Time varying skewness and kurtosis and a new model

Transcription

1 Time varying skewness and kurosis and a new model Jonahan Dark Deparmen of Economerics and Business Saisics, Monash Universiy, Ausralia Absrac This paper proposes a new model, he Hyperbolic Asymmeric Power ARCH (HYAPARCH) model. This flexible model ness a large number of ARCH models, allowing for long memory in volailiy and asymmeries. Following Hansen (99), he model is generalised o allow for ime varying skewness and kurosis. The finie sample properies of MLE for his class of model are examined. I is found ha ime varying skewness can be reliably esimaed even when asymmeries are presen in he condiional variance. Time varying kurosis however is very difficul o idenify in he presence of ARCH effecs. Applicaion of he HYAPARCH model wih ime varying skewness o a number of equiy markes illusraes he usefulness of he approach. Keywords: Long memory, asymmeries, covariance saionariy, ime varying skewness and kurosis. JEL classificaion: C, C, C5. The auhor hanks M. Silvapulle, R. Brooks and E. Senana for heir assisance

2 . Inroducion Tradiional finance heory ypically assumes a mean variance framework (see for example Sharpe, 964; Linner, 965; Ederingon, 979). This in urn is based on he assumpions of quadraic uiliy and reurn normaliy. However here is considerable heoreical and empirical evidence o refue such assumpions. Quadraic uiliy is highly implausible given ha i implies increasing absolue risk aversion (Arrow, 97). Furher, here is considerable empirical evidence suggesing ha financial marke reurn disribuions are no normal (see for example, Harvey and Siddique, 999; Adesi e al, 004; Premarane and Bera, 00). This paper addresses some of he issues ha are no usually addressed wihin he radiional finance lieraure. In paricular he modelling of financial daa can go beyond he mean variance framework and esimae ime varying skewness reliably using maximum likelihood. I is also found ha i is no easy o model ime varying kurosis wih realisic sample sizes. This is no surprising since ime varying variances will have accouned for much of he long ail behaviour. Recen developmens in opion pricing, porfolio selecion and asse pricing highligh he imporance of skewness (he hird momen) and kurosis (he fourh momen). Ceeris paribus, invesors should prefer asses wih a posiive skew o asses ha are negaively skewed. Kraus and Lizenberger (976), Harvey and Siddique (000b) and Smih (00) develop asse pricing models where skewness is priced. Fang and Lai (997) develop an asse pricing model where kurosis is priced. The opion pricing models of Heson and Nandi (000) and Baes (996) include he hird and fourh

3 momens. Skewness and kurosis are also likely o be imporan for he deerminaion of Value a Risk. Much of he economeric lieraure esimaes ime varying condiional mean and variance equaions. The GARCH class of processes wih condiionally normal innovaions produce excess kurosis in he uncondiional reurn disribuions, however he sandardised residuals commonly suffer from excess kurosis and skewness. To accommodae hese characerisics, condiional densiies like he Suden s (Bollerslev, 987), he generalised exponenial disribuion (Nelson 99) and he skewed Suden s disribuion (Peers, 00) have been employed. These mehods however do no explicily allow for ime variaion in he hird and fourh momens. Hansen (994) was he firs o exend he ARCH framework o esimae ime varying skewness and kurosis. Hansen (99) modelled he condiional skewness and degrees of freedom of a generalised skewed Suden s disribuion by imposing a quadraic law of moion on he condiioning informaion. This work has subsequenly been exended by Harvey and Siddique (999) and ohers who impose alernaive laws of moion, ypically equivalen o GARCH. Despie he wide use of hese models, he asympoic or finie sample properies of he esimaors are no well undersood. The conribuions of his paper are as follows. Firs, he paper inroduces a new model, he Hyperbolic Asymmeric Power ARCH (HYAPARCH) process. This flexible model allows for long memory in volailiy, asymmeries, is covariance saionary and les he daa deermine he power of he heeroscedasic equaion. The HYAPARCH process ness he HYGARCH (Davidson, 004), Fracionally Inegraed Asymmeric Power ARCH (FIAPARCH) (Tse, 998), FIGARCH (Baillie e al,

4 996), APARCH (Ding e al, 99), IGARCH (Engle and Bollerslev, 986) and GARCH (Bollerslev, 986) processes. The proposed model is imporan because i addresses some of he limiaions in previous long memory ARCH models. The FIGARCH model of Baillie e al (996) is no covariance saionary. The Hyperbolic GARCH (HYGARCH) model of Davidson (004) is covariance saionary, however boh he FIGARCH and HYGARCH models fail o allow for asymmeries. The proposed HYAPARCH process herefore augmens he HYGARCH model o allow for asymmeries using he asymmeric Power ARCH (APARCH) approach in Ding e al (99). Second, he proposed model is exended o allow for ime varying skewness and kurosis. Via mone-carlo simulaion, he finie sample properies of MLE for his class of model are examined. The resuls sugges ha ime varying skewness parameers can be well esimaed when employing samples in excess of 000 observaions. In conras, he esimaion of ime varying kurosis is unlikely o produce reliable esimaes in he presence of ARCH effecs. The resuls also indicae ha when esimaing ime varying skewness, he incorrec law of moion may provide esimaes ha appear reasonable. In pracice, boh laws of moion should be fied, and he log likelihood used o idenify he appropriae model. The plan of his paper is as follows. Secion will review he relevan long memory in volailiy lieraure. The secion will hen discuss he lieraure ha exends he ARCH class of processes by esimaing ime varying skewness and kurosis. Secion will presen he proposed HYAPARCH process and is exension o allow for ime varying skewness and kurosis. The secion will close wih a discussion of he esimaion mehod and he condiional densiies employed. Secion 4 will examine he finie sample properies of his class of models. I is shown ha in he presence of ARCH effecs, ime varying skewness can be reliably esimaed. Time varying 4

5 kurosis esimaion however is more difficul. Secion 5 applies he HYAPARCH process wih ime varying skewness o he Nikkei, S&P500 and he FTSE. The resuls suppor he use of fracionally inegraed volailiy processes wih a quadraic law of moion for he ime varying skewness. Secion 6 concludes.. Lieraure Review. Long memory processes The mos common definiion of a long memory process is one where he auocovariance funcion is no absoluely summable (Baillie, 996). Fracionally inegraed processes are long memory processes where he auocovariance funcion decays a he hypergeomeric rae k d- (0<d<0.5). Long memory in volailiy has been found in he Nikkei (Ding and Granger, 996), he S&P500 (Ding e al, 99; Bollerslev and Mikkelsen, 996; Ding and Granger, 996; Granger and Ding, 996; Andersen and Bollerslev, 997b; Lobao and Savin, 998; Liu, 000), he NYSE (Ding e al, 99), he DAX (Ding e al, 99) and a number of currencies (Dacorogna e al, 99; Baillie e al, 996; Andersen and Bollerslev, 997a, 997b, 998). Baillie e al (996) proposed he FIGARCH(p,d,q) process as one way of modelling long memory in volailiy. The FIGARCH (,d,) process can be expressed as ω ( φl)( L) σ β βl ω = + λ( L) ε β d = + ε () 5

6 where ε = zσ, z is an independenly and idenically disribued (i.i.d) process, E( z ) = 0, ( ) Var z =, σ is a ime varying posiive and measureable funcion of he informaion se a ime -, 0< d < and d d( d) d( d)( d) =. ()!! ( L) dl L L... Following Davidson (004) he ampliude (S) is defined as S λ ( ) =. A necessary and sufficien condiion for his class of models o be covariance saionary is S <. The FIGARCH process is no covariance saionary because S λ ( ) = = (Baillie e al, 996). Davidson (004) demonsraes how his arbirary resricion ( S λ ( ) = = ) explains he odd behaviour of he memory of he FIGARCH process. In paricular, Davidson (004) argues ha he lengh of he memory increases as d approaches zero. This is in conras o he convenional inerpreaion, which suggess ha he lengh of he memory increases as d increases. Davidson (004) develops he HYGARCH process as a generalisaion of he FIGARCH process. The HYGARCH process inroduces anoher parameer ς ino he lag polynomial. The HYGARCH(,d,) process can be expressed as d ( ) ( ) ( ) ω φl + ς L σ = + ε β βl. () Provided ha d > 0, he ampliude of he HYGARCH(,d,) process is φ S = ( ς ). (4) β The HYGARCH process herefore does no impose he resricion ha he ampliude of he process is equal o one (i.e. S λ ( ) = ). The HYGARCH process herefore 6

7 overcomes some of he limiaions of he FIGARCH process: i) he HYGARCH process is covariance saionary (as long as ς ); ii) he HYGARCH process is able o model he ampliude (S) and memory (d) separaely (unlike FIGARCH which models he memory subjec o S = ); and iii) he lengh of he memory increases as d increases. The FIGARCH and HYGARCH processes do no capure asymmeries in volailiy. Tse (998) exends he FIGARCH process o he FIAPARCH process using he asymmeric power ARCH approach in Ding e al (99). The FIAPARCH(,d,) process can be expressed as d δ ω ( φl)( L) σ = + ε γε β βl ( ) δ (5) where < γ < and δ > 0.When γ > 0, negaive shocks give rise o higher volailiy han posiive ones. The reverse applies if γ < 0. The FIAPARCH process herefore capures asymmeries in volailiy and les he daa deermine he power of he heeroscedasic equaion. The FIAPARCH process reduces o he FIGARCH process when γ = 0 and δ =.. Time varying skewness and kurosis Skewness and kurosis are likely o be imporan in a number of financial marke applicaions including he pricing of financial asses. Given ha he ARCH class of processes capure ime varying condiional variances, hey also capure ime varying condiional kurosis. Assuming a Gaussian densiy, Engle and Bollerslev (986) derive he condiional kurosis forecass as a funcion of he condiional variance. This is possible given ha he densiy is defined by is firs wo momens. However as 7

8 saed above, he Gaussian densiy is unable o capure he fa ails presen in he uncondiional disribuions of financial marke reurns. The GARCH class of processes have herefore been combined wih a number of disribuions wih ails faer han he normal, for example, he Suden s (Bollerslev, 987). Brooks e al (00) show ha if he errors are independenly disribued as a Suden s disribuion, hen he condiional variance and kurosis are no longer ied ogeher (as is he case wih Gaussian errors). The condiional variance and kurosis may herefore be allowed o vary freely over ime. To creae uncondiional densiies ha are asymmeric, GARCH models wih condiionally symmeric disribuions have been modified o allow for asymmeries in he condiional variance (for example, APARCH (Ding e al, 99) and EGARCH (Nelson, 99)). This approach however is ypically unable o fully capure he asymmeries in he uncondiional disribuion. To accoun for boh excess kurosis and asymmeries, GARCH models wih a large number of alernaive asymmeric condiional densiies have been employed, for example, he asymmeric sable densiy (Liu and Brorsen, 995) and he Gram-Charlier Expansion (Lee and Tse, 99). See Lamber and Lauren (00) for a review of his lieraure. Hansen (994) was he firs o allow for ime varying skewness and kurosis by exending he ARCH framework. Hansen (994) proposed he use of a condiional generalised skewed Suden s disribuion, where he skewness and degrees of freedom parameers were a funcion of condiioning informaion. Specifically le ( ) z ~ SKST 0,, ξ, υ, where SKST represens a skewed Suden s disribuion, ξ is 8

9 he skewness parameer and υ is he degrees of freedom. Hansen (994) proposed he following quadraic laws of moion ( ) ( ) ξ = ρ + ρ ε + ρ ε (6) * ( ) ( ) υ = λ + λ ε + λ ε (7) * where he logisic ransformaion is used o resric he skewness ( ξ ) and degrees of freedom ( υ ) esimaes o be wihin appropriaely defined lower and upper bounds. To illusrae, he condiional degrees of freedom esimaes υ are confined o be wihin upper and lower bounds (U and L respecively) via U L υ = L + (8) + exp * ( υ ) where he esimae υ * is modelled via equaion. The more recen lieraure adops GARCH specificaions for he higher order momens. Brooks e al (00) model he condiional kurosis ( κ ) direcly via 4 ( ) ( ) κ = λ + λ ε σ + λ κ (9) where all coefficiens are consrained o be posiive. Similarly, Harvey and Siddique (999) model he condiional skewness direcly via ( ) ( ) ξ = ρ + ρ ε + ρ ξ (0) where he consrains < ρ, ρ < and < ρ + ρ < are imposed. Table provides deails of he research examining ime varying skewness and kurosis across a number of markes. Time varying higher order momens are ypically modelled using approaches idenical or similar o hose above. For example, Leon e al (00) and Premarane and Bera (00) employ a GARCH specificaion 9

10 wih sandardised residuals as he condiioning variables, Brooks e al (00) also allow for asymmeries in he condiional kurosis via a GJR-GARCH specificaion, Jondeau and Rockinger (000) employ an AR() specificaion wih he reurns as he condiioning variables, Lamber and Lauren (00) employ a GARCH law of moion wih ime varying covariaes. The Table also illusraes ha a large number of condiional disribuions have been employed, for example he Pearson Type IV (Brannas and Nordman, 00; Premarane and Bera, 00) and he enropy densiy (Rockinger and Jondeau, 00). (Inser Table ) The effecs of emporal aggregaion are unclear. Jondeau and Rockinger (000) find he presence of ime varying skewness and kurosis in daily bu no weekly daa. This is consisen wih he empirical fac ha excess kurosis diminishes wih emporal aggregaion (Premarane and Bera, 00). Ohers (Bond and Pael, 00; Hansen, 994; Harvey and Siddique, 999, 000a) find evidence of ime varying higher order momens in weekly and monhly daa, whils Rockinger and Jondeau (00) do no. Much of he above research relies upon he significance of he esimaed skewness and kurosis parameers, along wih diagnosics on he sandardised residuals o draw conclusions abou he presence of ime varying skewness and kurosis. Unforunaely, when employing hese procedures he findings of ime varying higher order momens may be model specific. Brannas and Nordman (00) demonsrae ha he condiional disribuion employed may affec he resuls. They find ha he log-generalised gamma disribuion suppors ime varying skewness and kurosis, whils he Pearson Type IV disribuion does no. Jondeau and Rockinger (000) find ha he mehod used o consrain he ime varying skewness and kurosis esimaes is significan. Esimaes obained using a sequenial quadraic programming algorihm (which places 0

11 direc consrains on he parameers) produce very differen esimaes of ime varying skewness and kurosis han esimaes ha are logisically consrained. Bond and Pael (00) is he only paper ha considers boh he quadraic and GARCH laws of moion. They find ha for a given disribuion, he approaches in equaions and 5 produce very differen conclusions regarding he presence of ime varying skewness.their resuls suppor he quadraic law of moion given ha convergence was no always achieved when employing he GARCH specificaion. The remaining papers only consider one law of moion. This would appear o be a risky modelling sraegy given he very differen dynamics implied by he alernaive specificaions. The suppor for using a model ha capures asymmeries via he condiional disribuion and he condiional variance is also mixed. Harvey and Siddique (999), Premarane and Bera (00) and Peers (00) find ha he inclusion of skewness in he condiional disribuion decreases he significance of he asymmeries in he condiional variance. Jondeau and Rockinger (000) find ha for a number of currencies, here is lile suppor for a model ha incorporaes asymmeries in variance when ime varying skewness is capured via he condiional disribuion. However when modelling a number of equiies, heir resuls suppor he use of a model ha capures asymmeries in he variance and he condiional disribuion. This resul is consisen wih Peers (00) who uses he APARCH model wih a skewed Suden s on he FTSE. In conclusion, he resuls are mixed bu generally suppor he presence of ime varying skewness and kurosis. One mus exercise cauion in he inerpreaion of hese resuls given ha he findings of ime variaion in he higher order momens may depend

12 upon he daa frequency, he condiional disribuion, he law of moion employed and he mehod used o consrain he esimaes. Furher, as highlighed earlier, he asympoic and finie sample properies of he parameers are no very well undersood.. A New Model wih Time varying Skewness and Kurosis This secion proposes a new model ha ness a large number of models wihin he ARCH class of processes. The model is hen generalised o allow for ime varying skewness and kurosis.. HYAPARCH As noed above, he HYGARCH and FIAPARCH processes address limiaions in he FIGARCH process. A model combining he HYGARCH and FIAPARCH processes may herefore be useful. The proposed Hyperbolic Asymmeric Power ARCH (HYAPARCH) process does his by augmening he HYGARCH model o allow for asymmeries via he asymmeric Power ARCH in Ding e al (994) and Tse (998). The HYAPARCH(p,d,q) may be expressed as d ( ) φ( L) ( L) δ ω + ς σ = + ε γε β() β( L) ( ) δ () where ω > 0, 0< d <, < γ <, 0 p β L = β L β L... β L and δ >, ( ) q ( L) = L L... L, where all he roos of β ( L) and ( L) φ φ φ φ q p φ are consrained o be ouside he uni circle. The HYAPARCH (,d,) process may herefore be expressed as d ( ) ( L) ( L) δ ω φ + ς σ = + ε γε β βl ( ) δ. ()

13 Following Hansen (994) he HYAPARCH model is exended o allow for ime varying skewness and kurosis. Le ~ ( 0,,, ) skewed Suden s disribuion and * (). z SKST ξ υ, where SKST represens a ξ = f () * ( ) υ = f (4). where ( ) f denoes a funcion of he informaion se up o ime -, and he logisic. ransformaion is used o resric he ξ and υ esimaes o be wihin appropriaely defined lower and upper bounds. This process herefore ness he HYGARCH, FIAPARCH and FIGARCH processes. The process also ness a number of shor memory processes including he GARCH (Bollerslev, 986), IGARCH (Engle and Bollerslev, 986) and APARCH (Ding e al, 99) processes. Table deails he required resricions on he heeroscedasic equaion. Of course resricions on he skewness and kurosis may also be applied. The alernaive disribuional assumpions employed in his paper are discussed below. (Inser Table ). Esimaion This paper esimaes he proposed HYAPARCH process via Quasi Maximum Likelihood. Baillie e al (996) sugges ha Quasi Maximum Likelihood esimaion of he FIGARCH process assuming condiional normaliy will produce consisen and asympoically normal esimaes, even if he normaliy assumpion is violaed. Under he assumpion ha he random variable ~ ( 0,) can be expressed as L T T = ln z + + = z N, he log-likelihood funcion ( L T ) ( π) ln ( σ ) (5)

14 where T is he number of observaions. There are wo furher jusificaions for he use of densiies ha allow for skewness and kurosis. Firs, if he sandardised residuals are no normal, assuming ha he condiional mean and variance are correcly specified, GARCH esimaes are consisen bu asympoically inefficien, wih he degree of inefficiency increasing wih he degree of deparure from normaliy (Engle and Gonzalez-Rivera, 99). The skewed Suden s disribuion should herefore reduce he excess kurosis and skewness in he sandardised residuals and provide efficiency gains. Second, if he disribuion exhibis excess kurosis, he QML esimaes are consisen bu may be biased in finie samples. Cauion however is necessary given ha he asympoic properies of he esimaes for he long memory ARCH class of processes are no well esablished (Davidson, 004). Furhermore, he QML esimaor wih a Suden s disribuion is inconsisen when he innovaions are skewed (Newey and Seigerwald, 997). In order o capure some of he excess kurosis in he residuals, he Suden s disribuion is employed. If he random variable is ~ ( 0,, ) can be expressed as z ST υ, he log-likelihood L T υ+ υ = T ln Γ ln Γ ln ( ) π υ T z ln ( σ ) + ( + υ) ln + = υ (6) where < υ and Γ is he gamma funcion. The lower he value of υ he faer he ails of he densiy. As υ he Suden s approaches he Gaussian densiy, however values beyond 0 produce a densiy ha is approximaely Gaussian (Hansen, 994). To capure skewness and kurosis, his paper considers he skewed Suden s 4

15 disribuion proposed by Fernandez and Seel (998) and exended by Lamber and Lauren (00). If ~ ( 0,,, ) z SKST ξ υ, he likelihood funcion is L T T = l where = l T υ + υ = ln + ln Γ 0.5ln π ( υ ) ln Γ ξ + ξ ( ) sz + m ξ I s + ln 0.5( + υ ) ln + σ υ (7) wih ξ > 0, υ > and I m if z s = m if z < s (8) where he consans m= m( ξ, υ ) and s s ( ξ, υ ) deviaion of he skewed densiy, represened by = are he mean and sandard m s ( ξυ, ) ( ξυ) υ Γ υ = ξ υ ξ π Γ, = ξ + m ξ (9) Bauwens and Lauren (00) noe ha ξ is he raio of probabiliy masses above and below he mode, so ha ξ can be viewed as a measure of skewness. The ln ξ represens he esimaed skewness measure, if lnξ > 0 (<0), he densiy is righ (lef) skewed. The skewed Suden s densiy is equal o he Suden s densiy when ξ =. See Bauwens and Lauren (00) for furher deails. Furher, noe ha when ξ = ( ξ ), he mean(mode) and he variance(dispersion) are modelled. These erms will be used inerchangeably. 5

16 4. Simulaion As highlighed previously, he asympoic and finie sample properies of he ime varying skewness and kurosis esimaes are no well undersood. The asympoic properies of MLE for he long memory ARCH class of processes are also no well esablished. Baillie e al (996) demonsrae he suiabiliy of QMLE when esimaing univariae FIGARCH processes wih sample sizes of 500 and 000. The finie sample properies of he FIAPARCH and HYGARCH models have no been explored. 4. Experimen design The experimens consis of four Pars. Par A examines he finie sample properies of MLE for GARCH and APARCH models ha allow for ime varying skewness. Par A will herefore examine wheher i is possible o idenify a model ha capures ime varying skewness via he condiional densiy and he condiional variance. Par B examines he finie sample properies of MLE for GARCH models ha allow for ime varying kurosis. Par C will hen examine he implicaions of fiing an incorrec law of moion o he condiional skewness. Par D will examine he finie sample properies of MLE for he HYAPARCH model wih and wihou ime varying skewness. Table summarises he simulaions performed. (Inser Table ) For each experimen 500 replicaions are performed for samples of size n = 500, 000 and Tables 4 o 9 presen he rue parameer values of he DGPs which all assume y = ε. 4 MLE esimaion is performed using he sequenial quadraic programming algorihm of Lawrence and Tis (00) in Ox v.. To ensure ha he 6

17 ime varying skewness and degrees of freedom esimaes are well defined, he logisic ransformaion is employed where 0 < ξ < and < υ < 0. 5 The resuls repor he bias and RMSE for hose replicaions where srong convergence was achieved. The firs se of percenages a he boom of each able repor he percenage of simulaions where srong convergence was achieved. The brackeed percenages repor he percenage of simulaions where srong convergence was achieved and here were no parameer esimaes on he boundaries. For Table 7, he las hree rows represen he average increase in he log likelihood (LL) when he correc law of moion is fied. Coun represens he percenage of imes ha he LL from he correcly specified model is larger han he LL from he incorrecly specified model. (Inser Tables 4 o 9) 4. Discussion Par A ime varying skewness (Tables 4 and 5) The bias and RMSE of he esimaes are reasonable and appear / T consisen. 6 The condiional variance parameers are well esimaed, wih ˆ γ and ˆ δ similar o APARCH simulaions assuming condiional normaliy (resuls available on reques). The firs column of Figure presens kernel densiy esimaes of he simulaed finie sample densiies for ˆρ, ˆρ and ˆρ for experimen, he second column is for experimen 5. 7 (Inser Figure ) A comparison of ˆρ, ˆρ and ˆρ beween he GARCH and quadraic laws of moion reveals wo issues. Firs, for he GARCH law of moion, he finie sample densiies for ˆρ, ˆρ and ˆρ are negaively skewed when n = 500. In fac he kernel densiy esimaes for ˆρ and ˆρ when n = 500 have been rimmed. There is a endency for 7

18 he model o simulaneously decrease ˆρ and ˆρ, wih he correlaion beween hese wo esimaes being approximaely When n = 500, MLE finds i difficul o disinguish beween he conribuion from he consan and a persisen auoregressive componen. This problem is reduced as n increases wih MLE appearing reasonable for sample sizes of 000 observaions or more. In conras he finie sample densiies of ˆρ, ˆρ and ˆρ under he quadraic law of moion are more symmeric. Second, models wih GARCH laws of moion did no achieve convergence beween 5% and 7% of he ime. The quadraic law of moion was also superior in his regard wih srong convergence being achieved 00% of he ime. For all experimens, he inroducion of asymmeries ino he condiional variance slighly increased he bias and RMSE. For he GARCH law of moion, i also decreased he number of simulaions where srong convergence was achieved. When n = 500, approximaely 4% of simulaions in Experimens and 4 produced esimaes of ˆ γ on he upper boundary. This was no he case for larger sample sizes or when employing he quadraic law of moion. Overall, he simulaions sugges ha for n 000, MLE provides reasonable esimaes, even when ime varying skewness is in he condiional variance and he condiional disribuion. Par B ime varying kurosis (Table 6) The resuls for he variance parameers are very similar o hose presened in Tables 4 and 5. To conserve space hey have no been presened. The finie sample densiies for he condiional variance and skewness parameers are unaffeced by he inclusion of ime varying degrees of freedom. Even when n = 5000, he simulaed finie sample densiies for ˆλ, ˆλ and ˆλ are poor. To illusrae, Figure presens kernel densiy esimaes of ˆλ, ˆλ and ˆλ for experimens 8 and 0. 8

19 (Inser Figure ) When he GARCH law of moion is employed (experimens 8 and 9), he significan negaive bias and large RMSEs are consisen wih he negaively skewed disribuions in Figure. On a large number of occasions ˆλ would converge o a large negaive value and ˆ + λ 0 (he correlaion beween ˆλ and ˆλ is 0.9). On hese occasions λ would also be very imprecisely esimaed. In fac, mos of he boundary esimaes were λ on he upper and λ on he lower boundary. The resuls obained using he quadraic law of moion are equally poor, wih an even larger proporion of ˆλ and ˆλ on he lower and upper boundaries. These resuls illusrae ha he esimaion of ime varying kurosis is unlikely o provide reliable esimaes. For his reason, he remaining experimens only consider ime varying skewness. Par C fiing he incorrec law of moion (Table 7) The resuls illusrae he imporance of fiing boh laws of moion, given ha he incorrec law of moion may provide parameer esimaes which appear reasonable. If he DGP has a GARCH law of moion and boh laws of moion are fied; i) he LL correcly idenifies he GARCH law of moion mos of he ime, wih here being subsanial differences beween he LL values; and ii) he parameer esimaes for ˆρ are generally insignifican. In conras, if he DGP has a quadraic law of moion; i) he LL incorrecly idenifies he GARCH law of moion on a number of occasions (paricularly when n = 500); and ii) he parameers for he GARCH law of moion may appear significan. The resuls in he able for experimens 4 and 5 however are misleading because on many occasions ˆρ decreased and ˆρ converged o he lower boundary. (The correlaions beween ˆρ and ˆρ were 0.99 in experimens 4 and 5). For example, in experimen 5 his occurred 5% (n = 500), 8% (n = 9

20 000) and 48% (n = 5000) of he ime. This can be seen in Figure which displays he simulaed finie sample disribuions of ˆρ for experimens 4 and 5. (Inser Figure ) In summary, he resuls illusrae he imporance of fiing boh laws of moion and sugges ha he log likelihood may be used o assis in model selecion. Par D HYAPARCH (Tables 8 and 9) Assuming condiional normaliy, he resuls sugges ha: i) as n increases, he RMSEs decrease a approximaely he appropriae rae, and he simulaed densiies approach normaliy. (See Figure 4 where he kernel densiy esimaes of he simulaed finie sample densiies for ˆd and ˆ ς in experimens 6 and 8 are presened); ii) as d increases, he bias and RMSE on ˆd and ˆ ς decrease. When d is low (say 0.), MLE has greaer difficuly disinguishing beween he conribuion from he long memory componen capured via d, and he shor memory componen capured via ς. When d is overesimaed/(underesimaed) ς ends o be underesimaed/(overesimaed); 8 iii) when n = 500 a large number of simulaions had parameer esimaes (mainly d,φ or β ) on he boundary; iv) he inroducion of φ resuls in an increase in he RMSE of mos esimaes and a decrease in he number of replicaions ha achieved srong convergence wih no parameers on he boundary; v) he simulaed finie sample disribuions of ˆd for experimen 8 are close o he FIGARCH(,d,0) disribuions for d = 0.7 presened by Baillie e al (996). The addiional hree parameers (ς, γ, δ ) do no seem o impair he esimaion of d. In conclusion, he proposed HYAPARCH model seems o be reasonably well esimaed by MLE as long as large samples ( n 000 ) are employed. (Inser Figure 4) 0

21 For he HYAPARCH model wih ime varying skewness, he bias and RMSE are comparable o Table 8, excep for ς in experimen. When d is low he presence of ime varying skewness exacerbaes he difficulies encounered when disinguishing beween he conribuion from d and ς. Like Table 8, mos esimaes on he boundary were φ and β (on he lower boundary for experimen and ) and d (on he upper boundary for experimen ). Tables 4, 5 and 9 illusrae ha as he condiional variance becomes increasingly parameerised, he bias and RMSE of ˆρ, ˆρ and ˆρ increases. This is consisen wih he greaer negaive skew eviden in he simulaed finie sample densiies for ˆρ, ˆρ and ˆρ (kernel densiy esimaes no shown). Neverheless, for large samples he esimaes appear reasonable. Concluding remarks In conclusion he simulaions sugges ha when employing MLE wih samples in excess of 000 observaions; i) he HYAPARCH model wih ime invarian higher order momens can be reasonably well esimaed for low values of d, wih he esimaes improving as d increases; ii) ime varying skewness parameers can be well esimaed even when employing a heavily parameerised model like HYAPARCH. This resul suggess ha MLE is able o idenify asymmeries in he condiional variance and he condiional disribuion. Noneheless, as he model becomes more heavily parameerised, he bias and RMSE of ˆρ, ˆρ and ˆρ increases; iii) he esimaion of ime varying degrees of freedom/kurosis is likely o produce unreliable esimaes even when employing large samples. In paricular, when employing he GARCH law of moion, he degrees of freedom esimaes frequenly aribue oo much weigh o he consan erm ( ˆλ ) a he expense of he auoregressive parameer ( ˆλ ).

22 These general conclusions appear o be consisen wih he empirical resuls in he lieraure. Firs, Leon e al (005) and Harvey and Siddique (999) esimae ime varying skewness using a GARCH law of moion across a number of financial asses. The skewness parameers are quie similar across markes, suggesing ha he parameers for his class of model are well behaved. Their resuls also confirm he seleced parameer values employed in he simulaions. Second, he ime varying kurosis esimaes in Leon e al (005) vary considerably beween asses and appear o reflec he esimaion issues idenified in he simulaions. When he consan erm is small/(large), he auoregressive parameer ends o be quie large/(small). Brooks e al (00) find similar resuls for he S&P500, he FTSE00 and US and UK 0 year bond indices. Third, Bond and Pael (00) on modelling he condiional skewness of 0 lised UK propery companies, prefer he quadraic specificaion over he GARCH specificaion. This is because he GARCH specificaion proved difficul o esimae and did no converge in many insances. This resul is consisen wih he evidence in Tables 4, 6 and Empirical Example 4. Daa This paper employs hree daily daa ses: i) he Nikkei 5 from March 6, 985 o Augus 5, 00; ii) he S&P500 from January, 987 o Augus 0, 00 and iii) he FTSE from January 4, 984 o December, 00. All daa series are obained from IRESS and he choice of sample periods reflecs daa availabiliy. Coninuously

23 compounded reurns are calculaed as he difference beween he log of consecuive prices muliplied by 00. Table 0 presens descripive saisics. The Jarque Bera es clearly rejecs he normaliy of reurns for all hree markes. The uncondiional reurn disribuions in all markes exhibi excess kurosis. The Nikkei appears symmeric, wih he S&P500 and he FTSE exhibiing a negaive skew. (Inser Table 0) 4. Mehodology In he firs sage of analysis, ess for long memory in he reurns and squared reurns are performed using he modified R/S saisic (Lo, 99) and he KPSS saisic. Kwiakowski e al, (99) developed he KPSS saisic o es an I(0) null agains an I() alernaive. This es also has power agains saionary fracionally inegraed alernaives (Lee and Schmid, 996; Lee and Amsler, 997; Giriais e al 00). Specral densiy esimaes of he fracional differencing parameer (d) for he reurns and squared reurns are also obained using he procedure deailed in Robinson (994). 9 The second sage deermines he mos appropriae mean and variance specificaion esimaing he FIGARCH(,d,), FIAPARCH(,d,), HYGARCH(,d,) and HYAPARCH(,d,) processes assuming condiionally Gaussian innovaions. The models are esimaed using he QMLE procedures implemened in Baillie e al (996), where he pre-sample values are se equal o he uncondiional variance esimae wih a runcaion lag of 000 observaions. Likelihood raio ess are used o deermine he mos appropriae volailiy equaion. The seleced model is hen reesimaed using he skewed Suden s disribuion of Lamber and Lauren (00). The final sage exends he seleced models o allow for ime varying skewness. Boh he GARCH and quadraic laws of moion are considered.

24 4. Resuls assuming consan skewness For boh he Nikkei and S&P500 reurns, he R/S and KPSS saisics fail o rejec he null of shor memory a he 5% level of significance. For he FTSE reurns, he R/S saisic fails o rejec he null a 5%, he KPSS saisic fails o rejec he null a %. These resuls are consisen wih he specral esimaes of d which are close o zero for all models. Boh ess on he squared reurns indicae ha all markes exhibi long memory a he 5% level of significance. These resuls are also consisen wih he specral esimaes which range beween 0.0 o 0.6. These preliminary resuls suppor he esimaion of he fracionally inegraed volailiy processes below. (Inser Table ) The mos appropriae models assuming normaliy are: AR()-HYAPARCH(,d,) for he Nikkei; ARMA(,)-HYGARCH(,d,0) for he S&P500; and AR()- FIAPARCH(,d,) for he FTSE. 0 For all models, he parameer esimaes are consisen wih covariance saionariy. The firs row for each asse in Table presens he resuls for he seleced models wih skewed Suden s innovaions. For all models: i) he mean and variance esimaes were insensiive o he use of he Gaussian or he skewed Suden s disribuion; ii) Box Pierce saisics on he sandardised and squared sandardised residuals were saisfacory; iii) Jarque-Bera ess indicae ha he models suffer from non normaliy; and iv) Nyblom (989) ess indicae ha here is some evidence of parameer insabiliy. Cauion should herefore be exercised when inerpreing he resuls. (Inser Table ) For all models, likelihood raio ess suppor he skewed Suden s densiy over he normal densiy: χ () = 58 for he Nikkei, χ () = 4 for he S&P500, and χ () = 5 4

25 for he FTSE. The esimaes of ξ sugges ha all reurn disribuions exhibi a negaive skew. Table 0 indicaes ha his is he case for he S&P500 and he FTSE. For he Nikkei, he small difference beween he posiive skew of 0.0 and ln( ξ ) = seems reasonable. All models provide esimaes of d ha are saisically significan, furher supporing he findings of long memory in volailiy. The esimaes of d for he Nikkei and he FTSE are close o heir specral esimaes. For he S&P500, he QMLE esimae of d is less han he specral esimae and ς >. This resul is consisen wih he simulaion evidence and suggess ha QMLE may be aribuing oo much weigh o ς a he expense of d. Finally, he resuls on he Nikkei and he FTSE are consisen wih Peers (00) and Jondeau and Rockinger (000) who find ha here is validiy in developing models ha employ asymmeries in he condiional variance and he condiional disribuion. 4.4 Time variaion in skewness Table also presens he resuls where he seleced models were augmened o allow for ime varying skewness. The resuls are consisen wih much of he simulaion evidence. Firs, he mean and variance esimaes are insensiive o he inclusion of ime varying skewness. Second, he skewness parameer esimaes are consisen wih he simulaions. For he S&P500 and FTSE, he GARCH law of moion is consisen wih high degrees of persisence. The Nikkei would appear o be aribuing weigh o ˆρ a he expense of ˆρ. The esimaes for he quadraic law of moion are also similar o he simulaions, and appear sable given heir consisency across he hree markes. Furher, he ˆρ, ˆρ and ˆρ esimaes are significan for he quadraic bu no he GARCH law of moion for all models. LL values are also significanly beer for models assuming he quadraic law of moion. 5

26 An examinaion of Figures 5 and 6 reveals ha he differen laws of moion imply very differen behaviour. This resul is consisen wih Bond and Pael (00). The average ime varying skewness esimaes in he final column of Table however are comparable beween models. The resuls herefore suppor he quadraic law of moion when modelling ime varying skewness in all hree markes. (Inser Figures 5 and 6) The resuls also illusrae he imporance of fiing boh laws of moion as well as a model wih consan skewness. If only he GARCH law of moion was fied, likelihood raio ess would sugges ha only he Nikkei had ime varying skewness. These resuls are a odds wih he es for ime varying skewness of Bera and Lee (99), which sugges ha all markes exhibi ime varying skewness. Furher if only he Gaussian densiy and he skewed Suden s wih GARCH laws of moion were esimaed, likelihood raio ess would very srongly suppor he laer model. This is clearly no he case, paricularly wih he S&P500 and he FTSE where he inroducion of ime varying skewness sees no significan improvemen in he LL from he consan skewness model. 5. Conclusion This paper examined he finie sample properies of he MLE of he ARCH class of processes ha allow for ime varying skewness and kurosis. The paper also inroduced he HYAPARCH model, which augmens he HYGARCH model of Davidson (004) o allow for asymmeries. 6

27 Simulaions revealed ha he MLE for he proposed HYAPARCH model performs reasonably well for low values of d wih he esimaes improving as d increases. Time varying skewness can be reliably esimaed even when employing heavily parameerised models like HYAPARCH. The MLE of he GARCH law of moion is no as well behaved as he MLE of he quadraic law of moion. Convergence is no always achieved and he consan and auoregressive parameer esimaes exhibi a negaive skew in small samples. The MLE of ime varying kurosis esimaes is poor, suggesing ha i is difficul o disenangle he ime variaion in he second and fourh momens. Applicaion of he proposed HYAPARCH model wih ime varying skewness o he Nikkei, S&P500 and he FTSE illusraed he usefulness of he approach. By nesing a large number of models in he lieraure, a likelihood raio esing procedure could be implemened o choose he mos parsimonious model. The empirical resuls suppored he use of fracionally inegraed volailiy processes ha model ime varying skewness via he quadraic law of moion. These resuls imply ha invesors and porfolio managers need o pay aenion o he disribuion of reurns. The radiional approaches which employ he mean variance framework may no be suiable and alernaive approaches ha ake ino accoun higher order momens should be considered (eg Kraus and Lizenberger, 976; Fang and Lai, 997). The resuls also sugges ha asse pricing, opion pricing and porfolio selecion needs o be dynamic, allowing for he possibiliy of ime varying skewness. 7

28 Noes The relaionship beween he degrees of freedom (υ ) and kurosis (κ ) is ( 6) ( 4) ( 4) κ = υ υ = + υ. Using a dominance argumen, Baillie e al (996) argue ha he FIGARCH process is sricly saionary. This asserion has been criicised (Kirman and Teyssiere, 000). The sric saionariy of his model will no be enered ino here. Appendix A derives he condiions for covariance saionariy assuming normaliy. Non negaive condiional variance esimaes are imposed via a modificaion o he Bollerslev and Mikkelsen (996) condiions for he FIGARCH process; ω > 0, β ςd φ d, and ς d φ ( d) β( φ β + ςd). The following upper and lower bounds are also imposed; 0 < d, φ, β <, 0 < ς, δ < 5, and < γ <. 4 To avoid sar up problems he firs 7000 realisaions were discarded for each replicaion. ~ ( 0,,, ) z SKST ξ υ was generaed in he following manner. Firs, Ox. was used o generae disribued random variables, v. To generae disribued random variables for he FIGARCH process, Baillie e al (996) scale he innovaions by ( υ ) / υ. This scaling ensures ha he generaed DGP is covariance saionary (see Appendix B). This also applies o he FIAPARCH, HYGARCH and HYAPARCH processes. For consisency, he scaling is also applied o he disribued random variables for he GARCH and APARCH processes. Second, z was calculaed as z z m s * =, where z v ( sξ ) * = if * κ =, z ( v ξ) s = if κ = 0, where ~ binom(, ) κ ϑ wih ϑ ( ξ ) = +. 5 The upper value for υ is se a 0, given ha here is very lile difference beween he normal and Suden s disribuions for υ > 0. The following consrains are 8

29 imposed: < ρ <, < ρ <, 0< ρ < /( < ρ < ) GARCH/(quadraic), and 0 λ 0 < <, < λ <, 0< λ < /( < λ < ) GARCH/(quadraic). 6 To illusrae, he RMSE should decrease by 5000 / 000 when increasing he sample size from 000 o Kernel densiy esimaes were obained using Eviews 4.. A Gaussian densiy was implemened, wih he bandwidh being deermined auomaically via he echnique in Silverman (986) and he number of poins being seleced a To illusrae for samples of size 5000, he correlaion beween d and α is -0.8 for experimen 6, for experimen 7 and -0.0 for experimen 8. Similar resuls hold for he smaller samples. 9 The ess and specral densiy esimaes are performed using Long Memory Modelling version. 0 For he S&P500, Engle and Ng (99) ess indicaed ha FIGARCH and HYGARCH models suffered from asymmeries. HYAPARCH and FIAPARCH esimaes of γ were on he upper boundary, an APARCH(,) esimae of γ was The simulaions revealed ha on a small percenage of occasions, γ would converge o he upper boundary. The Power ARCH approach o modelling asymmeries herefore seems inappropriae for his sample of daa. The Exponenial GARCH approach may be more appropriae, bu will no be examined here. Nyblom (989) parameer sabiliy es saisics available on reques. The es has a null hypohesis of consan skewness and assumes an AR(p) process. The seleced models wih consan skewness and degrees of freedom were used o esimae he es saisics. For he S&P500 he seleced model was re-esimaed wih an AR() process for he mean. The esimaion resuls provided p values of for he Nikkei, for he FTSE and <0.000 for he S&P500. 9

30 Table Research examining ime varying skewness and kurosis Reference Daa Disribuion Skewness & kurosis laws Bond and Pael (00) 0 lised UK propery companies, monhly Suden s Generalised Skewed Suden s (Hansen, 994) of moion i) quadraic ii) GARCH -residual (skewness only) Resuls Quadraic approach suppored ime varying (TV) skewness for a few series. GARCH approach rarely converged. Brannas and Nordman (00) Brooks, Burke and Persand (00) Hansen (994) Harvey and Siddique (999) Harvey and Siddique (000a) Jondeau and Rockinger (000) Lamber and Lauren (00) Leon, Rubio and Serna (00) Premarane and Bera (00) Rockinger and Jondeau (00) NYSE, daily S&P500, FTSE 00, UK and US 0 year bond indices, daily US excess holding yield, monhly; USD/SF weekly 8 sock indices, daily, weekly, monhly US sock marke, annual; Value weighed NYSE, monhly; MSCI world index, monhly. 6 currencies, 5 sock indices, 8 ineres rae series, daily and weekly. DM/USD daily 5 exchange raes and 5 sock indices, daily Pearson Type IV (PIV) Log-generalised gamma (LGG) Suden s Suden s Skewed Suden s GJR-GARCH residual GJR-GARCH sd residuals (kurosis only) Quadraic PIV no TV skewness or kurosis, LGG TV for boh. PIV is preferred disribuion. TV kurosis, no evidence of asymmeries in kurosis. TV skewness and kurosis. Non cenral GARCH residuals TV skewness. Inclusion of condiional skewness affecs he persisence of he condiional variance and causes asymmeries in variance o disappear. disribuion Generalised Skewed Suden s (Hansen, 994) Skewed Suden s Skewed sable Gram-Charlier series expansion of he normal GARCH residuals (skewness only) AR() reurns GARCH model wih ime varying covariaes. GARCH sd residuals Skewness is TV and is imporan when explaining marke risk premia. Daily daa all asses excep shor erm ineres raes generally exhibi TV skewness and kurosis.weekly daa no TV skewness and kurosis. Evidence of asymmeries in variance and TV skewness. Mehod used o consrain esimaes is significan. TV skewness. TV skewness and kurosis. NYSE, daily Pearson Type IV GARCH sd residuals Skewness and kurosis are no TV S&P500, FT 00, Nikkei 5, weekly Key: TV = ime varying, sd residual = sandardised residual Enropy densiy Skew lagged residual & sd residual, Kurosis lagged absolue residual and absolue sd residual TV skewness and kurosis generally provides lile improvemen. 0

31 Table Models nesed wihin he HYAPARCH model Model Resricions HYGARCH δ =, γ = 0 FIAPARCH ς = FIGARCH δ =, γ = 0, ς = APARCH d =, ς < GARCH d =, ς <, δ =, γ = 0 IGARCH d =, ς =, δ =, γ = 0

32 Table Mone Carlo Experimens Experimen Condiional variance Condiional disribuion Skewness Degrees of freedom Table no Par A, GARCH ~ ( 0,,, ), 4 APARCH ~ ( 0,,, ) 5, 6 GARCH ~ ( 0,,, ) 7 APARCH ~ ( 0,,, ) Par B 8 GARCH ~ ( 0,, ) 9 GARCH ~ ( 0,,, ) 0 GARCH ~ ( 0,, ) GARCH ~ ( 0,,, ) z SKST ξ υ GARCH consan 4 z SKST ξ υ GARCH consan 4 z SKST ξ υ quadraic consan 5 z SKST ξ υ quadraic consan 5 z ST υ n/a GARCH 6 z SKST ξ υ GARCH GARCH 6 z ST υ n/a quadraic 6 z SKST ξ υ quadraic quadraic 6 Par C, GARCH ~ ( 0,,, ) z SKST ξ υ DGP: GARCH Fi: quadraic 4, 5 GARCH ~ ( 0,,, ) z SKST ξ υ DGP: quadraic Fi: GARCH consan 7 consan 7 Par D 6 o 9 HYAPARCH ~ ( 0,) 0 HYAPARCH ~ ( 0,,, ), HYAPARCH ~ ( 0,,, ) HYAPARCH ~ ( 0,,, ) Condiional variance equaions: GARCH σ = ω+ αε + βσ z N n/a n/a 8 z SKST ξ υ consan consan 8 z SKST ξ υ GARCH consan 9 z SKST ξ υ quadraic consan 9 δ δ δ = + + APARCH σ ω α( ε γε ) βσ δ d HYAPARCH σ = ω( β) + ( ( βl) ( φl) + ς (( L) ) )( ε γε) Condiional skewness equaions ( ξ is logisically consrained) * * ξ = ρ + ρ ε + ρ ξ GARCH ( ) ( ) * quadraic ξ = ρ+ ρ( ε ) + ρ( ε ) Condiional degrees of freedom equaions ( υ is logisically consrained) * * GARCH υ = λ+ λ( ε ) + λ( υ ) * quadraic υ = λ + λ ( ε ) + λ ( ε ) δ

33 Table 4 Esimaed bias and RMSE of he MLE of GARCH law of moion for ime varying skewness Parameer Sample size Experimen Variance: GARCH Skewness: GARCH Disribuion: Skewed Sudens True Mean RMSE value bias Experimen Variance: GARCH Skewness: GARCH Disribuion: Skewed Sudens True Mean RMSE value bias Experimen Variance: APARCH Skewness: GARCH Disribuion: Skewed Sudens True Mean RMSE value bias Experimen 4 Variance: APARCH Skewness: GARCH Disribuion: Skewed Sudens True Mean RMSE ω α β γ δ υ ρ ρ ρ Convergence % (94%) 97% (96%) 88% (85%) 88% (84%) 000 9% (9%) 97% (97%) 87% (87%) 8% (8%) % (9%) 95% (95%) 84% (84%) 84% (84%) See Table for furher deails on he DGPs value bias

34 Table 5 Esimaed bias and RMSE of he MLE of quadraic law of moion for ime varying skewness Parameer Sample size Experimen 5 Variance: GARCH Skewness: quadraic Disribuion: Skewed Sudens True Mean RMSE value bias Experimen 6 Variance: GARCH Skewness: quadraic Disribuion: Skewed Sudens True Mean RMSE value bias Experimen 7 Variance: APARCH Skewness: quadraic Disribuion: Skewed Sudens True Mean RMSE ω α β γ δ υ ρ ρ ρ Convergence % (00%) 00% (00%) 00% (99%) % (00%) 00% (00%) 00% (00%) % (00%) 00% (00%) 00% (00%) See Table for furher deails on he DGPs. value bias 4