Volatility Models* Manabu Asai Faculty of Economics Tokyo Metropolitan University

Transcription

1 Dynamic Leverage and Threshold Effecs in Sochasic Volailiy Models* Manabu Asai Faculy of Economics Tokyo Meropolian Universiy Michael McAleer School of Economics and Commerce Universiy of Wesern Ausralia Revised: April 2004 * The auhors wish o acknowledge 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. The firs auhor appreciaes he financial suppor of he 21s Cenury Cener-of-excellence Program of he Japanese MEXT. The second auhor is graeful for he financial suppor of he Ausralian Research Council. 1

2 Absrac In his paper we examine wo mehods for modelling asymmeries, namely dynamic leverage and hreshold effecs, in Sochasic Volailiy (SV) models, one based on he hreshold effecs (TE) indicaor funcion of Glosen, Jagannahan and Runkle (1992), and he oher on dynamic leverage (DL), or he negaive correlaion beween he innovaions in reurns and volailiy. A general dynamic leverage hreshold effecs (DLTE) SV model is also used o enable non-nesed ess of he wo asymmeric SV models agains each oher o be calculaed. The hree SV models are esimaed by he Mone Carlo likelihood (MCL) mehod proposed by Sandmann and Koopman (1998), and he finie sample properies of he esimaor are invesigaed using numerical simulaions. As he numerical simulaion resuls show ha he MCL esimaor is biased, a simple mehod for correcing he bias is suggesed and he performance of he bias-correced MCL esimaors is evaluaed. Four financial ime series are used o esimae he SV models, wih empirical asymmeric effecs found o be saisically significan in each case. The empirical resuls for S&P 500, TOPIX and Yen/USD reurns indicae ha dynamic leverage dominaes he hreshold effecs model for capuring asymmeric behaviour, while he resuls for USD/AUD reurns show ha boh he non-nesed dynamic leverage and hreshold effecs models are rejeced agains each oher. For he four daa series considered, he dynamic leverage model dominaes he hreshold effecs model in capuring asymmeric effecs. In all cases, here is significan evidence of asymmeries in he general DLTE model. Key words: Sochasic volailiy, asymmeric effecs, dynamic leverage, hreshold effecs, indicaor funcion, Mone Carlo likelihood, numerical simulaions, bias correcion, non-nesed models. 2

3 1 Inroducion I has long been recognized ha he reurns of financial asses are negaively correlaed wih changes in he volailiies of reurns (see Black (1976) and Chrisie (1982)) and, moreover, ha such volailiies end o change over ime. In he class of auoregressive condiional heeroskedasiciy (ARCH) models pioneered by Engle (1982), several auhors have proposed exensions of he ARCH model and found evidence of such negaive correlaion. For insance, Nelson (1991) proposed he exponenial generalized ARCH (EGARCH) model, while Glosen, Jagannahan and Runkle (1992) developed a hreshold indicaor funcion GARCH model, which is commonly called he GJR 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. 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 (1987) 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 (1986) have been analysed by Wiggins (1987), Chesney and Sco (1989), and Harvey and Shephard (1996) 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 dynamic leverage SV model o disinguish i from an alernaive model of asymmery. In addiion o he dynamic leverage model, his paper considers an alernaive asymmeric SV model using a hreshold effecs indicaor funcion, as developed by Glosen, Jagannahan and Runkle (1992) in he conex of ARCH models. We will refer o he asymmeric behaviour based on a zero hreshold indicaor funcion as he hreshold effecs SV model. These wo asymmeric SV models, as well as a more general model which incorporaes boh ypes of asymmeries, called he dynamic leverage hreshold effecs SV model, will be esimaed and esed for an opimal and pracical represenaion of asymmery. The general model also permis he non-nesed dynamic leverage and hreshold effecs SV models o be esed agains each oher. 3

4 The empirical analysis is concerned wih boh sock reurns and exchange rae reurns. Alhough Gallan, Hsieh and Tauchen (1991) 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 (1989), we observed asymmeries in he exchange rae daa based on he dynamic leverage and hreshold effecs SV models, even hough such asymmeries may no be capured adequaely using he ARCH approach. For esimaion of he SV model, recen developmens have been on he likelihood-oriened procedures (see Fridman and Harris (1998), Sandmann and Koopman (1998) and Waanabe (1999)), and on he Bayesian Markov Chain Mone Carlo (MCMC) echnique proposed by Jacquier, Polson and Rossi (1994) (see, among ohers, Chib, Nardari and Shephard (2002) and Shephard and Pi (1998)). The Mone Carlo resuls conduced by Fridman and Harris (1998), Sandmann and Koopman (1998) and Waanabe (1999) show ha he properies of hese mehods are very similar o hose of Jacquier, Polson and Rossi (1994). While he procedures proposed by Fridman and Harris (1998) and Waanabe (1999) are more compuaionally demanding han he MCMC echnique of Jacquier, Polson and Rossi (1994), he Mone Carlo likelihood mehod proposed by Sandmann and Koopman (1998) is much easier o implemen compuaionally. Wih regard o he Bayesian approach, Asai (2003) found some evidence ha he mehod of Chib, Nardari and Shephard (2002) was he bes wih regard o a numerical efficiency measure ha was proposed by Geweke (1992). The remainder of he paper is organized as follows. Secion 2 examines he dynamic leverage (DL), hreshold effecs (TE), and dynamic leverage hreshold effecs (DLTE) SV models, and invesigaes heir relaionships. A non-nesed esing procedure o discriminae beween he DL and TE models is also discussed. Secion 3 discusses some esimaion echniques for SV models, and Secion 4 presens he resuls of some Mone Carlo experimens regarding he finie sample performance of he esimaors of he alernaive SV models. As he numerical simulaion resuls show ha he MCL esimaor is biased, a simple mehod for correcing he bias is suggesed and he performance of he bias-correced MCL esimaors is evaluaed. In Secion 5, he wo asymmeric SV models and he DLTE 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 6 gives some concluding remarks. 4

5 2 Asymmeric Effecs in Sochasic Volailiy Models In his paper we consider wo ypes of asymmeric behaviour in SV models. Dynamic leverage capures asymmery by he negaive correlaion beween reurns and volailiy innovaions, as follows: y = σε exp( h / 2), ε ~ N(0,1), 1,..., T, (1) = h = h + + N E = (2) 2 1 φ, ~ (0, σ ), ( ε ) ρσ, where y = R µ 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, Chow and Kroner (1992), Bollerslev, Engle and Nelson (1994), and Li, Ling and McAleer (2002) for useful surveys), he esimaion of µ is no he subjec of ineres in his paper. We will refer o his ype of asymmery, namely when ρ 0, as he dynamic leverage (DL) SV model. When ρ = 0, here is no dynamic leverage, alhough alernaive asymmeries may be presen (such as γ 0 in equaion (3) below). 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 (1991). 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. Shephard (1996) suggesed a likely sufficien condiion for consisency of he quasi-maximum likelihood esimaor (QMLE). McAleer, Chan and Marinova (2002) noed ha a similar condiion was likely o be sufficien for he exisence of momens and for asympoic normaliy of he QMLE. A more frequenly used model of asymmeric behaviour in ARCH-ype models is he hreshold indicaor funcion ARCH (or GJR) model of Glosen, Jagannahan and Runkle (1992). The hreshold effec is ypically called asymmery when he hreshold is se o zero. Ling and McAleer (2002) esablished he necessary and sufficien condiions for he exisence of momens of he GJR(1,1) model, while McAleer, Chan and Marinova (2002) esablished he sufficien condiions for consisency and asympoic normaliy of he QMLE of GJR(1,1). In view of hese recen heoreical resuls, he developmen of an alernaive o he DL SV model will be based on he 5

6 hreshold model of Glosen, Jagannahan and Runkle (1992). In he model of asymmery based on hresholds, volailiy is affeced by he sign of he previous reurns innovaion, as follows: { } h = 1 φh + ξ, ξ = γ I( ε) + E[ I( ε)] +, (3) 2 where ~ N(0, σ ), E( ε ) = 0, and I ( ) is an indicaor funcion such ha I ( x) = 1 if x < 0 and I ( x) = 0 oherwise. Noe ha E[ ξ ] = 0 and E [ ξ ] σ γ 4 = +. In he following, we refer o his ype of asymmeric behaviour, ha is, when γ 0, as he hreshold effecs (TE) SV model. When γ = 0, here is no hreshold effec, alhough alernaive asymmeries may be presen (such as ρ 0 in equaion (2) above). When ρ 0 in (2) andγ 0 in (3), his yields he dynamic leverage hreshold effecs (DLTE) SV model. The general DLTE model may be inerpreed as eiher: (i) an asymmeric model which exhibis boh dynamic leverage and hreshold effecs; or (ii) an arifac which is used solely for purposes of esing he non-nesed DL and TE models agains each oher. In he laer case, he four possible oucomes of he non-nesed ess of he DL and TE models agains each oher are as follows: (i) ρ = 0 and γ = 0, which leads o rejecion of boh DL and TE; (ii) ρ 0 and γ = 0, which leads o rejecion of TE bu no DL; (iii) ρ = 0 and γ 0, which leads o rejecion of DL bu no TE; (iv) ρ 0 and γ 0, which leads o rejecion of neiher DL nor TE. Tess of non-nesed condiional volailiy models, specifically GARCH versus EGARCH, and GJR versus EGARCH, have been examined by Ling and McAleer (2000) and McAleer, Chan and Marinova (2002), respecively. For furher deails regarding non-nesed esing procedures in he conex of economeric ime series and regression models, see McAleer (1995). An alernaive comparison of he DL and TE models can be made as follows. Averaged over he whole sample, we can examine he relaionship beween he DL and TE models hrough he correlaion of ξ and ε under alernaive models. While Corr ξ, ε ) = ρ for he DL model, he correlaion coefficien is given by ( 6

7 Corr 2 2 ( ξ, ε) γ 2 π( σ γ /4) = + for he TE model, since Cov ( I( ε ), ε ) E ε / 2 = 1 2π. This resul indicaes ha an appropriae choice = of γ in he TE model would yield he same Cov( ξ, ε ) as in he DL model. By appropriae consrucion of he TE and DL models, if he y are generaed by he TE model when γ = γ 0 and σ = σ 0 bu he DL model is esimaed, he esimae of ρ will be approximaely γ 2 π( σ + γ /4). However, if he daa are generaed by he rue DL model bu he TE model is esimaed, he esimae of γ will be smaller han he values derived by ρ since he TE model can capure only he hreshold effecs. In he DLTE model, he correlaion coefficien of ξ and ε is given by ρσ π γ 2 Corr( ξ, ε) =. π σ + γ γρσ 2 2 ( /4) (4) This is also useful for esing he differences beween he DL and TE models. I should be noed ha Corr 2 2 ( ξ, ε) γ 2 π( σ γ /4) 2 π = + < = for he TE model, which implies ha his model is no appropriae for describing highly posiive or negaive correlaion. 3 Model Esimaion Before we examine he empirical performance of he wo non-nesed asymmeric SV models, as well as he general DLTE SV model, i will be useful o discuss esimaion of he SV model. There are hree caegories of esimaor: (i) sampling heory based on y ; (ii) sampling heory based on log y 2 ; (iii) Bayesian Markov Chain Mone Carlo (MCMC) mehods. The disincion beween he firs wo caegories is imporan, especially for likelihood-based inference. For insance, Kim, Shephard 7

8 and Chib (1998) proposed he non-nesed likelihood raio es of he SV model versus he GARCH model, which needs addiional esimaion of he likelihood of for he second caegory. The hird caegory is indifferen o he choice of y or log y 2 y because he poserior disribuions are invarian o he choice of disribuion of y or log y 2 if he priors are he same. For each of he firs wo caegories, here is an opimal mehod wih respec o he finie sample properies, namely he likelihood-oriened procedures of Fridman and Harris (1998), Sandmann and Koopman (1998) and Waanabe (1999). For he firs caegory, Fridman and Harris (1998) and Waanabe (1999) independenly applied he non-gaussian sae-space-model filering and smoohing procedure of Kiagawa (1987) o evaluae he likelihood hrough recursive numerical inegraion. I should be noed ha anoher mehod of esimaion is he efficien mehod of momens (EMM) procedure proposed by Gallan and Tauchen (1996), which maches he score of he auxiliary model hrough simulaion. Alhough he procedures based on mehod of momens are known o be subopimal relaive o he likelihood-based mehods, Gallan and Tauchen (1996) claim ha, if he auxiliary model is an accurae approximaion o he disribuion of he daa, hen EMM is as efficien as maximum likelihood. However, here is no mehod for esimaing he insananeous volailiy hroughou he sample, = 1, K,T, so ha an addiional form of esimaion, such as he Kalman filer based on log 2 y, is required. In he second caegory, Sandmann and Koopman (1998) proposed he Mone Carlo maximum likelihood mehod, for which he likelihood funcion can be approximaed arbirarily by decomposing i ino a Gaussian par, consruced by he Kalman filer, and a remainder funcion, for which he expecaion is evaluaed hrough simulaion. Mone Carlo resuls conduced by Fridman and Harris (1998), Sandmann and Koopman (1998) and Waanabe (1999) show ha heir mehods have properies which are very close o hose of Jacquier, Polson and Rossi (1994), which is an esimaion mehod from he hird caegory. While he procedures proposed by Fridman and Harris (1998) and Waanabe (1999) are more compuaionally demanding han he MCMC mehod of Jacquier, Polson and Rossi (1994), he Mone Carlo likelihood mehod proposed by Sandmann and Koopman (1998) is much easier o implemen compuaionally. Regarding he esimaion of he likelihood of y, one of several 8

9 compuaionally inefficien bu accurae mehods can be used as his requires no ieraions. For example, he auxiliary paricle filers proposed by Pi and Shephard (1999) would be suiable. A disinguishing feaure of he hird caegory is ha anoher measure of efficiency is required o compare he various mehods as all he approaches produce a single poserior. Jacquier, Polson and Rossi (1994) proposed a Bayesian approach for esimaing SV models using he MCMC echnique. Their mehod is called he single-move sampler since i requires sampling each h. Illusraive examples in de Jong and Shephard (1995), Shephard and Pi (1997) and Kim, Shephard and Chib (1998) regarding he normal SV model sugges ha he single-move sampler would produce a highly correlaed sample sequence when sae variables are highly auocorrelaed. The single-move sampler is, herefore, inefficien in he sense ha i needs o repea he sampling a large number of imes. Two mehods are more efficien han he single-move sampler, he firs of which is he muli-move sampler proposed by Shephard and Pi (1997). As he original muli-move sampler suffers from serious esimaion bias, Waanabe and Omori (2001) suggesed a correcion. A second mehod is he so-called inegraion sampler, which was proposed by Kim, Shephard and Chib (1998) and exended by Chib, Nardari and Shephard (2002). Based on simulaed daa, Asai (2003) found ha, by using an efficiency facor presened in Geweke (1992), he inegraion sampler always ouperformed he muli-move sampler for sampling ( φ, σ ), bu ha he former was generally less efficien han he laer for sampling σ and he laen volailiies. Using he Yen/Dollar exchange rae daa, Asai (2003) also showed ha here was an empirical case in which he inegraion sampler ouperformed he muli-move sampler for all he parameers ( σ, φ, σ ). Compared wih he Mone Carlo Likelihood (MCL) mehod of Sandmann and Koopman (1998), he Bayesian MCMC mehods are compuaionally demanding. The Mone Carlo resuls of Sandmann and Koopman (1998), which compare he MCL mehod wih he MCMC mehod of Jacquier, Polson and Rossi (1994), show ha MCL yields a larger bias han MCMC when he uncondiional variance of he ime-varying log-volailiy is relaively small. Since such an oucome suggess ha he volailiy is 9

10 no paricularly significan, in wha follows only he MCL mehod is used for esimaing he hree SV models. Reurning o he asymmeric SV models, as he MCL mehod can incorporae asymmery and explanaory variables ino he volailiy equaion, i is a sraighforward exension o esimae he DL and TE models. For he Bayesian MCMC mehod, we would use a sligh modificaion of he inegraion sampler of Chib, Nardari and Shephard (2002). I should be noed ha Jacquier, Polson and Rossi (2004) proposed a Bayesian MCMC echnique o esimae he DL model. However, his approach is based on Jacquier, Polson and Rossi (1994), which is less efficien han he mehod of Chib, Nardari and Shephard (2002) wih respec o numerical efficiency. Moreover, Yu (2004) showed ha i was no clear how o ensure or inerpre he leverage effec in he model of Jacquier, Polson and Rossi (2004). 4 Mone Carlo Experimens Simulaion experimens were conduced in order o assess he performance of he MCL esimaor. The range of parameer values θ = ( σ, φ, σ, γ )' was seleced as follows. Firs, he auoregressive parameer φ is se o 0.95, and ( σ, ρ, γ ) is seleced so ha he coefficien of variaion, namely CV = Var( h ) Var( ξ ) = exp 2 E( ) 1 h φ 2 1 akes he value of uniy in he DLTE model, wih a resricion ha he correlaion coefficien beween ξ and ε is or Specifically, we se he parameer vecor o be ( σ, ρ, γ ) = {(0.260, -0.30, 0), (0.241, 0, 0.195), (0.253, -0.15, 0.100)}, which represens he DL, TE and DLTE models, respecively. Noe ha for each parameer se, he value of Var( ξ ) is 0.260, he absolue value of ρ in DL is wice ha of ρ in DLTE, and he value of γ in TE is roughly wice ha of γ in DLTE. For he case Corr( ε, ξ ) = 0.60 we specify he parameer values o be ( σ, ρ, γ ) = {(0.260, -0.60, 0), (0.173, 0, 0.388), (0.224, -0.30, 0.225)}. 10

11 Again, he value of Var( ξ ) is for each parameer se. Second, he values of he locaion parameer, σ, are chosen such ha he expeced variance, namely 2 2 Var( ξ ) E ( y ) = σ exp, 2 2(1 φ ) is se o If he simulaed daa are regarded as weekly reurns, his corresponds o an approximae 22% annualized sandard deviaion. For convenience in esimaion, 2 we mapped σ ino α using α = logσ, which yields α = For each θ, we generaed a sample of size T = 1000, 2000 and 5000, and esimaed he DL, TE and DLTE models using he MCL mehod. I should be noed ha we exclude he case of highly negaive auocorrelaion, such as Corr( ε, ξ ) = 0.90, since i may exceed he bound for he TE model, as described in he previous secion. 4.1 Esimaes based on correc models Table 1 shows he resuls for ( ) Corr ε, ξ = 0.30 under various rue models and sample sizes. Table 1(a) repors he sample means and sandard deviaions of he MCL esimaes for K = 500 replicaions, while Table 1(b) presens he 95% coverage probabiliies of he Mone Carlo simulaions. Thus, for each replicaion i, a confidence inerval is compued as θ ˆ ± 1.96 Var(ˆ ), where Var(ˆ θ ) is he ij θ ij ij relevan elemen of he covariance marix of he esimaor θˆ ij. The coverage probabiliy, pˆ, corresponds o he number of imes he rue value of θ (which, for he purposes of he experimens, is assumed o be known), θ 0, falls wihin he confidence inerval for each replicaion, divided by he number of replicaions, K. Sandard errors are compued from he Bernoulli formula ( 1/ K) pˆ(1 pˆ ). Table 1 indicaes ha here is a small bias in he esimaor of ρ relaive o he rue value. The bias seems o increase as ρ becomes large, being around 0.03 for ρ = 0.15 and around 0.06 for ρ = This may explain he relaively low 11

12 coverage probabiliy for ρ compared wih γ, especially in DLTE. The bias in ρ remains even for sample size T = 5000, which may lead o he resul ha he coverage probabiliy for ρ decreases as he sample size increases for ρ = Esimaed Corr ε ξ and Var( ξ ) values of (, ), which are ρ and σ, respecively, in he DL model, also suppor he resuls regarding he bias. The absolue value of he esimae of ρ in DL is roughly wice ha of ρ in DLTE, and he esimae of γ in TE is wice ha of γ in DLTE, which are similar o he rue parameer values. In TE (where ρ = 0 ), he sample mean of he correlaion coefficien is close o he rue value, bu hose for ρ = 0.15 and ρ = 0.30 are abou and -0.37, respecively. The esimaes of ρ and γ are significan in DL and TE, respecively, bu neiher is significan in DLTE for T = 1000 and The sample mean of Var( ξ ) approaches he rue value as ρ approaches zero, and he bias seems o disappear as he sample size increases. Alhough i migh be expeced, he sandard deviaions in Table 1 are very close o hose in Table 3 of Sandmann and Koopman (1998), which repor he finie sample properies of he MCL esimaes when ρ = 0 and γ = 0. Table 2 presens he resuls for ( ) Corr ε, ξ = 0.60 under various rue models. The bias for ρ increases as ρ deviaes significanly from zero. While he bias is abou 0.05 for ρ = 0.30, i is abou 0.10 for ρ = Esimaes of γ are close o he rue values. In addiion o hese resuls, he evidence ha he sample mean of Var( ξ ) approaches he rue value as ρ approaches zero implies ha a lile bias for σ is affeced by he bias for ρ. Before we discuss a way of correcing such biases, i will be helpful o presen he resuls for misspecified models, as he main concern in he paper is o examine he relaionships among he DL, TE and DLTE models. 4.2 Esimaes based on misspecified models and LR ess Table 3 shows he esimaion resuls for wo models when a hird model is rue. The sample size for Table 3 is T = 2000, wih he resuls for T = 1000 and

13 being omied as hey are qualiaively similar. Table 3(a) presens he sample means and sandard deviaions of he MCL esimaes of TE and DLTE when he rue model is DL. As expeced, he esimaed correlaion coefficien beween ξ and ε is far from he rue value While he esimae of γ is significan in he incorrec TE bu no in DLTE, he esimae of ρ is significan in DLTE. Table 3(b) indicaes ha DL can capure he rue correlaion beween ξ and ε even when TE is he rue model. While he esimae of ρ is significan in he incorrec DL bu no in DLTE, he esimae of γ is significan in DLTE. Thus, Tables 3(a) and 3(b) show ha he non-nesed -es based on he general DLTE model has power o disinguish beween he DL and TE models by rejecing TE when DL is rue and also rejecing TE when DL is rue. Table 3(c) shows ha he esimaes of ρ and γ are significan in he incorrec DL and TE models, respecively, when DLTE is rue. Table 4 repors he rejecion frequencies of he likelihood raio (LR) ess of he respecive null hypoheses under alernaive rue models. Table 4(a) and (b) correspond o he cases for (, ) Corr ε ξ o be and -0.60, respecively. The LR es is based on log y 2 since i does no require any addiional compuaion. Table 4 indicaes ha he rejecion frequency of he LR es is close o he nominal size of 5% when he DL and TE models are rue, and ha he LR es has sufficien power o rejec he respecive null hypoheses when he general DLTE model is rue. Table 4(a) and (b)b also indicaes ha he rejecion frequency of he LR es when he DLTE is rue increase as he sample size increases. Comparing Table 4(a) wih (b), he rejecion frequency when he DLTE is rue for ( ) ha for ( ) Corr ε, ξ = 0.60, as expeced. Corr ε, ξ = 0.30 is relaively smaller han 4.3 Bias correcion As saed above, he esimaes of ρ have a small bias even for sample size T = 5000, wih he esimaes of σ being sensiive o he esimaes of ρ. In order o cope wih his problem, we propose an effecive mehod for correcing he bias. 13

14 Based on he response surface mehodology, we consider wo regressions as follows: ˆ ρ ρ = a + a ρ+ error, 1 2 ˆ σ σ = b + b ρ+ b / T + bσ / T + error, where a i (i = 1,2) and b i (i = 1,, 4) are coefficiens. The firs regression does no depend on sample sizes as he bias of ρ does no seem o be affeced by he sample sizes ha are ypically used in empirical analysis. The firs and second erms of he righ-hand side of he second regression follow from he same idea, wih he hird and fourh erms diminishing as T increases. In order o esimae hese models, we used welve observaions lised in Tables 1(a) and 2(a) for he DL and DLTE models. Table 5 shows he esimaion resuls. All parameers excep for he consan erm for ρ are significan a he five percen level. The p-value of he consan erm is Several oher specificaions were considered, including erms such as ρ / T, bu hese were all insignifican. Based on hese resuls, wo naural bias-correced esimaors are given as follows: ( ρ ) % ρ = ˆ , ˆ σ ( % ρ / T ) % σ = / T This mehod can be applied no only o he DL and DLTE models, bu also o he TE model, wih he laer obained upon seing % ρ = 0. We performed Mone Carlo experimens o invesigae he performance of he bias-correced MCL esimaors. Table 6 shows he resuls for T = 2000 and Corr( ξ, ε ) = 0.3. Several oher cases examined are qualiaively similar o he resuls in Table 6, and hence hey are omied. Compared wih he original esimaes in Table 1, Table 6 shows ha he biases in σ and ρ are reduced dramaically in he DL and DLTE models, and ha he coverage probabiliies abou ρ are much closer o 0.95 han hose repored in Table 1. The esimaes of he correlaion coefficiens beween ξ and ε, and he esimaes of Var( ξ ), are all close o he rue values of -0.3 and 0.260, respecively. More precise esimaes would require furher exensive 14

15 Mone Carlo experimens using response surface mehodology. The resuls in Table 6, however, indicae ha he simple bias-correcion mehod suggesed above can be quie effecive. 5 Empirical Resuls This secion examines he MCL esimaes of asymmeric behaviour in he hree SV models for four ses of empirical daa, namely Sandard and Poor's 500 Composie Index (S&P), he Tokyo sock price index (TOPIX), he US Dollar/Ausralian Dollar exchange rae (USD/AUD), and he Japanese Yen/US dollar exchange rae (YEN/USD). The sample period for S&P is 1/6/1986 o 12/4/2000, giving T = 3723 observaions, ha for TOPIX is 1/4/1990 o 9/30/1999, giving T = 2403 observaions, ha for USD/AUD is 1/6/1986 o 12/4/2000, giving T = 3723 observaions, and ha for Yen/USD is 1/4/1990 o 12/28/1999, giving T = 2467 observaions. We define reurns y as 100 {log P - log P -1 } minus he sample mean, where P is he closing price on day. Figure 1 shows he mean subraced reurns of all four series. There seems o be an oulying observaion early in he sample for S&P, and here are cluserings of volailiy in each series. For sock reurns, a negaive correlaion would be expeced beween he innovaions in reurns and volailiy. Table 7 shows he MCL esimaes for S&P. Alhough he esimaes of ρ and γ are significan, and have he expeced signs, a he five percen level in he DL and TE models, respecively, he non-nesed LR es based on he general DLTE model rejecs he TE null hypohesis bu does no rejec he DL null hypohesis. The resul ha DL is preferred o TE for S&P implies ha hreshold effecs are inadequae for capuring he asymmeric srucure of sock reurns, whereas dynamic leverage is appropriae. Table 8 for TOPIX reurns leads o a similar implicaion in ha he individual esimaes of ρ and γ are significan, and have he expeced signs, a he five percen level in he DL and TE models, respecively, bu he non-nesed LR es based on he general DLTE model rejecs TE in favour of DL. In boh Tables 7 and 8, here is significan evidence of asymmeries in he general DLTE model. The Mone Carlo resuls of Sandmann and Koopman (1998) show ha heir MCL mehod yields a larger bias han he Bayesian MCMC approach of Jacquier, Polson and Rossi (1994) when he uncondiional variance of he ime-varying log-volailiy is 15

16 relaively small, especially when he value of CV defined in he previous secion is close o When CV is equal o or greaer han one, he MCL esimaes are very close o he rue value, as are he Bayesian MCMC esimaes. For S&P he value of CV is 1.060, which will guaranee robusness of he resuls wih respec o he coefficien of variaion. Tables 9 and 10 presen he MCL esimaes for USD/AUD and Yen/USD reurns. In Table 9, he value of CV is 1.171, and he esimae of γ, hough having he expeced sign, is insignifican in he TE model, which corresponds o he resul of Gallan, Hsieh and Tauchen (1991) for he Briish Pound/US Dollar rae. However, he esimae of ρ is significan in he DL model, indicaing a negaive correlaion beween he USD/AUD reurns and volailiy innovaions. I is ineresing o noe ha he esimaes of boh ρ and γ are significan in he general DLTE model, which leads o rejecion of boh he non-nesed DL and TE models agains each oher. Thus, he non-nesed LR es does no rejec eiher model in favour of he oher. The resuls for Yen/USD reurns in Table 10 are similar o hose for USD/AUD reurns in Table 9, excep ha he esimae of γ is no significan in he general DLTE model, hereby leading o he rejecion of he TE model in favour of he DL model. In boh Tables 9 and 10, here is significan evidence of asymmeries in he general DLTE model. The mehod of Sandmann and Koopman (1998) is used o obain he smoohed volailiy esimaes of he DL and TE models for each of he four series, which are given in Figures 2 and 3, respecively. Apar from a spike early in he sample for S&P, which corresponds o an oulying observaion, here do no appear o be exreme volailiy esimaes elsewhere in he series. Sample correlaions beween he pairs of volailiy esimaes for he four series in Figures 2(a)-2(d) and 3(a)-3(d) are very similar a , , and , respecively. The differences in he smoohed volailiy esimaes in Figures 2 and 3 may be inerpreed as follows. Le Corr( ε, ξ ) be denoed ρ1 and ρ2 for DL and TE, respecively, and he corresponding volailiies be denoed h1 and h2, respecively. In he empirical analysis, i is always he case ha ρ1 < ρ2 < 0. There is a endency for h1 > h2 afer a large negaive price shock. For example, here is a large negaive shock around T = 500 in Figure 1(a), wih Figures 2(a) and 3(a) showing ha he corresponding volailiies saisfy h1 > h2. Similarly, here is a endency for h1 < h2 immediaely afer a large posiive shock. For example, here is a posiive shock in Figure 1(c) around T = 3300, wih Figures 2(c) and 3(c) showing ha he corresponding volailiies saisfy h1 < h2. 16

17 6 Conclusion In his paper we considered wo mehods for modelling asymmeries in Sochasic Volailiy (SV) models, namely he hreshold effecs (TE) model based on he indicaor funcion of Glosen, Jagannahan and Runkle (1992), and he dynamic leverage (DL) model based on he negaive correlaion beween he innovaions in reurns and volailiy. A general dynamic leverage hreshold effecs (DLTE) SV model, which could be inerpreed as eiher an asymmeric model which exhibis boh dynamic leverage and hreshold effecs; or as an arifac which is used solely for purposes of esing he non-nesed DL and TE models agains each oher, was also analysed. The hree SV models were esimaed by he Mone Carlo likelihood mehod proposed by Sandmann and Koopman (1998), and he finie sample properies of he esimaor were invesigaed using numerical simulaions. As he numerical simulaion resuls show ha he MCL esimaor is biased, a simple mehod for correcing he bias was suggesed and he performance of he bias-correced MCL esimaors was evaluaed. Four financial ime series were used o esimae he SV models, wih asymmeric effecs found o be saisically significan in each case. The empirical resuls for S&P 500, TOPIX and Yen/USD reurns indicaed ha he dynamic leverage model dominaed he hreshold effecs model for capuring asymmeric behaviour, while he resuls for USD/AUD reurns showed ha boh he non-nesed dynamic leverage and hreshold effecs models could be rejeced agains each oher. For he four daa series considered, he dynamic leverage SV model dominaed he hreshold effecs SV model in capuring asymmeric behaviour. In all cases, here was significan evidence of asymmeries in he general DLTE model. 17

18 References Asai, M. (2003), Comparison of MCMC Mehods for Esimaing Sochasic Volailiy Models, unpublished paper, Faculy of Economics, Tokyo Meropolian Universiy. Black, F. (1976), Sudies of Sock Marke Volailiy Changes, 1976 Proceedings of he American Saisical Associaion, Business and Economic Saisics Secion, pp Bollerslev, T., R.Y. Chow and K.F. Kroner (1992), ARCH Modelling in Finance: A Review of he Theory and Empirical Evidence, Journal of Economerics, 52, Bollerslev, T., R.F. Engle and D.B. Nelson (1994), ARCH Models, in R.F. Engle and D. McFadden (eds.), Handbook of Economerics, 4, Norh-Holland, Amserdam, pp Chesney, M., and L.O. Sco (1989), Pricing European Currency Opions: A Comparison of he Modified Black-Scholes Model and a Random Variance Model, Journal of Financial and Quaniaive Analysis, 24, Chib, S., F. Nardari and N. Shephard (2002), Markov Chain Mone Carlo Mehods for Sochasic Volailiy Models, Journal of Economerics, 108, Chrisie, A.A. (1982), The Sochasic Behavior of Common Sock Variances: Value, Leverage and Ineres Rae Effecs, Journal of Financial Economics, 10, de Jong, P., and N. Shephard (1995), The Simulaion Smooher for Time Series Models, Biomerika, 82, Engle, R.F. (1982), Auoregressive Condiional Heeroskedasiciy wih Esimaes of he Variance of Unied Kingdom Inflaion, Economerica, 50, Fridman, M., and L. Harris (1998), A Maximum Likelihood Approach for Non-Gaussian Sochasic Volailiy Models, Journal of Business and Economic Saisics, 16,

19 Gallan, A.R., and G. Tauchen (1989), Seminonparameric Esimaion of Condiional Consrained Heerogeneous Processes: Asse Pricing Applicaions, Economerica, 57, Gallan, A.R., and G. Tauchen (1996), Which Momens o Mach?, Economeric Theory, 12, Gallan, A.R., D.A. Hsieh and G. Tauchen (1991), On Fiing a Recalciran Series: he Pound/Dollar Exchange Rae, , in W.A. Barne, J. Powell, and G. Tauchen (eds.), Nonparameric and Semiparameric Mehods in Economerics and Saisics, Proceedings of he Fifh Inernaional Symposium in Economic Theory and Economerics, Cambridge Universiy Press, Cambridge, pp Geweke, J. (1992), Evaluaing he Accuracy of Sampling-Based Approaches o he Calculaion of Poserior Momens, in J.M. Bernardo, J.O. Berger, A.P. Dawid and A.F.M. Smih (eds.), Bayesian Saisics 4, Oxford Universiy Press, Oxford, pp Glosen, L., R. Jagannahan and D. Runkle (1992), On he Relaion Beween he Expeced Value and Volailiy of Nominal Excess Reurns on Socks, Journal of Finance, 46, Harvey, A.C., and N. Shephard (1996), Esimaion of an Asymmeric Sochasic Volailiy Model for Asse Reurns, Journal of Business and Economic Saisics, 14, Hull, J., and A. Whie (1987), The Pricing of Opions on Asses wih Sochasic Volailiy, Journal of Finance, 42, Jacquier, E., N.G. Polson and P.E. Rossi (1994), Bayesian Analysis of Sochasic Volailiy Models, Journal of Business and Economic Saisics, 12, Jacquier, E., N.G. Polson and P.E. Rossi (2004), Bayesian Analysis of Sochasic Volailiy Models wih Fa-ails and Correlaed Errors, o appear in Journal of Economerics. 19

20 Kim, S., N. Shephard and S. Chib (1998), Sochasic Volailiy: Likelihood Inference and Comparison wih ARCH Models, Review of Economic Sudies, 65, Kiagawa, G. (1987), Non-Gaussian Sae-Space Modeling of Nonsaionary Time Series (wih discussion), Journal of he American Saisical Associaion, 82, Li, W.K., S. Ling and M. McAleer (2002), Recen Theoreical Resuls for Time Series Models wih GARCH Errors, Journal of Economic Surveys, 16, Reprined in M. McAleer and L. Oxley (eds.), Conribuions o Financial Economerics: Theoreical and Pracical Issues, Blackwell, Oxford, 2002, pp Ling, S. and M. McAleer (2000), Tesing GARCH Versus EGARCH, in W.-S. Chan, W.K. Li and H. Tong (eds.), Saisics and Finance: An Inerface, Imperial College Press, London, pp Ling, S. and M. McAleer (2002), Saionariy and he Exisence of Momens of a Family of GARCH Processes, Journal of Economerics, 106, McAleer, M. (1995), The Significance of Tesing Empirical Non-nesed Models, Journal of Economerics, 67, McAleer, M., F. Chan and D. Marinova (2002), An Economeric Analysis of Asymmeric Volailiy: Theory and Applicaion o Paens, paper presened o he Ausralasian Meeing of he Economeric Sociey, Brisbane, July 2002; o appear in Journal of Economerics. Nelson, D.B. (1991), Condiional Heeroskedasiciy in Asse Reurns: A New Approach, Economerica, 59, Pi, M.K., and N. Shephard (1999), Filering via Simulaion: Auxiliary Paricle Filers, Journal of he American Saisical Associaion, 94, Sandmann, G., and S.J. Koopman (1998), Esimaion of Sochasic Volailiy Models via Mone Carlo Maximum Likelihood, Journal of Economerics, 87,

21 Shephard, N. (1996), Saisical Aspecs of ARCH and Sochasic Volailiy, in O.E. Barndorff-Nielsen, D.R. Cox and D.V. Hinkley (eds.), Saisical Models in Economerics, Finance and Oher Fields, Chapman & Hall, London, pp Shephard, N., and M.K. Pi (1997), Likelihood Analysis of Non-Gaussian Measuremen Time Series, Biomerika, 84, Taylor, S.J. (1986), Modelling Financial Time Series, Wiley, Chicheser. Waanabe, T. (1999), A Non-linear Filering Approach o Sochasic Volailiy Models wih an Applicaion o Daily Sock Reurns, Journal of Applied Economerics, 14, Waanabe, T., and Y. Omori (2001), A Noe on Muli-move Sampler for Esimaing Non-Gaussian Time Series Models: Commens on Shephard and Pi (1997), Research Paper Series No.25, Faculy of Economics, Tokyo Meropolian Universiy. Wiggins, J.B. (1987), Opion Values Under Sochasic Volailiy: Theory and Empirical Esimaes, Journal of Financial Economics, 19, Yu, J. (2004), On Leverage in a Sochasic Volailiy Model, o appear in Journal of Economerics. 21

22 Table 1: Simulaions for he MCL Esimaor Based on True Models for ( ) Corr ε, ξ = 0.30 (a) Sample means and sandard deviaions of he MCL esimaes Model T φ σ α ρ γ Var( ξ ) Corr. DL (0.018) (0.038) (0.156) (0.119) (0.012) (0.028) (0.123) (0.092) (0.007) (0.017) (0.076) (0.052) TE (0.018) (0.037) (0.152) (0.057) (0.037) (0.075) (0.012) (0.027) (0.121) (0.042) (0.027) (0.057) (0.007) (0.017) (0.074) (0.025) (0.017) (0.033) DLTE (0.018) (0.040) (0.157) (0.236) (0.105) (0.038) (0.133) (0.012) (0.028) (0.125) (0.164) (0.070) (0.027) (0.101) (0.007) (0.017) (0.076) (0.103) (0.044) (0.017) (0.059) Noe: `Corr.' denoes he correlaion coefficien beween ξ and ε given in equaion (4). 22

23 (b) 95% Coverage Probabiliies Model T φ σ α ρ γ DL (0.008) (0.010) (0.012) (0.015) (0.009) (0.012) (0.013) (0.016) (0.009) (0.014) (0.015) (0.018) TE (0.008) (0.007) (0.012) (0.009) (0.009) (0.010) (0.013) (0.010) (0.010) (0.010) (0.014) (0.009) DLTE (0.008) (0.008) (0.011) (0.014) (0.011) (0.007) (0.009) (0.013) (0.013) (0.010) (0.009) (0.011) (0.014) (0.012) (0.010) Noe: The coverage probabiliy is he fracion of imes ha he rue parameer values falls wihin he confidence inerval. Sandard errors are given in parenheses and are compued from he Bernoulli formula given on page

24 Table 2: Simulaions for he MCL Esimaor Based on True Models for ( ) Corr ε, ξ = 0.60 (a) Sample means and sandard deviaions of he MCL esimaes Model T φ σ α ρ γ Var( ξ ) Corr. DL (0.015) (0.037) (0.137) (0.081) (0.010) (0.027) (0.099) (0.059) (0.006) (0.016) (0.066) (0.036) TE (0.013) (0.031) (0.114) (0.052) (0.032) (0.050) (0.008) (0.022) (0.083) (0.034) (0.022) (0.036) (0.005) (0.014) (0.055) (0.023) (0.014) (0.022) DLTE (0.014) (0.039) (0.137) (0.231) (0.098) (0.035) (0.114) (0.009) (0.027) (0.101) (0.159) (0.065) (0.024) (0.084) (0.006) (0.017) (0.067) (0.098) (0.041) (0.015) (0.051) Noe: `Corr.' denoes he correlaion coefficien beween ξ and ε given in equaion (4). 24

25 (b) 95% Coverage Probabiliies Model T φ σ α ρ γ DL (0.008) (0.010) (0.012) (0.015) (0.009) (0.012) (0.013) (0.016) (0.009) (0.014) (0.015) (0.018) TE (0.008) (0.008) (0.010) (0.009) (0.009) (0.009) (0.012) (0.009) (0.010) (0.012) (0.014) (0.010) DLTE (0.009) (0.009) (0.011) (0.015) (0.012) (0.009) (0.010) (0.013) (0.014) (0.010) (0.009) (0.014) (0.014) (0.015) (0.010) Noe: The coverage probabiliy is he fracion of imes ha he rue parameer values falls wihin he confidence inerval. Sandard errors are given in parenheses and are compued from he Bernoulli formula given on page

26 Table 3: Simulaions for he MCL Esimaor Based on True Models (a) True model: DL Model φ σ α ρ γ Var( ξ ) Corr. TE (0.013) (0.028) (0.124) (0.042) (0.028) (0.060) DLTE (0.013) (0.031) (0.124) (0.145) (0.069) (0.029) (0.094) Noe: `Corr.' denoes he correlaion coefficien beween ξ and ε given in he ex. (b) True model: TE Model φ σ α ρ γ Var( ξ ) Corr. DL (0.012) (0.028) (0.124) (0.090) DLTE (0.012) (0.027) (0.121) (0.173) (0.069) (0.027) (0.105) Noe: `Corr.' denoes he correlaion coefficien beween ξ and ε given in he ex. (c) True model: DLTE Model φ σ α ρ γ Var( ξ ) Corr. DL (0.012) (0.028) (0.126) (0.091) TE (0.012) (0.027) (0.124) (0.042) (0.027) (0.058) Noe: `Corr.' denoes he correlaion coefficien beween ξ and ε given in he ex. 26

27 Table 4: Rejecion Frequencies (RF) of Likelihood Raio Tess Corr ε, ξ = 0.30 (a) Rejecion Frequencies for ( ) Null True RF hypohesis model T = 1000 T = 2000 T = 5000 H 0 : γ=0 DL DLTE H 0 : ρ=0 TE DLTE Corr ε, ξ = 0.60 (b) Rejecion Frequencies for ( ) Null True RF hypohesis model T = 1000 T = 2000 T = 5000 H 0 : γ=0 DL DLTE H 0 : ρ=0 TE DLTE Noe: The likelihood is based on he disribuion of log y 2. The nominal significance level is 5% and he corresponding value of he cumulaive disribuion funcion of χ 2 (1) is

28 Table 5: Response Surface Regressions for Biases Dependen Cons. ρ 1/T σ /T 2 S.E. R variable ˆρ ρ ˆ (-2.19) (8.47) [0.053] [0.000] σ σ (2.97) (-11.69) (-2.80) (3.26) [0.018] [0.000] [0.023] [0.011] Noe: -values are given in parenheses and p-values based on he -disribuion are given in brackes. Table 6: Simulaions for he Bias Correced MCL Esimaors Based on True Models for ( ) Corr ε, ξ = 0.30 and T = 2000 Sample means and sandard deviaions of he bias correced MCL esimaes 95% Coverage Probabiliies Model σ ρ Var( ξ ) Corr. σ ρ DL (0.025) (0.073) TE (0.024) (0.024) (0.053) DLTE (0.025) (0.144) (0.026) (0.081) Noe: `Corr.' denoes he correlaion coefficien beween ξ and given in equaion (4). The coverage probabiliy is he fracion of imes ha he rue parameer values falls wihin he confidence inerval. Sandard errors are given in parenheses and are compued from he Bernoulli formula given on page 11. ε 28

29 Table 7: MCL Esimaes for S&P 500 Reurns Model φ σ α ρ γ LogLike Corr. SV (0.0083) (0.0248) (0.1060) DL (0.0096) (0.0279) (0.0893) (0.0599) (0.06) TE (0.0090) (0.0258) (0.0993) (0.0329) (0.04) DLTE (0.0096) (0.0291) (0.0903) (0.0871) (0.0452) (0.06) Noe: Sandard errors are given in parenheses. Table 8: MCL Esimaes for TOPIX Reurns Model φ σ α ρ γ LogLike Corr. SV (0.0111) (0.0281) (0.1143) DL (0.0092) (0.0260) (0.1186) (0.0623) (0.06) TE (0.0087) (0.0245) (0.1190) (0.0333) (0.05) DLTE (0.0087) (0.0266) (0.6516) (0.1061) (0.0486) (0.07) Noe: Sandard errors are given in parenheses. 29

30 Table 9: MCL Esimaes for USD/AUD Reurns Model φ σ α ρ γ LogLike Corr. SV (0.0232) (0.0421) (0.0579) DL (0.0238) (0.0423) (0.0596) (0.0533) (0.05) TE (0.0238) (0.0423) (0.0596) (0.0387) (0.04) DLTE (0.0241) (0.0449) (0.0587) (0.0715) (0.0584) (0.05) Noe: Sandard errors are given in parenheses. Table 10: MCL Esimaes for YEN/USD Reurns Model φ σ α ρ γ LogLike Corr. SV (0.0153) (0.0351) (0.0953) DL (0.0173) (0.0379) (0.0875) (0.0800) (0.08) TE (0.0166) (0.0367) (0.0923) (0.0373) (0.05) DLTE (0.0168) (0.0395) (0.0874) (0.1203) (0.0624) (0.07) Noe: Sandard errors are given in parenheses. 30

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