A CONVERGENCE MODEL OF THE TERM STRUCTURE OF INTEREST RATES

Transcription

1 ISN VIKORS AJVSKIS KRISĪN VĪOLA A CONVRGNC MOL OF H RM SRUCUR OF INRS RAS WORKING PAPR 9 Lavja anka 9 h orc o b ndcad whn rprodcd.

2 A CONVRGNC MOL OF H RM SRUCUR OF INRS RAS CONNS Abrac Inrodcon 3. Modl Sp 5. Fxd Momn of m for Conry nry Ino ro Ara 5. Random Momn of m For Conry Accon o ro Ara 9.3 Implcaon of Modl. aa 3 3. maon 3 4. mprcal Rl 4 5. Conclon 7 Appndc 8. rvaon of cr-m Sa-Spac Modl Rprnaon 8. xndd Kalman Flr Sp blography 4 ARVIAIONS F lgan frank M Grman mark MU conomc and Monary Unon SP Spanh pa U ropan Unon FIM Fnnh mark FRF Frnch frank IL Ialan lra NLG ch gldn

3 A CONVRGNC MOL OF H RM SRUCUR OF INRS RAS ASRAC h papr dvlop a convrgnc modl of h rm rcr of nr ra n h conx of nrng h MU. Compard wh h ohr modl dvlopd o far n h fld or modl pcfcaon nr convrgnc of h domc hor-rm nr ra o h ro ara on. W achv h convrgnc by ang ha h prad bwn h domc and ro hor-rm nr ra follow h rownan brdg proc. W alo dvlop an conomrc conrpar of h horcal modl. o addr h problm of nonaonary and nonlnary of h modl h xndd Kalman flr for coffcn maon appld. Ky word: rm rcr of nr ra h rownan brdg h MU nonlnar Kalman flr. JL clafcaon cod: 43 F36 G G5 h vw xprd n h pblcaon ar ho of h ahor mploy of h Monary Polcy parmn of h ank of Lava. h ahor am rponbly for any rror and omon. h ahor wold lk o xpr hr apprcaon o Jpr Lnd for provdng h daa.

4 A CONVRGNC MOL OF H RM SRUCUR OF INRS RAS INROUCION In rcn yar wlv conr hav jond h U. All nw mmbr a ar rqrd o jon h MU and adop h ro a offcal crrncy. I mporan for cnral bank n MU candda conr o dvlop fnancal a prcng modl nc on h ba of h modl wold b pobl o a xpcaon of mark parcpan wh rpc o varo fr vn for xampl h xpcd nry da o h ro ara or probably of jonng h MU by fxd da. For fnancal mark parcpan h modl wold rv a an nrmn o proprly vala a prc. In nrnaonal rm rcr modl h yld prad bwn dffrn conr drmnd by xchang ra rk. Afr jonng h MU h yld prad bwn domc and h ro ara govrnmn bond hold dappar. 3 Sch vn a conry accon o h ro ara n h fr cranly affc prc of fnancal a. A long a any ncrany wh rpc o h vn x wll b rflcd n h rm rcr of nr ra. Jpr Lnd n h work analy h ffc of plannd MU mmbrhp on domc yld crv n ponal mmbr a.3 h mo nrng problm ar whn hr om ncrany abo h mmbr a of a monary non or h nal pobl nry da or boh; hrfor J. Lnd pay h man anon o h ca. J. Lnd dvlop an nrnaonal rm rcr modl whch ncld MU nry plan propo a parclar maon mhod and mak mprcal conclon. In h modl conry MU accon da a random varabl. In h mprcal analy ng nr ra daa J. Lnd appl h far o ma h mark mpld probably of jonng h MU for ach U mmbr a. Pal d Graw 4 and Carlo Ambrogo Favro al. 7 condr mlar da. avd S. a 3 mak a rvw of lrar and prn a dald comparon of dffrn mhod for h MU nry probably maon. In ordr o dcrb h dynamc of hor-rm prad bwn domc and ro nr ra ndr h ral probably mar J. Lnd appl h andard Vack proc Ornn-Uhlnbck whl xndng h rk-nral proc wh a cond facor.. h ochac prc of rk whch alo follow h Vack proc. I no dffcl o prov ha n ordr o avod an arbrag opporny h horrm prad hold convrg o zro by h m of nrng h crrncy ara. J. Lnd pcfcaon do no nr h flflmn of h condon. In h prn papr w dvlop a rm rcr modl of nr ra for a conry ha wll jon h ro ara n h fr. W dfn h pcfcaon of h hor-rm nr ra In May 4 Cypr h Czch Rpblc ona Hngary Lava Lhana Mala Poland Slovaka and Slovna jond h U. lgara and Romana jond h U n Janary 7. Slovna adopd h ro n Janary 7 Cypr and Mala n Janary 8. Slovaka adopd h ro n Janary 9. 3 I amd ha dffrnc n ch facor a lqdy ax and crd rk do no x. 3

5 A CONVRGNC MOL OF H RM SRUCUR OF INRS RAS prad dynamc by h rownan brdg proc 4; 5. h ochac proc ha a propry o convrg o zro a a pcfc m momn. hrfor h pcfcaon nr convrgnc of h hor-rm prad o zro by conry nry no h ro ara and h avod an arbrag opporny a h momn n m. In ordr o modl ncrany of h MU nranc da w follow J. Lnd 3 and xponnal drbon. h modl rcr alo rlad o work dvod o h modllng of crd rk dynamc 6;. W alo dvlop an conomrc conrpar of h horcal modl. o h nd w ranform h conno-m ym no a dcr-m on. o addr h problm of nonaonary and nonlnary of h modl w apply h xndd Kalman flr for coffcn maon. h mprcal rl vdnc ha h dvlopd modl f daa br han Lnd ngl-facor modl for lgm Franc Ialy and Span; h am for h Nhrland and lghly wor for Fnland. In addon or modl conan a mallr nmbr of paramr han h rval on 4 agan 5. Sbcon. of Scon provd horcal grond of arbrag-fr prcng and on h ba w dvlop h rm rcr convrgnc modl ndr h condon ha h da of conry nry no h ro ara cranly known. Sbcon. nrodc h modl xnon ndr h condon ha h ro ara nry da ncran and drv formla for h compaon of h modl rm rcr. Sbcon.3 foc on h mplcaon of h modl. Scon dcrb h daa Scon 3 prn conomrc mhod for coffcn maon of h dvlopd modl and Scon 4 conan mprcal rl. h la con concld. 4

6 A CONVRGNC MOL OF H RM SRUCUR OF INRS RAS. MOL SUP. Fxd Momn of m for Conry nry no ro Ara W condr wo crrnc h ro and domc crrncy. W dno h ro nr ra a m by R and h domc on by r. h h hor-rm ra n h rpcv conry af r R [] whr δ h local prad rlav o R. I amd ha R and δ ar drvn by ndpndn ochac proc. h followng ochac dffrnal qaon dcrb h dynamc of h local prad and ro hor-rm nr ra ndr h r probably mar: d f d dw [] dr f R d R dw [3] whr w and w ar wo ncorrlad andard rownan moon proc f and f ar drf coffcn for ach proc whra σ δ and σ R ar dffon rm.4 hr ar wo mark prc of rk λ λ ach corrpondng o h orc of ncrany w and w rpcvly. Now a clam wh a payoff n crrncy a m gvn by h fncon h condrd. Undr h no-arbrag prncpl prc a m n n of crrncy oby h followng xpron: r d h r = r or R [4] whr h rk-nral probably mar for crrncy and condonal xpcaon opraor ndr h probably mar.5 h For h prc a m of crrncy zro-copon bond marng a m n n of crrncy w hav r d [5]. Snc obvo ha afr h nry no h ro ara h rm rcr of domc mark and ro ara nr ra wll b h am horly bfor h ro ara accon da h domc zro-copon bond prc dnomnad n domc crrncy m almo concd wh h ro ara zro-copon bond prc dnomnad n ro. d h fac ndpndn of h xchang ra lvl a h momn of h ro ara nry. 5

7 A CONVRGNC MOL OF H RM SRUCUR OF INRS RAS L P b h m prc of a ro bond marng a m. From qaon [4] w g ha h crrn m domc bond prc for a gvn MU mmbrhp da dnod by P qal o P P d R [6] whr h xpcaon akn ndr h rk-nral probably mar for domc crrncy. From qaon [5] w oban h m prc of a ro bond marng a m d R d R P [7]. I akn no accon ha afr h ro ara nry m mar and m concd. h Radon-Nkodym drvav dfn h rlaonhp bwn wo probably mar.5 Amng ha h Radon-Nkodym drvav ndpndn of R w can wr h prc of a ro bond a d R d R d R d R d d d d d d d d P [8]. h follow ha w can oban h ro bond prc by akng xpcaon ndr hr h domc rk-nral probably mar or ro mar. Sbng qaon [7] no qaon [6] and ng h law of rad xpcaon w oban P P δd Rd δd Rd Rd d δ R Rd d δ R [9] whr 6

8 A CONVRGNC MOL OF H RM SRUCUR OF INRS RAS δd δ [] h local dcon facor rlang h domc and ro bond prc. If h m o mary < formla [] can b rwrn a d []. h bcrp and δ dno ha h xpcaon hall b akn gvn h followng dynamc of h facor: d [] f d dw [3] whr w a Wnr proc ndr mar. Formla [] h Fynman-Kac rprnaon of h olon of h followng parabolc paral dffrnal qaon: [4] f wh h rmnal condon. [5]. I frhr amd ha h drf rm n qaon [] ha h followng form: f [6] and σ δ = σ = con. h mpl h followng ochac dffrnal qaon ndr h r mar P: d d dw [7]. I amd ha h m of nry no h ro ara. h proc dfnd n [7] wll known n h ochac proc hory a h o-calld rownan brdg.4;5 h propry of h proc ha qal o zro a m wh probably. h prpo of h pcfcaon o nr convrgnc of h prad o zro by h da of nrng h ro ara whch garan qal domc and ro nr ra. A h r mar P and rk-nral mar ar qvaln h vn wh probably ndr mar P wll alo hav probably 7

9 8 A CONVRGNC MOL OF H RM SRUCUR OF INRS RAS ndr mar. hrfor probably of vn {δт = } wll b qal o alo ndr h rk-nral mar. 4 h movaon o choo ch pcfcaon ha f h prad wa non-zro a h momn of nry arbrag wold b pobl. h xchang ra alon wold b nffcn o accommoda h non-zro prad bca a h m h formr wold b fxd o om lvl known n advanc. I no pobl o garan a zro hor-rm ra prad j bfor jonng h ro ara ng h convnonal modl pcfcaon Vack Cox-Ingroll-Ro ml-facor affn c. Sbng qaon [6] no qaon [4] w g [8]. W alo allow h prc of rk λ o b conan. L condr now h ca of h affn rm rcr modl A xp [9] whr. hn h varo paral drvav of δ τ ar A A A [] A [] A []. Inrng qaon [] and [] no paral dffrnal qaon [8] gv h followng xpron: A. h cancllng of akng no accon ha = τ and rarrangng of h rm gv A. 4 Undr h rk-nral mar h hor-rm prad proc. dw d d I no dffcl o how ha h proc ha h followng rprnaon f λ conan: ln δ dw δ whr h fr and cond rm rprn h andard rownan brdg proc from δ o and h hrd rm drmnc fncon wh rmnal val a.

10 9 A CONVRGNC MOL OF H RM SRUCUR OF INRS RAS h la qaon hold for all δ and τ val and w can concld ha h wo rm n brack ar qal o zro. h rdc h problm o olvng h followng ym of wo ordnary dffrnal qaon: A [3] [4]. Solvng h ym of qaon [3] and [4] ndr nal condon ha A = and = gv h olon of affn rm rcr modl [9] [5] A ln [6].. Random Momn of m for Conry Accon o ro Ara h dcon on a conry accon o h ro ara and da mad by h U non. Pror o hr ncrany of om dgr wh rpc o h momn of m. o accomplh h modl ncary o dfn probably drbon of a m ndr h rk-nral mar. h probably ha a conry wll no jon h ro ara pror o m dfnd n h followng form: xp Pr d [7]. h probably dny fncon h drvav of drbon fncon wh rpc o xp Pr Pr d d d d d p [8] whr π a hazard fncon. Gvn h drbon of Т by compng h xpcd val of qaon [9] for all pobl ro ara accon da Т w can oban h bond prc

11 A CONVRGNC MOL OF H RM SRUCUR OF INRS RAS d p P d p P d p P P [9] whr xp A from qaon [9] b А and В follow from qaon [5] and [6] rpcvly. W dfn h hazard fncon n h followng form 3: [3] whr h momn of m whch corrpond o om cran m n h fr and h conan >. h pcfcaon of h modl do no adm accon o h ro ara bfor da. h fncon af h followng qaon:. h gvn h hazard fncon w can rwr h bond prc from qaon [9] a d d P d P P [3] or P = P F [3] whr d d d F [33]. akng no accon qaon [3] and [3] w can comp yld o mary for zro-copon bond:

12 A CONVRGNC MOL OF H RM SRUCUR OF INRS RAS log P log P F y y S y S log P log F [34] whr y τ h yld o mary for ro bond wh h rm o mary and S S yld prad of h rpcv rm o mary..3 Implcaon of Modl In Lnd modl yld prad for ra wh pr-nry da mar can no b affcd by h MU accon whch howvr no conn wh h daa Char 3. In h propod modl h MU affc all mar whch n or opnon mor ralc. o how ha h propod modl gv a plabl ocom w condr nmrcal xampl. h fr xampl dal wh a conry havng a rong probably of jonng h MU. For h conry coffcn = 4.. h probably of jonng h MU whn yar from da qal o p = =.98. h crrn val of a varabl δ =.. For h cond conry coffcn =... h probably of jonng h MU whn yar from da qal o p = =. or vry low. Char -yar nr ra prad on ranacon n F FRF and NLG o M

13 A CONVRGNC MOL OF H RM SRUCUR OF INRS RAS Char -yar nr ra prad on ranacon n F FRF and NLG o M Char 3 - and -yar nr ra prad on ranacon n SP FIM and IL o M Char 4 dplay h rm rcr of h prad for boh conr wh h 3-yar rm 3 and -yar rm o da. For h conry wh a hgh probably of jonng h MU h rm rcr of nr ra prad rapdly dcrang wh mary n boh ca; howvr h horr h rm o h pobl MU accon da h mor rapd h dcra. h hor-rm nr ra prad abo % whra h -yar prad 3 ba pon for h 3-yar rm and ba pon for -yar rm. For h low probably conry h rm rcr of nr ra prad dcrang vry gradally and almo ndpndn of rm. h -yar nr ra prad lghly xcd.6%.

14 A CONVRGNC MOL OF H RM SRUCUR OF INRS RAS Char 4 horcal rm rcr of nr ra prad For h conry wh rong probably of jonng h MU = 4.. h probably of jonng h MU whn yar from da qal o p = =.98 and h crrn val of a varabl δ =.. For h cond conry coffcn =... h probably of jonng h MU whn yar from da qal o p = =... vry low. rm o h pobl MU accon da.. AA For h maon of modl paramr followng J. Lnd papr 3 w mony mark and wap ra. h mony mark daa ncld nr ra wh mar of 3 6 and monh whra h wap mark daa ncld ra wh mar of 5 7 and yar. h followng crrnc wr condrd: F FRF FIM IL SP and NLG. All daa ar prad bwn h corrpondng naonal crrncy and M yld. h yld ar ampld wkly Wdnday from 4 Ag 996 o Ag 998. W choo ch a ampl d o prononcd obrvabl convrgnc of yld prad o zro n h prod wh h daa rvalng propr of h rownan brdg proc Char SIMAIONS L am ha n nr ra wh mar τ τ τ n x n h mark. h corrpondng prad wh rpc o ro nr ra ar S S n. In h ca akng no accon qaon [7] h modl dvlopd abov can b wrn a d d dw [35] S S = n [36] 3

15 A CONVRGNC MOL OF H RM SRUCUR OF INRS RAS log F whr S h horcal prad from qaon [34] S dpnd on δ nc follow from qaon [9] ha dpnd on δ N I h marmn rror h vcor of modl paramr and ψ = σ η σ δ λ. h ym of qaon [35] and [36] drmn a nonlnar conno-m apac ym whr qaon [35] a ranon qaon and qaon [36] a marmn qaon.8; 9 In ordr o ma h paramr vcor ψ w nd o ranform h ym of qaon [35] and [36] no a dcr-m form and h xndd Kalman flr 9 d o nonlnary of marmn qaon 5 Appndx for dal of h procdr. 4. MPIRICAL RSULS h maon rl oband wh h propod modl ar rpord n h abl. All paramr ar acally gnfcan. Paramraon of h hazard fncon n qaon [35] man ha h probably of jonng h MU whn on yar from Janary 999 qal o xp. h maon of paramr o om xn conn wh J. Lnd maon.3 hr wa a rong probably of jonng h MU for lgm and Franc. For Fnland Ialy and Span h probably wa mall. For h Nhrland paramr qal o.86 whch man ha h probably of jonng h MU whn on yar from Janary 999 qal o.58. h probably appar o b oo mall. I can b xpland by h fac ha for h Nhrland h prad for hor-rm mar dd no convrg o zro ovr h condrd ampl b only flcad arond Char. hrfor no pobl o capr h dynamc of h obrvabl hor prad wh h hlp of h rownan brdg proc. 5 W can no apply GMM Gnrald Mhod of Momn d o non-aonary of h modl. 4

16 A CONVRGNC MOL OF H RM SRUCUR OF INRS RAS abl Comparon of ma of rownan brdg modl wh Lnd ngl-facor modl 3 wh conan hazard fncon boh modl ar for yld prad o M Crrncy Modl σ η 3 σ δ 3 λ k F rownan brdg Lnd modl SP rownan brdg Lnd modl FIM rownan brdg Lnd modl FRF rownan brdg Lnd modl IL rownan brdg Lnd modl NLG rownan brdg Lnd modl h rownan brdg modl h xndd Kalman flr for coffcn maon for dal Appndc. Lnd modl a ngl-facor modl wh conan hazard fncon 3. h ampl ncld h prod from Ag 996 o Ag 998. σ η andard dvaon of marmn rror for obrvabl yld prad o M σ δ h nanano volaly of hor-rm prad λ h prc of rk k h man rvron paramr for Lnd modl. h hazard fncon π zro for and for whr h momn of m bfor whch h modl do no adm accon o h MU. h probably ha a conry wll no jon h MU bfor m xp d. Pr and for h da of jonng h MU. Sandard rror ar n parnh. In ordr o vala h n-ampl fng prformanc of h rownan brdg modl w compar andard dvaon of h marmn rror wh h on n a nglfacor vron of Lnd modl 3 wh h conan hazard fncon. In parclar ndr h ral mar P h dynamc of hor-rm yld prad govrnd by h Vack proc d k d dw. W h xndd Kalman flr o ma fv paramr of h ngl-facor vron of Lnd modl 3 wh conan hazard fncon k λ σ δ and σ η whr 5

17 A CONVRGNC MOL OF H RM SRUCUR OF INRS RAS k h man rvron paramr σ δ h nanano volaly of h hor-rm nr ra prad λ h prc of rk and σ η h andard dvaon of marmn rror. h maon procdr dcrbd n Appndx. In paramr maon of h modl w h am ampl a for h rownan brdg modl. h raonal o choo h pcfcaon of Vack modl who ncondonal man l n h propry of h condonal xpcaon of h proc o convrg o zro a m nd o nfny. h propry can capr convrgnc n h horcal m r. h abl how h maon rl for boh modl. h comparon of andard dvaon of marmn rror for h wo modl how ha h propod modl gv a br n-ampl fng for almo all conr xcp for h Nhrland and Fnland. Morovr conan a mallr nmbr of paramr han h rval on 4 agan 5. In h ca of h Nhrland h andard dvaon of marmn rror ar qal for boh modl. h can b xpland by h abovmnond ncomplly corrc pcfcaon of h rownan brdg proc for h hor-rm prad ovr h condrd ampl. For Fnland h rownan brdg modl gv a lghly grar andard dvaon of marmn rror whch can b accond for by h obrvd dvrgng rm rcr of yld prad Char 3. h lar mpl ha an addonal facor ndd o dcrb h bhavor of Fnnh yld prad corrcly. I hold b nod ha n Lnd modl h man rvron coffcn k ha a ngav gn for Fnland and h Nhrland. I gg ha h modl dynamc bcom nabl and h lack any conomc n. In h ca paramr k rv only for a br f. 6

18 A CONVRGNC MOL OF H RM SRUCUR OF INRS RAS 5. CONCLUSION In h papr w dvlop a rm rcr of nr ra convrgnc modl whn h no-arbrag prcng framwork. h modl dcrb h convrgnc of MU candda conr nr ra o h ro ara nr ra. W condr boh fxd and random MU nry da n h modl p. Compard wh ohr modl dvlopd o far n h fld 3; 3 h modl pcfcaon dvlopd n h papr nr convrgnc of domc hor-rm nr ra wh ro on by h nry da. I achvd by amng ha h facor h nr ra prad follow h rownan brdg proc. Sch an ampon avod arbrag opporny n h modl a h m of nrng h MU. h drvd formla lad o a mor corrc valaon of prc of nr ra nrmn and hnc a mor accra xracon of mark xpcaon from h prc. W alo dvlop an conomrc conrpar of h horcal modl. h papr provd a framwork of h conomrc maon for h modl ng h xndd Kalman flr. mprcal rl how plabl paramr val. W compar h andard dvaon of marmn rror for h propod modl and for J. Lnd vron of h Vack modl. h andard dvaon of marmn rror n or modl mallr han n rval on for lgm Franc Ialy and Span; h am for h Nhrland and lghly wor for Fnland. For h Nhrland h prad a hor mar dd no convrg o zro b only flcad arond ovr h condrd ampl mplyng ha h rownan brdg proc no a complly corrc pcfcaon for a hor-rm prad. For Fnland h obrvd rm rcr of yld prad dvrgng h an addonal facor ndd o dcrb corrcly. In addon or modl conan a mallr nmbr of paramr han h rval on 4 agan 5. h nrodcd modl can b appld o dvlop an nr ra rm rcr modl no only for a conry plannng o jon h ro ara b for any conry plannng o jon any crrncy ara. For xampl h modl cold alo b appld o Soha Aan conr ha ngoa a pobly o cra hr own crrncy ara. 7

19 A CONVRGNC MOL OF H RM SRUCUR OF INRS RAS APPNICS. rvaon of cr-m Sa-Spac Modl Rprnaon L am ha n nr ra wh mar τ τ τ n x n h mark. h corrpondng prad wh rpc o ro ra ar S S S n. In ch a ca akng no accon qaon [7] h modl dvlopd abov can b rwrn a d d dw [37] S S = n [38] whr log F S a horcal prad from qaon [34] S dpnd on δ nc follow from qaon [9] ha dpnd on δ N I h marmn rror h vcor of modl paramr and ψ = σ η σ δ λ. L dfn h vcor S S S... S n S S S... S and... n. n h ym of qaon [37] and [38] drmn a a-pac ym whr qaon [37] a ranon qaon and qaon [38] a gnal or marmn qaon.8; 9 An xcpon of h problm alo random drbon of n qaon [37]. Followng h convnonal approach of h ochac dffrnal qaon hory 5 w dfn h olon of qaon [37] a a fncon δ whch af h followng ngral qaon: d d dw. A chang n h ordr of ngraon n h fr ngral gv 8

20 9 A CONVRGNC MOL OF H RM SRUCUR OF INRS RAS dw d dw d d whr xp d x x a pcal fncon known a xponnal ngral. h la xpron qvaln o h followng ochac dffrnal qaon: dw dw d [39]. Snc h obrvaon ar dcrly ampld n ordr o prodc conomrc maon w nd o ranform h conno-m a-pac ym of qaon [37] and [38] no a dcr-m form. For h prpo w condr an xpron d δ whr a fndamnal olon of h followng ordnary homogno dffrnal qaon: d d d [4] wh nal condon = [4]. From qaon [39] w can oban [39] dw dw d d d d d [4]. Ingrang qaon [4] from o and akng no accon qaon [4] w oban dw [43] or dw [44]. h n ordr o oban a dcr-m vron of qaon [39] w nd o fnd fncon. For h prpo qaon [4] can b rwrn a d d [45]. Ingraon of qaon [45] from o akng no accon qaon [4] ng h dfnon of fncon and changng h ordr of ngraon rl n

21 A CONVRGNC MOL OF H RM SRUCUR OF INRS RAS 3 ln I I I d d d d d d d d d d d d d d [46] whr h fr ngral d d I [47]. h cond ngral compd n h am way a I [48]. For h hrd ngral w hav. ln ln 3 d d d I [49]. From qaon [46] [49] follow ha ln ln or akng h xponnal of boh d yld xp [5]. Amng ha ar known dcr-m momn = N and h m nrval bwn wo concv m momn gvn by follow from qaon [5] ha

22 A CONVRGNC MOL OF H RM SRUCUR OF INRS RAS xp xp xp [5]. h from qaon [44] and [5] w hav oband a rcrv dpndnc of on prvo val whr. σdw Δ Δ σdw σdw ε Δ - xp xp h ochac ngral ha normal drbon wh zro condonal xpcaon F whr nformaon avalabl a m momn. Condonal varanc of F follow from h propr of ochac ngral.4; 5. xp xp d F dw F F Var hrfor h ym of qaon [37] and [38] can b rwrn n a dcr-m a pac rprnaon a [5]

23 A CONVRGNC MOL OF H RM SRUCUR OF INRS RAS S S [53] whr drmnd by qaon [5] and S by qaon [34].. xndd Kalman Flr Sp h marmn qaon [53] nonlnar wh rpc o a varabl hnc w nd o h xndd Kalman flr.9 In conra o Lnd approach w h xndd Kalman flr b no h rav xndd Kalman flr. h nabl o nally pd p h flrng algorhm and conqnly maxmaon of h lklhood fncon. o oban h xndd Kalman flr w hall lnar qaon [53] a pon whr a on-p-ahad a prdcon a m momn S S S [54]. h componn of drvav vcor S can b drvd from qaon [33] [34] and [9]: F J J F d S whr d J d J and F δ ψ gvn by qaon [33]. Now l dfn h nnovaon vcor S S [55]. h varanc of prdcon of obrvabl H S S F [56]

24 A CONVRGNC MOL OF H RM SRUCUR OF INRS RAS whr h prdcon varanc of facor δ a m momn : [57] H h covaranc marx of vcor I whr I a n marx. H Ung nformaon a momn h a pdang qaon S F [58] whra h varanc of h maon gvn by S S H [59]. hrfor h rcrv maon chnq may b d n h am way a n h andard Kalman flr. Choong any nal val and and ng formla [54] [59] w can rcrvly comp nnovaon and hr covaranc marx F. Now l dfn h log-lklhood fncon L N log F F [6] and ng om nmrcal mhod calcla h paramr vcor whch maxm h lklhood fncon. h mo complcad ak n h procdr compng fncon S and drvav a ach p nc S dfnd by log from mpropr ngral n qaon [33] whch ar rahr nonlnar fncon. h of nmrcal mhod vally rqrd. 3

25 A CONVRGNC MOL OF H RM SRUCUR OF INRS RAS ILIOGRAPHY. ARAMOWIZ Mlon SGUN Irn A. Handbook of Mahmacal Fncon. ovr Pblcaon : Nw York AS avd S. Fnancal Mark Amn of MU. NR Workng Papr No Janary CORZO Sanamara ra SCHWARZ dardo S. Convrgnc whn h U: vdnc from Inr Ra. conomc No vol. 9 No. Jly pp GRAUW Pal. Forward Inr Ra a Prdcor of MU. CPR con Papr No. 395 May UFFI arrll. ynamc A Prcng hory. Prncon Unvry Pr : Prncon. 6. UFFI arrll SINGLON Knnh J. Crd Rk. Prncon Unvry Pr : Prncon FAVRO Carlo Ambrogo GIAVAZZI Francco IACON Fabrzo ALLINI Gdo. xracng Informaon from A Prc: h Mhodology of MU Calclaor. CPR con Papr No. 676 Jly HAMILON Jam. m Sr Analy. Prncon Unvry Pr : Prncon HARVY Andrw C. Forcang Srcral m Sr Modl and h Kalman Flr. Cambrdg Unvry Pr : Nw York KARAZAS Ioann SHRV Svn. rownan Moon and Sochac Calcl. Sprngr-Vrlag : rln 99.. LANO avd. On Cox Proc and Crd Rky Scr. Rvw of rvav Rarch vol. No. 3 cmbr 998 pp LUN Jpr. Fv ay n Fnancal conomrc and h rm Srcr of Inr Ra Ph.. raon. parmn of Fnanc h Aarh School of n Jly 997 [cd on 4 Ocobr 8]. Avalabl: hp:// 3. LUN Jpr. A Modl for Sdyng h ffc of MU on ropan Yld Crv. ropan Fnanc Rvw vol. No pp PROR Phlp. Sochac Ingraon and ffrnal qaon. Sprngr-Vrlag : rln RVUZ anl YOR Marc. Conno Marngal and rownan Moon Sprngr-Vrlag : rln VASICK Oldrch. An qlbrm Characrzaon of h rm Srcr. Jornal of Fnancal conomc vol. 5 Novmbr 977 pp