Testing of Markov Assumptions Based on the Dynamic Specification Test

Transcription

1 Aca olechnca Hungarca Vol. 8 o. 3 0 Teng of Markov Aumon Baed on he Dnamc Secfcaon Te Jana Lenčuchová Dearmen of Mahemac Facul of Cvl Engneerng Slovak Unver of Technolog Bralava Radkého Bralava Slovaka lencuchova@mah.k Abrac: An alernave aroach o eng ear agan Markov-wchng e nonear rooed. The man roblem of he clacal eng va he lkelhood rao e ha he e ac doe no have a andard drbuon. Therefore me-conumng mulaon mu be carred ou. Inead of he clacal e we ugge ung Hamlon dnamc ecfcaon e for he vald of Markov aumon. We how ha h new aroach rovde much faer calculaon. Wh he ame dea we calculae he e for remanng non-ear o comare -regme wh 3-regme model. We comare hee wo aroache wh 00 eleced real me ere from econom and fnance. Keword: Markov-wchng model Markov aumon eng ear dnamc ecfcaon e Inroducon Markov-wchng model have acheved a grea exanon n non-ear me ere modeg becaue of her grea decrve roere. The dea ha model arameer can acqure dfferen value. Th deend on he regme or ae he model n. The arameer wchng follow he dnamc behavor of economc and fnancal me ere que well. For nance one regme can exre an exanon and he econd one a receon. Regme change are caued b dramac occaonal and rare even lke war olcal nabl fnancal cre and o on. Such dcree hf n arameer can caue change n he execed value he varance or coeffcen of he model. In 993 Granger [] decrbed a rocedure whch hould be ke n cae of a nonear modeg. We acce he rncle from he ecfc o he more common. So we ar wh a mler ear model and hen afer mananng gven condon we go o more comlex non-ear model. Here are he e for uch modeg: 7

2 J. Lenčuchová Teng of Markov Aumon Baed on he Dnamc Secfcaon Te Chooe an arorae ear model ARq for he examned me ere e he null hohe abou model ear agan non-ear f he null hohe rejeced hen we connue o he nex e 3 emae he arameer of choen non-ear model 4 check he aroraene of he choen non-ear model b dagnoc e 5 modf he model f necear and 6 ue he model for he decron and redcon of he me ere. In he followng econ I brefl ummarze he ba of Markov-wchng model and he clacal eng ear agan Markov-wchng e nonear va mulaon. Then I nroduce he man dea abou he rooed eng whch uggeed a a faer alernave o he old one. In he end I reen m reul n comarng boh aroache. Markov-Swchng Model Markov-wchng model MSW model belong o regme-wchng model whoe regme deermned b an unobervable varable. Th mean ha we canno deermne n whch regme he roce exacl bu onl wh ome robabl. Suoe ha he regme occurrng n me decrbed b a random varable and f we dnguh oble regme he random varable can aan value from he e {... }. We can defne he ochac roce { } whch a equence of random varable. In 989 Hamlon [] rooed o ecf h ochac roce a a fr-order Markov roce. Th mean ha he roce ha o af h roer: j k... j. Thu he regme n me deend onl on he revou regme n me -. Such a decrbed roce called an -ae Markov chan and { } j... are called ranonal robable. The rereen he robabl of change ha he roce n he regme n me - followed b he regme j n me. Therefore hold ha for In h aer we uoe he followng MSW model form: 8

3 Aca olechnca Hungarca Vol. 8 o. 3 0 φ 0 + φ q φ q + ε for q +... T and I decrbed b he ARq model n arcular regme where he h obervaon q a model order q max{ q q... q } T he lengh of me ere and ε he..d. whe noe drbued wh 0. In h cae he MSW model den of condonal on he random varable and he hor of obervaon ha he follow-u form ϕ jx f j θ ex 4 πσ ε σ ε where ϕ j φ φ... φ a vecor of AR coeffcen for regme j 0 j j q j x... q and... he hor of obervaon. σ ε 3 The Clacal Aroach o Teng A we menoned n he nroducon one of he e we hould follow eng ear agan he Markov-wchng e non-ear. Sml we examne he uabl of a non-ear model nead of a ear model. The clacal aroach o uch eng he lkelhood rao e wh a null hohe H 0 : ϕ ϕ agan alernave H : φ φ for a lea one { 0... q} where ϕ φ0 φ... φq a vecor of AR coeffcen for he h regme. The null hohe rereen he ear model agan he alernave hohe of he -regme MSW non-ear model. The lkelhood rao e ac ha he followng form LR L MSW L AR 5 where L MSW and L AR are logarhm of lkelhood funcon he fr one for he uable -regme MSW model and he econd one for he be AR model. One erou roblem are now: he roblem of nuance arameer. We hould realze ha n he cae of he ear model we emae a lower amoun of arameer han n he cae of he MSW model where we have added ranonal robable o he arameer vecor. Hanen [6] roved n 99 ha h e ac 5 ha a non-andard robablc drbuon. Such a drbuon canno be exreed analcall and for calculang crcal value we need o carr ou a mulaon. The mulaon an exermen whch con of 9

4 J. Lenčuchová Teng of Markov Aumon Baed on he Dnamc Secfcaon Te * generang a large number a lea 5000 of arfcal me ere accordng o he model rereenng he null hohe. The nex e o emae he arameer of he be AR model and MSW model for each generaed me ere and o calculae he correondng lkelhood rao ac 5. Thu we ge crcal value and hen we are able o do he eng. I necear o do he mulaon for each me ere and each model order q dncvel. The bg dadvanage of h aroach he comuaon me. I ake hour o calculae a ngle mulaon. Of coure he comuaon me deend on he comuer erformance or he lengh of he me ere. In he la econ ung real daa I how he dfference beween he comuaon me comarng boh aroache o eng. Inead of uch a me conumng e we rooe ung he ewe-tauchen- Whe e [7 0 ] he core funcon and he ecfcaon e for he vald of Markov aumon whch wa rooed n Hamlon [4]. We decrbe he new eng n he nex econ. 4 A ew Aroach o Teng Lnear For h new aroach we need a core funcon he Whe e [] for eral correlaon whch ue condonal momen e rooed b ewe [7] and Tauchen [0] and he dnamc ecfcaon e rooed b Hamlon [4]. 4. Score Funcon The core funcon for he h obervaon defned a he vecor of aral dervaon of he logarhm of he condonal lkelhood funcon wh reec o he arameer vecor θ log f θ h θ 6 θ where he h obervaon and... a hor of obervaon. We have o calculae he core for he whole lengh T of he me ere. Hamlon n [4] derved he core funcon for uch a decrbed MSW model. The core funcon ha he form f α 0 θ f x θ α 7a 30

5 Aca olechnca Hungarca Vol. 8 o for and } { + + f f f α θ α θ α θ x x 7b for 3 T where he arameer vecor con of hee elemen - α θ. The vecor α nclude AR coeffcen for all regme and model redual deron whch... ε σ ϕ ϕ ϕ α. The vecor a vecor of ranonal robable wh omng redundan arameer whch we are able o exre b oher a follow Elemen of he core funcon derved wh reec o ranonal robable have he form ] [ ] [ ] [ j j j f θ 8a for T j and 0 f θ 8b for. The calculaon rocedure and more deal can be found n Hamlon [4]. 4. The Whe Te and he Dnamc Secfcaon Te A menoned above Whe rooed a e for eral correlaon b ung condonal momen e from ewe and Tauchen. For he conrucon of h e we need l vecor c θˆ whch ju a rereenave of examned roere from an ouer roduc of he h core funcon and one-lagged ha h core funcon - ] ˆ ˆ].[ [ θ θ h h. Th e baed on an aumon ha f daa are reall generaed from drbuon 0 θ f where 0 θ he

6 J. Lenčuchová Teng of Markov Aumon Baed on he Dnamc Secfcaon Te vecor of rue arameer hen he follow-u equaon E h θ0 0 mu be afed. Th mean ha f he model correcl ecfed he core funcon h θ 0 canno be redced on he ba of lagged value avalable a me -. So we r o confrm he ndeendence of he core funcon n me and -. l The e ac ha a χ amoc drbuon n uch a cae and he followng form T T ˆ ˆ [ ˆ ] / T / ˆ T c θ T c θ c θ T c θ χ l. 9 Hamlon n [4] ued h eng for MSW model where he derved auocorrelaon e nde and along regme ARCH effec e and he la one we are focung on he e for he vald of Markov aumon. The have he form j j j... 0 j j k j k.... The fr aumon mean ha ranonal robable would no be deenden on he obervable varable and he econd one rereen ml he fr-order Markov roer. In he vecor c θˆ we mu nclude he elemen correondng o he above menoned examned aumon f ˆ θ f φ 0 ˆ θ f ˆ θ f ˆ θ j... j The vecor c θˆ conan - elemen afer omng redundan arameer where he number of regme and hen he e ac ha a χ amoc drbuon. More deal can be found n Hamlon [4]. 4.3 Alcaon Frl we al h aroach o eng ear agan Markov-wchng e non-ear and hen we ue he ame dea for eng remanng non-ear where we comare a -regme model agan a 3-regme model. We are rng o 3

7 Aca olechnca Hungarca Vol. 8 o. 3 0 fnd ou f wo regme are enough for decron or f necear o add anoher one Teng Lnear agan Markov-Swchng Te on-lnear A null hohe rereened b he vald of Markov aumon n h cae. If a -regme model doe no confrm he examned aumon 0 and we rejec he null hohe. Th mean ha a ear model would be beer becaue h me ere doe no how Markov-wchng e non-ear. Frl we mu calculae he e ac 9 for he -regme model and fnd ou -value from χ 4 drbuon. If he -value le han he gven gnfcance level α he null hohe wll be rejeced and we wll conclude ha for h me ere beer o ue a ear model Teng Remanng on-lnear We ue he ame rncle for eng remanng non-ear. So f he revou eng ear doe no rejec he null hohe we can connue and e for remanng non-ear b comarng a -regme wh a 3-regme model. We examne he aroraene of a -regme model agan an alernave hohe abou a 3-regme model. Afer calculang he e ac 9 for he 3-regme model and correondng -value from χ drbuon for he -regme model h alread calculaed we comare hee -value wh he gven gnfcance level α. The followng alernave can occur: on-rejecng he null hohe for a -regme model rejecng null hohe for a 3-regme model. Th mean ha he -regme model arorae. on-rejecng he null hohe for a -regme model non-rejecng he null hohe for a 3-regme model. The model wh a greaer -value from eng he vald of Markov roere he arorae model. In h cae we can check hee reul wh oher creron uch a he BIC Baean Informaon Creron value for boh e of model he beer model ha a lower BIC value redual deron forecang error value reul for eng auocorrelaon and o on. 33

8 J. Lenčuchová Teng of Markov Aumon Baed on he Dnamc Secfcaon Te 5 Comarng he ew Teng wh Clacal Smulaon To uor our heor of new eng we calculae he clacal e va mulaon a decrbed n Secon 3. We choe 00 me ere from he fnancal and economc ecor. The bgge advanage of he new eng rocedure he much horer comuaon me. For nance ee Table. We needed onl 68.5 for he model order q 5 he lengh of he me ere wa 30 wh he new aroach eng ear. On he oher hand he mulaon wa comued n for he ame me ere and he model order. For he alernave e of remanng nonear we needed he ame me ere and he mulaon exermen ook To underand beer wh he mulaon ake o long he reader recommended o reurn back o Secon 3 where he mulaon exermen decrbed n deal. Table Comarng of comuaon me n boh aroache T30 q5 Comuaon me [] Teng ear Remanng non-ear Smulaon ew aroach The man reul are ha he ame concluon are reached b boh aroache. Th mean ha an aroraene of he ear model or non-ear model occurred n 7% of all cae. In he eng of remanng non-ear we obaned he ame concluon n 79% of all cae. A an examle of he e evaluaon we reen he reul of he eng for he Ruan rouble o Euro exchange rae me ere n Table. Table Reul of eng ear and eng remanng non-ear b boh aroache for Rouble/EUR exchange rae -value for Teng ear Teng remanng non-ear Rouble ew aroach Smulaon ew aroach Smulaon q < q <0.00 < q < We can ee ha for q ear wa clamed for he examned me ere b boh aroache. So we do no e remanng non-ear an furher. For q and q3 he ear hohe wa rejeced. Then we connued wh eng and he reul were ha for q he Markov aumon were no confrmed. The 34

9 Aca olechnca Hungarca Vol. 8 o. 3 0 mulaon alo confrmed he uffcenc of he -regme model. In he cae of q3 we can ee ha he vald of he Markov aumon wa confrmed n boh cae. We ake he model wh he hgher -value bu we hould alo uor wh oher creron a we menoned n he la ar of Secon 4. Concluon Our man goal for roong a new aroach of eng ear agan Markovwchng e non-ear wa o reduce he comuaon me becaue wa ver demandng o calculae b mulaon arcularl f more han one me ere anal wa carred ou. Even hough we canno clam ha boh aroache are exacl he ame e or a lea alwa ubuable he new one could be ver helful n avodng all mulaon. There are ll ome oen roblem. Afer hee reul we would lke o follow u and calculae he ower roere of boh aroache and comare hem wh oher e of non-ear e from [8]. Oher nereng dea for furher reearch would be o nvegae effcenc and o dcover how he mehod work f he heorecal model known. ex we hould e he redual and her ndeendence becaue correlaon on own no uffcen. We could ge earl uncorrelaed redual bu he can be deenden. Anoher ue wa menoned b ál Rákoncza n 008 [9] who uggeed ung auo-coula nead of he auocorrelaon funcon becaue he auocorrelaon funcon decrbe onl ear deendence and we need o decrbe non-ear deendence. Acknowledgemen The uor of he gran AVV o. L-0-09 announced. Reference [] Granger C. W. J.: Sraege for Modelg onear Tme-Sere Relaonh. The Economc Record o [] Hamlon J. D.: A ew Aroach o he Economc Anal of onaonar Tme Sere and he Bune Ccle Economerca o [3] Hamlon J. D.: Anal of Tme Sere Subjec o Change n Regme Journal of Economerc o [4] Hamlon J. D.: Secfcaon Teng n Markov-Swchng Tme Sere Model Journal of Economerc o [5] Hamlon J. D.: Tme Sere Anal rnceon Unver re

10 J. Lenčuchová Teng of Markov Aumon Baed on he Dnamc Secfcaon Te [6] Hanen B. E.: The Lkelhood Rao Te under onandard Aumon: Teng he Markov Swchng Model of G Journal of Aled Economerc o [7] ewe W. K.: Maxmum Lkelhood Secfcaon Teng and Condonal Momen Te Economerca o [8] aradak Z. Sagnolo.: ower roere of oner Te for Tme Sere wh Markov Regme Sude n onear Dnamc & Economerc o. 6 Iue 3 Arcle 00 [9] Rakoncza. Márku L. Zemlén A.: Goodne of F for Auo-Coula n Teng he Adequac of Tme Sere Model n roceedng of COMSTAT 008: Inernaonal Conference on Comuaonal Sac - Conrbued aer oro orugal 008 [0] Tauchen G.: Dagnoc Teng and Evaluaon of Maxmum Lkelhood Model Journal of Economerc o [] Whe H.: Secfcaon Teng n Dnamc Model. In Truman F. Bewle edor Advance n Economerc 5 h World Congre Vol. Cambrdge: Cambrdge Unver re