Pattern-based Expectations: International Experimental Evidence and Applications in Financial Economics

Transcription

1 Parn-basd Expcaions: Inrnaional Exprimnal Evidnc and Applicaions in Financial Economics Tobias F. Röhli* Dparmn of Economics Univrsiy of Erfur Nordhäusr Srass 63 PF D-9905 Erfur Grmany Tl: Fax: July 200 JEL-Classificaion: D84, C9, F3, G2 Ky Words: Parn xrapolaion, bhavioral modl of xpcaions, xprimnal conomics, sock pric drminaion, forward discoun on h forign xchang mark Absrac: W sudy how subjcs xrapola simpl parns in financial im sris in ordr o dvlop a dscripiv modl of acual agn bhavior. Th laboraory xprimn for his analysis was conducd in boh Grmany and Japan. Saisical analyss indica considrabl similariy in xpcaions formaion across culurs and documn ha agns xpcaions ar a varianc wih h noion of sandard rnd xrapolaion. Th papr hn proposs a mhod for compuing xpcaions for any conomic im sris basd on h xprimnal daa. Such parn-basd xpcaions ar shown o xplain sock prics and h dynamics of h forward discoun on h forign xchang mark. * For his projc I hav bnfid from h hospialiy and h suppor offrd by Tasuyoshi Saijo a Osaka Univrsiy and h ffors of Knju Akai and Kiko Aoki in hlping o run h xprimn wih Japans subjcs. I would lik o hank an anonymous rfr, John Conlisk, Robr Jung and Mark Machina for vry hlpful commns and discussions. Also, I would lik o acknowldg commns from paricipans of prsnaions a h 2 nd Inrnaional Nonlinar Scinc Confrnc, h 2006 SABE mings in Paris, h Univrsiy Auonoma of Madrid and h Univrsiy of Dormund (u). Enrico Schumann providd suppor wih programming asks.

2 . Inroducion 2 Th concp of raional xpcaions has bn h singl mos imporan conribuion o h modling of xpcaions in h pas 50 yars. Th ida ha conomic agns us h sam analysis and daa as conomricians for hir forcass has o a larg xn rplacd h various noions of xrapolaiv xpcaions ha had bn popular bfor Muh s (96) conribuion. Howvr, much mpirical rsarch invsigaing xpcaions licid hrough survy chniqus or hrough xprimns cass doub on h gnral mpirical validiy of h raional xpcaions hypohsis (s,.g. Psaran, 987, Manski, 2004 and Röhli, 2007). This aricl proposs a modl of xrapolaiv xpcaions basd on h noion ha agns for xampl invsors form xpcaions basd on h visual parn shown by h im sris o b projcd ino h fuur. Parn in his conx mans a spcific squnc of changs ovr h rcn pas of a im sris. Psychological rsarch documns h imporanc of rlying on parns (i.., shaps, forms) and shows how nworks of nurons can larn and gnra fas rsponss o a vas numbr of parns (s,.g., Pucci 974, Rumlhar al. 986, Posnr 989, and Lund 200). From h prspciv of voluion h abiliy of organisms o dc parns (i.., rcognizing similariis in siuaions) and o draw quick infrncs has a high survival valu (s,.g., Edlman and Rk, 990). Svral sudis documn ha subjcs rly on simpl visual parns whn forming xpcaions. In paricular, runs and zigzag movmns in im sris sand ou as parns on which subjcs rly whn forming on-sp-ahad xpcaions (Fldman, 97, Jons 97, Egglon, 982, and Röhli, 998). Ths conribuions sudy bhavior whn subjcs fac a binary sris. Svral rsarchrs hav alrady xprimnally sudid inuiiv im-sris forcasing using financial daa. Whil D Bond (993) shows subjcs hisorical daa, Bloomfild and Hals (2002) work wih sylizd financial daa. W follow h lar approach bu sudy subjcs rsponss ovr a broad spcrum of possibl circumsancs (i.., no jus for a fw inrsing squncs in a im sris). Th ulima goal of liciing such an array

3 3 of rsponss is h us of hs daa in a nw mhod for calculaing xpcd valus for any conomic im sris. W would lik o mak i clar ha i is a posiiv (i.., bhavioral) modl of xpcaions w propos, no a normaiv on. In an imporan horical conribuion Barbris, Shlifr, and Vishny (998, hrafr calld BSV) propos a bhavioral modl of invsors xpcaions or invsor snimn ha is abl o accoun for ass pric rgulariis which ar a varianc wih raional xpcaions. Thir conribuion is rlad o h prsn approach and w will compar is implicaions wih our xprimnal findings. In ordr o chck h gnraliy of parn-basd xpcaions w compar xpcaions daa collcd in Japan and Grmany. 2 Th aricl is srucurd as follows: scion 2 dscribs h xprimn. Scion 3 prsns h saisical analysis of h xprimnal daa and prsns sriking similariis in parn-basd xrapolaion across Grman and Japans subjcs. Scion 4 documns ha nihr h linar modl of rnd xrapolaion nor h BSV-modl of xpcaions adqualy dscribs h xpcaions licid xprimnally. Scion 5 shows how h xprimnal daa can b usd o compu hisorical xpcaions daa for any im sris. Scion 6 applis his approach o h conomric sudy of hisorical U.S. sock pric daa. Scion 7 documns ha h parn-basd xpcaions hlp o xplain h forward discoun on h forign xchang mark. Scion 8 concluds h aricl. 2. Th xprimnal dsign Our daa is licid using an xprimnal dsign inroducd by Röhli (2007). 3 Appndix provids h insrucions for h xprimn. Subjcs ar shown shor squncs in a im sris hy ar old o considr o b a financial im sris (lik a sock pric or an xchang ra). Thy ar informd ha hy ar going o s diffrn possibl cass of how his financial sris can volv ovr h cours of four priods and ha i is hir ask o assss h likly coninuaion of his sris. Th xprimn simplifis h possibl cours of h im

4 4 sris: h sris can only procd in sps of +2, +, 0, -, -2. Wih his rsricion hr xis 25 cass of how a sris can volv ovr four priods. W limi h numbr of cass o 63 on h basis of ss prsnd in Röhli (998) indicaing ha h hypohsis of symmry is no rjcd for a majoriy of subjcs (i.., agns forcass basd on a squnc of changs lik,.g., -, + is ypically h sam as ha basd on h squnc +, - muliplid by -). In principl, subjcs could b askd for rsponss o a widr array of possibl parns including changs of sps of siz hr or largr. Howvr, his widning of h s of obsrvaions would imply a vasly largr s of parns and a longr xprimn. Th numbr of asks (in our xprimn alrady a oal of 3x63=89) also dpnds on h lngh of h daa window shown in ach cas. Th lngh of our windows is chosn basd on arlir rsarch (s Röhli, 998) indicaing ha in a similar ask fw subjcs rly sysmaically on informaion ha rachs farhr back han h las hr sps of a im sris. Morovr, Carlson and Shu (2007) basd on a variy of daa documn ha i ypically aks popl hr obsrvaions o conclud ha a sris of oucoms forms a srak. To b clar, subjcs s 63 diffrn four-priod squncs and giv hir individual projcion wihou rciving fdback on how h sris coninus ino h fifh priod. Th insrucions sricly us of h rm cas and avoid h rm parn. A furhr poin concrns h display of h im sris. Givn h vidnc on h imporanc of visualizaion of informaion (s,.g. Chaomi and Czrwinski, 2000) w prsn subjcs wih graphs insad of numbrs and in accordanc wih arlir similar sudis (lik,.g., Bloomfild and Hals, 2002) w prsn h graphs in lvl form and no in h form of changs. Figur prsns on of h cass shown o subjcs (cas 6). In our rminology his is parn 6. W chos his paricular parn o illusra ha w lici xpcaions for many yps of squncs and no jus for a fw prominn parns lik sraigh rnds (parns 20 and 5) or zigzags (parns 30 and 6). Each cas is prsnd sparaly along wih h asks dscribd blow. Tabl documns h whol lis of h 63 parns shown in h xprimn. Th

5 5 xprimnal daa in h column wih h hading xpcd chang in his abl will b xplaind in scion 4. Tabl abou hr Figur abou hr Subjcs in h xprimn wr givn hr diffrn asks. Task (a) asks for an sima of h likly chang in h sris from priod 4 o priod 5 xprssd in probabiliy valus (in sps of 0.) for h diffrn possibl sps (+2, +, 0, -, -2). 4 Task (b) asks for an sima of h populaion man (i.., h avrag rspons ovr all subjcs) of h xpcd valus givn undr ask (a). All subjcs hr mans h 45 subjcs prsn in on (naional) locaion on h day of h xprimn. Hnc, his ask asks for an sima of h answrs of h ohr subjcs. Hr, subjcs wr askd o provid a singl valu bwn +2 and 2 down o on dcimal poin. Finally, ask (c) calls on subjcs o xprss hir confidnc in hir rspons o ask (b). Hr, subjcs b bwn 0 and 0 Euro cns (in Japan bwn 0 and 5 Yn) on h proposiion ha hir answr in (b) diffrs by no mor han 0.5 in absolu valu from h acual man of h xpcd valus compud ovr all (naional) subjcs. Th financial rwards in h wo counris wr adjusd o local hourly wags for sudn aids and for h xchang ra so as o mak financial incnivs as similar as possibl across counris. In concr rms, h show up f in Grmany was 4 Euro (in Japan 650 Yn). Morovr, subjcs rcivd 0 Euro cns in Grmany (5 Yn in Japan) pr cas for compling asks (a) and (b). This (rspciv) pr-cas amoun is h maximum ha can b nrd as an answr in ask (c). Th main purpos of ask (c) is o offr a financial incniv for dilign procssing of h asks a hand givn ha hr is no masur of forcas accuracy in ask (a). An analysis of h individual rsponss (mans and sandard dviaions) for ach of h 90 subjcs shows ha h dviaions of h individual rsponss rgarding hir own xpcaion (ask a) and hir assssmn of h collciv xpcaion (ask b) diffr

6 6 frqunly and subsanially. Hnc, nohing in h daa indicas ha subjcs would no ruhfully rpor hir xpcaions. Th 45 subjcs paricipaing in Grmany wr undrgradua sudns from h Univrsiy of Erfur who had compld a las on principal cours of conomics prior o h xprimn. Th 45 subjcs paricipaing in Japan wr undrgradua sudns from h Univrsiy of Osaka from h filds of conomics and social scinc. Subjcs (afr 5 minus of insrucions) had o alloca a minimum of 60 minus for compling h ask. Th maximum im allowd was 90 minus. During his im paricipans wr allowd o mak changs o any of hir answrs. Th avrag Grman (Japans) subjc arnd. 57 Euros (756 Yn). 3. Exprimnal findings Hr w documn in wha rspc parn-basd xpcaions of Grman and Japans subjcs ar similar and how hy diffr. W sar wih h xpcd chang as calculad from h probabiliis givn for ask (a) and compar h mans ovr h rspciv (naional) populaion of subjcs for ach of h 63 parns. Figur 2 shows h xpcd changs in h xprimnal sris licid from h Japans subjcs plod agains h man of h Grman subjcs. Spcifically, a singl poin in h scar plo shows h avrag of h Japans xpcd valus associad wih on parn agains h avrag of h Grman xpcd valus for h sam parn. Th display indicas a srong similariy in answrs across h wo naional pools of subjcs (wih a cofficin of corrlaion of 0.96). Th saisical analysis documns ha h slop of h rgrssion lin is no significanly diffrn from on a h on prcn lvl of significanc. Howvr, h inrcp rm of (saisically diffrn from zro a h on prcn lvl) indicas a small diffrnc in h xpcaions in h wo subjc pools. Hnc, h prdiciv bhavior in h wo subjcs groups is vry similar alhough on avrag Japans subjcs appar o prdic h nx valu of h sris o li slighly lowr han h valu prdicd by h Grman subjcs.

7 Figur 2 abou hr 7 Nx, w urn o ask (b) which addrsss h individual s sima of collciv assssmns. Hr, w obsrv inrsing diffrncs bwn h Grman and h Japans subjcs. For h Grman subjcs figur 3 shows h scar plo of h assssmn of h xpcd chang as aribud o h collciv h avrag of h answrs undr ask (b) for any of h 63 parns plod agains h acual xpcd chang ovr all Grman subjcs (h avrag of h man undr ask a). Figur 4 shows h sam display for Japan. In h Grman cas h saisical analysis indicas ha h rgrssion lin gos hrough h origin (h inrcp is no significanly diffrn from zro) and has a slop of.086, which is diffrn from on a h on prcn lvl significanc. Hnc, h Grman subjcs assss hmslvs as bing mor xrm (in h sns of xpcing a largr absolu chang in h variabl) han hy acually ar. Compar his o h Japans subjcs: h slop cofficin of h rgrssion lin (of 0.892) is significanly diffrn from on and h posiiv inrcp (of 0.038) is significanly diffrn from zro (boh assssd a h on prcn lvl). Hnc, in h cas of Japans subjcs h collciv slf-assssmn is (for mos parns) ovrly rsrvd. Figur 3 abou hr Figur 4 abou hr Summing up, w find ha in rms of h prsonal assssmns of h fuur cours of a financial sris Japans and h Grman subjcs mak vry similar prdicions. Only whn i coms o pondring collciv bhavior do w find inrsing diffrncs bwn h wo naional sampls.

8 4. Parn-basd xpcaions and alrnaiv modls 8 In his scion w documn ha nihr h modl of linar rnd xrapolaion or h BSV-modl adqualy capur h subjcs im sris xrapolaions. Wald ss (saisics rpord blow) indica ha in h simas shown blow hr is no significan diffrnc bwn h rsponss of Grman and Japans subjcs. Hnc, w pool h daa of all our subjcs. Th ndognous variabl in our analysis blow (dnod by X j, 5 X j, 4 ) is h xpcd chang (ha is h probabiliy wighd sum of possibl changs) of h sris from priod 4 o priod 5 as licid in ask (a) avragd ovr all 90 subjcs. This variabl is indxd by j wih j running from o 63. Hnc, w hav 63 obsrvaions. Tabl liss hs xpcd changs for all 63 diffrn parns. Th noion of linar rnd xrapolaion mans ha hs licid xpcaions should b xplainabl by a wighd avrag of h obsrvd laggd changs in h xprimnal sris whr h wighs of h laggd changs should b consan ovr all possibl hisoris of a sris. Th symbols X j, 4, X j, 3, X j, 2, X j, sand for h valus of h sris shown o h subjcs in priod 4, 3, 2 and. Ths valus ar also indxd bcaus ach of h j parns has a diffrn hisory of Xs. In ordr o assss whhr h xprimnally licid xpcaions can b capurd by linar rnd xrapolaion w sima h following rgrssion quaion: X ( X j,4 X j,3 ) + β 2 ( X j,3 X j,2 ) + β3 ( X j,2 X j, ) ε j j 5 X j,4 = 0 + β, β + () This is wha linar rnd xrapolaion mans: h wigh aachd o laggd changs in X varis only wih h lag of h chang (i.., β, β 2, β 3 can diffr). In his rgrssion h firs four rms mak up h par of h xprimnal daa ha can b capurd by h modl of rnd xrapolaion and ε j dnos h par of h licid xpcaion ha canno b xplaind by h linar xrapolaion schm. As hind a arlir a Wald s indicas (wih a p -valu of 0.6 basd on h Chi-squar saisic) ha h β cofficins for h Grman

9 9 daa do no saisically diffr from h β cofficins of h Japans daa. Hnc, w rpor h simas from h poold daa s. Similarly, a Wald s (wih a p valu of 0.54) indicas ha β 3 is no saisically diffrn from zro. Th rgrssion rsul incorporaing his paramr rsricion is h following: X ( X X ) + 0. ( X X ) j, 5 X j,4 = j,4 j,3 7 j,3 j,2 (2) R 2 = 0.864, SEE = (0.028) (0.02) (0.024) As a firs imprssion rnd xrapolaion appars as a rasonabl xplanaion of h xprimnal daa. Howvr, w wan o find ou whhr prominn yps of parns (ha is rnds and zigzags) lad o xpcaions formaion a odds wih h gnralizaion proposd in quaions () and (2). For his purpos w dfin wo dummy variabls: (i) Trnd D is whnvr X has changd in h sam dircion ovr h las hr priods, and 0 ohrwis and, (ii) Zigzag D is whnvr h changs in X hav a rvrsd sign a vry sp ovr h las hr priods, and 0 ohrwis. 5 Equaion (3) alrs quaion () by allowing h β cofficins o diffr across classs of parns. Spcifically, quaion (3) shows h rsuls whn zigzag parns and rnd parns and all ohr parns ar considrd as hr classs of parns: Zigzag [ ( X X ) + 0. ( X X )] D X 0 (3) j, 5 X j,4 = j,4 j,3 300 j,3 j, 2 (0.220) (0.0) (0.0) Trnd [ ( X X ) + 0. ( X X )] D + j, 4 j,3 28 j,3 j, 2 (0.220) (0.0) (0.0) Zigzag Trnd [ ( X X ) ( X X )] ( D ) ( D ) + R 2 = 0.950, SEE = j, 4 j,3 j,3 j, 2 (0.022) (0.06) (0.06) Hr w hav hr diffrn linar modls on for ach class of parns. Wald ss again indica ha h cofficin simas for Grmans and Japans subjcs do no diffr

10 0 significanly ( p valu of 0.8) and an inclusion of X 2 X is no jusifid ( p valu of 0.874). Furhrmor, h hypohsis ha h paramrs of h hr linar modls ar h sam across h hr diffrn classs of parns is rjcd a h on prcn lvl of significanc. Morovr, whn considring 2 R -valus compud from h hr subrgrssions of quaion (3) h modl of linar rnd xrapolaion capurs las wll h subjcs rsponss afr zigzag parns ( R 2 = 0. 33). Th linar modl dscribs h xpcaions br for h cas of rnds ( R 2 = 0. 78) and for h rmaining circumsancs ( R 2 = ). In summary, h simas indica ha h sandard modl of linar xrapolaion is no a saisfacory rprsnaion of agns im sris forcass. Nx w wan o assss whhr h xpcaions licid by our xprimns can b xplaind by h invsor snimn modl of Barbris al. (998). Th BSV-modl proposs ha agns bas hir xpcaions on h noion ha h sris o b projcd ino h fuur (arnings in hir analysis) is ihr gnrad by a rgim (modl ) whr changs ar man-rvring or a rgim (modl 2) whr hy ar rnding whn in fac h sris follows a random walk. Th probabiliis capuring h man-rvring or h rnding bhaviour ar assumd by agns and inrac wih a furhr procss whr a Markov chain drmins which of h wo rgims ruls. In his approach agns upda h probabiliis praining o ihr of h wo modls in a Baysian raional way. This ida of sophisicad updaing of probabiliis ss h BSV-modl clarly apar from h noion of parn-basd xrapolaion proposd hr. Y i is inrsing o invsiga whhr h xpcaions licid by our xprimns ar in accordanc wih h BSV-modl. For his purpos a subs of parns shown o subjcs is of rlvanc. Considr parn 2 (i.., 00, 00, 00, 0). Th subjcs rspons o his parn is an xpcd chang of 0.538, ha is, a rahr srong incras. Th BSV-modl in his cas would no y s a raional for changing xpcaions compard o parn (00, 00, 00, 00): sinc changs (posiiv or ngaiv) ar

11 announcd as possibl, a on-im chang dos no induc h Baysian larnr o alr his probabiliis whhr man-rvrsion or rnding is mor likly o gnra h nx obsrvaion. Th sam argumn holds for parns 4 and 4. A rvision of probabiliis should occur only afr a furhr chang, ha is, afr parn 5 (i.., 00, 00, 0, 02) or afr parn 7 (i.., 00, 00, 0, 00). Furhr, considr our subjcs rsponss o on of hs parns (parn 7). Hr, h invsor snimn yp agn would calcula a probabiliy of modl of abov 0.5 and hnc should projc a likly incras (i.., a rvrsion of h las dclin). In conras, h subjcs xpcaions in his cas clarly poin oward a dclin. Las, considr parn 30 (i.., 00, 0, 00, 0). This squnc of ups and downs should lad o an assssmn of incrasd probabiliy of chang-rvrsion and hus an xpcd dcras for h immdia fuur. Th xprimnal daa insad show a dclin vry clos o zro (-0.030). Aloghr, hs findings indica ha h xpcaions licid by our xprimns canno b xplaind by h probabiliy updaing ida of h BSV-modl. 6 Th findings in his scion documn ha h parn-basd xpcaions masurd hr ar disinc from ohr noions of xpcaions formaion. Hnc, w procd o show how hs xprimnal xpcaions daa can b ransformd o a modl of agns im sris xpcaions which can hn b usd in h conomric analysis of financial daa. 5. Applying h xprimnal daa o mpirical analysis This scion dscribs how h informaion in h xprimnal daa can b usd o compu a im-sris of xpcd valus for any paricular conomic variabl. Mor concrly, w us h xpcd changs (avragd ovr all subjcs in our sudy) of h sris as licid in ask (a) afr h 63 diffrn parns rpord in Tabl o compu a hisorical sris rflcing how agns forcas undr h assumpion ha, on avrag, agns funcion lik our subjcs. Th ky problm o b solvd hr is o scal h sylizd xprimnal parns o acual conomic im sris. Th procss of scaling h xprimnal daa o

12 2 hisorical daa consiss of svral sps. Firs, w spli h hisorical im sris ino rolling daa windows of four daa poins a a im. Scond, focusing on on daa window a a im corrlaion analysis (o b xplaind in mor dail) is usd o drmin h sylizd parn mos similar o h givn four daa poins in h hisorical sris. For his sp h lis of 63 parns has firs o b shornd o h 50 parns ha ar (in rms of h changs) linarly indpndn. Considr Tabl and no ha h changs ovr im in h 3 pairs of parns numbrd 2-3, 4-9, 5-, 7-3, 4-39, 5-4, 7-43, 9-49, 20-5, 22-53, 29-59, 30-6, ar rlad o ach ohr by a facor (dnod β ) of wo. Tha is, in rms of h changs of h xprimnal sris 3 of our parns show changs wic h siz of h changs in xacly on ohr parn. So, for xampl, in firs diffrncs parn 3 is wic parn 2, likwis parn 9 is wic parn 4. From h subjcs rsponss o h 3 hus rlad pairs w can driv an sima of how subjcs rac whn a parn is alrd in rms of h siz of h sps (i.., o a scaling). In ordr o quanify his scaling rlaion w rgrss h rpord xpcd changs (as shown in abl ) afr h 3 parns 3, 9,, 3, 39, 4, 43, 49, 5, 53, 59, 6, 63 o h xpcd changs afr parns 2, 4, 5, 7, 4, 5, 7, 9, 20, 22, 29, 30, 32. This yilds a rgrssion cofficin of.5 (wih sandard rror 0.). From his sima (and rjcing a h on prcn lvl h hypohsis ha h rgrssion cofficin is wo) i is clar ha agns rsponss do no simply vary in proporion o changs. Insad, basd on h sima w us 0.6 h scal ransformaion β, maning ha for valus of β of 2, 3, 4 c w will us facors of proporion.5,.93, 2.29, c. Figur 5 shows his scaling rlaion. Nx, w mach h hisorical sris (i.., four conscuiv obsrvaions a a im) o h sylizd parn mos similar o i. Hr, w ak h absolu lvl of h corrlaion cofficin bwn h logarihmical acual daa and ach of h 50 linarly indpndn parns as h cririon. Finally, h hus slcd mos similar sylizd parn is scald o h hisorical daa by linar rgrssion.

13 3 Figur 5 abou hr To illusra h dscribd procdur ak h firs four daa poins of h log of h dflad sock pric sris usd by Shillr (98) wih h daa poins 4.70 in 87, in 872, in 873, and in 874. Th parn wih h highs (absolu) corrlaion wih hs hisorical daa is parn 47, ha is, h squnc.00,.02,.03,.02. Th avrag subjc paricipaing in our sudy in his cas xpcs h xprimnal sris o chang by α.00β pr cn. Minimizing h sum ( ) 2 + ( α.02β ) 2 + ( α. 03 β ) 2 ( α.02β ) 2 + whil nforcing 4.70 = α +. 00β (making h acual and fid parn sar from h sam poin) yilds h cofficin simas α = and β = Figur 6 shows h fi of h approprialy scald parn 47 (i.., , , , ) o h firs four daa poins of h Shillr sock mark daa s. Basd on his procdur h compud xpcd sock pric for h yar 875 is xp 0.6 ( ) = By a sp-wis applicaion of h dscribd procdur w rach h xpcd valu on sp ahad for h whol hisorical im sris. Figur 6 abou hr 6. Sock Pric Expcaions In his scion w pursu h issu of sock pric xpcaions and swich o h us of monhly daa compild and updad by Robr Shillr (s Shillr, 2000, for a dscripion). W firs compu h sris of h xpcd sock pric saring in May 87 up o Dcmbr This sris of (parn-basd) xpcd sock pric valus can now b usd for furhr conomric analysis. L us firs subjc h xpcaions daa o a sandard s of raionaliy.

14 4 Equaion (4) shows h rsul whn w rgrss h sock pric on h parn-basd xpcd sock pric and a consan: 7 q = q (4) (0.007) (0.00) R 2 = 0.997, SEE = , DW =. 87 A Wald s (wih a p -valu of 0.393) dos no rjc h hypohsis ha h consan is zro and h cofficin of q is on. Hnc, parn-basd xpcaions hr do no viola a basic rquirmn of raionaliy. Appndix 2 documns various furhr fficincy ss applid o h parn-basd sock pric xpcaions and documns ha no clar rjcion of fficincy mrgs. Nx, considr an mpirical vrsion of h BSV-modl of invsor snimn as an alrnaiv bhavioral modl of xpcaions formaion. On way o compar ha modl wih h parn-basd xpcaions modl is o sima is paramrs using Shillr s monhly sock pric daa. Th BSV-modl is basd on four probabiliy paramrs and a paramr capuring h siz of a ypical shock o h sris: Paramr π L dnos h probabiliy ha h sris rnds in h man-rvrsion rgim (modl ), π H dnos h probabiliy ha h sris rnds in h rnding rgim (modl 2), λ is h probabiliy ha a swich occurs from modl o modl 2 and vic vrsa. 8 Th las paramr o b drmind is y which dscribs h (absolu siz) of h shock o h sris. Barbris al. (998, p. 323) documn formally how h xpcd chang of h sris o b projcd dpnds on hs paramrs and w sima h rlvan paramr valus by grid-sarching for h minimum of h sum of squard forcas rrors. Th paramrs hus drmind ar λ = 0. 49, π = 0. 50, π = 0.58, and y = Th BSV-modl givs a sum of squard rrors ovr h priod H L from 87 o 2006 of In comparison h parn-modl (assssing q q ) givs a lowr sum of squard forcas rrors of This is noabl considring ha for h formr

15 5 modl w vary four paramrs o find h vrsion ha forcass bs (x an) and for h lar w do no vary any paramr. Th dcisiv poin, howvr, is ha h paramrs of h BSV-modl simad from acual sock prics ar no in accordanc wih h sory of Barbris al. (998). In ordr for hir bhavioral modl o capur inrsing ass pric phnomna invsors should bas hir xpcaions, in paricular, on lowr valus of λ and π L. Hnc, i bcoms apparn ha minimizing h sum of squard rrors wih hisorical ralizaions of sock prics dos no yild paramr valus ha could b considrd bhaviorally snsibl. In a furhr sp w invsiga whhr parn-basd sock pric xpcaions can shd ligh on issus of sock pric drminaion. Spcifically, w ask whhr vidnc for comping modls of xpcaions can b found in prsn valu modls of sock prics (s Chow, 989). Th analysis sars wih h quilibrium condiion ( Q D ) = + Q δ +, (5) whr Q dnos h pric of a sock (or indx) and D dnos h dividnd paid o h ownr of h sock. W sar wih h hypohsis of raional xpcaions. In ordr o driv a sabl hypohsis in logs of obsrvabl variabls w formula raional xpcaions as Q = + η + Q + wih ln + = c + u+ η whr c is a consan and E u = + 0. Using Campbll s and Shillr s (988) approximaion ρq + + ( ρ) d + k for ( Q + + D ) [ ( )] ρ = + xp δ, δ is h man of q + h following rlaionship is implid ln whr d and k = ln [ + xp( δ )] δ xp( δ ) [ + xp( δ )] q ( lnδ + c k ) + ρ q ρ ( ρ) d + u = ρ ρ. (6) Esimaing his linar rlaionship bwn q on h on hand and q and d on h ohr hand by ordinary las squars w find

16 6 q = q d (7) (0.0062) (0.005) (0.008) R 2 = 0.997, SEE = , DW = According o a Wald s (wih a p -valu of 0.427) h simad cofficins of (7) agr wih h paramr rsricion on h cofficins of (6). 9 Nx, considr h hypohsis of adapiv xpcaions formaion. This is h prfrrd xpcaions hypohsis by Chow (989). According o Chow s analysis wih annual daa h noion of adapiv xpcaions is in accordanc wih h sock pric whil h noion of raional xpcaions is no. Th spcificaion o b simad undr adapiv xpcaions (Chow, 989, p. 382) is q λ ( k lnδ + c) λ λ( ρ) = + q + d λρ λρ λρ + ε, (8) λρ which in rms of h variabls includd is h sam as (6). Whn comparing h simad cofficins of (7) wih h modl paramrs of (8) w find ha h simad cofficins ar no consisn wih h horically prsumd paramr valus for λ and ρ bwn 0 and. This dos no lnd crdnc o h hypohsis of adapiv xpcaions in his conx. Finally, considr h hypohsis of parn-basd xpcaions. Basd on h Campbll and Shillr approximaion schm h sock pric quaion in log form in his cas bcoms ( ) d u q = k + lnδ + ρq + + ρ +, (9) whr now q + is h log of h parn-basd xpcaion of h sock pric formd a. Clarly, givn ha q + builds on q i is corrlad wih u and hnc his variabl nds o b insrumnd. Using h sima

17 7 q + = q 0.425q d d 2 (0) (0.009) (0.029) (0.029) (0.083) (0.082) R 2 = , SEE = , DW =. 968, ha dos no includ q as a rgrssor o compu h fid variabl insrumning for d along h sam lins) o sima qˆ +, w procd (afr q = qˆ d () (0.009) (0.004) (0.007) R 2 = , SEE = , DW = A Wald s (wih a p valu of 0.498) suppors h rsricion on h cofficins for ˆ + and q d suggsd by (9). Finally, a comparison wih h prvious sima undr raional xpcaions indicas ha h hypohsis of parn-basd xpcaions fars br whn assssd by h sandard rror of h sima. 7. Exchang Ra Expcaions This scion aks up h analysis of h drminans of h forward discoun on h forign xchang mark. To sar wih, w apply h procdur inroducd o compu h parn-basd xpcaions of h spo xchang ra of h pound srling vis-à-vis h U.S. dollar (dnod by s ). Th daa usd ar monhly daa (nd of monh daily numbrs) providd by h bank of England for h priod 979:0 o 2006:2. As a nx sp h s for raionaliy of xpcaions suggsd by Froo and Frankl (989) is conducd wih hs daa. This s avoids problms du o ponial rrors in h masurmn of xpcaions. I consiss of running a rgrssion whr h xpcaions rror is rgrssd on a consan and h forward discoun from h prvious priod. Th forward discoun xprssd in prcnag rms is h diffrnc bwn h log of h forward ra ( f ) for dlivry in h nx priod

18 8 drmind in a givn priod and h log of h spo ra ( s ) in ha priod. Hr is h rsul of his rgrssion: ( f s ) s s = (2) (0.002) (0.927) R 2 = , SEE = , DW = , whr s dnos h parn-basd xpcaion for im formd in. A Wald s (wih a p valu of 0.62) indicas ha w canno rjc unbiasdnss and fficincy (i.., h hypohsis ha h wo cofficins do no diffr from zro). Wih his rsul parn-basd xpcaions prform br han survy xpcaions of xchang ras (s Frankl and Froo, 989, and Cavaglia al. 994). Whn including on h righ hand sid of (2) laggd xpcaions rrors (wih lags of on and wo monhs) h assssmn of h fficincy of h parn-basd xpcaions has o b qualifid. In his spcificaion h Wald s rjcs h hypohsis of fficincy a h fiv prcn lvl of significanc. Hnc, whn x-pos valuaing parn-basd xchang ra xpcaions hy appar no o b fully fficin. Morovr, whn comparing our xpcaions daa agains random walk xpcaions (i.., saic xpcaions) parn xpcaions ar no suprior. Th roo-man squar forcasing rror of h parn xpcaions ovr h whol sampl is whras saic xpcaions yild a numbr of Hnc, in rms of prdiciv powr h parn-basd xpcaions ar no abl o ba h random walk modl for h xchang ra daa sudid hr. Howvr, h ky issu is no h prdiciv accuracy of parn-basd xchang ra xpcaions bu h qusion whhr his xpcaions hypohsis hlps o xplain imporan characrisics of xchang ras. In his rspc on of h mos holy dbad issus is h qusion how xpcaions rrors and h risk prmium affc h forward discoun (s,.g., Froo and Frankl, 989). This is addrssd hr by simaing h following rgrssion quaion

19 ( f s ) + 0. ( s s ) f s = (3) (0.000) (0.025) (0.004) R 2 = , SEE = , DW = No ha for h scond xplanaory variabl (h xpcd chang of h xchang ra) w insrumn s by s in ordr o avoid any possibl ndogniy bias. Whn applying a Wald s (wih a p valu of 0.249) w canno rjc h hypohsis ha h consan in quaion (3) is zro and ha h sum of h wo rmaining cofficin is on. Equaion (4) shows h oucom of simaing h rgrssion wih hs rsricions: ( f s ) + 0. ( s s ) f s = (4) (0.005) (0.005) R 2 = , SEE = , DW = Thus, h following picur mrgs: h forward xchang bias is highly prsisn and affcd by h xpcaions rm. In h long run an xpcd chang (h xpcaions rror if w undrsand s o b an insrumn for s ) in h xchang ra is fully rflcd in a corrsponding chang in h forward ra rlaiv o h spo ra. In h shor run, howvr, h forward ra adjuss slowly giving h apparanc of a im varying risk prmium. This inrpraion is in accordanc wih rsuls by Ziz (995) and Lohian and Wu (2003). 8. Conclusions This sudy licis parn-basd xpcaions in a comprhnsiv way in ordr o dvlop a dscripiv modl of acual agn bhavior. Th xprimnal vidnc indicas ha parn-basd xpcaions ar vry similar in Grmany and in Japan. This suggss ha h way humans xrapola im sris is lil affcd by diffrncs in culur and hisory. Economric analysis indicas ha h daa on xpcaions collcd canno b adqualy rprsnd ihr by h modl of linar rnd xrapolaion or by h invsor snimn

20 20 modl of Barbris al. (998). For xampl, xpcaions of changs afr zigzag movmns canno b undrsood as a wighd sum of pas changs. Th papr furhr inroducs a mhod for using h xpcaions daa gahrd for compuing xpcaions for any conomic im sris. Economric analysis of hisorical daa shows ha hus drivd sock pric xpcaions and xchang ra xpcaions hlp o xplain variaions in sock prics and in h forward discoun on h forign xchang mark. Hnc, h xprimnally informd modl of parn-basd im sris xrapolaion shows promis as a modl of xpcaions.

21 Appndix : Th insrucions for h xprimn (ranslad from Grman) 2 You ar paricipaing in an xprimn invsigaing h forming of xpcaions on financial marks. Hnc, hink of h daa shown o you in h xprimn as sock prics or xchang ras. In wha follows you ar prsnd wih 63 cass of how h pric of an ass (lik a sock or a currncy) can dvlop ovr four priods. Th xprimn is simplifid inasmuch as only h four following sps ar possibl: An incras by 2 (i.., a chang by +2) An incras by (i.., a chang by +) No chang (i.., a chang by 0) A dcras by (i.., a chang by ) A dcras by 2 (i.., a chang by 2) I is your ask in his xprimn o forcas h dvlopmn for h fifh priod for all 63 cass prsnd o you. This mans (his is ask a) ha you hav o assign probabiliy valus o h diffrn possibiliis of h coninuaion of h displayd pah (+2, +, 0, -, -2). Plas slc probabiliy valus in sps of 0. and no ha h sum of h probabiliis mus qual. By way of an xampl you s blow hr of many possibl answrs:. xampl 2. xampl 3. xampl In addiion, w would lik o obain your simaion of h avrag of h forcass of all s prsons aking par in h xprimn hr oday for ach of h 63 cass (his is ask b). This mans ha you ar askd o sima h forcass of h ohr s prsons as accuraly as possibl. Spcifically, w ar asking you for a singl valu bwn +2.0 and 2.0 down o on dcimal poin. Th following will giv you a hin of a possibl procdur o solv his

22 22 ask: Sar wih your own xpcd valu (your xpcd valu of h chang of h displayd variabl is h sum of h possibl changs wighd wih hir probabiliy valus as givn by you) in ask (a). In h hr xampls abov his would b 0 (x. ), + (x. 2) and 0.5 (x. 3). Now prdic h avrag of h xpcd valus of all s prsons in (a) and wri down his valu. If you, for xampl, in his posiion nr a valu of.2 whil your prsonal xpcd valu is 0 (as in h. xampl of possibl answrs) you judg h avrag of forcass o b significanly abov your prsonal forcas. You will rciv, as a financial compnsaion, a basic f of 4 Euros. For answrs o asks (a) and (b) you addiionally arn 0 Euro cns pr cas (i.., a maximum of 6.30 Euros). Morovr, w would lik o masur h dgr of your confidnc rgarding your answr in ask (b). In his par of h xprimn you can gain, or los, mony. Spcifically, (his is ask c) w wan o know how much (bwn 0 and 0 cns) you b on your assssmn in (b) bing no mor han 0.5 abov or blow h acual avrag of all subjcs xpcd valus in (a). If your assssmn is wihin a rang of 0.5 of h avrag, you will gain h amoun you nr in (c). Howvr, if your assssmn dvias by mor han 0.5 you los his amoun. Thus, your final payoff consiss of h paricipaion f of 4 Euros and bwn 0 cns and 20 cns pr cas.

23 Cas : (a) your prsonal probabiliy forcas (down o on dcimal poin) chang probabiliy (b) your assssmn of h avrag of h xpcd valus of all s prsons (down o on dcimal poin) (c) h amoun you b on your assssmn of h avrag forcas of h s prsons (figur wihou dcimal poins bwn 0 and 0) [Hr cass 2 o 63 follow]

24 24 Appndix 2: This appndix documns rsuls for fficincy ss for parn-basd monhly sock pric xpcaions Hr, w sar wih an fficincy s ovr h full daa s (i.., form 87:05 o 2006:2). Equaion (A) shows h rsul of rgrssing h xpcaions rror on laggd rms of h xpcaions rror: ( q q ) 0. ( q q ) q q = (A) (0.00) (0.037) (0.040) R 2 = 0.02, SEE = , DW = A Wald s rjcs h hypohsis ha h cofficins in his rgrssion ar joinly zro a h on-prcn lvl of significanc. Howvr, considr h rsuls from similar ss using rolling daa windows squnially covring wo dcads saring wih h sampl 87 o 889 procding o 890 o 909, 90 o 929 up o h las sampl 990 o Tabl A prsns h rsuls of simaing quaion (A) for hs shornd sampls. Th abl prsns h simad cofficins and h probabiliy lvl a which a Wald s rjcs h null hypohsis ha all cofficins ar zro. From his squnial analysis no clar rjcion of h fficincy of h parn-basd sock pric xpcaions mrgs. Tabl A2 givs basically h sam picur wih simas basd on nominal sock pric daa insad of dflad daa. Finally, h forcas prformanc of parn-basd xpcaions is compard wih saic xpcaions. Tabl A3 prsns roo-man squar rrors ovr h svn sub-sampls usd abov. Th numbrs indica ha in all sub-sampls random walk xpcaions wr infrior o parn-basd xpcaions. Th rsuls in Tabl A4 documn ha his conclusion is also warrand whn analyzing nominal sock pric variabls.

25 25 Tabl A: Efficincy ss ovr sub-sampls wih dflad sock pric daa probabiliy Sampl Consan q q q 2 q 2 for rjcion of fficincy 87M07 889M M0 909M M0 929M M0 949M M0 969M M0 989M M0 2006M Tabl A2: Efficincy ss ovr sub-sampls wih nominal sock pric daa probabiliy Sampl Consan q q q 2 q 2 for rjcion of fficincy 87M07 889M M0 909M M0 929M M0 949M M0 969M M0 989M M0 2006M

26 26 Tabl A3: Comparison of roo-man squar forcas rror of parn-basd xpcaions and saic xpcaions wih dflad sock pric daa Sampl Parn-basd Exp. Saic Exp. Raio 87M07 889M M0 909M M0 929M M0 949M M0 969M M0 989M M0 2006M Tabl A4: Comparison of roo-man squar forcas rror of parn-basd xpcaions and saic xpcaions wih nominal sock pric daa Sampl 87M07 889M2 890M0 909M2 90M0 929M2 930M0 949M2 950M0 969M2 970M0 989M2 990M0 2006M2 Parn-basd Exp. Saic Exp. Raio

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29 29 Röhli, Tobias F. (2007), Expcaions, Raionaliy and Economic Prformanc: Modls and Exprimns, Chlnham: Edward Elgar. Rumlhar, David E., Jams L. McCllland and h PDP Rsarch Group (986), Paralll Disribud Procssing: Exploraions in h Microsrucur of Cogniion, Volum : Foundaions, Cambridg, MA: MIT Prss. Shillr, Robr J. (98), Do Sock Prics Mov Too Much o b Jusifid by Subsqun Changs in Dividnds? Amrican Economic Rviw 7, Shillr, Robr J. (2000), Irraional Exubranc, Princon, Princon Univrsiy Prss. Usunir, Jan-C. (998), Inrnaional and Cross-Culural Managmn Rsarch, London: Sag. Ziz, Joachim (995), Som Evidnc on h Efficincy of h Forward Mark for Forign Exchang from Mon-Carlo Exprimns, Economic Journal 05,

30 30 Tabl : Avrag xpcd changs Expcd Expcd Parn Priods chang Parn Priods chang numbr o priod 5 numbr o priod

31 3 Figur : Visual display of cas

32 32 Figur 2: Expcd changs in Japan agains xpcd changs in Grmany (man xpcd changs ovr all paricipans for all 63 parns)

33 33 Figur 3: Assssmn of collciv xpcaions agains acual collciv xpcaion: h Grman daa

34 34 Figur 4: Assssmn of collciv xpcaions agains acual collciv xpcaion: h Japans daa

35 35 Figur 5: Scaling h licid xpcaions: xpcd chang as a funcion of β

36 36 Figur 6: Acual annual daa (naural logs) and fid parn numbr 47 for sock pric acual daa fid parn

37 37 Foonos Parn-basd xpcaions of ordinary popl should no b confusd wih h various mhods applid in so-calld chnical analysis as usd by som profssional financial forcasrs (s,.g. Brock al, 992). 2 Such an as-ws comparison is also inrsing givn h xisnc of a subsanial liraur addrssing culural diffrncs in conomic dcision making (s, Hofsd 997, Usunir 998, Gannon and Nwman 2002, and Maock and Bannon 2003). Wih rspc o xprimnal invsigaions in h fild of conomics auhors lik Cason al. (2002) and Brands al. (2004) hav prsnd rlvan inrnaional comparaiv sudis. Brands al. (2004) conclud from hir xprimn on volunary conribuion mchanisms ha hr ar only minor bhavioral diffrncs across Japan, h Nhrlands, Spain, and h U.S. In conras, Cason al. (2002) rpor significan diffrncs in h bhavior of Japans subjcs as compard o U.S. subjcs. Th main bhavioral rai invsigad in ha sudy is spiful bhavior. Concrning culural ffcs ha influnc h formaion of xpcaions hr is, o my knowldg, no rlvan vidnc. Wih rspc o forward looking bhavior Aggarwal and Mohany (2000) compar Japans and U.S. bhavior. Thir analysis of survy daa of macroconomic variabls dos no indica imporan sysmaic as-ws diffrncs. As o financial bhavior hr is a las on sudy ha indicas ha culural facors mak a diffrnc: Brown al. (2002) documn ffcs of Chins culur (and suprsiion) on ass prics. A diffrn (non-culural) naional variaion in ass pricing is documnd in Garr al. (2005): in a muli-counry comparison hs auhors documn ha variaions in h lngh of h day across counris can xplain som cross-naional diffrncs in h daa. 3 This rfrnc offrs a daild dscripiv analysis of h Grman daa in chapr 9. 4 Manski (2004) rcnly argud srongly for liciing xpcaions in h form of probabiliis.

38 38 5 Givn ha w only show 63 parns (i.., no parns saring wih a ngaiv chang) h condiion for h rnd dummy X ( X X ) 0 Trnd D o b is ( 4 X 3 ) > 0, ( X 3 X 2 ) > 0, 2 > and likwis h condiion for h zigzag dummy Zigzag D o b is ( X ) >, ( X X ) < 0, ( X X ) > X. 6 Wr i no for hs conradicions h xprimnal daa would lnd hmslvs o an mpirical drminaion of h paramrs of h BSV-modl. Rsponss o parns 5, 7, 20, 22, 30 and 32 would suffic o calcula valus for all fixd probabiliy valus of h BSVmodl. 7 Th sandard rrors rpord ar Whi hroskdasiciy-consisn simas. Thy ar no usd in any of h ss rpord. 8 W simplify hr by qualizing wo probabiliis, ha is, by sing λ λ = λ = 2. 9 Insrumning h dividnd variabl as suggsd by Chow (989) maks no noicabl diffrnc in h rsuls on h lvl of monhly daa usd hr.