E xplainingStoc hastic Volatility inAsset P ric es
¤
David L.K elly
Departm ent ofE c onom ic s
U niversity ofM iam i
B ox2 84 12 6
CoralG ab les,F L 33134
DouglasG .Steigerw ald
Departm ent ofE c onom ic s
U niversity ofCalif
ornia
Santa B arb ara,CA 9 310 6
J anuary 2 0 ,2 0 0 0
A bstract
W edevelop atheoreticalmodelthatreplicates threeobserved phenomenain secu-rities markets:serialcorrelation in trades;serialcorrelation in squared price changes (conditionalheteroskedasticity);and more persistentserialcorrelation in trades than in squared price changes. In the modelexogenous news is captured by signals that informed agents receive.A gents trade anonymouslythrough amarketspecialist,who does notreceive a signal.W e showthatentry and exitofinformed traders following the arrivalofnews produces serialcorrelation in thenumberoftrades and serialcor-relation in squaredpricechanges.B ecausethebid-askspread ofthemarketspecialist tends toshrinkas individuals tradeandrevealtheirinformation,theserialcorrelation in trades is morepersistentthan theserialcorrelation in squared pricechanges.
¤W e thank L awrence H arris, B ruce L ehman Steve L eR oy, Charles Stuart, and seminarparticipants at
U niversityofA arhus,U niversityofA rizona,U niversityofCaliforniaatSantaCruz,U niversityofCalifornia atSan D iego, Econometric Society 1 998 W interM eetings, FederalR eserve B ank ofKansas City, Federal R eserve B ankofSan Fransisco,and U niversity ofT exas,forcomments and suggestions.A llerrors are our own.
1
In
trod uc tion
M anyasset pric esexhib it c onditionalheterosked astic itythrough serialc orrelationinsquared pric e changes. Statistic alm od elsthat ¯t the ob served serialc orrelationinsquared pric e changesare now w id ely used inempiric al¯nance.1 M od eling serialc orrelationinsquared
pric e changeshasimportant implic ationsfor optionpric ingand c onditionalreturnforec ast-ing.Ac c urate spec i¯c ationofa statistic alm od elrequiresknow led ge ofanec onom ic m od el that explainsw hy serialc orrelationispresent. Although there isw id espread spec ulation that the arrivalofnew s in¯nancialm arkets has animportant im pac t onsquared pric e changes, w e know ofno ec onom ic m od elthat linksthe arrivalofnew sto serialc orrelation insquared pric e c hanges.W e provid e anec onomic m od elthat linksthe b ehavior oftrad ers ina ¯nancialm arket follow ingthe arrivalofnew s,to serialc orrelationinthe squared pric e changesthat arise from the m arket.
E m piric alanalysisof¯nanciald ata revealsseveralad d itionalfeaturesofthe d ata that anec onom ic mod elshould explain.F irst,there isextensive serialc orrelationinthe numb er oftrad es(Harris, 1987) and intotalsharestrad ed (Harris, 19 87; Andersen, 1996; B rock and LeB aron, 199 6). Sec ond , serialc orrelationintrad es is more persistent thanserial c orrelationinsquared pric e c hanges (Harris, 19 87; Andersen, 199 6; Steigerw ald, 19 97).2
G allant,Hsieh,and T auc hen(19 91) show that ifthe numb er oftrad esina c alend ar period isseriallyc orrelated ,thensquared pric e c hangesare seriallyc orrelated .Although the ¯nd ing ofG allant,Hsieh,and T auc henprovid esanim portant linkb etw eentrad esand squared pric e changes, anec onom ic m od elthat linksthe arrivalofnew sto the em piric alfeaturesofthe d ata through the ac tionsoftrad ersisneed ed to ac c urately spec ify a statistic alm od el.
W e d evelop anec onomic m od eloftrad e ina ¯nancialm arket that linksthe arrivalof new sto serialc orrelationintrad es, and hence, serialc orrelationinsquared pric e c hanges. T he exogenousarrivalofprivate inform ation(new s) isc aptured b y signalsthat inform ed agentsrec eive. Agentstrad e anonymously w ith a m arket spec ialist w ho d oesnot rec eive a signal. T he m arket spec ialist fac esanad verse selec tionprob lem b ec ause the spec ialist trad esw ith m ore informed agentsw ith positive prob ab ility.
T he arrivalofprivate signalshastw o im portant e®ec ts. F irst, inform ed trad ersenter the m arket, increasing the num b er oftrad esrelative to trad ing period sinw hic h there is no new s(suc h trad ing period sare c om m only referred to as\trad ing d ays").O ver trad ing period sshorter thana trad ing d ay, ifinform ed trad ershave aninform ationalad vantage,
1Forasurvey,see B ollerslev,Engle,and N elson(1 993).
2Similarly, T auchen, Z hang, and L iu (1 996)reportthata price change has more persistente®ects on
thenmost likely informed trad ersw illhave aninform ationalad vantage the next period as w ellsince inform ationisrevealed slow ly over tim e,d ue to the presence ofliquid ity trad ers. T husthe entry and exit ofinform ed trad ersim pliestrad esare serially c orrelated .Sec ond, the m arket spec ialist w id ensthe b id -askspread inresponse to the possib le ad verse selec tion prob lem .Astrad e oc c urs,the m arket spec ialist usesB ayesrule to upd ate b eliefsand hence the b id and ask.Asinform ed trad erstrad e and revealtheir inform ation,the b id -askspread d ec lines.B ec ause the squared (c alendar period ) pric e c hange isd eterm ined b y the numb er oftrad esinthe period and the variance ofthe pric e innovationfor eac h trad e,positive serial c orrelationintrad eslead sto positive serialc orrelationinsquared pric e c hanges.B ec ause the b id -ask spread b oundsthe variance oftrad e-b y-trad e pric e innovations, the d ec lining b id -ask spread red uc esthe serialc orrelationinsquared pric e c hangesw ithout a®ec tingthe serialc orrelationintrad es.T hus,serialc orrelationintrad esism ore persistent thanisserial c orrelationinsquared pric e c hanges.
T he entry and exit ofinformed trad ersafter the arrivalofprivate inform ationisa key c om ponent ofour mod el.T he im portance ofprivate inform ationasa d eterminant ofstock pric e volatility issupported b y French and R oll(1986), w ho c onclud e that revelationof private information(rather thanpub lic informationor pric ing errors) d rives stock pric e changes. O ur m od elisb ased onthe m arket m ic rostruc ture m od elofE asley and O 'Hara (19 92 ) w hich mod elsthe new sarrivalproc ess.M arket m ic rostruc ture m od elsw hic h d o not m od elthe new sarrivalproc essgenerally d o not exhib it serialc orrelationintrad es.G losten and M ilgrom (1985) c onsid er only a single new s event, so trad es are c onstant and thus serially uncorrelated . Sargent (1993) and B roc k and LeB aron(19 96) m od eltrad ersw ho rec eive noisy signals.B ec ause trad ersd o not d ec id e to leave the market,trad esare serially uncorrelated ,although volum e generally d ec linesthrough tim e.
Severalresearc herspropose alternative explanationsfor serialc orrelationinsquared pric e changes.T im m erm an(1996) c omb inesrare struc turalb reaksinthe d ivid end proc essw ith incomplete learning.Shorish and Spear (19 96) show how m oralhaz ard b etw eenthe ow ner and m anager ofa ¯rm generates serialc orrelationinsquared pric e c hanges ina Luc as asset pric ing m od el. DenHaanand Spear (199 7) show how agency c ostsand b orrow ing c onstraintsgive rise to w ealth e®ec tsthat yield serialc orrelationinsquared interest rate changes.Divid end b ased m od elsprovid e anim portant ¯rst step b yd irec tlyexplainingserial c orrelationinsquared pric e c hangesat low frequencies. Serialc orrelationinsuc h m od els d oesnot arise from the trad ingproc ess,since the \no trad e"theorem shold.Inc ontrast our m od elexplainshow new s(say ab out the d ivid end proc ess) generateshigh frequency serial c orrelationthrough the trad ingproc ess.
2
M arket M ic rostruc ture M od el
W e c onsid er a pure d ealership m arket.Inthisw ayw e rule out b rokerage servic esprovid ed b y the spec ialist,im plyingthat allord ersare m arket ord ers.3 T he spec ialist setsa b id and ask,
w hich are the pric esat w hic h he isw illingto b uy and sell,respec tively,one share ofstock. T he b id and askare d eterm ined so that the spec ialist earnsz ero expec ted pro¯tsfrom eac h trad e.T he z ero expec ted pro¯t c onditionisanequilib rium c ond ition,w hic h arisesfrom the potentialfree entry ofad d itionalm arket spec ialistsshould the b id and ask lead to positive expec ted pro¯tsfor the spec ialist.T hus,asinG lostenand M ilgrom (1985) and E asley and O 'Hara (19 92 ),w e assum e a B ertrand-style m arket.
T he informationstruc ture ofthe m arket isasfollow s.Inform ed trad erslearnthe true share value w ith positive prob ab ility b efore trad ing starts, w hile the spec ialist and unin-form ed trad ersd o not learnthe true share value b efore trad ingstarts.W e d e¯ne the interval over w hic h asym m etric informationispresent to b e a trad ing d ay, although w e rec ogniz e that the intervalneed not c orrespond to one c alendar d ay.At the b eginningofeac h trad ing d ay informed trad ersrec eive the signalSm , w here m indexestrad ing d ays.At the end of each trad ing d ay the signalisrevealed to uninform ed trad ersand to the spec ialist, and all trad ersagree uponthe share value.
O neach trad ing d ay the random d ollar value per share,Vm , takesone oftw o values
vLm < vHm w ithP(Vm = vLm) = ±. T o ensure the c ontinuity ofpric esover trad ing d ays,
EVm = vm¡1 ifthe inform ed learnthe true value ofthe stoc k ontrad ingd aym ¡1.Ifthe
inform ed d o not learnthe true value ofthe stoc k ontrad ing d aym ¡1, thenw e presum e the possib le share valuesare unchanged and vLm = vLm¡1 and vHm = vHm¡1.
T he signalsrec eived b y inform ed trad ersat the start ofa trad ing d ay are independent ac rosstrad ing d aysand id entic ally d istrib uted . T he signalSm takesthe value: sH ifthe inform ed rec eive the high signaland learnVm = vHm,sL ifthe inform ed rec eive the low
signaland learnVm = vLm, and s0 ifthe inform ed rec eive the uninform ative signaland
hence,no private information.T he prob ab ility that the inform ed learnthe true value ofthe stock through the signalisµ,so the prob ab ility that Sm takesthe valuesL is±µ.
T he signalc om pletely d eterm ines the trad ing d ec isionsofthe informed . Conditional onrec eiving the uninform ative signal, inform ed agentsd o not trad e b ec ause ofid entic al preferences.Ifinform ed trad ersrec eive signalsL, theninform ed trad ersalw ayssellaslong asthe spec ialist isuncertainthat the true value isvLm.Ifinform ed trad ersrec eive signal
sH , theninform ed trad ersalw aysb uy aslong asthe spec ialist isuncertainthat the true
3O urmarketspecialistdoes notkeep anorderbook.B ollerslevandD omowitz (1 991 )relatethevariance
ofprices directlytothespreadexistingintheorderbook.A ssuch,theyareabletoobtainheteroskedasticity withoutserialcorrelation inthenumberoftrades.
value isvHm.
Alltrad ersand the market spec ialist,are riskneutraland rational.T o induc e uninform ed rationaltrad ersto trad e, some d isparity ofpreferencesor end ow mentsac rosstrad ersmust exist.W e let !ib e the rate oftim e d iscount for theith trad er.AsinG lostenand M ilgrom each ind ivid ualassignsrandom utility to sharesofstock,s,and c urrent c onsumption,c,as
!sVm + c.4 T he larger the value of! the greater isthe d esire to invest and forego c urrent c onsum ption.W e set ! = 1 for the spec ialist and inform ed trad ers.T here are three types ofuninform ed trad ers, those w ith! = 1, w ho have id entic alpreferencesand d o not trad e, those w ith! = 0 , w ho alw ayssellthe stoc k, and those w ith ! = 1, w ho alw aysb uy the stock. Am ong the populationofuninform ed trad ers, the proportionw ith! = 1 is1¡", the proportionw ith! = 1 is(1¡°)", and the proportionw ith! = 0 is°".T he trad ing d ec isionsofthe uninform ed are d eterm ined c om pletely b y the value of! and d o not d epend onthe b id and ask.
T rad ersarrive to the m arket one at a tim e,so w e ind extrad ersb y their ord er ofarrival. T he prob ab ility that the arrivingtrad er isinform ed is® > 0 .A trad er arrives,ob servesthe b id and ask, and d ec id esw hether to b uy, sell,or not trad e.Let Cib e the random variab le that c orrespondsto the trad e d ec isionoftrad eri.T henCitakesone ofthree values:cA if theith trad er b uysone share at the ask,Ai;cB iftheith trad er sellsone share at the b id ,
Bi; and cN iftheith trad er elec tsnot to trad e.T he sequence oftrad ingd ec isionsispub lic inform ation.Let Z ib e the pub lic ly availab le inform ationset afteritrad ershave c om e to the m arket.T he inform ationset availab le to the spec ialist and the uninform ed isZ i.
B ec ause the spec ialist and the uninform ed have the sam e inform ationset, they have the same learning proc ess.Inw hat follow s, w e simply refer to the learning proc essfor the spec ialist, noting that the sam e proc essappliesto the uninformed .After the ac tionofthe trad er,the spec ialist revisesb eliefsab out the signalrec eived b yinform ed trad ers,and thence ab out the true value ofa share.After theith trad er hasc om e to the m arket,the spec ialist's b eliefthat informed trad ersrec eived a high signalis,
P(Sm = sH jZ i) = yi:
Correspondingly,the spec ialist'sb eliefthat informed trad ersrec eived a low signalis,
P(Sm = sLjZ i) = xi:
4W eassumeanin¯nitenumberoftraders sothattheprobabilityofanyplayerplayingmorethanonceis
zero.B ecauseVmis realized attheend ofthe tradingday,Vmis therandom sharevalue used toconstruct atrader's utilityattheendofatradingday.
B yc onstruc tion,the spec ialist'sb eliefthat inform ed trad ersrec eived anuninform ative signal is,
P(Sm = s0jZ i) = 1¡xi¡yi:
T he spec ialist'sb eliefsab out Sm translate d irec tly into b eliefsab out the value ofa share. Ifthe spec ialist b elievesSm = sH ,thenthe ac c urac y ofthe signalim pliesthat the spec ialist b elievesVm = vHm. Sim ilarly, ifthe spec ialist b elievesSm = sL, thenthe spec ialist also
b elievesVm = vLm . Ifthe spec ialist b elievesSm = s0, thenthe spec ialist assigns the
unconditionalprob ab ilitiesto the possib le valuesforVm.T o sum m ariz e,after theith trad er hasc om e to the m arket,the spec ialist'sc ond itionalprob ab ility that Vm = vHm is
P(Vm = vHmjZ i) = yi+ (1¡xi¡yi)(1¡±);
w hileP(Vm = vLmjZ i) = 1¡P(Vm = vHmjZ i).T he ac tionofeac h trad er,eventhe d ec ision
not to trad e,c onveysinform ationab out the signalrec eived b y inform ed trad ers.
2 .
1
Determ inationofAsk and B id
At the b eginning ofeac h trad ing d ay,x0 = µ± and y0 = µ(1¡±). Let A1 and B1 b e
the initialask and b id , respec tively.(T husA1 isthe ask that the ¯rst trad er fac es.) T he
equilib rium c ond itionthat the spec ialist earnz ero expec ted pro¯t from eac h trad e provid es the equationsthat d eterm ine the quoted pric es(B1;A1).Inessence, the quoted pric esset
the spec ialist'sexpec ted lossfrom trad e w ith aninform ed trad er equalto the spec ialist's expec ted gainfrom trad e w ith anuninform ed trad er.W e explic ilty d eriveA1 (d erivationof
B1 follow ssim ilar logic ).Ifthe ¯rst trad er trad esat the ask,thenthe spec ialist'sexpec ted
lossfrom trad e w ith aninform ed trad er is
® ¢y0 (A1¡vHm );
w herey0 (A1¡vHm) isthe expec ted lossifthe ¯rst trad er trad esat the ask,giventhat the
¯rst trad er isinform ed .Similarly, ifthe ¯rst trad er trad esat the ask, thenthe spec ialist's expec ted gainfrom trad e w ith anuninform ed trad er is
(1¡®)"(1¡°)f[x0 + ±(1¡x0 ¡y0)](A1¡vLm) + [y0 + (1¡±)(1¡x0 ¡y0)](Ai¡vHm )g:
Ifexpec ted pro¯tsequalzero,then
A1 =
® y0vHm + (1¡®)"(1¡°)E(Vm jZ 0)
® y0 + (1¡®)"(1¡°)
w hereE(VmjZ 0) = x0vLm + y0vHm + (1¡x0 ¡y0)EVm .Inparallelfashion
B1 =
® x0vLm + (1¡®)"°E(Vm jZ 0)
® x0 + (1¡®)"°
:
T he equationsfor (Bi;Ai) are sim plythe equationsfor (B1;A1) w ithy0 replac ed b yyi¡1 and
x0 replac ed b yxi¡1 (w hic h im pliesE(Vm jZ 0) isreplac ed b yE(Vm jZ i¡1)). Asone w ould
expec t,b oth the b id and ask increase w ithyi¡1 and d ec rease w ithxi¡1.
It iseasy to see that vLm ·Bi·Ai·vHm, w ith stric t inequality unlessthe spec ialist
isc ertainthe inform ed learned the true value ofVm (no ad verse selec tion).M athem atic ally, the spec ialist isc ertainthe informed learned the true value ofVm ifxi¡1 = 1 oryi¡1 = 1.
It is also easy to see that Bi·E(Vm jZi¡1) ·Ai, w hic h follow s d irec tly from vLm ·
E(Vm jZ i¡1)·vHm.
2 .
2
LearningR ul
es
Astrad ingoc c urs,inform ationac c ruesto the spec ialist.Inresponse,the spec ialist upd ates the prob ab ilities(xi;yi).W e b eginb y exam ining how the spec ialist learnsfrom the ac tion ofthe ¯rst trad er and explic itly d iscussonly upd atingofy1 (upd atingofx1 follow ssim ilar
logic ).5 T he keyparam etersthat governthe speed oflearningare® and ".Ifthe ¯rst trad er
trad esat the ask
y1 = y0
® + (1¡®)"(1¡°)
® y0 + (1¡®)"(1¡°)
:
Aslong asy0 < 1, a trad e at the ask increasesy1.If® = 1 or"= 0 only inform ed trad ers
trad e,so learningisim m ed iate and y1 = 1.Ifthe ¯rst trad er trad esat the b id
y1 = y0
(1¡®)"° ® x0 + (1¡®)"°
:
Aslong asx0 > 0 , a trad e at the b id d ec reasesy1. If® = 1 or "= 0 againlearning is
im m ed iate,so y1= 0 and x1 = 1.F inally,ifthe ¯rst trad er d oesnot trad e
y1 = y0
(1¡®)(1¡")
®(1¡x0 ¡y0) + (1¡®)(1¡")
:
Aslong as(1¡x0 ¡y0)> 0 , a d ec isionnot to trad e d ec reasesy1.If® = 1, or if"= 1 in
w hich c ase alluninformed trad erstrad e,thenlearningisim med iate w ithy1= 0 and x1 = 0 .
T he learningformulae foryiare sim ply the learningformulae fory1 w ithy0 replac ed b y
yi¡1.T he learningformulae forxiare:
xi= xi¡1
(1¡®)"(1¡°)
® yi¡1 + (1¡®)"(1¡°)
;
iftrad eritrad esat the ask;
xi= xi¡1
® + (1¡®)"° ® xi¡1+ (1¡®)"°
;
iftrad eritrad esat the b id ; and
xi= xi¡1
(1¡®)(1¡")
® (1¡xi¡1¡yi¡1) + (1¡®)(1¡")
;
iftrad erid oesnot trad e.
2 .
3 Consistency ofLearning
W e have posited that the signalisrevealed at the end ofa trad ing d ay, w hic h c onsists ofa ¯nite numb er oftrad er arrivals. T o ensure that the learning formulae w e d escrib ed ab ove are useful, w e estab lish that ifthere w ere anin¯nite numb er oftrad er arrivals, the spec ialist w ould learnthe value ofSm .Asa result,the b id and ask c onverge to the strong-form e± c ient value ofa share,inw hic h the b id and ask re°ec t b oth the pub lic and private inform ation. B ec ause transac tionpric esare d eterm ined b y the b id and ask, transac tion pric esalso c onverge to the strong-form e± c ient value ofa share.
T hree setsofb eliefsc apture the spec ialist'suncertainty ab out the value ofSm .T he ¯rst isthe spec ialist'sb eliefthat Sm = sH, w hic h isexpressed asthe sequence ofc onditional prob ab ilitiesfyig1i= 1.T he sec ond isthe spec ialist'sb eliefthat Sm = sL,w hic h isexpressed asthe sequence ofc onditionalprob ab ilitiesfxig1i= 1 , and ¯nally the third isthe b eliefthat
Sm = s0,w hic h isexpressed asthe sequencef1¡xi¡yig1i= 1 .
T heorem 1:T he sequence ofbid sand asks, and hence the sequence oftransac tionprices, converge alm ost surely to their strong-form e± c ient valuesat anexponentialrate.Form ally, asi! 1: IfSm = sH , thenxi!as 0,yi!as 1and Ai!as vHm,Bi as ! vHm. IfSm = sL, thenxi as ! 1,yi as ! 0 and Ai as ! vLm,Bi as ! vLm . IfSm = s0, thenxi!as 0,yi!as 0 and Ai!as EVm ,Bi!as EVm .
P roof:See Appendix.
Although the asym ptotic b ehavior ofpric esisstraightforw ard to d eterm ine,c alculating the serialc orrelationpropertiesrequiresknow led ge ofthe d istrib utionofshare pric esineac h tim e period ,a m ore d i± c ult task w hic h w e turnto next.
3 Cal
en
d ar P eriod Im pl
ic ations
W ith the learning rulesestab lished , w e now show that the mod elac c ountsfor the m ain empiric al¯ndingsd escrib ed inthe introd uc tion.W e ¯rst show that one im plic ationofthe m od elisthat the numb er oftrad esina c alend ar period isseriallyc orrelated .Asd escrib ed in the introd uc tion,suc h serialc orrelationlead sto serialc orrelationinsquared pric e c hanges. W e thenshow that the serialc orrelationinthe numb er oftrad esper c alendar period ism ore persistent thanisthe serialc orrelationinsquared pric e changes.
T o d erive c alendar period implic ations,w e must b e c lear ab out how trad ingopportunities are aggregated .A trad ing d ay c ontainskc alendar period s(suc h asanhour).A c alendar period , w hic h isind exed b yt, c ontains´ trad er arrivals, w hic h asab ove are ind exed b yi. (W e c anthink ofa trad er arrival,or trad ingopportunity,asa unit ofec onomic tim e.) T he sample period c onsistsofa large sam ple oftrad ingd ays.
Calendar Period Trades
F irst w e exam ine the c ovariance struc ture ofthe numb er oftrad esper c alendar period . Let the numb er oftrad esin(c alendar) period tb eIt:B ec ause´trad ersarrive eac h period ,It takesinteger valuesb etw een0 and ´.Infac tItisd istrib uted asa b inomialrand om variab le w here the numb er oftrad esc orrespondsto the numb er of\suc c esses"in´ \trials".For all period sontrad ingd aym the prob ab ility that a trad er d ec id esto trad e is
P(Ci6= cN jSm 6= s0) = ® + (1¡®)";
P(Ci6= cN jSm = s0) = (1¡®) ";
so the d istrib utionofItc onditionalonthe value ofSm is
Itj(Sm 6= s0) » B (´;® + "(1¡®));
Itj(Sm = s0) » B (´;"(1¡®)):
E[ItjSm 6= s0] = ¹1 = ´(® + "(1¡®)); E[ItjSm = s0] = ¹0 = ´"(1¡®); V ar[ItjSm 6= s0] = ¾12 = ´[® + "(1¡®)](1¡®)(1¡"); V ar[ItjSm = s0] = ¾02 = ´"(1¡®)[1¡"(1¡®)]: U nconditionally,w e have: E[It] = ¹ = µ¹1+ (1¡µ)¹0 (1) V ar[It] = ¾2 = µ¾12 + (1¡µ)¾02 + µ(1¡µ)(¹1¡¹0)2 (2 )
G iventhe ab ove struc ture for the numb er oftrad esina c alendar period ,w e c and erive the serialc orrelationpropertiesofthe numb er oftrad es.
T heorem 2 :Let r > 0.Ifr < k, thenIt¡r and Itare positively serially correlated .If
r¸k, thenIt¡r and Itare uncorrelated .Further for allr, the correlationbetw eenIt¡r and
Itisgivenby: C or(It¡r;It) = µ(1¡µ)(® ´)2 ¾2 " k¡m in(r;k) k #
P roof:See Appendix.
T heorem 2 givesthe exac t formula for the c orrelation.T herefore, it isstraightforw ard to estab lish c om parative static s,w hic h w e sum m arize inthe follow ingc orollary.
Corollary 3:Ifr < k, thenthe correlationbetw eenIt¡r and Itis d ec reasing inr,
increasingink, increasingin´ and increasingin®.
P roof:
Sub stitutinginthe d e¯nitionsofkand ¾2 into equation(7) and assum ingr < k,results
in: C or(It¡r;It) = µ¿¡r´ ¿ ¶ ´µ(1¡µ)®2 "(1¡®)[1¡"(1¡®)]+ µ® [(1¡®)(1¡2") + ® ´(1¡µ)]:
T he resultsthenfollow b y takingthe appropriate d erivatives.
T he c om parative static c alculationsinCorollary 3imply c ertainpatternsofserialc orre-lationintrad esac rossm arkets.W e ¯rst stud yhow the serialc orreorre-lationintrad esisa®ec ted b y changesinthe parametersc harac teriz ingaggregationover tim e.Asthe numb er ofperi-od sina trad ingd ay,k,increases,the im pac t ofthe entry and exit ofinformed trad ersgrow s and the serialc orrelationincreases.Asthe num b er oftrad er arrivalsina c alendar period ,
´, increases, the impac t ofinformed trad ersisagainreinforc ed and the serialc orrelation increases.T husone m ay expec t to see m ore pronounced serialc orrelationinasset m arkets inw hic h the revelationofprivate inform ationtakesa relatively longer period oftim e.Sim-ilarly,one m ay expec t to see m ore pronounced serialc orrelationinthic ker m arketsthanin thinner m arkets.
B oth ofthe prec ed ing c alculationsallow only one param eter to c hange; im plic it inour c om parisonofm arket thic knessisthe assum ptionthat the length ofthe trad ingd ayis¯xed . Y et for m any c om parsions,b othkand ´ are c hanging.A lead ingc ase w ould b e c om parsion ofinform ationgathered at tw o d i®erent c alendar period frequencies,say 5 m inute intervals versushourly intervals. B ec ause the d ata are gathered for the sam e asset, the numb er of trad er arrivalsina trad ing d ay,¿ = k´, isc onstant for b oth frequencies. T o understand the e®ec t onthe c orrelationc aused b y c hangingfrom 5 m inute intervalsto hourly intervals, w e sub stitute ¿k for ´ and take the d erivative w ith respec t to k. As the c hange from 5 m inute intervalsto hourly intervalssimultaneously d ec reaseskand increases´,w e have tw o c ountervailinge®ec tsonthe c orrelation.Ingeneral,the serialc orrelationc aneither increase or d ec rease w ith a c hange inc alendar period and,perhapsmost interestingly,the c hange is not c onstant ac rossr.B ec ause the m agnitud e ofthe e®ec t ofa c hange inkonthe c orrelation d ependsonr,it isfor longlagsthat w e w ould m ost likely see the serialc orrelationintrad es d ec line asw e m ove from 5 m inute d ata to hourly d ata.
T o understand how the serialc orrelationintrad esd ependsuponthe und erlying para-m etersofthe para-m arket para-m ic rostruc ture para-mod el, w e c and ec opara-m pose the c orrelationinto three terms.T he ¯rst term isthe d i®erence b etw eenthe num b er oftrad esona trad ingd ay w ith new sand ona trad ing d ay w ithout new s, w hic h is(¹1¡¹0)2. T he rem aining tw o term s
are the c onditionalvariancesona trad ing d ay w ith new s(¾2
1) and a trad ing d ay w ithout
new s(¾2
0),respec tively.Anincrease in(¹1¡¹0)2 increasesb oth the c ovariance oftrad esand
the variance oftrad es,w here (¹1 ¡¹0)2 entersthe variance through the c om ponent for the
variance ofthe c onditionalm eans, so the overallim pac t onthe serialc orrelationintrad es must b e c alculated .Anincrease to the c ond itionalvarianceslead sonly to anincrease inthe variance,so the overallimpac t isto red uc e the serialc orrelationintrad es.
increasing® hastw o e®ec tsonthe c orrelation. F irst, w ith a larger numb er ofinform ed trad ers,there isa w id er d i®erence b etw eenthe numb er oftrad esona trad ingd ay w ith new s and ona trad ing d ay w ithout new s. Sec ond , b ec ause the informed trad ersallmake the same trad ingd ec ision,anincrease inthe num b er ofinform ed trad ersd ec reasesat least one ofthe c ond itionalvariances.Asa result, the positive im pac t onthe c ovariance outw eighs the postive im pac t onthe variance.B ec ause the serialc orrelationintrad esisanincreasing functionof®, a m arket w ith many inform ed trad ershasmore serialc orrelationintrad es thand oesa m arket w ith few er inform ed trad ers.
Next,c onsid er the relationship b etw een"(the frac tionofuninform ed w ho d o not trad e) and the serialc orrelationintrad es. For spec i¯c param eter values the partiald erivative isd e¯nitively signed . If" issm all(prec isely, if" < 2 (11¡¡2® "®)), thenvirtually alltrad esare b y inform ed trad ersand increasing" d ilutesthe informed trad ersand red uc esthe serial c orrelationintrad es. If® islarge (prec isely if® µ ¸ 1
2), thenincreasing"increasesthe
variationintrad esac rossd aysand increasesthe serialc orrelationintrad es.
Insimilar fashion, increasingµ increasesthe c orrelationifµ issm allenough (prec isely (1¡µ2)> ¾2
1).B ec ause good and b ad new sare sym m etric inthe m od el,the serialc orrelation
isuna®ec ted b y changesto °or ±. Aninteresting im plic ationisthat our m od elpred ic ts c orrelationina variety ofmarkets.For exam ple,there isserialc orrelationintrad esinb oth liquid and illiquid markets.O ur ¯nd ingsofserialc orrelationeveninilliquid m arketsisalso supported em piric ally b y Lange (19 98).
O fc ourse, serialc orrelationinthe numb er oftrad esc ould b e arti¯c ially imposed b y c reating serialc orrelationinthe private inform ationarrivalproc ess. E ngle et al.(1990 ) ¯nd som e evid ence ofserialc orrelationinpub lic new s; although serialc orrelationinpub lic new sd oesnot im ply serialc orrelationinprivate new s. Appealing to serialc orrelationin private new sd oesnot reallyprovid e anec onom ic c ause for serialc orrelationinsquared pric e changesasit b egsthe questionasto w hat c ausesserialc orrelationinprivate new s.
Behavior of Individual Trader Price Changes
T o understand the serialc orrelationinsquared pric e changesper c alendar period , w e ¯rst stud y the b ehavior ofthe pric e c hangesthat follow the arrivalofeac h trad er. T he pric e change that resultsfrom the ac tionoftrad eriisUi= E(Vm jZ i)¡E(Vm jZ i¡1),w here
E(Vm jZ i) = xivLm + yivHm + (1¡xi¡yi)EVm .
6 T he d e¯nitionofU
iincorporatesthe arrival
6T he price is conditionalon public information and is hence theoretically observable to the
econome-trician.In reality the setofparameters mustbe estimated,resultingin an estimate ofthe price based on theestimatedparameters.H owever,inmostempiricalstudies ofserialcorrelationinsquaredpricechanges, econometricians usethe bid,ask,orlasttrade,which mayhavedi®erentproperties from theprice.
ofpub lic inform ationafter the d ec isionoftrad eri¡1 b ut b efore the d ec isionoftrad eri.T o relate d ec isionsinec onomic tim e givenb y our m od elto the c alendar period m easurem ents, w e w rite c alendar period pric e c hangesas
¢Pt= t´
X
i= (t¡1)´+ 1
Ui: (3)
P ric e c hangesinec onom ic tim e thusd rive c alendar pric e c hanges.Inturn,the inform ation c ontent oftrad es(or no trad es) d rive pric e c hangesinec onom ic time. T he inform ation c ontent ofa trad e or no trad e d ependsonthe history oftrad esand the param eter values. For exam ple, if" islarge, no trad esc onvey relatively m ore inform ation. If°islarge, a trad e at the askc onveysrelatively m ore inform ation.T rad esor no trad esat early ec onom ic tim e period sc onvey m ore informationthantrad esat later tim e intervals.Inthisw ay,serial c orrelationinsquared pric e c hangesare serially c orrelated .
T o provid e insight,w e stud y ind etailthe pric e c hange assoc iated w ith the arrivalofthe ¯rst trad er ontrad ingd aym .T here are three possib le valuesforU1, one c orrespond ingto
each ofthe possib le trad e d ec isions.IfC1 = cA,thenE(Vm jZ 1) = A1,and
U1 = ® y0 [vHm ¡E(Vm jZ 0)] P(C1 = cAjZ 0) : IfC1 = cB,thenE(Vm jZ 1) = B1 and U1 = ® x0 [vLm ¡E(Vm jZ 0)] P(C1 = cBjZ 0) : F inally,ifC1 = cN ,then E(Vm jZ 1) = ®(1¡x0 ¡y0)EVm + (1¡®)(1¡")E(Vm jZ 0) ®(1¡x0 ¡y0) + (1¡®)(1¡") and U1 = ®(1¡x0 ¡y0)[EVm ¡E(Vm jZ 0)] P(C1 = cN jZ 0) :
Ifinitialpriorsare logic ally c onsistent, so that ±y0 = (1¡±)x0, thenupd ating from the
B1 < E(Vm jZ 0;C1 = cN )< A1.W hile the inequality isgenerally satis¯ed for the rem aining
trad ers,it ispossib le forE(Vm jZ i¡1;Ci= cN ) to falloutsid e (Bi;Ai).7 T he m eanpric e c hange from trad er 1 is
E(U1jZ 0) = X j=A;B ;N P(C1 = cjjZ 0)U1(C1 = cj); w hich equals ® y0vHm + ® x0vLm + ®(1¡x0 ¡y0)EVm ¡® E(Vm jZ 0) = 0: B ec ause P(Ci= cAjsm )=6P(Ci= cAjZi)
for any ¯nite i, pric e c hangesare not m eanzero w ith respec t to the informationset ofthe inform ed .
T he variance ofthe pric e c hange from trad er 1 is
E³U2 1jZ 0 ´ = X j=A;B ;N P(C1 = cjjZ 0)U12(C1 = cj) w hich equals (® y0)2 [vHm ¡E(Vm jZ 0)] 2 P(C1 = cAjZ 0) + (® x0) 2 [vLm ¡E(VmjZ 0)] 2 P(C1 = cBjZ 0) + ® 2 (1¡x 0 ¡y0)2 [EVm ¡E(Vm jZ 0)]2 P(C1 = cN jZ 0) :
T o und erstand the impac t ofinform ed trad ersonthe b ehavior ofc alendar period squared pric e changes,w e must c om pare the variance ofU1 for Sm = s0 w ith the variance ofU1 for
Sm 6= s0. (Ingeneral, the c om parisonw illd epend onw hether the low or high signalw as
rec eived .If°= ±= :5,thenx0 = y0 and the variance ofU1 isid entic alfor the low and high
signals. Inthe rem ainder ofthe sec tionw e assume°= ± = :5 and so w e d o not need to d istinguish b etw eenthe low and high signals.) T he ad d itionofthe signalaltersthe variance ofU1 onlythrough the im pac t onthe prob ab ilityw ith w hic h eac h trad e outc ome isob served
E³U12jSm = sm ;Z 0
´
= X j=A;B ;N
P(C1= cjjSm = sm )U12(C1 = cj):
W e c om pare the prob ab ilitiesofeac h trad e outc om e forSm = sH w ithSm = s0:
7Forexample,if"is very large and® is very small(sothatthe rare notrade decisions are mostoften
P(C1 = cAjSm = sH ) = P(C1 = cAjSm = s0) + ®
P(C1 = cN jSm = sH ) = P(C1 = cN jSm = s0)¡® ;
w here the prob ab ilitythatC1 = cB isthe sam e for the tw o valuesofSm .T husE(U12jSm = sH ;Z 0)¡
E(U2 1jSm = s0;Z 0) equals ® ( (® y0)2 [vHm ¡E(Vm jZ 0)] 2 [P(C1 = cAjZ 0)]2 ¡® 2 (1¡x 0 ¡y0)2 [EVm ¡E(VmjZ0)]2 [P(C1 = cN jZ 0)]2 ) ;
w hich isgreater thanz ero b ec ause EVm = E(Vm jZ 0). B ec ause the term ispositive, the
pric e uncertainty from the ¯rst trad er ishigher ona d ay w ith new sthanona d ay w ithout new s. T he im pac t oftrad er 2 and follow ing trad ers is not im m ed iately signed b ec ause
EVm 6= E(Vm jZ i) for i> 0 .T o d etermine the signofthe d i®erence w e stud y the b ehavior ofUifor generali.
For generalithere are 3ipossib le valuesfor U
i, so d irec t c alculationofthe m om ents ofUiisted ious. R ather, w e c onstruc t analytic b oundsto the m omentsthat d escrib e the b ehavior ofthe d istrib utionofUi.Let
e
Ai¡Bei= m axfAi;E[Vm jZ i¡1;Ci= cN ]g ¡m infBi;E[Vm jZ i¡1;Ci= cN ]g
b e the \spread " or the d i®erence b etw eenthe m aximum pric e change and the m inimum pric e change.For m ost param eter values,the spread isequalto the fam iliar b id -askspread . How ever,asnoted earlier,for some param eter valuesa no trad e m ayind uc e larger or sm aller pric e c hangesthana trad e at the ask or b id ,respec tively.
Let fUigk´i= 1 b e the sequence oftrad er pric e c hanges(pric e c hangesinec onom ic tim e) for a trad ingd ay.W ith respec t to the pub lic inform ationset,the elem entsofthe sequence are uncorrelated b ut are d epend ent and not id entic ally d istrib uted .Spec i¯c ally, the trad er pric e c hangesare heterosked astic and the heterosked astic ity isautoregressive.
T heorem 4 :P rice c hangesineconom ic time satisfy:
1.E(UijZ i¡1) = 0 2 .E(UhUijZ i¡1) = 0 for h < i 3.[P(Ci= cA)P(Ci= cB)P(Ci= cN )] ³ e Ai¡Bei ´2 ·E(U2 ijZi¡1)· ³ e Ai¡Bei ´2 :
P roof:See Appendix.
T he ¯rst tw o partsofT heorem 4 d eliver the trad itionalresultsthat EUi= 0 and that
E(UiUj) = 0 ifid oesnot equalj.T he spread d rivesthe variance inUi.T heorem 4 and P roposition1 together im plythatE(U2
ijZ i¡1)! 0 asi! 1.Asthe market maker b ec om es
c ertainofthe true value ofthe share,the b id and askc onverge to the true value ofthe share and squared pric e changesgo to zero.
T o d eterm ine how the properties ofthe d istrib utionofUiare a®ec ted b y the signal rec eived b y informed trad ers,w e c ompare the variance ofUiifSm = s0 w ith the variance of
UiifSm 6= s0.P arallelto the c ase for trad er 1,E(Ui2jSm = sH ;Z i¡1)¡E(Ui2jSm = s0;Z i¡1)
equals ® ( (® yi¡1)2 [vHm ¡E(Vm jZ i¡1)] 2 [P(C1 = cAjZi¡1)]2 ¡® 2 (1¡x i¡1¡yi¡1)2 [EVm ¡E(Vm jZ i¡1)]2 [P(C1 = cN jZ i¡1)]2 ) :(4 ) T he d i®erence (4 ) d ependsonE(Vm jZ i¡1),w hic h inturnd ependson(xi¡1;yi¡1).
Ifthe proportionofinform ed trad ersishigh enough, thenlearning takesplac e quic kly and the entire d istrib utionofUic anb e d irec tly c alculated .For exam ple,if® = :9 ,thenthe b id -ask spread isred uc ed very c lose to z ero inonly 10 trad es.For® = :9 and °= ±= :5, the c olum nsofT ab le 1 c ontainthe valuesof(4 ) c orrespondingto trad er 1 through trad er 8 and the row sofT ab le 1 c orrespond to d i®erent valuesof".
Table 1 V alue of E(U2 ijSm = sH ;Z i¡1)¡E(Ui2jSm = s0;Z i¡1) ® = :9 T rad er: 1 2 3 4 5 6 7 8 "= :9 3.46 0 .64 0 .55 0 .14 0 .0 8 0 .03 0 .0 1 0 .00 "= :8 2 .89 0 .52 0 .4 6 0 .11 0 .0 7 0 .02 0 .0 1 0 .00 "= :7 2 .30 0 .4 7 0 .32 0 .09 0 .0 4 0 .01 0 .0 1 0 .00 "= :6 1.66 0 .4 2 0 .18 0 .07 0 .0 2 0 .01 0 .0 0 0 .00 "= :5 0 .97 0 .33 0 .0 4 0 .04 0 .0 0 0 .00 0 .0 0 0 .00 "= :4 0 .20 0 .18 -.0 5 -.01 0 .0 0 0 .00 0 .0 0 0 .00 "= :3 -.65 -.11 -.0 9 -.09 -.0 1 -.01 0 .0 0 0 .00 "= :2 -1.61 -.70 -.14 -.05 -.0 4 -.01 -.0 1 0 .00 "= :1 -2 .73 -2 .12 -.32 -.15 -.0 9 -.02 -.0 1 0 .00
T he entriesinT ab le 1 revealtw o im portant features. F irst, aslearning ac c um ulates (m oving ac rossa row ) the d i®erence insquared pric e c hangestendstow ard z ero.Sec ond,
the value of"playsa key role.For large valuesof",the b ehavior oftransac tionlevelpric e changesissuc h that the variance ishigher,uniform ly,for d aysinw hic h the inform ed trad e. As"d ec lines,the variance oftransac tionlevelpric e c hangesc anb e higher ond aysinw hic h the inform ed d o not trad e. W hy? W ith "low , very few ofthe uninform ed trad e and so there are very few trad esifSm = s0 (b ec ause the inform ed also d o not trad e).W ith so few
trad es, learning isslow ed enough that the slow learning c orrespond ing to Sm = s0 ac tually
outw eighsthe uncertainty assoc iated w ithSm 6= s0.
For smaller valuesof® learning oc c ursm ore slow ly, so red uc tionofthe b id -ask spread to zero takesmany m ore trad esand c alculationofthe exac t d istrib utionisc umb ersome. T o understand the b ehavior of(4 ) w ith a sm aller value of®, w e approxim ate the exac t d istrib utionw ith simulations. Inthe last ¯gure titled \mean: squared pric e c hanges" w e report the simulated d istrib utionfor® = :2 and"= :5and c onstruc t 30 0 0 sim ulations.T o b e prec ise,w e simulate 30 0 0 trad ingd ays.At the outset ofeac h trad ingd aythe prob ab ilitythat the inform ed rec eive a signal(µ) is.4 ,so slightlylessthanhalfofthe simulationsc orrespond to the line lab eled \New sDays".8 A trad ingd ay isassum ed to c onsist of9 60 trad er arrivals,
b ut asthe ¯gure revealsthe squared pric e c hange ise®ec tively z ero after the ¯rst 50 0 trad er arrivals.T he last attac hed ¯gure c ontainsE(U2
ijSm = sH ;Z i¡1)¡E(Ui2jSm = s0;Z i¡1) as
the d i®erence b etw eenthe linesc orrespond ing to \New sDays" and \No New sDays".As the ¯gure reveals, the d i®erence isgenerally positive and shrinksto z ero asthe numb er of trad ersincreases.(T he horiz ontalaxism easuresthe numb er oftrad ersthat have arrived , so learningisnearly c om plete after 50 0 trad ershave c om e to the m arket.)
Calendar Period Squared Price Changes
From the resultsab ove, w e form a struc ture for the expec tationofthe c alendar period squared pric e change. T he analytic and sim ulationresultsindic ate that (for reasonab ly large valuesof") ontrad ingd aysinw hic h the inform ed d o not trad e,the spec ialist'sinitial uncertainty ab out the signalisresolved fairly quic kly.Asa result,the variance inc alendar period pric e c hanges,w hic h isd rivenb y the random d ec isionsofthe uninform ed ,isc onstant over the c ourse ofthe trad ing d ay.Inc ontrast, ontrad ing d aysinw hich the inform ed d o trad e,the spec ialist'suncertainty isnot resolved quic kly,b ut rather d ec linesover the c ourse ofthe trad ing d ay.T o c apture these phenom ena m athem atic ally, let jindex the c alendar period sina trad ingd ay.Fort= 1;:::;jw e represent the struc ture as
Eh(¢Pt)2 jSm = s0
i
= ¾0
8T oensurethatourresults arestable,we¯nd littlevariationin results when the numberofsimulations
Eh(¢Pt)2 jSm 6= s0 i = k X j= 1 ¾j1 (t= j);
w here w e assum e¾1 > ¾2 > ¢¢¢> ¾k> ¾0.Ifc alendar period tisthe ¯rst period oftrad ing
d aym onw hic h Sm 6= s0,the expec ted squared pric e change is¾1.T he inequality¾k> ¾0
arisesfrom the ob servationthat the informationad vantage ofthe informed persistsuntilthe trad ing d ay ends.T he unconditionalexpec tationofc alendar period squared pric e c hanges is
Eh(¢Pt)2
i
= µ¾¹k+ (1¡µ)¾0;
w here ¹¾k= 1kPkj= 1¾j.
T o d erive the serialc orrelationproperties ofthe squared pric e c hange ina c alendar period ,anim portant c ond itionarisesthat ensure the c orrelationispositive.
For generalkthe c ovariance ofc alendar period squared pric e c hanges,C ovh(¢ Pt¡r)2 ;(¢ Pt)2
i , equals " k¡m in(r;k) k # fµ(1¡µ) k¡r X j= 1 (¾j¡¾0)(¾j+r ¡¾0) + µ2 k X j= 1 (¹¾k¡¾j)(¹¾k¡¾j+r)g; w here the ad d itionisw rapped at k.T hat is,ifj+ r > k,thenreplac ej+ r w ithj+ r¡k.
Asfor the c ovariance ofc alend ar period trad es,the c ovariance ofc alendar period squared pric e changesisz ero ifr ¸k.T o d eterm ine the signofthe c ovariance ifr < k, w e must examine each term ind etail. T he ¯rst term inb rac kets is the sum ofthe c onditional c ovariances, w hic h ispositive. T he sec ond term inb racketsisthe sum ofthe c ovariances ofthe c onditionalm eans, w hic h isgenerally negative. Determinationofthe signofthe c ovariance d ependsonthe relative magnitud esofthe tw o term s.
W e b eginour analytic d erivationsw ith a trad ingd ay inw hic h there are tw o period s,so
k= 2 .For thisc ase anim portant c ond itionem ergesthat isneed ed to ensure the c ovariance ispositive.
Condition1 issaid to hold for period j¤, w ith1< j¤·k, ifj¤isthe largest value ofj for w hic h
¾j> µ¾¹k+ (1¡µ)¾0:
Condition1 isperhapsmost intuitive for the c asek= 2 .(Ifa trad ingd ay c orresponded to a c alendar d ay, thenem piric alstud y ofm orningsversusafternoonsw ould yield k= 2 .) R ec allthat positive c ovariance b etw eentw o random variab les,w ith the sam e unconditional
m ean,im pliesthat ifone random variab le isb elow the unconditionalmean,thenthe other rand om variab le tendsto b e b elow the unconditionalm ean.Correspondingly,ifone rand om variab le isab ove the unconditionalm ean,thenthe other random variab le tendsto b e ab ove the unconditionalm ean.From the struc ture for the expec tationofc alendar period squared pric e changesit follow sthat ¾1 liesab ove the unconditionalm eanand ¾0 liesb elow the
unconditionalm ean.Let (¢Pt¡1)2 b e the ¯rst period oftrad ing d aym .IfSm = s0, then
inexpec tation(¢Pt¡1)2 equals¾0 and so tendsto b e b elow the unconditionalm ean. Y et
ifSm = s0, theninexpec tation(¢ Pt)2 also equals¾0 and so also tendsto b e b elow the
unconditionalmean.IfSm 6= s0,theninexpec tation(¢Pt¡1)2 equals¾1 and so tendsto b e
ab ove the unconditionalm ean.W ithSm 6= s0,theninexpec tation(¢Pt)2 equals¾2 and the
b ehavior ofthe c ovariance d ependsonthe relative magnitud e of¾2 and the c onditionalm ean.
IfCondition1 holds, then¾2 islarger thanthe unconditionalmean, so if(¢Pt¡1)2 tends
to b e ab ove the unconditionalmean, then(¢ Pt)2 also tendsto b e ab ove the unconditional m ean.
P rop osition5:Letr > 0.T he covariance ofcalendar period squared price c hangesis " k¡m in(r;k) k # fµ(1¡µ) k¡r X j= 1 (¾j¡¾0)(¾j+r ¡¾0) + µ2 k X j= 1 (¹¾k¡¾j)(¹¾k¡¾j+r)g;
w here the ad d itionisw rapped atk.T hat is, ifj+ r > k, thenreplacej+ r w ithj+ r¡k. Ifr < k= 2 and Condition1 holdsfor period 2 , then
C ovh(¢ Pt¡r)2 ;(¢Pt)2 i = " k¡r k # fµ(1¡µ)(¾1¡¾0)(¾2 ¡¾0) + µ2 2 (¾1 ¡¾2) 2 g ¸0:
P roof:See Appendix.
T o shed further light onthe b ehavior ofthe c ovariance ofc alendar period squared pric e changes, w e d erive analytic resultsfor a trad ing d ay w ith 3 period s. Ifk= 3, thenthe sec ond term inthe formula forC ovh(¢Pt¡r)2 ;(¢Pt)2
i
,w hic h c orrespondsto the c ovariance ofthe c onditionalm eans,isid entic alforr= 1 and r = 2 .Asthe c onditionalc ovariance for
r= 1 exc eed sthe c onditionalc ovariance forr= 2 ,
C ovh(¢ Pt¡1)2 ;(¢ Pt)2
i
> C ovh(¢ Pt¡2)2 ;(¢Pt)2
i :
W e b eginb y estab lishingund er w hat c ond itionC ovh(¢Pt¡2)2 ;(¢ Pt)2
i
P rop osition6:Let Condition1 hold for period 3 w ithk= 3.For r < kthe covariance ofcalendar period squared price c hangesispositive.
P roof:See Appendix.
W hile Cond ition1 holdsnaturally for period 2 , there isno suc h naturalintuitionfor extending Condition1 to hold for period 3. Ifthere are 3 period sina trad ing d ay, then inthe last period ofa trad ing d ay enough ofthe inform ationofthe trad ers m ay have b eenrevealed that the expec ted squared pric e c hange for that period need not exc eed the unconditionalexpec ted squared pric e c hange for a period . IfCondition1 holdsonly for period 2 ,thenthe d ec line ofthe c ovariance inc alend ar period squared pric e c hangesc anb e d ram atic ally rapid .
More Rapid Decay of Covariance for Calendar Period Squared Price Changes
As w e shallsee, the heterosked astic ity inU2
i that arises from the movem ents inthe expec ted b id -askspread play anim portant role inexplainingthe persistence puz z le.IfU2
iis assum ed to b e homosked astic ,asinG allant,Hsieh,and T auc hen(199 1),thenthe c ovariance ofc alend ar period squared pric e c hangesisd rivenexc lusively b y the c ovariance inc alendar period trad es, and the persistence inthe c ovariance intrad esshould b e m atc hed b y the persistence inthe c ovariance insquared pric e c hanges. O ur m od elb reaksthe persistence link b ec ause one pred ic tionofour m od elisthat the variance ofUiisnot c onstant.Infac t, the variance ofUid ec linesastrad esoc c ur b ec ause informationisrevealed and the b id -ask spread d ec linesover tim e.Ifthe variance ofUid ec lines,thenthe c ovariance insquared pric e changesw illeventually b e lessthanthe c ovariance inthe numb er oftrad es.W e show that evend uringthe period inw hic h alltrad ersare w illingto trad e,the variance ofUid ec linesso that the new sarrivalhasa m ore persistent e®ec t onthe num b er oftrad esthanonsquared pric e c hanges.
Close stud y ofthe c ase inw hic h k= 3 revealsmuch ab out the relative d ec ay ofthe c orrelationinc alend ar period trad esand squared pric e c hanges.O ne ofthe em piric alfeatures b rought forw ard inthe introd uc tionisthat the c orrelationofc alendar period trad esd ec ays m ore slow ly thand oesthe c orrelationofc alend ar period squared pric e c hanges.Close stud y ofthe c ase inw hic h k= 3 revealsw hy thisisso. Asthe variance ofeither quantity is c onstant asthe lag ofthe c orrelationc hanges, the d ec ay ofthe c orrelationisd rivenb y the d ec ay ofthe c ovariance.
Suppose Condition1 holdsfor period 3 so that C ovh(¢ Pt¡r)2 ;(¢ Pt)2
i
ispositive for
P rop osition7:Let Condition1 hold for period 3 w ith k= 3. T he covariance, and hence the correlation, ofcalendar period squared price c hangesd ecaysmore rapid ly thanthe covariance ofcalendar period trad es.
P roof:T he proportionald ec ay ratesare revealed b y d irec t c alculationfrom the c ovari-ances.For c alendar period trad es
C ov[It¡1;It]¡C ov[It¡2;It]
C ov[It¡1;It]
= 1 2 ; so the c ovariance d ec linesb y ¯fty perc ent.
For c alendar period squared pric e c hangesthe c orrespond ingquantity is 1 2 + 1 C ovh(¢Pt¡1)2 ;(¢Pt)2 i ½1 3µ(1¡µ)[(¾1¡¾0)(¾2 ¡¾3) + (¾2 ¡¾0)(¾3¡¾0)] ¾ :
B ec ause Condition1 holds for period 3, the c ovariance b etw een(¢Pt¡1)2 and (¢ Pt)2 is positive.B y d e¯nition¾1 > ¾2 > ¾3> ¾0,so the sec ond term ispositive and the c ovariance
d ec linesb y m ore than¯fty perc ent.
IfCondition1 holdsfor period 2 , it ispossib le that the c ovariance ofc alendar period squared pric e c hangesat lag 2 isnegative. Asthe c ovariance ofc alendar period trad esis alw aysnonnegative, suc h a ¯ndingfurther enforc esthe more rapid d ec ay ofthe c ovariance ofc alendar period squared pric e c hanges.
Corrolary 8:Let Condition1 hold for period 2 w ithk= 3.T here isanopensubset of parameter valuesfor w hic hC ovh(¢ Pt¡2)2 ;(¢Pt)2
i
isnegative.
P roof:W e need onlyestab lish that there exist param eter valuesfor w hic h the c ovariance is negative. Consid er the set f¾jg3j= 1 = f2 0;7;3g so ¹¾3 = 10:Let ¾0 = 1. From the
d e¯nitionofthe c ovariance ofc alendar period squared pric e c hanges,C ovh(¢Pt¡2)2 ;(¢Pt)2
i
isnegative ifµ > :33.Letµ= :4 ,so Condition1 holdsfor period 2 .AsCondition1 c ontinues to hold for period 2 for anopenset ofvaluesofµab ove .4,Corollary 8 isestab lished .
Note that for the set ofparam eter valuesthat estab lish Corollary 8, Condition1 d oes not hold for period 3,asmust b e the c ase from Lem m a 4 .
4
Simul
ations
T o provid e anid ea ofthe patternofserialc orrelationthat isimplied b y our mod el, w e simulate sequencesoftrad esand the assoc iated pric e changesover a period ofm any trad ing d ays.
Let the unit ofec onom ic tim e b e one sec ond and assum e that informationisrevealed at the end ofeac h d ay, so that there are¿ = 2 880 trad ing opportunitiesinan8 hour trad ing d ay.Note that the m od elc ould also b e interpreted asfor exam ple,w ith a unit ofec onom ic tim e b eing2 sec ondsw ith new srevealed at the end ofthe sec ond d ay.9
Suppose´ = 30 ,so that c alend ar tim e period sare 30 sec ondsinlength.O ur simulated sam ple c onsistsof10 0 trad ing d ays, eac h ofw hic h hasprob ab ility ofnew sµ = :4 . G iven new s,the prob ab ility ofgood new sis±= :55.T o ensure that asym m etriesinthe mod elare not d riving our resultsw e set °= "= 1
2, so that the uninform ed are equally likely to b uy
or sell.A key param eter that rem ainsisthe proportionoftrad ersw ith private inform ation. W e initially set ® = :2 ,b ut vary thisparam eter,alongw ith´ invarioussim ulations.
F igure 1 c ontainsthe b id and ask from a sam ple offour trad ingd ays.O nthe ¯rst d ay,
S= sL,so the inform ed trad ed .O nthe sec ond and third d ays,S= s0,so the inform ed d id
not trad e,w hile onthe fourth d ay,S= sH ,so againthe informed trad ed .T he w id e spread at the start ofeac h d ay re°ec tsthe ad verse selec tionfac ed b y the spec ialist uncertainofthe partic ipationofinformed trad ers.Squared pric e c hangesare thuslarge at the b egining of the d ay.AsseeninF igure 3, squared pric e c hangesd ec line exponentially w ith the b id -ask spread asnew sisrevealed . Further, squared pric e c hangesare large onnew sd aysversus no new sd ays.F igure 2 show sthe trad e proc ess.O nd ays1 and 4 the inform ed trad e, so there are m ore trad es(the m eanis1.9).O nno new sd aysthe inform ed d o not trad e and the m eanisc orrespondingly low er (the m eanis1.1).
F igure 4 d epic tsthe resultsover 10 0 trad ing d ays.F igure 4 show sthe autoc orrelation functionsfor b oth trad esand squared pric e c hanges.T he magnitud e ofthe serialc orrelation intrad esismuch larger and m ore persistent, b arely d ec lining evenafter 10 lags. Serial c orrelationinsquared pric e c hangesismuc h sm aller and d ec linesto nearly zero after 4 lags. Aninteresting exam ple isto use the estim ationresultsin[8]. Ina sim ilar m od el, [8] ob tainestim atesof® = :172 ,"= .33,µ = :75,¿ = 96, and ± = :50 2 . W e also set ´ = 4 , equivalent to 5 m inute c alendar intervals. R esults are d etailed inF igure 5. T he serial c orrelationintrad esissmallb ut quite persistent. T he serialc orrelationinsquared pric e changesissm alland b arely persistsfor tw o lags.
9EasleyandO 'H ara(1 993)inamodelsimilartoours,assumethatatradingopportunityis ¯veminutes
andatradingdayis onedayforA shlandO il,basedonthenumberoftrades observeddaily(amaximum of 73).
5 Con
c l
usions
Inthis paper w e provid e anec onom ic m od elthat generates serialc orrelationintrad es and serialc orrelationinsquared pric e c hanges.Further,serialc orrelationintrad esism ore persistent thanserialc orrelationinsquared pric e c hanges.W e propose that serialc orrelation intrad esarisessimply from the entry and exit ofinform ed trad ers, w ho rec eive a private signal.G iventhat informed trad ersare trad ing inthe c urrent period , inform ed trad er w ill m ost likely trad e inthe follow ing period , w hic h generatesserialc orrelationintrad es.T he serialc orrelationintrad esisquite strong and persistent. Inthe sim ulationsafter 30 lags the c orrelationw asstillab ove .8.
Inour m od elserialc orrelationintrad es generates serialc orrelationinsquared pric e changes. G iventhat the inform ed trad ersare trad ing, there ism ore variance insquared pric e changes sim ply b ec ause there are m ore trad es ina c alend ar period . M ore trad es im pliesthat the pric e c hange isthe sum ofm ore random trad es,w hich inturnimpliesthat the pric e change hasa larger variance.B ec ause there isserialc orrelationintrad es, there isserialc orrelationinsquared pric e c hanges.How ever, there isanad d itionale®ec t onthe serialc orrelationinsquared pric e c hanges,the d ec line inthe b id -ask spread .Alltrad esare at the b id -askspread ,hence expec ted pric e c hangesare b ounded b ythe b id -askspread .T he b id -askspread d ec linesaslearningproc eed s,w hic h red uc esthe variance and the persistence ofthe serialc orrelationinsquared pric e c hanges.G iventhere are m ore trad esina c alendar period , there are most likely more trad esinthe next c alendar period ,w hic h implieshigher variance inb oth period s. How ever, the trad esinthe sec ond c alendar period are from a rand om variab le w ith a sm aller variance,d ue to the sm aller b id -askspread .Hence the serial c orrelationis sm aller and less persistent. O ur simulations ind ic ate that the c orrelation c oe± c ient at one lagis.35,and d ec linesto .1 after 30 lags.Hence our m od elreplic atesthe ob served em piric alfeaturesofthe d ata and explainsserialc orrelationthrough the entryand exit ofinformed trad ersand the assoc iated revelationofinform ationinthe pric es.
W hat inform ationset should b e used to form c ond itionalexpec tationsof(¢Pt)2? T he ab ove resultsindic ate that pred ic tionofthe variance ofpric e c hangesd epend sonpred ic tion ofthe entryand exit ofinform ed trad ers.Spec i¯c ally,the c onditionalvariance ofstoc kpric es d ependsonthe previousnumb er oftrad es,b ut d oesso ina nonlinear w ay.T he prob ab ility ofinform ationarriving,µ,playsanintriguingrole.Serialc orrelationishighest inm arketsin w hich new sarrivesinfrequently,w here the arrivalofnew sc ausesa large numb er ofinform ed to enter.
Future researc h includ esexpand ing our m od elto the possib ility ofmultiple trad ingpe-riod sw ith randomly arriving private new sevents.M ultiple trad ing peingpe-riod sallow sfor the fully inform ed not to trad e (ifno new sisreleased ) hence the serialc orrelationintrad esw ill
R ef
erences
[1] Andersen, T ., 199 6, \R eturnVolatility and T rad ing Volume: AnInform ationF low InterpretationofStoc hastic Volatility"J ournalofF inance51,169-2 0 4 .
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[3] B ollerslev,T .,R .E ngle,and D.Nelson,19 93,\AR CH M od els"inHandbookofE cono-metric s,Volum e 4 ,Am sterd am : North-Holland.
[4 ] B rock, W .and B .LeB aron, 199 6, \A Dynam ic Struc turalM od elfor Stoc k R eturn Volatility and T rad ingVolume"R eview ofE conomic sand Statistic s78,9 4 -110 .
[5] Clark, P ., 19 73, \A Sub ord inated Stoc hastic P roc essM od elw ith F inite Variance for Spec ulative P ric es"E conom etrica4 1,135-159.
[6] DenHaan,W .and Spear,S.,199 7,\VolatilityClusteringinR ealInterest R ates: T heory and E vid ence"m anuscript,U .C.SanDiego.
[7] E asley,D.,and M .O 'Hara,1992 ,\T im e and the P roc essofSec urityP ric e Ad justm ent"
J ournalofF inance4 7,577-60 5.
[8] E asley, D., N.K iefer, and M .O 'Hara, 199 3, \O ne Day inthe Life ofa Very Com m on Stock"manuscript,CornellU niversity.
[9] E ngle,R .,and Ng,V.,and R othschild,M .,19 90 ,\Asset P ric ingW ith a Fac tor-AR CH Covariance Struc ture: E m piric alE stim atesFor T reasuryB ills"J ournalofE conometric s
4 5,2 13-37.
[10 ] French,D.and R .R oll,19 86,\Stoc kR eturnVariances: T he ArrivalofInformationand the R eac tionofT rad ers"J ournalofF inancialE conomic s17,5-2 6.
[11] G allant, R ., D.Hsieh, and G .T auc hen, 19 91, \O nF itting a R ec alcitrant Series: T he pound/d ollar E xc hange R ate,1974 -1983"inNonparam etric and Semiparametric M eth-od sinE conometric sand Statistic s,W .B arnett,J .P ow ell,and G .T auc hened s.,Cam-b rid ge: Cams.,Cam-b rid ge U niversity.
[12 ] G losten, L.and P .M ilgrom , 1985, \B id , Ask and T ransac tionP ric esina Spec ialist M arket w ith Heterogeneously Inform ed T rad ers"J ournalofF inancialE conomic s14 , 71-10 0 .
[13] Harris, L, 1987, \T ransac tionData T estsofthe M ixture ofDistrib utionsHypothesis"
J ournalofF inancialand Quantitative Analysis2 2 ,12 7-14 1.
[14 ] Lange,S.,19 98.\M od elingAsset M arket Volatility ina Sm allM arket: Ac c ountingfor Non-synchronousT rad ing E ®ec ts,"J ournalofInternationalF inancialM arkets, Insti-tutionsand M oney,9,1-18.
[15] Sargent,T .,19 93,B ounded R ationality inM ac roeconom ic s,ClarendonP ress,O xford . [16] Shorish,J .and Spear,S.,199 6\Shakingthe T ree: AnAgency-T heoretic M od elofAs-set P ric ing"T ec hnic alR eport 1996-E 1,G rad uate SchoolofIndustrialAd m inistration, Carnegie M ellonU niversity.
[17] Steigerw ald, D., 199 7, \M ixture M od els and ConditionalHeterosked astic ity" manu-script,U .C.Santa B arb ara.
[18] T auchen, G ., H.Zhang, and M .Liu, \Volume, Volatility, and Leverage: A Dynam ic Analysis"J ournalofE conometric s74 ,177-2 0 8.
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6 Appen
d ix
Derivation of Learning Formulae
W e explic ilty d erive the learningform ula foryigiventhat trad eritrad esat the ask.All other learningformulae follow the sam e logic .From B ayesrule
P(Sm = sH jZ i¡1;Ci= cA) = P(Sm = sH jZ i¡1)P(Ci= cAjSm = sH ) P j=sL;sH;s0 P(Sm = jjZ i¡1)P(Ci= cAjSm = j) :
W e must c alculateP(Ci= cAjSm = j) for j= sL;sH ;s0.Ifthe inform ed rec eive the signal
sH ,thenthe inform ed w illtrad e at the ask.Further,the frac tion"(1¡°) ofthe uninform ed w illalso trad e at the ask.Hence
P(Ci= cAjSm = sH ) = ® + (1¡®)"(1¡°):
Ifthe inform ed rec eive the signalsL or d o not rec eive a signal, thenthe inform ed w illnot trad e at the ask.B ec ause only the uninform ed trad e at the ask ifSm equalssL ors0,b oth
P(Ci= cAjSm = sL) and P(Ci= cAjSm = s0) equal(1¡®)"(1¡°).
Proof of Theorem 1
T he learning formulae for xiand yiare nonlinear in(xi¡1;yi¡1) and are not rec ursive,
w hich m ake it d i± c ult to d eterm ine the asym ptotic b ehavior ofxiand yi. B ec ause the d enom inator ofthe learningformula,c onditionalonthe d ec isionoftrad eri,isid entic alfor
xi,yiand 1¡xi¡yi, the learning formulae for ratiosofxiand yiare linear inratiosof (xi¡1;yi¡1) and rec ursive.W e w ork w ith ratiosxiand yiand b eginw ith the c aseSm = sH , for w hic h the relevant ratiosare xi
yiand
1¡xi¡yi
yi .Consid er
xi
yi.Iftrad eritrad esat the ask
xi yi = xi¡1 yi¡1 ¢ P(Ci= cAjSm = sL) P(Ci= cAjSm = sH ) :
Iftrad eritrad esat the b id ,thenthe expressionfor xi
yiisasab ove w ithCi= cA replac ed b y
Ci= cB.Iftrad erid oesnot trad e
xi
yi
= xi¡1
yi¡1
;
b ec auseP(Ci= cN jSm = sL) equalsP(Ci= cN jSm = sH ).W e have ln à xi yi ! = ln à x0 y0 ! + nAln " P(Ci= cAjSm = sL) P(Ci= cAjSm = sH ) # + nB ln " P(Ci= cBjSm = sL) P(Ci= cBjSm = sH ) # ;
w herenA isthe numb er ofthe ¯rst itrad ing opportunitiesfor w hic h there w asa trad e at the ask,nB isthe numb er ofthe ¯rst itrad ing opportunitiesfor w hic h there w asa trad e at the b id ,and nN isthe numb er ofthe ¯rst itrad ingopportunitiesfor w hic h there w asno trad e.
B ec ause the trad er arrivalproc essisi.i.d ., 1 iln à xi yi ! as ! X j=cA;cB P(Ci= jjSm = sH )ln " P(Ci= jjSm = sL) P(Ci= jjSm = sH ) # (5) asi! 1. T he right-hand sid e of(5), multiplied b y m inusone, isa m easure ofd istance b etw eenthe prob ab ility m easureP(¢jSm = sH ) and the prob ab ility m easure P(¢jSm = sL), w hich is term ed the entropy ofP(¢jSm = sH ) relative to P(¢jSm = sL) and is d enoted
J(sH ;sL).B yc onstruc tionthe entropyisnonnegative and equalsz ero onlyifthe prob ab ility m easuresd i®er solely ona set w ith m easure z ero.Hence
1 iln à xi yi ! as ! ¡J(sH ;sL)< 0 asi! 1,so that xi yib ehavesase ¡iJ(sH;sL).T husxi
yic onvergesalmost surely to z ero at the
exponentialrateiJ(sH ;sL).A sim ilar argum ent show sthat 1¡xyii¡yic onvergesalm ost surely to zero at the exponentialrateiJ(sH ;sN ).
IfSm = sH , thenxyii as ! 0 and 1¡xi¡yi yi as ! 0 asi! 1.
IfSm = sL, thenthe relevant ratiosare yxiiand 1¡xxii¡yi. IfSm = s0, thenthe relevant
ratiosare xi
1¡xi¡yi and
yi
1¡xi¡yi. Againthe fac t that the trad er arrivalproc essisi.i.d. is
su± c ient to estab lish that
ifSm = sL, thenyxii!as 0 and 1¡xxii¡yi!as 0, ifSm = s0, then1¡xxii¡yi as ! 0 and yi 1¡xi¡yi as ! 0, asi! 1.
From the c onvergence propertiesofthe ratios, w e c aneasily d ed uc e the c onvergence propertiesofxiand yi.W e c ontinue w ith the c aseSm = sH and note that similar argum ents hold forSm = sL and Sm = s0.T he statem ent 1¡xyii¡yi!as 0 isequivalently w rittenas
1 yi¡ xi yi¡ 1!as 0: (6) B ec ause xi yi as
! 0 ,the statem ent (6) isequivalent to 1
yi¡
w hich d irec tly im pliesyi!as 1.Ifyi!as 1, thenthe statem ent xyii!as 0 im pliesxi!as 0 .From the d e¯nitionofAiand Bi,ifxi
as ! 0 and yi as ! 1,thenAi as ! vHm and Bi as ! vHm. Proof of Theorem 2
T he proofisa straightforw ard ,b ut ted iousc alculationofthe c orrelation.B y d e¯nition, the c ovariance is
C ov(It¡r;It) = E(It¡rIt)¡EIt¡r ¢EIt:
Ifr¸k,thenthe independence ofthe signalproc essimpliesthat It¡r isindependent of
It,so E(It¡rIt) = EIt¡r ¢EItand the c ovariance iszero.
Ifr < k,thenthere are three possib le c onditionalexpec tationsof(It¡rIt).F irst, ifIt¡r and Itare m easured onthe sam e trad ingd ay the c onditionalexpec tationof(It¡rIt) is
µ¹21+ (1¡µ)¹20;
w hich oc c urs w ith prob ab ility k¡r
k . Sec ond, ifIt¡r and Itare m easured onc onsec utive trad ingd aysand Sm+ 16= s0,the c ond itionalexpec tationof(It¡rIt) is
µ¹21+ (1¡µ)¹0¹1;
w hic h oc c ursw ith prob ab ilityrkµ.T hird ,ifIt¡r and Itare m easured onc onsec utive trad ing d aysand Sm+ 1= s0,the c onditionalexpec tationof(It¡rIt) is
µ¹0¹1+ (1¡µ)¹20;
w hich oc c ursw ith prob ab ilityrk(1¡µ).W e c omb ine the three c onditionalexpec tationsto yield E(It¡rIt) = k¡r k h µ¹21+ (1¡µ)¹20i+ r k[µ¹1 + (1¡µ)¹0] 2 :
B ec ause the proc essfor c alendar period trad esisstationary,EIt¡r equalsEIt.Asnoted in the text EIt= µ¹1 + (1¡µ)¹0; so C ov(It¡r;It) = k¡r k µ(1¡µ)(¹1¡¹0) 2 = k¡r k µ(1¡µ)(® ´) 2 :
Comb iningthe tw o possib le c asesforr relative to kyields C ov(It¡r;It) = ( µ(1¡µ)(® ´)2 hk¡r k i r < k 0 r ¸k (7)
Comb ining the c ovariance and variance ofItgivenb y (1) gives the d esired c orrelation. B ec ause allterm sare postive forr < k,the c orrelationispositive.
Proof of Theorem 4
For the proofofT heorem 4 , let CN representCi= cN inthe c onditioning information set.W e have E(UijZ i¡1) = P(Ci= cA)(Ai¡E(Vm jZ i¡1)) + P(Ci= cB)(Bi¡E(Vm jZ i¡1)) +P(Ci= cN )¢(E[Vm jZ i¡1;CN ]¡E(Vm jZ i¡1)) = P(Ci= cA)Ai+ P(Ci= cB)Bi+ P(Ci= cN )E[Vm jZ i¡1;CN ]¡E(Vm jZ i¡1) = E(Vm jZ i¡1)¡E(Vm jZ i¡1) = 0:
Insimilar fashionw e ¯nd that Uiisa serially uncorrelated random variab le.Let h and
ib e d istinct valuesw ithh < i,
E(UhUijZ i¡1) = EZi¡1fUh [E(Vm jZ i¡1)¡E(Vm jZ i¡1)]g= 0:
R ec allE(U2
ijZ i¡1) equals
P(Ci= cA)(Ai¡E(Vm jZi¡1))2 + P(Ci= cB)(Bi¡E(Vm jZi¡1))2
+P(Ci= cN )(E[Vm jZ i¡1;CN ]¡E(Vm jZ i¡1))2:
T he upper b ound for the c onditionalvariance is
E³Ui2jZ i¡1 ´ · P(Ci= cA)( ~Ai¡E(Vm jZ i¡1))2 + P(Ci= cB)( ~Bi¡E(Vm jZ i¡1))2 +P(Ci= cN )(E[Vm jZ i¡1;CN ])¡E(Vm jZ i¡1))2 · [P(Ci= cA) + P(Ci= cN )]( ~Ai¡E(Vm jZ i¡1))2 + [P(Ci= cB) + P(Ci= cN )]( ~Bi¡E(Vm jZ i¡1))2 · ( ~Ai¡E(Vm jZ i¡1))2 + ( ~Bi¡E(Vm jZi¡1))2 · h( ~Ai¡E(VmjZi¡1))¡( ~Bi¡E(Vm jZ i¡1)) i2 = hAei¡Bei i2 ;
w here the ¯rst inequality follow sfrom the d e¯nitionof~Aiand ~Biand the fourth inequality follow sfrom Bi·E[Vm jZi¡1]·Ai.Note that the unconditionalvariance isimm ed iately ob tained from J ensen'sinequality
EUi2 ·E³Aei¡Bei
´2
·³EAei¡EBei
´2
:
T o ob tainthe low er b ound for the c onditionalvariance w e c onsid er three c ases. If
Bi·E[Vm jZ i¡1;CN ]·Ai,then E³Ui2jZ i¡1 ´ ¸ P(Ci= cA)(Ai¡E(Vm jZ i¡1))2 + P(Ci= cB)(Bi¡E(Vm jZ i¡1))2 ¸ P(Ci= cA)P(Ci= cB) ³ ~ Ai¡B~i ´2 ;
w here the sec ond inequality follow sfrom Lemm a 4 .1, w hic h isprovenb elow .IfBi·Ai·
E[Vm jZ i¡1;CN ],then E³Ui2jZ i¡1 ´ ¸ P(Ci= cN )(E[Vm jZ i¡1;CN ]¡E(Vm jZ i¡1))2 + P(Ci= cB)(Bi¡E(Vm jZ i¡1))2 ¸ P(Ci= cN )P(Ci= cB) ³ ~ Ai¡B~i ´2 ;
w here the sec ond inequality follow sfrom Lem m a 4 .1.IfE[Vm jZ i¡1;CN ]·Bi·Ai,then
E³Ui2jZ i¡1 ´ ¸ P(Ci= cA)(Ai¡E(Vm jZ i¡1))2 + P(Ci= cN )(E[Vm jZ i¡1;CN ]¡E(Vm jZ i¡1))2 ¸ P(Ci= cA)P(Ci= cN ) ³ ~ Ai¡B~i ´2 ;
w here the sec ond inequality follow sfrom Lem m a 4 .1. T he unconditionalvariance thussatis¯es: E · P(Ci= cA)P(Ci= cB)P(Ci= cN ) ³ e Ai¡Bei ´2¸ ·E³Ui2´
Lem m a 4 .1:Let c2[0;1].For any pair ofrealnumbersa and b c(1¡c)(a+ b)2 ·ca2 + (1¡c)b2:
P roof.T he left sid e ofthe inequality isc(1¡c)(a2 + b2 + 2ab), w hic h w hensub trac ted
from b oth sid esc onvertsthe inequality to
Proof of Proposition 5
W e ¯rst c arefully d eriveC ovh(¢ Pt¡1)2 ;(¢ Pt)2
i
fork= 2 .Let N = 1 ift¡1 isthe ¯rst period ofa trad ingd ay and let N = 2 ift¡1 isthe sec ond period .T hen
Eh(¢ Pt¡1)2 jN = 1 i = µ¾1+ (1¡µ)¾0 = E h (¢Pt)2 jN = 2 i ; Eh(¢ Pt¡1)2 jN = 2 i = µ¾2 + (1¡µ)¾0 = E h (¢Pt)2 jN = 1 i ; Eh(¢ Pt)2 i = µ³¾1 2 + ¾2 2 ´ + (1¡µ)¾0:
B ec auseN isequally likely to take the values1 or 2 ,the c onditionalc ovariance is
1 2fE h (¢Pt¡1)2 (¢ Pt)2 jN = 1 i ¡Eh(¢Pt¡1)2 jN = 1 i Eh(¢ Pt)2 jN = 1 i g + 12fEh(¢Pt¡1)2 (¢Pt)2 jN = 2 i ¡Eh(¢ Pt¡1)2 jN = 2 i Eh(¢Pt)2 jN = 2 i g: (A4:1) From the formulae for the expec ted c alendar period squared pric e c hange giventhe value of
N ,(A4:1) equals
1
2f[µ¾1¾2 + (1¡µ)¾02]¡(µ¾1+ (1¡µ)¾0)(µ¾2 + (1¡µ)¾0) + µ[µ¾2¾1+ (1¡µ)¾2¾0]
+ (1¡µ)[µ¾0¾1 + (1¡µ)¾02]¡(µ¾2 + (1¡µ)¾0)(µ¾1+ (1¡µ)¾0)g;
w hich issim pli¯ed as
1
2µ(1¡µ)(¾1¡¾0)(¾2 ¡¾0): (A4:2 )
T he c ovariance ofthe c onditionalmeansis
Ef³E(¢Pt¡1)2 ¡E h (¢ Pt¡1)2 jN i´³ E(¢ Pt)2 ¡E h (¢ Pt)2 jN i´ g; w hich equals P(N = 1)f³E(¢Pt¡1)2 ¡E h (¢Pt¡1)2 jN = 1 i´³ E(¢Pt)2 ¡E h (¢ Pt)2 jN = 1 i´ g +P(N = 2 )f³E(¢ Pt¡1)2 ¡E h (¢Pt¡1)2 jN = 2 i´³ E(¢Pt)2 ¡E h (¢Pt)2 jN = 2 i´ g: Note ³ E(¢Pt¡1)2 ¡E h (¢ Pt¡1)2 jN = 1 i´ = µ(¹¾2 ¡¾1); ³ E(¢Pt)2 ¡E h (¢Pt)2 jN = 1 i´ = µ(¹¾2 ¡¾2); ³ E(¢Pt¡1)2 ¡E h (¢ Pt¡1)2 jN = 2 i´ = µ(¹¾2 ¡¾2); ³ E(¢Pt)2 ¡E h (¢Pt)2 jN = 2 i´ = µ(¹¾2 ¡¾1):
h µ³¾1 2 + ¾2 2 ¡¾1 ´i h µ³¾1 2 + ¾2 2 ¡¾2 ´i = µ2 ³¾1 2 ¡ ¾2 2 ´³ ¾2 2 ¡ ¾1 2 ´ : (A4:3)
From (A4:2 ) and (A4:3) theC ovh(¢Pt¡1)2 ;(¢Pt)2
i equals 1 2µ(1¡µ)(¾1¡¾0)(¾2 ¡¾0) + µ 2 ³¾1 2 ¡ ¾2 2 ´³ ¾2 2 ¡ ¾1 2 ´ :
B ec ause ¾1 > ¾2, the sec ond term ofthe c ovariance is negative (w hile the ¯rst term is
positive) and the c ovariance ispositive if
(1¡µ)(¾1¡¾0)(¾2 ¡¾0)> µ2 (¾1¡¾2)2 : (A4:4 )
F irst,b y inspec tion (¾1¡¾0)> (¾1¡¾2):
T husto verify (A4:4 ),w e need only show (1¡µ)(¾2 ¡¾0)> µ2 (¾1¡¾2):
B ec ause µ2 (¾1 ¡¾2) = µ(¹¾2 ¡¾2),to verify the prec ed inginequality,w e must show
(1¡µ)(¾2 ¡¾0)¡µ(¹¾2 ¡¾2)> 0:
Condition1 implies
(1¡µ)(¾2 ¡¾0)> µ(1¡µ)(¹¾2 ¡¾0):
Hence
(1¡µ)(¾2 ¡¾0)¡µ(¹¾2 ¡¾2)> µ(1¡µ)(¹¾2 ¡¾0)¡µ(¹¾2 ¡¾2):
T he right-hand sid e ofthe prec ed inginequality equals
µ[(¾2 ¡¾0)¡µ(¹¾2 ¡¾0)];
and Cond ition1 implies
(¾2 ¡¾0)¡µ(¹¾2 ¡¾0)> 0:
6.
1
Proof of Proposition 6
Ifk= 3,thenthe c ovariance ofc alend ar period squared pric e c hangesislarger forr = 1 thanfor r = 2 , so P roposition6 isestab lished ifC ovh(¢Pt¡2)2 ;(¢Pt)2
i
haveC ovh(¢Pt¡2)2 ;(¢Pt)2
i
equals
1
3µf(1¡µ)(¾1¡¾0)(¾3¡¾0)+µ(¹¾3¡¾1)(¹¾3¡¾3)+µ(¹¾3¡¾2)(¹¾3¡¾1)+µ(¹¾3¡¾3)(¹¾3¡¾2)g
T he ¯rst term ispositive, the sec ond negative, and the rem aing tw o term sare opposite in signand d epend onthe signof(¹¾3¡¾2).W e c onsid er eac h ofthe three c ases: (¹¾3¡¾2)> 0 ,
(¹¾3¡¾2)< 0 ,and (¹¾3¡¾2) = 0 inturn.
Case 1: (¹¾3¡¾2)> 0
If(¹¾3¡¾2)> 0 , then¾1 > ¾¹3> ¾2 > ¾3> ¾0. De¯ne d1 = ¾1 ¡¾¹3,d2 = ¹¾3¡¾2,
d3= ¾2 ¡¾3,and d4 = ¾3¡¾0.T heC ov h (¢ Pt¡2)2 ;(¢Pt)2 i ispositive if (1¡µ)(¾1¡¾0)(¾3¡¾0)+µ(¹¾3¡¾3)(¹¾3¡¾2)> jµ(¹¾3¡¾1)(¹¾3¡¾3) + µ(¹¾3¡¾2)(¹¾3¡¾1)j;
w hich isequivalently expressed as (1¡µ)³P4j= 1dj ´ d4 + µ(d2 + d3)d2 > µd1(2d2 + d3): (A5:1) R ew rite (A5:1) as d1(1¡µ)d4 + (1¡µ)³P4j= 2 dj ´ d4 + µ(d2 + d3)d2 > d1µ(d2 + d3) + µd1d2:
IfCond ition1 holdsfor period 3,then (1¡µ)d4 > µ(d2 + d3);
and (A5:1) issatisi¯ed if
d2 (1¡µ)d4 + µd2 (d2 + d3) + (1¡µ)³P4j= 3dj
´
d4 > µd1d2: (A5:2 )
IfCond ition1 holdsfor period 3,then (1¡µ)d4 > µd2;
and (A5:2 ) issatis¯ed if
µd2 (2d2 + d3)¡µd1d2 = µd2 (2d2 + d3¡d1) = 0
From the d e¯nitionof¹¾3,P3j= 1(¾j¡¾¹3) = d1¡2d2 ¡d3,so
(2d2 + d3¡d1) = 0:
If(¹¾3¡¾2)< 0 , then¾1 > ¾2 > ¾¹3 > ¾3> ¾0. De¯ne d1 = ¾1 ¡¾2,d2 = ¾2 ¡¾¹3, d3= ¹¾3¡¾3,and d4 = ¾3¡¾0.T heC ov h (¢ Pt¡2)2 ;(¢Pt)2 i ispositive if (1¡µ)(¾1¡¾0)(¾3¡¾0)+µ(¹¾3¡¾1)(¹¾3¡¾2)> jµ(¹¾3¡¾1)(¹¾3¡¾3) + µ(¹¾3¡¾2)(¹¾3¡¾3)j;
w hich isequivalently expressed as (1¡µ)³P4j= 1dj
´
d4 + µ(d1+ d2)d2 > µd3(2d2 + d1): (A5:3)
From the d e¯nitionof¹¾3,
2d2 + d1 = d3;
so (A5:3) issatis¯ed if (1¡µ)d3d4 ¡µd23> 0:
Note (1¡µ)d3d4 ¡µd23= d3(d4 ¡µ(d3+ d4)).IfCondition1 holdsfor period 3
¾3¡¾0 > µ(¹¾3¡¾0);
w hich isequivalently expressed as
d4 > µ(d3+ d4): Case 3: (¹¾3¡¾2) = 0 T heC ovh(¢Pt¡2)2 ;(¢ Pt)2 i ispositive if (1¡µ)(¾1¡¾0)(¾3¡¾0)> jµ(¹¾3¡¾1)(¹¾3¡¾3)j:
F irst,b y inspec tion (¾1¡¾0)> (¾1¡¾¹3):
Sec ond,ifConditon1 holdsfor period 3 (1¡µ)(¾3¡¾0)> µ(¹¾3¡¾3):