Towards an Agent-Based Foundation of Financial Econometrics: An Approach Based on Genetic-Programming Artificial Markets

Approach Based on Genetic-Programming Articial Markets

Shu-HengChen

DepartmentofEconomics

NationalChengchiUniversity

Taipei,Taiwan11623

E-mail: chchen@nccu.edu.tw

Tzu-Wen Kuo

DepartmentofEconomics

NationalChengchiUniversity

E-mail: g7258502@grad.cc.nccu.edu.tw

Taipei,Taiwan11623

Abstract

Usingafewnonlineareconometrictools,this

paper examines some time-series properties

of GP-based articial markets. We nd

that GP-based articial markets are ableto

replicate several stylized features well

docu-mented in nancialeconometrics. In

partic-ular, the time series generated by the

GP-based articial markets are consistent with

the eÆcient market hypothesis in the

lin-ear sense. Furthermore, the emergence of

thesestylizedfeaturesmaybecausedbysome

institutional factors, such asposition limits

and transaction factors. Byintroducing the

complexityofevolvedGP-trees,abottom-up

analysisoftheimpactoftransactiontaxeson

GP-basedarticialmarketsisalsoprovided.

1 Motivation

One of the recent achievements made in nancial

econometrics is to identify several salient features

shared by almost all nancialmarkets. Features like

fat tails, volatility clusters, and nonlinear dependence

havebeenwelldocumentedinPagan(1996). Fattails

concern the fourth moment (kurtosis) of the

empir-ical distribution and refers to the presence of excess

kurtosis,whichisanindicatorthatthetimeseries

un-der study is not normally distributed. Volatility

clus-tering concerns the second moment (variance), more

precisely, the dynamics of conditional variance. As

Mandelbrot (1963) described, large changes tend to

be followed by large changes-of either sign-and small

changes by small changes. Innancial econometrics,

thisphenomenonisformalizedastheGARCHprocess,

where \GARCH" standsfor Generalized

AutoRegres-siveConditionalHeteroskedasticity. Thelastone,

non-linear dependence,indicates that,while nancialtime

series isnotpredictablein thelinearsense,itmaybe

statistical basis, a satisfactory economic explanation

remainstobeestablishedforthesestylizedfacts.

In this paper, a time-series econometric study of a

GP-based articial market constructed by Chen and

Yeh(1997)isconducted. Weattempttotestwhether

GP-basedarticialmarketscanactuallyreplicatethe

above-mentioned stylized features. If GP-based

arti-cial markets can, in eect, replicate those patterns,

then theexpressive power of GP-basedmarkets may

help us further explorethe possibleinstitutional

con-nections for those stylized features. Inparticular, in

thispaper,wewouldliketoidentifythesignicanceof

two institutional factors, namely, position limits and

transactiontaxes.

2 The Analytical Model

Giventhe above-mentionedpurpose, GP-based

arti-cial markets are employed to generate articial time

series of prices of an abstract commodity. The

GP-basedarticial market used in this paperis basedon

Chen andYeh(1996,1997),which isknownasa

cob-web model in economics. Before proceeding further,

let's brie y review this model. 1

Consider a

compet-itive market composed of n rms which produce the

same goods by employing the same technology and

whichface thesamecostfunction described in

Equa-tion(1): c i;t =xq i;t + 1 2 ynq 2 i;t (1) where q i;t

isthequantitysuppliedbyrmiattimet,

andx andy aretheparametersofthecostfunction.

Since at time t 1, the price of the goods at time

t, P

t

,is notavailable, thedecisionaboutoptimal q

i;t

mustbebasedontheexpectation(forecast)ofP

t ,i.e., P e i;t . GivenP e i;t

andthecostfunctionc

i;t

,theexpected

1

One can nd details in Chenand Yeh (1996, 1997),

 e i;t =P e i;t q i;t c i;t (2) Given P e i;t , q i;t

is chosen at the level such that  e

i;t

can be maximized and, according to the rst order

condition,isgivenby q i;t = 1 yn (P e i;t x) (3) Onceq i;t

isdecided,theaggregatesupplyofthegoods

at timet isxed andP

t

,whichsets demandequalto

supply,isdeterminedbythedemandfunction:

P t =A B n X i=1 q i;t ; (4)

whereAandBareparametersofthedemandfunction.

GivenP

t

, theactualprotofrmiattimetis:

 i;t =P t q i;t c i;t (5)

Inarepresentative-agentmodel,itcanbeshownthat

the rational expectations equilibrium price (P 

) and

quantity(Q 

)are(ChenandYeh,1996,p.449):

P  t = Ay+Bx B+y ; Q  t = A x B+y : (6)

Toextend themodel (Equations (1)-(6)) with

specu-lation, thebehaviorof speculatorshasto bespecied

rst. SupposeweletI

j;t

representtheinventoryofthe

jth speculatorat the end ofthe tth period, then the

prottoberealizedatthenextperiod t+1is

 j;t =I j;t (P t+1 P t ): (7)

Ofcourse,theactualprot

j;t

isunknownatthe

mo-mentwhentheinventoryplaniscarriedout;therefore,

likeproducers,speculatorstendtosettheinventoryup

to thelevelwhere speculators'expected utility Eu

j;t

orexpectedprot E

j;t

can bemaximized. Weshall

followMuth(1961)toassumethattheobjective

func-tion forspeculatorsis to maximizetheexpected

util-ityratherthantheexpectedprot. Withoutassuming

anyspecicformofutilityfunction,whatMuth(1961)

did was to approximate the general utility function

u

j;t (

t

)bytaking thesecond-orderTaylor'sseries

ex-pansionabouttheorigin:

u j;t ( t )( t )=(0)+ 0 (0) j;t + 1 2  00 (0) 2 jt (8)

Based on Equation (8), the approximate utility

de-pendson themomentsoftheprobabilitydistribution

of t ,i.e., Eu j;t (0)+ 0 (0)E j;t + 1  00 (0)E 2 j;t (9)

(9), we can rewrite the expected utility function as

follows. Eu j;t  (0)+ 0 (0)I j;t (P e j;t+1 P t ) + 1 2  00 (0)I 2 j;t [ 2 t;1 +(P e j;t+1 P t ) 2 ];(10) where P e j;t+1

is the conditional expectation E(P

t+1 j t )and 2 t;1

istheconditionalvariancevar(P

t+1 j t ) and t

is the -algebra(the largestinformation set)

generated byP

t ;P

t 1

;:::. Theoptimalpositionof the

inventorycanthenbederivedapproximatelyby

solv-ing therstorder conditionandtheoptimalposition

oftheinventoryI  j;t is givenby I j;t =(P e j;t+1 P t ); (11) where =  0 (0)  00 (0) 2 t;1

. Equation(11)explicitlyshows

that speculators' optimal decision about the level of

inventory depends on their expectations of the price

in thenextperiod, i.e.,P e

j;t+1 .

Now, if the market is composed of n producers and

m speculators, the equilibrium condition is given in

Equation(12), A B 1 B P t + m X j=1 (P e j;t+1 P t ) = n X i=1 1 yn (P e i;t x)+ m X j=1 (P e j;t P t 1 ): (12) 3 Experimental Designs

Chen andYeh (1997)replacedtheconditional

expec-tations appearing in Equations (12) by a GP-driven

learningprocesses,and simulated thepricedynamics

under thisnewsetup. Whiletheyshowedhow

specu-lators mayhaveadverse impactson market stability,

propertiesofthese pricedynamicswerelargelyleft

un-exploited. In this paper, weshall rst resimulate the

pricedynamicsofthismarketandthenconducta

rig-orous econometric analysis of thepricedynamics. In

particular,wewouldliketoseewhetherourGP-based

markets posses theeconometric propertieswidely

ex-isting in nancial time series. If so, how the

emer-gence of these properties can be possibly accounted

for by institutionalfactors, such astransaction taxes

andpositionlimits.

The cobweb markets are composed of two groupsof

adaptiveagents, producers and speculators. (At this

stage,consumptiondemandisgivenexogenouly;hence

the adaptive behavior of consumers is not explicitly

modeledatthis moment.) Theadaptivebehavior

ad-dressed here is exclusively restricted to the

on which the agent'sforecast and decision-makingis

based. Forproduceri,thismodelisaforecasting

func-tionemployedto forecastthenextperiod'sprice,i.e.,

P e

i;t

in Equation (2). For speculator j, this model is

an positionfunction, which is afunction of price

his-tory, i.e., I

j;t

in Equation (11). The evolving agents

canthenbeconsideredasthe evolutionofacollection

of models: POP 0 !POP 1 !POP 2 !:::!POP t !:::; (13) wherePOP t

denotesthepopulationofmodelsattime

periodt. Anaturalapproachtoimplementthe

evolu-tionaryprocessdepictedaboveisgeneticprogramming.

The end-user supplied control parameters for this

study is givenin Table1. Here,weconsider amodel

composedof300producersand100speculators.These

numbersarechosentoroughlymimicarealadvanced

economy, i.e., 25% of GDP is from the nancial

in-dustry and 75% of GDP is from the manufacturing

industry. Thefunction set denes theset of possible

mappings,i.e.,theset ofallpossibleformsofP e i;t and I e j;t

. As wemaynotice, thefunctions included in our

function set are very limited to only +; ;Sin;Cos.

This choice is based on our calibration described as

follows. According to Equation (6), the equilibrium

priceisdeterminedbyfourparametersA;B;xandy,

and is $1.12 given their values specied in Table 2.

Therefore,asimpleoperationofSin andCos isgood

enough to have arange covering this point, 1.12. In

otherwords,thefunctionsetchosenhereisaminimal

set tosatisfytheclosure property.

The other reason that we havethis limited choice is

due to positionlimits. Foreachspeculatorj,I

j;t can

be both positive (long position) and negative (short

position). However, these positions are restricted to

a limit s, i.e., s  I

j;t

 s;8t (See Table 2). So,

theinclusionofExp,R Log,andcaneasilymake

I

j;t

beyondthisboundaryandresultin anumber

be-ing either s or s. Therefore, while speculatorscan

be dierent in the genotype, but is identical in

phe-notype, and hence identical in tness. In this case,

the selection process may, in eect, proceed with an

almost uniform distribution, which is certainly not a

desirablefeature.

The terminal set includes the ephemeral random

oating-point constant R ranging over the interval

[-9.99, 9.99] and the price lagged up to 10 periods

P

t 1 ;:::;P

t 10

. Whilelittleguidanceisavailableto

de-cidewhathorizonshouldspeculatorsusetoformtheir

expectations,basedonafewpilotexperiments,we

be-lieve that most of our resultspresentedbelow would

notbesensitiveto alongerhorizon. Theterminalset

and thefunctionset togetherdetermineinputsof the

treesevolvedbyGP.Theoutputis P e i;t forproducers andI j;t forspeculators.

Table1: Tableau ofGP-BasedCobwebModel

Number of producers (n) 300 Number of speculators (m) 100

Number of trees

cre-atedbythefullmethod

30(P),10(S)

Number

of trees created by the

growmethod

30(P),10(S)

Functionset f+; ;Sin;Cosg

Terminalset fP t 1 ;P t 2 ;


;P t 10 ;R g

Selectionscheme Tournamentselection

Tournamentsize 2

Number of trees

cre-atedbyreproduction

30(P),10(S)

Number of trees

cre-atedbycrossover

210(P),70(S)

Number of trees

cre-atedbymutation

60(P),20(S)

MutationScheme TreeMutation

Probabilityofmutation 0.2

Maximumdepthoftree 17

Probability of leaf

se-lectionundercrossover 0.5

Numberof generations

(GEN=t)

9000

Maximum number in

thedomainofExp

1700

Criterion oftness Prot

\P"standsfortheproducersand\S"standsforthe

spec-ulators. Thenumberoftreescreatedbythefullmethodor

growmethodisthenumberoftreesinitializedin

Genera-tion0 withthedepthoftree being2, 3,4,5,and6. For

details,seeKoza(1992).

netic programming. Whenapplyinggenetic

program-mingtooptimization,theusermustnoticethat

dier-entselectionschemesmayhavedierentimplications

forthetnessvalue,selectionintensity,selection

vari-ance, and lossof diversity. Bythe sametoken,when

geneticprogrammingisappliedtosimulatingthe

evo-lution and learning of the economic system, we have

to keepin mind that dierent schemesmayhave

dif-ferent economic implications. From the viewpointof

matchingprocesses,proportionateselectionisproneto

aglobal networkandtournamentselectionis proneto

a local network. Since local interaction among

spec-ulators plays an extremely important role in nance

(Shiller, 1984), tournament selection is more

appro-priate thanproportionateselection.

Inthecontextofeconomics,protseemstobeavery

naturalmeasure for tness. Here, protis dened in

Equation(5)fortheproducerandinEquation(7)for

Table2: InstitutionalDesigns Scenario A B C D E F G H TaxRate () 0 0.01 Position Limit(s) 0.01 0.1 0.01 0.1 0.01 0.1 0.01 0.1 Parameter A 2.296 3.36 2.296 3.36 Parameter B 0.0168 0.032 0.0168 0.032 Parameter x 0 Parameter y 0.016 Cobweb Ratio (B=y) 1.05 2 1.05 2

For all cases,the timeconstraint for recoveringthe short

issettobe20.

ticial markets with the following two particular

in-stitutional designs. Oneconcernstradingrestrictions.

In addition to the above-mentionedposition limitss

and s, for those speculators who hold ashort

posi-tion(I

j;t

<0),thereisatimelimitdforrecoveringthe

short. Inthispaper,dissettobe20forallsimulations.

Inotherwords,whentheshortpositionhasremained

for20tradingdays,thespeculatorisforcedtorecover

theshort onthenexttradingday. Theotherinvolves

the transaction tax. The transaction tax considered

in thispaperisaproportionatetaxandisdenoted by

thetaxrate. Thetaxrateisimputedtospeculators

only andis imposed onbothdirectionsof trading,to

buy and to sell. These institutional parameters are

summarizedin Table2.

Here, weconsider twotax rates(0 and 1%)and two

position limits(0.01 and 0). Since transaction taxes

andpositionlimitsaretwomajorcomponentstoaect

speculators'potentialprots, changingthese two

pa-rameterswillhaveanin uenceonspeculators'motives

and hencetheir adaptive behavior. It would then be

interestingtoseehowthesechangesmayhavefurther

impactsonpricedynamicsintermsoftheir

economet-ric properties. Apart from thetax rateand position

limit,wealsoincludetwodierentcobwebratios(1.05

and2),whichisdenedastheratiooftheslopeofthe

demand curveto the slopeof thesupply curve, B=y.

While, by Equation (6), these two dierent cobweb

ratioshavethesamerationalexpectationsequilibrium

price($1.12),ahighercobwebratiotendstobemore

inherentlyunstable. (ChenandYeh, 1996)

Giventhedescriptionofouradaptivespeculatorsand t A B 1 B P t + m X j=1 I j;t = n X i=1 1 yn (P e i;t x)+ m X j=1 I j;t 1 ; (14) where s  I j;t

 s for all j;t, and 0  I

j;t  s if

I

j;t k

<0fork=1;:::20.

Everythingwehavedescribedisalsowellencapsulated

intoEquation(13). TheevolvingtargetsarefP e i;t g 300 i=1 and fI j;t g 100 j=1

. Atthe end of each trading day, P

t is

announced,andtnessofi;j,f

i;t g 300 i=1 andf j;t g 100 j=1 ,

can be calculated. Genetic operators are then

ap-plied to evolvethesetwopopulationsseparatelywith

thetournamentselectionscheme. Thenewgeneration

fP e

i;t+1

gand fI

j;t+1

gis thengenerated,and the

mar-ket is open again. The cycle goes onand on until it

meets theterminationcondition,whichisthenumber

ofgenerationsin thispaper.

\NumberofGenerations"insettobe9,000inall

simu-lations. Noticethatthenumberofgenerationsisalso

the time scale of the simulation, i.e., GEN = t. In

other words,wearesimultaneouslyevolvingthe

pop-ulationwhilederivingthemarket-clearingprice,P

t .

Finally, theprogram to implement all simulations in

this paper is called Speculators, which is available

from thewebsite:

http://econo.nccu.edu.tw/ai/sta/csh/Software.htm

4 Time Series Analysis of Price Series

There are totally eight scenarios simulated in this

study. For each scenario, weconducted ve

indepen-dentruns,with9000periods foreach. These resulted

in 5articialtimeseriesforeachscenario(40time

se-ries in total). Figures 1.1-1.4display the time series

fP

t

gforatypicalrunforsomescenarios.Table3

sum-marizesthebasicstatisticsofthesesimulations.Based

ontheseguresandstatistics,wecanseethatP

t

basi-cally uctuates around therational equilibrium price

(P 

=1.12). However,thevolatilityofP

t (

p

Var(P) )

depends on the institutional parameters, in

particu-lar,thecobwebratioandthepositionlimit. Generally

speaking,thehigherthecobwebratioandtheposition

limit,themorevolatiletheprice. Whatseemsalittle

counter-intuitive is that 1 percent tax rate does not

stabilize thepricemovementto asignicantdegree.

Wethenexaminedtheeconometricpropertiesofthese

articial time series. The statistical properties

un-der examination are motivated by the list of

styl-ized features documented by Pagan (1996). To

ini-Table3: MeanandVolatilityofPrices Scenario P p Var(P) Scenario P p Var(P) A 1.1189 0.0191 E 1.1291 0.0148 B 1.1181 0.0209 F 1.1181 0.0212 C 1.1183 0.0359 G 1.1184 0.0717 D 1.1190 0.0481 H 1.1197 0.0505 ThePand p

Var(P)reportedherearetheaverageofthe

meanandthestandarddeviationoftheveruns. Foreach

run, the mean and thestandard deviation are calculated

fromthelast6000observationsonly,i.e.,fP

t g

9000

t=3001 .

dropped from the original series. In other words, all

statisticalpropertiesareexaminedunderthesubseries

fP

t g

9000

t=3001

. The rstpropertyto examineis

normal-ity. Acommonstatisticemployedtotestnormalityis

the Jarque-Bera statistics. Based on these statistics,

the null hypothesis that the price series is normally

distributedisrejectedforall40seriesatthe5%

signi-cancelevel. Thisresultisconsistentwithawell-known

resultinempiricalnance: mostnancialreturnseries

are not normally distributed (the tailsare toofat as

opposedtothenormaldistribution).

Thenexteconometricpropertytoexamineisthe

IID-ness oftheseries, i.e., are price series identically and

independently distributed over time?. The most

fre-quently used test in this area is the celebrated BDS

test (Brock,Dechert andScheinkman,1996). Due to

thepagelimit,wedonotintendtogiveafullaccount

oftheBDStest, theinterestedreadercannddetails

and a program to run this statistic in the following

website:

http://econo.nccu.edu.tw/ai/sta/csh/course/naecon/

lec4/lec4.htm

The BDS test is frequently applied to testing

non-linear dependence. Hence, the linear process of the

time series has to be ltered out before

implement-ingthetest. WeappliedRissanen'spredictive

stochas-tic complexity (PSC) to lter out the linear process

of each fP t g 9000 t=3001 . (Rissanen, 1986). A detailed

description of PSC with many illustrated examples

andacomputerprogramcanbefoundinthewebsite:

http://econo.nccu.edu.tw/ai/sta/csh/

course/naecon/lect8/lect8.htm

One oftheby-productsof thePSC lteris to inform

us of the linear AutoRegressive-MovingAverage

pro-cess,i.e.,theAR MA(p;q)process,extractedfromthe

originalseries. Amongthe40seriesexamined,38have

no linearprocess at all, i.e., theyare allidentied as

AR MA(0;0). Theonlytwoexceptionsaretheonerun

under ScenarioC(AR MA(0;2))and theone run

un-der Scenario E (AR MA(1;0)). This result indicates

that the GP-based articialmarketissoeÆcientthat

there are hardly any linear signals left. To some

ex-WethenappliedtheBDStesttothelteredresiduals.

Theparameter\"intheBDStestisequaltoone

stan-dard deviation. (In fact, wealso triedotherepsilons,

buttheresultisnotsensitivetothechoiceofepsilon.)

The embedding dimensions considered are from 2 to

5. Following Barnett et al. (1997), if the absolute

value of all BDS statistics under various embedding

dimensions aregreaterthan 1.96,the nullhypothesis

of IIDness is rejected. In this case, nonlinear

depen-dence is detected. If all of them are less than 1.96,

thenone failstorejectthenullhypothesisofIIDness.

However,ifsomearegreaterthan1.96,and someare

lessthan1.96,thentheresultisambiguous. (Actually,

in Barnett'spaper, they used theword \strongly

re-ject" and\weaklyreject". Wedonotintendtomake

suchadistinctionhere.)

Basedonthiscriterion,thenullhypothesisofIIDness

is rejected 18times outof 20when transaction taxes

are imposed. However, it is rejected only 14 times

out of 20 whenthere is notransaction tax.

Further-more, whentheposition limitis relaxedfrom 0.01to

0.1, the number of rejection increases from 14 to 18

times. Therefore,imposing transaction taxesand

re-laxing position limits may weaken the degree of

sta-tisticalindependence ofthedata. Thersthalfofthe

resultisnothingsurprisingastransactioncostreduces

the chance of arbitragingand hence make the useof

information less eÆcient in terms of statistical

inde-pendence. Whatissurprisingisthesecondhalfofthe

result, for the relaxationof the position limit should

makeitmorerewardingfor speculatorsto extract

in-formation from the price series. The resultant time

series is anticipated to be moreeÆcientand is more

likelytobeIID. 2

So far, we have examined our simulated time series

with atest fornon-linear dependence. However,itis

wellknownthat mostofthenon-linearityin nancial

dataseemstobecontainedintheirsecondcomments.

Thevoluminous(G)ARCH (Generalized

AutoRegres-sive Conditional Heteroskedasticity) literature is the

outcomeoftheattempttocapturebyappropriatetime

seriesmodelstheregularitiesinthebehaviorof

volatil-ity. Inorder toproceed further,wecarryoutthe

La-grange multiplier test for the presence of ARCH

ef-fects. A detailed descriptionof ARCH and GARCH

andanassociatedSASprogramtorunthetestis

avail-ablefromthewebsite:

http://econo.nccu.edu.tw/ai/sta/csh/course/naecon/

lec6/lec6.htm

If the ARCH eect is rejected, we will further

iden-tify the GARCH structure of theseries byusing the

Akaike Information Criterion (AIC). The results are

exhibited inTable4.

2

However, as we shallsee later, relaxing position

Scenario A B C D E F G H Run 1 (2,1) (1,1)  (1,1) (2,1) (2,2) (1,2) (1,1) 2 (2,2) (2,2) (1,1) (2,1) (1,1) (2,1) (2,2) (1,1) 3 (2,1) (1,1)  (1,1)  (1,1) (2,1) (1,1) 4 (1,1) (1,1) (2,1) (2,1)  (1,1) (2,1) (2,2) 5 (2,1)  (2,1) (1,1) (1,1) (2,1) (2,2) (1,1)

The (p,q) within each bracket refers to the model

GARCH(p,q), while  means that there is no ARCH

ef-fect.

There are couple of points worth noting. First, we

canndthat volatility clusteringcharacterizedasthe

ARCH eect is quite ubiquitous. Out of the 40

se-ries,there areonly5serieswithouttheARCHeect.

Not surprisingly,all theseve seriesfail to reject the

nullhypothesis oftheBDStest. However,afew time

series which fail to reject the BDS test still have the

ARCHeect. Second,transactiontaxesseemto play

norole inaccountingfortheemergenceoftheARCH

eect. For example, 17 out of the 20 runs without

transaction taxes exhibit the ARCH eect, while 18

outofthe20runs withtransactiontaxeshavethe

ef-fect. Nevertheless, position limits may have certain

eects. Inoursimulations,thescenarioswithlow

po-sitionlimitsfailto rejecttheARCHeect in4outof

20runs,whiletheones withhigh position limitsonly

failonce. Therefore,onemayhypothesizethatposition

limits mayhave someconnectiontothe ARCHeect,

andrelaxingpositionlimitscanincreasethe likelihood

of the emergenceofthe ARCHeect.

5 The Complexity of Evolved

Strategies

In addition to the macro-phenomenon, i.e., the price

series, an equally important thing is the

micro-phenomenon, i.e., what happens for the individuals

whocollectively generatesuchacomplexnonlinear

dy-namics. Certainly, one may ask whether the

com-plex macro-phenomenon is coupled with the complex

micro-phenomenon; in other words, agents with

so-phisticated strategies collectively generated complex

macro-phenomenon.

Togiveananalysisof theconnectionbetweenbottom

andup,wegivetwodenitions ofthecomplexityofa

GP-tree. The rstdenition isbased on thenumber

of nodesappearing in thetree, while the second one

is basedonthedepth ofthetree. Atthe endof each

run,wehaveaproleoftheevolvedGP-treesfor300

producersand100speculators. Thecomplexityofeach

GP-treeiscomputed. Wethenaveragethecomplexity

of evolvedGP-treesbyproducersandbyspeculators.

ofNodes

Averageover300Producers

Scenario A B C D E F G H Run 1 1.05 1.21 1.01 1.01 1.00 1.01 1.02 1.02 2 1.02 1.01 1.01 1.03 1.02 1.05 1.01 1.00 3 1.10 1.01 1.03 1.05 1.02 1.02 1.06 1.03 4 1.03 1.06 1.02 1.01 1.01 1.06 1.06 1.05 5 1.02 1.03 1.02 1.05 1.03 1.01 1.02 1.09 mean 1.04 1.06 1.02 1.03 1.02 1.03 1.03 1.04 mean 1.03 1.03

Averageover100Speculators

Scenario A B C D E F G H Run 1 1.88 4.47 1.45 3.38 1.26 2.46 3.13 5.59 2 2.51 2.71 4.42 4.20 1.93 1.93 2.16 3.36 3 4.46 3.86 4.32 2.21 2.87 1.26 3.17 4.01 4 1.99 1.59 1.49 1.93 2.02 1.53 2.48 3.07 5 1.23 1.82 23.19 8.96 2.86 1.55 2.97 1.23 mean 2.41 2.89 6.97 4.14 2.19 1.75 2.78 3.45 mean 4.10 2.54

Atrstsight,itseemstobeveryremarkablethat

pro-ducers are so simple: many evolved GP-trees consist

of only one node. This seems to indicate that GPis

not \at work" at all or the problem is not

interest-ing. Acarefulexamination, however,informsus that

this is not the case. In fact, in each stage of

evolu-tion, very oftenthe ttest tree belonged to theclass

ofcomplexnonlinearmodels. However,thisgroupdid

notsuccessfullypropagate.

The reasonis that forproducerstheirprotsdepend

on only the ow of thequantity supplied, which is a

function ofthe rst moment of prices, E(P

t

). On the

average, this numberis constantly around1.12

(Fig-ures1.1-1.4andTable3). Furthermore,fromthe

pre-vious PSC-ltering results, there is not much linear

signalleftintherstmomentofthepriceseries. Asa

result, statisticians whostudy ourmarkets may

con-cludewiththefollowingmodel:

P t =1:12+ t ; (15) where E( t

)=0. Giventhismodel, theforecast

P e

i;t

=1:12: (16)

seemstobeverycompetitive,andthisisthemost

com-mon type of the one-node trees. Although there are

other nonlinear models which can outperform

Equa-tion(15),but theyareonlysuitablefor certaintypes

of nonlinearityandarenotrobustto thegeneral

non-linear properties of 

t

. Hence, while complexmodels

canfrequently havethechampionship, theyhave

dif-cultiestokeepitandbeprosperous. Eventually,the

majoritybelongstotherobustsimplemodels,suchas

taking the simpleaverage.

While evolving simple strategies may sound strange

Table6: Complexityof EvolvedStrategies: Depth of

Trees

Averageover300Producers

Scenario A B C D E F G H Run 1 1.03 1.10 1.01 1.01 1.00 1.01 1.02 1.01 2 1.01 1.01 1.01 1.02 1.02 1.04 1.01 1.00 3 1.05 1.01 1.02 1.03 1.01 1.01 1.03 1.02 4 1.02 1.02 1.01 1.00 1.01 1.02 1.03 1.03 5 1.01 1.02 1.01 1.02 1.02 1.01 1.01 1.05 mean 1.02 1.03 1.01 1.01 1.01 1.02 1.02 1.02 mean 1.02 1.02

Averageover100Speculators

Scenario A B C D E F G H Run 1 1.54 3.16 1.37 2.69 1.21 1.77 2.46 3.89 2 1.86 2.05 2.61 2.63 1.59 1.59 1.68 2.50 3 2.72 2.18 2.61 1.86 2.53 1.26 2.35 2.67 4 1.56 1.34 1.33 1.71 1.65 1.37 1.85 2.00 5 1.15 1.44 8.40 3.92 1.86 1.52 2.03 1.20 mean 1.77 2.03 3.26 2.56 1.77 1.50 2.07 2.45 mean 2.39 1.95

are not best (optimal) at verypointin time, but are

veryrobustfordierentenvironmentsandbecomethe

mostpopularrulesonwhichpeoplerely. Onefamous

example is that 65 to 70 percent of all mutual and

pension fund managers fail to beat the simple rule,

the market indices overthe longrun(Malkiel, 1996).

Sophisticated strategiescan beoptimal, but theyare

notrobustandhavetobeupdatedquickly. Whenthey

arenotproperlyupdated,theycanperformextremely

poorin averydynamicenvironment.

On the other hand, speculators have evolved more

complex strategies than producers, and this is true

forallscenarios. Forspeculators,theirprotsdepend

on both the ow and stock of the quantity supplied,

whichisafunctionofboththerstmomentand

high-ordermoments. Fromthe previousvolatilityanalysis

(GARCH analysis), most series have linearstructure

in thesecond moment;consequently, itrequires

spec-ulators to evolve more complex strategies to extract

thesesignals. Weplantoputallevolvedstrategieson

the website so that interested reader can have their

ownanalysis.

Theotherinterestingresultistheeectofthe

transac-tiontax onthe complexity of evolved strategies. From

Table5, it can been seenthat imposing the

transac-tion tax tends to evolvesimpler strategiesfor

specu-lators. This resultis consistentwiththeonefoundin

theBDStest. IntheBDStest, imposingthe

transac-tiontaxincreasestheprobabilityofrejectingIIDness,

i.e., increase the chance of leaving nonlinear signals

unexploited. As weconjectured earlier,this isdue to

a weaker incentive to extract information, as

impos-ingthetransactiontaxreducethechanceofarbitrage.

Here,fromthebottompart,weactuallyseethat

spec-ulators indeed become \more lazy" when transaction

Thispaperprovidesathoroughtime-seriesanalysisof

prices generated from genetic-programming articial

markets. Many stylized features well documented in

nancial econometrics can in principle be replicated

from the GP-based articial markets, which

includ-ingleptokutosis,non-IIDnessandvolatilityclustering.

Moreover,theGP-basedarticialmarketsallowusto

searchfor thebehavioral foundation ofthese stylized

features. The two institutional factors, transaction

taxesand position limits, may both contributeto the

emergence ofthosestylizedfeatures.

Asonemayexpectthattransactiontaxescanhave

ad-verseeectsonspeculativetrades. Ouranalysisof

GP-basedmarketspartiallysupportsthisviewpoint. From

the bottom part, transaction taxes reduce the chance

of arbitrage;hence,speculatorshavelessincentiveto

search. In particular, the GP-trees evolved get

sim-plerwhenthetransactiontaxisimposed.

Correspond-ingto thebottompart,whatwehaveexperienced on

theupperpartisalessunexploitedoramore

nonlin-ear dependentseries. Nevertheless, theemergenceof

volatilityclusteringmaybeaconsequenceofrelaxing

position limits and havelittleto dowith transaction

taxes.

The empiricalevidences accumulated from GP-based

markets' simulations are quite limited. At this

mo-ment, theycan beonlyuseful forthepurpose of

mo-tivating hypotheses. However,thepointofthispaper

is mainlyto showwhatGP-basedmarketscan

poten-tiallyservefor theadvancementoftheeconomic

the-ory. Inthefuture,itisexpectedthat alargerscaleof

simulationwill beconducted forgettingmorefruitful

results.

Acknowledgement

ResearchsupportfromNSCgrants

No.88-2415-H-004-008and No. 87-I-2911-15isgratefully acknowledged.

Theauthorsarealsogratefulforthehelpfulcomments

from threeanonymousreferees.

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