Time-Varying Liquidity in Foreign Exchange

Transcription

1 Tme-Varyng Lqudty n Foregn Excange Martn D. D. Evans Rcard K. Lyons 26 October 200 Abstract Ts paper addresses weter currency trades ave greater prce mpact durng perods of rapd publc nformaton flow. Central bankers often suggest tat expectatons are at tmes rpe for coordnated adjustment, and tat perods of rapd nformaton flow are suc a tme. We develop an optmzng model to account for te jont beavor of order flow and returns around announcements. Usng transacton data made avalable by electronc tradng, we estmate te prce mpact of trades n te DM/$ market precsely. We ten test weter trades durng perods wt macroeconomc announcements ave ger prce mpact. Tey do. We also test for dependence of lqudty on tradng volume and return volatlty (two oter promnent state varables n te lterature on lqudty varaton). We do not fnd any evdence tat lqudty depends on tese varables. Te fndngs provde polcy-makers wt gudance for te tmng and magntude nterventon. Correspondence Rcard K. Lyons Haas Scool of Busness, UC Berkeley Berkeley, CA Tel: , Fax: lyons@aas.berkeley.edu Respectve afflatons are Georgetown Unversty and NBER, and UC Berkeley and NBER. We tank te Natonal Scence Foundaton for fnancal assstance and semnar partcpants at UC Berkeley for comments.

2 Tme-Varyng Lqudty n Foregn Excange Ts paper addresses tme-varyng lqudty n te foregn excange market and ts relevance for polcy. Lqudty s defned by te prce mpact of trades: trades ave more prce mpact wen markets are less lqud, oter tngs equal. (Ts usage s standard n teoretcal work on tradng; see, e.g., Kyle 985.) Our partcular nterest s weter currency trades ave more prce mpact durng perods of rapd publc nformaton flow. Ts s mportant for polcy: f te answer s yes, t provdes a means of ncreasng te effcacy of nterventon (e.g., by selectvely tmng nterventon trades). Central bankers ave long suggested tat market expectatons are at tmes rpe for coordnated adjustment, partcularly wen new publc nformaton arrves. Implct n ts vew s te dea tat agents draw dfferent conclusons from common macroeconomc data, makng transactons at tese tmes especally valuable for measurng expectatons. We estmate te prce mpact of trades usng transacton data recently made avalable by electronc tradng. Tese data allow precse trackng of ow te market absorbs actual trades and any nformaton conveyed by tem. Toug te data represent prvate trades rater tan central bank trades, tey are relevant for measurng te prce mpact of central bank trades as long as nterventons are sterlzed and convey no monetary polcy sgnal. Tese condtons nsure tat central bank trades are uncorrelated wt current and future monetary polcy, wc s also a property of prvate trades (so long as excange rates are floatng). Our analyss s relevant to te prce mpact of tese trades per se, not to te prce mpact of accompanyng nterventon announcements. Incremental effects from nterventon announcements are best analyzed usng more tradtonal metods and data (see Domnguez and Frankel 993). We begn by developng a tradng model for structural gudance on te Toug nterventon n practce s often coupled wt an nterventon announcement, ts s not always te case;.e., stealt nterventon s defntely a polcy opton.

3 jont beavor of order flow and returns. 2 Te model s suffcently rc to allow estmaton usng te partcular type of order flow data avalable (data on dealerto-dealer trades). Two features of te model stand out n terms of testable mplcatons. Frst, t ncludes macroeconomc announcements wose mplcatons for excange rate fundamentals are partally gleaned from order flow. Ts allows order flow to medate part of tese announcements prce mpact and provdes a structural understandng for wy lqudty s reduced. Second, te model produces so-called ot potato tradng,.e., te passng of postons from dealer to dealer for rsk management purposes (Flood 994, Lyons 997). Hot potato tradng mposes testable restrctons on te jont beavor of order flow and returns tat we examne n te data. Our estmaton strategy as two stages. Frst, we estmate a lnear, constant-coeffcent model wt two equatons, one for returns and one for order flow. Tat te system ncludes an equaton for order flow (sgned trades) s mportant: macro announcements may affect te trades process tself (versus affectng te sze of a gven trade s prce mpact). Toug te constant-coeffcent model does not allow for state-dependent lqudty, t does allow testng of several of our tradng model s predctons. For example, t allows testng of predctons about wc varables sould drve bot returns and order flow (ncludng te relevance of varous lags). Te second stage of our estmaton strategy examnes nonparametrc (kernel) regressons of our two tradng model equatons. Kernel estmaton allows us to address drectly te state dependence of lqudty. (Kernel estmaton s feasble ere due to te large quantty of avalable transacton data; one could not effect smlar analyss wt actual nterventon trades due to muc smaller samples.) Te state varable tat s our prmary focus s te flow of publc macroeconomc nformaton, measured ere from te flow of macro announce- 2 Order flow s a term from mcrostructure fnance tat refers to te net of buyer- and seller-ntated trades. (It s terefore not synonymous wt tradng volume.) In markets organzed lke foregn excange (.e., dealer markets), te ntatng sde of te trade s te sde tat decdes weter to trade at te dealer s posted quote. (Ts measure as no counterpart n ratonal expectatons models of tradng because n tat settng all trades are symmetrc tere s no ntator. ) As an emprcal matter, order flow plays a central role n aggregatng dspersed nformaton n many dfferent securtes markets (e.g., equtes, bonds, and foregn excange; see Lyons 200). 2

4 ments. Te kernel results (usng ourly data from te largest spot market, DM/$) sow a clear postve mpact of announcement flow on market lqudty (.e., te elastcty of prce wt respect to order flow). Tus, we fnd tat order flow does ndeed medate te prce mpact of announcements, n keepng wt our tradng model. We also test for dependence of lqudty on tradng volume and return volatlty (two oter promnent state varables n te lterature on lqudty varaton). We do not fnd any evdence tat lqudty depends on tese varables. Fnally, we fnd evdence of ot potato tradng. Specfcally, tere s momentum n flows (postve flows follow postve flows) and prce mpact s lmted to ntal flow nnovatons. Our approac to analyzng te prce mpact of unannounced nterventon llumnates a promsng drecton for future researc: central banks wt precse knowledge of ter own trades e.g., tme of day, metod of executon (e.g., brokers versus dealers), stealt level, etc. can estmate te nfluence of tese varous parameter settngs. Wt te types of data now avalable, central banks can learn exactly ow tradng s affected, ncludng te nduced flow of orders on eac sde of te market and te process by wc prce adjusts. It s sometng lke a doctor wo determnes ow a dgestve system s functonng by avng a patent ngest blue dye te wole process becomes transparent. Suc s te future of emprcal work on ts topc. Te remander of te paper s n fve sectons. Te next secton develops a tradng model for understandng te jont beavor of dealer-to-dealer order flow and excange rate returns. Secton 2 descrbes te dealer-to-dealer tradng data. Secton 3 presents our emprcal results for bot te constant-coeffcent model and te kernel regresson model. Secton 4 revsts te drecton of causalty n te model. Secton 5 concludes.. Model Our model as two man features tat dstngus t from earler emprcal models of currency order flow (e.g. Evans and Lyons 999). Frst, te model ncludes macroeconomc announcements from wc mplcatons for te excange rate are not nferable from te announcement alone. Ts allows order flow to convey ncremental nformaton about agents cangng expectatons. 3

5 Second, te model allows for ot potato tradng. If ot potato tradng s present n te data, falng to account for t at te modelng stage wll result n mscaracterzng te jont beavor of order flow and returns. Before ntroducng specfcs, let us provde a bref sketc of te model. At te begnnng of eac day, te publc (.e., non-dealers) place orders n te foregn excange market. Tese orders are stocastc and are not publcly observed. Intally, dealers take te oter sde of tese trades sftng ter portfolos accordngly. To compensate te (rsk-averse) dealers for te rsk tey bear, an ntraday rsk premum arses, producng some ntraday mean reverson n prce. At te end of eac tradng day dealers unload ntraday postons back onto te publc, wose greater rsk bearng capacty (n aggregate) gves tem comparatve advantage n oldng overngt postons. Because te publc s (non-stocastc) demand at te end of te day s not perfectly elastc, te publc s orders at te begnnng of te day ave portfolo-balance effects tat persst beyond te day. (Tnk of te publc s orders at te begnnng of te day as representng, say, sfts n edgng demand, wereas publc demand at te end of te day s purely speculatve, beng drven by suffcent expected return per unt rsk borne.) Te portfolo balance effects arse because even wen tese rsky postons are sared marketwde, ter prce mpact s not dversfed away. 3 Specfcs Consder an nfntely lved, pure-excange economy wt two assets, one rskless and one rsky, te latter representng foregn excange. 4 Eac day, foregn excange earns a payoff R, publcly observed, wc s composed of a seres of random ncrements: () R t R = t = 2 Te ncrements R are..d. Normal(0, σ R ). We nterpret te ncrements as 3 Note tat te sze of te order flows te DM/$ spot market needs to absorb are on average more tan 0,000 tmes tose absorbed n a representatve U.S. stock (e.g., te average daly volume on ndvdual NYSE stocks n 998 was about $9 mllon, wereas te average daly volume n DM/$ spot was about $300 bllon). 4 Some of te model s basc structure s sared wt te model n Evans and Lyons (999), so we omt some detals ere were overlap s strongest. 4

6 nterest rate canges. Te foregn excange market as two partcpant types, customers and dealers. Tere s a contnuum of customers, ndexed by z [0,], eac customer avng constant absolute rsk averson (CARA) and maxmzng utlty of te followng form: s Ut = Et δ exp( θct+ s) s = 0 were Et s te expectatons operator condtonal on te customers nformaton at tme t, and ct+s s consumpton n perod t+s. We assume tat all customers ave te same tme dscount factor δ and rsk averson parameter θ. Tere are N dealers, ndexed by. Dealers also ave CARA utlty. Te problem dealers solve s descrbed below, followng specfcaton of te tradng envronment. Wtn eac day t tere are four rounds of tradng: Round : Dealers trade wt customers (te publc). Round 2: Dealers trade among temselves (to sare nventory rsk). Round 3: Rt s realzed and dealers trade among temselves a second tme. Round 4: Dealers trade agan wt te publc (to sare rsk more broadly). Te tmng of events wtn eac day s summarzed below, along wt some notaton. Daly Tmng Round Round 2 Round 3 Round 4 Announ. dealers publc dealers nter- order dealers payoff nter- order dealers trades: quote trades quote dealer flow quote realzed dealer flow quote publc trade observed trade observed A P C P 2 T 2 x2 3 P R 3 T x3 P 4 C 4 5

7 Tradng Round At te begnnng of eac day, nature cooses weter to produce a macro announcement. We denote ts announcement wt te ndcator varable At, wc equals f tere s an announcement, and 0 f not. To some customers, te announcement conveys nformaton about te payoff ncrement to be realzed later tat day, Rt. 5 Ts s mportant to te model: t mples tat te order flow nduced by observaton of ts announcement wll be nformatve of future payoffs (and dealers wll factor ts nto ter processng of te order flow sgnals). Te next event eac day s dealer quotng. Eac dealer smultaneously and ndependently quotes a scalar prce to te publc. 6 We denote ts round- prce of dealer as P. (We suppress notaton for day t; as we sall see, t s te wtnday rounds te subscrpts tat capture te model s economcs.) Ts prce s condtoned on all nformaton avalable to dealer. Te fnal event eac day n round s customer (.e., publc) tradng. Eac of te N dealers receves a customer order C unobservable to te rest of te market tat as two components. (Te dealer sees only s own customer order n total, not te components separately.) Bot of tese components are executed at s quoted prce P. (Let C <0 denote net customer sellng dealer buyng.) Te second component s non-zero only f a macro announcement as occurred (ndcator varable At=): ˆ C = C + C wt ˆ C = 0 f A = 0 t Te frst component C s dstrbuted Normal(0, σ ). Te frst component s also 2 C 5 It s mportant tat dealers learn about ts nformaton from order flow, rater tan from te announcement tself. Te specfcaton s most sensble n an envronment were te data-generatng process s tme varyng. Note tat dfferng assessments need not be rratonal: n a world were te true model s not obvous, model formulaton wll nvolve costs, leadng to ratonal dsagreement n equlbrum, despte avng observed te same macroeconomc nformaton. To formalze ts, consder a settng n wc agents are able to process a common sgnal more precsely f tey pay a fxed cost to observe a better model (n leu of payng a fxed cost to observe te sgnal tself, as n Grossman and Stgltz 980). 6 Wle t s true tat a bd-ask spread of zero would not nduce entry nto dealng, ntroducng a bdoffer spread (or prce scedule) n round one to endogenze te number of dealers s a stragtforward but dstractng extenson of our model. Te smultaneous-move nature of te model s n te sprt of smultaneous-move games more generally (versus sequental-move games). 6

8 uncorrelated across dealers and uncorrelated wt te payoff ncrement R at all leads and lags. (Ts frst component reflects portfolo sfts of te non-dealer publc; toug we leave ter precse source unspecfed, tey could come from cangng edgng demands, cangng transactonal demands, or cangng rsk preferences a la Evans and Lyons 999.) On announcement days, te second component of te customer order C ˆ s dstrbuted Normal(0, σ 2 Ĉ ). It s postvely correlated wt te payoff ncrement to be realzed later tat day, Rt, but s uncorrelated wt te frst component of customer order flow C (across all dealers). For te analyss below, t s useful to defne te aggregate customer demand n round as: C N = C = Tradng Round 2 Round 2 s te frst of two nterdealer tradng rounds. Eac dealer smultaneously and ndependently quotes a scalar prce to oter dealers at wc e agrees to buy and sell (any amount), denoted P 2. Tese nterdealer quotes are observable and avalable to all dealers. Dealers ten smultaneously and ndependently trade on oter dealers quotes. If more tan one dealer posts a quote at wc a dealer wants trade, te dealer s desred trade s allocated to te dealer wose ndex s te nearest precedng ndex (lettng ndex wrap to ndex N). 7 Let T 2 denote te net nterdealer trade ntated by dealer n round two. At te close of round 2, all agents observe a nosy sgnal of nterdealer order flow from tat perod: (2) x2 = T2 + ν N = 2 were ν s dstrbuted Normal(0, σ ν ), ndependently across days. Te model s dfference n transparency across trade types corresponds well to nsttutonal 7 Ts trade allocaton rule s mportant for generatng ot potato tradng. Te ot potato result does not requre tat a dealer s trade be routed to only one oter dealer at te gven prce; t requres only tat te dealer s trade s not evenly splt across all oter dealers at te gven prce. 7

9 realty: customer-dealer trades n major foregn-excange markets (round ) are not generally observable, wereas nterdealer trades do generate sgnals of order flow tan can be observed publcly. 8 Tradng Round 3 Round 3 s te second of te two nterdealer tradng rounds. At te outset of round 3 te payoff ncrement Rt s realzed and te daly payoff Rt s pad (bot observable publcly). Lke n round 2, eac dealer ten smultaneously and ndependently quotes a scalar prce to oter dealers at wc e agrees to buy and sell (any amount), denoted P 3. Tese nterdealer quotes are observable and avalable to all dealers n te market. Eac dealer ten smultaneously and ndependently trades on oter dealers quotes. If more tan one dealer posts a quote at wc a dealer wants trade, te allocaton rule s te same as tat for te frst nterdealer tradng round. Let T 3 denote te net nterdealer trade ntated by dealer n round 3. At te close of round 3, all agents observe nterdealer order flow from tat perod: (3) x3 T3 N = = Ts specfcaton wt noseless observaton of round-tree order flow captures dealer learnng and te ncreasng precson of ter order-flow belefs. Tradng Round 4 In round 4, dealers sare overngt rsk wt te non-dealer publc. Unlke round, te publc s tradng n round 4 s non-stocastc. Intally, eac dealer smultaneously and ndependently quotes a scalar prce P 4 at wc e agrees to buy and sell any amount. Tese quotes are observable and avalable to te publc. Te mass of customers on te nterval [0,] s large (n a convergence sense) relatve to te N dealers. Ts mples tat te dealers capacty for 8 Te screens of nterdealer brokers (suc as EBS) are an mportant source of tese nterdealer order- 8

10 bearng overngt rsk s small relatve to te publc s capacty. Wt ts assumpton, dealers set prces optmally suc tat te publc wllngly absorbs dealer postons, and eac dealer ends te day wt no poston (wc s common practce among actual spot foregn-excange dealers). Tese round-4 prces are condtoned on te nterdealer order flow x3, descrbed n equaton (3). We sall see tat ts nterdealer order flow nforms dealers of te sze of te total poston tat te publc needs to absorb (to brng te dealers back to a poston of zero). To determne te round-4 prce te prce at wc te publc wllngly absorbs te dealers aggregate poston dealers need to know te rsk-bearng capacty of te publc. We assume t s fnte. Specfcally, gven tat customers ntertemporal utlty s CARA (coupled wt daly returns beng..d.), te publc s total demand for foregn excange n round-4 of day t, denoted C 4, s proportonal to te expected return on foregn excange condtonal on publc nformaton: (4) γ ( E[ P + R Ω P ) C 4 = 4, t+ t+ 4, t ] 4, t were te postve coeffcent γ captures te aggregate rsk-bearng capacty of te publc (γ= s nfntely elastc demand), and Ω 4, t ncludes all publc nformaton avalable for tradng n round 4 of day t. Te Dealer s Problem quotes Te dealer s problem s defned over sx coce varables, te four scalar P, P 2, P 3, and P 4, and te two dealer s nterdealer trades T 2 and Te appendx provdes te full specfcaton of te dealer s problem and model s soluton. Here we provde some ntuton. Consder te four quotes P, P 2, P 3, and T 3. P 4. No arbtrage ensures tat at any gven tme all dealers quote a common prce: quotes are executable by flow sgnals. 9

11 multple counterpartes, so any dfference across dealers would provde an arbtrage opportunty. Hereafter, we wrte P, P 2, P 3, and P 4 n leu of P 3, and P, P 2, P 4. It must also be te case tat f all dealers quote a common prce, ten tat prce must be condtoned on common nformaton only. Common nformaton arses at te end of round 2 (nterdealer order flow x2 ), at te begnnng of round 3 (payoff ncrement R), and at te end of round 3 (nterdealer order flow x 3 ). Te prce for round-4 tradng, P 4, reflects te nformaton n all tree of tese sources. (Recall tat te customer orders engendered by macro announcements are not common knowledge.) Gven our model s analytcally smlar to tat n Evans and Lyons (999), we relegate soluton detals to te appendx. Te resultng prce canges and nterdealer flows (end our - to end our ) for a representatve our can be wrtten as: P = ( β + β A ) x β P + η p (5) 2 3 x = β x + β P + η x (6) 4 5 p x were η = β6 R, η = ( β7 + β8a) C and te constants β troug β8 are postve. (Te varable C denotes te aggregate round-one order flow n our.) Te order flow coeffcents n te prce equaton, β and β2, depend on γ (te publc s aggregate rsk-bearng capacty from equaton 4), te varances 2 2 σ R, σ C, te correlaton between announcement engendered order flow C ˆ and R. σ 2 Ĉ, and Te ntuton for eac of te terms n equatons (5) and (6) s as follows. Te frst term n te return equaton s te prce mpact of order flow. Te contemporaneous prce mpact of order flow ncreases wen a macro announcement mmedately precedes t (pcked up by te ndcator varable A). Te second term n te return equaton reflects te fact tat prce effects from order flow ave a mean revertng component, due to te transtory ntraday rsk prema 0

12 tat arse n te model. Te resdual n te return equaton reflects te flow of unmeasured macroeconomc nformaton (ortogonal to order flow x). Te frst two terms n te order flow equaton capture ot potato tradng: nterdealer order flow n te model s postvely autocorrelated as postons are passed from dealer to dealer for rsk management purposes. Hot potato tradng covares postvely wt lagged returns as well (due to contemporaneous prce mpact) and because order flow s measured wt nose. Te resdual n te order flow equaton reflects te flow of unmeasured order flow from customers (contemporaneous). 2. Data Te dataset contans tme-stamped, tck-by-tck observatons on actual transactons for te largest spot market DM/$ over a four-mont perod, May to August 3, 996. Tese data are te same as tose used by Evans (200), and we refer readers to tat paper for addtonal detal. Te data were collected from te Reuters Dealng system va an electronc feed customzed for te purpose. Accordng to Reuters, over 90 percent of te world's drect nterdealer transactons took place troug te system. 9 All trades on ts system take te form of blateral electronc conversatons. Te conversaton s ntated wen a dealer uses te system to call anoter dealer to request a quote. Users are expected to provde a fast two-way quote wt a tgt spread, wc s n turn dealt or declned quckly (.e., wtn seconds). To settle dsputes, Reuters keeps a temporary record of all blateral conversatons. Ts record s te source of our data. (Reuters would not provde te dentty of te tradng partners for confdentalty reasons.) For every trade executed on D2000-, our data set ncludes a tme-stamped record of te transacton prce and a bougt/sold ndcator. Te bougt/sold ndcator allows us to sgn trades for measurng order flow. Ts s a major 9 At te tme of our sample, nterdealer transactons accounted for about 75 percent of total tradng n major spot markets. Ts 75 percent from nterdealer tradng breaks nto two transacton types drect and brokered. Drect tradng accounted for about 60 percent of nterdealer trade and brokered tradng accounted for about 40 percent. For more detal on te Reuters Dealng System see Lyons (200) and Evans (200).

13 advantage: we do not ave to use te nosy algortms used elsewere n te lterature for sgnng trades. One drawback s tat t s not possble to dentfy te sze of ndvdual transactons. For model estmaton, order flow x t s terefore measured as te dfference between te number of buyer-ntated and seller-ntated trades. Te varables n our emprcal model are measured ourly. We take te spot rate, as te last purcase-transacton prce (DM/$) n our, P. (Wt rougly mllon transactons per day, te last purcase transacton s generally wtn a few seconds of te end of te our. Usng purcase transactons elmnates bd-ask bounce.) Order flow, x, s te dfference between te number of buyer- and seller-ntated trades (n tousands, negatve sgn denotes net dollar sales) durng our. We also make use of tree furter varables to measure te state of te market: te number of macroeconomc announcements a; tradng ntensty n, measured by te gross number of trades durng our ; and prce volatlty σ, measured by te standard devaton of all transactons prces durng our. Te macroeconomc announcements comprse all tose reported over te Reuter s News servce tat relate to macroeconomc data for te U.S. or Germany (Money Market Headlne News). Te source s Olsen Assocates (Zurc); for detals, see, e.g., Andersen and Bollerslev (998). Altoug tradng can take place on te D2000- system 24 ours a day, 7 days a week, te vast majorty of transactons n te DM/$ take place between 6 am and 6 pm, London tme, Monday to Frday. Te results we report below are based on ts sub-sample. (Tey are smlar to results based on te 24-our tradng day.) Ts sub-sample ncludes a vast number of trades, provdng us wt consderable power to testng te state dependence of lqudty. 3. Results Stage : Constant Coeffcent Model Table presents results for our frst-stage estmaton, te constantcoeffcent model wt two equatons, one for returns and one for order flow 2

14 (ourly data). In te returns equaton, bot of te varables our model predcts sould be relevant are sgnfcant and correctly sgned. Te magntude of te coeffcent on order flow x mples tat te contemporaneous mpact of order flow on prce s about 60 bass ponts per $ bllon. 0 Te coeffcent on lagged returns mples tat about tree quarters of order flow s mpact effect perssts ndefntely. Two oter mportant facets of te returns-equaton results warrant attenton. Frst, note from row () tat tere s no mean reverson n prce uncondtonally. Ts s consstent wt te excange rate followng a martngale n ourly data. It s only wen order flow s ncluded tat prce exbts some condtonal mean reverson. (Ts condtonal mean reverson result s not a volaton of market effcency, owever, because our data were not avalable to market partcpants n real tme.) Second, note from row (v) tat lags beyond tose predcted by our model are not sgnfcant n te returns equaton. (Ts olds for furter lags of bot varables as well; not reported.) Estmates for te order flow equaton accord wt te model as well. Bot varables predcted to be relevant are sgnfcant and properly sgned, wereas lags beyond tose predcted are not sgnfcant. In ts equaton, as n te returns equaton, tere s evdence of eteroskedastcty, so our standard errors are adjusted for ts. Tese results ndcate tat tere s momentum n order flows: postve flows follow postve flows, wc s consstent wt te presence of ot potato tradng. Stage 2: Kernel Regresson Estmates of State Dependent Lqudty To test for state dependence n lqudty, we consder nonparametrc regressons of te form: p, p = π ( x, p, s ) + η x, x = µ ( x, p, s ) + η 0 Ts s based on an average trade sze n our sample of $3.9 mllon. (Ts average trade sze s avalable despte ndvdual trade szes not beng avalable.) We use log prce cange as our dependent 3

15 were π(.) and µ(.) are arbtrary fxed, unknown, and nonlnear functons of te varables sown, and ηt s a mean zero..d. error. (Te vector s s a vector of state varables, specfed below.) Our strategy ere s to estmate tese functons by kernel regresson and ten test weter our estmates of order flow s prce mpact are nfluenced by te flow of macroeconomc announcements. As noted n te ntroducton, te perod followng announcements s commonly tougt to be a tme wen market expectatons are rpe for coordnated adjustment (and our tradng model provdes a structural understandng of ow ts mgt work). To nsure tat our estmates for te announcement state varable are robust, we consder two addtonal potental state varables suggested by teory (see, e.g., Easley and O Hara 992): prce volatlty and tradng volume. Tables 2 and 3 present our kernel regresson results for te returns and order flow equatons, respectvely (ourly data). For te returns equaton (Table 2), π j denotes te dervatve of te estmated functon π ˆ(.) wt respect to j t varable. Tus, te frst tree rows ndcate ow te prce mpact of order flow (te frst argument n te ˆ π functon) vares wt te varables sown. Note te sgnfcant effect from te announcements state varable a-. Ts varable s defned as te number of macro announcements n our -. Accordngly, t mples tat te prce mpact of order flow s about 0 percent ger wt eac macro announcement n te prevous our (0.024/0.26 beng about 0 percent, were 0.26 comes from te lnear prce mpact estmate n Table ). Tus, lqudty n ts market depends on te pace of publc nformaton flow Te oter two potental state varables do not appear to matter for te prce mpact of order flow: σ- and n- are nsgnfcant. (σ- s te standard devaton of all transacton prces n te prevous our and n- s te number of transactons n te prevous our.) Note too tat te tme-of-day dummes are nsgnfcant (see table varable because ts s a more common measure of returns n te emprcal lterature. Use of te raw (unlogged) prce cange as no qualtatve effect on our results. Te announcement and volume state varables are also consdered n te emprcal analyss of central bank nterventon by Domnguez (200). Usng nterventon trade data, se fnds tat nterventons tat occur near macro announcements and durng eavy tradng volume are te most lkely to ave large effects. In er analyss, eavy volume means occurrng wen bot London and New York are tradng, and near macro announcements means tat te publc announcement of nterventons occurred near te tme of te macro announcement. 4

16 notes for defnton); tus, wtn te 6 am to 6 pm London tme perod, tme does not appear to defne dstnct states once we control for te oter varables n our specfcaton. Fnally, note tat tere s no evdence of non-lnearty n order flow s prce effect, once tese state varables are ncluded. From te lower panel of Table 2, one sees tat all tree of te state varables nfluence te persstence of prce effects. Hger tradng volume ncreases te persstence of ourly prce movements, wereas prce volatlty and announcement flow decrease te persstence of ourly prce movements. From ts lower panel, note too te sgnfcant non-lnearty n te p- varable. Ts mples tat larger moves are less persstent (controllng for te oter varables). Table 3 presents kernel regresson results for te order flow equaton. Te tree state varables and te tme-of-day dummes are nsgnfcant across te board n ts case. Tere s some slgt evdence of non-lnearty of n te x- varable, but te magntude of te effect s not large economcally. Bottom lne: te order flow process does not appear to be state or tme dependent as specfed ere. Impulse Responses Impulse responses provde an effcent way to summarze te effects of announcement flow on te dynamc, two-equaton system examned n Tables 2 and 3. Tese mpulse responses are presented n Fgures -2. Fgure llustrates te effect of order flow socks on bot returns and subsequent order flow, ncludng ow announcement flow nfluences te dynamcs. Fgure 2 llustrates te effect of return socks on bot returns and subsequent order flow, also ncludng ow announcement flow nfluences te dynamcs. Tese fgures complement te evdence n Tables 2 and 3 n tat tey provde an llustraton of te effects of announcements over tme. For completeness, Fgures 3-6 address te effects of te two addtonal state varables, tradng volume and prce volatlty. (See te appendx for computaton detals.) Te strongest state-varable effects are manfest n fgures and 2, wc llustrate te effects of announcement flow. (Tat tese effects sould be strongest s consstent wt te kernel regresson results reported above, gven ter 5

17 complementarty.) Te experment n fgure s te followng. Suppose te announcement flow n te prevous our s g (one devaton above te sample average) and s expected to reman g forever: ow does ts cange te effect of an nnovaton n order flow on prce and subsequent order flow? Te ncreased prce mpact from announcement flow s clear from te upper panel of Fgure. Note too tat te order flow and return responses persst. (Ts s condtonal on te announcement flow persstng, so t sould not be vewed as an ndcaton of a non-statonary system.) Note too te prce sock effect on te next our s order flow from Fgure 2. Ts s consstent wt ot potato tradng,.e., one average muc of prce varaton s due to order flow nnovatons, wc on average are followed by subsequent flows n te same drecton (toug te prce mpact of tose subsequent flows s ndstngusable from zero). Te mpulse responses n fgures 3-6 are broadly consstent wt te results on lqudty state dependence from te kernel regressons. Te two addtonal state varables of prce volatlty and tradng volume ave lttle effect on te prce mpact of order flow nnovatons. Te order flow process s more sgnfcantly affected. Fgure 3, for example, exbts dfferental effects on order flow due to g tradng volume (number of trades). Hg tradng volume also sgnfcantly alters te order flow response to prce socks, as sown n fgures 4 and Bas Analyss Toug te drecton of causalty n our model runs from order flow to prce (as s true of mcrostructure teory generally), tere s a popular alternatve ypotess tat nvolves reverse causalty, namely feedback tradng. Ts secton examnes weter feedback tradng can account for our results. We begn wt some perspectve. Most models of feedback tradng are based on non-ratonal beavor of some knd, makng tem less appealng to many economsts on a pror grounds. Models of feedback tradng tat do not rely on non-ratonal beavor generally requre tat returns be forecastable usng te frst lag of returns, wc s not a property of major floatng excange rates (and s not a property of our ourly data eter see Table, row ). Accordngly, te 6

18 class of feedback tradng models tat mgt be relevant ere s te non-ratonal class. 2 Exstng emprcal evdence on feedback tradng n foregn excange s scant. Vald nstruments for dentfyng returns-casng order flow ave not been employed and t s not clear wc varables would qualfy. One pece of relevant evdence s provded by Klleen et al. (200). Usng daly data on foregn excange order flow, tey fnd tat order flow Granger causes returns but returns do not Granger cause order flow. Ts evdence s purely statstcal, owever, and apples at te daly frequency, so ts message (toug suggestve) s not defntve for te ssue n ts paper. Our approac ere s to pose and address te followng queston: Suppose ntra-our (.e., contemporaneous n ourly data) postve-feedback tradng s present, under wat condtons could t account for te key moments of our data? To address ts queston, we decompose measured order flow x nto two components: * fb (7) x = x + x were x * denotes exogenous order flow from portfolo sfts as dentfed n our model, and x fb denotes contemporaneous order flow due to feedback tradng, were: fb (8) x = φ p. Te sgn of te parameter φ tat most people ave n mnd for explanng our results s postve,.e., postve feedback tradng (based on te postve coeffcents on contemporaneous order flow n te prce-cange equatons n Table ). Next, suppose te true structural model can be wrtten as: 2 Weter te non-ratonal class s ntellectually appealng s not an ssue we could ope to resolve ere. We smply offer te fact tat mmense amounts of money are at stake wen dealng n foregn excange at major banks (te source of our data). Tese banks take te evaluaton of traders performance and decson makng very serously. 7

19 * p 2 (9) p = β x + β p + η * * x 2 22 x = β x + β p + η Notce tat te equatons n (9) are vald reduced-forms from our model tat could be estmated by OLS f one ad data on x *. However, f feedback tradng s present (.e., φ 0 ), estmates of (9) usng measured order flow, x, wll suffer from smultanety bas. We can evaluate te sze of ts bas be estmatng (7) (9) as a wole system of equatons. Specfcally, we can combne (7) (9) nto a bvarate system for prce canges and measured order flow. Te dynamcs of ts system depend on seven parameters: φ, β, β2, β2, β 22, and te varances of η p and η x. Tese parameters can be estmated by GMM usng sample estmates of te covarance matrx for te vector [ p, x, p, x ], as descrbed more fully n te appendx. In broad terms, te feedback tradng alternatve predcts tat () te coeffcent β on exogenous order flow wll be smaller tan tose n Table, f not zero, and (2) te coeffcent on feedback tradng φ wll be postve and sgnfcant. Te GMM estmates are reported n Table 4. Te last row of te table reports te estmate of te feedback parameter, φ : te estmate s negatve and statstcally nsgnfcant. Tus, nsofar as tere s any emprcal evdence of feedback tradng n our data, t ponts to te presence of negatve rater tan postve feedback tradng. Moreover, estmates from te prce equaton sow tat our causal nterpretaton of te order flow s mpact on prce (based on our model) remans ntact: te estmate of β s slgtly larger tat tose n Table and remans gly statstcally sgnfcant, n contrast to wat te feedback tradng alternatve predcts. 5. Conclusons We are, for te frst tme, at te pont were we can measure lqudty n te FX market, and wy t vares over tme. Our results sow tat lqudty n 8

20 currency markets depends on te pace of publc nformaton flow. Lqudty does not, owever, appear to depend on tradng volume or return volatlty (two oter promnent state varables n te lterature on lqudty varaton). Tese results provde polcymakers wt concrete gudance for ncreasng te effcacy of nterventon by selectvely tmng ter trades. Te tradng model we develop provdes a structural nterpretaton for wy te effect from publc nformaton flow sould be present n te data. In te model, te mplcatons of new macroeconomc data for te excange rate are not nferable from te macro data alone, wc allows order flow to convey ncremental nformaton about agents cangng expectatons. Our tradng model provdes testable mplcatons beyond tose nvolvng publc nformaton flow. For example, t provdes a structural account for te jont beavor of order flow and returns. Emprcally, te model accords well wt te data. Te varables te model predcts sould be relevant are ndeed sgnfcant. Te varables (and lags of varables) te model predcts sould be nsgnfcant are nsgnfcant. Our estmate of te contemporaneous mpact of order flow on prce s about 60 bass ponts per $ bllon. Of ts 60 bass ponts, rougly 80 percent perssts ndefntely. Our model also produces ot potato tradng, and te data accord well wt predctons n ts respect as well. Specfcally, we fnd momentum n order flows (postve flows follow postve flows n ourly data), toug only te ntal nnovatons n flow appear to ave mpact on prces. Toug our polcy focus n ts paper as been central bank nterventon, order flow analyss s relevant for oter polcy ssues as well. Consder two examples. Te frst s te lqudty ole tat occurred n te dollar-yen market n October 998 (te mmedate aftermat of te LTCM crss). Wtn about a day, te yen/$ rate fell from about 32 to about 7 and bd-ask spreads rose to nearly one yen (.e., rose to about 30 tmes ter typcal sze n te nterbank market). Ts was not a trval event n terms of resource allocaton: te new lower level of te excange rate was persstent. Wy market lqudty dred up so fast and so drastcally s stll a puzzle, one tat order flow analyss may elp to resolve (see, e.g., te analyss of Ctbank s customer trades troug October 998 n Lyons 200, ncludng tose for edge funds). A second polcy area were order flow 9

21 analyss may prove valuable s currency market desgn, especally n emergng markets. Wt approprate data, one could estmate ow prce mpact n tese markets canges as a functon of te market state (devaluaton lkelood, etc.). Also, one could determne weter customer forward trades ave te same prce mpact as customer spot trades of smlar sze. If not, one could quantfy te dfference. (Many developng countres restrct or even forbd forward tradng on te belef tat suc tradng s more speculatve n nature tan spot tradng and s terefore more destablzng.) One mgt also compare prce mpact across countres, n an effort to determne wc nsttutonal structures are better at promotng lqudty. 3 Te ssue of prce mpact s related to te ssue of market stablty. Polcymakers n some developng countres appear to beleve tat addtonal lqudty s destablzng. In teory, t s less lqudty tat s destablzng, not more lqudty: te less te lqudty, te larger te prce mpact, and te more prces move (oter tngs equal). To make te case tat oter tngs are not equal, n a way tat mgt reverse te relatonsp between lqudty and stablty, one could use te dscplne of mcrostructure tradng models to dentfy te countervalng forces. 3 One paper tat addresses speculatve attacks n Mexco usng a mcrostructure approac s Carrera (999). For teoretcal work on te desgn of currency markets n developng countres see Krlenko (997). 20

22 Appendx A: Kernel Regressons We consder nonparametrc regressons of te form: y = mz ( ) + η t t t were m(.) s an arbtrary fxed but unknown nonlnear functon of te varables n te vector zt, and ηt s a mean zero..d. error. An estmate of te m(.) functon s estmated by kernel regresson as: mz ( ) = t T j= 0, j t T j= 0, j t K ( z z ) y b t j j K ( z z ) b t j were Kb(u)=b - K(u/b) wt K ( u) 0 and K ( u) du =. In ts applcaton, we use te multvarate Gaussan kernel Ku d/2 ( ) (2 π ) exp( uu ' /2) = were d = dm( u). Te bandwdt parameter, b, s cosen by cross-valdaton. Tat s to say, b mnmzes: T ( y ( )) 2 t t mzt wt T were wt s a wegtng functon tat cuts off 5% of te data at eac end of te data nterval as n Hardle (990), p. 62. We follow te common practce of ncludng te standardzed value of eac of tese varables n te Gaussan kernel (.e., eac element of zt s dvded by ts sample standard devaton). Asymptotc teory for kernel regressons n te tme seres context appear n Berens (983) and Robnson (983). Robnson sows tat consstency and asymptotc normalty of te estmator can be establsed wen te data satsfy α- δ t 2/ mxng wt mxng coeffcents α(k) tat obey te condton T α ( k) = O() δ and E yt <, wt δ > 2. 2

23 Appendx B: Impulse Response Functons We use our kernel estmates of te prce cange and order flow equatons p, p = π ( x, p, s ) + η x, x = µ ( x, p, s ) + η were s = [ a, σ, n ] s te vector of state varables, to compute two sets of mpulse responses as follows. Responses to order flow socks Te mpulse response to an order flow sock of δ n our, s defned as I p( τ) = E[ p+ τ Ω, x = δ] E[ p+ τ Ω, x = 0 ], I x ( τ ) = E[ x+ τ Ω, x = δ ] E[ x+ τ Ω, x = 0 ], were Ω s te nformaton set contanng te story of prce canges, order flows, and te state varables untl te start of our : Ω = { x, x,..., p, p..., s, s,...} I p ( τ) dentfes te cange n expectatons regardng p +τ f order flow n our canges from zero to δ, gven te partcular story of past order flow, prce canges, and te state varables n Ω. Smlarly I x ( τ ) dentfes te cange n expectatons regardng x +τ f order flow n our canges from zero to δ, gven Ω. Notce tat gven te specfcaton of te prce and order flow equatons, I x ( 0 ) = δ and I p( 0) = π ( δ, p, s ) π ( 0, p, s ) so te mpact of te order flow sock can be drectly calculated from te kernel regresson estmates. Calculatng te cange n expectatons for τ > 0 s muc more complcated because current prce canges and order flow depend nonlnearly on past prces and order flow. Specfcally, consder te one-perod-aead prce cange forecast. By terated expectatons, ts forecast can be expressed as E[ p+ Ω, x = δ ] = E E[ p Ω, x = δ, x, p ] Ω, x = δ. + + Te nner condtonal expectaton can be calculated from te kernel estmates of π ( x +, p, s) gven realzatons of x,,and + p s. To compute E[ p + Ω, ] x= δ we terefore ave to calculate te expected value of π ( x +, p, s) usng te jont dstrbuton of x,,and + p s condtonal on { Ω, x = δ}. Wen π (.) s lnear, ts s a stragtforward calculaton. Wen π (.) s nonlnear, te expectaton must be calculated by numercal smulaton (descrbed below). 22

24 Hstory dependency also complcates te calculaton of mpulse response functons n nonlnear models. Notce tat te mpulse responses specfy bot te sock and story Ω. In lnear models, te affect of a sock does not depend on te story of past socks so t s unnecessary to specfy Ω. Here we must specfy Ω because te prce mpact of order flow n our may (n prncple) vary accordng to te value of p, and te state varables, z. Te results presented n te paper assume a partcular story Ω, were p = 0, x = 0 and s equals a vector of constants, s 0 (specfed below). Wt ts story, 0 0 E[ p+ τ Ω, x = 0 ] = E[ x+ τ Ω, x = 0] = 0. Te Impulse Responses are computed as follows:. Compute te vector of prce and order flow resduals from te kernel regressons: η = p π ( x, p, s ), and η = x µ ( x, p, s ). Tese p x resdual vectors ave mean zero Set x = δ, and compute E[ p Ω, x = δ ] = π ( δ, 0, s ) from te kernel estmates and I x ( 0 ) = δ. Compute ~ 0 p E[ p Ω, x ] ~ p = k = δ + η were η ~ p s a random drawng from te vector of prce resduals. 3. Add te realzatons { p ~, δ } to te data set and compute new kernel estmates of te π (.) and µ (.) functons usng te optmzed bandwdt parameters calculated n te data sample, π ~ (.) and µ ~ (.). 4. Compute E[ x Ω 0 ] ~ ( x ~, p ~, 0 µ z ) and ~ 0 x E[ x Ω ] ~ x = + η were η ~ x + = s a random drawng from te vector of order flow resduals, and 0 0 Ω = Ω { p ~, δ }. 5. Compute E[ p Ω 0, x ~ ] ~ ( x ~, p ~, 0 = π z ) and ~ 0 p E[ p Ω, ~ x ] ~ p = + η were ~ η p+ s a random drawng from te vector of prce resduals. 6. Add te realzatons { ~ p, ~ + x + } to te data set and compute new kernel estmates of te π (.) and µ (.) functons usng te optmzed bandwdt parameters calculated n te data sample, π ~ + (.) and µ ~ + (.). 7. Repeat steps 4, 5, and 6, τ tmes. We now ave a τ -perod story of prce canges and order flow drven by te δ sock to order flow n perod and subsequent random prce and order flow socks. We also ave te sets of forecasts E[ p Ω 0, x ~ ] 0 and E[ x + Ω + ]. 8. Calculate 000 τ -perod stores of prce canges and order flow usng j te steps - 7 above and save te forecasts as E[ p Ω 0, x ~ ] and 0 j E[ x + Ω + ] togeter wt te resduals [ ~ p j η + ] and [ ~ x j η + ] for j =,2, Compute j = E[ p+ Ω, x = δ ] = E E[ p Ω, x ] Ω, x δ 23

25 j = E[ x+ Ω, x = δ ] = E E[ x Ω ] Ω, x for =,2, τ. An estmate of te expectaton n te frst equaton of step 9 j s found as te constant n a regresson of E[ p Ω 0, x ~ ] on [ ~ p j η + ] for j=,2, An estmate of te expectaton n te second equaton of 0 j step 9 s smlarly found as te constant n a regresson of E[ x + Ω + ] on [ ~ x j η + ]. In bot regressons, realzatons of resduals are used as control varates to obtan more precse estmates of te expectatons tan would j be obtaned by smply averagng over E[ p Ω 0, x ~ ] 0 j or E[ x + Ω + ]. (For a dscusson of te use of control varates, see Davdson and MacKnnon 993, capter 2.) Te OLS standard error for te constant estmates te standard error of te smulated expectaton. Te procedure descrbed above allows us to calculate te mpulse response and a confdence band (.e., ±.96 standard errors) for an order flow sock of δ gven a partcular set of values for te state varables, s 0. In our baselne case, we set s 0 equal to te sample average for s. We compare ts baselne case aganst alternatves n wc one element of s 0 s canged. δ Responses to prce socks Te mpulse response to a prce sock of δ n our, s defned as I p( τ) = E[ p+ τ Ω, x = 0, p = δ] E[ p + τ Ω, x = 0, p = 0 ], I x ( τ ) = E[ x+ τ Ω, p = δ ] E[ x+ τ Ω, p = 0 ], Gven te tmng of order flow and prce canges n our model, I x ( 0) = 0 and I p ( 0 ) = δ. Te mpulse responses for τ > 0 are calculated by smulaton n te same manner as te order flow responses. 24

26 Appendx C: Bas Analyss Estmates Let α represent te vector of parameters to be estmated (.e., φ, β, β 2, β2, β 22, and te varances of η p and η x ). Combnng equatons (7) (9), we can wrte te dynamcs of returns and measured order flow as Y = AY + Bη were Y = [ p, x ]', η = [ η, η ]', p x β + ββ22 φββ2 ββ2 A = 2 φ( β ββ22) β22 φ ββ2 φβ2 φββ2 β, β B = φ φβ, + and Θ = Cov( η, η ' ), a dagonal matrx. We can terefore compute te covarance of returns and measured order flow, as a functon of te model parameters α ; Γ( k; α ) = Cov( Y, Y ' ) Y, as k wt Γ( k; α ) = AΓ( k ; α ) c Γ( 0; α ) = vec e A A) vec( Θ) j. Te GMM estmates of α are based on ortogonalty condtons of te form were E[ m ( k; α )] = 0 m ( k; α ) = D( k) vec Y Y Γ( k; α ) k and D( k) s a vector of ones and zeros tat selects te unque elements n Γ( k; α ). Te results n Table 4 use k = 0 and, for a total of seven moments. b g 25

27 Appendx D: Te Dealer s Problem and Model Soluton (Incomplete) Wtn a gven day t, let usng te conventon tat W j denote te end-of-round j wealt of dealer, W 0 denotes wealt at te end of day t-. (We suppress notaton to reflect te day t were clarty permts.) Wt ts notaton, and normalzng te gross return on te rskless asset to one, we can defne te dealers problem over te sx coce varables descrbed n secton 2, namely, te four scalar quotes trades, T 2 and T 3 : P, one for eac round j, and te two outgong nterdealer j (A),, { P P, P P, T, T } Max E exp( θ W4 Ω ) s.t. ( ) ( ) ( ) ( 2 )( 3 2) ( 3 2 2)( T C P P T T C T P4 P3) W = W + C P P + T P P + T P P Dealer s wealt over te four-round tradng day s affected by postons taken two ways: ncomng random orders and outgong (delberate) orders. Te ncomng random orders nclude te publc order C and te ncomng nterdealer ~ ~ orders and (tlde dstnguses ncomng nterdealer orders and prces T 2 T 3 from outgong). Te outgong orders are te two nterdealer trades T 2 and P T 3. j denotes an ncomng nterdealer quote receved by dealer n round j. As an example, te second term n te budget constrant reflects te poston from te publc order C receved n round one at dealer s own quote P and subsequently unwound at te ncomng nterdealer quote n round-two. (Recall ~ tat te sgn of dealer s poston s opposte tat of dealer f te publc order P 2 C, so a fallng prce s good for C s a buy,.e., postve. Te dealer s speculatve postonng based on nformaton n C s reflected n te fnal two terms of te budget constrant.) Terms tree and four reflectng te ncomng (random) dealer orders are analogous. Terms fve and sx of te budget constrant reflect te dealer s speculatve and edgng demands. Te outgong nterdealer trade n round 2 as tree components: T = C + D + ET [ Ω ] (A2) T 26

28 were D 2 s dealer s speculatve demand n round 2, and ET [ 2 Ω 2T ] s te dealer s edge aganst ncomng orders from oter dealers (ts term s zero n equlbrum gven te dstrbuton of te C s). Te dealer s total demand (speculatve plus edgng) can be wrtten as follows: D + ET [ Ω ] = T C 2 2 2T 2 wc corresponds to te poston n term fve of te budget constrant. Te sxt term n te budget constrant s analogous: te dealer s total demand n round tree s s total trade n round tree ( T 3 ) plus s total demand n round two ( T 2 C ) less te random nterdealer order e receved n round two ( T 2 ). Te condtonng nformaton Ω at eac decson node (4 quotes and 2 outgong orders) s summarzed below (see also te daly tmng n te text). t { { R, x, x, P, P, P, P, A, A k= } Ω = Ω Ω P k 2k 3k k 2k 3k k4 k t {, P, C } {, P } {, x } {, P, R } {, x } = Ω 2P P t = Ω 2T 2P 2t Ω = Ω 3P 2T 2t Ω = Ω 3T 3P 3t t Ω = Ω 4P 3T 3t At ts stage t s necessary to treat eac of te prces n tese nformaton sets as a vector tat contans te prce of eac ndvdual dealer (toug n equlbrum eac of tese prces s a scalar, as sown below). Equlbrum Te equlbrum concept we use s Bayesan-Nas Equlbrum, or BNE. Under BNE, Bayes rule s used to update belefs and strateges are sequentally ratonal gven belefs. To solve for te symmetrc BNE, frst consder optmal quotng strateges. PROPOSITION A: A quotng strategy s consstent wt symmetrc BNE only f quotes wtn any sngle tradng round are common across dealers. PROPOSITION A2: A quotng strategy s consstent wt symmetrc BNE only f P=P2 and tese prces are equal to te fnal round prce P4 from te prevous day. PROPOSITION A3: A quotng strategy s consstent wt symmetrc BNE only f te common round-tree quote s: P = P + λ x