Evoluon wh Indvdual and Socal Learnng n an Agen-Based Sock Marke Ryuch YAMAMOTO Brandes Unversy Second Draf May 30, 005 Absrac An agen-based sock marke evolves as agens learn from he pas experence and adap her behavor o he evolvng marke. Ths paper nroduces a learnng and adapaon mechansm whch allows agens o choose one rule among a se of deas updaed hrough boh ndvdual and socal learnng. I hen examnes how he learnng mechansm affecs he dynamcs of he arfcal sock marke.
. Inroducon An agen-based sock marke consss of a se of neracng heerogeneous agens. The marke evolves as agens learn from he pas experence and adap her behavor o he evolvng envronmen. Such learnng and adapve behavor of he agens s usually modeled wh approaches of an evoluonary compuaon. Recen research has shown a varey of compuaonal echnques o descrbe he evoluon n an arfcal sock marke. One can dsngush he echnques based on a whch level he learnng of agens s modeled. The prevous leraure descrbes learnng a eher ndvdual or socal level. The learnng a ndvdual level s usually called ndvdual learnng where agens updae her behavoral rules from her own pas performance. The learnng a socal level s called socal learnng where agens updae her rules hrough drecly neracng wh oher agens. In he prevous leraure, he level of learnng s exogenously gven, and agens nvolve only a parcular level of learnng when hey updae her rules. Bu such a seng doesn say anyhng abou why agens choose a parcular level of learnng o updae her radng rules. Ths paper nroduces a learnng mechansm whch allows agens o choose one rule a each perod among a se of deas updaed hrough boh ndvdual and socal learnng. A radng sraegy performed well n he pas s more lkely o be seleced by agens regardless s creaed a ndvdual or socal level. Ths framework allows agens o choose a decson rule endogenously among a wder se of deas. Wh such evoluon, he followng wo quesons are examned. Frs, snce agens who have a wder se of deas o choose are more nellgen, a queson would arse f he me seres from such an economy would move around more closely o a homogeneous raonal expecaon equlbrum han n an economy wh only one level of learnng. A convergence propery o he raonal expecaon equlbrum (hereafer REE) s nvesgaed n LeBaron (000) and Arhur e al. (996). They fnd when agens adap her forecass very slowly o new observaons he marke converges o he REE. The more nformaon from he marke agens ge before updang her rules, he more closely hey behave raonally. They deal wh such a convergence propery by lookng a dfferen me-horzon. Bu hs paper nvesgaes a convergence propery by lookng a dfferen levels of nellgence gven a me horzon. Can a marke reach he REE f agens have many deas o process he marke nformaon alhough hey adap her behavor
quckly o new observaons? Would agens be able o behave raonally when hey are nellgen? However, he resul n hs paper shows ha he economy wh more nellgen agens canno reach he REE. Agens don behave raonally when hey are nellgen. Inellgen agens are no raonal. The second nvesgaes whch level of learnng s lkely o domnae n he marke. Ths s analyzed by nvesgang who chooses whch level of learnng and wha proporon of he agens ofen uses ndvdual or socal learnng. Agens are allowed o choose endogenously a beer dea creaed from ndvdual and socal learnng. Some agens wh beer deas would use her own dea more ofen han he ohers do, whle some wh less successful deas would rely on he deas from oher agens. Who has beer deas and who doesn? In hs paper, s consdered as a beer dea ha could produce hgher wealh over a parcular pas me span. Agens are more lkely o pck an dea whch produces hgher wealh n he pas. So, we would hypohesze ha an agen who accumulaes more wealh n he pas s more lkely o pck an dea from her own (from ndvdual learnng) han ohers do, whle some who are poor are more lkely o adop an dea from ohers (from socal learnng). Then, a queson arses wha proporon of he agens use her prvae deas and mae ohers. Ths paper shows ha mos of he agens follow he herd, and only agens wh very hgh wealh would possbly rely on prvae deas. So, concludes ha he socal learnng domnaes he marke. Agens would be beer off n an ex ane welfare sense by consranng he use of her own deas. The second par of he paper evenually ndcaes ha he agen-based sock marke n hs paper could possbly explan he mechansm of herdng behavor n he real world. The res of he paper proceeds as follows. Secon descrbes prevous leraure. Secon 3 presens he marke srucure. Secon 4 gves he resuls from he compuer expermens, and he las secon concludes.. Comparng wh Prevous Leraure In he prevous leraure abou an agen-based marke, a varey of compuaonal echnques are used for evoluon n he marke. In parcular, agens radng or forecasng sraeges are evolved as agens learn from he pas and adap her behavor o a marke. Learnng and adapve behavor of he agens s ofen descrbed a eher ndvdual or 3
socal level. A marke wh socal learnng s consdered o be a sngle populaon conssng of drecly neracng heerogeneous agens Fgure represens a socal learnng. The symbol characerzes he drec neracon. Fgure : Socal Learnng: Marke In socal learnng, nvesors behavor s nfluenced by oher nvesors. Invesors mee, for example, a some conferences, communcae each oher, and exchange her opnons abou he prce predcon. Then based on such neracons, hey would updae her radng sraeges. In a marke wh ndvdual learnng, each agen has a se of her prvae deas. The deas of each agen are no dsclosed o oher agens so ha here s no mave behavor. Agens learn from her own pas experence and updae her behavoral rules by hemselves. There are no drec exchanges of he deas among agens n hs learnng. Reacons o oher agens behavor only occur ndrecly hrough prces. Fgure represens an ndvdual learnng. Agan, he symbol characerzes he drec neracon, and he symbol = shows he ndrec neracon. So, n hs seng, populaon ndrecly neracs hrough he marke. Mechansms of learnng and adapve behavor of agens are dfferen n ndvdual and socal learnng. However, mos of he prevous papers don explan why agens choose Chen e al. (00) clarfes he dsncon beween ndvdual and socal learnngs. Here I follow her argumens. 4
a parcular level of learnng mechansm o updae her radng rules. For example, papers Fgure : Indvdual Learnng Marke relaed o he Sana Fe Arfcal Sock Marke Model adop an ndvdual learnng. Some papers allow drec neracon among agens whch s represened as socal learnng. 3 Only Vrend (000) and Yeh and Chen (000) are he papers whch clarfy he dsncon. Only Yeh and Chen (000) movaes why agens choose a parcular level of learnng. Vrend (000) compares he smulaon resuls of a smple Courno model wh ndvdual learnng and socal learnng hrough he same dencal genec algorhm (hereafer GA) for exacly he same dencal underlyng economc model. The resul shows ha he dfference s essenal. For example, he GA wh ndvdual learnng moves close o he Courno-Nash oupu level, whereas he GA wh socal learnng converges o he compeve Walrasan oupu level. In addon, he socal learnng GA shows qucker convergence o an equlbrum han he ndvdual learnng GA does whle he socal learnng GA reaches hgher oupu levels han he ndvdual learnng GA does. So, snce he resuls dffer accordng o he dfferen learnng, hs resul ndcaes ha he choce of he compuaonal modelng beween ndvdual and socal learnng algorhms should be made carefully. Vrend clarfes he dsncon bu doesn menon why and under wha condon agens choose a parcular level of learnng. Yeh and Chen (000) consruc an arfcal sock marke whch negraes boh socal learnng and ndvdual learnng wh he genec programmng (GP) framework. These papers nclude LeBaron, Arhur, and Palmer (999), Tay and Lnn (00), Arhur, Holland, Lebaron, Palmer, and Tayler (996), LeBaron (00), LeBaron (00b). 3 Those papers nclude Arfovc (996), Arfovc and Gencay (000), Arfovc (00a), Arfovc (00b), and Arfovc (00). However, Vrend (000) and Yeh and Chen (000) concern he dfference of he levels of learnng. 5
Snce agens n her marke have more deas o creae radng sraeges han he sngle populaon GP based marke, hey ask how he degree of raders nellgence nfluences he economy. In parcular, frs, he economerc properes of me seres are examned under dfferen degrees of nellgence. Second, her expermen examnes whch ypes of raders are more lkely o survve beween predcon accuracy and prof orened raders. Ther resuls show ha prof orened raders are more adapve and easer o survve whle hey don ge much dfference of he me seres properes n dfferen levels of nellgence. Ther paper s he only one whch clarfes he dsncon beween ndvdual and socal learnng and why agens choose a parcular level of learnng. Ths paper dffers on some pons from each of he above wo papers. Frs, hs paper concerns a model of sock marke whle Vrend (000) deals wh a Courno model. Second, alhough Yeh and Chen (000) negrae ndvdual and socal learnng no one model, he agens choose deas produced eher from ndvdual or socal learnng. Agens selec an dea afer hey go o eher level of learnng. So, a each sage of decson-makng, each agen s allowed o pck one dea only from a parcular level of learnng. Bu hs paper allows agens o choose one dea from a se of deas updaed from boh levels of learnng. Each agen has her own deas and updaes hem n her mnd whle she has a se of deas whch evolve wh oher agens. A a me of decson makng, she refers o a se of deas evolved a ndvdual and socal levels. The second expermen of hs paper consders more nellgen agens. Then wha would happen o he me seres from such an economy? Does move around more closely o he REE han n an economy wh only one level of learnng. Ths nvesgaon dffers from works of LeBaron (000) and Arhur e al. (996). They deal wh he REE propery by lookng a dfferen me-horzon, and conclude ha a marke wh more observaons abou he prces can converge o he REE. Bu hs paper asks he convergence propery wh more nellgen economy. The las par of he second expermen nvesgaes whch level of learnng domnaes he marke, and shows ha mos of he agens ofen follow he herd, and only small poron of agens wh very hgh wealh are more lkely o use prvae deas. Ths concluson s conssen wh he herdng leraure 4 and he evdence n he acual sock marke (Graham (999)). 4 Bkhchandan e al. (998) and Devenow e al. (996) survey he herdng leraure. 6
3. Marke Srucure Ths secon descrbes an arfcal sock marke based on he one oulned n LeBaron e al. (999). In he followng secons, he expermens are conduced on an arfcal sock marke wh dfferen syles of learnng. Bu he marke srucure s exacly dencal for all expermens. I s presened as follows. The arfcal sock marke has wo radable asses, a rsky sock and rsk free bond. The rsk-free bond s n nfne supply and pays a consan neres rae, r f =0%. The rsky sock pays a hghly perssen and sochasc dvdend whch follows an AR() mean-reverng dvdend process: () wh d = d + ρ( d d ) + µ. d = 0, ρ = 0.95, and µ ~ N(0, σ µ ) The number of shares of he sock s 30, whch equals he number of agens n he marke. The marke consss of many heerogeneous neracng agens who have dfferen mehods of predcon on he sock prce and dvdend. They make predcons abou he fuure sock prce and dvdend each perod based on he marke nformaon relaed o he prce and dvdend. Usng predcons, each agen se her demand for shares. Takng he overall marke demand and supply no accoun, he sock prce s deermned. The more dealed seps of how evens n hs arfcal marke proceed are as follows:. Informaon se: A me, agens observe he pas prce and dvdend, and calculae echncal ndcaors. They form a se of nformaon, z whch s used by agens o predc fuure prces. Followng LeBaron (00), he echncal rules are based on exponenal movng averages 5 formed as ( ) k, ρ k mk, + ( ρ k m = ) p where k= and. 5 Exponenal movng average s a ype of movng average ha s smlar o a Smple Movng Average, bu pus more wegh o he laes daa. 7
ρ =0.8 for m, and ρ =0.99 for m,. The nformaon se, z, ncludes: p + d p. r = p. d 3. d log p Agens use he prce and dvdend nformaon o calculae he followng wo echncal ndcaors. 4. 5. p log m p log m,, A me, dvdend, d, s revealed and pad.. Predcon: Agens process he pas nformaon and make predcons on he fuure prce and dvdend. In parcular, agen forecass he fuure prce and dvdend accordng o: E ˆ + b ( 3) ( p + + d + ) = a ( p + d ) Ê denoes he bes forecas of agen a me. Each agen decdes he forecas parameers, a and b, accordng o he pas nformaon se, z. They are expressed wh a parcular funcon n z whch s descrbed as follows. 6 The funconal form used o generae he wo forecasng parameers s assumed o be a feedforward neural nework wh a sngle hdden-un wh resrced npus, whch s 6 The lnear forecasng model n (3) s opmal when agens beleve ha prces are a lnear funcon of dvdends and a homogeneous raonal expecaon equlbrum obans. Bu here here s no such resrcon. 8
used n LeBaron (00a) as follows. (4) (5) (6) h = g( ω z k λ ( z l, k, k ) = 0.5*( + g( ω + g( u) = anh( u) + ω 0, k ) 5 k = ω 3, k h k )) l = a, b. Neural nework s a parcular ype of funconal form ofen used n he feld of bologcal nervous sysems. Here snce he nformaon se, z,consss of 5 varables, hose are frs combned wh weghs ( ω0k and ω k for k=,,5) and ransformed n a hdden layer, whch s expressed as equaon (4), and produce sgnals, h. Snce he nformaon se has 5 varables, hdden layers produce 5 sgnals n oal ( h k for k=,,5). Those sgnals are conneced wh weghs ( ω and ω 3k for k=,,5) and produce a sgnal, λ, whch s expressed as equaon (5). λ les beween 0 and by consrucon. We call feedforward snce he drecon of he sgnal s jus one way from npu, z, o oupu, λ. Fgure 3 s a pcure for hs neural nework, equaon (4) o (6). Permng o range λ wh he allowable bounds for a and b n LeBaron (999), ha s, a [0.7,.] and b [ 0,9] 7, he forecas parameers gven by ( 7) a =.* λa ( z ) + 0.7 *( λa ( z )). ( 8) b = 9* λb ( z ) + ( 0) *( λb ( z )). 8 a and b are Agens are heerogeneous n erms of her expecaon snce each has dfferen values of weghs n her own neural ne. Here agens buld her forecas usng he neural nework. Ths s an exenson from he fnancal marke n LeBaron e al. (999). The agens n LeBaron e al. (999) forecas usng wha are called condon-forecas rules. 9 7 Those ranges are gven o be cenered around he raonal expecaon equlbrum values. 8 Each agen has wo neural neworks snce he has wo forecasng parameers. Snce each has 6 parameers n hs neural nework, he has 3 n oal. 9 In LeBaron e al. (999), he basc dea s ha he rules wll mach ceran saes of he world whch are 9
3. Sraegy makng: Based on he predcon, each agen ses hs demand for share as: (9) ˆ E ( p s * = + + d+ ) ( + γσˆ p+ d, rf ) p. 4. Prce deermnaon: The new equlbrum prce, p, s deermned accordng o he marke equlbrum condon as: N N a ( p + d ) + b ( + rf ) p (0) s ( p ) = = = γσˆ marke (=30). = N where N s he number of agens n he 5. Volume deermnaon and updang varance esmaes: Afer revealng he prce, forecasng parameers a and b are updaed accordng o he feedforward neural nework, (4)-(6) o ge a b + and +, and radng volume s recorded. The prce a me s derved by solvng equaon (0). Afer he prce s se, agens updae her porfolo and radng volume s recorded. 0 Once he equlbrum prce p s revealed, agens change her forecasng rule, a and b. Then he marke demand and supply become unbalanced. So, p clears he marke only emporally. In hs sense equlbrum prce (LeBaron (00b)). p s a emporary Here supply and demand are somehow balancng n some unspecfed marke nsuon (Arhur e al (996)). Agens calculae her desred holdngs and subm her decsons o he marke specals who funcons as a marke maker. The specals collecs defned endogenously. These saes map no a forecas for he fuure prce and dvdend whch s hen convered o share demand hrough he agen s demand funcon. 0 Snce here s one marke maker and 30 raders, he radng volume s defned as 30 30 v = = y + = y, where y s rader s radng volume. Ths model doesn say anyhng ha wealher people can affec he prce. Usually radng by wealher people could affec he prce. 0
bds and offers from agens, and announces a prce ha clears he marke. So, hs nsuon deals wh buyng and sellng n real me. An nvesor, who wans o buy sock, can always buy sock whle seller can always sell he sock n hs marke. Wealh, w, for ndvdual s evolved accordng o: ( ) w = s ( p + + d + ) + ( + r )( w p s + f. Each agen s nally allocaed 0,000 uns of cash. ) 6. Genec Algorhm: Seps -5 are repeaed for S (=5) perods. Then genec algorhm (GA) s nvoked o updae her forecasng parameers. The seps -5 wh GA are repeaed 500 mes every 5 perods. The GA manpulaes he parameers, ω, n he neural nework, equaon (4)-(6), o mprove he performance accordng o a fness creron. Here he fness creron s wealh-based uly of he pas 5 perods whch s gven as: ( ) S = 5 V = ( ) = U w+ where U w ) exp( γw ) 3 ( + = + γ s a consan absolue rsk averson coeffcen and assumed o be 0.5. The varance esmae s updaed accordng o an exponenally weghed average of squared forecas error, (3) ˆ σ ( ) ˆ σ τ τ [ ] {( p + d ) a ( p + d ) b } p+ d,, = p+ d,, + + where τ s fxed a 75. All agens nvolve GA smulaneously. The more dealed seps of he GA are nroduced nex. Seps of GA Implemenaon For an nroducon o GA, see Mchell(996), Haup and Haup (998), Muhlenben and Schlerkamp-Voosen (993), and Toknaga (000). Jankow and Mchalewcz (99) emprcally compare floang pon and bnary based GAs usng dynamc conrol problem. Wrgh (99) apples genec algorhms o opmzaon problems over several real parameers. 3 As smulaon proceeds, he wealh s always normalzed o be n 5-dg number dvdng by 0 f one of he wealh exceeds 00,000. When we evaluae he uly, he wealh s dvded by,000,000 snce he uly funcon s negave exponenal.
GA mplemens o he parameers, ω, n he neural nework, equaon (4) o (6). Ths paper consders he markes wh ndvdual and socal learnng. So, GA s run a he ndvdual and socal levels. Bu he seps o mplemen GA are exacly he same for exacly he dencal marke srucure for socal and ndvdual learnng. So, he followng explanaon o mplemen GA can be appled o marke srucures boh wh socal and ndvdual learnng. A GA consss of a se of operaons whch manpulae a gven populaon. There s one populaon n a socal learnng economy whch consss of 30 agens whle an ndvdual learnng marke has 30 populaons each of whch represens agen who has 30 deas n her mnd. In an ndvdual (socal) learnng marke, each dea (agen) has a se of 3 parameers n her neural nework. In ndvdual learnng, deas nerac whn each of he agen s mnd whle agens nerac n socal learnng. More specfc explanaons abou GA are as follows. ) Inalzaon of populaon: The nal ses of parameers for each dea (agen) are chosen randomly from he range [-,] 4. Snce here are 30 deas (agens), he marx of parameers s 30x3. ) Rankng and Selecon: In each generaon n he agens face 5 (=S) porfolo decsons. The forecasng rules ha dd well accordng o he fness measure wll be more lkely o be coped han a rule wh a lower fness. The forecasng rule for dea (agen) s o be coped wh he probably: (4) where V = P = U ( w N = 30 j= S = 5 = + / V / V ) j. 3) Crossover and Muaon: The GA hen nroduces new rules hrough wo genec operaors ha manpulae some parameers of he populaon. These wo operaons are called crossover and muaon. The 4 Ths range s defned n LeBaron (00).
algorhm chooses beween crossover and muaon wh equal probably. Afer eher of crossover and muaon s seleced, he algorhm randomly chooses one of he nformaon varables. Then all of he parameers relaed o ha nformaon varable are subjec o he crossover or muaon. For example, n Fgure 3, when he algorhm chooses crossover (muaon) and nformaon varable, r, hen he parameers, ω, ω0, ω3, ω, n he seleced par of parens are subjec o he crossover (muaon). 5 A seleced se of he parameers n he neural nework, whch are relaed o one of he nformaon varables, s crossed over. Inuvely, n socal learnng, for he crossover on forecasng parameers, wo nvesors who have smlar predcon sklls are more lkely o mee and exchange her opnons for he sock prce predcon. They alk abou a parcular nformaon varable for he predcon. Afer ha, hey updae her predcon rule of ha varable based on he dscusson wh oher agens. For he crossover n ndvdual learnng, agens updae her own deas nsde her mnd. Here he crossover for he real-valued GA s as follows. A new parameer (offsprng) s produced by combnng wo parameers (parens) as: (5) offsprng=paren + α * (paren paren ). where α s a scalng facor chosen unformly a random n he nerval [-0.5,.5]. A new α s generaed for each par of parens combned ogeher (Muhlenben and Schlerkamp-Voosen (993)). For muaon on he real-valued GA, a parameer ω s seleced wh probably p m (=0.08) for muaon, and are added a small perurbaon. 6 5 A par of parameers undergo crossover wh a consan probably of 40% as n Leau (997). 6 A value ou of an nerval, ] s added o he seleced varable. The range s defned as range range [ 0.5 * [-,]. The new value ω * s compued accordng o ω * = ω ± range δ. The + or sgn s chosen wh probably 0.5. δ s compued from a dsrbuon ha prefers small values. Ths s realzed as follows: m δ = η. = 0 η = wh probably /m, else 0. Here m=0(muhlenben and Schlerkamp-Voosen (993)). 3
4) Renseron: Afer crossover or muaon s conduced, agens do back-esng. The fness funcon () s calculaed wh updaed parameers and observed prces. A se of he updaed fness s compared wh he old fness. The algorhm conducs nseron of he updaed parameers no he curren se of parameers when he new ones could produce hgher fness han he old ones do. Offsprngs replace leas f parens. 7 4. Expermens wh Inellgen Agens Ths secon consss of wo nvesgaons n a marke wh evoluon whch allows agens o choose one rule a each perod among a se of updaed deas produced hrough ndvdual and socal learnng. Frs, he convergence propery o he REE s examned wh an nellgen economy, and concludes ha he nellgen economy canno converge o he REE snce nellgen agens are no raonal. Second, s examned whch level of learnng domnaes he marke. I shows ha a wealhy agen pcks an dea from ndvdual learnng more ofen han ohers do whle poor agens mae ohers. More deals of he evoluon are as follows. When agens updae her radng rules, hey evaluae he pas performance of a se of he deas whch are creaed n boh ndvdual and socal learnng, and choose one dea from hem. Each agen has a se of 30 deas whch are updaed n her mnd. In addon, here are 30 agens n he marke, each of whom has one dea for socal learnng. So, here are 30 deas n oal for socal learnng. Afer a GA s conduced, each agen s able o selec one dea from 60 deas by akng wo seps. A he frs sage, an agen pcks wo deas, one of whch s aken from ndvdual learnng and anoher from socal learnng. The deas are evaluaed accordng o he wealh-based fness creron (equaon ()). A he frs sage, each agen s more lkely o choose one dea whch performed beer n he pas. In he second sage, she selecs one from he seleced wo by comparng absolue values of he fness. The deas used n ndvdual and socal learnng are oally solaed n hs seng. On he one hand, each agen has her prvae deas, and never reveals hem o ohers. These deas are evolved only wh ndvdual learnng. On he oher hand, agens have some deas, and are wllng o reveal hem, n order o ge new nformaon from oher agens. So, n 7 Ths s equvalen o he elecon operaor n a seres of Arfovc s papers. 4
hs seng, each agen has wo ypes of deas, one of whch s oally prvae whle anoher s for socal neracon. Under hs seng, he choce of he level of learnng s no exogenous bu endogenous. In addon, snce each agen has a wder se of dea o choose n hs economy, hey are consdered o be more nellgen han he agens n he prevous secons. Ths secon deals wh such an economy wh nellgen agens. 4. Are nellgen agens raonal? Smulaons are repeaed for 0 mes under dfferen random seeds o collec cross-seconal sascs. The seres of sock prce, dvdend, and volumes are recorded for he las 5,000 perods, and he followng sascs for reurns and volumes are calculaed, and compared wh he economes whch have only one level of learnng,.e., ndvdual or socal learnng. In eher ndvdual or socal learnng economy, each agen can choose one rule from 30 rules whle an economy wh nellgen agens allows hem o choose one from 60 rules creaed from boh levels of learnng. The sascs for reurns seres are he sandard devaon, excess kuross, volaly cluserng, and nonlnear dependence whle for he volume seres, he averages are calculaed. For he reurn seres sascs, he followng regresson s frs conduced wh he smulaed daa: (6) p + + d + = α + β ( p + d ) + ε. Followng LeBaron e al. (999), he esmaed resdual seres εˆ are analyzed, and he resuls are n Table. Under he homogeneous raonal expecaon equlbrum, he resdual me seres follow ndependen and dencal dsrbuon wh he sandard devaon of (N(0,4)). The sandard devaons are 4.495 n an economy wh nellgen agens, 3.5494 n ndvdual learnng and 3.856 n socal learnng. These show hgher varably han should be n he homogeneous raonal expecaon equlbrum. The ARCH es deals wh he volaly cluserng for he reurn seres (Engle (98)). I ess he null hypohess ha a me seres of sample resduals s..d. Gaussan dsurbances (.e., no ARCH effecs exs). The numbers repored are he means of he es 5
sascs. The numbers n he brackes are he fracon of runs ha rejeced 'no ARCH' a he 95% confdence level. 8 When an economy wh nellgen agens s n he raonal expecaon equlbrum, here should be no ARCH phenomena. Bu he no ARCH null hypohess s rejeced 8 mes n an nellgen agen economy. The resul ndcaes he ARCH dependence n he resduals n he economy. Table : Summary sascs Descrpon Inellgen Economy Indvdual Learnng Socal Learnng Sd. ARCH() BDS Tradng Volume 4.495 (0.9379) 3.878 [0.7].948 [].968 (0.759) 3.5494 (0.3065) 0.45 [0] -0.509 [0].5637 (0.0943) 3.856 (0.7087) 5.385 [0.8] 5.0855 [0.9] 3.03 (0.660) Noe for Table : Means over 0 runs. Numbers n parenhess are sandard errors esmaed usng he 0 runs. Numbers n brackes are he fracon of ess rejecng he no ARCH, or ndependen dencally dsrbued null hypohess for he ARCH and BDS ess, respecvely, a he 95% confdence level. Nonlnear dependence asks f he reurn seres are dencally and ndependenly dsrbued over me. I s esed wh he BDS (Brock, Decher, and Schenkman) sasc (Brock, Decher, Schenkman, and LeBaron (996)). The BDS es s conduced wh he null hypohess ha a me seres sample comes from an IID daa generang process. 9 Under he homogeneous raonal expecaon equlbrum, he null on BDS es canno be rejeced. However, he IID null hypohess s rejeced 9 mes n an economy wh nellgen agens. The resul shows ha here are some correlaed paerns on daa generang process n he economy wh nellgen agens. 8 The es procedure s o run he OLS regresson and save he resduals. Then we regress he squared resduals on a consan and p lags. The asympoc es sasc s M * R, where M s he number of squared resduals ncluded n he regresson and R s he sample mulple correlaon coeffcen. I s asympocally Ch-Square dsrbued wh p degrees of freedom under he null hypohess. 9 The alernave hypohess s no specfed. Bu hs es has good power agans nonlnear alernaves. I s dsrbued asympocally sandard normal. 6
The ffh row s he resuls for he radng volume. In sandard effcen marke fnancal heory, dencal nvesors share raonal expecaons of an asse s fuure prce. Takng no accoun all marke nformaon, he nvesors make an nvesmen decson whch clears he marke. So, n he sandard fnancal heory, here s no opporunes lef open for speculave prof excep by luck. In hs case, he radng volume s low or zero 0. The resuls show ha he average radng volumes over he fnal 5000 perods are no zero n all economes. The resuls are oally oppose o wha he homogeneous raonal expecaon equlbrum says. The economy wh more nellgen agens canno reach he REE as n an economy wh eher ndvdual or socal learnng. Agens don behave raonally when hey are nellgen. Inellgen agens are no raonal. Why doesn he marke wh nellgen agens converge o he REE? Each agen has 60 deas, each 30 of whch are from ndvdual and socal learnng. Agens updae her deas every 5 perods, and choose one whch could acheve hgher wealh-based fness. When he sock prce moves much hgher level han ha of he REE durng he 5 perods, agens ake he nformaon of he hgher level of he prces no accoun o updae her deas. Agens choose deas o ge hgher level of wealh, bu hose deas reflec such hgher prce levels n hose perods. Durng he perod of hgher varaon of he sock prce han he REE seres, he deas would reflec such hgher varaons. So, as far as he prce seres behave dfferenly from he REE, he deas are also far from he REE. As a resul, he economy never converges o he REE. I would be he only way o reach he REE ha agens adap her forecass very slowly o new observaons as n LeBaron (000) and Arhur e al. (996). In he real world, nvesors n he sock marke have become more nellgen han hose n, for example, 40 years ago. They have now more sophscaed ways n analyzng he sock marke and forecasng fuure prces, and so on. However, does ha mean ha he recen sock marke behaves raonally? Defnely, doesn. The sock marke behaves n oppose ways o wha he raonal expecaon heory says. The marke wh nellgen nvesors s no relaed o he raonaly a all. 4. Whch Level of Learnng Domnaes he Marke? 0 For lqudy purpose, agens may lqudae (rade) some amoun of her own asses. In hs case, radng volume s no zero. 7
Agens n he second expermen can choose deas from boh levels of learnng. They bascally choose an dea whch performed well n he pas. Snce he fness funcon s he wealh-based uly, s consdered as a beer dea ha could produce hgher wealh over a parcular pas me span. Ths secon nvesgaes a hypohess ha agens who accumulae more wealh n he pas are more lkely o pck an dea from ndvdual learnng han ohers do whle some who have less wealh are more lkely o adop an dea from socal learnng. Mos of he agens follow he herd, and only agens wh very hgh wealh would rely on prvae deas. The marx on he choce of learnng levels and he marx on wealh n he las 5000 perods of 0 smulaons are used for he nvesgaon, boh of whch are descrbed as follows. Daa: A each generaon n a smulaon wh 500 generaons, he choces of he level of learnng by agens are sored wh 0 f she chose ndvdual learnng and f s socal learnng. There are 00 generaons n he las 5000 perods. Snce we have 30 agens n a marke, he marx on he choces s evenually 30x00. For he marx on wealh, he wealh of each agen over a generaon s summed up (he sum of he 5 perods wealh). Snce we have 00 generaons n he las 5000 perods, he dmenson of he marx on wealh s 30x00. A each generaon, akng he marx on he choces as a dependen varable and he marx on wealh as an ndependen varable, he parameers n he followng model (equaon (7)) s esmaed. Model: Snce he dependen varable consss of only 0 and, s convenen o use he bnary choce model whch lnks he decson by agens o he wealh varable. The hypohess s ha less wealhy agens are more lkely o choose socal learnng. Usng he logsc funcon, he followng probably o choose socal learnng s useful o analyze he hypohess. β + β WEALTH (7) P(y= wealh) = β + β WEALTH e 0 + e 0. 8
A varable, WEALTH, needs o be sandardzed o have small numbers for hs ype of probably funcon. The parameers, β 0 and β, of hs log model are esmaed by he maxmum lkelhood mehod. Analyses: There are 00 ses of choce and wealh varables for 30 agens. The parameers are esmaed generaon by generaon o ge 00 ses of β 0 and β. Wh he parameer esmaes, he esmaed probables are calculaed for each agen and for all generaons. Our hypohess whch poorer agens are more lkely o choose deas from socal learnng suggess ha here s a negave relaon beween he probably o choose socal learnng and he wealh level. So, he esmaed probables and he wealh levels are compared for all 30 agens and for all 00 generaons. As a frs nvesgaon, a each generaon agens are caegorzed no he ones who have wealh more han average a he generaon and he ones who have wealh less han average. The esmaed probables are caegorzed no more han 0.5 and less han 0.5. If he probables are more han 0.5, ha agen s more lkely o pck an dea from socal learnng. The followng four cases are nvesgaed. (Case ): Agens wh more han average wealh are more lkely o choose deas from ndvdual learnng whle agens wh less han average wealh are more lkely o choose deas from ndvdual learnng also. (Case ): Agens wh more han average wealh are more lkely o choose deas from ndvdual learnng whle agens wh less han average wealh are more lkely o choose deas from socal learnng. (Case 3): Agens wh more han average wealh are more lkely o choose deas from socal learnng whle agens wh less han average wealh are more lkely o choose deas from ndvdual learnng. (Case 4): Agens wh more han average wealh are more lkely o choose deas from socal learnng whle agens wh less han average wealh are more lkely o choose deas The sum of he wealh over 5 perods s dvded by 000000 for he sandardzaon. 9
from socal learnng also. The numbers n Table shows he fracons of agens n whole perods correspondng n each case. The numbers n parenhess are he sandard devaons across 0 smulaons. Table : Who chooses whch level of learnng? More han average Indvdual Learnng Socal Learnng Less han average Indvdual Learnng 0.04 0.56 (0.0587) (0.043) Socal Learnng 0.4884 0.7986 (0.043) (0.0587) The number n he upper-lef corner s he fracon represenng wo ypes of agens. Those are he agens, who have more han average wealh and nvolve ndvdual learnng, and he ones, who have less han average wealh and assocaed wh ndvdual learnng. Around 0% of all agens n whole perods correspond o hs case. The number n down-lef corner s he fracon assocaed wh Case whch s abou 49% whle ha n upper-rgh corner represens Case 3 whch s abou 5%. The number s down-rgh corner shows he hghes. Ths s he fracon sasfyng he followng wo ypes of agens. Those are he agens who have more han average wealh and nvolve socal learnng, and hose who have less han average wealh wh socal learnng. The resul shows ha abou 80% of he agens are more lkely o choose deas from socal learnng over he whole perods. The analyss so far caegorzed he wealh levels no only wo,.e., more han average and less han average. So he behavor of he agens who have really hgh Each of he fracon s he average across 0 smulaons. 0
wealh s no clear from he analyss. The followng examnes he behavor of agens who have hghes wealh and are he hree hghes wealhy a each generaon. Two of he mporan resuls n Table 3 are as follows. (Resul ): Agens wh hgh wealh n each generaon are more lkely o choose deas from ndvdual learnng whle oher agens are more lkely o choose deas from socal learnng. (Resul ): Agens who have hgh wealh n each generaon are more lkely o choose deas from socal learnng whle oher agens are more lkely o choose deas from socal learnng also. Table 3: Who chooses whch level of learnng? Indvdual Learnng Socal Learnng Hghes wealh 3 hghes wealhy Hghes wealh 3 hghes wealhy Ohers Indvdual Learnng 0.04 0.04 0.44 0.440 (0.0587) (0.0587) (0.055) (0.047) Socal Learnng 0.7856 0.7560 0.7986 0.7986 (0.055) (0.047) (0.0587) (0.0587) For example, abou 79% of agens are he ones, some of whom wh he hghes wealh a a parcular generaon choose deas from ndvdual learnng (or socal learnng), and some of he ohers nvolve socal learnng. For he wealhy agen, he numbers are que hgh regardless of he levels of learnng. Bu he deas n ndvdual learnng s more frequenly seleced by he wealhy han he ohers do. So, we would conclude ha when agens have really hgh wealh, hey are more lkely o choose her own deas han oher agens do. The oher agens are more lkely o nvolve socal learnng.
These wo resuls are he same for he cases of he hree hghes wealhy agens. Bu he fracon for he wealhy agens usng ndvdual learnng s decreasng as akng more wealhy agens no accoun, gven ha he ohers choose socal learnng. Agens use her prvae deas more ofen han he ohers do only when hey have really hgh wealh. In oher words, mos of he agens follow socal learnng snce he socal learnng would possbly produce beer deas. The socal learnng domnaes he marke. The resuls ndcae ha he wealh level of each agen s mporan for choosng a level of learnng. I would be because he fness funcon n hs paper s wealh-based uly. However, f he fness funcon s based on, for example, predcon accuracy, we would reach a concluson ha an agen wh more accurae forecasng mehods s more lkely o choose ndvdual learnng han ohers do. Regardless of he fness funcon we specfy, would be concluded ha he socal learnng domnaes he marke, and mos agens would be beer off by consranng he use of her own deas. 5. Concluson Ths paper nroduces a learnng and adapaon mechansm whch allows agens o choose one rule among a se of deas updaed hrough boh ndvdual and socal learnng, and manly showed he followng wo resuls. Frs, he me seres from an economy wh nellgen agens doesn show a convergence o a homogeneous raonal expecaon equlbrum. The second nvesgaes whch level of learnng s lkely o domnae n he marke. I concludes ha he socal learnng domnaes he marke. Agens would be beer off n an ex ane welfare sense by consranng he use of her own deas. References Arfovc, Jasmna. The Behavor of he Exchange Rae n he Genec Algorhm and Expermenal Economes. Journal of Polcal Economy vol.04 no.3 (996): 50-54. Arfovc, Jasmna, and Gencay, Ramazan. Sascal Properes of Genec Learnng n a Model of Exchange rae. Journal of Economc Dynamcs and Conrol 4 (000): 98-005. Arfovc, Jasmna. Evoluonary Dynamcs of Currency Subsuon. Journal of Economc Dynamcs and Conrol 5 (00a): 395-47. Arfovc, Jasmna. Performance of Raonal and Boundedly Raonal Agens In a Model Wh
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Fgure 3: A Feedforward Neural Nework wh a Sngle Hdden-Un wh 5 npus (Equaon(4), (5), and (6) Inpu Hdden Layer Oupu Layer r = z d = z p log = z d log = z p m p,, log = z m 3 5 4 ω u ω 0 ω u ω 0 ω 3 u 3 ω 03 ω 4 u 4 ω 04 0 ω 5 u 5 ω 05 Tanh( u ) Tanh( u ) Tanh( u 3 ) Tanh( u 4 ) Tanh( u 5 ) h 4 h 3 h h h 5 ω 3 ω 33 ω 34 ω 3 ω 35 ω s Tanh( s ) s s 3 0.5 = anh( ω k z k + 0 ) k h k ω k =,...,5 λ = 0.5( + anh( ω + k = ω 3 kh k 5 )) λ 5