Forecasting Correlation and Covariance with a. Range-Based Dynamic Conditional Correlation Model

Transcription

1 Forecsing Correlion nd Covrince wih Rnge-Bsed Dynmic Condiionl Correlion Model Ry Y Chou * Insiue of Economics, Acdemi Sinic Nhn Liu Deprmen of Mngemen Science, Nionl Chio-ung Universiy Chun-Chou Wu Deprmen of Inernionl rde, Chung Yun Chrisin Universiy Jnury, 005 * Corresponding uhor. Conc ddress: Insiue of Economics, Acdemi Sinic, #8, Sec, Yen-Jio-Yun Rod, Nnkng, ipei, iwn. elephone: ex.-, fx: , emil: rchou@econ.sinic.edu.w

2 Asrc his pper proposes rnge-sed Dynmic Condiionl Correlion (DCC) model, which is n exension of Engle s (00) DCC model. he efficiency of he rnge d in voliliy esimion is documened in Prkinson (980), Alideh, Brnd, nd Dieold (00), Brnd nd Jones (00), nd Chou (004, ), mong ohers. I is hence nurl o consider he implicion of his resul in he esimion of mulivrie GARCH models. In he DCC model, he condiionl correlion coefficiens re esimed y dynmic model for he produc of he pir-wise reurn series wih ech normlied y heir condiionl sndrd deviions. he condiionl sndrd deviion is clculed y using univrie GARCH for he reurn series. We use he Condiionl Auoregressive Rnge (CARR) model of Chou (004), s n lernive o he univrie GARCH in he DCC firs-sep esimion. We herefore consruc rnge-sed DCC model. he susnil gin in efficiency in he voliliy esimion cn induce n efficiency gin in he esimion of he series of he ime-vrying correlion coefficien nd covrince. For comprison we esime he generlied reurn-sed DCC model s enchmrk o gin insighs ino he difference of hese mehods. We use hree d ses for empiricl nlyses: he sock indices of S&P500 nd Nsd, nd he 0-yer resury ond yield. Boh in-smple nd ou-of-smple resuls indice h our rgumen is suppored in erms of he precision in esiming nd forecsing he correlion nd covrince mrices. Keywords: DCC, CARR, rnge, dynmic correlion, covrince, voliliy

3 I. Inroducion I is of primry impornce in he prcice of porfolio mngemen, sse llocion, nd risk mngemen o hve n ccure esime of he covrince mrices for sse prices. When vluing derivives, forecss of voliliies nd correlions over he whole life of he derivive re usully reuired. he univrie ARCH/GARCH fmily of models provides effecive ools o esime he voliliies of individul sse prices. ilored o he needs of differen sse clsses, hese vrious models hve chieved remrkle success. For survey of his vs lierure, see Bollerslev, Chou, nd Kroner (99), nd Engle (004). I is, however, sill n cive reserch issue in esiming he covrince or correlion mrices of muliple, especilly lrge ses of sse prices. Erly emps include he VECH model of Bollerslev, Engle, nd Wooldridge (988), he BEKK model of Engle nd Kroner (995), nd he consn correlion model of Bollerslev (990), mong ohers. he consn correlion model is oo resricive s i imposes he sringen consrin h he dynmic srucure of covrince is compleely deermined y he individul voliliies. he VECH nd he BEKK models re more flexile in llowing ime-vrying correlions. he BEKK prmeeriion for ivrie model involves prmeers, only wo more hn he VECH prmeeriion, u for higher-dimensionl sysems, he exr numer of prmeers in he BEKK model increses, nd compleely free esimion ecomes very difficul indeed. In series of ppers, Engle nd Shepprd (00), Engle (00), nd Engle, Cppiello, nd Shepprd (00) provide soluion o his prolem y using model eniled he Dynmic Condiionl Correlion Mulivrie GARCH (henceforh DCC). he condiionl covrince esimion prolem is simplified y esiming univrie GARCH models for ech sse s vrince process. Crrying on y using he rnsformed sndrdied residuls from he firs sge, nd esiming ime-vrying condiionl correlion esimor in he second sge, he DCC model is no liner, u cn e esimed simply wih he wo-sge mehods sed on he mximum likelihood mehod. A meningful nd srong performnce of his model is repored in hese sudies especilly considering he ese of implemenion of he esimor. Oher mehods for esiming he ime-vrying correlion re proposed y sy (00) nd he n-dimensionl VECH model is wrien s vech(h )=AB vech( )C vech(h -), where H is he condiionl covrince mrix ime nd vech(h ) is he vecor h scks ll he elemens of he covrince mrix. A generl prmeeriion h involves he minimum numer of prmeers while imposing no cross euion resricions nd ensuring posiive definieness for ny prmeer vlue is he BEKK model, nmed fer B, Engle, Krf, nd Kroner who wroe he preliminry version of Engle nd Kroner (995). ξ ' ξ

4 y se nd sui (00). In his pper, we consider refinemen of he DCC model y uiliing he high/low rnge d of sse prices. In esiming he voliliy of sse prices, here is growing wreness of he fc h he rnge d of sse prices cn provide shrper esimes nd forecss hn he reurn d sed on close-o-close prices. Sudies of supporing evidence include Prkinson (980), Grmn nd Klss (980), Wiggins (99), Rogers nd Schell (99), Kuniomo (99), nd more recenly Glln, Hsu, nd uchen (999), Yng nd Zhng (000), Alideh, Brnd, nd Dieold (00), Brnd nd Jones (00), Chou (004, 004) nd Chou, Wu, nd Liu (004). Chou (004) proposed he Condiionl Auoregressive Rnge (henceforh CARR) model where cn cpure he dynmicl voliliy process nd oined some insighful evidence in rel d. In oher words, rnge-sed voliliy model is n lernive mnner ou of he reurn-sed voliliy model. In ligh of he success of he rnge-sed univrie voliliy models, i is nurl o inuire wheher his esimion efficiency cn e exended o mulivrie frmework, in his cse of he DCC model. he reminder of he sudy proceeds in he following mnner. Secion inroduces he frmework of he ivrie models o esime he correlion nd covrince process, especilly for he reurn-sed nd he rnge-sed DCC models. Secion descries he empiricl d used nd gives discussion of he empiricl resuls. he conclusion nd direcions for fuure sudies re given in secion 4. 4

5 II. Correlion/Covrince Esimion nd he DCC Model Our ojecive is o esime he curren level of covrince nd correlion. rdiionlly, he condiionl covrince nd correlion eween wo rndom vriles r nd r wih ero mens re defined y: COV, (,, = E r r ), () ρ, E ( r r ),, =. () E ( r, ) E ( r, ) In his definiion, he condiionl covrince nd correlion re decided y previous informion. his mehod hs wo prolems, nmely, oo erly d re used nd eul weighs re ssigned for every previous lg. o overcome he firs prolem, we inroduce he moving verge ype wih 00-week window, MA(00): = r, s s 00 COV, = r, s, () 00 r, sr, s ˆ ρ = s= 00. (4) ( r, s s= 00 )( r, s s= 00 ) I mkes sense o give more weigh o recen d. From his poin of view, we inroduce n exponenilly-weighed moving verge (EWMA) model where he weighs decrese exponenilly s we move ck hrough ime. he exponenillyweighed moving verge model hs he rcive feure h relively lile d need o e sored. Exponenil verges ssign he mos weigh o he mos recen oservion, wih weighs declining exponenilly wih ime. Hence, he EWMA model for covrince nd correlion cn e illusred s follows. COV, = ( ) s= ( λ λ r r, (5) )( s, s, s s λ r, sr, s s= ˆ ρ =. (6) s= λ r s, s s= λ r ) s, s 5

6 he vlue of λ governs how responsive he esime of he curren voliliy is o he mos recen period s percenge chnge. As o he coefficien λ is usully clled he exponenil smooher in his model y RiskMerics M. he RiskMerics M pproch uses exponenil moving verges o esime fuure voliliy, ecuse i elieves he mehod responds rpidly o mrke shocks. Bollerslev (990) proposes he Consn Correlion Coefficien (henceforh CCC) model, which specifies h H = D RD, (7) where R is he smple correlion mrix nd D is he k k digonl mrix of ime-vrying sndrd deviions from univrie GARCH models wih i h digonl, where h i, on he h i, is he sure roo of he esimed vrince. Under such siuion, we cn oin he esime of condiionl covrince y he informion of he fixed correlion nd he produc of he wo condiionl sndrd deviions. Alhough he CCC model is meningful, he seing of consn condiionl correlions cn e oo resricive. Engle (00) exends he CCC model o he DCC model. he DCC model is new form of he mulivrie GARCH h is priculrly convenien for complex sysems nd suile for ime-vrying condiionl correlions. he DCC model differs from he CCC model only in llowing R o e chnged over ime. hus, he DCC model cn e shown s follows. H = D R D, (8) / / R = dig{ Q } Q dig{ Q}. (9) Here, D is defined like euion (7) nd Q S o ιι A B A Z Z B Q. (0) = ( ' ) o ' o he RiskMerics M dse uses he exponenilly-weighed moving verge model wih λ =0.94 for upding dily voliliy esimes. J.P. Morgn found h, cross vrin mrke vriles, his vlue of λ gives forecss of he voliliy h come closes o he relied voliliy. Following J.P. Morgn s suggesion, he vrile λ euls 0.94 for he ime eing in he ler empiricl discussion. 6

7 In euion (0), A nd B re prmeers nd o denoes he Hdmrd mrix produc operor, i.e., elemen-wise muliplicion. he symol ι is vecor of ones nd S is he uncondiionl covrince of he sndrdied residuls. Finlly, Z = D r re he sndrdied u correled residuls. he vrile r represens he reurns of sses. he reurns cn e eiher men ero or he residuls from filered ime series, i.e. r ~ (0, ) I N H. () he condiionl vrinces of he componens of Z re, in oher words, eul o, u he condiionl correlion mrix is given y he vrile of R. If A nd B re ero, hen we oin he resuls of he CCC model. I is imporn o recognie h lhough he dynmic of he D mrix hs usully een srucured s sndrd univrie GARCH model, i cn exend o mny oher ypes. For insnce, one could dop he EGARCH model o cpure he symmeric effecs in he voliliy process or he FIGARCH model o llow for he long memory voliliy processes. Ler on, we shll propose o use he Condiionl Auoregressive Rnge (CARR) model of Chou (004) s n lernive. he deils will e given in he ler pr of he secion. As for prmeers A nd B, i is shown h if A, B, nd ( ιι ' A B) re posiive semi-definie, hen Q will e posiive semi-definie. If ny one of he mrices is posiive definie, hen Q will lso e so. For he ij h elemen of R, he condiionl ij, correlion mrix is given y. As o he condiionl covrince, i cn ii, jj, hen e expressed using he produc of condiionl correlion eween hese wo vriles nd heir individul condiionl sndrd deviions. Engle nd Shepprd (00) show resuls h simplify finding he necessry condiions for R o e posiive definie nd hence correlion mrix wih rel, symmeric posiive semi-definie mrix, wih ones on is digonl line. he log-likelihood of his esimor cn e wrien s: 7

8 L = = = ( k log(π ) log H r ' H r ) = ( k log(π ) log D R D r ' D R D r ) = ( k log(π ) log D log R Z ' R Z ) = () Here, Z ~ N(0,R ) re he univrie GARCH sndrdied residuls. Bsed on Engle (00) s rgumen, we cn perform he esimion in wo seps. his esimor will no longer e efficien, u sill consisen (lso see Hfner nd Frnses (00)). Le he prmeers in D e denoed θ nd he ddiionl prmeers in R will e denoed y φ. he log-likelihood funcion cn e spli ino wo respecive prs: ( θ, φ) L V ( θ ) L ( θ, φ) L =. () C he former erm expresses he voliliy pr: L V ( ) = ( n log(π ) log D r ' D r ) θ. (4) he ler erm is he correlion componen: L C (, φ) = ( log R Z ' R Z Z ' Z ) θ. (5) A he firs sep, euion (4) is mximied wih respec o θ. A he second sep, euion (5) is mximied wih respec o θ nd φ. We use his wo-sep esimion procedure in our empiricl sudy. he voliliy pr of he likelihood is he sum of he individul GARCH likelihood if D is deermined y GARCH specificion. L V k r i, θ log(π ) log( h i, ). (6) i= hi, ( ) = his cn e joinly mximied y seprely mximiing ech erm. If D is deermined y CARR specificion, hen he likelihood funcion of he voliliy 8

9 erm is L V k r * i, θ log(π ) log( λ i, ), (7) * i= λi, ( ) = where * λ i, is he condiionl sndrd deviion s compued from scled expeced rnge from he CARR model. he second pr of he likelihood will e used o esime he correlion prmeers. As he sured residuls re no dependen on hese prmeers, hey will no ener he firs-order condiions nd cn e ignored. he wo-sep pproch o mximiing he likelihood is o find { L ( θ )} ˆ θ = rg mx, (8) V nd hen ke his vlue s given in he second sge, φ { ( ˆ, θ φ) } mx L. (9) C I is shown in Engle nd Shepprd (00) h under resonle regulriy condiions, consisency of he firs sep will ensure consisency of he second sep. he mximum of he second sep will e funcion of he firs-sep prmeer esimes, nd so if he firs sep is consisen, hen he second sep will e oo s long s he funcion is coninuous in neighorhood of he rue prmeers. hese condiions re similr o hose given in Whie (994) where he sympoic normliy nd he consisency of he wo-sep QMLE esimor re eslished. Anoher heoreicl jusificion of he ove resul is ppered in Engle (00). He referred o he work of Newey nd McFdden (994) wherey in heorem 6., formuled wo-sep GMM prolem cn e pplied o his model. Consider he L θ =0. he momen momen condiion corresponding o he firs sep s { ( )} condiion corresponding o he second sep is { ( ˆ θ φ) } φ L C, θ v. Under sndrd regulriy condiions which re given s condiions i) o v) in heorem.4 of Newey nd McFdden, he prmeer esimes will e consisen, nd sympoiclly norml, wih covrince mrix of fmilir form. his mrix is he produc of wo invered Hessins round n ouer produc of scores. Deils of his proof cn e found in Engle nd Shepprd (00) 9

10 he DCC model is new ype of mulivrie nd cn fi he GARCH or CARR model in he firs sge, which is priculrly convenien for complex sysems. he DCC mehod firs esimes voliliies for ech sse nd compues he sndrdied residuls. For ivrie cses, we use he following GARCH nd CARR srucures o perform he firs sep, respecively. he covrinces re hen esimed eween hese using mximum likelihood crierion nd one of severl models for he correlions. For he GARCH voliliy srucure (reurn-sed condiionl voliliy model): r k, ε k, h = ε ~ N(0, h ), k=, k, I k, = k, k α kε k, i β k hk, ω, (0) =. k, rk, / hk, If he voliliy model is CARR rnge-sed condiionl voliliy model : R u I ~ exp(; ), k=, k, = u k, k, k, = ωk α krk, β kλk, λ, () * = r λ where λ k, = dj k λk, dj k c * k, k, / k, σ =, ˆ λ k where R k, is he high/low rnge in logrihm, of he k h sse during ime inervl, σ nd λˆ k re respecively he uncondiionl vrince of he reurn series nd he smple men of he esimed condiionl rnge of he series k. his is specil cse of he muliplicive error model of Engle (00). he specificion of he exponenil disriuion of he disurnce erm provides consisen lhough inefficien esimor for he prmeers. For specific discussions lso see Chou (004). In he following nlysis, we use wo lernive versions of DCC. he firs one 0

11 is he sndrd DCC wih men reversion (henceforh MR_DCC), discussed in Engle (00). he second one is he inegred DCC (henceforh I_DCC). Boh of hese wo models re simplified versions of he generl expression in euion (8). For he ivrie cse, he MR_DCC is consruced y he following euion. ' ) ' ( = Q B Z Z A B A S Q o o o ιι, or =,,,,,,,,,, o o,,,, o, () where = =,,. For I_DCC, he dynmic srucure simplifies o: ) ( = Q A Z Z A Q o o or =,,,,,,,,,,,,,, o o. () Like he specific propery of voliliies, he correlion nd covrince mrices re lso unoservle. We use dily d o consruc he proxies for he weekly-relied covrince/correlion oservions. he purpose of such doing is o exrc hese so-clled mesured covrince/correlion, denoed MCOV/MCORR respecively, s one kind of enchmrk in deermining he relive performnce of he reurn-sed DCC model nd he rnge-sed DCC model for he ime eing. On he oher side, we perform he ilor-mde regression frmework proposed y Mincer nd Zrnowi (969) for he in-smple comprison. We demonsre he regression expression elow: reurn MCORR, 0 ˆ ε ρ γ γ = rnge MCORR, 0 ˆ ε ρ γ γ = (4)

12 reurn rnge MCORR γ 0 γ ˆ ρ ˆ γ ρ ε, =. he mjor focus here is o check he significnce of coefficiens γ nd γ. he sisicl inuiion here is similr o he convenionl OLS frmework. Similrly, we consruc he sysem of covrince in (5): MCOV = φ COV, reurn 0 φ ε rnge MCOV = φ 0 φcov ε, (5) reurn rnge MCOV φ 0 φcov φcov ε, =, where = ˆ ˆ σ k, re sndrd deviions esimed from he COV ρ ˆ σ ˆ, σ, reurn-sed DCC model or he rnge-sed DCC model. In consrucing he comprison of in-smple d in our suseuen empiricl nlysis ou correlion nd covrince, severl reled models re included, such s MA(00), EWMA wih λ = 0. 94, nd he CCC models. However, we exclude he correlion coefficien shown for he CCC model, due o he consn resricion in nure 4. Here one jus uses he esimed correlion regression on he relied correlion, nd he correlion is similr in he sme mnner. For simple regression, he R-sured cn e used s rough proxy for he model s performnce. For compleeness, we lso perform ou-of-smple forecs comprisons. I is very srighforwrd o derive he formulion in compuing he ou-of-smple condiionl correlion for MR_DCC specificion. Given s he smple sie, he 4 From expression (4), ny one of he explnory vriles is significnly differen from ero in sisics, which will rejec he null hypohesis for he correlion eing consn.

13 s oservion is oined y: =,,,,,,,,,, o o,,,, o, (6) where,,, / = ρ. For he period of h, wih h, he correlion is: =,,,, h h h h o,,,, h h h h o. (7) Since he vlues for he ou-of-smple correlion forecs derived from he I_DCC model re consns, we ridge he redundn explnion. In ddiion o he rnge-sed nd reurn-sed DCC models, he MA(00) nd he CCC models re inroduced for n ou-of-smple predicive comprison 5. I is lso worh noing h he correlion is consn for he ou-of-smple forecs sed on he CCC model, nd so we focus on he performnce of he models in forecsing he condiionl covrince. Empiriclly speking, we sill ke he vlue of R-sured s n indicion for he comprison of preciseness. 5 I is lso inuiively cler h he ou-of-smple forecss for he correlion nd covrince re oh consn in he EWMA model. hus, we ignore he reled discussion here.

14 III. Evluion of Condiionl Correlion nd Covrince Forecss he d employed in his sudy comprises 85 weekly oservions on he S&P500 Composie (henceforh S&P500), he Nsd sock mrke index, nd he yield for 0-yer resury ond (ond) spnning he period Jnury 4, 988 o Jnury, 004. In ddiion, dily oservions re used o consruc he series of mesured or so-clled he relied covrince nd correlion in he reled lierure. We rerieve he rnges nd reurns d for he enire period from Yhoo s dse. I is worh king look some descripive sisics. Pnels A, B, nd C in Figure demonsre he weekly d perns for he ime-series of he S&P500 sock mrke index, he Nsd index nd he yield o muriy for he 0-yer ond over he smple period. Addiionlly, le provides summry sisics for weekly coninuously compounded reurns nd weekly rnges for hese indices. Le MCORR nd MCOV represen he mesured or relied correlion coefficien nd mesured covrince, respecively. he MCORR is defined s τ i MCORR = ρ, (8) τ i= where τ denoes he rding dys during he week nd ρ i is he correlion coefficien he i h rding dy of he week. his series is oined from using he dily reurns d nd fiing hem wih MR_DCC model. By he sme wy, he mesured covrince (MCOV ) cn e expressed s follows. τ i i MCOV = ( r r ), (9) i= where i r k represens he dily reurns of index k he i h rding dy during week. his expression is direc exension of he concep of he relied voliliy of Andersen, Bollerslev, Dieold, nd Lys (000). Figure shows he grphs of MCORR nd MCOV for he S&P500 nd Nsd indices. We lloce hem Pnels A nd B in Figure seprely. I is ineresing o noe h for he ls wo yers, he wo series were highly posiively correled, nd moved more smoohly. king ond reurns o e minus he chnges in he 0-yer 4

15 enchmrk yield o muriy s in Engle (00), he correlion nd covrince eween he ond mrke nd sock mrke re rced. As o Pnels A nd B in Figure, we repor he ime series of MCORR nd MCOV for S&P500 nd ond yields. On he sme picure we lso show he MCORR nd MCOV series eween he Nsd index nd S&P500 in Figure 4. Judging from Figures nd 4, we find h he correlion perns eween he ond mrke nd sock mrkes pper o show revering phenomenon round pproximely he yer 000. I cn lso e deermined h he covrince processes re more volile fer he yer 000. A. In-smple forecs comprison In his secion we presen he resuls of using he in-smple d; h is, he forecs performnces re consruced nd mesured using he sme dse. wo vrin DCC forms re discussed, nmely, he men revering DCC form nd he inegred DCC form (i.e., MR_DCC nd I_DCC models for shorhnd). Pnel A in le descries he in-smple forecsing performnce for MCORR. I is very consisen o recognie h he inercive regression model fiing o he relionship eween differen clss mrkes is more suile hn he sme clss of sock mrkes judged from he R-sured index. Wih he excepion of he correlion eween he S&P500 nd Nsd indices, ll of he esimes of R-sured for he rnge DCC model re sisiclly preferred o he reurn DCC model. As o he in-smple forecs performnce, here re no cler dominn dvnges eiher for he MR_DCC or I_DCC model. Whever DCC model is chosen nd wheher he reurn-sed pproch or rnge-sed pproch or he moving verge model nd exponenil smoohing model re used, here re sufficien resons o infer h he correlion is ime-vrying vrile from he significn coefficien in he -vlue. Here, he -vlue in our regression model is shown fer djusmen of he Whie heeroskedsiciy-consisen sndrd errors. Surprisingly, he exponenil smoohing mehod for he correlion fiing under he in-smple forecsing scenrio is eer hn he ohers, regrdless of which wo mrkes inercion is disseced, especilly in he S&P500 nd Nsd indices. he exponenil smoohing model seems o perform well in he smple for correlion forecsing, u poorly for he ou-of-smple forecsing. echniclly, sed on he exponenil smoohing model, i oins consn vlue when we predic he ou-of-smple correlion eween ny wo vriles. In oher words, we cnno cpure he erm srucure of correlion for he ou-of-smple period. Finlly, 5

16 no mer wh mrke d is used, he MA00 is he wors model in our in-smple correlion forecs performnce comprison. Ech nlysis nd inference herein is consuled fer he inroducion of he relied correlion proxy in expression (8).As o oher oservions from Pnel A in le, he rnge-sed prmeer ˆ γ under he MR_DCC model is no significnly differen from ero in sisics when oh of he wo independen vriles (i.e. ˆ ρ reurn nd ˆ ρ rnge ) re simulneously fied. From Pnel B in le, we find h he ˆ γ coefficien ppers negive under he MR_DCC model when he wo correlion esimes re independen vriles he sme ime. However, he inference is indifferen from ero y he viewpoin of convenionl sisics. Due o he vlue of R-sured lmos sying he sme level, i seems h when he in-smple MCORR forecsing is evlued, he reurn-sed proxy derived y he GARCH ype voliliy is domined y he rnge-sed proxy derived y he CARR ype voliliy sed on he S&P500 index nd ond yield d. As o he in-smple forecsing of MCORR for he Nsd sock index nd ond yield for Pnel C in le, we find h he iliy o explin he relied correlion vrile s chnges is eer nd cn e shown when he significnce of he reurn-sed prmeer esime of ˆ γ is reduced. Conrsing o oh independen vriles h re incorpored he sme ime, we chieve he nlogous resul illusred in Pnel B. I is pprenly h he rnge proxy fiing o he in-smple forecss of correlion is gin eer on he seing of he DCC model. From Pnel A in le we rrnge he in-smple MCOV forecsing performnce comprison for severl useful models. Regrdless of which DCC model is seleced, he vlues of R-sured for he rnge-sed models re ll lrger hn hose of he reurn-sed models. Noneheless, model fiing o he covrince pern is slighly poorer hn o corresponding correlion pern. Clues come from he overll informion of R-sured doped y les nd. I is ineresed h he vlue of R-sured drops y significn moun when we discuss he covrince ehvior. o our knowledge, he chnge for he covrince pern migh e more volile hn he corresponding correlion series nd cnno esily cpure he movemen, nd we cn shown he R-sured vlue is lower hn he corresponding correlion h evidence from le. I is lso worh seeing h when he wo independen vriles re incorpored 6

17 simulneously in he regression model, hen he reurn-sed coefficien esime ˆ φ is no more significnly differen from ero under he MR_DCC or I_DCC model. Evidenly, he proxy vrile of he rnge-sed model for voliliy is more powerful hn he reurn-sed one in explining he vrin of he relied covrince vrile once more. he focus on Pnel B in le menions ou he covrince relionship eween he S&P500 nd ond yield series. Judging from he R-sured vlue nd he significnce of he coefficien, we cn see he fc h he rnge-sed proxy produces superior in-smple forecss for MCOV relive o he reurn-sed one when he wo independen vriles re fied in he sme regression euion. No mer if he MR_DCC model or he I_DCC model is exrced, he conclusion is consisen. As o Pnel C in le compres he in-smple covrince forecsing eween he Nsd sock index nd ond yield for lernive models. he inference is nlogous o he Nsd sock index nd ond yield when he issue of correlion ehvior is explored. Nmely, i is highly possile for he rnge-sed vrile for voliliy o replce nd domine he reurn-sed vrile for voliliy when he perns of MCOV nd MCORR re cpured. From le, we oin oher informion for covrince when he lower pr in ech pnel is checked. When using rdiionl nd convenionl mnners o descrie he covrince civiies eween vriles, for insnce, moving verge pproch, exponenil smoohing pproch, or consn condiionl correlion model proposed y Bollerslev (990), he forecsing iliy for he in-smple covrince hs no cler dvnge direcion mong hem. However, he only inference is h he DCC-sed fmily is eer hn he rdiionl mehodologies in he performnce of he in-smple forecsing for he covrince vrile. B. Ou-of-smple forecs comprison o ssess he relive performnces for he ou-of-smple correlion nd covrince forecsing, we dop he procedure of rolling smple o esime he ou-of-smple forecss using MR_DCC nd I_DCC specificions for oh he reurn-sed nd rnge-sed models. For ech individul model, we compue he ou-of-smple forecss for he horions of,,, nd 4 weeks. In ll cses, we 7

18 re-esimed he esimes 00 imes. We hen use simple regression o compre he explnory power of hese vrious forecss on he relied covrinces or correlions. le 4 repors he vlue of R from liner regression of MCORR on ech of hese ou-of-smple forecs series. he overll resul is consisen whever ou-of-smple horion is chosen. We chieve confirmion h he DCC-rnge model is more powerful hen DCC-reurn model when he forecsing inervls of correlion re during one-monh period. Wih he excepion of he relionship eween he Nsd index nd he ond yield for he four weeks ou of smple predicion, ll of he esimes of R-sured for ny oher mrke d winess he inference. le 5 shows he resuls for he comprison of ou-of-smple forecss for he covrince vrile. A cler lueprin emerges immediely from he implicion of his le. I is imporn o noe h he DCC model wih he rnge-sed frmework is significnly dominn hn he reurn-sed DCC model. Judging from he R-sured index, no mer wh mrkes rding d re shown in his period. he rnge-sed DCC model ouperforms he ohers in ll of he cses for ou-of-smple covrince forecsing, oo. here re some differences from he correlion predicion we oined from le 4. We oin h he reurn-sed DCC model no longer significnly eer hn he convenionl moving verge pproch in forecsing for he covrince vrile. A he exreme, he reurned-sed DCC is worse hn he moving verge pproch when he Nsd sock index nd ond yield re discussed. As o he CCC model, i cnno del wih he propery of he ime-vrying correlion nurlly nd is performnce for he ou-of-smple forecsing ou he covrince vrile is poor for one-monh period. In fc, s o he performnce for ou-of-smple forecsing in correlion nd covrince, he ler does no previl gins he former. Wih he excepion of he one-week ime horion for covrince forecsing sed on he rnge-dcc model, oher susiue models hrdly cpure he ouline for he ou-of-smple covrince vrince. I is possile o infer h he covrince pern is no esily cpured nd he chrcerisic of he mrke rding d is noher suile reson. Due o he voliliy of correlion eing fler hn he covrince, he R-sured eing higher hn he corresponding covrince model is uie recognile. 8

19 IV. Conclusions In his pper, new esimor of he ime-vrying correlion/covrince mrices is proposed uiliing he rnge d y comining he CARR model proposed y Chou (004) nd he frmework of Engle (00) s DCC model. he dvnge of his rnge-sed DCC model ouperforming he sndrd reurn-sed DCC model hinges on he relive efficiency of he rnge over he reurn d in esiming voliliies. Using weekly reurns of S&P500, Nsd nd 0-yer resury ond res, we find consisen resuls h he rnge-sed DCC model ouperforms he reurn-sed models in esiming nd forecsing covrince nd correlion mrices, oh in-smple nd ou-of-smple. Alhough we pply his esimor o he ivrie sysems, i cn e pplied o lrger sysems in mnner similr o he pplicion of he reurn-sed DCC model srucures h is demonsred in Engle nd Shepprd (00). Fuure reserch will e useful in doping more dignosic sisics or ess sed on vlue risk clculions s is proposed y Engle nd Mngnelli (999). Applicions o he esimion of opiml porfolio weighing mrices nd he clculion of he dynmic hedge rio in he fuures mrke will lso e fruiful. 9

20 REFERENCES Alideh, Sssn, Michel Brnd nd Frncis Dieold. (00) Rnge-sed esimion of sochsic voliliy models or exchnge re dynmics re more ineresing hn you hink, Journl of Finnce, 57: Andersen, oren, im Bollerslev, Frncis Dieold nd Pul Lys. (000). he disriuion of exchnge re voliliy, Journl of Americn Sisicl Associion, 96, Bollerslev, im (990) Modeling he coherence in shor-run nominl exchnge res: mulivrie generlied ARCH Model, Review of Economics nd Sisics, 7: Bollerslev, im, Ry Y. Chou, nd Kenneh Kroner. (99) ARCH modeling in finnce: review of he heory nd empiricl evidence. Journl of Economerics, 5: Bollerslev, im, Roer Engle nd J.M. Wooldridge (988) A cpil sse pricing model wih ime vrying covrinces, Journl of Poliicl Economy 96: 6-. Brnd, Michel, nd Chrisofer Jones. (00) Voliliy forecsing wih rnge-sed EGARCH models, mnuscrip, Whron School, U. Penn. Chou, Ry Y. (004) Forecsing finncil voliliies wih exreme vlues: he condiionl uoregressive rnge (CARR) model, Journl of Money Credi nd Bnking, forhcoming Chou, Ry Y. (004) Modeling he symmery of sock movemens using price rnges, Advnces in Economerics, forhcoming. Chou, Ry Y., Chun-Chou Wu nd Nhn Liu, (004) A comprison nd empiricl sudy in forecsing iliies of dynmic voliliy models, Journl of Finncil Sudies.Vol. No.: -5. Engle, Roer (00) Dynmic Condiionl Correlion - A Simple Clss of Mulivrie GARCH Models, Journl of Business nd Economic Sisics, Vol. 0 No. :9-50. Engle, Roer (00) New Froniers in ARCH Models, Journl of Applied Economerics, Vol. 7 No. 5 Engle, Roer (004) Risk nd Voliliy: Economeric Models nd Finncil Prcice, Noel Lecure, Americn Economic Review Vol. 94, No., Engle, Roer nd K. Kroner (995) Mulivrie Simulneous GARCH, Economeric heory : -50. Engle, Roer, Loreno Cppiello nd Kevin Shepprd (00) Asymmeric Dynmics in he Correlions of Glol Euiy nd Bond Reurns, mimeo, 0

21 NYU working pper Engle, Roer nd Kevin Shepprd, (00) heoreicl nd empiricl properies of dynmic condiionl correlion mulivrie GARCH, NBER Working Pper Engle, Roer nd Simone Mngnelli (999) CAViR: Condiionl Vlue A Risk By Regression Quniles, NBER Working Pper 74. Glln, Ronld, Chien-e Hsu, nd George uchen. (999) Clculing voliliy diffusions nd exrcing inegred voliliy, Review of Economics nd Sisics, 8: Grmn, Mrk, nd Michel Klss. (980) On he esimion of securiy price voliliies from hisoricl d, Journl of Business, 5: Kuniomo, Noo (99) Improving he Prkinson mehod of esiming securiy price voliliies, Journl of Business, 65: Mincer, Jco nd Vicor Zrnowi. (969) he evluion of Economic Forecss, in J. Mincer, ed, Economic Forecss nd Expecions (NBER) -46. Newey, Whiney nd Dniel McFdden (994) Lrge smple esimion nd hypohesis esing, Chper 6 in Hndook of Economerics Volume IV, (ed. Engle nd McFdden), Elsevier Science B.V. pp-45. Prkinson, Michel (980) he exreme vlue mehod for esiming he vrince of he re of reurn, Journl of Business, 5: sy, Ruey (00) Anlysis of Finncil ime Series, J. Wiley pulicions. se,yiu Kuen nd Aler K.C. sui (00) A mulivrie GARCH model wih ime-vrying correlions, Journl of Business nd Economic Sisics, 0:5-6. Wiggins, Jmes (99) Empiricl ess of he is nd efficiency of he exreme-vlue vrince esimor for common socks, Journl of Business, 99, 64, : Yng, Dennis nd Qing Zhng. (000) Drif independen voliliy esimion sed on high, low, open, nd close prices, Journl of Business, 7, :

22 Figure : S&P500, Nsd Indices, nd ond Yield Weekly Prices, /4/988-//004 Pnel A: S&P500 Sock index weekly closing prices /04/88 /04/9 9/04/95 7/05/99 5/05/0 SP_CLOSE 6000 Pnel B: Nsd sock index weekly closing prices /04/88 /04/9 9/04/95 7/05/99 5/05/0 NASDAQ_CLOSE 0 Pnel C: Yield o muriy for 0-yer ond weekly closing yields /04/88 /04/9 9/04/95 7/05/99 5/05/0 BOND_CLOSE

23 .0 Figure : MCORR nd MCOV for S&P500 nd Nsd Indices, /4/988-//004 Pnel A: Correlion series eween S&P500 nd Nsd Indices /04/88 /04/9 9/04/95 7/05/99 5/05/0 CORR_SP_NQ Pnel B: Covrince series eween S&P500 nd Nsd Indices /04/88 /04/9 9/04/95 7/05/99 5/05/0 COV_SP_NQ

24 Figure : MCORR nd MCOV for S&P500 Index nd ond Yield, /4/988-//004 Pnel A: Correlion series eween S&P500 index nd ond yield /04/88 /04/9 9/04/95 7/05/99 5/05/0 CORR_SP_BOND Pnel B: Covrince series eween S&P500 index nd ond yield /04/88 /04/9 9/04/95 7/05/99 5/05/0 COV_SP_BOND 4

25 Figure 4: MCORR nd MCOV for Nsd Index nd ond Yield, /4/988-//004 Pnel A: Correlion series eween Nsd index nd ond yield /04/88 /04/9 9/04/95 7/05/99 5/05/0 CORR_NQ_BOND Pnel B: Covrince series eween Nsd index nd ond yield /04/88 /04/9 9/04/95 7/05/99 5/05/0 COV_NQ_BOND 5

26 le : Summry Sisics for he Reurns nd Rnges of Weekly S&P500, Nsd Indices, nd ond Yield, /4/988-//004 high low Rnges nd reurns for sock indices re compued y 00 log( p / p ) nd close 00 log( p / ), respecively. Rnges nd reurns for he 0-yer close p high low close close (resury)ond re inferred y 00 log( p / p ) nd -00 log( p / ), respecively. Jrue-Ber is he sisic for normliy. here re 85 weekly smple oservions. All d re ken from Yhoo! Finnce. he compuion of he reurns of he ond yield follows Engle (00). p Index S&P500 Nsd 0-yer ond ype rnge reurn rnge reurn rnge reurn Men Medin Mximum Minimum Sd. Dev Skewness Kurosis Jrue-Ber

27 le : In-smple Forecsing for Correlions eween he S&P500 nd Nsd, S&P500 nd ond, nd Nsd nd ond, /4/988-//004 γ ˆ 0 γ ρ ε reurn MCORR =, rnge MCORR = γ 0 γ ˆ ρ ε, reurn rnge = 0 γ ˆ ˆ ρ γ ρ MCORR γ ε, his le repors he R from liner regression of he mesured correlion (MCORR) on he correlion forecss of he reurn-sed DCC model, he rnge-sed DCC model, models sed on moving-verge mehods (MA00) nd on exponenil smoohing mehods (Exp Smoohing). he vlues wih Heeroskedsiciy-Auocorrelion-Consisen sndrd errors for he regression coefficiens re in prenheses. here re 85 weekly smple oservions. Pnel A: S&P500 nd Nsd MCORR ˆ γ 0 ˆ γ ˆ γ R-sured Reurn 0.5 (0.45) (4.69) MR _DCC Rnge 0.47 (6.97) (4.59) 0.89 BOH 0.57 (0.54) (7.45) 0.58 (5.77) 0.45 I_DCC Reurn 0.50 (7.806) (8.) 0.6 Rnge 0.80 (0.676) 0.67 (.554) 0.50 BOH 0.7 (5.689) 0.6 (6.545) (0.) 0.9 MA (9.4) (.) 0.0 Exp. Smoohing 0.75 (0.745) (5.7) Pnel B: S&P500 nd ond Reurn 0.00 (4.54) (57.8) MR _DCC Rnge 0.0 (.789) (70.08) BOH 0.0 (.6) (-.5).56 (.066) 0.80 I_DCC Reurn 0.0 (.59) (59.05) Rnge (.64) (7.45) 0.8 BOH (.458) 0.8 (.6) 0.8 (7.70) 0.8 MA (-.704) (6.65) Exp. Smoohing 0.05 (.947) (67.70) 0.79 Pnel C: Nsd nd ond Reurn 0.06 (5.55) 0.85 (57.09) MR _DCC Rnge 0.09 (6.574) 0.88 (67.500) 0.86 BOH 0.0 (7.) (-5.57). (6.445) 0.8 I_DCC Reurn 0.0 (6.888) (57.549) 0.79 Rnge 0.08 (.987) (60.008) 0.80 BOH 0.0 (4.970) 0.5 (.944) 0.57 (6.777) MA (-.707) 0.87 (4.4) Exp. Smoohing 0.00 (6.8) (6.599)

28 le : In-Smple Forecsing for Covrinces eween S&P500 nd Nsd, S&P500 nd ond, nd Nsd nd ond, /4/988-//004 reurn MCOV = φ 0 φcov ε, rnge MCOV = φ0 φcov ε, reurn rnge MCOV = φ0 φcov φcov ε, his le repors he R from liner regression of he mesured covrince (MCOV) on he covrince forecss of he reurn-sed DCC model, he rnge-sed DCC model, models sed on moving-verge mehods (MA00), on exponenil smoohing mehods (Exp Smoohing) nd he consn condiionl correlion (CCC) model. he vlues wih Heeroskedsiciy-Auocorrelion-Consisen sndrd errors for he regression coefficiens re in prenheses. here re 85 weekly smple oservions. Pnel A: S&P500 nd Nsd MCOV φ 0 φ φ R-sured Reurn 0.75 (.968) 0.99 (0.78) 0. MR _DCC Rnge 0.9 (0.750) (9.48) 0.54 BOH (.89) -0. (-.09) (5.55) 0.57 Reurn.48 (.96) 0.86 (9.57) 0.0 I_DCC Rnge 0.84 (0.87) (9.7) 0.5 BOH 0.69 (.86) (-.06) 0.96 (5.78) 0.54 MA (.906).0 (0.56) 0.0 Exp. Smooh.0 (.577) 0.86 (9.855) 0.00 CCC (.690) 0.97 (9.994) 0.5 Pnel B: S&P500 nd ond Reurn -0.9 (-.654) (0.4) 0.0 MR _DCC Rnge (-0.676) (0.) 0.09 BOH (0.056) (-.05).6 (4.864) 0.0 Reurn (-.04).0 (0.850) 0. I_DCC Rnge (-.489) (0.) 0.09 BOH (-.9) (-.57).6 (4.775) 0.5 MA (-.84).070 (8.85) 0.77 Exp. Smooh -0.0 (-.4) (0.040) 0.7 CCC.57 (9.8) -.98 (0.0) 0.0 Pnel C: Nsd nd ond Reurn (-.6) 0.94 (9.68) 0.6 MR _DCC Rnge (-0.49) 0.70 (9.749) 0.0 BOH (0.04) -0.0 (-.57) (4.48) 0. Reurn (-.556).064 (9.9) 0.68 I_DCC Rnge -0.7 (-.004) 0.78 (9.847) 0.8 BOH (-.00) (-0.4) (4.59) 0.8 MA (-.7) (8.7) 0.5 Exp. Smooh (-.66) 0.84 (9.8) 0.6 CCC.696 (5.7) (-5.789)

29 le 4: Ou-of-Smple Forecsing for he Correlions eween S&P500 nd Nsd, S&P500 nd ond, nd Nsd nd ond, /4/988-//004 MCORR = γ γ ˆ ρ ε 0 his le repors he R from liner regression of he mesured correlion (MCORR) on he correlion forecss of he reurn-sed DCC model, he rnge-sed DCC model, nd he moving-verge mehods (MA00). he regressions re conduced using 00 oservions sed on he rolling smple mehod. For ech model, 700 oservions re used for in-smple esimion nd ou-of-smple forecss of horions,, nd 4 weeks re mde. Pnel A: S&P500 nd Nsd R-sured MR_DCC Forecs horion reurn-sed rnge-sed MA Pnel B: S&P500 nd ond R-sured MR_DCC Forecs horion reurn-sed rnge-sed MA Pnel C: Nsd nd ond R-sured MR_DCC Forecs horion reurn-sed rnge-sed MA

30 le 5: Ou-of-Smple Forecsing, for he MCOV eween S&P500 nd Nsd, S&P500 nd ond, nd Nsd nd ond, /4/988-//004 MCOV = φ φcov ε 0 his le repors he R from liner regression of he mesured covrince (MCOV) on he covrince forecss of he reurn-sed DCC model, he rnge-sed DCC model, he moving-verge mehod (MA00), nd consn condiionl correlion (CCC) model. he regressions re conduced using 00 oservions sed on he rolling smple mehod. For ech model, 700 oservions re used for in-smple esimion nd ou-of-smple forecss of horions,, nd 4 weeks re mde. Pnel A: S&P500 nd Nsd R-sured MR_DCC Forecs horion reurn-sed rnge-sed MA(00) CCC Pnel B: S&P500 nd ond R-sured MR_DCC Forecs horion reurn-sed rnge-sed MA(00) CCC Pnel C: Nsd nd ond R-sured MR_DCC Forecs horion reurn-sed rnge-sed MA(00) CCC