A State Space Approach to Estimating Integrated Variance and Microstructure Noise

Transcription

1 A Sae Space Approach o Esang Inegrae Varance an Mcrosrcre Nose Daske Nagakra an Toshak Waanabe Ocober, 8 Absrac In hs paper, we conser esaon of negrae varance (IV) as well as crosrcre nose by sng a sae space eho. Or eho s base on he resl obane n Meah (3) an Barnorff-Nelsen an Shephar (), naely, when he nerlyng re log-prce process follows a ceran class of connos-e sochasc volaly (SV) oels, he IV follows an ARMA(, ) process. Assng ha an observe log-prce s he s of he re log-prce an an... crosrcre nose, we show ha he realze varance (RV) calclae fro he observe log-prces, whch we call nose-conanae RV (NCRV), follows an ARMA(, ) process. We represen he NCRC by a sae space for showng ha he oel paraeers are enfable. We esaes he IV an crosrcre noses by applyng he Kalan flerng o he sae space for. The propose eho s apple o yen/ollar exchange rae aa. We fn ha he agne of a crosrcre nose s, n average, abo 3% 3% of he NCRV. Key Wors: Realze Varance; Inegrae Varance; Mcrosrcre Nose; Sae Space Ths research s parally sppore by Gran-n-A for Scenfc Research 839 fro he Japanese Mnsry of Ecaon, Scence, Spors, Clre an Technology. Vews expresse n hs paper are hose of ahors an o no necessarly reflec he offcal vews of he Bank of Japan Inse for Moneary an Econoc Ses, Bank of Japan, E-al: aske.nagakra@boj.or.jp Inse of Econoc Research, Hosbash Unversy, E-al: waanabe@er.h.ac.jp

2 Inrocon The varance of fnancal asse rerns are known o change over e. More specfcally, he varance, or volaly, he sqare roo of he varance, ens o be large (sall) followng sccessve large (sall) varances n prevos peros. Ths phenoenon s known as volaly clserng. The oel, known as he sochasc volaly (SV) oel, n whch he volaly s changng over e, have been se for oelng hs kn of volaly ynacs of fnancal asses. Base on he SV oel wh esae oel paraeers, researchers can esae he changng varances. See, for exaple, Ghysels, Harvey an Renal (996) for he revews of soe of he oler papers on SV oels an Shephar (5) for a ls of selece papers n he SV lerare. Recenly, a new class of esaors for he changng varances has been evelope by Barnorff-Nelsen an Shephar (, ), Anersen, Bollerslev, Debol an Ebens () an Anersen, Bollerslev, Debol an Labys (). The esaor s calle he realze varance (RV). The RV eploys very hgh freqency fnancal aa sch as ne-by-ne rern aa or enre recors of qoe or ransacon prce aa. Uner ceran asspons, he RV s a conssen esaor for he negrae varance (IV). The foral efnon of IV wll be gven n Secon. Invely, s a easre of a varably of fnancal asses over a pero specfe by researchers (for exaple, a ay). The RV s a oel-free esaor n he sense ha we o no have o specfy he volaly ynacs. One of he key asspons neee for he conssency of he RV s ha here are no easreen errors n observe log-prces. The easreen error s parclarly calle crosrcre nose an s e o, for exaple, screeness of prces, bask bonce an rreglar rang. When hs asspon s volae, he RV s no longer a conssen esaor of he IV. I can be shown ha ner he exsence of crosrcre noses, he RV verges as he saplng freqency ens o be hgh. Several alernave esaors of he RV, ha are conssen even ner he exsence of crosrcre noses, have been propose by Zho (996), Zhang, Myklan an A l-sahala (5), Hansen an Lne (6) an Ban an Rssell (6). See also Ban an Rssell (8) whch conser a ean-sqare-error opal saplng heory for recng he effec of crosrcre nose. In hs paper, we propose a new approach for esang he IV ner he exsence of crosrcre noses. Or eho s base on he resl obane n Meah (3) an Barnorff-Nelsen an Shephar (), naely, when he nerlyng re log-prce process follows a ceran class of connos-e SV oels, he IV follows an ARMA(, ) process. Assng ha an observe log-prce s he s of he re log-prce an an... crosrcre nose, we show ha he RV calclae fro he observe log-prces, whch we call nose-conanae RV (NCRV), follows an ARMA(, ) process. Then, we represen he NCRC by a sae space for showng ha he oel paraeers are enfable. We esaes he IV as well as crosrcre noses by applyng he Kalan flerng o he sae space for. The propose eho s apple o yen/ollar spo exchange rae aa. We fn ha he agne of a crosrcre nose s, n average, abo 3% 3% of he NCRV for each ay epenng on he saplng freqency. The res of he paper s organze as follows. In he nex secon, we nroce

3 a class of SV oels eploye n hs paper. We also efne forally he RV, IV, an crosrcre nose n hs secon. In Secon 3, we explan or approach for esang IV an crosrcre noses n eals. Secon 4 proves an eprcal analyss applyng or eho o yen/ollar spo exchange rae. The las secon proves sary an conclng rearks. Connos Te SV oel an RV. One facor SR-SARV oel Le p() be he log of (effcen) spo prce a e. We sppose ha p() follows he class of connos-e SV oel consere n Meah (3), whch s ere as one facor sqare-roo sochasc aoregressve varance (SR-SARV) oel. In hs class of oels, p() follows he ffson process efne as p() = σ()w (), σ () = σ + ω P (f()), () where f() s a sae-varable process an he fncon P ( ) s efne so ha EP (f()) =, varp (f()) =, EP (f( + h)) f(s), p(s), s = exp( λ h)p (f()), h >. () Noe ha Eσ (s) = σ an varσ (s) = ω. Ths specfcaon of σ () ncles any known oels sch as consan elascy of volaly (CEV) processes, GARCH ffson oels (Nelson, 99), egenfncon sochasc volaly (ESV) oels (Meah, ) an posve Ornsen-Uhlenbeck Levy-rven oels (Barnorff-Nelsen an Shephar, ).. Inegrae an realze varance The negrae varance (IV) s efne as IV σ (s)s, =,,..., (3) where he n of s eerne epenng on he objecve of research. For exaple, f he researcher s nerese n changes n he varances of aly rerns, s nerpree as a ay. Uner ceran conons, IV can be conssenly esae by he esaor known as he realze varance (RV), whch s efne as RV () = r (), (4) + where r () p() p( ) = σ(s)w (s) an s a posve neger. Here, an hereafer, he noaon () ples ha s vale epens on he saplng freqency. For exaple, f enoes a ay an we ake observaons every fve nes, hen = 88 an r (88) enoes a fve-ne rern. I s well known ha, as, RV () p IV. 3

4 Meah (3, Proposon 3.) shows ha f he re process of p() belongs o he one-facor SR-SARV oel nroce n he prevos secon, hen IV follows an ARMA(, ) process: IV = c IV + φiv + η + θ IV η, (5) where η s a whe nose wh var(η ) = σ η an cov(η, () s ) =, () RV () IV, for all an s. The ARMA(, ) oel paraeers, c IV, φ, θ IV an σ η are expresse, n ers of he one-facor SR-SARV oel paraeers, σ, ω an λ, as φ = exp( λ ), c IV = ( φ)σ, θ IV = 4ρ, ρ = ρ φ + corriv, IV + φ φcorriv, IV, (6) ση = ( + φ )variv φcoviv, IV, + θiv where var(iv ), cov(iv, IV ) an corr(iv, IV ) are gven as variv = ω (log φ) (φ log φ ), coviv, IV = ω (log φ) ( φ), corriv, IV = ( φ) (φ log φ ). (7) Also, σ () var () s gven as ( ) σ () = σ4 + 4ω φ (log φ) log φ,. (8) See Meah (3) for he above resls. Meah (3, Proposon 3.) shows ha RV () = IV + () also follows an ARMA(, ) process. Noe ha he for ARMA(, ) oel paraeers, c IV, φ, θ IV an ση, are copleely eerne as fncons of he hree one-facor SR-SARV oel paraeers, σ, ω an λ..3 Mcrosrcre nose Now sppose ha he observe log-prce p () s conanae by a easreen error or a crosrcre nose so ha p () = p() + ε(). (9) We asse he followng properes on he crosrcre nose ε(). Asspon (a) ε()...(, σ ε) wh Eε 4 () <. (b) ε() s nepenen fro p() for all s an. 4

5 We o no asse any srbon for ε(). The observe rern s efne as r () p () p ( ) () = r + e (), () where e () = ε() ε( ). I s easy o show ha E e () =, var e () = σε an cov e (), e () = { σ ε, =,. Noe ha vare () an cove (), e () o no epen on. We efne he noseconanae RV (NCRV), enoe by RV (), as RV () r (). Wre + = where RV () = () = = RV () = ( ) r () + e () + +, + (). r () e () = () () e (). (3) + Noe ha, nlke RV () an RV (), () s no necessarly posve snce he frs er of () can be negave. We call () crosrcre nose n NCRV where we nee o sngsh fro ε(). Oherwse, where here s no abgy, we sply call () crosrcre nose. Ths crosrcre nose wll be esae n he laer secon. In he Appenx, we show ha E = σε, an cov () (), () s = 8σεσ + ( )ωε + 4σε 4 = s, ωε = s ±, oherwse, where ωε = varε () = Eε 4 () σε. 4 Ths, () has he aocovarance srcre of a MA() process. Sppose ha he MA() process s represene as (4) () = c () + () + θ (), () () W N(, σ () ). (5) The ean an aocovarances of () E = c (), an cov () (), () s = ( + θ () )σ () θ () n ers of c (), θ () an σ () are = s, σ () = s ±, oherwse. (6) Laer, we lze hese wo fferen expressons of he oens of () relaonshps aong paraeers. 5 o erve he

6 3 Sae Space Approach Or approach s n he sae spr of he sae space eho se n Barnorff- Nelsen an Shephar (), where hey conser he saon who crosrcre nose. The exsence of crosrcre noses reqres aonal effors for checkng enfcaon of sae space oel paraeers. In hs secon, we escrbe he enfcaon proble n esang he IV fro he NCRC by a sae space eho. 3. Sae space for of NCRV Snce RV () RV () = IV + (), we have = IV + () + (). (7) Fro (5), (5) an (7), we have he followng sae space for of RV () : (Observaon eqaon) = RV () IV () α β + (), (8) (Sae eqaon) IV () α = β c IV c () + φ θ IV θ () IV () α β + η, (9) where () η, σ () σ η σ (). () Gven he vales of φ, c IV, θ IV, ση, c (), θ (), σ () an σ (), we can esae IV an () by applyng he Kalan flerng o he sae space for. One proble of he sae space for s how o esae hose paraeers. One ay sply hnk ha we can esae he fro he sae space for by, e.g., qas-ax lkelhoo esaon ner Gassan nose asspon. Ths s; however, no always possble. I s known ha n general he paraeers of sae space for are no necessarly enfe (see, for exaple, Halon, 994, p.388). More precsely, here ay be nfnely any cobnaons of sae space oel paraeers ha gve he sae aocovarance srcre. We have o confr wheher he oel paraeers are nqely enfe before paraeer esaon. We conser hs proble n he nex secon. In fac, we show ha he above paraeers n he sae space for canno be nqely enfe. 6

7 3. Ienfcaon of oel paraeers Snce RV () s he s of hree coponens: IV (an ARMA(, ) process), () (a whe nose process) an () (an MA() process), RV () self follows an ARMA(, ) process (see Granger an Morrs, 976) so ha s expresse as ( φl)rv () = c () + ( + δ () L + δ () L )τ, τ W N(, σ () τ ). () Noe ha he AR coeffcen φ s he sae as ha of IV n (5). The ARMA oel of a sae space represenaon s coonly referre o as a rece for or ARMA rece for. Noe ha he paraeers of any ARMA rece for are always enfe (an hence can be esae). Fro (5), (7) an (5), we have ( φl)rv () = ( φl)iv + ( φl) () = ( φ)σ + η + θ IV η + () +( φ)c () + (θ () + ( φl) () φ) φθ () φ () +. () These wo expressons on he rgh-han ses n () an () are of he sae oel an hence her ean an aocovarances s be encal. The aocovarances of he MA pars n () are gven as γ () = ( + δ () + δ () )στ (), γ () = (δ () + δ () δ () )στ (), γ () = δ () στ (), an γ j = for j 3. I s shown n he Appenx ha he aocovarance fncons of he MA pars n () are γ () = ( + θiv )ση + ( + φ )σ () + + (θ () γ () = θ IV ση φσ () + (θ () φ φθ () γ () = φθ () φ) + φ θ () σ (), (3a) + φ θ () )σ () (3b) σ (), (3c) an γ j = for j 3. By eqang he eans of he MA pars n () an (), we oban c () = ( φ) ( ) σ + c (). (3) Gven ARMA(, ) oel paraeers n (), we can oban c (), φ, an γ () j, j =,,. Then, nknown paraeers n he eqaons n (3a) (3) are θ IV, σ, ση, c (), θ (), σ () an σ (). Observe ha he nber of nknown paraeers are ore han he nber of eqaons. Hence, we canno nqely enfy hese paraeers fro hese eqaons. In oher wor, for a gven ARMA(, ) oel, here are nfnely any cobnaons of paraeers θ IV, σ, ση, c (), θ (), σ () an σ () ha gve he sae aocovarance srcre as he ARMA(, ) oel. 3.3 Esaon We rece he nber of nknown paraeers as follows. Eqaons (6) an (7) ply ha θ IV, ση an σ are fncons of φ, σ, ω. Below we show ha c (), θ () an σ () can also be expresse as fncons of σ, ωε an σε. Sbsng hese They also epen on as he noaon ples. 7

8 fncons no (3a) (3), we can rece he nber of nknown paraeers. To erve he relaonshps aong hose paraeers, we lze he expressons gven n (4) an (6). In vews of (4) an (6), we oban he followng eqaons: c () = σ ε, (4a) ( + θ () )σ () = 8σσ ε + ( )ωε + 4σε, 4 (4b) θ () σ () = ωε. (4c) Assng he MA paraeer sasfes he nverbly conon,.e., θ () <, we can solve he eqaons (4a) (4c) for c (), θ () an σ () as c () = σ ε, σ () = ω ε θ () an θ () = A A, (5) where A = 4 σ σ ε ωε Appenx. Noe ha < θ () + + σ4 ε. The eal of he calclaon s gven n he ωε < snce A >. Fro (6), (7) an (5), we see ha c IV, θ IV, ση, c (), θ (), σ () an σ () can be expresse as fncons of φ, σ, ω, σε an ωε. To ephasze hs, we ay wre he as c IV = c IV (φ, σ ), θ IV = θ IV (φ), σ η = σ η(φ, ω ), c () θ () = θ () (σ, σε), σ () = σ () (φ, σ, ω) an σ () = c () (σε), = σ () (σ, σε, ωε). (6) Noe ha θ IV s a fncon of only φ, hence can be asse o be known (snce φ s enfe fro he rece for). Sbsng he expressons n (6) no he eqaons n (3a) (3), evenally, we have for eqaons for he for nknown paraeers σ, ω, ωε an σε. Hence, he orer conon for enfcaon s sasfe. However, hs resl only oes no necessarly ean ha for a gven aocovarance srcre, we can nqely enfy σ, ω, ωε an σε. For he nqeness of he enfcaon, we s check f he rank conon s also sasfe. To show he nqeness of he enfcaon, we explcly erve he solons for he paraeers n ers of c, φ, γ () j j =,...,. In he Appenx, we show ha, gven c, φ, γ () j j =,..., an (6), he eqaons n (3a) (3) can be nqely solve for σ, ω, σε an ωε as ωε = γ() φ, σ ε = ω = (log φ) () φγ + ( + φ )γ () ( φ) 3 ( + φ) c ( φ) ( )γ() φ γ() More precsely, ner he conon σ ε >. + +φ4 φ γ(), (7a) ωd γ (), (7b) 4( + φ ) 8

9 an σ = c φ σ ε, (7c) where D = B + ( + φ )C, B φ ( + φ ) log φ (log φ) an C ( ) φ log φ (log φ). (7) The eqaons n (7a) (7) show ha he for paraeers, σ, ω, σ ε an ω ε can be nqely enfe fro he ARMA(, ) rece for of he sae space for n (8) an (9). I shol be ephasze ha hs oes no ply ha we can recly esae he sae space oel paraeers n (8) an (9), b ples ha we can esae he above for paraeers recly by replacng he sae space oel paraeers wh he fncons of he for paraeers. We esae hese for paraeers by he qas-ax lkelhoo esaon of ARMA(, ) oel, where nnovaons are (convenenly) asse o be Gassan. Here, we sarze how o calclae he lkelhoo. (Sary on how o calclae he lkelhoo) () For a gven, calclae RV (). () Gven φ, σ, ω, σε an ωε, calclae c IV, θ IV, ση, c (), θ (), σ () accorng o (6), (7) an (5). an σ (), θ (), σ (), j =,...3 an c () n (3a) (3). (3) Wh c IV, θ IV, σ η, c () γ () j an σ () obane n Sep (), cope (4) Applyng a resl n Meah (), (represenaons of MA() paraeers n ers of he aocovarance fncons) calclae corresponng ARMA(, ) paraeers, c (), δ (), δ () an σ () fro γ () j, j =,...3 obane n Sep (3). (5) Calclae he Gassan ARMA(, ) log-lkelhoo for RV δ () an σ (). wh φ, c () δ (), We can oban he esaes of σ, ω, σ ε an ω ε by axzng he above log-lkelhoo wh respec o he for paraeers. The eho proves conssen esaors for he for paraeers. Usng he propose ehoology, we conc an eprcal analyss for exchange rae aa n he nex secon. 4 Eprcal Analyss 4. Daa escrpon The yen/ollar spo exchange rae seres we se are he -qoe prces observe every ne, whch are obane fro Olsen an Assocaes. The fll saple 9

10 covers he pero fro Janary o Deceber 3 6. Fgre plos he aly rerns calclae fro he aa over he pero. We nerpolae ssng prces by he prevos ck eho,.e., nerpolang he prevos prces. Also, followng Anersen, Bollerslev, Debol an Labys (), we reove he aa of nacve rang ays. Whenever we so, we always c fro : GMT on one ngh o : he nex evenng. For eals on he ovaon of hs efnon of ay, see Anersen, Bollerslev, Debol an Labys (), Anersen an Bollerslev (998) an Bollerslev an Doowz (993). We c he aa accorng o he followng crerons, whch s slar o he crerons se n Bene e al. (7): () he ays ha ss ore han 5 prce observaons, () he ays where, n oal, here are ore han nes of zero rerns (3) he ays where he prce no change for ore han 35 nes. By hese crerons, we can reove all weeken aa. However, he ays sch as U.S. holays ha Anersen, Bollerslev, Debol an Labys () an Bene e al. (7) reove are no necessarly c by hese crerons. Ths s becase even when he U.S. arke s close, he ransacons are ae n oher arkes. Evenally, we are lef wh 89 coplee ays, or = 6496 prce observaons fro whch we calclae he -n an 5-n rerns. Wh hese rerns, we calclae he seres of aly NCRVs. We call he NCRVs -n NCRV ( = 44) or 5-n NCRV ( = 88) epenng on he nervals of he rerns. Table repors he escrpve sascs of -n an 5-n NCRVs. Fgre plos hese wo NCRVs. The ean of -n NCRV s hgher han ha of 5-n NCRV. Ths can be e o crosrcre noses snce he ean of NCRV ncreases as he saplng freqency ens o be hgh, or, as shown n (3) an (4a). The frs orer aocorrelaon of NCRVs are soewha lower han sally expece for varances of fnancal e seres (.4794 for -n NCRV an.477 for 5-n RV). 4. Esaon of paraeers, IV an crosrcre nose For hese wo seres of NCRVs, we esae paraeers, φ, σ, ω, σ ε an ω ε, as escrbe n he prevos secon. Noe ha, n general, he vales of -n an 5 n NCRVs are fferen an hs he esaes fro hese wo NCRVs are fferen. Table shows he esaes of he above paraeers an he vales of sae space for paraeers n (8) an (9) cope fro he esaes. The noaon () ples ha hose vales epen on he vale of. Noce ha even hogh we se he seres of NCRVs wh fferen s for he esaons, he esae vales of paraeers ha o no epen on are very slar. For exaple, he esae vale of σ, he ncononal ean of he IV, s.358 for = 44 an.378 for = 88. The esae vale of ω, he ncononal varance of IV, s.3 for for = 44 an.79 for = 88. The esaes of he frs orer aocorrelaon of IV are sgnfcanly hgher han hose of NCRVs. Ths sggess ha he observe low aocorrelaons of he NCRVs are e o crosrcre noses. One neresng observaon s ha alhogh we o no asse any srbon for

11 he crosrcre nose ε(), he kross of ε(), calclae fro he esae varances of σε = varε() an ωε = varε(), are close o 3 n he case of 5-n NCRV. We splay he esaes of he IV by Kalan soohng n Fgre 3(a) for 5-n NCRV, 3(b) for -n NCRV an 3(c) for he boh. Noe ha boh are he esae of he sae IV seres. These wo seres of IV esaes are very slar as seen n Fgre 3(c). Fgre 4 plos he soohe esaes of crosrcre nose s. The resls are naral. We fn ha -n NCRV has a larger bas han 5-n NCRV. The ean of he crosrcre noses for -n NCRV s hgher han ha for 5-n NCRV. The sall bases of 5-n NCRV s conssen wh he se of 5-n NCRV n he prevos lerare. Las, we calclae he rao of crosrcre noses o he NCRVs,.e., R() = () û () () /RV () for each, =,..., 89, where û () s he soohe esaes of. The ax an n vales are, respecvely,.697 an 4.79 for R(88)(5 n NCRV) an.783 an.7 for R(44)( n NCRV). We also calclae he average agne of crosrcre nose () s as he ean of R() an R(). The vales of he eans of R() an R() are, respecvely,.458 an.34 for = 88, an.373 an.373 for = Sary an Conclng Rearks In hs paper, we propose a new approach, sng a sae space for of he realze varance, o esang he negrae varance. Or eho s base on he resl n Meah (), whch shows ha when he prces follows a ceran class of sochasc volaly oels, he negrae varance follows a ARMA(, ) oel. We showe ha ner he exsence of crosrcre noses, he observe realze varance follows a ARMA(,) oel. We represene he ARMA(, ) oel by a sae space for an esablshe he nqeness of he enfcaon of sae space for paraeers. The propose eho s apple o yen/ollar exchange rae aa, where we fn ha he RV calclae wh 5-n rerns are less base han wh -n rerns. The wo seres of IV esaes obane by he propose eho wh -n rerns an 5-n rerns are very slar. The eho was also se for esang crosrcre noses. In he esaon, we consrce he lkelhoo sng only eher -n or 5- n RV. I s ore esrable o se boh RVs for esang coon paraeers. Invesgang he opal way for cobnng RVs of fferen scales s a sbjec of fre research. I s also neresng o apply or eho o sock rern aa an exane he effecs of crosrcre noses.

12 Appenx Hereafer, we sppress () n he noaons r (), () Dervaon of (4) an e (), for splcy. Frs, we erve E. Snce r an e are nepenen by Asspon an var(e ) = σε, we have E = E = = E = = σ ε. r + r + e + E e + + E = + var(e ) e + The followng resls are se for ervng var n (34) an cov, n (35) below: For any s an, we have (8) 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. (9) Ths, when = s we have cov r s e s, r e = σεe r = σεe( σ(s)w / (s)) = σεe / σ (s)s = σεe σ (s)s = σ εσ. The hr eqaly coes fro he Io soery. When s, we have cov r s e s, r e = E e s e E r s r E r s E e s E r E e =. (3a) (3b) Nex, we erve cov e, e s. When = s, we have cove, e = vare = Ee 4 (Ee ) = E ε 4 4ε 3 ε + 6ε ε 4ε ε 3 + ε 4 4σ ε 4 = Eε 4 + σε. 4 When = s ±, we have cov e s, e s = cov = cov = varε s = ω ε. e, e s+ s ε ε s+ s+ ε s + ε s, ε s ε s ε s + ε s (3) (3)

13 When = s ± for, we have cove, e s =. We have covr e, e s = for any an s snce covr e, e s = Er e e s Er e Ee s = Er Ee e s Er Ee Ee s =. (33) Fro (3) (33) we have. var = an var = = 4var r + r + = +4cov = = 4cov = +4cov = = 4 cov = j= +4 = j= e + e var = e + r + e +, e + = r + cov r + e +, r + = r + e +, e + = e +, r + j r + e +, e + = e + e + e + j + cov e, e + + = = + = 8σ εσ + (Eε 4 + σ 4 ε) + ( )(Eε 4 σ 4 ε) = 8σ εσ + ( )ω ε + 4σ 4 ε, cov, + = cov = = 4cov r + e + + e + = r + = +cov = +cov = = cov e, e + = ωε. e +, r + = r + e +, r + = e + e +, e + = e + = j=, = + cov cov e, e + + j (34) r + e + + e + = e, e + + = = I s easy o check cov, ± = for, an hence we have (4). (35) 3

14 Dervaon of (3a) (3c) γ = cov{η + θ IV η + φ + + (θ φ) φθ, η + θ IV η + φ + + (θ φ) φθ } = σ η + θ IV σ η + σ + φ σ + σ + (θ φ) σ + φ θ σ = ( + θ IV )σ η + ( + φ )σ + + (θ φ) + φ θ σ, γ = cov{η + θ IV η + φ + + (θ φ) φθ, η + θ IV η + φ + + (θ φ) φθ 3 } = θ IV σ η φσ + (θ φ φθ + φ θ )σ γ = cov{η + θ IV η + φ + + (θ φ) φθ, η + θ IV η 3 + φ (θ φ) 3 φθ 4 } = φθ σ. Dervaon of (5) Fro (4c), we have σ = ω ε/θ. Sbsng hs no (4b), we have ( + θ) ω ε θ = 8σσ ε + ( )ωε + 4σε 4 ωεθ 8σσ ε + ( )ωε + 4σεθ 4 + ωε = θ 4 σ σ ε + + σ4 ωε ε θ ωε + = θ = A ± A, (36) where A = 4 σ σε + + σ4 ωε ε. Snce A > for, we have A+ A ω >. ε Assng ha θ sasfes he nverbly conon, we oban θ n (5). Dervaon of (7a) an (7c) Fro (3c) an (4c), we have ωε = γ. Fro (6) an (7), we have φ ση = ω B an σ + θiv = σ4 + ω C, (37) where B an C are gven n (7). Fro ωε = θ σ n (4c), we have ( ) ( + θ θ φ + φ + φ θ)σ = θ + θ φ + φ θ + φ θ ωε ( ) = θ + θ ( + φ ) φ ωε, an ( ) (θ φ φθ + φ θ )σ = φ θ φθ + φ ωε ( ) = + φ θ + θ φ ωε. Sbsng (37), (38) an (39) no (3a) an (3b), we have ( ) γ = ωd + σ 4 + φ + + θ ( + φ ) φ ω θ ε, (38) (39) (4a) 4

15 an γ = ωe σ 4 φ where D = B + ( + φ )C an E = an θ IV + θ IV ρb = Hence, we have or ( ) + θ φ ( + φ ) ω θ ε, θ IV +θ IV B φc. Fro (6), we have ρ( 4ρ = ) 4ρ + ( 4ρ ) ρ( 4ρ = )( + 4ρ ) 4ρ ( + 4ρ ) + ( 4ρ ) ( + 4ρ ) = ρ, (4b) (4) φ(φ log φ ) + ( φ) (log φ) (4) φγ + ( + φ )γ = φd + ( + φ )E ω + ( + φ ) φ ω ε, = φ + ( + φ )ρ Bω + ( + φ 4 )ω ε, = ( φ)3 ( + φ) (log φ) ω + ( + φ 4 )ω ε, (43) ω = (log φ) φγ + ( + φ )γ ( + φ 4 )ωε. (44) ( φ) 3 ( + φ) Sbsng ωε = γ, we oban (7a). Nex, noe ha fro (5), we have φ + θ = + θ θ θ = + (A A ) A A = A + A + (A A ) (A + A ) (A A )(A + A ) = A. Fro (3) an (4a), we have c = ( φ) ( σ + σ ε Sbsng σε n (46) no A, we have ( ) A = 4σ c ( φ)σ ( φ) ) (45) or σ ε = c ( φ)σ ( φ). (46) ω ε + + ( c ( φ)σ ( φ) ) ω ε = cσ ( φ)ω ε σ4 ω ε ( ) c + + ( φ)cσ +( φ) σ 4 ( φ) ωε (47) = = 4cσ ( φ)ω ε cσ ( φ)ω ε 4σ4 ω ε 3σ4 ω ε + ( ) + c ( φ) ω ε c + ( ) +. ( φ) ωε cσ ( φ)ω ε + σ4 ω ε 5

16 Fro (4), (45) an (47), we have γ = ω D ( + φ ) σ4 + ( + φ )c ( φ) σ +( )(+φ )ω ε + ( + φ )c ( φ) φω ε. Mlplyng boh ses n (48) by /( + φ ) an arrangng, we have σ 4 (48) c c φ σ ( φ) + (γ ωd + φωε) ( )ω + φ ε =. (49) Solvng hs qarac eqaon for σ, we have σ = c φ ± c ( φ) + ( )ω ε (γ ωd + φωε). (5) ( + φ ) Fro σε c > an (46), we s have > φ σ. Hence, he sgn of he secon er n (5) s negave. Fro (46), we have σε = c ( φ) + ( )ω ε (γ ωd + φωε). (5) ( + φ ) Fro he above resls, we oban (7b) an (7c). 6

17 References Anersen, T. G., T. Bollerslev, (998) Deche Mark-Dollar volaly: nraay acvy paerns, acroeconoc annonceens, an longer rn epenences, Jornal of Fnance, Anersen, T. G., T. Bollerslev, F. X. Debol an H. Ebens () The srbon of realse sochk rern volaly, Jornal of Fnancal Econocs, 6, Anersen, T. G., T. Bollerslev, F. X. Debol an P. Labys () The srbon of exchange rae volaly, Jornal of he Aercan Sascal Assocaon, 9, Barnorff-Nelsen, O.E. an N. Shephar (), Non-Gassan OU base oels an soe of her ses n fnancal econocs, Jornal of he Royal Sascal Socey, B 63, Barnorff-Nelsen, O.E. an N. Shephar (), Econoerc analyss of realze volaly an s se n esang sochasc volaly oels, Jornal of Royal Sascal Assocaon, Seres B, 64, Par, Ban, F. M. an J. R. Rssell (6), Separang crosrcre nose fro volaly, Jornal of Fnancal Econocs, 79, Ban, F. M. an J. R. Rssell (8), Mcrosrcre nose, realze varance, an opal saplng Revew of Econoc Ses, 75, Bene, M., J. Lahaye, S. Laren, C.J. Neely, an F.C. Pal (7) Cenral bank nervenon an exchange rae volaly, s connos an jp coponens, Inernaonal jornal of fnance an econocs,, 3. Bollerslev, T., an I. Doowz (993), Trang Paerns an Prces n he Inerbank Foregn Exchange Marke, Jornal of Fnance, 48, Granger, W. J. C. an M. J. Morrs (976) Te Seres Moellng An Inerpreaon, Jornal of he Royal Sascal Socey. Seres A (General) Vol. 39, Ghysels, E. A. C. Harvey, an E. Renal (996), Sochasc Volaly, 9-9, n hanbook of sascs 4, ee by G.S. Maala an C.R. Rao. Halon DJ Te seres analyss. Prnceon Unversy Press: Prnceon, NJ. Hansen, P. R. an A. Lne (6) Realze varance an arke crosrcre nose, Jornal of Bsness & Econoc Sascs, 4, 7 6. Meah, N. (), An egenfncon approach for volaly oelng, CIRANO Workng Paper, s 7. Meah, N. (), ARMA represenaon of wo-facor oels. CIRANO workng paper, s 9. 7

18 Meah, N. (3), ARMA represenaon of negrae an realze varances, Econoercs Jornal, 6, Nelson, D.B. (99), ARCH oels as ffson approxaons, Jornal of Econoercs 45, Shephar, N. (5), Sochasc Volaly, Selece Reangs. Zhang, L., P. Myklan an Y. A l-sahala (5), A ale of wo e scales: eernng negrae volaly wh nosy hgh-freqency aa, Jornal of he Aercan Sascal Assocaon,, Zho, B. (996), Hgh-freqency aa an volaly n foregn-exchange raes, Jornal of Bsness & Econoc Sascs, 4,

19 Table : Descrpve sascs of he NCRV -n NCRV 5-n NCRV Mean Varance.69.6 S.D SACORR() SACORR() SACORR(3) SACORR(4) SACORR(5) Noe: he able repors he ean, varance an sanar evaon of RV calclae wh fferen. SACORR(k) enoes he saple aocorrelaon of orer k. 9

20 Table : Esaes of paraeers -n NCRV 5-n NCRV φ (.56) (.39) σ (.895) (.55) ω.3.79 (.84) (.34) σ ε ( ) ( ) ω ε ( ) ( ) L corr(iv, IV ) ĉ IV θ IV σ η ĉ () θ () σ () σ () variv variv + var () + var () var () variv + var () + var () σ η σ η + σ σ σ η.74. Eε 4 /σ 4 ε Noe: he robs sanar errors are nse he parenhess. L s he log-lkelhoo.

21 Fgre : Daly rerns of yen/ollar exchange rae 3 RETURN (%) YEAR Fgre : -n NCRV an 5 n NCRV n NCRV 5 n NCRV VARIANCE (%) YEAR

22 Fgre 3: Soohe IV esaes for n an 5 n NCRVs (a) 5 n Soohe IV (5 n) 5 n NCRV VARIANCE (%) YEAR (b) n Soohe IV ( n) n NCRV VARIANCE (%) YEAR

23 (c) n an 5 n Soohe IV ( n) Soohe IV (5n) VARIANCE (%) YEAR Fgre 4: Esae of crosrcre nose for n an 5 n NCRVs MICROSTRUCTURE NOISE(%) Mcrosrcre nose ( n) Mcrosrcre nose (5 n) YEAR 3