Nws and Expcaions in Financial Marks: An Exprimnal Sudy Gordon Douglas Mnzis School of Financ and Economics, Univrsiy of Tchnology, Sydny CAMA, Ausralian Naional Univrsiy Danil John Zizzo School of Economics and CBESS, Univrsiy of Eas Anglia CAMA, Ausralian Naional Univrsiy Absrac W considr an xprimnal sing whr agns rciv on sylizd pic of informaion a a im abou h valu of a financial ass. W find ha Knighian uncrainy abou h prior disribuion of ru financial ass valus dos no hampr dcision making by agns and marks. On a man squard rror cririon, Baysian updaing is closr han simpl avraging in prdicing mark prics and individual bids and offrs, vn in ramns wih uncrainy whr Baysian updaing should no b fasibl givn h limid informaion s. Baysian updaing also ouprforms adapiv xpcaions in rlaion o mark prics. Kywords: uncrainy, xpcaions, informaion, framing. JEL Classificaion Cods: C9, D83, D84, F3, G. Tl: +6--954778; fax +6--95477. Email addrss: gordon.mnzis@us.du.au Tl: +44-603-593668; fax: +44-603-45659. Email addrss: d.zizzo@ua.ac.uk W ar graful o h Briish Acadmy for financial suppor, o paricipans o prsnaions in Lyon, Norwich and Sydny for hlpful fdback, and o Jams Wason, Nicol Huang, Ki Tsusui and Maya Ellio for rsarch assisanc. Th usual disclaimr applis. Th xprimnal insrucions can b found in an onlin appndix.
. Inroducion This papr considrs a simpl xprimnal sing whr agns rciv on sylizd pic of informaion a a im abou h valu of a financial ass. W considr whhr, in a mark sing, asss will b pricd diffrnly if hr is only limid informaion abou h naur of h financial ass. Rcn vns bar ou h cnraliy of shars and forign xchang for conomic prformanc and for h ransmission of global shocks. W hrfor considr h pricing of shars and forign xchang, afr ach pic of informaion is rcivd. Our spcific qusion is: dos Baysian updaing, which rquirs full informaion abou h financial ass, prform wors han mploying h maximum liklihood simaor, which dos no? To xpand, h ky qusion w ry o answr is whhr h ssnial uncrainy (as opposd o risk, following Knigh s, 9, disincion) inrinsic in h informaion s availabl o radrs maks hm upda hir blifs diffrnly, rsuling in a diffrn rajcory of mark prics. W isola h ffc of uncrainy by having a ramn whr h disribuion of ru financial ass valu is vry limid, and on in which i is fully known. This is of inrs bcaus boh individual choic and mark xprimns, (as in, Ellsbrg, 96, and Sarin and Wbr, 993) show ha subjcs dislik ambiguiy, i.. Knighian uncrainy. Whil our papr is no abou ambiguiy avrsion pr s, his liraur shows inrinsic ambiguiy migh mar whn agns procss informaion and rad as a rsul. Idnifying whhr uncrainy mars is imporan for anohr, and mor gnral, rason: if agns do no hav sufficin informaion abou h prior disribuion of h valu of h financial ass (including h sandard dviaion), Baysian signal xracion is no fasibl in principl on h par of raional xpcaions agns, unlss hy appnd ducad gusss
abou h prior disribuion o hir infrnc procdur. If hy rfrain from hs auxiliary assumpions, h raional procdur is maximum liklihood simaion (MLE) basd on an avrag of h signals rcivd. Thrfor, how w modl xpcaion formaion is ponially a funcion of how agns procss h inrinsic ambiguiy in h mark nvironmn. Our ky and surprising finding is ha Baysian modls of xpcaion formaion nd o ouprform h corrsponding MLE modls, vn in h no prior disribuion knowldg ramns. W prform our xprimn in wo diffrn frams for h mark (an xchang ra mark fram vs. sock mark fram), as w ar conscious ha subjcs can somims mor or lss clos o raional choics dpnding on h fram usd (.g., Cosmids and Tooby, 99), and w find ha h ky finding of ouprformanc by Baysian xpcaions is robus bwn frams. 3 Scion prsns h horical sup and soluions undr diffrn xpcaions rgims. Scions 3 and 4 dscrib h xprimnal dsign and rsuls, whil scion 5 concluds.. Thorical Sup and Soluions for Diffrn Expcaion Assumpions. Sup Th horical sup is on in which agns mus infr h valu of a singl paramr drawn from a disribuion: ~ ( Λ, σ ) In priod zro, agns rad wihou any sampl informaion. Thrafr, hy rad on h basis of a squnc of noisy signals + ~ (0, σ ) which conain informaion abou his unknown paramr: 3 Our analysis also includs a sandard adapiv xpcaions (ADE) spcificaion, wih a wigh β assignd o h prvious priod s obsrvaion in forming h currn xpcaion. ADE ar includd as hy ar ofn purpord o b good daa dscripors dspi a lack of horical microfoundaion (.g., Mankiw, 00) or vn any modl consisncy rquirmn.
3 x x x + + +... and hy mus guss afr ach x i is rvald. W now oulin wo soluions.. Th Maximum Liklihood Soluion (MLE Expcaions) From priod on onwards, h simpls soluion is simply o avrag h signals: () ˆ x i i i i M + Condiional on, his has all h dsirabl las-squars and maximum liklihood propris V E M M ) ˆ ( ) ˆ ( σ and h Gauss-Markov horm holds. 4 Nvrhlss, h simaor dos no us all informaion. Thr is prior informaion on h disribuion of, namly Λ and σ..3 Th Baysian Signal Exracion Soluion (Baysian xpcaions) Considr a class of soluions of h form: () ) ( ˆ ˆ ) ( ˆ B B B x x θ θ θ θ + Λ + 4 Tha is, condiional on i is h minimum varianc simaor among h class of all linar unbiasd simaors.
4 In ohr words, h firs guss is basd on a wighd avrag of h known man of, namly Λ, and h signal, wih h opimal wighs o b drmind prsnly. Thrafr, agns us a rcursion wih h sam wighs. In gnral, opimal wighs for () dpnd on h im horizon for which signals ar availabl. Bu h soluion rquirs numrical opimizaion, so w driv h opimal wigh θ for h firs priod, and assum i prvails in h soluion. 5 Considr h firs priod: ˆB θ Λ + ( θ ) x θ Λ + ( θ )( + ) ( θ ) + + θ ( Λ ) ˆB ( θ ) + θ ( Λ ) W now sk θ ha will minimiz h xpcd valu of h man squard rror (MSE). Whn aking h xpcaion, w do no condiion on, bcaus w ar sking an opimal wigh ha aks accoun of is disribuion. W firs driv h uncondiional man and varianc (V[.]): ˆB E( ) ˆB V ( ) E( θ ) + θ E( Λ ) 0 V (( θ ) ) + V ( θ ( Λ )) (( θ ) σ + θ σ ) W now minimiz h MSE: 5 Equaion () can b wrin as θ )[ + θ +... + θ ] + + θ ( Λ ). An Opimal θ which minimiz i ( E( ˆ -) /, as in h x, rquirs a numrical soluion, as i is a funcion of. Whn his B xprssion is opimizd for 8, as in our xprimn, h opimal θ is 0.7.
5 E[( ˆ ) B ] V ( ˆ ) + { E( ˆ )} B ˆB de[( ) ] ( θ ) σ + θ σ 0 dθ σ θ σ + σ B (( θ ) σ + θ σ ) + 0 Our soluion is calld B ˆ bcaus his is a classic Baysian signal xracion soluion. 6.4 Adapiv Expcaions (ADE) Undr adapiv xpcaions, agns ar consrvaiv in h sns ha hy anchor hir sima of on h las priod s pric of h financial ass, calld p -. Thus, ˆ A β p + rror. Thr is no horical valu for β, so i mus b simad. I is obaind by Las-Squars whr w proxy p β p rror. + A ˆ wih h currn pric p. Tha is, w run h rgrssion 3. Exprimnal Dsign Th xprimn was conducd bwn January and Jun 008 a h univrsiy of h scond auhor. 7 Apar from h xprimnal insrucions and a conrol qusionnair, h xprimn was fully compurizd. A oal of 40 subjcs paricipad in h 48 sssions: fiv subjcs paricipad o ach sssion, and hy paricipad o on of four ramns (discussd blow), giving a oal of indpndn obsrvaions pr ramn. Subjcs wr randomly sad in h laboraory. Compur rminals wr pariiond o avoid communicaion by visual or vrbal mans. Subjcs rad h xprimnal insrucions and answrd a conrol qusionnair bfor bing allowd o procd wih h asks. 6 On subl diffrnc is ha w do no rquir normal disribuions, vn hough, in fac, w do us normal disribuions for our xprimn. 7 Th xprimnal insrucions ar providd in lcronic appndix A.
6 Exprimnal suprvisors individually advisd subjcs wih incorrc answrs in h qusionnairs. Th xprimn lasd up o hours and was dividd ino four sags, ach dividd in urn in 9 rading priods. Th priods wr dividd ino priod 0, whn agns didn rciv a signal, and priods o 8, whr hy did, as dscribd blow. Th asss radd wr ihr forign currncy (in an xchang ra fram) or shars (in a financial fram). Subjcs did no know xacly wha h inrinsic valu of h financial ass was, hough hy knw ha asss raind h sam valu hroughou ach sag hough no across sags. In h limid prior knowldg ramns, subjcs only knw ha h financial ass valu was drawn from a prior symmric disribuion wih man (i.., ha on avrag a uni of h financial ass could b ransfrrd on-for-on ino pounds a h nd of h sag), bu hy did no know h xac shap or, crucially, sandard dviaion of h prior disribuion. As a rsul, Baysian updaing was no fasibl unlss w assum ha subjc bhavd as if hy mad auxiliary assumpions. Tchnically, avraging is h only fasibl soluion. In h prior knowldg disribuion, subjcs wr providd informaion abou h prior disribuion in h form of a abl of frquncis (ffcivly, a hisogram); informaion was providd as a abl of frquncis rahr han of probabiliis sinc i is known ha subjcs procss informaion mor fficinly in rms of h formr han in rm of h lar (.g. Hrwig and Gigrnzr, 999). As a rsul, subjcs could infr h sandard dviaion of h prior disribuion, and Baysian updaing bcom fasibl. Th disribuion acually usd was h sam in boh h limid prior knowldg and h prior knowldg ramns. I was normal wih man and sandard dviaion 0.5: only h xn of knowldg abou h disribuion diffrd across ramns. Ovrall, h xprimn had a facorial dsign crossing h fram mployd (sock mark vs. xchang ra mark) and h knowldg abou h prior disribuion. W had four
7 ramns ovrall, AE (knowldg, xchang ra fram), BE (limid knowldg, xchang ra fram), AS (knowldg, sock mark fram) and BS (limid knowldg, sock mark fram). (Insr Tabl abou hr.) A h bginning of ach priod subjcs rcivd 0 unis of ach of wo asss (forign and hom currncy in on fram, cash and shars in h ohr fram), wih only imprfc informaion abou hir inrinsic valu a which h financial asss would ulimaly b convrd o pounds. Th informaion ha subjcs had diffrd dpnding on h ramn, and is dscribd blow, bu was common across subjcs of any givn sssion. Trad occurrd according o a Walrasian claringhous aucion mchanism, wih ach subjc bing askd o provid a buying pric and a slling pric. Th xclln fficincy propris of Walrasian aucions ar wll-known, and clos o ha of doubl aucions (.g., Smih al., 98). 8 Trad could only ak plac using h ndowmns rcivd a h bginning of h rlvan rading priod. A gnric xprimnal sssion is rprsnd in h diagram blow. (Insr Figur abou hr.) A h bginning of ach sag a nw valu for h ru valu of h financial ass was drawn applicabl o h sag, and subjcs knw ha. From h sar of priod onwards, subjcs rcivd an indpndn noisy signal abou h valu of h financial ass of h kind discussd in scion, i.. of h form + ~ (0, σ ), o 8. Subjcs rcivd a abl of frquncis illusraing h (normal) disribuion of h signal around is ru valu (s h appndix). In AE and AS, hy wr also shown a abl of frquncis of. 8 A Walrasian aucion has h addiional advanags of making crain ha an quilibrium mark pric (xchang ra) would b formd (as bids and offrs wr rquird on h par of ach radr), ha hr was a singl valu of h xchang ra pr sag and ha all rad was compld, wih all bids and offrs bing licid, wihin a shorr amoun of im han a doubl aucion (in an xprimn ha could alrady las ovr wo hours, im was vry much a a prmium).
8 In all ramns, o hlp wih undrsanding, subjcs also rcivd an xampl sh showing xampls of how h conomy may work in ach sag. A h nd of ach sag, h unis of h financial ass obaind wr convrd ino hom currncy (xchang ra fram) or cash (sock mark fram) using h ru valu. A h nd of h xprimn, on priod was chosn a random by h compur and ach uni of hom currncy or cash arnd in ha priod (whhr dircly or afr h convrsion using ) was ranslad ino U.K. pounds a h ra of pound pr uni. Man xprimnal paymn wr -3 UK pounds (roughly 40 US dollars) pr subjc for bwn ½ and hours of work. 4. Exprimnal Rsuls 9 4. Gnral Rsuls Figur shows xampls of financial ass pric dynamics in on sssion ach from ach ramn. 0 Our focus blow is on priods hrough 8, sinc prdicions of Baysian and MLE xpcaions can only divrg afr agns rciv signals, and ADE prdicions can only b idnifid whn hr is a lag. For ADE, w sima h laggd dpndn variabl cofficin using las-squars on mark prics, individual buying prics, and individual slling prics). Th cofficins for h diffrn ramns, dnod β, ar givn in Tabl. (Insr Figur and Tabl abou hr.) Subjcs gnrally appard o hav a good undrsanding of h xprimn, and financial ass prics wr almos always in h 0.5 o.5 rang wih a mod around, which is wha w would xpc wih our N (, 0.5) disribuion of ru financial ass valus (s Figur 3). (Insr Figur 3 abou hr.) 9 All P valus rpord in his papr ar wo aild. All ss ar a h sssion lvl o guaran h non indpndnc of obsrvaions. 0 Elcronic appndix B conains similar graphs for all sssions. Tha bing said, no rsul would chang if w wr o includ priod 0.
9 Subjcs radd financial ass unis 6% of h ims (bwn 59% and 64% dpnding on h ramn). Tabl 3 includs avrag, varianc, skwnss and kurosis of financial ass prics, individual buying prics, individual slling prics and quaniis radd in all four ramns. W provid boh man and mdian informaion, sinc, paricularly in rlaion o slling prics and buying prics, mdian informaion hlps filr ou noisy oulirs. No surprisingly, mark prics nd o b mor sabl han ihr slling prics or buying prics. Tha bing said, a comparison of AE and AS vs. BE and BS shows ha h inroducion of uncrainy had no saisically significan ffc no only on mark prics, bu also on slling prics, buying prics and quaniy radd in Mann Whiny ss; h closs w g o significanc is in mark prics bing marginally highr undr uncrainy (.00) han undr full informaion (0.967), which yilds P 0.09, bu Tabl 3 shows ha his small ffc appars fram dpndn. Th fram manipulaion appars o mar mor, wih h man mark pric (P 0.05), quaniy bough (P 0.04), skwnss and kurosis of quaniy bough (P < 0.00 in boh cass) all rsuling in saisically significan ss. Mark prics and quaniis radd ar slighly lowr undr a sock mark fram han undr an xchang ra fram, wih h disribuion of quaniis bough bing howvr a lil mor skwd owards highr valus and kurosis bing marginally highr. RESULT. Uncrainy gnrally dos no affc h avrag, varianc, skwnss and kurosis of h disribuions of mark prics, buying prics, slling prics and quaniis radd. Framing mars mor han uncrainy dos. 4. Comparing h goodnss of fi of diffrn classs of modls W now us h sum of squard rrors (SSE) bwn prdicd and obsrvd xchang ras o compar h goodnss of fi of h diffrn classs of modls: Baysian, MLE and
0 adapiv xpcaions. Figur 4 compars h goodnss of fi of diffrn xpcaion modls in prdicing financial ass valus. (Insr Figur 4 abou hr.) Baysian xpcaions consisnly hav br prdiciv powr han h simpl avraging implid by MLE xpcaions, as rflcd in a lowr sum of squard rrors in all ramns, including h BE and BS ramns wih limid knowldg abou h prior disribuion. This suprioriy is unquivocally confirmd in Wilcoxon ss (P 0.005), and is also confirmd if w look a SSE in rlaion o individual buying or slling prics. Across ramn diffrncs in SSE ar nvr saisically significan in Kruskal Wallis or Mann Whiny ss. RESULT. Baysian xpcaion modls hav br prdiciv powr han MLE xpcaion modls in all ramns, including hos wih limid knowldg abou h prior disribuion. Th SSE in rlaion o adapiv xpcaions is lowr han ha in rlaion o Baysian updaing in ramn AE, bu no saisically significanly so. I is highr in h ohr ramns. Ovrall, a h mark lvl hr is modraly srong vidnc ha h Baysian modls ouprform adapiv xpcaions, (P 0.055 in a Wilcoxon s). Ths rsuls conras wih h suprioriy of adapiv xpcaions ovr Baysian updaing a h lvl of individual buying and slling prics. Thr is a sns, hrfor, ha marks ar mor raional (in rms of Baysian fi rlaiv o adapiv xpcaions) han individuals. RESULT 3. A h individual lvl, adapiv xpcaions ouprform Baysian xpcaions. A h mark lvl, Baysian xpcaions globally ouprform adapiv xpcaions.
5. Conclusion This papr bgan wih a spcific qusion: dos Baysian updaing, which rquirs full informaion abou h financial ass, prform wors han mploying h maximum liklihood simaor, which dos no? Our answr, which is a qualifid no, has wo sub-pars. Th firs is ha xplici knowldg of h prior disribuion of ru financial ass valus is no ncssarily crucial for subjcs o bhav raionally. Th scond is ha, in our sup, h Baysian signal xracion framwork dscribs bhavior jus as wll wihou such knowldg as i did wih i. Inrsingly, allowing for simaing on dgr of frdom in an adapiv xpcaion modl did no nabl adapiv xpcaions o ouprform Baysian signal xracion modls a h lvl of mark prics. Th qualificaion is ha our subjcs do bhav a lil diffrnly whn rading forign xchang and shars. In paricular, hy rad highr quaniis of h formr. Whil h main rsuls ar robus across h wo frams, i migh b inrsing, and prudn, o furhr xplor h diffrncs in blif formaion in hs wo marks bfor dclaring gnraliy. Our rsuls migh sm o b ou of sp wih h groundswll agains raional xpcaions as a modlling ool, paricularly sinc h Global Financial Crisis (Akrlof and Shillr 009). In fac, w would no wish o dny ha uncrainy dos mar in a numbr of conxs. Howvr, if our rsuls sand h scruiny of furhr sing hy suggs ha hr migh b a las som mark sups whr conomiss may b abl o modl bhavior as if agns know h prior disribuion whn hy rad shars or forign xchang. Rfrncs Akrlof, G. A. & Shillr, R. J. (009) Animal Spiris: How Human Psychology Drivs h Economy, and Why I Mars for Global Capialism, Princon: Princon Univrsiy Prss. Cosmids, L., Tooby, J. (99). Cogniiv adapaions for social xchang. In J.H. Barkow, L. Cosmids & J. Tooby (Eds.), Th adapd mind: Evoluionary psychology and h
gnraion of culur (pp. 63-9). Nw York and Oxford: Oxford Univrsiy Prss. Ellsbrg, D. (96). Risk, ambiguiy, and h Savag axioms. Quarrly Journal of Economics, 75, 643-669. Hrwig, R., & Gigrnzr, G. (999). Th conjuncion fallacy rvisid: How inllign infrncs look lik rasoning rrors. Journal of Bhavioral Dcision Making,, 75-305. Knigh, F. H. (9). Risk, Uncrainy, and Profi. Boson, MA: Har, Schaffnr & Marx. hp://www.conlib.org/library/knigh/knrup.hml Mankiw, N. G. (00). Th inxorabl and mysrious radoff bwn inflaion and unmploymn. Economic Journal,, C45-6. Sarin, R. K., & Wbr, M. (993). Effcs of ambiguiy in mark xprimns. Managmn Scinc, 39, 60-65. Smih, V. L., Williams, A. W., Braon, K. W., & Vannoni, M. G. (98). Compiiv mark insiuions: Doubl aucions vs. sald bid-offr aucions. Amrican Economic Rviw, 7: 58-77. Figur. Timlin of Exprimnal Sssions draw 3 4 0 3 4 5 6 7 8 0 3 4 5 6 7 8 0 3 4 5 6 7 8 0 3 4 5 6 7 8 rval x x x x x x 3 x x 4 Sag Sag Sag 3 Sag 4
3 Figur. Obsrvd Prics and Prdicions in Sampl Sssions No: h adapiv xpcaions prdicion is qual o h obsrvd pric in h prvious priod.
4 Figur 3. Hisograms of Financial Ass Valus No: Basd on ass prics (i.. pric of forign xchang or shars) in priods -8.
5 Figur 4. Goodnss of Fi of Diffrn Expcaion Modls Nos: SSE: sum of squars rror; AE: adapiv xpcaions; MLE: maximum liklihood xpcaions; Baysian: Baysian xpcaions.
6 Tabl. Exprimnal Tramns Tramn AE Knowldg of Prior Disribuion, Exchang Ra Fram Tramn AS Knowldg of Prior Disribuion, Sock Mark Fram Tramn BE No Knowldg of Prior Disribuion Exchang Ra Fram Tramn BS No Knowldg of Prior Disribuion Sock Mark Fram No: sssions wr run in ach ramn. Tabl. Esimad Man β Valus by Typ of Prics and Tramn Tramn AE BE AS BS Ovrall β Mark prics 0.987 0.97 0.965 0.974 0.974 Buying prics 0.99 0.98 0.903 0.89 0.96 Slling prics 0.93 0.899 0.884 0.904 0.905 Nos: Th adapiv xpcaions paramr β was simad, in ach cas, by minimizing h sum of squars rror bwn acual and prdicd valus.
7 Tabl 3. Disribuions of Mark Prics, Slling Prics, Buying Prics and Quaniy Tradd Avrag Varianc Mark Slling Buying Quaniy Mark Slling Buying Quaniy Tramn Pric Pric Pric Tradd Pric Pric Pric Tradd AE Man.03.09.9.795 0.036 79.78.96 3.57 Mdian.08.7 0.966.94 0.06 5.36 0. 3.808 BE Man 0.985 3.79.88.7 0.03.7 0.054 3.435 Mdian 0.99.8 0.996.70 0.03 0.79 0.6 4.06 AS Man 0.979 5.368.68.83 0.04 87.5 64.94 4.358 Mdian 0.957.75 0.995.339 0.038 0.37 0.09 3.937 BS Man 0.949.394 0.967.68 0.037 55.793 487.394 3.709 Mdian 0.939.088 0.88.696 0.09 0.54 0.4 3.57 Toal Man 0.984 3.46.577.6 0.036 897.56 96.087 3.768 Mdian 0.973.4 0.954.649 0.03 0.34 0. 3.85 Skwnss Kurosis Mark Slling Buying Quaniy Mark Slling Buying Quaniy Pric Pric Pric Tradd Pric Pric Pric Tradd AE Man -0.37 3.39.76 0.776-0.7 5.597 5.386-0.84 Mdian -0.493.8 0.65 0.776-0.55.68.65-0.9 BE Man 0..8.636 0.89 0.7 4.3 5.965-0.57 Mdian -0.05.455.63 0.788 0.54 8.87 4.09 -.087 AS Man -0.073 3.59..5-0.05 33.868 8.47 0.33 Mdian -0.095.44 0.458.06-0.08 6.945.4 0.04 BS Man 0.64 3.5.363.9-0.077 30.468 6.3 0.097 Mdian -0.33.88 0..63-0.06 4.407 4.04-0.47 Toal Man 0.0 3.3.993.00-0.03 8.564 8.977-0.5 Mdian -0.33.03 0.34 0.95-0.069 6.33.703-0.54