DYNAMIC ECONOMETRIC MODELS VOL. 7 NICOLAUS COPERNICUS UNIVERSITY IN TORUŃ. Jacek Kwiatkowski Nicholas Copernicus University in Toruń

Transcription

1 DYAMIC ECOOMERIC MODELS VOL. 7 ICOLAUS COPERICUS UIVERSIY I ORUŃ Jacek Kwakowsk cholas Coerncus Unvers n oruń A Baesan esmaon and esng of SUR models wh alcaon o Polsh fnancal me-seres. Inroducon One of he basc assumons n he (heorecal) fnance s ha he logarhmc rces of fnancal seres of asses or exchange raes dsla random walk-e behavor. Economerc radon has been o ncororae ARIMA models o caure he dnamcs of economc me seres. However recen emrcal es resuls for fnance seres sugges ha he are ofen rocesses ha have a roo ha s no consan bu s sochasc. hese rocesses are known as sochasc un roo rocess (SUR). One of her moran roer s ha he have a roo ha s me-varng around un herefore he can be saonar or exlosve. Man emrcal resuls on he denfcaon of SUR rocesses are encounered n Lebourne McCabe and remane (996) Granger and Swanson (997) Jones and Marro (999) Solls Lebourne and ewbold (000) Kwakowsk and Osńska (004) Kwakowsk (005a and 005b). he am of hs aer s o resen wh he Baesan esmaon and esng of SUR rocesses where he random arameer follows frs-order saonar auoregressve rocess. Probabl he frs aem o emlo he Baesan nference was resened n Jones and Marro (999). In her aer he have used Granger and Swanson (997) model o derve oseror margnals and summar sascs. hs aer s concerned wh he SUR model nroduced b Lebourne McCabe and remane (996) whch s comuaonall less hs research was suored b Polsh Commee of Scence gran H0B 05 5; e-mal: jkwa@un.orun.l.

2 Jacek Kwakowsk demandng and eas o mlemen. he margnal oserors of arameers and summar sascs can be obaned b Gbbs samler. he aer s organzed as follows. Secon resens he sochasc un roo model as well as s Baesan esmaon and esng.. Secon 3 rovdes an emrcal alcaon o he sock reurns and exchange raes of zlo for weekl samlng frequences. Secon 4 concludes. Deals on he mlemenaon of Gbbs samler for SUR models are rovded n he aendx.. he Model and Baesan nference he SUR (sochasc un roos) rocesses are resened b Lebourne McCabe and remane (996) and Granger and Swanson (997). Consder he followng SUR model: ( ) ε (.) where denoes an observed rocess a me and saonar auoregressve rocess: s a frs-order ( ) η. (.) Parameer s he auoregresson coeffcen whch s a number beween - and. Here ε and η are whe nose rocesses havng zero mean and resecve varances and. We also assume ha ε and η are muuall ndeenden. When 0 0 and 0 follows he random walk rocess. For > 0 and free ( ) we have a rocess wh a un roo n mean called a sochasc un roo rocess. he arameers follow an auoregressve mechansm so he orgnal seres ends o ossess one un roo n he long run bu n sub-erods ma have saonar or exlosve roos. Le assume ha s a rocess wh saonar frs dfferences : ε (.3) ( ) η (.4) where denoes frs dfferences of he observed rocess a me. Values for auoregressve arameer le n he saonar regon;.

3 A Baesan esmaon and esng of SUR... 3 Sochasc rocesses ε ~ (0 ) and η ~ (0 ) are assumed o be ndeenden. Due o he ormal of he unobserved random rocesses and ε we can exress he model n he followng srucure: ~ ( ) ( ( ) ) ~. (.5) For he secal case of (.5) n whch he sochasc un roo rocess for follows an for d rocess we can nference of arameers of neres b ung 0. herefore he samlng dsrbuon s: ( θ ) f ( ) ( ) f ( ) 0 (.6) where (... )' ( 0... )' θ ( )' R R R and (... )' R and f x c w denoes ormal dsrbuon wh denoes number of observaons mean c and varance w. he ror nformaon abou all arameers s refleced b he followng dens: ( θ ) f ( µ ) f ( µ ) f ( a b ) f ( a b ) where f ( x a b) InvGam InvGam (.7) Inv Gam means Inverse Gamma dsrbuon wh shae arameer a and scale arameer b. Snce he arameer s a ar of he model we can assume ha all nformaon abou s ncluded n he lkelhood (Jones and Marro 999; Josova and Phlov 005). For he auoregresson coeffcen he ror dens s runcaed o he saonar regon ( ). Under hs ror srucure (.7) he jon oseror dens of he arameers s: he sascal dsrbuons used n hs aer are resened e.g. n Gelman Carln Sern and Rubn (997).

4 4 Jacek Kwakowsk 0 ( ) f ( ) ( θ ) f ( ) f ( µ ) f ( µ ) f ( a b ) f ( a b ). InvGam InvGam (.8) In order o oban he oseror margnals and summar sascs for hem we could emlo Gbbs samler algorhm. For SUR model s que eas because he roer ror denses (.7) leads o sandard condonal oserors. he deals of Gbbs samler for SUR model are ncluded n he aendx. In he Baesan aroach o comarng models s consdered useful o emlo robables o reresen degree of belef assocaed wh alernave models. For he SUR model we can es wheher he random rocess follows he frs-order auoregressve rocess or he whe nose rocess. We can also es wheher he daa can be consdered as generaed b he SUR or he exac un roos rocess. Usng Baes s heorem he oseror odds rao for hs roblem s gven b: where ( M ) ( M ) j M and ( M ) ( M ) j ( M ) ( M ) j M j are he wo models we are comarng. Assgnng equal ror model robables ( M ) ( M ) 0. 5 be summarzed b he Baes facor: B j ( M ) ( M ) j j comarson of he models can. (.9) If hs rao s larger hen one we can sa ha he daa suors model M over model M j. he raccal dffcul n mlemenng oseror odds rao s he. For he SUR model comuaon of he margnal daa dens value hs negral s no analcall racable. One of smle numercal aroaches s o consder ewon and Rafer s (994) harmonc mean esmaor: ( M ) K ( k M ) K k M θ (.0)

5 A Baesan esmaon and esng of SUR... 5 where he ( k θ ) are drawn from he oseror usng he Markov chan Mone Carlo (MCMC) mehods. hs esmaor s eas o mlemen bu can be que unsable because fals o obe he Gaussan cenral lm heorem (Carln and Lous 000). Alhough for man alcaons he ewon and Rafer (-R) esmaor s sable enough and close o he rue value of margnal daa dens (Osewalsk and Peń 004). In he SUR case he -R esmaor s unsable because he small condonal lkelhood values overl nfluence he harmonc mean values. herefore we can onl es SUR rocess wh random arameer whch follows whe nose. In ha case we can negrae ou analcall he dens (.6) wh resec o. he condonal dsrbuon of a me s ormal wh mean and varance. Smlar aroach s used b cholls and Qunn (98) for he lkelhood of RCA models. In order o es auoregresson of he random arameer we can use less formal aroach namel he hghes robabl dens (HPD) nerval. hs nerval conans all a oseror mos lkel values of. 3. Alcaon o Polsh fnancal me-seres We al he SUR model o weekl reurns on sock and sock ndexes lsed a he Sock Exchange n Warsaw. We also esmae he SUR model o weekl exchange raes of foregn currences n zlos. he weekl sock and exchange raes reurns cover almos 5-ear samle erods from Januar 000 unl Seember 005. I gves aroxmael 9 observaon. We use he logarhmc ransformaons of he orgnal seres ln. P comued as P Dffused bu roer jon ror dsrbuons reflecs he lack of nformaon abou arameers. Values for auoregressve arameer le n he saonar regon beween - and. Hence for hese arameers we selec a runcaed- ormal ror wh mean 0 and large varance equal o 0. For he varance and we use an Inverse Gamma ror wh shae and scale arameers equal o 0.0. For he uncondonal mean arameer ormal ror wh mean 0 and varance equal o s seleced. Jon ror srucure s exressed b equaon (.7). All models have equal ror robables. We al he Baesan mehodolog for wo muuall exclusve and ndeenden models from each oher: RW : ε W : ε. η

6 6 Jacek Kwakowsk able. Decmal logs of Baes facor n favor of random walk aroxmaed b ewon Rafer esmaor for ndexes WIG WIG0 MIDWIG ECHWIG and for sock reurns. Weekl reurns Random walk SUR wh W Rank log 0 ( B RWRW ) Rank log 0 ( B RWW ) WIG WIG MIDWIG ECHWIG AMAOR BRE BZWBK DEBICA HADLOWY MIESZKO MILLEIUM OPIMUS PROCHIK PSA WAWEL oes: Column headed Rank conans he rank of he resecve models accordng o Baes facor. he Gbbs samler for he Baesan analss of he SUR model s resened n aendx. he logarhms of he Baes facor n favor of random walk comued b ewon-rafer for he sock and ndexes reurns are gven n able. able conans logs of Baes facor n favor of random walk for weekl exchange raes. In order o rovde necessar level of accurac of ewon Rafer esmaor we smulaed draws. Boh ables also show he rankng obaned usng hs aroxmaon. he resuls n ables and show ha here s no subsanal evdence for he resence of sochasc un roo. oce ha onl for he hree sock reurns namel MIESZKO. MILLEIUM and OPIMUS Baes facor suors SUR model over random walk. he resuls n able sugges ha s oor evdence of SUR o weekl exchange rae reurns. Afer esmang he SUR model urns ou ha random un roo model s no ver oular for seleced fnancal seres. In order o examne auoregressve behavor of random arameer we have o analze oseror dsrbuon of arameer. he oseror quanle nformaon and oher characerscs are summarzed n able 3. In he case of

7 A Baesan esmaon and esng of SUR... 7 hese hree seres here s no evdence ha random arameer follows auoregressve rocess. able. Decmal logs of Baes facor n favor of random walk. aroxmaed b ewon Rafer esmaor for exchange raes: Ausralan dollar (AUD) Canadan dollar (CAD) Swss franc (CHF) Czech koruna (CZK) Dansh crone (DKK) Euro (EUR) Pound serlng (GBP) Jaanese en (JPY) and US dollar (USD). Weekl reurns Random walk SUR wh W Rank log Rank log 0 B RWRW 0 B RWW AUD CAD CHF CZK DKK EUR GBP JPY USD oes: Column headed Rank conans he rank of he resecve models accordng o Baes facor. able 3. Poseror summares for auoregresson arameer MIESZKO MILLEIUM and OPIMUS calculaed for Poseror quanle Seres P( > 0 ) Poseror Mean Sandard devaon MIESZKO MILLEIUM OPIMUS able 4 resens he oseror means and sandard devaons (n arenhess) for SUR arameers and calculaed for MIESZKO MILLEIUM and OPIMUS where random arameer follows d.

8 8 Jacek Kwakowsk able 4. Poseror means and sandard devaons n (arenheses) of he coeffcen esmaes of SUR models wh random arameer whch follows d. rocess. Parameers Seres MIESZKO MILLEIUM OPIMUS (0.006) (0.45) 0.00 (0.0847) 0.00 (0.000) (0.0003) (0.000) 0.00 (0.0005) 0.00 (0.0005) (0.0005) 4. Concluson he aer resens a Baesan esmaon of he sochasc un roo model where random arameer follows whe nose or frs-order auoregressve rocess. he resuls se ou n ables and demonsrae ha he SUR model does no mrove uon a random walk model eher for weekl reurns on sock or exchange raes. Onl hree of wen four seres exhb random un roo behavor. References Box G.E.P. Jenkns G.M. (976) me Seres Analss: Forecasng and Conrol San Francsco Holden-Da. Carln B.P. Lous.A. (000) Baes and Emrcal Baes Mehods for Daa Analss ew York Chaman & Hall/CRC. Gelman A. Carln J. Sern H. Rubn D. (997) Baesan Daa Analss. London Chaman & Hall. Granger C.W.J. Swanson.R. (997) An Inroducon o Sochasc Un roo Process Journal of Economercs Jones C.R. Marro J.M. (999) A Baesan analss of sochasc un roo models Baesan Sascs Josova G. Phlov A. (005) Baesan analss of sochasc beas. Journal of Fnancal and Quanave Analss Kwakowsk J. (005a) Maxmum lkelhood esmaon of sochasc un roo models wh GARCH dsurbances. Forecasng Fnancal Markes heor and Alcaons Lodz

9 A Baesan esmaon and esng of SUR... 9 Kwakowsk J. (005b) Baesan analss of SUR models workng aer verson. Kwakowsk J. Osńska M. (005) Forecasng SUR rocesses. A comarson o hreshold and GARCH models. Aca Unversas Lodzenss Fola Oeconomca 90 Lodz Lebourne S.J. McCabe B.P.M. Mlls.C. (996) Randomzed un roo rocesses for modellng and forecasng fnancal me seres: heor and alcaons Journal of Forecasng Lebourne S.J. McCabe B.P.M. remane A.R. (996) Can economc me seres be dfferenced o saonar? Journal of Busness and Economc Sascs cholls D.F. Qunn B.G. (98) Random Coeffcen Auoregressve Models: An Inroducon ew York Srnger-Verlag. ewon M.A. Rafer A.E. (994) Aroxmae Baesan nference b he weghed lkelhood boosra (wh dscusson). Journal of he Roal Sascal Soce B Osewalsk J. Peń M. (004) Baesan comarson of bvarae ARCH-e models for man exchange raes n Poland Journal of Economercs Solls R. Lebourne S.J. ewbold P. (000) Sochasc un roos modelng of sock rce ndces Aled Fnancal Economcs Aendx he Gbbs samler and oseror denses for SUR model he Gbbs samler s a Markov chan Mone Carlo mehod for drawng from a jon oseror dsrbuon b samlng from he condonal dsrbuon. (Gelman Carln Sern and Rubn 997). I consss of samlng random varaes from Markov chan such ha s saonar dsrbuon s he oseror dsrbuon of he arameer of neres. For our urose SUR model s reresened b equaons (.3)-(.4). o al hs aroach we need all condonal oseror dsrbuons gven arorae ror dsrbuon. B assumng ror ndeendence and sandard dsrbuons (ormal and Inverse Gamma) for all unknown arameers he jon ror dsrbuon s gven b: ( θ ) f ( µ ) f ( µ ) f ( a b ) f ( a b ). InvGam InvGam (A.) Havng defned jon ror dsrbuon all condonal oseror dsrbuons have Inverse Ch-square or ormal dsrbuon. Due o sandard form of all condonals s ver eas o samle from oseror dsrbuon because we can draw drecl from Inverse Ch-square and ormal dsrbuon. Alng

10 Jacek Kwakowsk 0 Baes heorem we can derve followng condonal oseror dsrbuons: 0 a b Inv a f χ (A.) 0 a b Inv a f χ (A.3) 0 µ f (A.4) [ ] 0 µ f (A.5) where s x f Inv ν χ means scaled Inverse Ch-square dsrbuons wh > 0 v degrees of freedom and scale 0 > s. Due o saonar of random rocess condonal oseror dsrbuon of auoregresson coeffcen s runcaed o saonar regon. he full condonal dens for a me s ormal and can be wren as: [ ] f θ (A.6) for... and for he las observaon [ ] f θ. (A.7) hese condonals are smlar o condonal oseror dsrbuons derved b Josova and Phlov (005) for smle regresson lnear model wh random arameer. her model has been used o descrbe he evoluon of sochasc beas for US ndusr orfolos.