reurns Model LogLike AIC BIC SV (0.0035) (0.053) (0.536) SV-L (0.003) (0.05) (0.048) (0.033) SV-S (0.003) (0.0360) (0.357) (0.0535) SV-LS (0.000) (0.003) (0.34) (0.06) (0.097) Noe: Sandard errors are given in parenheses. Table 4: EIS Esimaes for TOPIX Reurns Model LogLike AIC BIC SV (0.0084) (0.003) (0.46) SV-L (0.08) (0.06) (0.0800) (0.007) SV-S (0.008) (0.04) (0.97) (0.054) SV-LS (0.0) (0.033) (0.0804) (0.007) (0.053) Noe: Sandard errors are given in parenheses.

23 Table 5: EIS Esimaes for USD/AUD Reurns Model LogLike AIC BIC SV (0.0096) (0.0) (0.0997) SV-L (0.008) (0.03) (0.0875) (0.06) SV-S (0.0094) (0.0) (0.07) (0.0486) SV-LS (0.00) (0.04) (0.087) (0.063) (0.0493) Noe: Sandard errors are given in parenheses. Table 6: EIS Esimaes for YEN/USD Reurns Model LogLike AIC BIC SV (0.0) (0.08) (0.07) SV-L (0.04) (0.08) (0.0954) (0.065) SV-S (0.05) (0.03) (0.64) (0.0440) SV-LS (0.03) (0.0309) (0.8) (0.058) (0.0447) Noe: Sandard errors are given in parenheses. 3

24 Table 7: Diagnosic Saisics for SV-LS Models Diagnosics S&P 500 TOPIX USD/AUD YEN/USD Skewness [0.000] [0.00] [0.068] [0.003] Kurosis [0.000] [0.000] [0.000] [0.000] LB(0) [0.48] [0.709] [0.465] [0.866] Noe: The p-values are in brackes. For skewness and kurosis, we employ he Jarque-Bera ess. LB(0) denoes he Ljung-Box pormaneau es for serial correlaion based on weny lags. 4

25 Table 8: EIS Esimaes for SV--LS Models (a) Paramer Esimaes Model v S&P 500 TOPIX USD/AUD YEN/USD (0.000) (0.0057) (0.37) (0.07) (0.08) (0.0053) (0.03) (0.034) (0.0804) (0.008) (0.053) (0.000) (0.007) (0.0) (0.0884) (0.06) (0.0500) (0.0066) (0.0073) (0.090) (0.456) (0.049) (0.075) (0.036) Noe: Sandard errors are given in parenheses. (b) Log-Likelihood and Informaion Crierions Model LogLike AIC BIC S&P TOPIX USD/AUD YEN/USD (c) Diagnosic Saisics Diagnosics S&P 500 TOPIX USD/AUD YEN/USD Skewness [0.075] [0.09] [0.] [0.078] Kurosis [0.000] [0.000] [0.000] [0.000] LB(0) [0.30] [0.703] [0.43] [0.876] Noe: The p-values are in brackes. We employ he Jarque-Bera es for excess kurosis. Wih respec o skewness, we use he es proposed by Godfrey and Orme (99) for asymmery under heavy-ailed disribuions. LB(0) denoes he Ljung-Box pormaneau es for serial correlaion based on weny lags. 5