High frequency analysis of lead-lag relationships between financial markets de Jong, Frank; Nijman, Theo

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1 Tlburg Unversy Hgh frequency analyss of lead-lag relaonshps beween fnancal markes de Jong, Frank; Nman, Theo Publcaon dae: 1995 Lnk o publcaon Caon for publshed verson (APA): de Jong, F. C. J. M., & Nman, T. E. (1995). Hgh frequency analyss of lead-lag relaonshps beween fnancal markes. (CenER Dscusson Paper; Vol ). Unknown Publsher. General rghs Copyrgh and moral rghs for he publcaons made accessble n he publc poral are reaned by he auhors and/or oher copyrgh owners and s a condon of accessng publcaons ha users recognse and abde by he legal requremens assocaed wh hese rghs. - Users may download and prn one copy of any publcaon from he publc poral for he purpose of prvae sudy or research - You may no furher dsrbue he maeral or use for any prof-makng acvy or commercal gan - You may freely dsrbue he URL denfyng he publcaon n he publc poral Take down polcy If you beleve ha hs documen breaches copyrgh, please conac us provdng deals, and we wll remove access o he work mmedaely and nvesgae your clam. Download dae: 17. sep. 2018

2 HIGH FREQUENCY ANALYSIS OF LEAD-LAG RELATIONSHIPS BETWEEN FINANCIAL MARKETS. Frank de Jong Theo Nman Tlburg Unversy February 1995 Malng address: Frank de Jong Deparmen of Economercs Tlburg Unversy PO BOX LE TILBURG The Neherlands phone: fax: Emal: Thanks are due o Peer Bossaers, Peer Schoman and Bas Werker for useful commens. Of course, he auhors reman responsble for all errors.

3 Absrac Hgh frequency daa are ofen observed a rregular nervals, whch complcaes he analyss of lead-lag relaonshps beween fnancal markes. Frequenly, esmaors have been used ha are based on observaons a regular nervals, whch are adaped o he rregular observaons case by gnorng some observaons and mpung ohers. In hs paper we propose an esmaor ha avods mpuaon and uses all avalable ransacons o calculae (cross) covarances. Ths creaes he possbly o analyze lead-lag relaonshps a arbrarly hgh frequences whou addonal mpuaon bas, as long as weak denfably condons are sasfed. We also provde an emprcal applcaon o he lead-lag relaonshp beween he SP500 ndex and fuures wren on. 1

4 1. Inroducon. Lead-lag relaonshps have been analyzed beween many fnancal markes. A prme example s he lnk beween he ndex fuures and he cash marke, where many researchers have found ha he fuures marke leads he cash marke (see e.g. Kawaller, Koch and Koch (1987), Soll and Whaley (1990), Chan (1992) and Grünblcher, Longsaff and Schwarz (1994)). Ohers consdered he relaonshp beween he sock marke and he opon marke. Sephan and Whaley (1990) fnd ha he sock marke leads he opon marke; hs phenomenon s explaned by Chan, Chung and Johnson (1993). Also, an ncreasng number of secures s raded on more han one fnancal marke e.g. secures from many European counres ousde he UK are raded on London s SEAQ Inernaonal marke n he domesc currency. Analyss of he lead-lag relaonshp beween hese markes would be ye anoher example. In order o analyze nformaon flows beween markes on shor me nervals, hgh frequency daa are requred. Typcally, all ransacons for some sample perod are avalable for analyss. However, he sascal analyss of ransacons daa s ofen hampered by he fac ha he clock me nerval beween such observaons s varyng. For some research quesons, such as mos mcro-srucure ssues, he dfferences n clock me nerval are no very mporan and one reles on esmang models n ransacon me. However, for he analyss of nformaon flows beween markes he clock me s of umos mporance. The usual approach o ackle he problem of rregularly spaced observaons s o spl he me axs n fxed lengh nervals of, say, 5 mnues, and use he las observaon recorded n ha nerval n he sascal analyss. Ths approach has wo mporan drawbacks, however: 2

5 () If he nervals are small and radng s no very frequen, some nervals may conan no observaon. Ths s referred o as he nonradng or non-synchronous radng problem. Anoher cause of mssng observaons are mperfecons n daa collecon, e.g. errors on he daa fle, whch somemes cause a loss of observaons. In boh cases, ad hoc procedures o deal wh mssng observaons mus hen be nvoked. () On he oher hand, n perods where radng s busy, a lo of observaons are hrown away. Ths makes he sascal analyss less effcen. The loss of effcency s an especally serous problem f busy radng s assocaed wh large prce changes, whch s usually he case. In hs paper we propose an esmaor ha avods arbrary mpuaon mehods. Ths creaes he possbly o analyze lead-lag relaonshps a arbrarly hgh frequences whou addonal mpuaon bas, as long as weak denfably condons are sasfed. The plan of he paper s as follows. In secon 2 we nroduce a conssen esmaor of he covarances and correlaons of neres from rregularly spaced daa. In secon 3 we derve he large sample dsrbuon of hese esmaors. In secon 4 we dscuss some poenal exensons of he mehod. Secon 5 conans an emprcal applcaon o he lead-lag relaonshp beween he S&P500 ndex and he fuures on hs ndex. Secon 6 concludes. Techncal deals are dscussed n he appendces. 2. Esmaon of correlaons n real me wh rregularly spaced observaons. In hs secon, we presen a mehod for esmang correlaons beween reurns from rregularly spaced ransacons daa. The underlyng model s a dscree 3

6 me process a an arbrary me nerval, no a connuous me process. We frs consder he case where he reurns have zero mean and here are no deermnsc componens n he model 1. Le p and q denoe he (logarhm of) levels of he wo prce seres under consderaon, where s he clock-me ndex. The prce levels are assumed o be non-saonary processes, whch are saonary afer dfferencng. Denoe he cross-covarance funcon of he underlyng reurns (one-perod prce changes) by (1) γ = Cov( p, q ), p p-p, q q-q. k -k -1-1 If he prce levels were observed a every pon, he covarances γ k could be esmaed effcenly by he usual expressons. However, when usng ransacons daa here are poenally a lo of me nervals wh no new observaon on he prce level. One way o solve hs problem s o mpue a zero reurn for hs nerval, bu ha wll bas he usual covarance esmaors owards zero. In order o oban an unbased covarance esmaor, we use he dfferences beween observaons on he prce level over more han one nerval. We hen nfer he covarances of he underlyng bu unobserved one-perod reurns from he cross-producs of hese more-perod reurns. In hs secon, we dscuss hs mehod n some deal. We ndex he observaons on p by he ndex and he observaons on q by he ndex j, and denoe he oal number of observaons by N and M, respecvely. The dfferences beween wo observed prce levels can be expressed as sums of he reurns of he unobserved underlyng prce process + 1 (2) p -p = p +1 = +1 4

7 where denoes he clock-me ndex of he h observaon. The cross produc of prce changes on he wo markes can hus be wren as + 1 j+ 1 (3) y (p -p )(q -q ) = p q. +1 j+1 j = +1 s= +1 s j The expecaon of hs cross-produc s a lnear combnaon of he crosscovarances γ k of he underlyng processes +1 j & * j+1 (4) E(y ) = E p q = γ(-s), 7 s8 = +1 s= +1 = +1 s= +1 j j where he expecaon n (4) s condonal on he observed ransacon mes (, j, +1, j+1). Le x (k) denoe he number of mes ha γ(k) appears n hs expresson. In appendx A he followng expresson for he x (k) s derved: (5) x (k) = max(0, mn(, -k) - max(, +k)). +1 j+1 j An mporan propery of he x s s ha hey are funcons of he ransacon mes only, no of he observed prces. Therefore, we replace he condonng on he ransacon mes by a condonng on he x s and wre E(y ) as a lnear combnaon of he covarances γ(k), k=-k,..,k, as follows (6) E(y 1x ) = K x (k)γ(k). k=-k Our esmaon mehod s based on he fac ha equaon (6) can be consdered as a regresson equaon wh he unknown cross-covarances γ k as parameers and 5

8 he coeffcens x as explanaory varables. In vecor noaon, he regresson equaon reads (7) y x γ +e The covarances can hen be esmaed by ordnary leas squares on he observaons of y and he consruced x s 2. In prncple, all possble dfferences beween observed prces can be used o consruc an x and y. However, we can confne ourselves o dfferences of adjacen observaons. The reason for hs s ha dfferences of non-adjacen observaons always can be wren as exac lnear combnaons of dfferences of adjacen observaons. For example, consder (8) (p -p )(q -q ) = +2 j+1 j (p -p )(q -q ) + (p -p )(q -q ) = y +1,j + y j+1 j +1 j+1 j For hs reason, non-adjacen observaons do no add nformaon and can be omed. All n all, N mes M cross-producs y are avalable for he analyss. I s no necessary o use all of hem, however, f he number of nonzero cross-covarances s lmed, say o K. In ha case, all cross producs where 1-1 K and 1-1 K can be omed because here wll be no +1 j j+1 non-zero elemens n x n ha case. Le all useful observaons be conaned n he desgn marx X and vecor of observaons y as follows 6

9 ( ) ( ) 2x 11(-K).. x 11(K) 2y11 x (-K) x (K) y (9) X = , y = 2 2 x 1M(-K) x 1M(K) y 1M... x (-K).. x (K) y 2 2 NM NM 2 2 NM Followng Cohen e al. (1983) and Lo and MacKnlay (1990,1991), we assume ha he order arrval process s ndependen of he prce process. Under hs assumpon, and f X X s nverble and weak regulary condons are ^ -1 sasfed, he OLS esmaor γ (X X) X y s a conssen esmaor for he uncondonal covarances of γ =(γ,..,γ ). A necessary condon for -K K conssency s ha all omed covarances (.e. of order > K) are ndeed equal o zero. If hese covarances are no equal o zero he regresson model wll suffer from an omed varables bas. Hence, even f one wans o esmae, say, only he frs order correlaon, one should esmae he whole vecor of non-zero covarances. The proposed esmaor s more general han he models proposed by Cohen e al. (1983) and Lo and MacKnlay (1991), because we do no assume a parcular process for he order arrval. As long as he order arrval process s exogenous o he prce changes our esmaor yelds conssen esmaes of he covarances n clock-me. We shall now dscuss some specal cases of our esmaor. Example 1. Prces observed n every perod. The frs case we dscuss s he sandard case where p and q are observed n every perod. In hs case, only he usual frs dfferences p and q need o be consdered. When esmang cross-covarances -K o K, he desgn marx becomes 7

10 ( ) ( ) ( ) 2y1,1-K 2 p1 q1-k y p q ,2-K -K y1,1+k p q X=2 2, y = 2 2 = K y2,2-k p q 2 2-K y2 N,N+K22 2 pn qn+k Obvously, he X X marx s a dagonal marx N I, and X y s a 2K+1 vecor wh ypcal elemens p q +k, so ha he OLS esmaor s equal o he usual covarance esmaor, ^γ = N-1 k p q +k. Example 2. Regularly mssng observaons. Now suppose ha he prces are no observed n every perod, bu on regularly spaced nervals. To choose he smples example, suppose ha p and q are observed every second perod. The useful cross-producs hen are y = (p -p )(q -q ) = ( p + p )( q + q ) -1 -k -k-1-2 -k -k-2 E(y ) = γ +2γ +γ k k-1 k-2 The desgn marx X n he smples case ha K=1 akes he form ( ) X=

11 I s clear ha he frs and second column are exac mulples of each oher, so ha here s exreme mulcollneary. Therefore X X s sngular and no all cross-covarances can be esmaed. Ths argumen can be generalzed o he saemen ha eher complee observaons or some rregularly mssng observaons are necessary o esmae all covarances. Throughou hs paper we shall assume ha such denfyng condons are sasfed. From he esmaes of he auocovarances, esmaes of he auocorrelaons can be compued n he usual way. The cross-correlaon funcon s defned as he cross-covarances, scaled by he square roo of he produc of he esmaed varances of p and q (10) ρ(k) ^ = ^γ(k) ^ ^ 1/2 [γ p (0)γ q (0)] 3. Large sample dsrbuon of he esmaors. In hs secon we derve he large sample dsrbuon of he esmaors derved n he prevous secon. Ths large sample dsrbuon can be used o es for he sgnfcance of lead-lag effecs. We sar from he usual resul ha he regresson esmaor s asympocally normal and ha s varancecovarance marx can be expressed as -1-1 (11) Ω =(X X) X E(ee )X(X X). 9

12 Two esmaors of Ω wll be consdered. Under srong addonal assumpons s possble o oban an analyc expresson for Σ=E(ee ). Subsequenly we presen a more robus, Whe-ype esmaor of Ω. In order o derve he frs esmaor of Ω, assume ha p and q are generaed by he same nnovaons ε, bu wh dfferen MA coeffcens (12) p = φε +φε +φε + = φ ε =0 - q = θε +θε +θε + = θ ε =0 - Noe ha hs assumpon mples ha he level varables p and q are conegraed 3. In Appendx B s shown ha he elemens of Σ can be expressed as (13) σ =x γ x γ +x γ x γ +(µ-3σ 4 4 )f(θ,φ),gh g p jh q h jg where γ p and γ q denoe he auo-covarances of {p } and {q }, respecvely, µ 4 he fourh momen of he nnovaons, and f(θ,φ) s an expresson n he MA coeffcens. If he errors are non-normal, he MA coeffcens and he fourh momen of he nnovaon have o be calculaed n order o esmae he (µ -3σ 4 4 )f(θ,φ) erm. Ths makes emprcal applcaon of hs resul cumbersome. An alernave and more robus way o calculae sandard errors s a Whe (1980) ype esmaor, where he expecaon of X ee X s esmaed by a summaon over all observaons for whch E(e e gh ) s non-zero. Thus, he esmaor of Ω becomes -1-1 (14) Ω =(X X) ( gh x x gh e e gh I(σ,gh 0))(X X), 10

13 where I(.) s an ndcaor funcon whch equals one f σ,gh 0, and zero elsewhere. The laer propery s easly checked from equaon (13). Ths esmaor wll be conssen for Ω under much weaker assumpons on he daa generang process. For example, we do no need normaly, nor he resrcve assumpon ha boh seres (p and q) are generaed by he same nnovaons. Noe ha he number of non-zero covarances used o calculae he sandard errors n (13) or o calculae he ndcaor n (14) can be smaller han he number of covarances acually esmaed. For example, we can esmae 10 covarances, bu calculae he sandard errors under he hypohess ha all bu he frs are zero. Ths wll smplfy and speed up he calculaons of he sandard errors consderably. 4. Exensons of he mehod. In hs secon, we dscuss wo poenal exensons of our mehod o esmae covarances on rregularly spaced daa. The frs exenson s he ncluson of a laen bd-ask spread, whch s very relevan for he applcaons o fnancal me seres. The second exenson s he ncluson of addonal observed explanaory varables. To sar wh he frs exenson, suppose ha he observed prces can be decomposed n an equlbrum prce π plus or mnus a fxed bd-ask spread, δ=s/2. We are neresed n esmang he auocorrelaons of π. Defne a bnomal ndcaor b, whch can ake values +1 and -1, such ha (15) p = π + b δ, so ha he observed prce dfferences can be wren as 11

14 + 1 (16) p -p = π -π +(b +1 -b)δ= π +(b +1 -b)δ = +1 Frs, consder he case where we do no know wheher he ransacon s a he bd or a he ask, hence b s unobserved. We now nroduce some srong assumpons on he bd-ask ndcaor: b has expecaon zero and s uncorrelaed wh boh s own pas and wh he prce and ransacon me processes. Noe ha hese are bascally Roll s (1984) assumpons, and herefore our mehod can be seen as an adapon of Roll s esmaor. Under hese assumpons, he expecaon of he cross-produc of prce dfferences s (17) E[(p -p )(p -p )] = x γ + E[(b -b )(b -b )]δ 2 =x γ +dδ 2 +1 j+1 j +1 j+1 j where γ now s he vecor of covarances of he equlbrum prce changes π, and he he new regressor d s defned as follows: d =2 f =j, d =-1 f j=+1 or j=-1, and d =0 oherwse. Noe ha he values of d do no depend on he me of he ransacons, only on he sequencng. Equaon (17) s a sraghforward exenson of he orgnal model (5), and he esmaors and sandard errors descrbed n he prevous secons can be appled o hs model mmedaely. Twce he square roo of he esmaed coeffcen of d can be used as an esmaor for he realzed bd-ask spread. Ths esmaor of he bd-ask spread s smlar n spr o he one proposed by Roll (1984) and Rchardson and Smh (1991), who use a GMM esmaor o esmae he mean, varance and bd-ask spread on seres of overlappng reurns. Our esmaor s more general han Roll s esmaor and Rchardson and Smh s esmaor because allows for seral correlaon n he equlbrum prce process and for rregular radng nervals. However, he spread esmaor suffers from he same weaknesses as Roll s esmaor: needs he assumpon 12

15 ha he bd-ask bounce s ndependen of he prce process. Marke mcrosrucure heory suggess ha hs s a very unrealsc assumpon. For example, n he Glosen-Mlgrom (1985) model wh only asymmerc nformaon here s a bd-ask spread, bu he seral correlaon n observed prces s zero, hence Roll s and our esmaor wll esmae a zero spread. The second exenson s he ncluson of observed regressors oher han he x s. Concepually, hs s rval as exends he model o (18) y = x γ + z β + e. As long as he z s are uncorrelaed wh he error erm, nohng changes and he OLS esmaors and he robus sandard errors wll be conssen. Ths exenson s useful f he bd-ask ndcaor b s observed. In ha case, he observed cross-producs (b +1 -b )(b j+1 -b j ) can be added o he model as addonal regressors: 2 (19) E(y ) = x γ + (b -b )(b -b ) δ +e. +1 j+1 j In hs case, here wll be no bas n he effecve spread esmaes even f he b seres s serally correlaed or depends on prevous prce changes. 5. Emprcal applcaon. In hs secon we presen an emprcal applcaon of he proposed esmaor o he lead-lag relaonshp beween he S&P 500 sock ndex and fuures on hs ndex. As saed n he nroducon, hs s a well-suded relaonshp, wh he general concluson ha he fuures marke leads he cash marke. Typcally, researchers have used fve mnue nervals, where few observaons are mssng. 13

16 In hs secon, we also presen resuls a he one mnue nerval, a whch more nervals whou rade occur n he fuures marke. Snce he sock marke ndex s adjused every mnue here are no mssng daa pons on he ndex unless he frequency a whch he daa are analyzed s even hgher han one mnue. Followng Soll and Whaley (1990), he relaon beween cash ndex prces and fuures prces can be expressed smply as (20) F = S exp[(r-d)(t-)], where F denoes he fuures prce, S he cash prce, (r-d) he neres rae mnus he convenence yeld (dvdends), assumed consan, and T he expraon dae of he fuures conrac. From (20) s easly seen ha here s an exac heorecal relaon beween he logarhmc reurns on he cash ndex and he fuures: F (21) R = (r-d) + R. S In pracce, he equaly does no always hold exacly. An obvous cause of hese devaons are measuremen errors and he effec of he bd-ask spread. Anoher explanaon, whch for he purpose of our paper s more neresng, s gven by poenal dfferences n he speed a whch nformaon s dssemnaed o boh markes or he lmed ably of ndex arbrage, whch nvolves radng n a large number of asses. Therefore, s neresng o assess wheher he reurns on one marke are predcable from he reurns n he oher marke. Soll and Whaley (1990) nvesgae hs queson for he US ndexes. Soll and Whaley use observaons on all ransacons or quoe changes of he 14

17 S&P 500 ndex and he Major Marke Index (MMI) and he fuures on hese ndces. The radng day s dvded no nervals of 5 mnues. The frs prces o be observed n hese nervals are hen used o consruc 5-mnue reurns n boh he cash ndex and fuures markes. Ths creaes some problems f here are no ransacons n some nerval. Usually, a zero reurn for hese perods s mposed. Soll and Whaley s emprcal mehodology s n wo seps. Frs, hey calculae he auo- and cross-correlaons of R and R. The SP500 cash ndex reurns show srong posve seral correlaon. The fuures reurns are almos serally uncorrelaed. Indvdual sock reurns end o be negavely serally correlaed due o he bd-ask bounce. These resuls are exacly n he drecon predced by Lo and MacKnlay (1991), who show ha he reurns of a connuously radng marke mus lead he observed reurns from a marke wh a posve probably of non-radng. However, he magnude of he correlaons found by Soll and Whaley canno be explaned by he acually observed probably of non-radng. Chan (1992) corroboraes hese conclusons on he Major Marke Index, whch consss of 20 large socks and s herefore less prone o non-radng problems. The fuures reurns lead he MMI ndex reurn by 15 mnues and also end o lead ndvdual sock reurns. Especally marke-wde nformaon seems o be processed faser n he fuures marke. The concluson of he leraure herefore s ha he fuures marke processes new nformaon faser han he cash ndex marke. In hs paper we shall nvesgae hs proposon usng he covarance esmaors developed n he prevous secons. The esmaor deals naurally wh nervals whou new observaons on he ndex or fuures prce. Therefore, he analyss can be 4 performed on a hgher frequency han he usual 5 mnues. Our daa concern spo and fuures prces of he S&P 500 ndex, obaned from he ISSM. The sample s from he las quarer of 1993 S F 5. The ndex prces 15

18 are me samped exacly a he full mnue, whereas he mng of he fuures prces s exac up o one second. The daa are dscrezed by akng he las rade or ndex repor n a gven nerval as he value of he level varable for ha nerval. If here s no sngle rade n an nerval, hs observaon s mssng. We consder observaons on he fuures ha expre n December 1993 (before 15/12) and March 1994 (afer 15/12). As usual when dealng wh nraday daa, we exclude overngh reurns from he analyss, as hese canno be expeced o have he same covarance srucure as whn-day reurns, see French and Roll (1986). We have nearly complee observaons for he ndex. However, for he fuures here are nervals whou ransacons. For example, a he one mnue frequency, 13% of he nervals does no conan a new observaon. As a frs sep n he analyss, we esmae he auocorrelaons of he fuures prce changes and he ndex changes. Table 1 repors he auocorrelaon esmaes of he ndex and Table 2 hose of he fuures reurns. We consder me nervals of en and fve mnues, as well as a one mnue nerval. In all emprcal resuls, he varance-covarance marx of he esmaes s calculaed under he assumpon ha only he varance and he frs covarances of he reurns are non-zero. Frs, we consder he resuls on a fve and en mnue nerval. Followng Chan (1992), he maxmum order of correlaon consdered s sx. The ndex reurns show lle seral correlaon on a en mnue nerval, and posve frs order correlaon on a fve mnue nerval, bu furher lags are no sgnfcan. The fuures reurns are serally uncorrelaed a boh he fve and en mnue nerval. If we ncrease he frequency of observaon o one mnue, a dfferen paern emerges. For he ndex, he seral correlaons are sgnfcanly posve, up o order egh. The esmaed auocorrelaons are smaller han he esmaes n Harrs e al. (1994), probably as a resul of he dfferen sample perod used. The frs order auocorrelaon n he fuures reurns s sgnfcanly negave. Ths s very lkely due o he bd-ask bounce 16

19 of he fuures conrac. There s no sgnfcan hgher order seral correlaon n he fuures reurns, whch shows ha all relevan nformaon s mmedaely refleced n he fuures prces, even on such a hgh frequency as one mnue. We now urn o he lead-lag srucure of cash and fuures prce changes. The cross-correlaons beween fuures and ndex reurns are repored n Table 3. These are defned as he cross-covarances, Cov(R x,r f -k ), dvded by he sandard devaon of he ndex and fuures reurn on he same nerval. A posve correlaon for k>0 ndcaes ha he fuures reurns have predcve ably for he ndex reurns. The resuls of hs able are unambguous: a all nervals, he fuures reurns sgnfcanly lead he ndex reurns. The me span of hs correlaon s a leas en mnues, gven he sgnfcan frs order cross-correlaon a he en mnue nerval. A he one mnue frequency, up o en lead correlaons of he fuures are sgnfcan. Ths concluson s confrmed by he jon sgnfcance ess of all lead coeffcens n Table 4. On he oher hand, here s no evdence ha he ndex reurns lead he fuures reurns by more han fve mnues, because he cross-correlaons for k<0 are nsgnfcan a he fve and en mnue nervals. A he one mnue nerval, here s some lead correlaon from he ndex o he fuures reurns, bu only up o wo mnues. The cross-correlaons are sronger han s predced by he auocovarances n he ndex alone (cf. Boudoukh, Rchardson and Whelaw (1994)). Hence, he correlaon canno be due solely o hn and nonsychronous radng n he ndex alone. An alernave explanaon, pu forward by Chan (1993) and Bossaers (1993) s based on dfferenal nformaon n markes. If frm specfc nformaon canno be separaed from marke wde nformaon n he ndvdual sock markes, ndex reurns wll be posvely serally correlaed, despe he fac ha he ndvdual sock reurns are serally 17

20 uncorrelaed. If he fuures marke reflecs only marke wde nformaon wll lead he reurns on he cash ndex. 6. Conclusons. In hs paper we have developed a mehod for esmang covarances of nonsaonary me seres wh rregularly spaced observaons. Under weak condons, hs esmaor s conssen under any paern of mssng observaons. Several exensons o nclude laen or deermnsc varables are developed. We apply he mehod o he lead-lag relaon beween sock marke ndex reurns and ndex fuures reurns. An analyss on a one mnue frequency reveals ha he fuures lead he cash ndex by a leas en mnues, whereas he cash ndex leads he fuures by a mos wo mnues. Anoher applcaon of our esmaor can be found n De Jong, Maheu and Schoman (1995). In ha paper, we apply he proposed mehods o exchange raes. In parcular, we sudy lead-lag paerns beween he acual Yen/Dmark exchange rae and he exchange rae mpled by cross-arbrage va he US dollar exchange raes. 18

21 Appendx A. An expresson for x. Recall he defnon of x n (4). In hs appendx we show how o smplfy he calculaons necessary o oban he elemens of x. By changng he ndex of summaon from -j o k and workng ou he resulng expresson we oban + 1 j j j - 1 x γ γ(-s) = γ(k) = x (k)γ(k), = +1 s= j+1 = +1 k= -j+1 k= - j+1+1 Wha remans o be deermned s he coeffcen x (k) of γ(k). To faclae he dervaon of hs number, n Fgure A he nervals [, +1] and [ j, j+1] are graphed. Fgure A. Overlappng nervals beween wo pars of observaons. q===========================================================================================================================================================================================e 2 + 2j k k k k l j k k k k k l 2 j j+122 z===========================================================================================================================================================================================c The number of correlaons γ(k) beween he prce changes over hese nervals can be deermned by shfng he [, ] nerval by k perods o he rgh, j j+1 o oban [ +k, +k]. The coeffcen of γ(k) s exacly equal o he number j j+1 of perods n he overlap of he nervals [, ] and [ +k, +k]. If he +1 j j+1 se of overlappng perods s no empy, he me ndex of he upper bound of he overlappng nerval s mn(, +k), and he me ndex of he lower +1 j+1 bound of he overlappng nerval s max(, j+k)). The number of covarances γ(k) s hus equal o he dfference beween he upper and lower bounds of hs nerval. If he nervals do no overlap, γ(k) s by defnon equal o 0. The upsho of hs analyss s he followng expresson 19

22 x (k) = max(0, mn(, +k) - max(, +k)). +1 j+1 j If he maxmal order of correlaon s resrced a pror, so ha γ(k)=0 for 1k1>K, hen he summaon over k s runcaed beween -K and K, as follows x γ = K x (k)γ(k), k=-k where he defnon of x (k) remans unchanged. Usng hs expresson for x reduces he compuaon me subsanally because double summaons are avoded. In he case of esmang auo-covarances, he coeffcens x (-k) should be added o x (k) for all k=1,..,k. Noe ha x (0) s no changed. The dmenson of he regresson model s hus reduced o K+1. 20

23 Appendx B. The covarance srucure of he error erms. Le p and q have he followng Wold represenaons, drven by he same nnovaons ε bu wh dfferen MA parameers {φ } and {ϑ } p = K Σ φ ε =0 - q = K Σ ϑ ε =0 - The error erms of he regresson equaon (5) are + 1 j j+ 1 e = y - E(y ) = ( p)( q)- s γ(-s) = +1 s= +1 = +1 s= +1 j j The covarance beween wo such errors s +1 j+1 g+1 & h+1 * E(e e gh ) = E ( p)( q)( p)( q ) - 7 s u v 8 = +1 s= +1 u= +1 v= +1 j g h +1 j+1 g+1 & *& h+1 * γ(-s) γ(u-v) = +1 s= +1 u= +1 v= +1 j g h + 1 j+ 1 g+ 1 h+1 & * = E( p q p q ) - γ(-s)γ(u-v) 7 s u v 8 = +1 s= +1 u= +1 v= +1 j g h By applcaon of he expresson gven n Brockwell and Davs (1987, p.220), for he expecaon of he four-fold produc p q s p u q v we oban 21

24 + 1 j+ 1 g+ 1 h+ 1 & E(e e gh ) = γ p(-u)γ q(s-v) + γ(-v)γ(u-s) + 7 = +1 s= j+1 u= g+1 v= h+1 K 4 * (µ 4-3σ ) Σ ϑ ϑ +s- φ +u- φ +v-8 =0 where γp and γq denoe he auo-covarances of p and q, respecvely, and σ 2 and µ denoe he second and fourh momen of he nnovaons ε 6 4. The expresson for he covarance consderably smplfes f he nnovaons ε are normally dsrbued. In ha case, he (µ -3σ 4 4 ) erm vanshes and he resulng expresson conans only auo- and cross covarances and he fourfold summaon can be spl no producs of double summaons + 1 j+ 1 g+ 1 h+ 1 & * E(e e gh ) = γ p(-u)γ q(s-v) + γ(-v)γ(u-s) = 7 8 = +1 s= +1 u= +1 v= +1 j g h + 1 g+ 1 j+ 1 & *& h+1 * γ p(-u) γ q(s-v) = +1 u= +1 s= +1 v= +1 g j h + 1 h+ 1 g+ 1 & *& j+1 * γ(-v) γ(u-s) = +1 v= +1 u= +1 s= +1 h g j In shorhand, usng he defnon of x, hs can be wren as (13). 22

25 References. Bossaers, Peer (1993), "Transacon prces when nsders rade porfolos", Fnance 14, Boudoukh, Jacob, Mahew Rchardson and Rober Whelaw (1994), "A ale of hree schools: Insghs on auocorrelaons of shor-horzon sock reurns", Revew of Fnancal Sudes 7, Chan, Kalok (1992), "A furher analyss of he lead-lag relaonshp beween he cash marke and he sock ndex fuures marke", Revew of Fnancal Sudes 5, Chan, Kalok (1993), "Imperfec nformaon and cross-auocorrelaon among sock prces", Journal of Fnance 48, Chan, K., Y.P. Chung and H. Johnson (1993), "Why opon prces lag sock prces: a radng based explanaon", Journal of Fnance 48, Cohen, K., G. Hawawm, S. Maer, R. Schwarz and D. Whcomb (1983), "Frcon n he radng process and he esmaon of sysemac rsk", Journal of Fnancal Economcs 12, De Jong, Frank, Ronald Maheu and Peer Schoman (1995), "The dynamcs of he acual and dollar mpled Yen/Dmark cross exchange rae", n preparaon. French, K. and R. Roll (1986), "Sock reurn varances: The arrval of nformaon and he reacon of raders", Journal of Fnancal Economcs 17, Grünblcher, A., F. Longsaff and E. Schwarz (1994), "Elecronc screen radng and he ransmsson of nformaon: An emprcal examnaon", Journal of Fnancal Inermedaon 3, Hannan, E.J. (1960), Tme Seres Analyss, Mehuen, London. Harrs, Lawrence, George Sofanos and James Shapro (1994), "Program radng and nraday volaly", Revew of Fnancal Sudes 7,

26 Kawaller, I., P. Koch and T. Koch (1987), "The emporal relaonshp beween S&P 500 fuures and he S&P 500 ndex", Journal of Fnance 42, Lo,Andrew and A. Crag MacKnlay (1990), "When are conraran profs due o sock marke overreacon?", Revew of Fnancal Sudes, 3, Lo,Andrew and A. Crag MacKnlay (1991), "An economerc analyss of nfrequen radng", Journal of Economercs, 45, Mller, Meron, Jayaram Muhuswamy and Rober Whaley (1994), "Mean reverson of Sandard and Poor s 500 ndex bass changes: Arbrage nduced or sascal lluson?", Journal of Fnance 49 (2), Rchardson, Mahew and Tom Smh (1991), "Tess of fnancal models n he presence of overlappng observaons", Revew of Fnancal Sudes, 4, Roll, Rchard (1984), "A smple mplc measure of he effecve bd-ask spread n an effcen marke", Journal of Fnance 39, Sephan, Jens and Rober Whaley (1990), "Inraday prce change and radng volume relaons n he sock and sock opon markes", Journal of Fnance 45, Soll, Hans and Rober Whaley (1990), "The dynamcs of sock ndex and sock ndex fuures reurns", Journal of Fnancal and Quanave Analyss 25, Whe, Hal (1990), "A heeroskedascy-conssen covarance marx esmaor and a drec es for heeroskedascy", Economerca 48,

27 Table 1. Auocorrelaons of ndex reurns. q===========================================================================================================================================================================================================================================================================e p p p lag1 10 mnues1 5 mnues1 1 mnue2 [ ] (7.53) (9.03) (19.83) (1.37) * (5.09) * (5.84) (0.50) (0.82) * (7.08) (0.19) (0.50) * (7.16) (0.51) (0.31) * (7.44) (0.44) (0.14) * (5.87) (0.59) (0.23) * (4.66) * (2.23) * (2.97) (0.74) (0.44) (0.19) (0.22) (0.32) (0.23) (0.16)2 [ ] nobs %mssng1 (0)1 (0)1 (0)2 z ===========================================================================================================================================================================================================================================================================c $ $ $ Noe: lag 0 denoes he varance of he seres, oher numbers are correlaons. The numbers n parenheses are heeroskedascy and seral correlaon conssen -sascs (calculaed wh one lag and lead wndow). 25

28 Table 2. Auocorrelaons of fuures reurns. q===========================================================================================================================================================================================================================================================================e p p p lag1 10 mnues1 5 mnues1 1 mnue2 [ ] (8.68) (12.57) (29.45) (0.08) (0.86) *(13.39) (0.26) (0.49) (1.52) (0.11) (0.32) (0.29) (0.00) (0.05) (0.58) (0.15) (0.27) (0.41) (0.32) (0.52) (0.24) (0.61) (0.06) (0.29) (0.21) (0.73) (0.53) (0.49) (0.05) (0.47)2 [ ] nobs %mssng1 (0)1 (1)1 (14)2 z ===========================================================================================================================================================================================================================================================================c $ $ $ Noes: see able 1. 26

29 Table 3. Correlaons beween ndex and fuure reurns. q=============================================================================================================================================================================================================e p p p 2lag1 10 mnues1 5 mnues1 1 mnue2 [ ] (0.25) (0.36) (0.94) (2.40) (0.60) (0.04) (0.17) (0.83) (0.14) (1.06) (0.25) (0.76) (1.30) (0.49) (0.87) (0.57) (0.69) (0.98) (0.48) (0.29) (0.47) (0.10) (0.04) * (3.15) (0.12) (1.46) * (9.19) * (6.09) * (7.43) * (5.07) * (4.43) * (6.82) * (7.23) (0.50) * (3.17) * (7.11) (0.33) (1.25) * (7.81) (0.08) (0.28) * (6.35) (0.12) (0.08) * (6.37) (0.41) (0.38) * (4.32) * (3.57) (1.44) * (3.83) (1.25) * (2.42) (1.57) (0.66) * (2.41) (0.08)2 z =============================================================================================================================================================================================================c $ $ $ Noe: he enres n hs able are esmaes of he cross-correlaons,.e. Cov( s, f ) dvded by he sandard devaon of s and f. -k The numbers n parenheses are heeroskedascy conssen -sascs. 27

30 Table 4. Jon sgnfcance of 6 lead or lag covarances. q============================================================================================================================================================e p p p 110 mnues15 mnues11 mnue2 [ ] lag lead z ============================================================================================================================================================c $ $ $ Noes: he enres are Wald (F-)sascs for he jon hypohess ha he lag (k<0) or lead (k>0) covarances are all equal o zero. The asympoc 2 dsrbuon of hs sasc s χ (6) These wll be nroduced n he model n secon 4. 2 In order o calculae auo-covarances of a me seres wh rregularly spaced observaons, we have o change he defnon of x slghly because n ha case γ(k)=γ(-k). 3 For he emprcal example n secon 5, where we esmae cross-correlaons beween a sock ndex and ndex fuures, hs s a reasonable assumpon. 4 The only sudy (o our bes knowledge) whch uses one mnue reurns s Harrs e al. (1994). However, hey calculae only auocorrelaons and no lead-lag correlaons beween ndex and fuures reurns. 5 No all radng days were repored on he ape. In oal, we have only 19 complee radng days avalable. The maxmum number of observaons for he ndex seres and he fuures seres are dfferen because he radng day for fuures s usually shorer han he perod for whch he ndex s repored. 6 Ths resul corresponds o ha found n Hannan (1960, p.39). 28