A THESIS SUBMITTED TO THE GRADUATE SCHOOL OF APPLIED AND NATURAL SCIENCES OF MIDDLE EAST TECHNICAL UNIVERSITY SERDAR ASLAN

Transcription

1 NONLINEAR ESTIMATION TECHNIQUES APPLIED TO ECONOMETRIC PROBLEMS A THESIS SUBMITTED TO THE GRADUATE SCHOOL OF APPLIED AND NATURAL SCIENCES OF MIDDLE EAST TECHNICAL UNIVERSITY BY SERDAR ASLAN IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF SCIENCE IN ELECTRICAL AND ELECTRONICS ENGINEERING NOVEMBER 004

2 Aroval of Gradua School of Naural and Ald Scncs. Prof. Dr. Canan ÖZGEN Drcor I crfy ha hs hss sasfs all h rqurmns as a hss for h dgr of Masr of Scnc. Prof. Dr. İSMET ERKMEN Had of Darmn Ths s o crfy ha w hav rad hs hss and ha n our onon s fully adqua, n sco and qualy, as a hss for h dgr of Masr of Scnc. Prof. Dr. Krm DEMİRBAŞ Survsor Eamnng Comm Mmbrs Prof. Dr. Mübccl DEMİREKLER METU,EEE Prof. Dr. Krm DEMİRBAŞ METU,EEE Prof. Dr. Kmal LEBLEBİCİOĞLU METU,EEE Asss. Prof. Dr. Üm Özlal BİLKENT,ECON Asss. Prof. Dr. Çağaay CANDAN METU,EEE

3 I hrby dclar ha all nformaon n hs documn has bn oband and rsnd n accordanc wh acadmc ruls and hcal conduc. I also dclar ha, as rqurd by hs ruls and conduc, I hav fully cd and rfrncd all maral and rsuls ha ar no orgnal o hs wor. Nam, Las Nam : Srdar Aslan Sgnaur :

4 ABSTRACT NONLINEAR ESTIMATION TECHNIQUES APPLIED TO ECONOMETRIC Aslan, Srdar M. Sc., Darmn of Elcrcal and Elcroncs Engnrng Survsor: Prof. Dr. Krm Dmrbaş Novmbr 004, 6 ags Ths hss consdrs h flrng and rdcon roblms of nonlnar nosy conomrc sysms. As a flr/rdcor, h sandard ool Endd Kalman Flr and nw aroachs Dscr Quanzaon Flr and Squnal Imoranc Rsamlng Flr ar usd. Th algorhms ar comard by usng Mon Carlo Smulaon chnqu. Th advanags of h nw algorhms ovr Endd Kalman Flr ar shown. Kywords: Dscr Quanzaon Flr, Squnal Imoranc Rsamlng Flr, Endd Kalman Flr, Sochasc Calculus, Mon Carlo v

5 ÖZ DOĞRUSAL OLMAYAN KESTİRME ALGORİTMALARININ EKONOMETRİK PROBLEMLERE UYGULANMASI Aslan, Srdar M. Sc., Darmn of Elcrcal and Elcroncs Engnrng Survsor: Prof. Dr. Krm Dmrbaş Novmbr 004, 6 sayfa Bu z doğrusal olmayan onomr gürülülü ssmlrd gürülüdn arındırma v ahmn yama üzrndr. Flr v öngörücü olara sandar grç olan Glşrlmş Kalman Algorması v yn yalaşımlar olan Zamanda Ayrı Ncmlm Flrs v Ardışı Önm Trarlı Örnlm Flrs ullanıldı. Algormalar brbryl Mon Carlo Smlasyon nğyl arşılaşırıldı. Algormaların Glşrlmş Kalman Algormasından avanajlı yanları gösrld. Anahar Klmlr: Glşrlmş Kalman Flrs, Zamanda Ayrı Ncmlm Flrs, Ardışı Önm Trarlı Örnlm Flrs, Olasılısal Analz, Mon Carlo. v

6 ACKNOWLEDGEMENTS I would l o han o my advsor, Profssor Krm Dmrbaş, for hs asssanc hroughou h hss. My hss was a combnaon of dffrn scnc branchs. In hs rsc I hav an clln fdbac from dffrn darmns such as: Mahmacs, Economcs, and Sascs. In hs rsc, I would l o han Profssor Hayr Körzloğlu for hs suggsons on sochasc calculus. H also drcd m hrough choosng h nonlnar dynamc sysms from h fld of conomcs. I am also graful o Asssan Profssor Üm Özlal for hs varous advcs on h conomc sysms bng usd. Rgardng h Malab cods, I would l o han Mura Tgöz, Hüsyn Yğlr and Srdar Suay for hr nd hl. Mura Tgöz hld o m o summarz h cods. Hüsyn Yğlr and Srdar Suay wro h cods n an alrnav way such ha I could comar my rogram ouu wh hrs. I am also graful o Umu Orgunr for asssng m n h hory of Endd Kalman Flr and n Mon Carlo Smulaons. I would also l o han Asssan Profssor Çağaay Candan and Asaf Bhza Şahn for hr valuabl advcs. Asssan Profssor Candan and Assoca Profssor Sncr Koç also hld m on h numrcal comuaon chnqus of solvng nonlnar quaons. Scal hans go o Prof. Dr. Mübccl Dmrlr for hr hl o connu my masr sudy. v

7 TABLE OF CONTENTS ABSTRACT v ACKNOWLEDGEMENTS v TABLE OF CONTENTS v LIST OF TABLES LIST OF FIGURES. INTRODUCTION. FIRST MODEL FROM CONTINUOUS TIME TO DISCRETE TIME 4. Dscrzaon of h Modl 5 3. STOCHASTIC CALCULUS and APPLICATION TO THE CASE STUDY 7 3. STOCHASTIC DIFFERENTIAL EQUATIONS 8 3. Th Rmann Ingral Th Rmann-Sljs Ingral L P Convrgnc and Convrgnc n Man Squar 3.5 Sml Procsss 3.6 Ouln for h Dfnon of h Io Sochasc Ingral 3.7 Th Io Sochasc Ingral of Sml Procsss 3.8 Th Io Sochasc Ingral of Gnral Procsss Io s Lmma Soluon of h Sochasc Dffrnal Equaon 6 3. Unqunss of h Soluon of h Sochasc Dffrnal Equaon 9 4. DQF ALGORITHM 4. DQF 4.. Obanng a Trlls Dagram 4.. Assgnng Mrcs o h Nods of h Trlls Dagram 4..3 Choosng h Hghs Mrc for h Drmnaon of h Bs Pah 4 4. Quanzaon of h random varabl 5 5. EKF 7 6. APPLICATION OF EKF TO ECONOMETRICS 9 7. PARTICLE FILTERS-SIR ALGORITHM 3 7. Drvaon of h SIS and SIR algorhm SIMULATIONS OF FILTERING AND PREDICTION EXPERIMENTS Gnral Ouln Flrng Tools Prdcon Tools Error Crra and Mon Carlo Smulaons 38 v

8 8.4. Flrng Error Error n R N Sacs: Errors n h Ermns Prdcon Error Smulaons Flrng Prdcon SIMULATIONS OF FILTERING EXPERIMENTS FOR STOCHASTIC GROWTH MODELS CONCLUSION 5. REFERENCES 54. APPENDIX 57 v

9 LIST OF TABLES TABLES Tabl Mon Carlo rsuls-errors of h algorhms vrsus chang of h varanc of w 43 Tabl Mon Carlo rsuls-errors of h algorhms vrsus chang of h varanc of w 44 Tabl 3 Mon Carlo rsuls-errors of h algorhms vrsus chang of h varanc of w 44 Tabl 4 Mon Carlo rsuls-errors of h algorhms for h varanc of w=5 46 Tabl 5 Mon Carlo rsuls-errors of h algorhms SIR and EKF 48 Tabl 6 Mon Carlo rsuls-errors of h algorhms SIR and EKF 5

10 LIST OF FIGURES Fgur... 45

11 CHAPTER INTRODUCTION In lcrcal and lcroncs ngnrng, hr ar many dynamcal sysm rrsnaons, whch nclud nos comonns. In such cass, on of h rmary goals s o rmov h nos comonn from h masurmn and/or ry o rdc h n sa of h sysm. If h roblm a hand can b wrn n a lnar Gaussan sysm form, hn h Kalman Flr mrgs as h omal mhodology. Howvr, whn h sysm has nonlnar characrscs, hn h Endd Kalman Flr EKF hncforh should b mloyd. Howvr, h smaon of nonlnar sysms should by no mans rsrcd o usng only EKF. As dscussd n [3] and [5] n dals, Dscr Quanzaon Flr and Squnal Imoranc Rsamlng SIR flrs can convnnly b usd Basd on h abov dscusson, hr ar hr faurs of hs sudy: Conrol hory aroachs ar ald o h fld of conomrcs: Sochasc Calculus has bn nroducd whn h con of a sochasc Io rocss, whch s usd bascally n h ass rcng modls 3 Two rlavly unnown algorhms, DQF and SIR flr ar nroducd and comard wh h EKF. As wll b clar, som cass whr hs wo alrnavs ar suror o EKF ar rsnd. To bgn wh, sa sac rrsnaons ar convnn o dscrb h dynamcal sysms. I s sraghforward o oban h dynamcs of h modl from h dffrnal quaon rrsnaon of h nonnosy sysms. [4] Howvr, n nosy sysms, h dscron of h sysm s gnrally s nroducd whn h sa sac rrsnaon. In such sysms, h nos s assumd o b gnrally addv. Bu hans o h sochasc calculus, w hav a mor gnral rscv. For a dald dscusson on Kalman Flr s [] and []

12 Th assumons of h nos srucur can b mad drcly n a dffrnal quaon form. Thn, h dffrnal quaons wh nos ar solvd by mloyng sochasc calculus. For a gnral ramn of h subjc sochasc calculus, h radr s rfrrd o [5]. Consqunly, h sudy brfly dscrbd abov s comosd of h followng ars: Inroducon Frs Modl From Connuous Tm o Dscr Tm 3 Sochasc Calculus and Alcaon o h Cas Sudy 4 DQF Algorhm 5 EKF Algorhm 6 Alcaon of EKF o h Economrc Problm 7 Parcl Flrs-SIR Algorhm 8 Smulaons of h Frs modl 9 Smulaons of h Scond Modl Th scond scon lans how h dscr-m nonlnar nosy sysm rrsnaon of h frs modl s oband from connuous m rrsnaon. Th dynamcal sysm s basd on obanng h rsson for h rc of on rsy ass soc. Snc h rmary goal of hs sudy s no o focus on h dals of h drvaon, jus a succnc nroducon s gvn n hs scon. In h hrd scon namd Sochasc Calculus and Alcaon o h Cas Sudy, h ncssary bacground o undrsand h bascs of Sochasc Calculus s land. Thn s shown, how h connuous m soluon has bn oband from h sochasc dffrnal quaons. In h fourh scon, namd DQF Algorhm, h bascs of h algorhms roosd by Dmrbaş [3] ar land. Alhough no wdly nown, hs algorhm s caabl o lad o br rsuls han h sandard EKF algorhm. Th n wo scons brfly dscuss h EKF algorhm. In h formr, EKF for dscr m cass s land. In h lar scon, h algorhm s ald o h nonlnar dynamc sysm of h modl drv n h rvous scons. Th svnh scon laboras h arcl flrs. I gvs h drvaon of h Squnal Imoranc Samlng SIS flr along wh s roof. Thn, h SIR flr, whch s also mloyd n hs sudy, s asly drvd from SIS flr.

13 Th ghh scon dslays h smulaon rsuls of h flrng and rdcon algorhms of h bnchmar modl nroducd bfor. Th n scon rsns h smulaon rsuls of h flrng algorhm of h scond modl, whch bascally amns roducvy shocs and caal accumulaon whn a sochasc growh modl. Th fnal scon concluds. 3

14 CHAPTER FIRST MODEL FROM CONTINUOUS TIME TO DISCRETE TIME In a convnonal ass rcng modl, h rc of on shar of h rsy ass soc s dscrbd by h followng sochasc dffrnal quaon.. dx = c X d + X db. X : Th rc X s calld rsy ass. I s also namd as soc. c : Th consan c s h man ra of rurn. : Th aramr dnos volaly. B : B = B, 0 rrsns h Brownan moon. [6] Th dffrnal formula n quaon. s smlar o h rssons usd n dffrnal calculus wh h moran con ha hr s a rm db, whch rrsns h sochasc ar. Such a rm mas h usag of h sochasc calculus ncssary, whch has also bn wdly usd n h fld of conomrcs. In h hrd scon, a ncssary bacground for a horough undrsandng of h abov mnond subjc s gvn. As h hrd scon shows n dals, h soluon o h abov gvn formula for X s gvn by: X c0.5 +B = f, B = X 0.. s an onnal funcon whch s randomly rurbd. I s also namd as Gomrc Brownan Moon. X 0 s assumd o b ndndn of B. If h 4

15 5 volaly rm 0 =, hn. rducs o h usual drmnsc onnal funcon, whch s gvn by.3: c X X 0 =.3. Dscrzaon of h Modl In hs scon, h connuous m rsson scfd n. s ransformd no h dscr m doman. On basc aroach s o saml. wh unform m nrval T. Thn a rcursv rlaon bwn T X + and X s oband as follows:.5 0 T B T o c X T X = T B T o c X + + = * T B T c c X + = * T B T c c X + = B -B * B T B T c B c X + = * ] [ B T B T c B c X + = * ] [ B T B T c B c X + + = 0.5 ] [ B T B T c X + = 0.5 B T B T c X + = 0.5 B T B T c X + = ] [ 0.5 B T B T c X + = 0.5 T w T c X = Whr T w rrsns Gaussan nos wh varanc T. Hnc.4 can b wrn as.

16 X + T = c0.5 T w T.4 X In ordr o oban.4, h fac ha h random varabls B B s hav a normal dsrbuon N 0, s for s <, s usd n h las quaon. [8] Summarzng h soluon.4, quaon.5 can b oband as: X + T = LX w T.5 whr L = c0.5 T, whch s assumd o b consan. Hnc sarng a X 0 and usng.5 s ossbl o oban h valus of { X T, X T, X 3 T, X T, }. Th noaons X T, X T, X 3 T, X T ar wrn as X, X, X 3, X rscvly. As a rsul.5 has h alrna rrsnaon as: X LX w + =.6 whr w sll hav L = c0.5 T. Snc h n sa of h sysm s calculad accordng o.6, mrgs as h sa quaon of h sysm. Th masurmn quaon, on h ohr hand s gvn by.7: Z = X + v.7 whr v rrsns h sculaors nos or radrs nos. Th abov drvaon and rsnaon of h ass rcng modl s farly sandard and can b convnnly usd o valua h rformancs of svral flrs dsgnd o b usd n non-lnar cass. 6

17 CHAPTER 3 STOCHASTIC CALCULUS and APPLICATION TO THE CASE STUDY Ths scon brfly nroducs sochasc dffrnal calculus. Bfor dscussng h mlcaons should b mnond ha, h radr, who s no famlar wh sochasc rocsss s rfrrd o [9]. Frs, s moran o rmnd h dfnon and h nrraon of sochasc dffrnal quaons. Th ngral ha s wrn n h form of 3. 0 b s, X s db s 3. s calld Io Ingral. Ths ngral rrsnaon dffrs n wo ways from s counrars: W hav dbs nsad of ds. B s no a drmnsc funcon bu a sochasc on. Th formal dfnon of 3. s gvn n h subscon 3.8. Th subscon 3. Th Rmann Ingral rmnds h radr h formal dfnon of Rmann Ingral. Th subscon 3.3 namd Th Rmann-Sljs Ingral s a sml nson of Rmann Ingral. Th rason o nroduc hs ngral y s h conncon bwn h Io Ingral and Rmann-Sljs Ingral Th subscons 3.4 and 3.5, whch ar L Convrgnc and Man Squar Convrgnc and Sml Procsss rscvly, dfn convrgnc ys of sochasc rocsss, whch ar usd n h dfnon of Io ngral, 7

18 Th subscon 3.7, namly h Io Ingral of Sml Procsss frs dfns h Io Ingral for a rsrcd class of rocsss. Thn h subscon 3.8 Io Ingral of Gnral Procsss nlargs h dfnon of h Io Ingral for a wdr class of rocsss. A hs sag, h horcal mlcaon of h Io ngral s concludd wh a brf dscusson. Afr hs scon, raccal ascs of sochasc calculus ar nroducd and ald o conomrcs, whch was land n h scond scon. Th subscon 3.9 Io Lmma s h basc ool, whch s ndd o solv h sochasc dffrnal quaons. Th subscon 3.0 s an alcaon of h Io Lmma o h cas sudy, whch s a sandard sochasc growh modl. Fnally, h subscon 3. dscusss h unqunss roblm of h soluon, whch s oband n h scon STOCHASTIC DIFFERENTIAL EQUATIONS In sochasc calculus, a basc rsson as h followng form: dx = a, X d + b, db 3. whr s h ndndn varabl, a, and b, ar drmnsc funcons, and B = B, 0 s h Brownan moon. I s asy o s ha, for b, = 0, 3. rducs o h usual drmnsc dffrnal quaon. In h sochasc calculus 3., can also b nrrd as: X = X 0 + a s, X s ds + b s, X s db s, 0 T In hs noaon, 0 a s, X s ds s h usual Rmann ngral. Th ngrand funcon b s, X s and h ngrang funcon B s ar sochasc rocsss. 8

19 Ingrals of h form 0 b s, X s db s should b nrrd dffrnly. Io rocss offrs such an nrraon. Snc Io nrraon s h mos commonly usd rocss, wll b mloyd hroughou h sudy. Howvr, should b rmndd ha hr ar also ohr nrraons l h Sraonovch nrraon. In som cass, s br o us Sraonovch nrraon, as rssd n [0] Th n scon Rmann ngral s dfnd formally. 3. Th Rmann Ingral Th classcal ngraon chnqu s h Rmann ngral. Th sandard noaon s shown n 3.4. = T =0 f d 3.4 whr dnos h ndndn varabl. f s h funcon o b ngrad. 0 and T ar h lms of ngraon. Th dfnon of Rmann Ingral s as follows: Frs h nrval [ 0, T ] s arond as: L τ n : 0 = 0 < <... < n < n = T b h mnond aron of h nrval [ 0, T ]. Choos y, y for =,..., n. n 3 L S = f y n = 4 If lm = S S n n ss, for som S and for ach ossbl aron τ n and yas 0, hn S = f d. = T = 0 Whl Rmann ngraon chnqu s usd nsvly n svral aras s no arora for analyzng h dynamcs of h sysms wh random rms. For ha uros, Io s Sochasc Ingral offrs a soluon. 9

20 Th Rmann-Sljs Ingral, whch s a sml nson of Rmann Ingral, s dscussd n h n scon. 3.3 Th Rmann-Sljs Ingral Dfnon [] L τ n : 0 = 0 < <... < n < n = T b a aron of h nrval [0,]. Choos y, y for =,..., n. 3 L f and g b wo ral-valud funcons on [0,] and dfn 3.5 g = g g, =,,n Th Rmann-Sljs sum corrsondng o τ n and s gvn by 3.6 n S n = S n τ n = f y g = f y [ g g ]. = n = If h lm lm S n ss and S s ndndn of h choc of h n arons τ n and yas 0, hn S s calld h Rmann-Sljs ngral of f wh rsc o g on [0,]. Symbolcally, s wrn as 3.7. S = 0 f dg. 3.7 Rmann-Sljs ngral s jus an nson of h classcal Rmann ngral. I s rducd o h Rmann ngral for g=. Ths ngraon chnqu s also usd n h robably hory. In hs cass g s o b chosn as F, whch s h dsrbuon funcon of a random varabl. For aml, 3.8 rrsns such a form: P { A} = df 3.8 A 0

21 Alhough Io s sochasc ngral s dffrn from h Rmann-Sljs ngral, as h rocdng scons wll show, hr ar conncons as wll. Ths conncons ar shown afr h dfnon of h Io sochasc ngral. In h n scon, som mahmacal rms, whch ar ncssary o undrsand h dfnon of h Io ngral, ar nroducd. 3.4 L P Convrgnc and Convrgnc n Man Squar If X n and X blong o L and X n X 0 E, hn X } s sad o { n convrg n h man o X. For =, s smly calld convrgnc n man; for =, on h ohr hand, h rocss s gnrally rmd as convrgnc n man squar or n quadrac man.[] Ths rocss s mloyd for h dfnon of Io Ingral. In h abov dfnon, E s h usual caons oraor whl sac of funcons, for whch h L norm s fn. L s h

22 3.5 Sml Procsss Th sochasc rocss C = C, [0, T] s sad o b sml rocss f sasfs h followng rors: Thr ss a aron: τ n : 0 = 0 < <... < n < n = Z, =,..., n of random varabls such ha: [3] C = f = T, Z n C = f <, Z =,..., n. T, and a squnc 3.6 Ouln for h Dfnon of h Io Sochasc Ingral As mnond bfor, h Io Ingral mloys h man squar convrgnc. For h dfnon of h Io sochasc ngral, h followng way wll b racd:. Th ngrand funcon s frs assumd o b a sml rocss.. Th Io ngral of h sml rocsss s gvn. 3. Th ngrand funcon s assumd o b a gnral rocss. 4. Th Io ngral of h gnral rocss s dfnd va h sml rocsss. 3.7 Th Io Sochasc Ingral of Sml Procsss Th Io sochasc ngral of a sml rocss C on [ 0, T ] s gvn by 3.9. T n C s db s : = C B B = = = 0 n Z B 3.9 whr Z C and B = B B. =

23 Th Io sochasc ngral of a sml rocss C on [0,],, s gvn by T C s db s : = C s I[0, ] s db s = Z B + Z B B 0 = 3.0 whr, s I for s [ 0,] [ 0, ] = and I s 0 ohrws. [ 0, ] = Whn comarng 3.9 wh h Rmann-Sljs sum noaon 3.6, s asly sn ha y =. Furhrmor, g corrsonds o Brownan moon. Hnc h Io sochasc ngral s h Rmann-Sljs sum, whr h nrmda ons y ar chosn a h lf nd ons of h nrvals, ]. 0 = Z B = 0 [ Ths concluds h dfnon of h Io ngral for sml rocsss, whch can b rgardd as h cornr son for h gnral sochasc ngraon. 3.8 Th Io Sochasc Ingral of Gnral Procsss Th Io sochasc ngral s dfnd for sml rocsss n scon 3.7. In hs scon, howvr, h ngrand rocss wll b from a wdr class of funcon sac. Bu h followng rsrcons ar u for hs class: Assumons on h Ingrand Procss C:. C s adad o Brownan moon on [0,T],.. C s a funcon of Bs s.. Th ngral 3. s fn. T 0 EC s ds 3. L C b a rocss sasfyng h abov addonal assumons. Thn, on can fnd a squnc C n of sml rocsss such ha h followng samn holds. S [4], for h roof of hs samn. 3

24 T 0 E[ C s C s n ] ds 0 3. Hnc, h sml rocsss n C convrg n man squar sns o h n ngrand rocss C. Usng 3.0, w can calcula h Io ngrals I C of hs sml rocsss. Hnc w hav a nw squnc { I C, C, }. Ths nw squnc convrgs o a unqu rocss, whch van b convnnly calld as I C on [0,T] Thn, w can wr 3.3 as[5]: I n E su[ I C I C ] T Th man squar lm I C s calld h Io sochasc ngral of C. I s dnod by 3.4[4] I C = C s db s, [ 0, T ]. o 3.4 Ths nds h formal dfnon for h Io sochasc ngral. In ordr o aly h ngral, howvr, h Io Lmma should b nroducd. 3.9 Io s Lmma In hs subscon, h Io s Lmma s nroducd n ordr o aly h sochasc ngral ha s dfnd abov. Io s Lmma: L f, b a funcon whos scond ordr aral drvavs ar connuous. Thn 3.5 s ru. f, B f s, B s = [ f, B + f, B ] d + f s s 4, B db 3.5

25 whr s <. For h dals of h lmma, las rfr o [6]. [ f, B + f s, B ] d 3.6 s f, B db 3.7 Th frs ngral 3.6 s a sml drmnsc ngral. Th scond ngral 3.7 s h aformnond sochasc ngral n scon 3.8. Furhrmor f, f and f ar h aral drvavs. 5

26 3.0 Soluon of h Sochasc Dffrnal Equaon In subscon 3., s land ha 3.8 s quvaln o h followng rrsnaon of 3.9. dx = a, X d + b, X db 3.8 X = X 0 + a s, X s ds + b s, X s db s, 0 T Tha s, 3.8 s nrrd as 3.9 n sochasc calculus. As s mnond n h scond char, h rc of on shar of h rsy ass soc s dscrbd by h sochasc dffrnal quaon 3.0. dx = cx d + X db 3.0 whr X : Th rc X s calld rsy ass. I s also calld as soc. c : Th consan c s calld man ra of rurn : Th s calld volaly. B : B = B, 0 s Brownan moon. Whn h quaons 3.0 and 3.8 ar comard, s sn ha 3.0 s a scal cas of 3.8 wh a, X = cx and b, X = X. Hnc 3.0 s quvaln o h followng: X = X 0 + c X s ds + X s db s, 0 T Mor gnrally, h sochasc dffrnal quaons n h followng form of 3., whr c and σ =, ar consans, can b rmd as lnar sochasc dffrnal quaons. Hnc 3.0 s from h famly of lnar sochasc dffrnal quaons. 6

27 c X s + c ds + X s + X = X 0 + σ σ db s, 0 T Th nroducd sochasc ngral n 3. can b solvd accordng o h Io s Lmma, whr h soluon s gvn n [7]. Soluon: Suos X s dscrbd as 3.3. X = f, B = c 0.5 +B 3.3 whr c and > 0 ar consans. I s rovd blow ha 3.3 s ndd a soluon. Accordng o 3.3, quaons 3.4 o 3.7 ar vald. f, B = c 0.5 +B 3.4 f, B = c 0.5 f, 3.5 B f f, B = f, 3.6 B, B = f, 3.7 B whr f, f and f ar agan h aral drvavs wh rsc o frs and scond varabls. L f, b a funcon whos scond ordr aral drvavs ar connuous. Thn 3.8 s ru. f, B f s, B s = [ f, B + f, B ] d + f s s, s <, B db 3.8 7

28 Usng h quaons 3.4 o 3.7, quaon 3.9 s oband. f, B f s, B s = c X y dy + X y db y s s 3.9 If on subsus s = 0 n 3.9, and uss X = f, B, 3.30 s oband, whch s nohng bu 3.. X = X 0 + c X y dy X y db y 3.30 Hnc o summarz, s shown ha c0.5 +B X = s a soluon for 3.. 8

29 3. Unqunss of h Soluon of h Sochasc Dffrnal Equaon In h soluon of h sochasc dffrnal quaon, was rovn ha 3.3 s a soluon for h sochasc ngral quaon 3.3. X c0.5 +B = 3.3 X = X 0 + c X s ds X s db s 3.3 Ths scon rovs ha 3.3 s also h unqu soluon. Th followng Lmma s gvn whch can b ald o our cas. Lmma Assum ha h gnral sochasc ngral quaon s dscrbd by X = X 0 + a s, X s ds b s, X s db s 3.33 Assum ha h nal condon X0 has a fn scond momn: E[X0 ] <, and s ndndn of B, 0. Assum ha, for all ε [ 0, T ] and,y ε R, h coffcn funcons a, and b, sasfy h followng condons: Thy ar connuous. Thy sasfy a Lschz condon 3.34 wh rsc o h scond varabl: a, a, y + b, b, y K y 3.34 Thn h Io sochasc dffrnal quaon 3.33 has a unqu srong soluon X on [ 0, T ]. [8] For mor nformaon on h srongnss of h soluon, h radr can rfr o [9]. 9

30 Alcaon of h Lmma Afr dfnng h Io s Lmma, s sraghforward o aly h abov samn o 3.3 Snc a, = c and b, =, hn a, and b, ar connuous. Furhrmor: a, a, y b, b, y = c = y y a, a, y + b, b, y = c + y 3.35 Comarng 3.34 and 3.35, for K = c + +, a, and b, sasfy h Lschz condon. As a rsul, 3.3 has a unqu srong soluon X on [0,T]. Ths unqu srong soluon s gvn n

31 CHAPTER 4 DQF ALGORITHM 4. DQF L h sysm b rrsnd by 4. and 4.. X + = f X,, w 4. Z = g X,, v 4. whr X s sa, X + s n sa, Z s masurmn, v, w ar noss. Fnally, f and g ar nonlnar funcons n gnral. Th roblm can b sad as: Gvn Z,.., Z, fnd X,.., X n h flrng algorhm Gvn Z,.., Z, fnd X + n h rdcon algorhm Ths roblm dos no hav omal soluons. Howvr, h EKF mrgs as a subomal soluon o h roblms gvn abov. Th da of DQF algorhm roosd by Dmrbaş [3] s o quanz h sa sac and quanz h connuous random varabls. Th quanzd sa sac rrsnaon s analyzd n ordr o fnd an omal soluon. Two aromaon chnqus ar usd: Th sac wll b sarad by gas. All h lmns n h gas wll b aromad wh h cnr of h gas. X 0, w 0, w, wll b aromad by dscr random varabls. Usng hs wo assumons, Dmrbaş obans a rlls dagram for h sas. Th algorhm comoss of 3 man ss.

32 Obanng a rlls dagram Assgnng mrcs o h nods of h rlls dagram 3 Choosng h hghs mrc for h drmnaon of h bs ah 4.. Obanng a Trlls Dagram Th abov mnond algorhm sas ha h frs s s o oban a Trlls Dagram. No ha h algorhm uss dscr random varabls nsad of h connuous random varabls. Tha s X 0, w, w,... ar h dscr vrsons of h orgnal connuous random varabls. Th rlls dagram s h s of all ossbl valus of { X, X, }. X = f X 0,0, w0 4.3 Pung all ossbl valus X 0 and w 0 o 4.3, all ossbl valus of X ar oband. Th n s s smlar: X = f X,, w 4.4 Smlarly ung all ossbl valus X and w o 4.4, all ossbl valus of X ar oband. As a rsul, all ossbl valus of X, X,, X N ar oband va 4.4. Th rlls dagram s jus h comoson of hs valus. Th dagram consss of columns, ach of whch ndcas h m 0,,... N. Also, ach column comoss from h ossbl valus of h sa X 0, X,.., X N 4.. Assgnng Mrcs o h Nods of h Trlls Dagram Th scond s of h algorhm s h mos crucal on. Frs, h mrcs of h frs column ar assgnd. L h ossbl valus of X 0 b X q0, X q0, X q30,.., X qm0

33 L M X q0, M X q0, M X q30,, M X qm0 mrcs of X q0, X q0, X qm0 rscvly. Th valus of h Mrcs ar calculad accordng o 4.5. dno h M X q 0 = ln P X 0 X 0 whr {,,, m } 4.5 q Th n s s o drmn h mrcs of h scond column nods. Thr ar manly hr ss o b followd o fnd hs mrcs. Transon mrcs wll b calculad from ach nod of h frs column o ach nod of h n column. For aml, M X q 0 X q dnos h ranson mrc from h h nod of h frs column o h l h nod of h scond column. Th calculaon of hs ranson mrc s accordng o 4.9, whch s land n h subscon 4... For h drmnaon of h mrcs of h scond column nods, h followng rocdur mus b followd: L X q dno h l h nod of h n column. Th sum 4.6 mus b calculad for ach nod of h frs column. M X q = M X q 0 + M X q 0 X q Th ral Mrc M s h largs among h calculad mrcs X q accordng o 4.6. Th calculaon of h mrcs of h hrd column jus rqurs h ulzaon of h sam rocdur Calculaon of h ranson mrc: L h sysm b dscrbd as 4.7 and 4.8. X + = f, X, w 4.7 Z = g, X + v 4.8 3

34 whr X 0 s a Gaussan random vcor wh man m0and covaranc R 0 ; Z s h masurmn daa; w s a Gaussan dsurbanc nos vcor wh zro man and varancσ ; f and g ar nonlnar funcons n gnral; v s a Gaussan w nos wh zro man and varanc σ. Morovr, h random varabls X 0, v w j, w, v l and v m ar assumd o b ndndn for all j,, l and m. Thn h ranson mrc s calculad accordng o h followng samn 4.9: M{ X X } = InΠ q [ z g, X q In{ π v } v q ] 4.9 whr Π s h ranson robably from h nod. For h calculaon of ranson mrc of h muldmnsonal sysm, s [0] Choosng h Hghs Mrc for h Drmnaon of h Bs Pah Usng h rocdur dscrbd n scon 4.., h followng ls s oband. A rlls dagram,. Each nods mrcs has bn calculad 3. Th rvous nod from whch w cam o ha nod Thn h ouu of h algorhm s found as follows:. Loo a h las column. Fnd h largs mrc 3. Loong a h assocad rvous nod, rac on s bac 4. Go u o h frs column 5. Th ouu s jus h comoson of hs nods. 4

35 4. Quanzaon of h random varabl DQF algorhm uss h quanzd vrson of connuous random varabls. In hs sudy, snc h noss ar assumd o b Gaussan, h robably dnsy funcon of h Gaussan random varabl wh man µ and varanc s dscrbd by 4.0: f µ = π 4.0 Whr 4.0 s aromad accordng o 4.. f y n = P y = 4. whr y s ar h dscr ossbl valus of h connuous random varabl and P s ar h assocad robably valus. L F and F y dno h robably dsrbuon funcons of h connuous and dscr m random varabls. Thn F = y for < y+ y, whr = P =. In ordr o fnd y and, h followng cos funcon s dfnd: [ F a Fy a ] J F. = da y 4. Th bs dscr candda for h robably dsrbuon funcon s assumd o mnmz h cos funcon dscrbd n 4.. 5

36 Accordng o Dmrbaş [6], mnmzaon of 4. s quvaln o solvng h sysm of nonlnar quaons from 4.3 o 4.6. = F 4.3 y F y, =,,3,..., n = + n n + = F y y y = F a da =,,3,..., n + y y 4.6 whr F dnos h robably dsrbuon funcon. In hs sudy, a rcursv Malab rogram s wrn n ordr o fnd y and. Th algorhm of h rogram s as follows: Assum an nal s of robabls Us 4.3 o 4.5 o fnd y 3 Uda usng Go o s Ingral n h quaon 4.6 s calculad numrcally usng h razodal rul. Usng hs algorhm, h sysm s solvd u o n = 40. 6

37 CHAPTER 5 EKF Th Kalman Flr aroach o h flrng and smaon roblms s on of h sandard ools n smaon hory. In h lnar cas, Kalman Flr wors n h omal sns. Bu n nonlnar roblms, whr h n sa s a nonlnar funcon of currn sa and nos, Endd Kalman Flr s usd. L h sysm b dscrbd as: X + = Φ, + Γ, w 5. Z = h, + v 5. L E X 0 = µ 0, Varanc { X 0} = V 0 Cov{ w, w j} = Vw [ j] Ew = Ev = 0 Cov{ v, v j} = Vv [ j] Cov { v, w j} = Cov{ v, X } = Cov{ w, X 0} = 0 In hs sag + and z wll b aromad accordng o 5.3 and 5.4. φ X, 5.3 X + Φ X P, + [ X ] + Γ X, w X h X, Z h X P, + [ X ] + v X 5.4 7

38 Snc 5.3 and 5.4 consu a lnar sysm, h sandard KF can b ulzd. As a rsul h followng quaons ar usd for flrng and rdcon []: V~ + = φ X X, T φ X, ~ + X V 5.5 Τ Γ X, V Γ X, w X + = Φ [, ] 5.6 X 3 V~ + = V~ + - V~ + h X T X + +, { h X X +, + + h * ~ + * V T X X +, V v + } - * h X X +, + + V + 4 K + = V~ + T h X +, + [ V v + ] X X + = X + + K + { z + h X +, +} 5.9 In h abov quaons, h nal condons ar E X 0 = µ 0 and V ~ 0 = V0. 8

39 CHAPTER 6 APPLICATION OF EKF TO ECONOMETRICS Th nonlnar sysm s dscrbd wh h followng wo quaons, whch ar 6. and 6.. X w + = LX 6. whr L = c0.5 T, whch s a consan. Z = X + v 6. In h ffh scon, h sysm rrsnaon of EKF was: X + = Φ X + Γ X, w 6.3 Z = h X, + v 6.4 Comarng 6.3 and 6.4 wh 6. and 6.,w g h followng rsul: h X, = X 6.5 Equaon 6. and 6.3 canno b qual for any choc of Γ and Φ. In ordr o us EKF, h lnarzaon chnqu wll b usd for 6.. w s rrsnd n h Taylor anson form

40 LX w [ w ] 6.6 = + w w T LX [ w ] = LX [ + w + w +...] = LX + LX w +... srs Equaon 6.7 s oband by ang a frs ordr aromaon for Taylor LX w T LX + L w T X 6.7 Insad 6., quaon 6.7 s usd o b abl o aly EKF. Comarng 6.7 wh 6.3, quaons 6.8 and 6.9 ar oband. Φ X, = LX 6.8 Γ X, = L X 6.9 Rlacng h valus of h, φ, Γ quaons s oband. n 5.5 o 5.9, h followng s of V~ + = L V~ L + [ L X ] V w [ L X ] 6.0 X + = LX 6. 3 ~ + = ~ + - ~ + * { ~ + + V V V V V v +} - * 6. V + 4 K + = V + [ V v + ] X + = X + + K + { z + X + } 6.4 Fnally, wh h hl of h quaons 6.0 o 6.4, s ossbl o smooh h nosy daa and ma a rdcon. 30

41 CHAPTER 7 PARTICLE FILTERS-SIR ALGORITHM SIR algorhm s drvd from a grou of algorhms, whch ar nown as arcl flrs. Th algorhm s a Mon Carlo mhod ha can b ald o rcursv Baysan flrng roblms. I s frs roosd by Gordon.al. [3]. SIR algorhm can b ald o sysms rrsnd by 7., 7. + = f, w 7. z = h, v 7. Th condons for h sa and masurmn quaons ar wa. Th sa and masurmn noss ar wh nos squncs, whr f and h ar nonlnar funcons n gnral. Th as s o uda 7.3 rcursvly. z,..., z 7.3 Th man da s o aroma h condonal robably dnsy funcon accordng o 7.4. N s : ω = z 7.4 3

42 Hr, rrsns h randomly gnrad arcls and w s ar h assocad robably valus. s h usual muls funcon. z : dnos h s {, z,..., z } : z. Also dnos h s {,,..., } Th SIR flr s comosd of hr ss:. Prdcon Uda 3 Rsamlng Prdcon A h rdcon s, gvn, z h as s o oban z : Each arcl wll b udad usng h followng quaon 7.5. : = f, w 7.5 Each w s drawn from h robably dnsy funcon df of. w Uda Th wghs of n 7.4 ar calculad accordng o 7.6. w = N s j= z z j 7.6 3

43 Rsamlng arcls Whn h uda s comld, z : from z: s achvd. Th and h assocad robably valus w rrsn h condonal valu n h sns of 7.4. If h rsamlng s was o b sd, hn w would bcom mor and mor swd. Afr a whl, only on arcl wh nonzro robably valu wll s [5]. To avod hs dgnracy, h rsamlng s s roosd. As a rsul, h nw df s rrsnd by 7.7. z N : = N N = 7.7 whr N = N = N 7.8 For mor nformaon abou h rsamlng sag, h radr s rfrrd o [4]. 7. Drvaon of h SIS and SIR algorhm Th squnal moranc samlng SIS algorhm s a Mon Carlo MC mhod ha forms h bass for mos squnal MC flrs dvlod ovr h as dcads. SIR s also on of hs flrs. Suos π, s a robably dnsy funcon from whch s dffcul o draw samls, bu for whch π can b valuad [as wll as u o rooronaly]. In addon, l q,,.., ~ = N b samls ha ar asly s gnrad from a roosal q calld moranc dnsy. Thn, a wghd aromaon o h dnsy s gvn by N s w =

44 whr w π q s h normalzd wgh of h h arcl [4]. If h samls ar drawn from q, hn h wghs wll b 7.. z 0: : w q 0: 0: z z : : 7. L h moranc dnsy b chosn as: q z q, z q z 7. 0: : = 0: : 0: : Usng h Bays rul: 0: z : = z 0:, z z : z : 0: z : 7.3 0: z : = z 0:, z : z 0: z, z : : 0: z : 7.4 nos Snc h sa s acually a Marov rocss du o whnss of h rocss, z 7.5 0: : = 34

45 35 and snc h masurmns ar a sac funcon of h las sa and h masurmn nos s wh, : : 0 z z z = 7.6 : : 0: : 0: = z z z z z 7.7 : 0: : : 0 z z z 7.8 Pung 7. and 7.8 no 7., h rsulng quaons 7.9 and 7.0 ar oband., : 0: : 0: : 0: z q z q z z w 7.9, : 0: z q z w w 7.0 Hnc, h gnral framwor for SIS flrs s concludd. Th SIR flr s asly drvd by choosng, : 0: z q = 7. Thn, z w w 7.

46 Hnc h wghs ar udad accordng o h 7. and accordng o h 7.. s ar udad 36

47 CHAPTER 8 SIMULATIONS OF FILTERING AND PREDICTION EXPERIMENTS 8. Gnral Ouln In hs scon, h conomrc sysm s analyzd from svral ascs. Th sysm was dscrbd by 8. and 8.. X w + = LX 8. whr L = c0.5 T, whch s a consan. 8. s h sandard sa quaon. In ral world alcaons, corrsonds o h ral rc uda n rms of currn ral rc and nos. Th obsrvaon quaon s dfnd by 8.. Z = X + v 8. whr v s h masurmn nos or sculaors nos or, alrnavly, h radrs nos. 8. and 8. consu oghr h dynamc sysm rrsnaon. In hs scon, s assumd ha h masurmn daa s gvn and h as s o flr h nos and c h currn and h n ral rc 37

48 8. Flrng Tools Th rvous lraur on flrng gnrally vws h Endd Kalman Flr as a, sandard analyss ool for flrng of nos for nonlnar dynamc sysms. In hs sudy, an alrnav mhod DQF, as mnond bfor n dals, s usd. Alhough h hory of DQF s dscussd n scon 4., h alcaons of DQF o svral cass nd o b word ou. Thrfor, hs sudy s an am o aly DQF o a nw ara, Mahmacal Fnanc, whch bcam ncrasngly oular n h rcn yars. 8.3 Prdcon Tools Ndlss o mnon, rdcon s crucally moran n conomc sysms. In hs con, rdcng h mos robabl n valu of h soc mrgs as an nrsng quson o answr. In ordr o rdc h n valu of h soc, whch s dscrbd by h sysm rrsnaon 8. and 8., SIR flr and EKF ar usd. 8.4 Error Crra and Mon Carlo Smulaons Whn hr s gra dal of dffculy n analyzng h sysms n svral branchs of scnc, Mon Carlo smulaons ar mloyd. Thrfor, n hs sudy, o comar h rformancs of EKF and DQF, hs mhod s rfrrd. Th mhod can smly b summarzd as mang many rmns and usng h rsuls of hs rmns o rdc h rformanc of h sysm. In ach smulaon rmn, h followd rocdur s smlar: On aramr s vard or chosn a scfc valu. For ach valu of h aramr, 0 rmns ar don. 3 Errors for ach rmn ar calculad for o EKF ouu 38

49 o DQF ouu 4 Avrag Error EKF and DQF ar calculad Flrng Error Error n R N Sacs: L N R b dfnd by 8.3. = [,,..., N] 8.3 L norm of s calculad accordng o 8.4 N = [ ] = 8.4 L N, y R, hn h dsanc bwn h vcors and y s gvn by 8.6 y 8.6 Th dsanc conc dfnd n 8.6 s usd for h dfnon of rror. L b h ru valu hn h rror y s dfnd by 8.7 rror y = y Errors n h Ermns In ordr o calcula h rrors, h noaons 8.8 o 8. ar usd o rrsn ral daa, EKF ouu, and DQF ouu rscvly. 39

50 X = [ X, X,, X N] 8.8 EKF = [EKF, EKF,, EKF N] 8.0 DQF = [DQF, DQF,, DQF N] 8. Th corrsondng rror valus of EKF and DQF ar calculad accordng o 8. and 8.3 rror EKF = EKF X = N [ EKF = X ] 8. rror DQF = DQF X = N [ DQF = X ] 8.3 Havng oband h rrors for ach rmn, h avrag of h rrors of boh EKF and DQF ar calculad accordng o h followng formula n 8.4. avrag _ rror = K rror 8.4 K = whr K s h numbr of rmns don n h Mon Carlo smulaon Prdcon Error For a sngl rmn, h rror calculaons of rdcon algorhms of nrs, whch ar EKF and SIR arcl flr, ar 8.5 and 8.6. rror EKF = EKF N + X N rror SIR = SIR N + X N Havng oband h rrors for ach rmn, hn h avrag of h rrors of boh EKF and SIR arcl flr ar calculad accordng o h formula

51 avrag _ rror = K rror 8.7 K = whr K s h numbr of rmns don n on Mon Carlo smulaon. 8.5 Smulaons Afr h smulaon rsuls ar oband, h followng rgulars hav bn obsrvd: Th rncl of EKF algorhm s h lnarzaon of h sysm quaons. Whn h lnarzaon condon s volad, h rror rformanc of EKF bcoms oor. In ordr o oban good rror rformanc for DQF algorhm, h quanzaon lvls mus b adqualy chosn. Whn hs condon s sasfd, DQF rforms br comard o EKF. 3 Th comuaon m rformanc of h DQF algorhm s bad whn s comard o EKF. Quanzaon Lvl As mld abov and h rsuls show, h DQF algorhm has onnal comly, whch sands ou as s bggs dsadvanag. I s hard o mlmn n ral m alcaons, whr h comuaon m rformanc s h crucal on. In conomrcs, hr ar svral cass whr h comuaon m s no h rmary concrn. Thrfor, f h sac mus s adqualy quanzd, hn h DQF algorhm may b rfrrd. Howvr, h sa quaon 8. conans an onnal rm 8.8 ω 8.8 Wh h rm 8.8, s mor dffcul o adqualy dscrb h sysm n h quanzd doman. I rqurs a hug numbr of quanzaon lvl. 4

52 Lnarzaon Condon 8.8 s rssd n Taylor anson form as 8.9. w = + w ] w + [ ] Thn, a frs ordr aromaon 8.0 has bn mad. w + w 8.0 In ordr 8.0 o b an adqua aromaon of 8.9, 8. mus b sasfd. [ w ] << w s also quvaln o 8.. w << 8. 4

53 8.5. Flrng Whn h smulaons ar run and h rsuls ar oband, Tabl shows h rror rformancs vrsus h chang of varanc of h sa quaon nos. For ach varanc valu, 0 rmns ar don. Inal condons ar: quanzaon lvl of X 0 and w ar 3, Varanc of v s 4, s, man of X0 s, Varanc of X0 s 0., and m nd N s. Th varanc of w s changd from o 6. c s 0.. Tabl Mon Carlo rsuls-errors of h algorhms vrsus chang of h varanc of w EKF DQF I s clar from Tabl ha whl h varanc of w s ncrasd, h rrors of EKF and DQF algorhms ar ncrasd. Ths s cd, snc accordng o 8., h aromaon of h onnal funcon by frs ordr rms s no rgh. Hnc h lnarzaon of h sysm wh h frs ordr s no adqua. Th lnarzaon of h sysm s h basc assumon of EKF, whch s smly fals. DQF s assumd o b an omal soluon whn h ga sz aroachs o zro and quanzaon lvl aroachs o nfny. From h rrors, s also concludd ha h quanzaon lvl s no hgh nough Th rsuls ha ar dslayd n h rcdng scon s no sasfacory n rms of h h rrors of h DQF. I was assumd ha h quanzaon lvls wr no hgh nough. For ha uros, n hs rmn, h numbr of quanzaon lvl of w s an as 40 and h quanzaon lvl of X0 s an as 0. Th rrors of DQF algorhm ar dcrasd drascally, bu h crucal on s ha, rformd br han h EKF. For aml, whn h varanc of w s 5, rformd 43

54 aromaly 8 ms br han EKF. Tabl shows h rmnal rsuls of h scond Mon Carlo smulaon. Th scond obsrvaon from Tabl s ha, wh h ncras of varanc of w, h rror of DQF s also ncrasd. Ths s also cd snc h rang of ossbl valus of w s ncrasd, bu for ach varanc valu of w, h sam quanzaon lvl s usd. Tabl Mon Carlo rsuls-errors of h algorhms vrsus chang of h varanc of w EKF DQF In hs rmn, h rror rformancs of h algorhms ar shown whn h volaly consan s dcrasd o 0.. All nal condons ar sam wh h rvous rmn c h consan volaly. Agan h varanc of w s vard bwn and 6. Tabl 3 shows h Mon Carlo Smulaon rsuls. Tabl 3 Mon Carlo rsuls-errors of h algorhms vrsus chang of h varanc of w DQF EKF As s sn from Tabl 3, EKF s sabl n hs rmn. Ths s also cd, snc h rm w s dcrasd 0 ms for h sam valus of w comard o h rmn DQF s also sabl. Ths s also cd snc h ncras du o onnal rm s now lmd. Th rang of ossbl valus of X s dcrasd. Ths rang s also adqualy quanzd wh h chosn quanzaon lvls. 44

55 In h rmn 8.5.., was shown ha wh h ncras of h quanzaon lvl, h rror rformanc of DQF ncrasd subsanally. In hs rmn, h ncras of rformanc s sudd n dals. Inal condons ar sam wh h scond rmn, bu h quanzaon lvl of w s vard from o 40. Th quanzaon lvl of X 0 s assumd o b 0 n all smulaons. Th nos varancs ar assumd o b 5 and 4, for w and v rscvly. Fgur, whch dslays h lo of quanzaon lvl vrsus h rror rms shows h mrovmn of rrors whn h quanzaon lvl s ncrasd. I aroachs o a lmng valu as can b asly sn from h fgur. Fgur 45

56 As mnond abov, on moran roblm wh h DQF and DQP was h comly of h algorhms. Comard o EKF, comuaon m rformanc of h algorhm s also wors for larg quanzaon lvls. In hs rmn h m nd wll b ncrmnd on lvl. To a rasonabl rror rformanc and comuaon m, quanzaon lvl s chosn as 0 for w and 0 for X0. All ohr nal condons ar sam as h scond rmn bu N s. Conssn wh h rvous rmn, 0 rmns ar rformd for Mon Carlo smulaon. Tabl 4 Mon Carlo rsuls-errors of h algorhms for h varanc of w=5 EKF DQF.7439 Th rsuls ar rmarabl, whch ndca ha h rror rformanc of h algorhm s aromaly 5 ms br han h convnonal EKF. Bu should b nod ha comuaon m of DQF ncrass. 46

57 8.5. Prdcon Th rdcon ar dffrs from h smoohng ar. Th sysm dscrbd by 8.3 and 8.4 s ransformd o anohr quvaln form. Th rason for ha s h numbr of rmns n h Mon Carlo smulaon rqurd o comar DQP and EKF s oo hgh. Howvr, afr ransformng o h nw quvaln form, br rsuls hav bn oband. Furhrmor, h m nd could b ncrmnd o vn hghr numbrs, whch was no ossbl bfor for h cas of DQF algorhm. X w + = LX 8.3 Z = X + v 8.4 Whn ang logarhm of boh sds, 8.3 s rwrn as 8.5 ln X + = ln L + ln + w 8.5 Furhrmor f Y = ln X 8.6 Y + = ln L + Y + w 8.7 As a rsul h nw s of quaons ar 8.7 and 8.8. Y Z = + v 8.8 Th quaon 7.7 s lnar, bu h quaon 7.8 s nonlnar. Furhrmor h robably dnsy funcon of Y 0 s no Gaussan. In fac, s oband from h Gaussan nos by h ransformaon 8.6. Hnc h sandard assumon of EKF for h nal nos dos no hold.evn n hs cas, h nw s of nonlnar quaon s mlmnd by SIR and EKF algorhm. 47

58 Man and Varanc of Y0 Y can a valus 0 and ngav, so Y 0 can hav coml valus accordng o 8.6. For ha rason, all h valus for Y 0 s aromad by Y = Ths mas sns, snc h man of X 0 s and s varanc s an as 0.. Undr hs assumons, man and varanc of Y 0 s oband rmnally. For ha uros 8.9 and 8.30 ar usd. s Y 0 µ = Y 0 N = N = = N Y Y µ N Y whr Y s h randomly gnrad numbr. Ths smas ar nown o convrg o h ru man and varanc valus as N. Ermns In hs rmn, s, h man of X0 s, h varanc of X0 s 0., and h m nd N s 5. Th varanc of w and v ar s o uny. For hs smulaon, 000 rmns ar don. Tabl 5 Mon Carlo rsuls-errors of h algorhms SIR and EKF EKF SIR.976 As can b sn from Tabl 5, SIR rror rformanc s br han h on for EKF. 48

59 CHAPTER 9 SIMULATIONS OF FILTERING EXPERIMENTS FOR STOCHASTIC GROWTH MODELS Bgnnng wh h sandard sochasc growh modls, for whch h ycal soluons ar offrd n Hansn [7],[8], modlng boh roducvy shocs and caal soc accumulaon bcam crcal ssus. Alhough, h rlaonsh bwn h wo varabls can b land n a sandard way by usng h Solow rsduals, hr ar also som rcn suds, whch lan h dynamcs whn h con of mor comac modls. Howvr, hr ar only a fw of hs suds ha am o lor h ssu n a non-lnar framwor. Howvr, as s vdnly dscussd n Novals al [9], h roducvy shocs may carry a non-lnar naur, for whch h sandard Kalman flrng algorhms fal o b arora. In hs con, h non-lnar sa sac modl can b solvd hr by ndd Kalman flr or Parcl flr. Followng Novals al [9], h sa sac modl can b wrn as: log θ = ρ log θ + ε 9. = c + αθ + N 9. Th frs quaon 9. modls h roducvy shocs as bng frs ordr auorgrssv rocss, whr h dsurbanc rm s assumd o b ndndn and dncally dsrbud. If h aramr rho gs closr o uny, hn h roducvy shocs follow random wal rocsss, whr any shoc o h quaon wll hav rmann ffcs. Th scond quaon 9. rlas h moranc of hs roducvy shocs on h chang of h caal soc. Basd on h no-classcal growh modls, an ncras n h roducvy wll ncras h rurn o h caal, whch wll cra 49

60 an ra ncnv for h frms o accumula furhr caal. Usng h aramrs ha ar oband from h calbraon of h mcro-basd fundamnals, h followng valus for h aramrs ar usd. c = α = ρ = 9.5 In rms of smaon, h frs quaon, whch s h sa quaon n h modl, has a non-lnar naur, whr h rformanc of ndd Kalman flr and Parcl flr can b sd. By us of h ransformaon 9.6, h quaons 9.7 and 9.8 ar oband. X = ln θ 9.6 X = X + ε 9.7 X = c + α + N 9.8 Th man and varanc ar calculad rmnally. Man and Varanc of X0 As bfor, θ 0 can a valus 0 and ngav, so X 0 can hav coml valus accordng o 9.6. For ha rason, all h valus for θ 0 0 s aromad by θ 0 = Ths mas sns, snc h man of θ 0 s and s varanc s an as 0.. Undr hs assumons, h man and h varanc of X 0 s oband rmnally. µ X 0 N = = N X

61 s X 0 = N = X µ N X whr X s h randomly gnrad numbr. Ths smas ar nown o convrg o h ru man and varanc valus as N. Ermn In hs rmn, flrng rformanc of h algorhms of SIR and EKF ar comard. Man of X0 s, varanc of X0 s 0., and h m nd N s 5. Th varanc of w s. Th varanc of v s. For hs smulaon, agan, 000 rmns ar don. Tabl 6 Mon Carlo rsuls-errors of h algorhms SIR and EKF EKF.0875 SIR.836 As sn from Tabl 6, rror rformanc of h SIR algorhm s br han h EKF algorhm. Thrfor, h las wo rmns clarly show ha whn h SIR algorhm s comard wh h convnonally usd Endd Kalman Flr, has br rformancs. Such a concluson ndcas ha SIR can b a comng alrnav o EKF n comuaon rocss of h conomrcs roblm. 5

62 CHAPTER 0 CONCLUSION Ths sudy nds h rvous wor on h nonlnar smaon roblms. For ha uros, Endd Kalman Flr EKF, Dscr Quanzaon Flr and Squnal Imoranc Rsamlng SIR Flr ar mloyd.. Snc h alrady dns lraur on nonlnar smaon has no valuad h las wo flrs, hs sudy can b vwd as a conrbuon n offrng wo alrnav algorhms. Anohr rmary concrn n hs hss s o show h advanags of hs wo algorhms ovr EKF, whch s h convnonally usd algorhm n h fld. Th man da of DQF s o quanz h random varabls. If suffcn quanzaon of h random varabls ar mad, hn DQF rforms br han h EKF. Howvr, s major dsadvanag s h comuaon m. Hnc hr s a rad-off for rformanc vrsus comuaon m. SIR flr s from a grou of flrs ha ar nown as arcl flrs. I s h mlmnaon of Mon Carlo chnqus o smaon roblms. Error rformancs of h SIR flr wr br comard o EKF. Th comuaon m rformanc was also rasonabl comard o DQF. Thrfor, h flr can b sn as a good alrnav o hekf. Th cas suds wr chosn from h fld of conomrcs. Hnc, h conrol hory chnqus hav bn ald o a dffrn scnc branch. Ohr han valuang h rformanc of h abov mnond algorhms, anohr rmary concrn n hs sudy s o romo h us of sochasc calculus, whch nabls us o hav a mor gnral rscv for nonlnar dynamcal sysms wh nos. Th classcal aroach n h sysm hory s o ma assumons for nos drcly n h sa sac form. Howvr, wh h us of sochasc calculus, w can ma assumons also n h dffrnal quaons form. 5

63 Alhough h hory bhnd hs subjc rqurs undrsandng advancd mahmacal concs, s usag s farly sml. 53

64 REFERENCES [] Mohndr S.Grwal and Angus P. Andrws, Kalman Flrng Thory and Pracc usng Malab,John Wly,Nw Yor, 00 [] C.K.Chu and G.Chn, Kalman Flrng wh Ral-Tm Alcaons, Srngr, Brln; Nw Yor, 999 [3] Krm Dmrbaş, Informaon Thorc Smoohng Algorhms for Dynamc Sysms wh or whou Inrfrnc, Conrol and Dynamc Sysms, [4] Thomas Kalah, Lnar Sysms, Prnc Hall, Englwood Clffs, N. J., 980 [5] Zdzslaw Brzzna and Tomasz Zasawna, Basc Sochasc Procsss, Srngr-Vrlag, London Lmd, 999,.79- [6] Thomas Mosch, Elmnary Sochasc Calculus, World Scnfc, Sngaor; Rvr Edg, N. J.,. 80, 998 [7] Thomas Mosch, Elmnary Sochasc Calculus, World Scnfc, Sngaor; Rvr Edg, N. J.,. 68, 998 [8] Thomas Mosch, Elmnary Sochasc Calculus, World Scnfc, Sngaor; Rvr Edg, N. J.,. 35, 998 [9] Hayr Körzloğlu, Azz Basıyalı Hayfav, Elmns of Probably Thory, ODTÜ, Anara, 00 [0] Brn Osndal, Sochasc Dffrnal Equaons, An Inroducon wh Alcaons, Srngr, Brln; Nw Yor,. 36, 995 [] Thomas Mosch, Elmnary Sochasc Calculus, World Scnfc, Sngaor; Rvr Edg, N. J.,. 93, 998 [] Ludwg Arnold, Sochasc Dffrnal Equaons Thory and Alcaons, Wly, Nw Yor,. 3, 974 [3] Thomas Mosch, Elmnary Sochasc Calculus, World Scnfc, Sngaor; Rvr Edg, N. J.,. 0, 998 [4] Thomas Mosch, Elmnary Sochasc Calculus, World Scnfc, Sngaor; Rvr Edg, N. J.,. 90,

65 [5] Thomas Mosch, Elmnary Sochasc Calculus, World Scnfc, Sngaor; Rvr Edg, N. J., , 998 [6] Thomas Mosch, Elmnary Sochasc Calculus, World Scnfc, Sngaor; Rvr Edg, N. J.,. 7, 998 [7] Thomas Mosch, Elmnary Sochasc Calculus, World Scnfc, Sngaor; Rvr Edg, N. J.,. 8-9, 998 [8] Thomas Mosch, Elmnary Sochasc Calculus, World Scnfc, Sngaor; Rvr Edg, N. J.,. 38, 998 [9] Thomas Mosch, Elmnary Sochasc Calculus, World Scnfc, Sngaor; Rvr Edg, N. J.,. 37, 998 [0] Krm Dmrbaş, Informaon Thorc Smoohng Algorhms for Dynamc Sysms wh or whou Inrfrnc, Conrol and Dynamc Sysms,. 48 [] Krm Dmrbaş, Informaon Thorc Smoohng Algorhms for Dynamc Sysms wh or whou Inrfrnc, Conrol and Dynamc Sysms,. 9 [] A.P. Sag and J.L. Mlsa, Esmaon Thory wh Alcaons o Communcaons and Conrol, McGraw-Hll, Nw Yor,. 97 [3] N.Gordon, D. Salmond, and A.F.M. Smh, Novl Aroach o nonlnar and non-gaussan Baysan sa smaon, Proc. Ins. Elc. Eng., F, vol. 40,.07-3,993 [4] M. Sanjv Arulamalam, Smon Masll, Nl Gordon, and Tm Cla, A Tuoral on Parcl Flrs for Onln Nonlnar/Non-Gaussan Baysan Tracng, IEEE Transacons On Sgnal Procssng, Vol. 50, No., Fbruary 00 [5] Arnoud Douc, Nando d Fras, Nl Gordon, Squnal Mon Carlo Mhods n Pracc, Srngr-Vrlag 00,.0 [6] Krm Dmrbaş, Informaon Thorc Smoohng Algorhms for Dynamc Sysms wh or whou Inrfrnc, Conrol and Dynamc Sysms,.9 [7] Hansn, G.D. 985, "Indvsbl Labor and Th Busnss Cycl", Journal of Monary Economcs, 6, [8] Hansn, G.D. 997, "Tchncal Progrss and Aggrga Flucuaons", Journal of Economc Dynamcs and Conrol,,

66 [9] Novals A., E. Domnguz, J.J. Prz and J. Ruz 999, "Solvng Nonlnar Raonal Ecaons Modls By Egnvalu-Egnvcor Dcomosons", n R. Marmon and A. Sco ds., Comuaonal Mhods For Th Sudy Of Dynamc Economs, Oford Unvrsy Prss, Oford, UK. 56

67 APPENDIX Comma Srad Form for h quanzaon of h Gaussan random varabl uo 40 ons. Quanzd valus for h random varabl wh man 0 and varanc. :0 :-0.675, ,0, :.90, ,0.3550,.90 5:.3760,-0.590,0,0.590, :-.4990, ,-0.40,0.40,0.7670, :.5990, ,-0.430,0,0.430,0.9050, :-.6830,-.080, ,-0.830,0.830,0.5670,.080, :-.758,-.73, ,-0.330,-0.000,0.3309,0.688,.7, :-.8,-.005, , ,-0.48,0.480,0.453,0.7898,.003,.89 :-.8790,-.736, , ,-0.79,-0.000,0.76,0.5575,0.8778,.733,.8788 :-.930,-.3385, , , ,-0.43,0.39,0.3773,0.648,0.955,.338,.999 3:-.9765,-.3969,-.045,-0.785, ,-0.308,-0.000,0.303,0.4694,0.780,.04,.3966,.976 4:-.089,-.4499,-.0867, ,-0.555,-0.339,-0.07,0.065,0.333,0.5509, ,.086,.4495,.086 5:-.0580,-.4984,-.43, ,-0.645, ,-0.007, ,0.999,0.4058, 0.638,0.8638,.46,.4978, :-.094,-.5430,-.949,-0.93, , ,-0.838,-0.094,0.0933, 0.89,0.4797,0.6897,0.93,.94,.543, :-.77,-.584,-.45, , , , ,-0.776, ,0.766, ,0.5465,0.7497,0.9759,.46,.5834,.7 8:-.59,-.66,-.865,-.064, , ,-0.46,-0.57,-0.084, 0.089,0.55,0.449,0.6075,0.8047,.053,.856,.67,.584 9:-.886,-.6584,-.375,-.073, , , ,-0.308,-0.594, , 0.580,0.394,0.4866,0.6634,0.8554,.07,.364,.6573,.877 0:-.64,-.690,-.3659,-.5, ,-0.767, ,-0.383,-0.78, , ,0.6,0.385,0.543,0.75,0.904,.36,.3645,.6908,.53 :-.46,-.736,-.408,-.55, , , , ,-0.905, ,-0.000,0.48,0.886,0.4386,0.5956,0.7630,0.946,.534,.4003,.7,.43 :-.674,-.7534,-.4356,-.96,-0.989, , , ,-0.348, , ,0.0675,0.054,0.346,0.495,0.644,0.8077,0.987,.907,.4339,.758,

68 3:-.909,-.787,-.4675,-.79,-.077,-0.858,-0.699, ,-0.407, ,-0.36,-0.00,0.30,0.63,0.3993,0.5406,0.6896,0.8496,.056,.58,.4656,.7799,.894 4:-.333,-.8085,-.4978,-.63,-.064,-0.894, ,-0.589,-0.455, ,-0.908, ,0.065,0.880,0.366,0.4488,0.5865,0.730,0.8888,.067,.589,.4956,.8065,.36 5:-.3347,-.8339,-.565,-.99,-.0986,-0.986, ,-0.634,-0.498, ,-0.448,-0.6,-0.006,0.94,0.47,0.3664,0.4950,0.695,0.770,0.958,.0959,.903,.540,.838,.338 6:-.355,-.858,-.5537,-.38,-.3, ,-0.88,-0.673,-0.548, ,-0.95,-0.767, ,0.0564,0.73,0.97,0.49,0.5384,0.6699,0.8096,0.9607,.8,.300,.55,.8558, :-.3747,-.884,-.5797,-.354,-.6,-0.997, ,-0.76,-0.589, ,-0.34,-0.7,-0.4, ,0.0,0.33,0.3384,0.4565,0.579,0.7079,0.845,0.9938,.589,.348,.5768,.8788,.374 8:-.3934,-.9036,-.6046,-.3786,-.97,-.090,-0.887, ,-0.67, , ,-0.744,-0.647, ,0.059,0.604,0.70,0.38,0.4976,0.676,0.7439,0.8788,.05,.880,.375,.603,.9007, :-.44,-.949,-.684,-.4046,-.98,-.059,-0.950,-0.784, , ,-0.48,-0.389,-0.,-0.068, ,0.0,0.074,0.343,0.436,0.5364,0.654,0.778,0.907,.055,.58,.4008,.648,.97, :-.488,-.9453,-.65,-.495,-.466,-.0880, ,-0.853, , , , ,-0.566,-0.543, ,0.0479,0.49,0.56,0.3557,0.466,0.573,0.6886,0.805,0.94,.0836,.44,.454,.6473,.949,.458, 3:-.4454,-.9650,-.6730,-.4533,-.73,-.55,-0.975, ,-0.766,- 0.63, , ,-0.987,-0.990,-0.006,-0.008,0.0950,0.935,0.93,0.3950,0.4996,0.6080,0.74,0.844,0.970,.07,.677,.4489,.6689,.96,.44 3:-.465,-.9838,-.6940,-.476,-.970,-.49,-.003, ,-0.758, , , , ,-0.4,-0.453, ,0.0444,0.394,0.35,0.336, 0.43,0.5346,0.64,0.756,0.8709,0.9979,.367,.90,.474,.6895,.9798, :-.4770,-.000,-.74,-.498,-.306,-.67,-.030, , , ,-0.574, ,-0.376,-0.8,-0.876,-0.095, ,0.0887,0.8,0.746,