The Econometrics of High Frequency Data

Transcription

1 The Economercs of Hgh Frequency Daa Jeffrey R. Russell Prepared for he Hgh Frequency Daa Tuoral Organzed by Mahemacal Scences, Naonal Unversy of Sngapore and School of Economcs and Socal Scences, Sngapore Managemen Unversy Tenave Plan Day Inroducon o hgh frequency daa Daa feaures Marke mcrosrucure. Fxed Inerval Analyss Day 2 A pon process approach he ACD model Day 3 Marked pon process modelng. General approach Volaly Models Dscree prce movemens he ACM-ACD model Day 4 Measurng and modelng ransacon cos (sldes forhcomng) Exsng measures Full nformaon ransacon cos Decomposng he spread wh unobserved componens. 2

2 Inroducon Characerscs of Hgh Frequency Daa Irregularly Spaced Durnal Paerns Separae prces for buyng and sellng Dscree prces Hghly dependen We wll use he Argas sock raded on he NYSE hroughou he week. 3 2 hours of ransacon prce daa Prce :00 0:30 :00 :30 2:00 Tme 4 2

3 Auocorrelaon n prces ACF Md quoe Trans. Prce Lag n ransacon me 5 Durnal Paerns Cens/Seconds Duraon Sd Dev :00 :00 2:00 3:00 4:00 5:00 6:00 Tme of Day 6 3

4 Auocorrelaons n duraons Auocorrelaons n (log) volume

5 Quoes Transacon prces Volume Tme samps Orders Economc Daa 9 Economc Quesons Lqudy/Qualy of Execuon Forecasng Order Flow Hedgng/Prcng Prce Dscovery Volaly/Rsk Correlaon 0 5

6 Economerc Framework All economc evens can be descrbed by when hey occur and a ls of characerscs. Somemes he mng of he evens s fxed, bu wh hgh frequency daa s no. For example, Transacon evens and prces Lm order submsson and srke Quoe revsons Transacon Pon Process Prce Tme of day 2 6

7 Prce Change Tme of day Number of Transacons Tme of day 3 Noaon Le N() denoe he number of evens ha have occurred by me [ 0,T ]. Le denoe he h arrval me. These arrval mes are referred o as a pon process. where 0= 0 < < 2 < < N(T) =T Le x = - - denoe he h duraon. Le y denoe a vecor of characerscs assocaed wh he h arrval me. Jonly, he sequence of arrval mes and marks are referred o as a marked pon process. 4 7

8 Jon Densy The jon densy of he arrval mes and he marks gven he nal values for 0 and y 0 s he objec of neres. Ths can be recursvely decomposed: f N ( T ) ( ) ~ y = ( ~ N ( T ), yn ( T ),..., y, N ( T ), N ( T ),..., y0, 0 f y, y, ) where ~ = and ~ y y, y,..., y = = {, } { },..., How should such daa be analyzed? Conver pon process o fxed nervals and use radonal me seres echnques. Famlar ground los of avalable echnques. Model n ck me. Model as a pon process. When wll he nex even occur? When wll he nex even of a specfc ype occur? Pon process models for he marks. Jon modelng of prces and duraons. 6 8

9 Fxed nerval analyss. Mos me seres economercs s based on fxed nerval analyss. When he marks are of prmary neres, here s a naural endency o conver rregularly spaced daa o fxed me nervals. I.e. model prces over 5 mnue nervals. 7 Numerous examples n he leraure Andersen and Bollerslev (998) esmae GARCH models for 5-mnue FX reurns consruced from he mdquoes. Sophscaed modelng of durnal paern. Imporan o accoun for news announcemens. 8 9

10 Hasbrouck (2000), Zhang Russell and Tsay (200) use 5 mnue quoe daa o sudy quoe dynamcs. Hasbrouck(995) (200) use second nervals. 9 Mehods of converng o fxed nervals For some of he pons n me defnng he fxed nervals here wll no be a correspondng even. Wha value should be used for he mark? Les focus on he prce as he mark. 20 0

11 Three man mehods Denoe he log of he prce a me by *. Then le p p * * ( λp + ( λ) p+ ) where < + = I. Use he prevalng quoe: λ= II. Inerpolae: λ = + 2 III. A hrd possbly s adoped by Hasbrouck (2002). Snce me of a rade s usually recorded o he neares second, hen f he fxed nerval s aken o be one second here s a mos one observaon per me perod. A each second he prce s se eher o hs prce or o he prevous perod prce. 22

12 Prevalng prce (λ=): means and varances are no always reaned. Le he reurns be denoed by: * * y* = p p ransacon me reurn (rregularly spaced) y = p p nerpolaed value If here s never more han one rade per calendar nerval hen means and varances are preserved: N( T) T N( T) T 2 2 y* = y, and y* = y = = = = 23 If he nervals are larger so ha more han one rade can occur hen means are preserved, bu no varances. 2 N( T) N( T) T T * * 2 y y, and y = y = = = mulple = rades If he hgh frequency reurns are a marngale hen he expecaon of he cross producs are zero and he expeced value of he varances are he same. 24 2

13 When prces are nerpolaed hese relaons no longer hold. The sum of he squared nerpolaed seres wll be: N( T) N( T) 2 2 * * * * [ y ] = λp ( λ) p λjpj ( λj) p + j = = N( T) = λ + j + j j j = * * * * * * ( p p ) ( p p ) ( λ )( p p ) where and j are he evens jus afer he wo endpons of he fxed nerval Mean wll be approxmaely rgh. If he reurns form a marngale dfference sequence hen he expeced varance and s probably lm wll be less han he varance of he process. Auocorrelaon s nduced no he fxed nerval reurns Bvarae fxed nerval analyss Consder he case ha he fxed nerval s se o he shores nerval over whch he daa can evolve (ofen hs s one second). Unlke aggregang o longer fxed nervals, no nformaon s los If he prevalng prce s used, los of redundan values for he prce are creaed. The mplcaon s ha many zero reurns are creaed. Does he creaon of zeros nduce bas? 26 3

14 Smple dscree me example Le z be he log prce of an asse ha s connuously observed a =,2,,T. We assume he reurns z are d. Le y be he log prce of a second asse ha s observed a N(T) random arrval mes, 2,, N(T) for =,2,,T Le d denoe an ndcaor for wheher he prce of y s observed a me akng he value wh probably p. Then defne he prce a me for asse y as y y f N() = N(-) = y f N() > N(-) The reurn seres for y wll be zero f no prce s observed a me and wll be non-zero when a prce s observed a me. 27 Goal: Regress y on z We consder wo choces: Regresson usng he reurns assocaed wh he N(T) random me nervals. Regresson usng he T fxed nerval reurns. Prce Tme 28 4

15 Resuls: Suppose he rue relaonshp beween he wo seres s or, equvalenly y = β z + ε y = β z + ε If we use he T fxed nerval reurns and regress y on z Denoe he esmae usng he T fxed nerval reurns by ˆT β 29 Condonal on wheher he prce for y s observed a me yelds he OLS esmaor: hen T ( ' ) ( ) ˆ T β = z z z z β + ε d = Takng expecaons of he condonal momens of he esmaor yelds: plm ( ˆ β ) T ˆ T = pβ The resulng esmae β s downward based. The bas s drven by he probably of observng he prce of asse y. 30 5

16 Now suppose nsead ha we regress y on z,z -,,z -K. T z ( zβ + ε) d = T z ( z β + ε) d( d ) = T ˆ T β = ( z' z) z 2( z 2β + ε) d ( d )( d 2) = T z z + d d d d = ( β ε) ( )( ) ( ) k k 2 k+ 3 Agan, akng expecaons over he condonal momen Then ( ˆ T k β ) ( ) k = plm p p β ˆT β k The esmaes decay exponenally n p. Addonally, for he nfne lag model lm ˆ T p βk = β lm K More generally, f p s no consan, he probably lm wll be deermned by T k Ed ( z' z) z k z kd ( d j) β = j= 32 6

17 Inerpreaon s suspec however When we perform regresson wh he T equally spaced observaons he resul s slowly decayng parameer esmaes. Inerpreng hs as long range dependence, or predcably s wrong. Ths long range dependence s purely an arfac of he esmaon echnque. Resuls are no useful for dynamc hedgng or prcng of asse y. 33 Now suppose ha z s measured randomly Probably of observng y n any perod s p Probably of observng z n any perod s q ( ) 2 y observed a E y z k = pqβσ E overlap and z observed a -k ( ) = βσ 2 ( ) E y z pq p k k pq p + q pq 34 7

18 Regresson Coeffcens k pq plm βk = β ( p) pq p + q pq and he sum of he coeffcens n an nfne lag model s pq p βq p + q pq K lm βk = KT, k = 35 IMPLICATIONS If q=, hen he sum of he coeffcens wll be he rue response However, he apparen lag shape does no ndcae marke neffcency or causaly If p= bu q<, hen he sum of he coeffcens s q 2 so even he sum s undersaed If p< and q< hen s worse sll. 36 8

19 When wll he nex even occur? - Examples of Pon Processes A pon process can be descrbed eher n erms of he sequence of arrval mes or he sequence of duraons x. Engle and Russell (998) propose he Auoregressve Condonal Duraon (ACD) o model he dsrbuon of wang mes x condonal on he hsory of arrval mes. Many pon processes have been used n oher felds of sascs 37 ACD The ACD model assumes ha any dependence n he arrval rae can be summarzed by: where ψ ( x x, y ) s d wh E(ε )= and z z,. z,... z ~ ~ = E x = ψ ε { } 38 9

20 Flexbly of he ACD The flexbly of he ACD model les n he poenal models for he mean ψ and he choce of he nnovaon dsrbuon ε. Engle and Russell propose usng he lnear p q parameerzaon ψ = ω + Ths s referred o as an ACD(p,q) snce conans p lags of x and q lags of ψ. Ofen low order models such as he (2,2) model are capable of descrbng he emporal dependence n he arrval raes. j= α jx j + j= β ψ j j 39 Inerpreaon of lnear model Le η = x ψ whch wll be a margngale dfference sequence by consrucon. Then he ACD(p,q) model can be wren as an ARMA(p *,q) model n he duraons where p * =max(p,q). x p * ( ) = ω + α + β x + β η + η j j j j j j= j= q 40 20

21 Bauwens and Go (2000), Russell and Engle (2004), Engle and Lunde (2003) consder he Nelson From ACD model. p q ( ψ) = ω + α j ( ε j) + β j ( ψ j) ln ln ln j= j= Zhang Russell and Tsay (2003) consder a nonlnear specfcaon for he expeced duraon. ω + αx + βψ f x a ψ = ω + α x + βψ f a < x a ω3 + α3x + βψ 3 f a2 < x Fernandez and Grammg (2003) consder a famly of ACD models consruced from he Box-Cox Transformaon. 42 2

22 Choce of he error erm Suggesons for he dsrbuon of ε nclude he exponenal, webul, (generalzed) Gamma, and he Burr dsrbuons. Implcaons of he choce of he dsrbuon of ε are mos easly seen by examnng he condonal nensy funcon. 43 Condonal Inensy Funcon Recall ha N() denoes he counng funcon. Then he condonal nensy funcon s gven by: λ ( N,..., 0 ) ( N ( + ) > N N,..., 0 ) Pr ( ) ( ),, ( ),, = lm 0 So he condonal nensy funcon characerzes he nsananeous probably of an even occurrng gven he hsory of he process. Tha s, he probably ha an even occurs over he nex small me nerval s approxmaely gven by ( ( ),,,..., 0) λ N 44 22

23 Baselne hazard funcon ( ) Le p ε; φ denoe he densy funcon of ε. Defne he baselne hazard assocaed wh ε as: p ( ε; φ ) λ0 ( ε) = S ( ε; φ ) where S( ε; φ) = p( ε; φ) ε funcon. s he survvor 45 Condonal Inensy for ACD p ( ε; φ ) From he baselne hazard λ0 ( ε) = S ( ε; φ ) we oban he condonal nensy funcon. xn() Perform he change of varable ε = N() ψ N() Then x dε x λ( N( ),, ) = λ ( εn ) = λ = λ N() N(),..., 0 0 ( ) 0 0 ψ N() dx ψ N() ψ N() Hence s he shape of he dsrbuon of ε ha deermnes how he nsananeous probably of an even occurrng evolves n he absence of a new even

24 For example, he exponenal dsrbuon mples he well known fla baselne hazard: ψ N () The Webull, on he oher hand, allows for monoonc behavor of he hazard: γ λ( N (),..., 0 ) = ( Γ + ) N () + ( N ()) γ ψ The Gamma and Generalzed Gamma and Burr dsrbuons allow for a rch class of hump shape hazards. γ γ 47 Durnal Paerns Durnal paerns n radng raes are well documened. We mgh consder formulang he condonal expecaon as he produc of a sochasc and a deermnsc componen E ( x ) = φ( ; θ ) ψ ( x,.., x ; θ ) N() N() φ Where φ( ; θ φ ) s smply he expecaon of he wang me condoned on he me of day ha he duraon sars. Two sep and jon esmaon are possble. ψ 48 24

25 QMLE Resuls The EACD(,) model s clearly very smlar o he GARCH(,) specfcaon. The smlary s even closer han wha you mgh hnk. Consder he lkelhood for he EACD(,) model: N( T) x L = logψ + ψ = Ths s dencal o he lkelhood funcon for he GARCH(,) where y = x Hence, QMLE resuls carry over from he GARCH leraure (Lumsdane (996) or Lee and Hansen (994)). If sasfes he condons for y n her heorems hen he EACD(,) model s a QMLE. 49 QMLE Implcaons We can use sandard GARCH sofware o esmae parameers (ω and he α s and β s) of he condonal expeced duraon for he lnear ACD(,) model. Gven conssen esmaes of hese parameers one can non-paramercally esmae he baselne hazard model usng: x ˆ ε = ψˆ 50 25

26 Argas Sock Example DUR Seres: DUR Sample Observaons Mean Medan Maxmum Mnmum Sd. Dev Skewness Kuross Jarque-Bera Probably

27 DUR 53 ARG duraon auocorrelaons

28 ACD Esmaes for ARG Example usng GARCH esmaon code Coeffcen Robus Sd. Err. ω α α α β β Observed and Predced Duraons DUR PSI 56 28

29 Model Evaluaon Any nhomogeneous Posson process can be convered o a homogeneous Posson process by a deermnsc ransformaon of he me scale (see for example Snyder and Mller Sprnger). ( ( ),,,..., 0 ) u = λ s N ds s= The duraons measured on he u me scale should be a homogeneous Posson wh un nensy. Clearly hs can be esed gven an esmaed model. 57 Example For he exponenal model: x u = λ ( s N( ),,,..., 0 ) ds = ds = = ψ ψ ψ s= s= N() N() N() For he Webull model: γ ( (),..., ) 0 ( ) () + ( ) N N s= s= γ γ ( ) ψ ( ) N ( ) () γ + N() + γ u = λ = Γ + ψ γ s ds γ x = Γ + = Γ + γ ψ γ 58 29

30 u Dagnoscs Are he ˆ uncorrelaed? Perform seral correlaon es such as Ljung Box Are hey approxmaely un exponenal? sd ( u) NT ( ) 8 should have a lmng sandard Normal dsrbuon. Alernavely, ou of sample predcon could be examned or predcve dsrbuons (Debold Gunher and Tay (998) 59 Resdual Auocorrelaons for ARG

31 Does look lke he Exponenal dsrbuon assumpon s vald? Seres: EHAT Sample Observaons Mean Medan Maxmum Mnmum Sd. Dev Skewness.5027 Kuross Jarque-Bera Probably σ u = = Non-Paramerc Esmae of Baselne Hazard

32 There may be several ypes of evens Model only hese evens (hnnng) Buld a jon model o deermne he arrval probables of dfferen ypes of evens (more laer). 63 When wll he nex even of a parcular ype occur? Prces: How long wll ake for he prce o move more han an amoun c? Execuon: How long unl he a ransacon nvolvng a lm order s execued? f k () ψ k Tme ll even of ype k occurs 64 32

33 Modelng he marks n Tck Tme. Models he marks y as a me seres where ndexes he h even arrval. Snce here s no aggregaon o fxed nervals no nformaon s los. Ths modelng approach has proven useful n he analyss of a sngle sock. The model operaes n he me scale ha new nformaon occurs. 65 Examples nclude Hasbrouck 99 VAR s used o model he bvarae sysem of prces and volume for a gven sock. Where m s he prevalng prce (defned as he mdpon of he bd and ask) a he me of he h rade. w s he volume assocaed wh he h ransacon. m = a m + b w + ε j j j j j j w = c m + d w + ε j j j j 2 j j 66 33

34 Wha s he expeced prce mpac of a rade? Snce order flow s correlaed makes more sense o ask wha s he expeced prce mpac of he unexpeced poron of a rade?. Hasbrouck argues ha he marke srucure dcaes a specfc orderng n he mpulse response funcons. Namely, he argues, ha marke orders h exsng prces n he marke

35 VAR for Argas We esmae a VAR for mdquoe prces and a buy/sell ndcaor. Needs los of lags (abou 0). 69 Accumulaed Response o Cholesky One S.D. Innovaons ± 2 S.E. Accumulaed Response of DMIDPRICE o TRADE.04 Accumulaed Response of DMIDPRICE o DMIDPRICE Accumulaed Response of TRADE o TRADE Accumulaed Response of TRADE o DMIDPRICE

36 Tck me models are dffcul o apply o mulvarae daa. For example, consder jon modelng of wo sock prce reurn seres. Two dfferen me scales, how should hey be combned? Fxed nerval models clearly have an advanage here n ha once fxed nerval daa s obaned all he usual economercs ools (VARs ec) can be appled. Fnally, f he properes of he me seres depend on he spacng of he daa he ck me models may be mspecfed. 7 Marked pon process approach Whou loss of generaly we can always decompose he jon dsrbuon no he produc of a condonal and a margnal: (,, ; θ ) = (, θ ) (, θ ) f y y g y y h y 2 Dsrbuon of he Mark Here g denoes he condonal densy of he mark gven hsorcal nformaon as well as he conemporaneous duraon 72 36

37 Ineresng Marks Consdered Transacon prce reurns. Engle (2000), Ghysels and Jasak (998) GARCH models. Russell and Engle (2004), Rydberg and Shephard (2000) and many oher recen conrbuons model dscree ransacon prce moves. Order ype (lm, marke, markeable lm order, cancellaon ec ). 73 Ulra-Hgh Frequency Volaly Models Engle (2000) proposes a GARCH model for ransacons daa. The dea s ha he volaly per un me follows a GARCH process. The volaly per rade wll lkely depend on he me nerval

38 UHF GARCH se up Le r denoe he reurn from ransacon - o ransacon. Denoe he volaly per rade by: (, ) h Var r x r = Denoe he volaly per un me by: 2 r σ = Var x, r x 75 The volaly per un me s hypoheszed o follow a GARCH process. Afer modelng he mean reurn of r le e denoe he nnovaon. Then ( x ) σ = ω+ αe + βσ + γ Usng jon ACD model Engle proposes x σ = ω+ αe + βσ + γx + γ2 + γ3ψ + γ4ξ ψ Long run volaly measure exponenal smoohng 76 38

39 77 Models for dscree prce changes Russell and Engle (2004) propose he ACM model for dscree prce changes. When used jonly wh he ACD model for he duraons s referred o as he ACM- ACD model 78 39

40 SPECIFYING THE PROBABILITY STRUCTURE LET ~ x be he kx vecors ndcang he sae and ~ π observed and he condonal probably of all k saes respecvely. Tha s, akes he j h column of he kxk deny marx f he j h x~ sae occurred. A frs order markov chan () π = Px lnks hese wh a ranson probably marx P wh he properes ha a) all elemens are non-negave b) all columns sum o uny 79 In a more general seng P wll be he condonal ranson marx and wll vary wh nformaon avalable a me -. In hs conex hs wll nclude longer lags on x, π, and he me snce he las ransacon as well as oher parameers of he mng of rades, and economc varables such as spreads, volume and oher measures of marke lqudy. The resrcons on P are drecly sasfed by smple esmaors n he case of a consan ranson marx bu are dffcul o mpose n smple lnear exensons

41 Here we propose an nverse logsc ransformaon whch mposes such condons drecly for any se of covaraes. log( ~ π m = = / ~ π k ) = log k j= k j= log P * mj ( P / P ) x mj ( ) j k j= kj + c P ~ x m mj ~ x ( ) j ( ) j log k j= P ~ x kj ( ) j 8 Rewrng he k- log funcons as h() hs can be wren n smple form as: h ( π ) = P * x c + where P * s an unresrced (k-)x(k-) marx c s an unresrced (k-)x vecor and x s a he (k-)x sae vecor. 82 4

42 From esmaes of P * and he vecor c, we fnd ha P mn * exp[ Pmn + cm ] k * Pjn j j= = + exp[ + c ] so ha all probables are posve ncludng he probables of sae k whch are obaned from condon b). 83 Now by generalzng o allow for more dynamcs, we are generalzng he ranson marx o allow he condonal ranson probables o vary. For a frs order model wh predeermned or weakly exogenous varables z ha wll generally conan a consan, p ( ) ( ) ( ) h π = A x π + B h π + χz j j j j j j= j= q 84 42

43 An expresson for he probably of observng a sae can smlarly be expressed n erms of he pas hsory of he process: π * = exp Px + c ιπ ' * exp[ Px + c] π = * + ι 'exp Px + c where exp[p * * ] s nerpreed as a marx wh elemens P ι s a vecor of ones. exp[ ] mn, and 85 More generally, we defne he Auoregressve Condonal Mulnomal (ACM) model as: p ( ) ( ) ( ) h π = A x π + B h π + χz j j j j j j= j= q Where h () : ( K ) ( K ) s he nverse logsc funcon. Z mgh conan, a consan erm, a deermnsc funcon of me, or perhaps oher weakly exogenous varables. We call hs an ACM(p,q) model

44 Les consder he ARG ransacon prce changes. Hsogram of Prce Changes Percen < >2 Tcks 87 We herefore consder a 5 sae model defned as [,0,0,0 ] f p -2 cks [ 0,,0,0 ] f p = ck x = [ 0,0,0,0 ] f p=0 [ 0,0,,0 ] f p =+ ck [ 0,0,0, ] f p + 2 cks I s neresng o consder he sample cross correlogram of he sae vecor x

45 Sample cross correlaons of x s= up up down down up 2 up down down Are here deermnsc paerns n he prce movemens? Deermnsc paerns n fxed nerval volaly. Deermnsc paerns n duraons. Sochasc volaly has been found o be explaned by sochasc ransacon raes. Relaed queson s wheher deermnsc paerns n ransacon raes are drven by large per rade prce changes or smply faser radng

46 Deermnsc Regresson Resuls Cons. d d 2 d 3 d 4 d 5 d 6 F sa p-value Duraons % (.83) (7.02) (4.24) (5.03) (5.0) (4.39) (7.69) Down % (.0053) (.0076) (.0064) (.0067) (.0067) (.0064) (.00795) Down % (.07) (.068) (.040) (.048) (.049) (.042) (.075) Up % (.07) (.069) (.04) (.049) (.049) (.042) (.075) Up (.0052) (.0075).006 (.0063) (.0066) (.0067).0053 (.0064).003 (.0078) 23.3% 9 We propose he followng model for he ACM p q r h( π ) = c + Aj ( x j π j ) + B jh( π j ) + χ j ln( τ j+ ) j= j= j= The ACD s assumed o follow he Nelson form ACD wh exponenal error pas prce changes poenally nfluencng fuure duraons: u v w 2 ln( ψ ) = ω + α jε j + β j ln( ψ j ) + ( ρ j y j + ζ j y j ) j= j= j= 92 46

47 47 93 Lkelhood The log lkelhood for he ACM par looks lke Of course, he jon lkelhood for he ACM-ACD model s obaned by summng he ACM and he ACD log lkelhoods. The recursve srucure of he model perms closed form evaluaon of he lkelhood funcon subjec o nal condons. ( ) ( ) ( ) = = = = = N N K j j x j x L ~ log ~ ~ log ~ π π 94 Several models are esmaed n he paper. A smple o general model selecon procedure suggess an ACM(3,3,3)-ACD(2,2). There are a lo of parameers esmaed so I won show hem here. I U VU v U v = = where * ( ) v I = 0 E ( ) I = I v v E LB: x =23,324 LB: v =423 x v πˆ * = = s

48 Impled relaonshp beween rade nervals and prce changes Tme mean varance rw varance 95 Concludng Remarks Three general approaches o analyzng hgh frequency fnancal daa. Pon process Tck me Fxed nerval Choce of approach s drven by he goal/queson Relaonshp beween approaches s underdeveloped. Bases may be presen Inerpreaon may be suspec

49 Measurng Transacons Cos Marke mcrosrucure seeks o undersand he workngs of fnancal markes. One of he mos fundamenal feaures of a marke s he qualy of execuon of orders. An deal marke s one n whch marke parcpans can ransac as coslessly as possble. How should we measure ransacon coss? Ideally, we would lke o examne ransacon coss ha allow for a varey of radng sraeges. There s a radeoff beween he cos of mmedacy and he rsk of paenly radng over a longer perod of me. Typcally, he larger he quany raded a once, he worse he prce obaned. Breakng he rade up no small chunks decreases he expeced cos of he rade, bu exposes he rader o rsk of movemens n he underlyng. Any measure of ransacon cos, however, necessarly requres measurng he cos of any ndvdual ransacon

50 Wha s he cos of a sngle rade? Bd-ask spreads are one measure of he cos of a sngle rade. However, n many markes here s room for prce mprovemen so ha rades ofen occur srcly nsde he bd ask spread. An alernave measure ha accouns for prce mprovemen s he effecve cos of rade. m Noaon Le denoe he effcen prce. Ths s he prce ha would preval n equlbrum n absence of any marke frcons (e f a any pon n me here were a sngle prce a whch ransacons occur). Le p denoe he h ransacon prce. Le Q denoe an ndcaor for wheher he h rades s a marke buy or sell order akng he values and - respecvely. 2

51 Then he effecve spread s defned as: ( ) Q p m E m Problems: only n rare daa ses o we observe Q. We don observe he effcen prce. Soluon Use an algorhm o assgn rades as buyer or seller naed. Use mdpon of bd ask o proxy for effcen prce The vercal dsances represen he cos o he rader. 3

52 Roll s measure Le he reurns be gven by: Pr(Q =)=Pr(Q =-)=.5 and Q s d. p = mη Furher assume ha ln ( ) η = s Q 2 Then he reurn s gven by: ln ( ) ln ( ) ln ( ) ln s s p p ( ) = m m + Q Q 2 2 r e r ε r = r + ε e If we addonally assume ha he effcen prce follows a marngale dfference sequence and s uncorrelaed wh he nose reurn we ge: cov, e e ( r r ) = E ( r + ε )( r + ε ) = E( εε ) s s s s = E Q Q Q Q s 2 s = E Q = 4 4 So ( ) s = 2 cov r, r 4

53 Summary of Roll s measure Doesn requre a reference prce lke he effcen prce only requres ransacon prces. Can be esmaed based on daly daa. However Somemes nconssen wh he daa: posve covarances are ofen obaned. Srong assumpons regardng dependence srucure of effcen prce and nose process. Band and Russell (2004) propose an esmaor ha, lke Roll s model only requres ransacons prces. Unlke Roll s model here can be arbrary dependence n he cos dynamcs and unresrced dependence beween he effcen prce and he cos of rade. 5

54 Russell Tsay and Zhang (2002) Economerc Modelng Goals Model he dscree bd and ask prces. Tme varyng volaly Tme varyng lqudy Durnal paerns Address economc quesons regardng prce dscovery and lqudy. Tme seres models Bollerslev and Melvn (994) propose modelng FX daa va a GARCH process and feedng he GARCH volales no an ordered prob model for he bd ask spread. Engle and Paon (2000) propose an error correcon model for he bd and ask prces where he spread s he correcon varable. 6

55 Decomposon models for dscree bd and ask prces Le m denoe he log of he rue effcen prce. Le and denoe he observed ask and bd prce. Le and denoe he cos of exposure on he ask and bd sde respecvely. m + v a b a = m = round = round a b b α > 0 > 0 ( m + α ) ( m β ) β v 2 ~ N(0, σ ) Inerpreaons of he cos I wll refer o he cos of bd exposure and he cos of ask exposure. On he NYSE he specals chooses bd and ask prces a whch a maxmum quany can be raded. Hence he cos of exposure s he amoun he specals s compensaed for fxed cos and rsk. Large cos low lqudy and vce versa. Movaon n oher sudes o consder he spread as a measure of lqudy. 7

56 Specal Cases 2 2 If α = β = c, σ = σ, and round a =round b =round o he neares ck we ge he model of Harrs (990) If here are no coss (c=0) and round a =round b =round o he neares ck we ge he model for ransacon prces proposed by Roll (984) These smple models focused on he effecs of dscree prce observaons on volaly esmaes for he effcen prce. Tme varyng cos and volaly More generally, we would lke o le he cos funcons α and β be me varyng denong varably n he cos funcons faced by raders. A realsc model should also allow for me varyng volaly (or GARCH effecs). 8

57 Roundng Cos b a ln ln Our Model = Floor( m β ) = Celng( m + α ) α α ( α ) = µ + θ + θ + φ( ln( α ) µ ) + ν β β ( β ) = µ + θ + θ + φ( ln( β ) µ ) + ν m = m + δ + ε ε = σ z Effcen Prce 2 2 ln( σ ) = η + ς + ϕ( ln( σ ) η ) z + γ z ~ d GED µ and η are deermnsc funcons of me of day. θ are varables wh a common mpac on boh cos funcons α β θ and θ are varables ha affec he ask and bd sdes only respecvely δ s he mpac of varables on he effcen prce ν α ν β, and are d varables and muually ndependen, z Advanages of Decomposon The decomposon models have he advanage of allowng for, and poenally provdng quanave measures of separae cos funcons for he bd sde and he ask sde (he spread measures he sum of he wo cos funcons plus roundng nose) The permanen mpac of rade characerscs on he effcen prce m can be assessed. (prce dscovery process) We can esmae a volaly model for he effcen prce (no conamnaed by dscree measuremen). 9

58 Prce a α m β {{ b Tme Relaed model Hasbrouck (999a, 999b) proposes and esmaes a smlar model. Hs goal s o show ha he cos funcons are asymmerc and me varyng boh sochascally and deermnscally. Here we are neresed n how characerscs of he marke mpac he dynamcs of he cos funcons and he effcen prce. In dong so we wll learn abou marke lqudy and prce dscovery. We pay parcular aenon o he effecs of order flow. 0

59 Esmaon θ θ 2 s Parameers of he cos model Parameers of he Nelson EGARCH model Sae vecor of unobserved componens ncludng m, α, and β. The lkelhood nvolves mu-dmensonal negraon whch mus be solved numercally. We follow Manrque and Sheppard (997) and Hasbrouck (999b) and use MCMC mehods wh unnformave prors. Esmaon deals are n he appendx of he paper. The daa We exrac rades from he TORQ daa se spannng he hree Monhs Nov 90 o Jan % of he bd and ask changes are of ck or are unchanged. For racably we follow Hasbrouck and use quoes observed jus pror o he end of fxed 5 mnue nervals when he model s esmaed.

60 Marke Daa LogSpread = log of mean bd-ask spreads over 5 mnue nerval LogTPrceVar= log of varance of rade by rade prces over nerval LogVolume+= log of cumulave posvely sgned volume over nerval LogVolume-= log of cumulave negavely sgned volume over nerval LogAskDeph= log of mean ask deph over nerval LogBuyDeph= log of mean buy deph over nerval The sgned volume s obaned usng he Lee and Ready rule. Processng of Marke Daa All marke varables conan me of day effecs. All of he marke varables are hghly perssen (auocorrelaed). To ad nerpreaon of he model s useful o decompose he marke varables no her predcable componen and he surprse or nnovaon. The quoe updaes may reflec only he surprse elemen of he marke varable. In hs spr afer subracng off s deermnsc componen we decompose each marke varable no s predcable and surprse componens usng ARMA(p,q) models. 2

61 We are lef wh expeced and unexpeced componens of each of he marke varables. We defne wo more volume varables: Level: oal unancpaed=surprse buyer naed volume + surprse seller naed volume. Pressure: Unancpaed order mbalance=surprse buyer naed volume surprse seller naed volume. Pressure Deermnsc componens: Specfcaon open open open close µ = k + k2 exp( k3 τ ) + k2 exp( k3 close τ close open open open close close close ( I open = 0) + l + l2 exp( l3 τ ) + l2 exp( l3 τ ) ngh η = η τ ) The GED al parameer can be dfferen overngh as well. 3

62 Cos equaons conan expeced and unexpeced componens as well as he level and pressure varables. The volaly equaon conans expeced and unexpeced componens. The effcen prce dynamcs conan unexpeced componens only. I s herefore unforcasable and remans a random walk. Expecaons The cos equaons denoe compensaon o he specals for exposure o rade. Volume effecs mxed. Volaly ncreased rsk => ncreased cos of exposure. Lkely order mbalance wll be mos nformave for he drecon of movemens of he effcen prce. Volume and spreads should be nformave abou he volaly of he effcen prce. 4

63 Model Buldng Sarng wh boh he predced and unexpeced componens enerng unresrced no he wo cos models and he EGARCH model we use a general o smple model selecon approach. The predced componens drop ou wh he excepon of deph wh has a sgnfcan coeffcen on he predcable componen, bu nsgnfcan unexpeced componen. Symmery n he Cos Funcons Symmery n he bd and ask dynamcs are esed. Transacon prce volaly eners symmercally n cos. Sgned volume effecs are asymmerc n cos models. 5

64 Some Model Dagnoscs Somewha exensve model dagnoscs presened n he paper sugges no addonal lags are needed n he cos model bu 2 lags are needed n he volaly model. The fnal model 6

65 Fnal Model esmaes

66 cos(bea) cos(alpha) More on Volume (a) Level 0%le Level 25%le Level 50%le Level 75%le Level 90%le pressure (b) Level 0%le Level 25%le Level 50%le Level 75%le Level 90%le pressure 8

67 Order mbalance nerpreaons Unexpecedly large buyer naed volume leads o an ncrease n he ask cos and a decrease n he bd cos. Unexpecedly large seller naed volume leads o an ncrease n he bd cos and a decrease n he ask cos. The ncrease n he bd cos s more han wce ha of he ncrease n he ask cos above. Shor sale consrans and volume Shor sales consrans say ha an asse canno be shored followng a downck. Suppose here s bad news abou he asse ha a handful of raders know abou. Afer he frs nformed agen sells no oher agen can shor he sock o capalze on her beer nformaon. Hence sell volume may represen only a fracon of he volume ha raders would lke o have ransaced f here were no shor sales consrans. Concluson: each un of seller naed volume should have a larger mpac han each un of buyer naed volume. 9

68 Some Caveas We have nerpreed he resuls as he varables nfluencng he cos of exposure. In fac could be he oher way around. Low cos could nduce raders and hence ncrease overall volume. A movemen n he effcen prce could nduce volume pressure. Conclusons Cos Funcon Hgh volaly ncreases boh he cos of purchasng and sellng shares. Volume effecs on he cos are more complex. Unexpecedly hgh volume generally s assocaed wh lower cos. Imbalance n unexpeced volume has asymmerc affecs on he cos. Excess buyer volume ncreases he cos on he ask sde and decreases he cos on he bd sde. Excess seller volume ncreases he cos on he bd sde a lo and decreases he cos on he ask sde. 20

69 Conclusons Effcen Prce Unexpecedly large spreads are assocaed wh hgher volaly. Unexpeced volume on eher he buy sde or he sell sde ncreases he volaly. Unexpeced seller naed volume ends o have a larger mmedae mpac on volaly. Unexpeced buyer or seller naed volume changes he effcen prce n he way expeced. 2