Price trends and patterns in technical analysis: A theoretical and empirical examination

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1 Prce rends and paerns n echncal analyss: A heorecal and emprcal examnaon Geoffrey C. Fresen a*, Paul A. Weller b, Lee M. Dunham c a Deparmen of Fnance, College of Busness, Unversy of Nebraska Lncoln, Lncoln, NE 68508, USA b Deparmen of Fnance, enry B. Tppe College of Busness, Unversy of Iowa, Iowa Cy, IA 52242, USA c Deparmen of Fnance, College of Busness, Creghon Unversy, Omaha, NE 68178, USA Ths verson: December 5, 2008 Whle many echncal radng rules are based upon paerns n asse prces, we lack convncng explanaons of how and why hese paerns arse, and why radng rules based on echncal analyss are profable. Ths paper provdes a model ha explans he success of ceran radng rules ha are based on paerns n pas prces. We pon o he mporance of confrmaon bas, whch has been shown o play a key role n oher ypes of decson makng. Traders who acqure nformaon and rade on he bass of ha nformaon end o bas her nerpreaon of subsequen nformaon n he drecon of her orgnal vew. Ths produces auocorrelaons and paerns of prce movemen ha can predc fuure prces, such as he head-and-shoulders and double-op paerns. The model also predcs ha sequenal prce jumps for a parcular sock wll be posvely auocorrelaed. We es hs predcon and fnd ha jumps exhb sascally and economcally sgnfcan posve auocorrelaons. JEL classfcaon: G11; G12; G14 Keywords: Techncal analyss; ead-and-shoulders; Tradng rules; Confrmaon bas; Jump auocorrelaons We hank semnar parcpans a Iowa Sae Unversy and he 2007 FMA Annual Conference. * Correspondng auhor. Tel ; fax: E-mal addresses: gfresen2@unl.edu (G. Fresen), paul-weller@uowa.edu (P. Weller), leedunham@creghon.edu (L. Dunham). 1

2 1. Inroducon There s now convncng evdence ha sock prces dsplay shor-erm momenum over perods of sx monhs o a year and longer-erm mean reverson (De Bond and Thaler, 1985; Chopra, Lakonshok and Rer, 1992; Jegadeesh and Tman, 1993). There s also evdence of economcally sgnfcan prce reversals over shor me horzons of a week o a monh (Jegadeesh, 1990; Lehmann, 1990; Jegadeesh and Tman, 1995; Guerrez and Kelley, 2008). 1 Ths evdence provdes suppor for radng rules desgned o deec perssen rends n asse prces. Research has shown ha such rules have predcve power n equy markes (Brock, Lakonshok and LeBaron, 1992) and n foregn exchange markes (Dooley and Shafer, 1983; Sweeney, 1986; Levch and Thomas, 1993; Neely, Weller and Dmar, 1997; Dueker and Neely, 2007). The use of echncal sgnals based on prce paerns has receved less academc aenon, despe he fac ha hese sgnals are wdely used by praconers (Taylor and Allen, 1992; Lu and Mole, 1998; Cheung and Chnn, 2001). A presen, we lack heorecal models ha can explan he presence of paern-based radng rules, hough several emprcal sudes sugges ha such rules may be profable. Chang and Osler (1999) examne he profably of usng he head-and-shoulders paern n he foregn exchange marke o predc changes of rend, and fnd evdence of excess reurns for some currences bu no ohers. Lo, Mamaysky and Wang (2000) develop a paern deecon algorhm based on kernel regresson. They apply hs mehodology o denfy a varey of echncal prce paerns ncludng head-and-shoulders n he U.S. sock marke over he perod They fnd sascal evdence ha here s poenally useful nformaon conaned n mos of he paerns hey consder. Savn, Weller and 1 Conrad, Kaul and Nmalendran (1991) demonsrae ha bd-ask bounce explans some of hs reurn reversal. Cooper (1999) and Subrahmanyam (2005) fnd ha mcrosrucure ssues canno fully explan he documened reurn reversal. 2

3 Zvngels (2007) show ha a modfed verson of he algorhm of Lo, Mamaysky and Wang appled o he head-and-shoulders paern has subsanal predcve power for U.S. sock reurns over perods of one o hree monhs. 2 The objecve of hs paper s o presen a heorecal model ha provdes an explanaon for he observed auocorrelaon paerns n asse reurns and for he documened success of boh rend-followng and paern-based echncal radng rules. We do hs by nroducng a sngle cognve bas no he model, ha of confrmaon bas. The bas s a phenomenon ha has been exensvely documened n expermenal sudes. I refers o he search for, or he nerpreaon of evdence n ways ha favor exsng belefs or expecaons. I has been descrbed as perhaps he bes known and mos wdely acceped noon of nferenal error o come ou of he leraure on human reasonng. (Evans, 1989, p.41 quoed n Nckerson, 1998). In our model, nformaon arrval s modeled wh sgnals of varous magnudes, arrvng a dfferng frequences. Large, nfrequenly observed sgnals are nerpreed raonally by nvesors. owever, nvesors nerpreaon of less nformave sgnals (whch arrve more frequenly) s based by he recenly observed large sgnals. The model generaes prce paerns, mos noably he head-and-shoulders paern, ha have he predcve power for fuure sock reurns clamed by echncal analyss. The model hus provdes a heorecal foundaon for 2 The value of usng echncal radng rules based upon pas prces s sll an open emprcal queson. Jegadeesh (2000), n hs dscusson of Lo e al. (2000) pons ou ha here s no evdence of sgnfcan one-day reurns afer he denfcaon of echncal ndcaors. Ths fndng s confrmed for he UK sock marke n Dawson and Seeley (2003), whle Marshall, Young and Rose (2006) fnd ha candlesck radng sraeges do no have value for Dow Jones Indusral Average Socks. owever, Savn, Weller and Zvngels (2007) fnd ha wh longer holdng perods evdence of subsanal predcve power emerges. Bessembnder and Chan (1998) and Allen and Karjalanen (1999) sugges ha gross profs are avalable from echncal radng bu are nsuffcen o cover ransacons coss. Osler (2003) documens ha order cluserng of sop-loss and ake-prof orders a round numbers provdes a jusfcaon for nraday echncal analyss n he currency markes. Kavajecz and Odders-Whe (2004) conclude ha nraday echncal analyss capures changes n he sae of he lm order book and may add value by locang lqudy n he lm order book ha allows raders o place sraegc lm orders wh beer execuon and lower ransacon coss. In conras, Marshall, Cahan and Cahan (2008) nvesgae he profably of over 5,000 echncal radng rules usng nraday daa on Sandard and Poor s Deposory Receps (SPDRs) and conclude ha echncal analyss s no profable afer daa snoopng bas s aken no accoun. 3

4 several prce paerns commonly used by echncal analyss. The model also produces he welldocumened paern of prce momenum whch can be exploed by rend-followng echncal rules such as hose based on he comparson of shor- and long-run movng averages. In addon, our model makes several predcons. Frs, reurn auocorrelaons are negave over very shor horzons, posve over nermedae horzons, and become negave agan over long horzons. Ths feaure of he model conforms o he emprcal properes of U.S. equy prces descrbed above. To our knowledge, our model s he frs o smulaneously capure all hree of hese paerns n reurn auocorrelaons, and provdes a smple alernave o he mcrosrucure-based explanaon for negave shor-horzon auocorrelaons. Our model also produces a sharp predcon ha he me seres of jumps n he prce seres should be posvely auocorrelaed. So far as we know, hs s a new and unesed emprcal predcon. We provde emprcal evdence ha confrms he predcon of our model ha sequenal prce jumps n equy prces are posvely auocorrelaed. Specfcally, we ulze he sascal b-power varaon esmaon echnque o denfy all sascally sgnfcan jumps n he daly prce seres of he ndvdual componen socks of he S&P 100 Index over he sample perod We fnd ha sequenal prce jumps exhb sascally and economcally sgnfcan posve auocorrelaons, and ha hese auocorrelaons decay a a rae ha s also conssen wh he model. Our model presens an alernave momenum explanaon o he gradual nformaon dffuson hypohess of ong and Sen (1999). In her model, newswachers rade on fundamenal nformaon whle momenum raders make rades based on pas prce movemens. Fundamenal nformaon dffuses gradually across he newswachers and hs causes prces o underreac and dsplay posve auocorrelaon. The auocorrelaon provdes ncenves for 4

5 momenum raders whose smple radng sraeges based on pas prces evenually drve prces above fundamenal value, leadng o negave auocorrelaons over longer horzons. The agens n her model are boundedly raonal n ha her decsons do no make use of all relevan nformaon. Our approach s closer n spr o ha of Danel, rshlefer and Subrahmanyam (1998) n ha we assume ha decsons are affeced by a psychologcal bas. Our model seup s dfferen and begns wh he arrval of a large pece of nformaon ha s mmedaely and raonally mpounded no prces. Ths news bases nvesors nerpreaon of laer nformaon, and so n a sense s dffusve n ha connues o affec fuure prce changes. Lke ong and Sen (1999) our model predcs ha ceran radng sraeges based on pas prces can be profable. Two noable dfferences are ha our model predcs negave auocorrelaon n he very shor-run, and also explans why ceran echncal prce paerns forecas fuure reurns. Our model and emprcal ess also complemen he recen emprcal work of Guerrez and Kelley (2008). They documen negave weekly auocorrelaons mmedaely afer exreme nformaon evens, bu fnd ha momenum profs emerge several weeks afer an exreme reurn and perss over he remander of he year. Moreover, hs momenum easly offses he bref and nal reurn reversal. Our model produces predcons conssen wh hs fndng. They also fnd ha markes reac smlarly o explc (publc) and mplc (prvae) news, and noe ha many behavoral models requre nvesors o reac dfferenly o dfferen ypes of news. In conras, our model makes no dsncon beween publc and prvae news. Zhu and Zhou (forhcomng) offer a raher dfferen perspecve on he advanages of usng echncal analyss. They fnd ha when here s uncerany abou he degree of predcably of he sock prce, addng a echncal Movng-Average (MA) componen o he sraegy ha nvess a fxed percenage of wealh n socks may ncrease nvesor uly. Ths s 5

6 because opmal dynamc sraeges depend upon nvesors pror belefs and learnng abou unknown model-specfc parameers, whle MA sraeges are more robus o model and parameer msspecfcaon. 3 Whle Zhu and Zhou focus on he effecs of echncal sraeges on nvesor uly, one of our man objecves s o develop a model ha capures he underlyng phenomena ha gve rse o specfc prce paerns such as he head-and-shoulders or double-op paerns. The res of he paper s organzed as follows: Secons 2 and 3 presen he model. Secon 4 descrbes varous radng rules and relaes hem o he model. In Secon 5 we descrbe our jump deecon mehodology and presen emprcal resuls. Secon 6 concludes. 2. The confrmaon bas 2.1. Exsng leraure on he confrmaon bas As noed above, he confrmaon bas refers o he search for, or he nerpreaon of evdence n ways ha favor exsng belefs or expecaons. A relaed phenomenon has been exensvely nvesgaed n he managemen leraure under he headng of escalaon of commmen. Ths research seeks o provde explanaons for commmen whn organzaons o losng courses of acon. Theorecal explanaons ofen focus on he heory of cognve dssonance (Fesnger, 1957). I s argued ha people who are responsble for poor decsons seek o raonalze hem by basng her nerpreaon of nformaon relevan for assessng he oucome of he decsons. A sudy of he bankng ndusry found ha bank execuve urnover predced boh provsons for loan losses and he wre-off of bad loans (Saw, Barsade and Kopu, 1997). The mplcaon of hese fndngs s ha hose ndvduals responsble for makng 3 For a smlar argumen, see Blanche-Scalle, Dop, Gbson, Talay and Tanré (2007). 6

7 he orgnal loan decsons exhbed sysemac bas n her nerpreaon of nformaon abou he saus of he loans. A specfc example of how confrmaon bas s recognzed as a poenal source of neffcency whn he nvesmen communy s provded by Camerer and Loewensen (2004, p.17). They repor how an nvesmen banker had descrbed he way n whch hs frm combaed he effecs of raders emoonal aachmen o her pas rades by perodcally forcng raders o swch posons wh each oher. In a sudy lookng a dssonance effecs n he conex of muual fund nvesmen, Goezmann and Peles (1997) found ha even well-nformed nvesors had a endency o favorably dsor her percepons of he pas performance of funds ha hey held. Ths may explan he observed asymmery beween nvesmen flows no wnnng funds and ou of losng funds (Ippolo, 1992). Confrmaon bas has also been shown o manfes self n group decson makng (Schulz-ard, Frey, Lühgens and Moscovc, 2000). Usng a sample of mddle managers from banks and ndusral companes, he expermen nvolved analyss of a case sudy n whch a company has o decde wheher or no o proceed wh a large nvesmen. Subjecs were requred o come o a prelmnary concluson ndvdually before beng combned no groups. A hs pon hey were gven access o addonal nformaon. Groups ha agreed n her prelmnary conclusons showed a srong preference for accessng supporng raher han conflcng nformaon. Ths fndng s of parcular neres n he presen conex, snce many porfolo nvesmen decsons are he oucome of group delberaons The basc model wh a sngle low-frequency sgnal The process by whch nformaon s revealed and ncorporaed no prces s consruced o capure he mporan feaures of a jump-dffuson process n a dscree-me framework. The 7

8 jump-dffuson model of sock reurns has a long hsory (Meron, 1976) and recen work by Barndorff-Nelsen and Shephard (2004) ndcaes ha jumps n equy prces conrbue a sgnfcan proporon of oal prce volaly. Research on emprcal opon prcng has also found ha nroducng jump componens no he underlyng prce seres allevaes some of he prcng bases found n sandard models (Baes, 2003). We suppose ha here are low-frequency sgnals ha are more nformave han hghfrequency sgnals. One can hnk of he low-frequency sgnals as generang he jumps n he prce seres, and he hgh-frequency sgnals as generang he dffuson. There are wo groups of agens. One group s subjec o cognve bas, whereas he oher s no. We assume ha hose subjec o bas are rsk neural, bu ha hose who are raonal are rsk averse. Ths s a smplfcaon smlar o ha made by Danel, rshlefer and Subrahmanyam (1998). I allows us o concenrae exclusvely on he role of based raders n seng prces, snce s her expecaons ha deermne prces. Thus n wha follows, all expecaons wll be hose of he group subjec o bas. The agens are endowed wh shares of a rsky secury and of a rsk free asse. They observe a low-frequency sgnal (L-sgnal) a dae 0 abou he lqudaon value of a secury. A subsequen daes, a sequence of hgh-frequency sgnals (-sgnals) s observed. A dae T, all nformaon abou secury value s revealed and he nvesors receve s lqudaon value. The supply of he asse s fxed, and hus he prce a any pon n me s equal o he based nvesors expeced lqudaon value, gven he avalable nformaon. The rsky secury has a lqudaon value V T =, whch has a pror ha s normally 2 dsrbued wh mean zero and varance σ. The sgnal a dae 0 s L = ε (1) 8

9 2 where ε ~ N (0, ). I s hs sgnal ha deermnes he belefs ha generae subsequen σ ε confrmaon bas n he rsk neural group of nvesors. The nal prce of he asse (before he L-sgnal s observed) s deermned by he pror mean of, whch s zero. Gven rsk neuraly, he prce a dae 0 s gven by he expecaon of condonal on L. 2 2 σ σ P0 = E0 = σ σ σ σ ε ( L) = E( ) L w L ε 2 σ where w 0 =. The frs erm n he expresson for P drops ou because we have σ σ ε assumed he pror mean of, E ( ), o be zero. The nformaon assocaed wh he L-sgnal s suffcenly nformave ha produces a jump n he prce a me =0 equal o L-sgnal s followed by a sequence of -sgnals 2 where δ N( 0, σ ) ~ δ ε J 0 = 0 w L. The = δ, = 1,, T. (2). We nroduce cognve bas no he model by makng a dsncon beween objecve and perceved sgnals. Whle he nvesor s nerpreaon of he L-sgnal s always unbased, he L-sgnal deermnes a se of belefs whch nfluences he percepon of he subsequen -sgnals. We assume ha he perceved sgnal Ĥ akes he form ˆ = d( w ) 0 L, δ, = 1,, T. (3) The value of he perceved -sgnal s shfed by he value of he funcon d ( w0l, ), whch we call he confrmaon bas funcon. Ths funcon s assumed o depend on: (a) he weghed L- sgnal, w 0 L and (b) he me elapsed snce he L-sgnal s observed. We assume ha hs funcon akes he mulplcavely separable form 9

10 The properes of ( w L) ( w L) m( ) d( w0 L, ) f 0 (4) f 0 are: P1. f ( 0 ) = 0 P2. f ( w0 L) > 0 P3. ( w L) = f ( w L) f 0 0 P1 saes ha when he L-sgnal s neher favorable nor unfavorable ( L = 0 ) here s no subsequen bas n he percepon of he -sgnals. P2 saes ha when he L-sgnal s favorable (unfavorable), here s a posve (negave) bas n he percepon of he -sgnals, and ha hs bas ncreases wh he (absolue) value of he L-sgnal. P3 mposes he requremen ha he bas be symmerc for favorable and unfavorable sgnals. The properes of m() are P4. m () = 1, = 1; m ( ) > 0 for all, P5. m < 0, > 1, P6. lm m( ) = 0. The propery P4 of he funcon m ( ) s a sraghforward normalzaon. P5 and P6 are nended o capure he fac ha confrmaon bas does no perss ndefnely, bu dmnshes over me, and evenually dsappears. A suffcen sasc for a gven sequence of objecve - sgnals s gven by he average A 1 = τ = 1 τ 1 = δτ (5) τ = 1 A suffcen sasc for a gven sequence of perceved -sgnals s gven by he average A ˆ 1 = ˆ 1 1 τ = f ( w0l) m( τ ) δτ. (6) τ = 1 τ = 1 τ = 1 Snce he bas n he percepon of he publc sgnals amouns o an addve shf n he mean, he perceved varances are no affeced and are equal o he rue values. From now on s more convenen o work wh precsons raher han varances, and we nroduce he followng noaon: 10

11 1 1 1 ; 2 ε ; 2 δ. 2 σ σ ε σ δ A The precson of Ĥ s δ. The equlbrum secury prce s gven by he expecaon of s lqudaon value condonal on L- and -sgnals. where A P = w ˆ w L (7) δ w (8) δ ε w ε (9) δ ε Noe ha he weghs here are he raonal Bayesan weghs. Bas arses only because ˆ. A A Ths conrass wh he approach aken n behavoral models based on overconfdence, where sgnal precson s ncorrecly perceved. The raonal prce s gven by R A P = w w L (10) and allows us o sae he followng: Proposon 1 If nvesors msperceve -sgnals, as n (3), hen (a) (b) If P > hen P > P R 0 0,, = 1, 2, If P < hen P < P R 0 0,, = 1, 2, R (c) lm ( P ) = 0 P. The proof s sraghforward and so omed. The nequales n (a) and (b) ndcae ha prce always overreacs o he L-sgnal, bu no mmedaely, snce he mmedae prce jump ha occurs when he L-sgnal arrves s always raonal. Snce we have normalzed he nal prce o zero, P 0 > 0 represens a (raonal) posve prce response generaed by a favorable sgnal L > 0. Ths s followed by subsequen prces ha are greaer han he fully raonal prce. If he sgnal s unfavorable, he reverse s rue. 11

12 Par (c) mples ha he exen of he overreacon a some pon sars o declne and ha he asse prce evenually converges o he raonal prce Overreacon and auocorrelaons Nex we consder he evoluon of prce overreacon over me condonal on he realzaon of he L-sgnal: R f ( w0l) P P = w m( ) τ = 1 We need o specfy funconal forms for f ( w0 L) and m ( ), ε and for m () :. We specfy ( w L) w L δ f 0 τ. (11), and o choose parameer values for 0 = for smplcy, and choose a (reverse) sgmod form 1 m( ) (12) e R We se = 0. 5, ε = L = 1, and plo P P for varous values of δ, he precson of he - sgnal, n Fgure 1. The paern of overreacon s one ha nally ncreases, reaches a maxmum and hen declnes. As he precson of he -sgnals ncreases, he magnude of overreacon ncreases. Ths happens because he hgher precson leads o greaer wegh beng placed on he based percepon of he -sgnals. We examne nex he paern of reurn auocorrelaons mpled by he model. The uncondonal auocorrelaon funcon s gven by [ ΔP, ΔP k ] [ ΔP ] var[ ΔP ]. T 1 cov [, ] k ρ ΔP ΔP k = (13) T = 1 var k We evaluae he auocorrelaon funcon n he followng case: = 0. 5, = 1 and δ = 0.1. The resul s ploed n Fgure 2 and shows a paern of posve auocorrelaons a shor horzons followed by negave auocorrelaons a longer horzons. I s herefore conssen wh he emprcal evdence documenng shor-horzon momenum and long-horzon reversal. ε 12

13 As one mgh expec from he resuls n Fgure 1, qualavely smlar auocorrelaon paerns emerge for oher values of δ. The resuls derved from he model of hs secon are qualavely he same as hose obaned by Danel, rshlefer and Subrahmanyam (1998) (henceforh DS) from he mulperod verson of her model n whch hey nroduce overconfdence and based selfarbuon. 4 I s also rue ha confrmaon bas has been denfed as a source of overconfdence. Bu he way n whch we model he effecs of he bas s dsnc from ha followed by DS. They show ha based self-arbuon can generae me-varyng overconfdence wh respec o a prvae sgnal, and assume ha publc sgnals are correcly nerpreed. Bu as we noed above, Guerrez and Kelley (2008) fnd ha markes reac smlarly o publc and prvae nformaon. We make no dsncon beween publc and prvae sgnals, bu only beween he frequency (and hence he nformaveness) of he sgnals. Ths urns ou o generae new predcons n a model where we nroduce sgnals of nermedae frequency. We examne hs case n he nex secon. 3. A model wh low, medum and hgh-frequency sgnals The resuls of he basc model assume ha low-frequency L-sgnals generae prce jumps, and ha hese jumps are all..d draws from he same normal dsrbuon. In Secon 5 (below) we presen emprcal resuls suggesng ha hs assumpon s oo smple o adequaely descrbe he process ha gves rse o jumps. To prevew hose emprcal resuls, when we examne hghfrequency nra-day reurns for ndvdual S&P 100 socks, we fnd ha each sock experences a jump (on average) once every egh days. Mos of hese jumps are relavely small, wh an 4 For evdence supporng he hypohess ha equy nvesors are overconfden, see Chuang and Lee (2006) 13

14 average absolue jump sze of abou 1.4%. owever, abou wo percen of all jumps are more han hree sandard devaons on eher sde of he mean (greaer han 5% n absolue value), and occasonal jumps of 10% o 30% are denfed. Thus, jumps appear o be more conssen wh a mxure of a leas wo normal dsrbuons: one whch generaes relavely small, frequen jumps; and a second from whch an occasonal bu very large jump occurs. To more accuraely capure hese emprcal jump properes n our model, we nroduce a sequence of sgnals of nermedae frequency (M-sgnals) ha provde addonal fundamenal nformaon abou he value of he secury. These sgnals are less nformave and occur more frequenly han he low-frequency L-sgnals. A he same me, hey are nformave enough o generae small prce jumps, and hus are desgned o correspond o he smaller, more frequen jumps jus descrbed The model wh m-sgnals We assume ha he lqudaon value of he secury s gven by V T σ λ, N = λ (14) = λ σ 2 where λ ~ N(0, ) σ <, and, λ are ndependen. The random varable s realzed as before a me zero, and λ s realzed a me. We rean he srucure of he model analyzed n he prevous secon, bu now nroduce a sequence of sgnals of nermedae frequency (M-sgnals) ha provde nformaon abou he new fundamenal varables λ : 2 M = λ η = 1,, N; η ~ N(0, σ η ) (15) occurrng a me. Each M-sgnal M s followed by s own sequence of -sgnals provdng nformaon abou he same componen of he ermnal value of he asse. Thus 14

15 15 ν λ =, = 1,, N; = 1,, T. (16) where ) (0, ~ 2 σ ν ν N. The percepon of hese sgnals s also affeced by confrmaon bas. Thus M L w d L w d M ν λ η λ = = ), ( ˆ ), ( ˆ 0 0 (17) The bas funcons are assumed o have he same mulplcavely separable form. ) ( ) ( ), ( 0 0 k k m L w f a L w d =. M, k = (18) The consans k a allow us o scale he magnude of bas accordng o he frequency of he sgnal. We assume M a a >. Jus as n he analyss of he prevous secon, we can represen he nformaon conaned n he publc messages as = = A 1 ˆ ) ( 1 ˆ τ τ, ;,, 1 N K = = 1,, T. (19) We denoe he precson of he varables λ, η and ν by λ, η and ν respecvely. Then he asse prce s gven by = = j A A M L P 1 ) ( ˆ ˆ ) ( ˆ η λ ν η ν ε δ ε δ, j <. (20) The -h componen n he summaon becomes non-zero a, he me a whch he assocaed M- sgnal s observed. Inroducng he noaon ( ) η λ ν ν = ) ( w ; > η λ ν η = ) ( M w,, we can wre (20) more succncly as ( ) = = j M A A M w w w L w P 1 ˆ ˆ ˆ, j. (21) There are now wo channels hrough whch confrmaon bas causes he observed prce P o dverge from s raonal value. The frs, as before, arses because A A ˆ. -sgnals abou are msperceved as a resul of he bas generaed by he L-sgnal. The second arses

16 because nformaon abou he fundamenal componens λ s also affeced by he same source of bas Effec of M-sgnals on auocorrelaon paerns The frs queson we examne s how exendng he model o ncorporae sgnals occurrng a nermedae frequency affecs he paern of reurn auocorrelaons. We examne hs wh he use of numercal smulaons. The parameer values chosen are as follows: = 0.5 ; ε = 1; δ = 0. 1; λ = 4 ; η = 20 ; ν = 10 ; a M = 0. 5; a = 0. 5 We smulae prce pahs over 100 perods, and assume ha he me of arrval of he M-sgnals s random. The number of perods beween successve sgnals s assumed o be lognormally dsrbued. I s chosen o be he exponenal of a normal dsrbuon wh mean 2.3 and sandard devaon 0.3, whch generaes a lognormal dsrbuon wh mean 10.4 and sandard devaon 3.2. So on average en M-sgnals wll occur n each smulaon. The L-sgnal occurs a me 1, and he M- sgnals ha follow occur a random nervals. The uncondonal auocorrelaons are ploed n Fgure 3. We see ha n conras o he plo n Fgure 2, a very shor horzons he reurn auocorrelaons are negave. The paern s one of reversal followed by connuaon and hen agan reversal. Ths maches he paern of auocorrelaons documened n a number of sudes (Jegadeesh, 1990; Lehmann, 1990; Lo and MacKnlay 1990; Jegadeesh and Tman, 1995; Guerrez and Kelley, 2008) Furher explanaon of auocorrelaon paerns In he model wh L-sgnals and -sgnals only, we observe a paern of auocorrelaons ha are posve a shor lags and hen become negave a longer lags. owever, when M-sgnals 5 We choose a = 0 for smplcy. All we requre for our qualave resuls s ha a be sgnfcanly smaller han a.e. bas s reduced n absolue magnude for more frequenly observed sgnals. M 16

17 are nroduced, very shor lag auocorrelaons are negave, become posve a nermedae lags and hen agan urn negave a long lags. To help undersand he sources of hese auocorrelaon paerns, noe ha whenever here s a shock o an asse prce ha moves away from fundamenal value, hs wll nduce negave auocorrelaon a a horzon ha s dependen on he speed wh whch he devaon s correced. In he model wh L-sgnals and -sgnals only, he more nformave L-sgnal nduces a drf away from fundamenal value when he frs -sgnal s realzed. Ths devaon from fundamenal value s correced only relavely slowly as he effecs of confrmaon bas dsspae. The move away from fundamenal value generaed by he effecs of he bas causes he nal phase of posve auocorrelaon. Prce correcon occurs relavely slowly, producng negave auocorrelaons a long horzons. M-sgnals convey addonal fundamenal nformaon, bu are nerpreed n a based manner, whch resuls n a devaon from fundamenal value. Bu learnng abou he fundamenal underlyng he M-sgnals s more rapd and a shor horzons generaes negave auocorrelaon ha ouweghs he posve auocorrelaons nduced by he L-sgnals. In effec, he model overlays wo paerns of overreacon and reversal, one over a relavely long horzon (relaed o he L-sgnal and s own -sgnals) and he oher over a much shorer horzon (relaed o each M-sgnal and s assocaed -sgnals). The parameerzaon of he model s such ha we ge he nal nvered-u shape o he auocorrelaon funcon. Clearly, he auocorrelaons are dependen on he parcular parameer values chosen, bu hey do provde an explanaon for wo apparenly unrelaed asse prcng phenomena, namely he predcve power of ceran prce paerns used by echncal raders and he paern of reurn auocorrelaons. In addon, he model produces several sharp predcons abou he 17

18 auocorrelaon n he me seres of jumps n he prce seres. By assumpon, prce jumps are assocaed wh he occurrence of L-sgnals and M-sgnals. If we denoe he -h jump by J we may summarze he properes of he auocorrelaon n he jump seres as: Proposon 2 The correlaons beween jumps n he me seres: (a) are posve,.e. ρ [ J, J j ] >0. (b) declne over me,.e. [, ] ρ > ρ [, ] J J j J J k, j<k. (c) are ncreasng n he mporance of he nformaon assocaed wh he L- 2 sgnal as measured by σ. (Proof avalable upon reques). Properes (a) and (b) are self-explanaory. Propery (c) makes a dsncon beween he 2 mporance of he nformaon assocaed wh he L-sgnal, σ, and he precson of he L- sgnal self (capured by 2 2 σ ε ). An ncrease n he value of σ corresponds o an ncreased probably of larger realzaons of, and herefore a correspondng ncrease n he probably of larger jumps assocaed wh he L-sgnal. Thus, larger nal jumps wll lead o sronger correlaons wh subsequen jumps. Laer n Secon 5, we es he emprcal predcons of Proposon 2 on jumps n U.S. equy prces. Alhough he qualave feaures of our argumen are no sensve o precse parameer values, here are ceran relave magnudes ha are mporan. The low frequency fundamenal s chosen o have hgher varance han he hgher frequency fundamenal varables λ. I wll have on average a larger mpac on he value of he secury, whch hen leads naurally o our assumpon ha s nformaon abou he low frequency componen of fundamenals ha s he source of confrmaon bas. We also need o assume ha sgnals abou he hgher frequency fundamenal varables have hgher precson so ha learnng abou he sgnfcance of hese 18

19 realzaons s more rapd. A framework whch would render hs assumpon plausble s one where he low frequency fundamenal varable s concerned wh nformaon ha has a hgh degree of ambguy or nangbly ( sof nformaon) whereas he hgh frequency varables relae more o hard nformaon. 4. Techncal prce paerns An mporan componen of echncal analyss s he use of prce paerns as ndcaors of changes n a prce rend. Alhough a large number of such ndcaors are used, some are regarded as more relable han ohers, and are consequenly more wdely used by echncans. The occurrence of a parcular prce paern s ypcally aken as an ndcaor of a change n a prce rend, and herefore as a buy or sell sgnal. We wll llusrae he use of such ndcaors n he conex of wo commonly used paerns. Fgure 4 provdes a schemac llusraon of a prce seres dsplayng he head-and-shoulders paern. There s farly general agreemen n books and manuals on echncal analyss ha he mporan characerscs of he head-and-shoulders paern are he followng (see, for example Edwards and Magee, 1992; Bulkowsk, 2000): 1. The head should be sgnfcanly aller han he shoulders. 2. The op and boom of he shoulders should be of roughly equal hegh. 3. The overall paern should be farly symmerc.e. he spacng beween lef shoulder and head should be approxmaely he same as ha beween head and rgh shoulder. The necklne n he fgure s a sragh lne connecng he roughs beween he wo shoulders and he head. I s used o deermne he pon a whch a rade s naed, whch s where he necklne nersecs he prce seres afer he rgh shoulder. The paern sgnals an mmnen prce declne, so he echncal analys execues a shor sale. Fgure 5 provdes an example of he head- 19

20 and-shoulders paern usng a daly prce seres from Archer Danels Mdland (ADM) durng he perod. The lef shoulder appears n July 1997, he head appears n Sepember 1997, and he rgh shoulder appears n Aprl The nverse head-and-shoulders paern s smply he paern n Fgure 4 vewed upsde down. In hs case he paern predcs a rse n prce, and he echncal analys buys he sock. The double op paern s llusraed n Fgure 6. ere he paern s denfed by he appearance of wo local maxma of approxmaely equal value. As wh head-and-shoulders, he paern s nerpreed as a sgnal of fuure prce declne. The double boom s an nvered double op, and sgnals a fuure rse n prce. There are several sudes whch provde evdence ha prce paerns conan nformaon ha may be relevan for predcng fuure prces. Lo, Mamaysky and Wang (2000), henceforh LMW, develop an algorhm for denfyng a number of paerns ncludng he four descrbed above. They fnd ha he dsrbuon of prces condonal on a paern occurrence s sgnfcanly dfferen from he uncondonal dsrbuon. Savn, Weller and Zvngels (2007) use a modfed verson of he LMW algorhm o show ha he head-and-shoulders paern has sgnfcan predcve power for fuure ndvdual sock reurns over horzons of one o hree monhs. An explanaon for he predcve power of prce paerns emerges from he model wh sgnals of nermedae frequency. We consder he expeced pah of prces condonal on he nal sgnal L, whch can be wren as E j j A M A [ P L] = w L w E[ ˆ L] w E[ Mˆ L] w E[ ˆ L] j (22) = 1 We represen a ypcal prce paern by examnng he expeced pah of prces condonal on an nal L-sgnal. We use he same parameer values as for he auocorrelaon = 1 20

21 plo n Fgure 3, and choose L = owever, we assume ha he mng of M-sgnals s fxed and deermnsc. The same paern characerscs wll be recognzable n dsnc prce pahs even wh random varaon n sgnal mng, bu f we were o average over hese separae pahs hs would end o obscure he feaures of he paern. Fgure 7 shows a graph of he expeced pah of prces condonal on L. The head-andshoulders paern emerges clearly, wh boh shoulders and he head occurrng as a resul of he arrval of M-sgnals. In fgure 7, an nal L-sgnal has already arrved a me zero causng a posve jump (whch can be nferred by nong ha he expeced prce a me zero s posve). Subsequen -sgnals are nerpreed n a based manner, producng he upward drf. The based nerpreaon and subsequen correcon of he frs M-sgnal produces he lef shoulder; he response o he second M-sgnal generaes he head, and he response o he hrd M-sgnal produces he rgh shoulder. Thus boh he head and he shoulders n he prce paern can be vewed as analogues o he broader emplae of momenum and reversal excep ha hey occur a hgher frequency. Momenum bulds up and dsspaes relavely slowly, producng he nvered U prce pah. I resuls from overreacon o he low-frequency L-sgnal and s gradually correced. On he oher hand, overreacon o M-sgnals generaes a sequence of nvered V prce pahs a hgher frequency. The basc mechansm generang he nvered V pahs s he same overreacon followed by correcon as he mpac of he confrmaon bas dsappears. The head-andshoulders paern emerges when one overlays he successve epsodes of overreacon and correcon on o he longer erm phenomenon of momenum and reversal. The denfyng feaures of he paern are evden n Fgure 7. In addon, s clear ha he model confrms he predcve conen of he paern and s conssen wh he fndngs n 21

22 Savn, Weller and Zvngels (2007). The paern appears as he momenum phase ermnaes and s followed by reversal. In oher words, sgnals an mmnen prce declne. Some echncal analyss specfy also ha f he paern s o be a relable gude o radng should occur afer a perod of sgnfcan prce ncrease. Ths characersc s also conssen wh he pah shown n Fgure 7. If he prce pah s calculaed condonal on L = 0. 5, wh all oher parameers gven he same values, hen we oban he resul llusraed n Fgure 8. Ths paern sasfes all he requremens for he nvered head-and-shoulders prce paern. I s a predcor of fuure prce apprecaon. Expermenaon reveals ha boh he head-and-shoulders and he nvered head-andshoulders paerns are surprsngly robus, n ha hey wll appear for dfferen parameer values, dfferen nervals beween sgnals, and are also no sensve o changes n he speed of decay capured by he funcon m (). Ths accords wh he clams of echncal analyss ha hese parcular paerns are among he mos relable. Fgures 9 and 10 llusrae double op and double boom paerns. These paerns appear when he synchronzaon of M-sgnals wh momenum and reversal phases s shfed somewha. Agan, he prce paerns predc fuure prce movemens conssen wh he echncal analyss leraure. In summary, he prce dynamcs from he model wh sgnals of nermedae frequency are conssen wh he presence and valdy of several of he mos common echncal prce paern sraeges. The explanaon for he predcve power of prce paerns n he model les n he combnaon of momenum and reversal wh prce jumps ha are posvely correlaed wh he L-sgnal ha generaes he nal prce jump and he momenum phase. Whou he posve 22

23 correlaon resulng from confrmaon bas, subsequen jumps would be equally lkely o be posve or negave. A key feaure of he prce paerns we examne s he presence of a successon of jumps wh he same sgn. For he paerns o emerge s also necessary ha prces generaed by M-sgnals rever farly rapdly o fundamenal value. I s hs ha generaes he peaks n he prce seres characerzng head-and-shoulders and double-op paerns. In oher words, he prce jumps assocaed wh M-sgnals produce msvaluaon ha s quckly correced. Ths n urn produces he shor-horzon negave auocorrelaon conssen wh he fndngs of Guerrez and Kelley (2008) and ohers. Oher researchers have proposed explanaons for he phenomena of momenum and reversal n asse reurns ha make no use of behavoral assumpons. Lewellen and Shanken (2002) and Brav and eaon (2002), for example, show ha parameer uncerany and learnng may produce hese knds of effecs. George and wang (2007) presen evdence o sugges ha long-erm reurn reversals resul from he mpac of capal gan axes. The srengh of our approach les n s ably o provde a unfed framework o explan no only shor-erm momenum and long-erm reversal, bu also reversals over very shor horzons and a number of commonly used echncal prce paerns. 5. Jump deecon mehodology and ess We now urn o examnng he model s predcon ha sequenal prce jumps for a parcular sock wll be posvely auocorrelaed. Recenly, researchers have developed economerc echnques ha can effecvely separae he connuous and jump componens of he underlyng prce process by ulzng hgh-frequency rade-by-rade daa (Anderson, Bollerslev 23

24 and Debold (2007), Barndorff-Nelsen and Shephard (2004), Tauchen and Zhou (2006)). These researchers effecvely demonsrae ha he dfference beween realzed volaly (RV), whch approxmaes he oal daly reurn varance, and b-power varaon (BV), whch esmaes he varance due o he connuous reurn componen, s a conssen esmaor of he reurn varance due o he jump reurn componen. Jumps occur when he sasc (RV BV) s sgnfcanly dfferen from zero, and we assume ha a mos one jump occurs per day. I should be noed ha we are no usng he hgh-frequency daa o examne nraday reurns for he paerns descrbed n he model. Raher, he economerc echnques requre hghfrequency nraday daa as an npu o denfy when jumps occur and o esmae he sze of each jump. Even hough denfed jumps occur almos nsananeously, he oupu from applyng he jump-deecon mehodology s a me-seres of daly reurns, wh each daly reurn decomposed no a connuous and jump componen. These daly reurns are used o examne auocorrelaon paerns over longer ner-day horzons Daa descrpon To es he predcons of he model, we are neresed n obanng a reasonably large cross-secon of frms. A he same me, he daa-nensy of hgh-frequency rade daa creaes praccal lms on he number of frms n our sample. To balance hese ssues, we use he sample consrucon mehodology of Dunham and Fresen (2007), who analyze ndvdual sock daa for he componen socks of he S&P 100 Index as of July 1, Ths lms our sample o a manageable number of frms, ye capures a large percenage of he overall U.S. equy marke capalzaon. The daa are colleced from he NYSE s Trade and Quoe (TAQ) daabase for he sxyear sample perod January 1, 1999 December 31, 2005, usng only quoes from NYSE, 24

25 AMEX and NASDAQ exchanges. 6 Two oher frms, Uned Posal Servce and Goldman Sachs, wen publc durng he sample perod and for hese frms we use daa from he IPO dae hrough he end of he sample perod. We use he bd-ask mdpon for each ransacon o mgae bd-ask bounce, and also apply several flers o elmnae erroneous observaons. Frs, he offer/bd rao mus be less han 1.10 for he quoe o be ncluded. 7 Second, we apply a sandwch fler o elmnae quoes ha are 10% or furher n absolue value from surroundng quoes on boh sdes. Ths fler elmnaes he followng erroneous ype of quoe sequence: a frs quoe ha s mmedaely followed by a sgnfcanly hgher (or lower) second quoe ha s subsequenly followed by a hrd quoe whch s conssen wh he frs quoe n he sequence. 8 Vsual nspecon reveals numerous nsances of such spurous quoes sandwched beween wo oherwse conssen quoes, and our fler elmnaes he erroneous quoe. Whou hs fler, our esmaon model mgh ncorrecly denfy a sgnfcan jump when prces jump o he erroneous quoe and hen jump back o he correc prce. We only nclude quoes durng regular radng hours, segmen each radng day no fvemnue nervals and calculae nerval reurns usng he bd-offer mdpon. For he frs radng nerval of each day, we ulze he openng daly bd-offer mdpon and calculae he frs nerval reurn usng he bd-offer mdpon calculaed a he end of he frs nerval. We conrol 3 The only S&P 100 Index componen sock no ncluded n he sudy s CBS Corporaon, whch replaced Vacom on he S&P 100 Index on January 1, 2006, a dae jus ousde our sample perod. 7 For example, on 4/07/00 Amercan Arlnes (cker: AA) has a TAQ record conssng of a bd quoe of $ and an offer quoe of $80. All oher bd-offer spreads for he day were much narrower, ypcally less han $0.25, and hus our fler elmnaed hs bd and offer quoe. We also found occurrences n he TAQ quoe daa where a quoaon appeared o be a ypographcal error or seemed nconssen wh surroundng quoes. For example, a bd quoe of $ for CA on 8/17/00 s followed by a quoe of $33.375, and surrounded by bd quoes of $33 or greaer hroughou he radng day. 8 For example, a sequence of 3 mdpon quoes for Amercan Arlnes (cker: AA) on 12/07/2000 are as follows; $31.03 for nerval 1, $84.91 for nerval 2, and $30.66 for nerval 3. An examnaon of all oher quoes for AA on 12/07/2000 suggesed ha he $84.91 quoe for nerval 2 was nvald. 25

26 for sock spls by elmnang any daly nerval wh a fve-mnue reurn greaer han 50%. In shor, our daa se ncludes all of he componen socks of he S&P 100 Index as of July 1, 2006 over he sample perod , bu excludes CBS Corporaon (added o he ndex ou of sample on January 1, 2006), resulng n a fnal sample of 99 ndvdual frms Emprcal jump properes We frs repor cross-seconal summary sascs on realzed volaly and b-power varaon n Table 1 for all 99 componen frms n he S&P 100 Index over he sample perod. The rao of b-power varaon o realzed volaly (BV/RV), or he square roo of hs rao, whch can be nerpreed as a sandard devaon measure, has been used elsewhere n he leraure o measure he fracon of oal volaly generaed by he connuous reurn componen (Tauchen and Zhou, 2006). Panel (a) of Table 1 reveals ha approxmaely 90% of oal reurn varance for he average frm n he sample s arbuable o connuous reurns. Toal reurns are decomposed no her jump and connuous componens n Panel (b) of Table 1 n he form of sample varances. 9 Toal rsk, connuous rsk and jump rsk are calculaed as he varance of oal daly reurns, varance of connuous reurns, and varance of jump reurns, respecvely. Panel (b) of Table 1 repors ha jumps conrbue beween 5% and 10% of he oal varance n he average frm, measured as he rao of jump rsk o oal rsk. Noe, however, ha hese are uncondonal varance measures ha nclude all radng days n he sample, many of whch have no jump. In Table 2, we repor dsrbuonal sascs for he cross-secon of equy jumps for all 99 frms n he sample, condonal upon a jump occurrng. To do hs, we frs calculae he average value for each sasc separaely for each frm n he sample. Table 2 repors he mean, 9 We decompose oal varance, and no sandard devaon, no s componen pars snce he varance componens are addve whle he sandard devaon componens are no. 26

27 medan, mnmum and maxmum values of he frm-level averages. Panel (a) of Table 2 repors jump frequency, measured as he number of days wh a jump dvded by he oal number of days n he sample, and shows ha he average frm n he sample experences a daly jump approxmaely welve percen of he me, or abou once every egh radng days. We also repor an absolue jump sze o provde an ndcaon of he magnude of jumps when hey occur. Panel (a) of Table 2 repors a mean (medan) absolue jump sze 1.39% (1.36%). To shed lgh on he meanngfulness of hs sasc, we repor summary sascs on he absolue daly reurn n he hrd row of Panel (a). The rao of absolue jump sze o absolue daly reurn mples ha on days when jumps occur, he jump componen represens nearly 90% of he oal reurn, on average. Lasly, Panel (a) of Table 2 also repors jump varance and oal varance. The rao of jump varance o oal varance, whch s analogous o he rao (BV/RV) descrbed above, shows ha jumps conrbue nearly 60% of oal rsk for ndvdual socks ( / ). Thus, whle he uncondonal conrbuon of jumps o reurns and rsk s relavely small, hey accoun for he majory of he reurn and varance on he days when hey do occur. Panel (b) of Table 2 repors he same sascs as Panel (a), excep ha Panel (b) uses only equy jumps ha occur on hgh realzed volaly days, whch we defne as days where an ndvdual sock s realzed volaly s above s medan realzed volaly calculaed over he enre sample perod. As mgh be expeced on hgher volaly days, he mean (medan) absolue jump sze subsanally ncreases o 1.91% (1.88%) bu he jump frequency remans sable a around 12%. Whle he percenage of oal rsk arbuable o jumps s margnally hgher a 62% n Panel (b), he small dfference suggess ha he jump conrbuon o oal rsk s farly robus o wheher or no days of low volaly are ncluded or excluded. 27

28 5.3. Emprcal suppor for M- and L-frequency jumps Table 3 repors several ses of sascs ha are conssen wh he presence of boh M- and L-frequency sgnals n he model. Panel (a) presens some basc sascs on he pooled sample of jumps whch ndcae ha he jumps exhb slgh negave skewness and excess kuross. Panel (b) repors emprcal quanles for he sample of jumps, along wh correspondng quanles for a normal dsrbuon wh mean and varance equal o he emprcal jump sample mean and varance. Relave o he normal dsrbuon, he acual jumps are more ghly clusered abou he mean beween he 10 h and 90 h percenles, bu also exhb much longer als han he normal dsrbuon would predc. Panel (c) provdes several formal ess ha also rejec he assumpon of normaly. Whle hese sascs do no provde drec suppor for he specfc form of M- and L-sgnals we model, hey are conssen wh he noon ha jumps come from a leas wo dsrbuons: one whch generaes smaller, more frequen jumps and a second whch generaes occasonal bu very large jumps Emprcal auocorrelaons We nex examne he naure of he auocorrelaon beween sequenal equy jumps. Proposon 2 ndcaes ha sequenal prce jumps wll be posvely auocorrelaed. We sar by examnng correlaons beween sequenal equy jumps a me, 1, 2 and 3 n he frs column of Table 4. Usng only hose radng days on whch a jump occurs, we fnd an average auocorrelaon of nearly beween sequenal prce jumps. The auocorrelaons beween a jump a me and s subsequen hree jumps are all posve and sascally sgnfcan, hough he level of sgnfcance beween a jump a me and 2 and beween a jump a me and 3 s 28

29 slghly weaker a 5% and 10%, respecvely. 10 To provde furher nsgh, we also examne he auocorrelaons on days of hgh realzed volaly (as defned above n Table 2). These correlaons are repored n he second column of Table 4, and are all posve and sascally sgnfcan a he 1% level. 11 In he conex of he model, jumps assocaed wh separae L-frequency evens are hemselves uncorrelaed. The predced correlaon n jumps resuls from he jumps assocaed wh M-frequency sgnals ha follow a parcular L-frequency even. Thus, he correlaons repored n Table 4 can be refned by denfyng L-frequency jumps, and esmang correlaons only beween he L-jumps and he jumps mmedaely followng hem. Because our emprcal jump deecon mehodology smply denfes sgnfcan jumps, canno explcly denfy L- frequency jumps. owever, n he model L-frequency jumps are larger n magnude han M- frequency jumps, and Proposon 2 ndcaes ha he larger he value of he L-frequency jumps 2 (as capured by σ ), he sronger he correlaon wh subsequen M-frequency jumps. Thus, we sor each frm s jumps no decles based upon he absolue jump sze, and use he jumps n he larges one or wo decles as a proxy for ha frm s L-frequency jumps. Panel (a) of Table 5 repors cross-seconal sascs for each decle of sze-sored jumps, and Panels (b) and (c) repor correlaons beween L-frequency jumps and he jumps ha follow hem. In 10 Two poenal concerns are ha he posve jump correlaon we documen may be due eher o correlaed errors or nvesor underreacon. Frs, f jumps are esmaed wh error, hen any correlaon n errors may be pcked up as correlaon n jumps. owever, hs s unlkely o be drvng our resuls snce he jumps we esmae occur n a specfc fve-mnue nerval. Gven ha on average here are 624 fve-mnue nervals beween jumps (78 x 8 days), any error n he nal jump esmae s more lkely o be correlaed wh he fve-mnue reurns ha mmedaely follow han wh a subsequen jump. A second concern s ha any me nvesors underreac o new nformaon, hs wll lead o posve auocorrelaons beween he nal reurn and subsequen perod reurns. Ths can produce a posve correlaon n jumps f boh he nal and subsequen prce correcons ake he form of a jump. Ths s also unlkely o be drvng our resuls, snce would requre ha he nal underreacon was, on average, 624 perods before correcng wh a second dscree jump. 11 We have repeaed our correlaon analyss for jumps beween 3 and 10 lags, and fnd ha auocorrelaons beyond 3 lags are almos unformly posve bu sascally nsgnfcan. These esmaon resuls are omed for brevy bu avalable from he auhors. 29

30 Panel (b), L-frequency jumps are denfed as jumps n he larges wo decles, whle n panel (c) we use only he larges decle of absolue jumps. Condonng on L-frequency jumps n hs manner, we fnd ha he correlaons are wo o hree mes larger han hose repored n Table 4. In parcular, he lag-1 correlaon s n panel (b) and n panel (c), and boh correlaons are sascally sgnfcan a he 1% level. Whle hese correlaons are small n absolue value, he sze of he observed correlaon n jumps depends upon he magnude of he underlyng nvesor bas. Few sudes have explcly quanfed he magnude of he confrmaon bas n ndvduals, alhough Fresen and Weller (2006) measure he magnude of he closely relaed cognve dssonance bas n fnancal analys forecass. They fnd ha whle he cognve dssonance bas s presen, s relavely small n magnude. Specfcally, her esmaes sugges ha cognve dssonance nroduces a mean-shf bas no analys forecass of beween 5% and 10% of he lagged forecas error. In lgh of her fndng, he correlaons of 0.06 and 0.09 repored n Panels (b) and (c) of Table 5 seem reasonable. 6. Concluson Ths paper develops a heorecal framework ha can accoun for he apparen success of boh rend-followng and paern-based echncal radng rules. Our model nroduces a sngle cognve bas, whch has been exensvely documened n he psychologcal leraure and descrbes he endency of ndvduals o search for and nerpre nformaon selecvely o conform o a gven se of belefs. In he model, nformaon arrval s modeled wh sgnals of varous magnudes, arrvng a dfferng frequences. Large, nfrequenly observed sgnals are nerpreed raonally by 30