IMES DISCUSSION PAPER SERIES

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1 IMES DISCUSSION PAPER SERIES A Sae Space Approach o Esang he Inegraed Varance and Mcrosrcre Nose Coponen Daske Nagakra and Toshak Waanabe Dscsson Paper No. 9-E- INSTITUTE FOR MONETARY AND ECONOMIC STUDIES BANK OF JAPAN -- NIHONBASHI-HONGOKUCHO CHUO-KU, TOKYO -866 JAPAN Yo can download hs and oher papers a he IMES Web se: hp:// Do no reprn or reprodce who persson.

2 NOTE: IMES Dscsson Paper Seres s crclaed n order o slae dscsson and coens. Vews expressed n Dscsson Paper Seres are hose of ahors and do no necessarly reflec hose of he Bank of Japan or he Inse for Moneary and Econoc Sdes.

3 IMES Dscsson Paper Seres 9-E- March 9 A Sae Space Approach o Esang he Inegraed Varance and Mcrosrcre Nose Coponen Daske Nagakra* and Toshak Waanabe** Absrac We call he realzed varance (RV) calclaed wh observed prces conanaed by crosrcre noses (MNs) he nose-conanaed RV (NCRV) and refer o he coponen n he NCRV assocaed wh he MNs as he MN coponen. Ths paper develops a ehod for esang he negraed varance (IV) and MN coponen slaneosly, exendng he sae space ehod proposed by Barndorff-Nelsen and Shephard (). Or exenson s based on he resl obaned n Meddah (), naely, when he re log-prce process follows a general class of connos-e sochasc volaly (SV) odels, he IV follows an ARMA process. We represen he NCRV by a sae space for and show ha he sae space for paraeers are no denfable; however, hey can be expressed as fncons of fewer denfable paraeers. We llsrae how o esae hese paraeers. The proposed ehod s appled o yen/dollar exchange rae daa. We fnd ha he agnde of he MN coponen s, on average, abo %-48 % of he NCRV, dependng on he saplng freqency. Keywords: Realzed Varance; Inegraed Varance; Mcrosrcre Nose JEL classfcaon: C, G *Econos, Inse for Moneary and Econoc Sdes, Bank of Japan (E-al: daske.nagakra@boj.or.jp) **Professor, Inse of Econoc Research, Hosbash Unversy, and Inse for Moneary and Econoc Sdes, Bank of Japan (E-al: waanabe@h-.ac.jp, oshak.waanabe@boj.or.jp) The ahors hank he parcpans n Inernaonal Conference Hgh Freqency Daa Analyss n Fnancal Markes, Conference Recen Developen n Fnance and Econoercs and Senar a Hrosha Unversy of Econocs for her sefl coens. Fnancal sppor fro he Mnsry of Edcaon, Clre, Spors, Scence and Technology of he Japanese Governen hrogh Grans-n-Ad for Scenfc Research (No. 89) and he Global COE progra Research Un for Sascal and Eprcal Analyss n Socal Scences a Hosbash Unversy s graeflly acknowledged. Vews expressed n hs paper are hose of he ahors and do no necessarly reflec he offcal vews of he Bank of Japan.

4 Inrodcon The varance of nancal asse rerns s known o change over e. More speccally, he varance, or he sqare roo of he varance (volaly), ends o be large (sall) followng sccessve large (sall) varances n prevos perods. Ths phenoenon s known as \volaly clserng". A hge nber of researchers have red o esae hese changng varances becase her vales are crcally poran for opon prcng, rsk anageen, opal porfolo consrcon, ec. There are wo poplar classes of odels for hs sor of volaly dynacs, naely, generalzed aoregressve condonal heeroskedasc (GARCH) odels and sochasc volaly (SV) odels. Based on GARCH or SV odels wh esaed odel paraeers, one can esae he changng varances. See, for exaple, Bollerslev, Engle and Nelson (994), Pal (996) and Zvo (8) for coprehensve srveys on GARCH odels, Ghysels, Harvey and Renal (996) for a revew of soe of he older papers on SV odels and Shephard (5) for a ls of seleced papers n he SV lerare. Recenly, a new class of esaors for changng varances, or negraed varance (IV), has been developed by Barndor-Nelsen and Shephard (, ), Andersen, Bollerslev, Debold and Ebens () and Andersen, Bollerslev, Debold and Labys (). The IV s a easre of he varably of nancal asse rerns over a speced perod, for exaple, a day (a foral denon of IV wll be gven n Secon ). The esaor s called he realzed varance (RV). The RV eploys hgh freqency nancal e seres daa sch as ne-by-ne rern daa or enre records of qoe or ransacon prce daa. The RV s a odel-free esaor n he sense ha we do no have o specfy he volaly dynacs. Under oderae asspons, he RV converges n probably o he IV, as he saplng freqency ends o be hgh. One of he key asspons needed for he conssency of he RV s ha here are no easreen errors n observed log-prces. The easreen error s called crosrcre nose (MN) and eerges becase of, for exaple, dscreeness of prces, bd{ask bonce and rreglar radng. When hs asspon s volaed, he RV s no longer a conssen esaor for he IV. I can be shown ha, nder he exsence of MN, he RV dverges as he saplng freqency ncreases. Several alernave esaors of he RV, whch are conssen even nder he exsence of MN, have been proposed by Zho (996), Zhang, Mykland and Al-Sahala (5), Hansen and Lnde (6) and Band and Rssell (6). See also Band and Rssell (8), who consder a ean-sqared-error opal saplng heory for redcng he eec of MN. We call he RV calclaed wh observed log-prces conanaed by MN he noseconanaed RV (NCRV) and refer o he coponen n he NCRV assocaed wh he MN as he MN coponen (a foral denon of he NCRV and MN coponen s gven n Secon.). The objecve of hs paper s o esae he IV and MN coponens slaneosly. Or approach s an exenson of he sae space ehod proposed by Barndor-Nelsen and Shephard (), who consder a saon wh no MN. In hs saon, Barndor-Nelsen and Shephard () show ha he IV follows an ARMA process for soe specc connos{e SV odels. Ths enables s o represen he RV n a sae space for, naely, he s of he IV and an nobservable whe nose (dscrezaon error). Then, gven he sae space for We nerchangeably se he er \ARMA process" and \ARMA odel" n hs paper.

5 paraeers, we can esae he IV by applyng he Kalan ler. Ths resl s frher developed by Meddah (), who shows ha he IV follows an ARMA process for a general class of connos{e SV odels, whch s called he sqare roo sochasc aoregressve varance (SR{SARV) odel (Andersen, 994; Meddah and Renal, 4). Meddah () derves explc relaonshps beween he ARMA odel paraeers and he SV odel paraeers. We develop he sae space ehod by Barndor-Nelsen and Shephard () for dealng wh he proble of MN. We asse ha an observed log-prce s he s of he re log-prce and an..d. MN. We represen he NCRV by a sae space for n ha he NCRV s he s of hree nobserved coponens: he IV, whch follows an ARMA process, a whe nose (dscrezaon error) and a MN coponen, whch follows a MA() process. By applyng he resls of Granger and Morrs (976), we show ha he s of hese hree coponens, naely, he NCRV, follows an ARMA process. Ths ARMA process can be regarded as he (nqe) redced for of he sae space for. The exsence of MN coponen nrodces any coplexes n he dencaon of he sae space for paraeers. We show ha he nber of sae space for paraeers of he NCRV s ore han he aocovarance srcre of he NCRV can nqely deerne. In oher words, he sae space for paraeers of he NCRV are no eecvely dened n he sense ha deren ses of paraeer vales can gve he sae aocovarance srcre (.e., he sae aocovarance generang fncon). See Secon 4 for ore deals. We show ha he sae space for paraeers can be expressed as fncons of he ncondonal ean and varance paraeers of he nderlyng connos{ e SV odel and paraeers regardng MN (he varances of he MN and s sqare). Then, we prove ha hese paraeers are nqely dened. We llsrae how o esae hese denable paraeers and he sae space for paraeers. Wh esaes of he sae space for paraeers, we can esae he IV and MN coponens slaneosly by applyng he Kalan ler o he sae space for. One advanage of or ehod s ha can ler o no only he MN coponens b also he dscrezaon errors. The proposed ehod s appled o yen/dollar spo exchange rae daa. We nd ha he agnde of he (daly) MN coponen s, on average, abo % { 48% of he (daly) NCRV, dependng on he saplng freqency. The res of he paper s organzed as follows. In he nex secon, we nrodce he class of SV odels eployed n hs paper and dene forally he RV, IV, MN and MN coponen. In Secon, we brey sarze he resls n Meddah () on he ARMA represenaon of he IV. In Secon 4, we explan or sae space approach n deal. In Secon 5, we condc an eprcal analyss applyng or ehod o he yen/dollar spo exchange rae. The las secon provdes a sary and concldng rearks. Appendx A provdes deals on he dervaons of he eqaons n he ex. Soe resls are presened n Appendx B.

6 SR-SARV odel, IV, RV and MN. Sqare roo sochasc aoregressve varance (SR-SARV) odel Le p() be he log of he (ecen) spo prce a e. Throgho he paper, we asse ha p() follows he SR{SARV odel consdered n Meddah (), whch s gven by he followng class of connos{e SV odels: dp() = ()dw ; () = +! P (f()) +! P (f()); () where f() s a sae-varable process and he fncons P ( ) and P ( ) are dened so ha: E[P (f())] = E[P (f())] = ; var[p (f())] = var[p (f())] = ; cov[p (f()); P (f())] = ; E[P (f( + h))jf(s); p(s); s ] = exp( h)p (f()); E[P (f( + h))jf(s); p(s); s ] = exp( h)p (f()); 8h > ; () where and are posve real nbers. The ncondonal ean and varance of () are E[ (s)] = and var[ (s)] =! +!, respecvely. Le = exp( ) and = exp( ). In he res of he paper, we work anly wh and nsead of and becase s ore convenen for descrbng or resls. Ths, he odel has a oal of ve free paraeers:,!,!, and. The odel gven n () and () s called he \wo{facor SR{SARV odel". When! =, he odel s refereed o as he \one{facor SR{SARV odel". The SR-SARV odel ncldes any known odels, sch as consan elascy of volaly processes, GARCH dson odels (Nelson, 99), egenfncon sochasc volaly odels (Meddah, ) and posve Ornsen{Uhlenbeck Levy-drven odels (Barndor- Nelsen and Shephard, ). See Meddah () for ore deals.. Inegraed and realzed varances Gven he process of (), he IV s dened as IV Z (s)ds; = ; ; :::; where he n of s deerned dependng on he research objecve. For exaple, f he researcher s neresed n changes n varances of daly (weekly) rerns, s nerpreed as a day (week). Under oderae asspons, we can conssenly esae he IV by he esaor known as he RV, whch s dened as RV () X = r () ; + R where r () p() p( ) = (s)dw (s), and s a posve neger. Here, and hereafer, he noaon \()" ples ha s vale depends on he saplng

7 freqency. For exaple, f denoes a day and we ake observaons every ve nes, hen = 88. In hs case, r (88) denoes a ve-ne rern, becase one day s 5 88 nes. I s well known ha, as!, RV () p! IV (see, e.g., Barndor-Nelsen and Shephard, ). For he wo{facor SR{SARV odel, he varance and aocovarances of IV are expressed n ers of he nderlyng SV odel paraeers as: var[iv ] =! ( log ) (log ) +! ( log ) (log ) ; cov[iv ; IV ] =! ( ) +! ( ) ; and (log ) (log ) () Le d () cov[iv ; IV ] =! ( ) +! ( ) : (log ) (log ) () d RV () IV and () d = 4 + 4! (log ) I can be shown ha () d. MN coponen var[d () ]. For, we have: log + 4! (log ) log : (4)! as!. See Meddah () for he above resls. Now asse ha he observed log-prce p () s conanaed by a easreen error or MN so ha: p () = p() + "(): We asse he followng properes of MN "(). Asspon (a) "() ::d:(; ") wh! " var[" ()] <. (b) "() s ndependen of p(s) for all s and. We do no asse any specc dsrbon for "(). The observed rern r () dened as: s r () p () p = r () where e () "() "( ). I s easy o show ha E h e () = ; var h e () + e () ; (5) = " and cov h e () ; e () = " ; = ; : 4

8 Noe ha var[e () ] and cov[e () ; e () ] do no depend on. We dene he NCRV, P denoed by RV (), as RV () r (). We wre + where RV () = () X = X = r () + r () + = + e () + e () + + X = = RV () e () : + + () ; (6) Noe ha, nlke RV (), () s no necessarly posve becase he rs er of () ay be negave. We call () an MN coponen. We propose a way of esang he MN coponen as well as he IV n a laer secon. In Appendx A, we show ha: E h cov () h () = " and ; () s = 8 < : 8 " + ( )! " " = s;! " = s ; oherwse: Ths, () has he aocovarance srcre of a MA() process. Asse ha he MA() process s expressed as: () = c () + () + () () ; () W N(; () ): (8) (7) The ean and aocovarances of () E h cov () h () = c () ; () s = and 8 >< >: ( + () ) () (), n ers of c (), () = s; () = s ; oherwse: and (), are: (9) Laer, we lze hese wo deren expressons of he oens of () plc relaonshps aong he SV and MA() process paraeers. o derve he ARMA Represenaon of IV In hs secon, we brey sarze he resls n Meddah () on an ARMA represenaon of IV for SR-SARV odels.. One{facor case Meddah (, Proposon.) shows ha f he re process of p() follows a one{ facor SR{SARV odel, hen IV follows an ARMA(, ) process: IV = c IV + IV + + ; () 5

9 where s dened as n he saeen below (), s a whe nose process wh var( ) = and cov( ; d () s ) = for all and s. Oher ARMA(, ) odel paraeers c IV, and are expressed n ers of he one{facor SR-SARV odel paraeers,! and as: p c IV = ( ) ; = 4 ; = ( + )var[iv ] cov[iv ; IV ] ; + () where + corr[iv ; IV ] + corr[iv ; IV ] : I can be shown ha s eqal o he rs order aocorrelaon of he MA process n (),.e., =( ). The corr[iv ; IV ] s gven by corr[iv ; IV ] = ( ) ( log ) : Noe ha corr[iv ; IV ] s a fncon of and does no depend on oher SV odel paraeers, whch, n rn, ples ha s also a fncon of only. Ths s no re for he wo{facor case. Ths sbsanally sples he dencaon proble of he sae space for of he NCRV, as we wll see n Secon 4.. Two{facor case Meddah (, Proposon.) shows ha f he re process of p() belongs o he wo{facor SR{SARV odel, hen IV follows an ARMA(, ) process: IV = c IV + ( + )IV IV ; () where and are dened as n he saeen below Eqaon (), s a whe nose process wh var( ) = and cov( ; d () s ) = for all and s. Le = + and =. Oher ARMA(, ) odel paraeers n () c IV,, and are expressed n ers of he wo{facor SR{SARV odel paraeers,!,!, and as: c IV = ( ) ; = p 4s + ; = = var[iv ] cov[iv ; IV ] cov[iv ; IV ] ; + + p 4s + s ; s () We can rewre he ARMA(, ) for n () wh a ore falar paraeerzaon,.e., IV = c IV + IV + IV p p. The expressons of and n ers of and are gven as = + +4 and = +4, respecvely. 6

10 where = + + ; = ( ); s 4 + sgn( ) s + 5 ; ( ) + ( + )corr[iv ; IV ] corr[iv ; IV ] ( + + ) ( )corr[iv ; IV ] corr[iv ; IV ] ; (4) corr[iv ; IV ] + corr[iv ; IV ] ( + + ) ( )corr[iv ; IV ] corr[iv ; IV ] ; and sgn( ) = f > and sgn( ) = f <. We asse ha 6=, whch ples ha 6=. As n he one{facor case, we can show ha = ( + )=( + + ) and = =( + + ),.e., and are he rs and second order aocorrelaons of he MA process n (), respecvely. See Meddah (, ) for ore deals. 4 Sae Space Approach In hs secon, we explan or sae space approach n deal. Or sae space approach s n he sae spr as he sae space ehod sed n Barndor-Nelsen and Shephard (), who consder he saon who MN. Frs, we gve a sae space for of he NCRV n Secon 4.. The exsence of MN coponens reqres addonal eors for checkng he dencaon of he sae space for. In Secon 4., we show ha he sae space for paraeers are no denable; however, hey can be expressed as fncons of fewer denable paraeers. We llsrae how o esae hese denable paraeers n Secon 4.. In wha follows, we asse ha! = for ease of exposon. Correspondng resls for he wo{facor case can be derved n a slar anner and are sarzed n he Appendx B. 4. Sae space for of he NCRV Sbsng RV () = IV + d () no (6), we have: RV () = IV + d () + () : (5) Fro (8), () and (5), we have he followng sae space for of RV () : Observaon eqaon = 6 4 RV () IV () () d() ; (6a) 7

11 Sae eqaon 6 4 IV () () 7 5 = 6 4 c IV c () () IV () 5 + () () ; (6b) where 4 d() () ; 6 d () 4 () 7 5 C A : (6c) Gven he vales of c IV,,,, c (), (), () and () d, we can esae IV and () by applyng he Kalan ler o he sae space for. One proble of he sae space for s how o esae hose paraeers. One ay sply hnk ha we cold esae he drecly fro he sae space for by, for exaple, qasax lkelhood (QML) esaon nder Gassan nose asspon. We show; however, ha hs approach s no applcable for he sae space for gven n (6). In general, paraeers of a sae space for are no necessarly dened (see, for exaple, Halon, 994, p.88). More precsely, hey are no dened n he sense ha here are nnely any cobnaons of he paraeers ha gve he sae aocovarance srcre. Ths, we have o check wheher sae space for paraeers are nqely dened before proceedng o her esaon. We consder hs proble n he nex sbsecon. In fac, we show ha he above paraeers n he sae space for canno be nqely dened. 4. Idencaon of odel paraeers Becase RV () s he s of hree coponens, IV (an ARMA(, ) process), d () (a whe nose process) and () (an MA() process), RV () self follows an ARMA(, ) process (see Granger and Morrs, 976) so ha s expressed as: ( L)RV () = c () RV + ( + () L + () L ) () ; () W N(; () ): (7) Noe ha he AR coecen s he sae as ha of he IV n (). The ARMA odel represenaon of a sae space for s coonly referred o as a redced for or ARMA redced for. Paraeers of he ARMA redced for are denable. Fro (8), () and (5), we have ( L)RV () = ( L)IV + ( L)d () + ( L) () = c IV d () d () + () +( )c () + ( () ) () () () : (8) Noe ha here and do no follow a Gassan dsrbon. In hs case, he Kalan ler provdes he bes lnear esaor. See Drbn and Koopan () for ore deals on he Kalaan ler. 8

12 The wo expressons on he rgh-hand sdes n (7) and (8) are of he sae process and hence her eans and aocovarances s be dencal. The aocovarances of he MA process n (7) are gven as () = ( + () + () ) () ; () = ( () + () () ) () ; () = () () and j = for j : (9) I s shown n he Appendx A ha he aocovarances of he MA process n (8) are () = ( + ) + ( + ) () d + [ + ( () ) + () () = () d + ( () () () = () () ] () ; (a) + () ) () ; (b) ; (c) and j = for j. By eqang he eans of he MA processes n (7) and (8), we have c () RV = c IV + ( )c () : (d) Gven he ARMA(, ) odel paraeers, c (), RV,, and (), we can calclae, j = ; ;. Then, nknown paraeers n he eqaons (a)(d) are only () j he sae space for paraeers, c IV,,, c (), (), () and () d. Observe ha here seven nknown paraeers and only for eqaons. Hence, we canno nqely denfy hese paraeers fro hese eqaons. In oher words, for a gven ARMA(, ) redced for, here are nnely any ses of vales of c IV,,, c (), (), () and () d ha gve he sae aocovarance srcre as he ARMA(, ) redced for. In vew of (7) and (9), we oban he followng eqaons: c () = "; (a) ( + () ) () = 8 " + ( )! " + 4"; 4 (b) () () =!": (c) Assng ha he MA paraeer sases he nverbly condon,.e., j () j <, we can solve he eqaons (a) (c) for c (), () and () as: c () = "; () =! " () and () = A p A ; () where A = 4 "!" Appendx A. Noe ha < () ". The deals of he calclaon s gven n he!" < becase A >. Fro (), () and (), we see ha c IV,,, c (), (), () and () d are expressed as fncons of,,!, " and!". 4 To ephasze hese relaonshps, 4 They depend also on, as he noaon ples. 9

13 we denoe he as: c IV ( ; ); ( ); ( ;! ); c () ( "); () ( ; ";! "); () d ( ; ;! ) and () ( ; ";! "): () Noe ha s a fncon of only and hence can be assed o be known (becase s dened fro he redced for). Sbsng he expressons n () no Eqaons (a)(d), we have for eqaons for he for nknown paraeers,!, " and!". Hence, he order condon for dencaon s sased. However, hs resl does no ply ha one can nqely denfy,!, " and!". To show he nqeness of he dencaon, we explcly derve he represenaons of,!, " and!" n ers of c (), RV, () j, j = ; :::;. In Appendx A, we show ha, gven c (), RV, () j, j = ; :::; and (), Eqaons (a)(d) are nqely 5 solved for,!, " and!" as: and! " = " = s () ;! = (log ) [ c () RV ( ) () ( ) () + ( + ) () ( ) ( + ) () + +4 () ] ; (4a) D! () ; (4b) 4( + ) = c() RV "; where D = B + ( + )C; (4c) B ( + ) log (log ) and C log (log ) : (4d) These resls ply ha he for paraeers,,!, " and! " are nqely dened fro he ARMA(, ) redced for n (7). Hence, n prncple, we can esae he. Agan, shold be ephaszed ha hese resls do no ply ha one can drecly esae he sae space for paraeers b raher ha one can esae he above for paraeers by replacng he sae space for paraeers wh he fncons of he for paraeers. The esaes of he sae space for paraeers are obaned by sbsng he esaes of he for paraeers no hese fncons. 4. Esaon of odel paraeers We llsrae how o esae he for paraeers. There are wo possble approaches: drec and ndrec. Below, we llsrae rs he ndrec and hen he drec approach. In boh approaches, we apply QML esaon assng Gassan nnovaons. We showed n (4) ha hese for paraeers have explc expressons n ers of he ARMA(, ) redced for paraeers. Ths sggess he followng ndrec 5 More precsely, nder he condon " >.

14 approach for esang hese for paraeers. Sary of he ndrec approach Sep For a gven, calclae RV (). Sep Esae he nresrced ARMA(, ) odel n (7) by QML esaon assng Gassan nnovaons. Sep Gven he esaes of c (), RV, (), () and obaned n sep, calclae he rs hree aocovarances of he MA process, naely, () j, j = as n (9). Sep 4 Gven he esaes of c (), RV and () j, j = obaned n seps and, esae!", ",! and applyng he resls n (4). Ths approach s sple and easy o pleen; however, does no garanee ha he reslng paraeer esaes are posve becase of he nevable ncerany of he ARMA odel esaon. For exaple, f () >, hen he esae of!" by hs approach s negave becase > by asspon. Alernavely, one can drecly esae hese for paraeers. In hs approach, one calclaes he log-lkelhood drecly fro he for paraeers sng he resls n (4) and axzes wh respec o he for paraeers. Ths, we can easly pose he posvy of he for paraeers. Below, we sarze how o oban he QML esaes by hs approach. Sary of he drec approach Sep For a gven, calclae RV (). Sep Gven,,!, " and!", calclae c IV,,, c (), (), () accordng o (), () and (). and () d Sep Wh he c IV,,, c (), (), () and () d obaned n sep, calclae he Gassan log-lkelhood of he sae space for n (6) for RV Sep 4 Maxze he log-lkelhood obaned n sep wh respec o he ve paraeers,,!, " and! " o oban he QML esaes. Ths approach provdes conssen esaors for he for paraeers. Before closng hs secon, shold be noed ha f we can oban esaes properly by he ndrec approach, we do no need o proceed o he drec approach, becase boh approaches wll gve he dencal esaes n hs case. 5 Eprcal Analyss In hs secon, we condc an eprcal analyss wh exchange rae daa sng he proposed sae space ehod..

15 5. Daa descrpon The yen/dollar spo exchange rae seres we se are he d-qoe prces observed every one ne, whch are obaned fro Olsen and Assocaes. The fll saple covers he perod fro Janary, o Deceber, 6. Fgre plos he daly rerns calclaed fro he prce daa over he perod. Prce daa are no avalable for each ne. When prce daa are ssng we apply he prevos ck ehod,.e., we nerpolae he os recen observed prce. Frherore, followng Andersen, Bollerslev, Debold and Labys (), we reove he daa of nacve radng days. Whenever we do so, we always reove fro : GMT on one ngh o : he nex evenng. For deals on he ovaon behnd hs denon of \day", see Andersen, Bollerslev, Debold and Labys (), Andersen and Bollerslev (998) and Bollerslev and Doowz (99). We c he daa accordng o he followng crera, whch are slar o he crera adaped n Bene e al. (7). Speccally, we c () he days where here are ore han 5 ssng prce observaons, () he days where, n oal, here are ore han nes of zero rerns () he days where he prce does no change for ore han 5 nes. By hese crera, we cold reove all weekend daa. However, he days sch as US holdays ha Andersen, Bollerslev, Debold and Labys () and Bene e al. (7) reove are no necessarly reoved by hese crera. Ths s becase even when he US arke s closed, ransacons are ade n oher arkes. Evenally, we are lef wh 89 coplee days, or = 6496 prce observaons, fro whch we calclae he one{ne and ve{ne rerns. Wh hese rerns, we calclae wo seres of daly NCRV, naely, one{ne NCRV ( = 44) and ve{ne NCRV ( = 88). Table repors he descrpve sascs of hese wo seres of NCRV, and Fgre plos he. The saple ean of he one{ne NCRV s greaer han ha of he ve{ne NCRV. Ths s conssen wh he exsence of MN becase he ean of he NCRV ncreases as he saplng freqency ncreases, or! nder he exsence of MN (see (d) and (a) ). The rs order aocorrelaons of hese wo seres of NCRV are soewha lower han sally expeced for varances of nancal e seres: hey are :4794 for he one{ne NCRV and :477 for he ve{ne NCRV. Ths ay be becase of he exsence of MN. In fac, n he nex sbsecon, we show ha esaes of he rs order aocorrelaon of he IV are sgncanly hgher han hese vales. 5. Esaon of paraeers, IV and MN coponen For hese wo seres of he NCRV, we esae he paraeers of he one{ and wo{ facor SV odels by he ehod descrbed n Secon 4. (and n Appendx B for he wo{facor case). Noe ha, n general, he vales of hese wo NCRV seres are deren alhogh hey boh are esaes of he sae IV seres. Conseqenly, he esaes of he SV odel paraeers are deren, dependng on whch NCRV seres s sed. We repor only he resls by he drec approach becase he ndrec approach does no provde posve varance esaes. Table dsplays he

16 esaes of he SV odel paraeers. Narally, he esaed vales of he SV odel paraeers for one{ne and ve{ne NCRV seres are very slar. In boh he one{ and wo{facor cases, esaes wh he ve{ne NCRV seres are slghly ore ecen han hose wh he one{ne NCRV seres accordng o he robs sandard errors. 6 The esaes of he perssence paraeers for wo{facor SV odel (.e., b and b ) ply ha here are wo facors wh sgncanly deren levels of perssence. One of he s very perssen and he oher s oderaely perssen, alhogh her ncondonal varances are no sgncanly deren. For he one{facor SV odel, he perssence of hese wo facors s be capred by only one paraeer,. As a resl, he esae of n he one{facor case s soewha lower han ha n he wo{facor case. The esaes of sae space for paraeers n (6) (and n (4) for he wo{ facor case) are coped fro he esaes of he SV odel paraeers. They are shown n Table. Agan, he esaes of he coon paraeers, whch do no depend on, are very slar. We nd ha he esaes of he ean of he MN coponen, denoed by bc (), n one{ne NCRV seres s greaer han ha n ve-ne NCRV seres, whch ples ha he one{ne NCRV seres has a larger bas han he ve{ne NCRV seres. Ths s conssen wh he heory. The agnde of bas of he one{ne NCRV s abo for es larger han ha of he ve{ne NCRV. Table 4 repors he esaes of soe poran vales ncldng he aocorrelaons of he IV. In boh he one{ and wo{facor cases, he esaes of he rs order aocorrelaon of IV are sgncanly hgher han hose of he wo NCRV seres. Ths resl sggess ha he exsence of MN lowers he aocorrelaons of he NCRV seres. The esaes of he rao of he ncondonal varance of he MN coponen o he ncondonal varance of he NCRV ply ha abo half of he aggregae caons of he NCRV seres s becase of he MN coponen. We dsplay he esaes of he IV seres by Kalan soohng for he ve{ ne and one{ne NCRV seres n Fgres (a) and (b), respecvely. Fgre (44) (88) () (c) s he derence beween he, or IV c c IV, where IV c s he esae of IV wh a gven. Noe ha hese esaes are he esaes of he sae IV seres and hs are very slar. The IV esaes n he one{facor case see sooher han hose n he wo{facor case. Ths s becase of he resl ha he (esaed) aocorrelaons of he IV seres are lower for he wo{facor case and hs hey are relavely closer o whe nose copared wh he IV seres obaned for he one{facor case. Fgre 4 (a), (b) and (c) plo he soohed esaes of he ve-n and one-ne MN coponen seres and her derences, respecvely. We can see ha he MN coponen occasonally akes a large vale. Fgre 5 (a) and (b) dsplay he esaes of he dscrezaon errors by Kalan soohng for 6 To oban he QML esaes of he SV odel paraeers, rs, we calclae he QML esaes of he ransfored ones, sch as log( ), by applyng an nconsraned axzaon procedre. Then, he QML esae of, for exaple, s obaned by log(b), where b s he QML esae of. The robs sandard errors of he SV odel paraeer esaes are calclaed as follows. Frs, generae saples fro he asypoc noral dsrbon of he esaors of he ransfored paraeers wh her robs asypoc covarance arx esaes. Nex, for each saple, calclae he esaes of he SV odel paraeers. Lasly, calclae he saple sandard devaons of hese SV odel paraeer esaes, whch are or robs sandard errors.

17 ve{ne and one{ne NCRV, respecvely. The dscrezaon error esaes for he one{ne NCRV seres s qe sall han hose for he ve{ne NCRV seres, whch s agan conssen wh he heorecal resl. Correspondng gres for he wo{facor case are gven n Fgres 6{8. They are very slar o hose for he one{facor case. Fnally, we calclae he raos of he MN coponen o he NCRV R b(). They are gven by R b() = b () =RV (), = ; :::; 89, where b () s he esae of () by Kalan soohng. The resls are shown n Table 5. In he one{facor (wo{facor) case, he ax and n vales of R b() are, respecvely, :84 (:6574) and :584 ( :7) for he ve{ne NCRV seres and :857 (:454) and :584 ( :49) for he one{ne NCRV seres. We also calclae he average agnde of he MN coponen as he ean of jr () j (he average of R () s also repored n Table 5). In he one{facor (wo{facor) case, he vale of he ean s :4659 (:8) for he ve{ne NCRV seres and :478 (:477) for he onene NCRV seres. Fro hese resls, we conclde ha he average agnde of he MN coponen n he daly NCRV ranges fro % o 48% of NCRV, dependng on he saplng freqency. 6 Sary and Concldng Rearks In hs paper, we proposed a sae space approach o esang he IV and MN coponens slaneosly. Or ehod s based on he resl n Meddah (), who shows ha when he re log-prces follow a general class of connos{e SV odels, he IV follows an ARMA process. We showed ha nder he exsence of MN, he observed RV, or he NCRV, also follows an ARMA process. We represened he NCRV by a sae space for and esablshed he nqeness of he dencaon of he sae space for paraeers. The proposed ehod was appled o yen/dollar exchange rae daa, where we fond ha he NCRV calclaed wh ve{ne rerns s less based han wh one{ne rerns. The wo seres of IV esaes by he proposed ehod wh one-ne and ve{ne rerns are very slar. The ehod was also sed for esang he MN coponen. In he esaon, we consrced he log-lkelhood sng only eher he one{ ne or ve{ne NCRV seres. I s ore desrable o se boh NCRV seres for esang he coon paraeers. I wold be possble o oban ore ecen esaors by cobnng he one{ and ve{ne NCRV seres. Ths s a sbjec for fre research. I s also poran o relax he asspon ha here s no leverage eec n order o or ehod o sock rern daa. 4

18 Appendx A: Dervaons of Eqaons and e (), and le " de- Hereafer, we sppress \()" n he noaons r (), () noe "() for noaonal splcy. Dervaon of (7) Becase var(e ) = " and r s ndependen of e by Asspon, we have: P E[ ] = P h E hr + e + + E e + = = P = E hr + E he + + var(e ) = = ": To derve var[ ] and cov[ ; = s, we have: ], we calclae cov[r s e s ; r e ] and cov[e ; e s]. When cov [r e ; r e ] = E [r e ] (E [r e ]) = E [e ] E R [r ] (E [r ]) (E [e ]) = "E[( R (s)dw = (s)) ] = "E[ = (s)ds] = " : The forh eqaly coes fro he Io soery. When 6= s, we have: cov [r s e s ; r e ] = E [r s e s r e ] E [r s e s ] E [r e ] = E [e s e ] E [r s ] [r ] E [r s ] E [e s ] E [r ] E [e ] = : When = s, we have: cov[e ; e ] = var[e ] = E[e h 4 ] (E[e ]) = E " 4 4" " + 6" " = E[" 4 ] + ": 4 4" " + " " (5) (6) (7) When = s, we have: h cov e s; e = cov s = cov h = var[" s] =! ": e s+ h " s+ ; e s " s+ " s + " s; " s " s " s + " s (8) When = s for, we have cov[e ; e s ] =. Frherore, we have cov[r e ; e s] = for any and s becase: cov[r e ; e s] = E[r e e s] E[r e ]E[e s] = E[r ]E[e e s] E[r ]E[e ]E[e s] = : 5 (9)

19 Fro (5) (9), we have: var [ ] = var = 4var P = 4 P P r + e + + P = r + e + = P h cov P + var P e = r + e + ; r + j h = e + e + + j = j= P +4 cov r + e + ; e + = j= = 8" + (E[" 4 ] + ") 4 + ( )(E[" 4 ] ") 4 = 8" + ( )! " + 4"; 4 P + 4cov = P P + = j= cov P r + e + ; h e ; e + + j = e + and P cov[ ; + ] = cov = 4cov +cov = cov =! ": P = P = h e ; e + = P r + e + ; r + e + ; P r + e + + P = = e + = e + r + e + + cov + cov P e = P = P ; P r + e + + e + = = P r + e + ; P ; e + + = = e + I s easy o check ha cov[ ; ] = for, and hence we have (7). Dervaon of (a)(c) Here, we derve he aocovarances of he MA process n (8). by They are gven = covf + + d d + + ( ) ; + + d d + + ( ) g = + + d + d + + ( ) + = ( + ) + ( + ) d + [ + ( ) + ] ; = covf + + d d + + ( ) ; + + d d + + ( ) g = d + ( + ) = covf + + d d + + ( ) ; + + d d + + ( ) 4 g = : Becase follows an MA() process, he aocovarances of he order greaer han s zero. Dervaon of () 6

20 Fro (c), we have =! "=. Sbsng hs no (b), we have: ( + )! " = 8 " + ( )! " ": Mlplyng boh sdes by =! " and rearrangng, we have: 4 " " + = :!"!" The wo solons of hs qadrac eqaon for are gven by = A p A ; where A = 4 "! " Becase A > for, we have A + p A nverbly condon, we oban n () " :!" >. Assng ha sases he Dervaon of (4a) and (4c) Fro (c) and (c), we have!" = (4) and (), we have, whch s he rs resl n (4a). Fro (), = B! + and d = 4 + C! ; () where B and C are as gven n (4d). Fro!" = n (c), we have: ( ) = + + +!" h = + ( + )!"; and ( + ) = +!" h = + +!": Sbsng (), () and () no (a) and (b), we have: and = D! = E! 4 () () + ( + )! "; (a) + ( + )!"; (b) where D = B + ( + )C, E = B C and = =( ). Fro (), we have: + ( + ) = [ D + ( + )E]! + [( + ) ]! "; = [ + ( + )] B! + ( + 4 )! "; = ( ) ( + ) (log )! + ( + 4 )! "; 7 (4)

21 where, o oban he hrd eqaly, we se he alernave expresson of explaned below (). Fro (4), we have:! = (log ) [ + ( + ) ( + 4 )! "] : ( ) ( + ) Sbsng!" = (), we have:, we oban he second resl n (4a). Nex, noe ha fro + = + = + (A p A ) p A A = A + p p A + (A A ) (A + p A ) p (A A )(A + p A ) = A: (5) Fro (d) and (a), we have: c RV = ( ) + " ; or " = c RV ( ) ( ) : (6) Sbsng " n (6) no A n (), we have: A = = = 4! " h c RV ( ) c RV ( )! " 4c RV ( )! " ( ) 4 + +! " c RV ( ) ( )! " + + c RV ( )c RV +( ) 4 4! " + ( ) + c RV ( )! " ( )! " 4 c RV ( )! " + 4! " (7) = c RV ( )! " 4! " + ( ) + c RV ( )! " : Fro (a), (5) and (7), we have: = D! +( ( + ) 4 + ( + )c RV ( ) )( + )! " + ( + )c RV ( )! ": (8) Mlplyng boh sdes n (8) by =( + ) and rearrangng, we have: 4 c RV c RV ( ) + ( D! +!") + ( )! " = : Solvng hs qadrac eqaon for, we have: = c RV s c RV ( ) + ( )! " ( D! +!") : (9) ( + ) 8

22 Fro " >, < and (6), we s have c RV >. Hence, he sgn of he second er n (9) s negave. Fro (6) and (9), we have: " = s c RV ( ) + ( )! " Fro (9) and (4), we oban (4b) and (4c). ( D! +!") : (4) ( + ) 9

23 Appendx B: Resls for he Two{facor Case Le = +, =, = + +, = ( ) and! 6= hrogho Appendx B. Fro (8), () and (5), we can express he NCRV n he followng sae space for: Observaon eqaon RV = 6 4 IV IV d ; (4a) Sae eqaon 6 4 IV IV 7 5 = 6 4 c IV c IV IV ; (4b) where he ean vecor and varance arx of (d ; ; ) are as gven n (6c). Aocovarance fncons In he wo{facor case, by applyng he resls n Granger and Morrs (976), we can show ha he NCRV follows an ARMA(, ) process: ( L L )RV = c RV + ( + L + L + L ) ; W N(; ): (4) The sae RV can alernavely be expressed as: ( L L )RV = c IV d d d +( )c + + ( ) ( + ) ; (4) The aocovarance fncons of he MA process n (4) are gven as: = ( ) ; = ( + + ) ; = ( + ) ; = ; (44)

24 and j =, for j 4. Frherore, soe calclaons lead s o he followng aocovarance fncons of he MA process n (4): = ( + + ) + d + = ( + ) d + = = ; d + + ; + + ; ; (45a) (45b) (45c) (45d) and j =, for j 4. By eqang he eans of he MA process n (4) and (4), we oban: c RV = c IV + ( )c : (45e) As n he one{facor case, he nber of sae space for paraeers s greaer han he nber of ARMA redced for paraeers. Ths ples ha he sae space for n (4) s no dened. However, we show ha he sae space for paraeers are expressed as fncons of he nderlyng connos SV odel paraeers,!,!, " and! ", whch are nqely dened fro he ARMA redced for n (4). Idencaon of sae space for paraeers Here we show ha he paraeers,!,!, " and! " are nqely dened fro he redced for paraeers c RV ;,, and j for j =. As n he one{facor case, we can nqely solve Eqaons (45a)(45e) wh respec o,!,!, " and! " as:! " = ;! = (log ) [( ) ( + )] ; (46a) ( ) ( )( ) and! = (log ) [( ) + ( + )] ; (46b) ( ) ( )( + ) " = s c RV ( ) + ( )! " + H; (46c) where = c RV "; (46d) = + [ ( ) ]! "; = ( + ) ; = + ; = + ( + )( + )! "; = ; (47)

25 H = " + X j= ( C ;j + C ;j C ;j + C 4;j )! j +! " C ;j j log j (log j ) ; C ;j ( j) (log j ) ; C ;j j( j ) (log j ) ; and C 4;j = ( j log j ) (log j ) for j = ; : # ; (48) In wha follows, we derve he resls n (46). Fro =! " n (c) and = n (45d), we have! " =, whch s he rs resl n (46a). Frherore, fro (), (4) and (), afer soe calclaons, follows ha: (49) where = B! B! + + and d = 4 + C 4;! + C 4;! ; (5) B j = C ;j C ;j C ;j for j = ; : (5) Sbsng (5) no he aocovarance fncons n (45) and rearrangng, we have: = D! + D! = E! + E! 4 = F! + F! 4 +! "; (5a) + ( )! "; (5b) + +! "; (5c) where D j = B j + C 4;j, E j = B j C 4;j, F j = B j C 4;j for j = ;, = ( + )=( + + ) and = =( + + ). Hence, we have and + = ( + ) (B! + B! ) + [ ( ) ]! "; (5a) = ( ) (B! + B! ) [ ( ) + ]! ": (5b) Nong ha and can be expressed as n (4) (see he explanaons below ()), we have: and = var[iv ] + ( + )cov[iv ; IV ] cov[iv ; IV ] ( + + ) = P j= [ C ;j + ( + )C ;j C ;j ]! j B! + B! = var[iv ] cov[iv ; IV ] + cov[iv ; IV ] ( + + ) = P j= ( C ;j C ;j + C ;j )! j B! + B! : ; (54a) (54b)

26 Sbsng B j n (5), and n (54) no (5), we have: P + = ( C ;j C ;j C ; )! j j= P + [ C ;j + ( + )C ;j C ;j ]! j j= +[ ( ) ]! " P = f[ + ( + )]C ;j ( + )C ;j g! j j= +[ ( ) ]! "; (55) and = P j= j= ( C ;j C ;j + C ;j )! j P [ C ;j + ( + )C ;j C ;j ]! j [ ( ) + ]! " P = f[ ( + )]C ;j + ( + )C ;j g! j j= [ ( ) + ]! " P = f( )C ;j + C ;j g! j j= ( + )( + )! ": (56) We can regard (55) and (56) as he followng syse of wo eqaons for! and! : = ( C ; C ; )! + ( C ; C ; )! = ( C ; + C ; )! + ( C ; + C ; )! ; (57) where,,, and are as gven n (47). Solvng (57), we have! = ( )C ; ( + )C ; ( + )C ; C ; ( + )C ; C ; = (log ) [( ) ( + )] ; ( ) ( )( + ) and! = ( )C ; + ( + )C ; ( + )C ; C ; ( + )C ; C ; = (log ) [( ) + ( + )] : ( ) ( )( + ) Fro (45e) and (a), we have: " = c RV ( ) ( ) : (58)

27 Sbsng " n (58) no A n (), we have: A = c RV ( )! " 4! " c RV + ( ) + : (59) ( )! " Fro (5), (5c) and (59), we have: =! ( B C 4; ) +! ( B C 4; ) 4 h c n RV 4 + ( ) + c ( )!"! " RV ( )! " + o! " = (B! + B! ) (! C 4; +! C 4; ) + 4 c RV ( )! "( ) P = ( C ;j C ;j + C ;j C 4;j )! j j= + 4 c RV ( )! "( ) c RV ( )! " c RV ( )! ": (6) Mlplyng boh sdes n (6) by = and rearrangng, we have: 4 c RV c RV ( )! "( ) H; where H s as gven n (48). Solvng he qadrac eqaon for, and by he sae argen as sed n (4), we have: s c RV c RV = ( ) + ( )! " + H; (6) and s " = c RV ( ) + ( )! " + H: (6) Fro (6) and (6), we have (46c) and (46d). Fnally, we sarze drec and ndrec approaches for esang he paraeers n he wo facor case. Sary of he ndrec approach Sep For a gven, calclae RV (). Sep Esae he nresrced ARMA(, ) odel n (4) by QML esaon assng Gassan nnovaons. Sep Gven he esaes of c (), RV,, 7 (), (), () and obaned n sep, calclae he rs for aocovarances of he MA process, naely, () j, j = as n (44). 7 These can be obaned fro he esaes of and. See foonoe. 4

28 Sep 4 Gven he esaes of c () RV,, and () j, j =, obaned n seps and, esae! ", ",!,! and, applyng he resls n (46). Sary of he drec approach Sep For a gven, calclae RV (). Sep Gven,,,!,!, " and! ", calclae c IV,,,, c () and () d accordng o (), () and ()., (), () Sep Wh he c IV,,,, c (), (), () and () d obaned n sep, calclae he Gassan log-lkelhood of he sae space for n (4), for RV Spe 4 Maxze he log-lkelhood obaned n Spe wh respec o he seven paraeers,,,,!,!, " and! " o oban he QML esaes.. 5

29 References Andersen, T. G., (994), Sochasc Aoregressve Volaly: Volaly Modelng, Maheacal Fnace, 4, 75{. A Fraework for Andersen, T. G., and T. Bollerslev, (998), Deche Mark{Dollar Volaly: Inraday Acvy Paerns, Macroeconoc Annonceens, and Longer Rn Dependences, Jornal of Fnance, 5, 9{65. Andersen, T. G., T. Bollerslev, F. X. Debold, and H. Ebens, (), The Dsrbon of Sock Rern Volaly, Jornal of Fnancal Econocs, 6, 4{76. Andersen, T. G., T. Bollerslev, F. X. Debold, and P. Labys, (), The Dsrbon of Exchange Rae Volaly, Jornal of he Aercan Sascal Assocaon, 9, 4{55. Band, F. M., and J. R. Rssell, (6), Separang Mcrosrcre Nose fro Volaly, Jornal of Fnancal Econocs, 79, 655{69. Band, F. M., and J. R. Rssell, (8), Mcrosrcre Nose, Realzed Varance, and Opal Saplng, Revew of Econoc Sdes, 75, 9{69. Barndor-Nelsen, O. E., and N. Shephard, (), Non-Gassan OU Based Models and Soe of Ther Uses n Fnancal Econocs, Jornal of he Royal Sascal Socey, Seres B (Sascal Mehodology), 6, 67{4. Barndor-Nelsen, O. E., and N. Shephard, (), Econoerc Analyss of Realzed Volaly and Is Use n Esang Sochasc Volaly Models, Jornal of Royal Sascal Assocaon Seres B (Sascal Mehodology), 64, 5{8. Bene, M., J. Lahaye, S. Laren, C. J. Neely, and F. C. Pal, (7), Cenral Bank Inervenon and Exchange Rae Volaly, Is Connos and Jp Coponens, Inernaonal Jornal of Fnance and Econocs,, {. Bollerslev, T., and I. Doowz, (99), Tradng Paerns and Prces n he Inerbank Foregn Exchange Marke, Jornal of Fnance, 48, Bollerslev, T., R. F. Engle, and D. B. Nelson (994), ARCH Models, 959-8, n Handbook of Econoercs 4, eded by R. F. Engle, D. L. McFadden. Elsever, NY. Drbn, J., and S. J. Koopan, (), Te Seres Analyss by Sae Space Mehods. Oxford Unversy Press, NY. Ghysels, E., A. C. Harvey, and E. Renal, (996), Sochasc Volaly, 9-9, n Handbook of Sascs 4, eded by G. S. Maddala and C. R. Rao. Elsever, NY. Granger, W. J. C., and M. J. Morrs, (976), Te Seres Modellng And Inerpreaon, Jornal of he Royal Sascal Socey. Seres A (General), 9, 46{57. Halon D. J., (994), Te Seres Analyss. Prnceon Unversy Press, NJ. 6

30 Hansen, P. R. and A. Lnde, (6), Realzed Varance and Marke Mcrosrcre Nose, Jornal of Bsness and Econoc Sascs, 4, 7{6. Meddah, N., (), An Egenfncon Approach for Volaly Modelng, CIRANO Workng Paper, s{7. Meddah, N., (), ARMA Represenaon of Two-facor Models, CIRANO Workng Paper, s{9. Meddah, N., (), ARMA Represenaon of Inegraed and Realzed Varances, Econoercs Jornal, 6, 5{56. Meddah, N., and E. Renal, (4), Teporal Aggregaon of Volaly Models, Jornal of Econoercs, 9, Nelson, D. B., (99), ARCH Models as Dson Approxaons, Jornal of Econoercs 45, 7{9. Pal, F. C., (996), GARCH Models of Volaly, 9-4, n Handbook of Sascs 4, eded by G. S. Maddala and C. R. Rao. Elsever, NY. Shephard, N., (5), Sochasc Volaly, Seleced Readngs. Oxford Unversy Press, Oxford. Zhang, L., P. Mykland, and Y. Al-Sahala, (5), A Tale of Two Te Scales: Deernng Inegraed Volaly wh Nosy Hgh-freqency Daa, Jornal of he Aercan Sascal Assocaon,, 94{4. Zho, B., (996), Hgh-freqency Daa and Volaly n Foregn-exchange Raes, Jornal of Bsness and Econoc Sascs, 4, 45{5. Zvo, E., (8), Praccal Isses n he Analyss of GARCH Models, forhcong n Handbook of Fnancal Te Seres, eded by T.G. Andersen, R.A. Davs, J-P Kress and T. Mkosch. Sprnger. 7

31 Table : Descrpve sascs of he NCRV One{ne NCRV Fve{ne NCRV Mean :57 :49 Varance :69 :6 SD :57 :49 AC() :4794 :477 AC() :68 :9 AC() :6 :89 AC(4) :94 :595 AC(5) :46 :577 Noe: The able repors he saple ean (Mean), saple varance (Varance) and saple sandard devaon (SD) of he RV seres calclaed wh deren, where s he nber of nervals for each NCRV seres. AC(k) denoes he saple aocorrelaon of order k. Table : Esaes of SV odel paraeers One{facor case Two{facor case b One{ne Fve{ne One{ne Fve{ne b (.45) (.4) (.87) (.4) b - - (.) (.45) (.47) (.78) (.47) (.5) b! (.) (.8) (.46) (.64) b! (.) (.67) b " (.) (.9) (.57) (.6) b! " (.9) (.4) (.9) (.48) L Noe: L s he log-lkelhood. The robs sandard errors are n parenheses. 8

32 Table : Esaes of sae space odel paraeers One{facor case Two{facor case One{n Fve{ne One{ne Fve{ne bc IV b b b b b bc () b () b () b () d.... Table 4: Esaes of soe poran vales One{facor case Two{facor case One{ne Fve{ne One{ne Fve{ne \ var[iv ] \ corr[iv ; IV ] \ corr[iv ; IV ] \ var[ () ] \ \ var[iv ]= var[rv () ] \ var[ () ]= b =(b + b () \ var[rv () ] b () d ) b () =(b + b () + b () d ) Noe: \ var[rv () ] = var[iv \ ] + var[ \() ] + var[d \() ]. 9

33 Table 5: Mean, ax and n of he raos of MN coponens o NCRV One{facor case Two{facor case One{ne Fve{ne One{ne Fve{ne ean of b R () ean of j b R () j axf b R () g nf b R () g axfj b R () jg nfj b R () jg

34 Fgre : Daly rerns of he yen/dollar exchange rae Daly Rern (%) YEAR

35 NCRV (%) NCRV (%) Dfferences Fgre : -n and 5-n NCRV seres (a) 5-n NCRV seres YEAR (b) -n NCRV seres YEAR (c) Dfferences beween -n and 5-n NCRV seres YEAR Noe: Fgre (c) dsplays -n NCRV seres ns 5-n NCRV seres.

36 IV (%) IV (%) Dfferences (%) Fgre : Soohed seres of IV n he one-facor case (a) 5-n IV seres YEAR (b) -n IV seres YEAR (c) Dfferences beween -n and 5-n IV seres YEAR Noe: The fgre (c) dsplays -n IV seres ns 5-n IV seres.

37 MN Coponen (%) MN Coponen (%) Fgre 4: Soohed seres of MN coponen n he one-facor case (a) 5-n MN coponen seres YEAR (b) -n MN coponen seres YEAR Dfferences (%) (c) Dfference beween -n and 5-n MN Coponen seres YEAR Noe: Fgre 4(c) dsplays -n MN coponen seres ns 5-n MN coponen seres. 4

38 Fgre 5: Soohed seres of dscrezaon error n he one-facor case Dscrezaon Error (%) Dscrezaon Error (%) (a) 5-n dscrezaon error seres YEAR (b) -n dscrezaon error seres YEAR 5

39 IV (%) IV (%) Dfferences (%) Fgre 6: Soohed seres of IV n he wo-facor case (a) 5-n IV seres YEAR (b) -n IV seres YEAR (c) Dfferences beween -n and 5-n IV seres YEAR Noe: Fgre 6(c) dsplays -n NCRV seres ns 5-n NCRV seres. 6

40 Fgre 7: Soohed seres of MN coponen n he wo-facor case MN Coponen (%) MN Coponen (%) (a) 5-n MN coponen YEAR (b) -n MN coponen seres YEAR Dfferences (%) (c) Dfference beween -n and 5-n MN Coponen seres YEAR Noe: Fgre 7(c) dsplays -n MN coponen seres ns 5-n MN coponen seres n he wo facor case. 7

41 Fgre 8: Soohed seres of dscrezaon error n he wo-facor case Dscrezaon Error (%) Dscrezaon Error (%) (a) 5-n dscrezaon error seres YEAR (b) -n dscrezaon error seres YEAR 8