Bayesian Inference of the GARCH model with Rational Errors

Transcription

1 0 Inernaonal Conference on Economcs, Busness and Markeng Managemen IPEDR vol.9 (0) (0) IACSIT Press, Sngapore Bayesan Inference of he GARCH model wh Raonal Errors Tesuya Takash + and Tng Tng Chen Hroshma Unversy of Economcs, Hroshma Japan Faculy of Inegraed Ars and Scences, Hroshma Unversy, Hgash-Hroshma , Japan Absrac. We propose o use Padé approxmans whch consruc raonal funcons for he error erms of he GARCH model. Frs n order o perform he Bayesan nference we develop a Markov chan Mone Carlo mehod for he GARCH model wh he raonal errors (GARCH-RE). The Markov chan Mone Carlo mehod s performed by he Meropols-Hasngs algorhm wh he Suden s -dsrbuon. We confrm ha he Meropols-Hasngs algorhm wh he Suden s -dsrbuon s an effcen mehod for he Bayesan nference of he GARCH-RE model. We hen emprcally analyze he GARCH-RE model wh USD/JPY exchange rae reurn daa and fnd ha he GARCH-RE model s superor o he GARCH model wh normal errors. Thus he GARCH-RE model s consdered o be an alernave GARCH-ype model for fnancal me seres analyss. Keywords: Tme seres analyss, Markov Chan Mone Carlo, Bayesan nference, GARCH model, Pade approxmans. Inroducon In fnancal applcaons volaly plays a cenral role for rsk measuremen of fnancal asses. I has been recognzed ha volaly of asse reurns changes hrough me. Such me-varyng volaly s well modelled by he GARCH model[]. In he GARCH model asse reurns r are expressed as r = σ ε, where σ s he me-changng volaly whch s a funcon of pas reurns and volales. ε s an ndependen and dencally dsrbued random varable wh mean 0 and varance. In he orgnal GARCH model he normal dsrbuon was used for ε errors. I s well known ha he uncondonal reurn dsrbuons show faer als han he normal dsrbuons. Alhough he GARCH model wh he normal errors (GARCH-N) also shows he fa al dsrbuons, s recognzed ha he model does no suffcenly accoun for he lepokuross of he fnancal reurn daa. One way o crcumven he problem s he nroducon of fa al dsrbuons for ε errors such as suden - dsrbuon[]. Usng fa-aled dsrbuons for ε errors he non-normaly of he model can be mproved. However here s no consensus for he dsrbuonal assumpon of ε errors. One could also choose oher fa-aled dsrbuons. In hs paper we apply Padé approxmans whch conss of raonal funcons for ε errors. The Pade approxmans are flexble o approxmae a funcon n a ceran doman. In Ref.[3] Padé approxmans are used o descrbe he neres rae reurn dsrbuons. The emprcal neres rae reurn dsrbuons a als are usually faer han he normal dsrbuon. Ths fa-aled naure s successfully descrbed by a raonal funcon from he Padé approxmans. Here we apply he Padé approxmans for he error dsrbuon of he GARCH model and emprcally nvesgae wheher such a model performs beer han he sandard GARCH model wh normal errors. + T.Takash. Tel.: E-mal address: -aka@hue.ac.jp 303

2 In order o nfer model parameers from fnancal daa we employ he Markov Chan Mone Carlo (MCMC) mehod based on he Bayesan nference. There have been varous algorhms proposed o mplemen he MCMC mehod for GARCH models. In hs sudy we use he Meropols-Hasngs algorhm[4,5] wh an adapve Suden s -proposal densy whch has been shown o be effecve for GARCH-ype models[6, 7, 8, 9, 0,,]. We develop hs algorhm for he GARACH model wh he raonal errors (GARCH-RE) and hen make an emprcal sudy wh USD/JPY exchange rae reurns by he GARCH-RE model.. GARCH Model The GARCH(l,m) model[] s defned by y = σ ε, () where y corresponds o an asse reurn and l = σ = ω + α y + β σ, () where he GARCH parameers are resrced o ω > 0, α 0 and β 0 o ensure a posve volaly, l m and he saonary condon α + = < β = s also requred. The error erm ε s an ndependen varable generaed by a specfc probably dsrbuon. Our man dea s o use a raonal error dsrbuon for he error ermε. In hs sudy we focus on GARCH(,) model where he volaly process σ s gven by σ = + + ω αy βσ, (3) and we smply denoe he GARCH model. 3. Raonal Error Dsrbuon We use Padé approxmans for he probably dsrbuon of he error erm ε. Padé approxmans M, are gven as a raonal funcon of varable x wh wo polynomals T M and B N, TM PM, N =. (4) B P N N M and N sand for he degrees of he polynomal T M and B N respecvely. Snce P M, N s supposed o be a probably dsrbuon, mus be posve. We furher mpose on he condons ha P M, N akes a maxmum value a he orgn and s symmerc o he x=0 axs,.e. PM, N = PM, N ( x). In Ref.[3] possble normalzable dsrbuons wh fne varances are derved o approxmae he neres rae dsrbuons. The smples normalzed dsrbuon wh unable parameers s gven by a P0,4 =, (5) π 4 ( + ( a + a ) x + a x ) where a and a are unable parameers. The varance of hs P 0,4( x) s gven by -/ a. We se he varance of he error dsrbuon o one. Thus for P ) we also se a =. The fnal form of P wh q a s wren as 4. Bayesan Inference P 0,4( x q x) = π ( + ( q ) x + x ). (6) ( 4 304

3 The GARCH-RE model has 4 parameers α, β, ω and q whch have o be deermned so ha he model maches he gven daa,.e. here fnancal daa. We deermne he parameers by he Bayesan nference performed by he MCMC. Le us consder he poseror densy of he GARCH-RE model wh n fnancal daa denoed by y = (y, y,..., y n ). From he Bayes heorem he poseror densy π(θ y) s gven by π(θ y) L(y θ) π(θ), where θ (θ, θ, θ 3, θ 4 ) = (α, β, ω, q) and L(y θ) sands for he lkelhood funcon of he GARCH-RE model. π(θ) s he pror densy for θ. In hs sudy we assume ha he pror densy π(θ) s consan. The lkelhood funcon of he GARCH model s gven by n q L y θ ) = = 4 πσ ( + ( q ) y / σ + y / σ ). (7) ( 4 Usng he poseror densy π(θ y) he GARCH parameers α, β, ω and he raonal error parameer q are nferred as he expecaon values as where Z s a normalzaon consan, θ = θπ ( θ y dθ, (8) Z ) Z = π ( θ y) dθ. In order o evaluae eq.(8) we employ he MH algorhm[4,5]. In Refs.[6, 7], a mulvarae Suden s - dsrbuon s used as he proposal densy of he MH algorhm for he GARCH parameer esmaon and s shown ha he MH algorhm wh a mulvarae Suden s -dsrbuon ( MH-STD ) gves a good performance for GARCH-ype models[6,7,8,9,0,,]. In hs sudy we also use mulvarae Suden s - dsrbuons for he proposal densy of he MH algorhm and examne he effecveness of he MH-STD algorhm for he GARCH-RE model. 5. Mulvarae Proposal Densy Our MH algorhm uses (p-dmensonal) mulvarae Suden s -dsrbuons gven by ( ν + p) / Γ(( ν + p) / ) / Γ( ν / ) ( θ M ) Σ ( θ M ) g( θ ) = / p / +, (9) de Σ ( νπ ) ν where θ and M are column vecors, θ = (θ,..., θ p ) and M = (M,...,M p ), and M = E(θ ). Σ s he covarance marx defned as νσ = E[( θ M )( θ M ) ]. (0) ν ν s a parameer o une he shape of he Suden s -dsrbuon. Noe ha when ν he Suden s - dsrbuon goes o a Gaussan dsrbuon. Snce he GARCH-RE model has 4 parameers: θ =(θ, θ, θ 3, θ 4 ) = (α, β, ω,q), he dmenson p of he Suden s -dsrbuon s Numercal Tes wh Arfcal Daa In hs secon we nvesgae he effecveness of he MH-STD algorhm for he GARCH model wh he raonal error by usng arfcal GARCH daa generaed wh known parameers. Frs we se he GARCH parameers and he raonal error parameer o α = 0.05, β = 0.9, ω = 0.05 and q =.8, and generae 5000 daa by he GARCH process wh hese parameers. Then we esmae he values of he parameers only from he daa by he Bayesan nference and check wheher our MCMC algorhm correcly reproduces he parameer values or no. We performed he MCMC smulaons as follows. In order o esmae he parameers of he Suden s - dsrbuon frs we make a shor plo run by a sandard Meropols algorhm. We dscarded he frs 5000 samples as burn-n process or hermalzaon and accumulaed 000 daa for esmaon of M and Σ. The esmaed values of M and Σ are subsued o he Suden s -dsrbuon g(θ). Then we swch he MCMC 305

4 algorhm o he MH algorhm wh g(θ) and connue he MCMC smulaons. Snce a he frs sage of he smulaons he parameers of he Suden s -dsrbuon may no be accuraely esmaed we re-calculae M and Σ every 000 updaes and subsue he parameers o g(θ). Thus he parameers of g(θ) are updaed adapvely durng he MCMC smulaon. Afer accumulang daa we evaluae values of he GARCH and raonal error parameers. In order o compare he effecveness of he MH-STD algorhm, we also perform he smulaon by a sandard Meropols algorhm. In Table we ls he resuls of he parameers nferred by he MH-STD algorhm and he Meropols algorhm. We see ha boh algorhms well reproduce he values of he npu parameers whn he sandard devaon. Ths observaon smply means ha boh algorhms worked correcly. However he dfference arses n he auocorrelaon me. The auocorrelaon mes of he MH-STD algorhm are much smaller han hose of he Meropols algorhm, whch ndcaes ha he performance of he MH-STD algorhm s superor o he Meropols algorhm. Table : Values of esmaed parameers and auocorrelaon mes. SD and SE sand for sandard devaon and sascal error respecvely. α β ω q rue MH+STD SD τ.0 ± 0..3 ± 0..3 ± 0..0 ± 0. Meropols SD τ 400 ± ± ± ± 3 7. Emprcal Resuls In hs secon we apply he GARCH-RE model for he fnancal daa obaned from he real fnancal markes and nvesgae how he model f o he fnancal daa. We used USD/JPY exchange rae reurns from 4 Jan. 999 o 9 Dec As n he prevous secon we performed he MCMC smulaons for he GARCH-RE model by he MH-STD algorhm. For comparson we also performed he smulaons for he GARCH model wh normal errors. The parameers deermned by he MCMC smulaons are lsed n Table. To compare he wo models we calculae AIC[3] and DIC[4] crerons whch evaluae he goodness-off of he models. Here AIC and DIC are defned so ha smaller values ndcae beer goodness. We fnd ha boh AIC and DIC favor he GARCH-RE model (See Table ). Table : Values of esmaed parameers, auocorrelaon mes, AIC and DIC. α β ω q GARCH-RE SD τ 6.5 ± ± ± ± 7. 5 AIC DIC GARCH-N AD τ 4.7 ± ± ± 3. 6 AIC DIC

5 8. Concluson We proposed o use a raonal funcon for he error erm of he GARCH model and consruced he GARCH-RE model. To perform he Bayesan nference of he GARCH-RE model we developed he MH algorhm wh he Suden s -dsrbuons and found ha by esng he MH algorhm wh arfcal daa he MH-STD algorhm works well for he GARCH-RE model. We appled he GARCH-RE model for USD/JPY exchange rae reurns and found ha he GARCH-RE model s superor o he GARCH model wh normal errors. Therefore he GARCH-RE model may serve as an alernave model of he GARCH-ype models. I mgh be neresng o see wheher he GARCH-RE model performs well also for oher asse reurns. 9. Acknowledgemens Numercal calculaons n hs work were carred ou a he Yukawa Insue Compuer Facly and he facles of he Insue of Sascal Mahemacs. Ths work was suppored by Gran-n-Ad for Scenfc Research (C) (No.50067). 0. References [] T.Bollerslev, Generalzed Auoregressve Condonal Heeroskedascy. Journal of Economercs 3 (986) [] T. Bollerslev, A condonal heeroskedasc me seres model for speculave prces and raes of reurns, Revew of Economcs and Sascs 69 (987) [3] J.Nuys, I.Plaen, Phenomenology of he erm srucure of neres raes wh Padé Approxmans, Physca A 99 (00) [4] N. Meropols, A.W. Rosenbluh, M.N. Rosenbluh, A.H. Teller, E. Teller, Equaons of Sae Calculaons by Fas Compung Machnes. J. of Chem. Phys. (953) [5] W.K. Hasngs, Mone Carlo Samplng Mehods Usng Markov Chans and Ther Applcaons. Bomerka 57 (970) [6] H.Msu, T.Waanabe, Bayesan analyss of GARCH opon prcng models. J. Japan Sas. Soc. (Japanese Issue) 33 (003) [7] M. Asa, Comparson of MCMC Mehods for Esmang GARCH Models. J. Japan Sas. Soc. 36 (006) 99. [8] T.Takash, An Adapve Markov Chan Mone Carlo Mehod for GARCH Model. Lecure Noes of he Insue for Compuer Scences, Socal Informacs and Telecommuncaons Engneerng. Complex Scences, vol. 5 (009) [9] T.Takash, Bayesan Esmaon of GARCH Model wh an Adapve Proposal Densy. New Advances n Inellgen Decson Technologes, Sudes n Compuaonal Inellgence vol. 99 (009) [0] T. Takash, Bayesan Inference on QGARCH Model Usng he Adapve Consrucon Scheme. Proceedngs of 8h IEEE/ACIS Inernaonal Conference on Compuer and Informaon Scence, (009) [] T.Takash, Markov Chan Mone Carlo on Asymmerc GARCH Model Usng he Adapve Consrucon Scheme Lecure Noes n Compuer Scence, Volume 5754 (009) -. [] T.Takash, Bayesan nference wh an adapve proposal densy for GARCH models, J. Phys.: Conf. Ser. (00) 00. [3] H.Akake, Informaon heory and an exenon of he maxmum lkelhood prncple, nd Inernaonal Symposum on Informaon Theory, Perov, B. N., and Csak, F. (eds.), Akadma Kado, Budapes(973) [4] D.J.Spegelhaler e al., Bayesan Measures of Model Complexy and F, Journal of he Royal Sascal Socey 64(4) (00)