Learning-To-Forecast with Genetic

Algorithms

Working paper

Version February 2013 with appendices

Mikhail Anufrievy,Cars Hommesz and Tomasz Makarewiczz

yUniversity of Technology,Sydney

zCenter for Nonlinear Dynamics in Economics and Finance,University of Amsterdam;

Tinbergen Institute

Amsterdam { February 27,2013

Learning to Forecast with Genetic Algorithms

Working paper | version February 2013

Mikhail Anufriev,Cars Hommes and Tomasz Makarewicz

February 27,2013

Abstract

In this paper we study a model in which agents independently optimize rst order

price forecasting rule with Genetic Algorithms.This agent-based model (inspired by

Hommes and Lux,2011) allows for explicit individual heterogeneity and learning.We

show that it replicates individual behavior in various Learning-to-Forecast experiments.

In these,the subjects are asked to predict prices,which in turn depend on the predictions.

We use the data from Heemeijer et al.(2009) to ne tune our GA model.Furthermore

we investigate three other LtF experiments:with shocks to the fundamental price (Bao

et al.,2012),cobweb economy(vd Velden,2001;Hommes et al.,2007) and two-period

ahead nonlinear asset pricing market (Hommes et al.,2005).We perform a Monte Carlo

exercise with 50-period ahead simulations and use Auxiliary Particle Filter to study one-

period ahead forecasting performance of the model on the individual level,a novelty in

the literature on agents-based models.Our model is robust against these complicated

settings and outperforms many homogenous models,including Rational Expectations.

1 Introduction

Price expectations are a cornerstone of many economic models,because the economic agents

often operate in a dynamic context.Consumers have to organize their life-time work and con-

sumption paths,while companies decide on how to build up future production capabilities.In

either case,the agents want to know how the uncertain future may unfold.What makes mod-

eling predictions dicult is that they typically form a feedback with the realizations through

agents decisions.For instance,if everybody expects an increased price of a consumption good,

consumers are likely to save less,while rms rise production.This implies lower market clear-

ing price in the future { scenario not anticipated by the agents.It is therefore likely that they

would alter their predictions,leading to a new realized price.

Even if the agents know the structure of the economy,the price-expectation feedback can

to lead to non-trivial dynamics (Tuinstra and Weddepohl,1999).This picture becomes more

complicated if the agents furthermore have to learn this structure (Grandmont,1998).Agents

do want to form ne price expectations,but how would they cope with this complexity?

1

The traditional literature (after Muth,1961) emphasizes Rational Expectations (RE) hy-

pothesis,which states that in the equilibrium the predictions have to be model consistent.

Most economists would interpret RE as an`as-if'approximation { real markets behave as if

their representative agents were perfectly rational,because the real people are rational enough

to learn to avoid systematic,correlated errors.

1

However,this is not conrmed by the data.

Recent important example comes from the housing market in US before the latest economic

crisis,where people systematically misjudged the long-termvalue of their houses (Bentez-Silva

et al.,2008;Case and Shiller,2003;Goodman Jr.and Ittner,1992).In a broader context,

the in ation expectations formed by the`Jones'are far from the RE predictions (Charness

et al.,2007) and can be subject to cognitive biases (Malmendier and Nagel,2009).Many

rms similarly fail to use RE (see Nunes,2010,for an example on Phillips Curve and survey

expectations).

The failure of RE made many economists look for an alternative model with explicit

learning.They faced the so called`wilderness of bounded rationality'problem:there is

a myriad of possible learning mechanisms with varied restrictions on human memory and

computational capabilities.These range from simple linear heuristic models (see Evans and

Ramey,2006,for a discussion of adaptive expectations),through econometric learning (Evans

and Honkapohja,2001),through heuristic switching type of models (Brock and Hommes,

1997) to evolutionary learning mechanisms (Arifovic et al.,2012).Moreover these mechanisms

can lead to dierent dynamics.For example,Bullard (1994) and Tuinstra and Wagener

(2007) show that for a standard OLG economy,where the agents use OLS learning for price

forecasting,the choice of the learned variable (level of prices or in ation) makes a dierence

between stable and chaotic dynamics.

Learning-to-Forecast (LtF) experiments (Hommes,2011) oer a simple testing ground for

learning mechanisms.In these,the controlled experimental economies are simple and have

a straightforward fundamental (RE) equilibrium.Just as in the case of the real markets,

the subjects observe the realized prices and their past individual predictions,but not the

history of other subjects'predictions,and are not directly informed about the quantitative

law of motion of the economy.Many LtF experiments contradict the RE hypothesis.The

subjects can coordinate on oscillating and serially correlated time series.Convergence to the

fundamental equilibrium happens only under severe restrictions on the experimental economy

(Hommes,2011).Another important nding in the experiments is heterogeneity:within the

same experimental group,subject tend to give dierent predictions with dierent dynamic

structure and reliance on the past prices,which cannot be fully explained by the type of the

experimental economy.

The most successful attempt to explain the LtF experiments comes with the so-called

Heuristic Switching Model (HSM;Brock and Hommes,1997).The basic idea of the model

is that the agents have a set of simple forecasting heuristics (rules of thumb like adaptive or

1

One interesting and straightforward explication of this approach can be found in the concluding section

of Blundell and Stoker (2005).

2

trend extrapolating expectations) and choose those that had a better past performance.The

model replicates the stylized,aggregate dierence between the treatments in Heemeijer et al.

(2009) (henceforth HHST09).Nevertheless,the authors consider a limited set of heuristics

and so cannot fully account for the individual heterogeneity.Moreover they are unable to

explain the mechanism,with which the subjects would learn those heuristics.For instance,

Anufriev and Hommes (2012) use HMS to explain the experimental from a non-linear asset

pricing experiment (Hommes et al.,2005) (henceforth HSTV05),but only with a broader set

of forecasting rules.

In our paper we would like to reinforce the original HSMso that it will be able to replicate

the individual learning and heterogeneity in various experimental settings.To do so,we will

use Genetic Algorithms (GA).GA are a exible optimization procedure,thus the GA-based

model retains basic economic interpretation.Agents,who use GA,have to rely on a second-

best forecasting rules.Nevertheless,they learn to use them eciently.For example,if there

is a signicant trend in the data,the agents may want to harvest speculative trade revenues.

To do so,they will update their forecasting rule's parameters with GA,making it more trend

extrapolative.

GA was already used in its social learning form to explore stylized facts from experimental

data,outperforming RE hypothesis (Arifovic,1995),with the examples of the exchange rate

volatility (Arifovic,1996;Lux and Schornstein,2005) or quantitative choices in a cobweb

producers economy (Dawid and Kopel,1998).More mature use of GAcan be found in Hommes

and Lux (2011).In their setting the agents use GA to optimize a forecasting heuristic (instead

of directly optimizing a prediction) and,much like the actual subjects in LtF experiments,

cannot observe each others behavior or strategies.With this versatile model,Hommes and Lux

(2011) replicate the distribution of the predictions and prices of the cobweb experiments by

Hommes et al.(2007) and van de Velden (2001) (henceforth HSTV07 and V01 respectively).

In our paper,we want to build upon the GA-based individual learning introduced by

Hommes and Lux (2011) to explain the LtF experiments.The novelty of our paper is threefold.

The rst is that we will use a dierent than Hommes and Lux (2011) heuristic space,based on

the so called rst order rule,which is a mixture of adaptive and trend extrapolating heuristics.

This gives the model better micro-foundations,as HHST09 nd this rule to describe well the

individual expectations in their experiment.

The second novelty is that our model allows for a simultaneous explanation of dierent LtF

experiments,based on markets with breaks in the fundamental price or a highly non-linear

price expectations feedback.This challenge will prove the generality of our model.Finally,

the third novelty is that we explain the individual,not just the aggregate results of the LtF

experiments.We will use Auxiliary Particle Filter (Johansen and Doucet,2008),technique

based on Sequential Importance Sampling,to show that our model replicates the behavior of

the individual subjects.This is a major contribution to the literature,which usually focuses

on a model's t to the aggregate stylized facts,even if that model is agent-based by design.

The structure of the paper is the following.In the second section,we explain in detail the

3

Learning to Forecast experiments and also comment on the insight brought by the Heuristic

Switching Model.In the third section,we introduce the Genetic Algorithm model.We

calibrate it with the simple,linear feedback system from HHST09.We also comment on how

to use the Auxiliary Particle Filter to evaluate the t of the model to the individual predictions.

In the fourth section,we will use our model to describe the following experimental economies:

linear price-expectations feedback system with unexpected shifts to the fundamental price

(Bao et al.,2012) (henceforth BHST12);cobweb producers economy (HSTV07;V01) used also

by Hommes and Lux (2011);and nally a highly non-linear positive feedback asset pricing

economy,where the subjects are asked for two-period ahead predictions (HSTV05).The last

section will summarize our paper.For the sake of brevity and clarity,we decided not to

include too many technical details in the main text,including the full formal denition of

GA,or many robustness checks against the model specication.These can be found in the

appendix.

2 Learning to Forecast and Heuristic Switching

Consider a market with a set of subjects i 2 f1;:::;Ig,who are asked at each period t to

forecast price of a certain good.The subjects are explicitly informed that they are asked

only for and rewarded only for the accuracy of the predictions.Their role is of a forecasting

consultants for rms.These rms in turn will use subjects'predictions to optimize their

behavior,which determines the next market price.The subjects have no other in uence on

the realized price.The feedback relationship between the prices and predictions is summarized

by a reduced form law of motion in the form of

(1) p

t

= F(p

e

1;t

;:::;p

e

I;t

);

where the p

t

denotes the price and p

e

i

is the agent's i expectation of p

t

and F() results from

aggregating rms optimal choices.This is one-period ahead type of feedback,in the sense that

the price at period t depends on the predictions which were formulated after the previous price

p

t1

becomes known to the subjects.Dene the fundamental price p

f

as the RE outcome,the

self-consistent prediction:p

f

= E

F

p

f

;:::;p

f

.

Unlike the RE agents,subjects in the experiment have a limited information about the

market.They are told that their predictions in uence the average market mood,which in turn

determines the realized price,but they are given only a qualitative story about this feedback.

Moreover,they are not explicitly informed about the fundamental price.

2

One important example was investigated by HHST09,who use a linear version of (1):

(2) p

t

= A+B

P

I

i=1

p

e

i;t

I

A

!

= A+B(p

e

t

A);

2

Usually it is possible to infer it from the experiment instructions.Anecdotal evidence suggests that even

economics students,including graduate students,fail to realize it.

4

where p

e

t

=

P

I

i=1

p

e

i;t

I

is the average prediction at period t and A = p

f

is the fundamental

price.The two cases are with B > 0 (positive feedback) and B < 0 (negative feedback).A

typical example of the positive feedback market is a stock exchange.The investors will buy a

certain stock if they are optimistic and expect it to become more expensive.But this increased

demand means that the stock's price will indeed go up.In this way the investor sentiments

are self-fullling.the contrary case is a producers economy,who face a lag in the production.

If the producers expect high price in the future,they will increase the production and so the

future price must be low for the markets to clear.Here,there is a negative feedback between

predictions and prices.

In the experiment,the authors used two specic treatments for I = 6 subjects:

Positive feedback:p

t

=

20

21

(3 + p

e

t

) +"

t

;(3)

Negative feedback:p

t

=

20

21

(123 p

e

t

) +"

t

;(4)

where"

t

NID(0;0:25) is a small disturbance and the experiment run for 50 periods for

each group in both treatments.

The two treatments are symmetric.They have the same unique fundamental price p

f

= 60.

Also,the dumping factor jBj =

20

21

is the same in absolute terms.It was chosen so that under

naive expectations (p

e

t

= p

t1

),the fundamental price for both treatments is a unique and

stable steady state,but the system would still require some time to converge.

The two feedback treatments resulted in very dierent dynamics,see Figure 1a and Fig-

ure 1b for typical time series for the negative and positive feedback treatments respectively.

The price under the negative feedback would jump around the fundamental for a handful of

periods and hence converge to it.Only after that would the subjects converge to the fun-

damental as well,and before that their behavior was volatile.Most of the groups under the

positive feedback resulted in systematic oscillations as seen on Figure 1b,where the price twice

overshoots and once undershoots p

f

;and if the price actually converged to the fundamental,

it did so only towards the end of the experiment (which happened for two out of seven cases).

In spite of that,the subjects would coordinate in around three periods and remain so until

the end of the experiment.This means that the price oscillations were caused by systematic

prediction errors (in respect to the RE outcome),which were highly correlated between the

subjects.

To describe the subjects'behavior,HHST09 focus on the rst-order rule (FOR):

(5) p

e

i;t

=

1

p

t1

+

2

p

e

i;t1

+

3

60 +(p

t1

p

t2

);

for

1

;

2

;

3

> 0,

1

+

2

+

3

= 1, 2 [1;1].The authors estimated this rule separately

for each subject,based on their predictions from the last 40 periods.Notice that the third

term,the fundamental which is associated with the

3

coecient,was used for the sake of

5

stationarity of the estimation,and to test the RE hypothesis.

3

HHST09 nd that the individual forecasting rules are varied between the subjects,even

within the same treatment.The authors also report signicant dierence between the two

treatments.Under the positive feedback,subjects tended to focus on trend extrapolation and

estimated

3

fundamental price coecients were insignicant.Under the negative feedback,

the reverse holds:trend extrapolation is barely used,while the weight for the fundamental

price is signicant.This shows that a model with a homogeneous forecasting rule (RE,but

also linear heuristics like trend extrapolation or naive expectations) may explain one of the

two treatments,but not both at the same time.Moreover,it cannot explain the signicant

dierences between the subjects within each treatment.

This led Anufriev et al.(2012) to investigate the Heuristic Switching Model (HSM),in

which the subjects are endowed with two prediction heuristics:

adaptive:p

e

i;t

= p

i;t1

+(1 )p

e

i;t1

for 2 [0;1],

trend extrapolation:p

e

i;t

= p

i;t1

+(p

t1

p

t2

) for 2 [1;1],

where the authors have used = 0:75 and = 1.Both heuristics are a special case of

the rst-order rule.The adaptive heuristic is the FOR with

3

= = 0,while the trend

extrapolation is the FOR with

2

=

3

= 0.The idea of the model is that people adapt their

behavior to the circumstances.Subjects can at any time use any of the two heuristics,but

will focus on the one with higher relative past performance.Under the positive feedback,the

agents easily coordinate their predictions below the fundamental,but (by the construction of

the feedback equation) the realized price is slightly higher than the average prediction.Trend

extrapolation heuristic captures this gradual increase of the initial prices and so becomes

more popular among the agents.This reinforces the trend,as well as the performance of

the heuristic itself.Reverse story holds for the negative feedback:there is no possibility of

coordination unless the agents converge to the fundamental price,otherwise the realized price

is in contrast with the average market expectation.In this case adaptive expectations can

have a better performance,as they facilitate the agents to converge to the fundamental.

HSMcaptures the essence of the aggregated predictions behavior and successfully replicates

the results of HHST09 in a stylized fashion.The drawback of the model is that the authors

assume a limited space of heuristics.At rst glance this is not a problem,since the model can

be easily extended to include a broader portfolio of predicting rules (Anufriev and Hommes,

2012).Nevertheless,the model cannot explain two things.First issue is where do those

heuristics come from,that is,how the subjects are able to learn the two heuristics in the rst

place.Second issue with HSMis that it cannot account for the heterogeneity of rules between

the subjects and explain the experiment on the individual level.To answer these two issues,

we will introdcue a model with explicit individual learning through Genetic Algorithms.

3

Under RE,FOR in (5) should be specied with

1

=

2

= = 0,which implies that the subjects always

predict the fundamental price,p

e

i;t

= 60.

6

3 The model

3.1 Genetic Algorithms

Genetic Algorithms (GA) is a class of numerical stochastic maximization procedure,which

mimics the evolutionary operations with which DNA of biological organisms adapts to the

environment.GA were introduced to solve`hard'optimization problems,which may involve

non-continuities or high dimensionality with complicated interrelations between the argument.

They are exible and ecient and so found many successful applications in computer sciences

and engineering (Haupt and Haupt,2004).

GA routine starts with a population of random arguments that are possible,trial solutions

to the problem.Individual trial arguments are encoded as binary strings (strings of ones or

zeros),or chromosomes.They are retained into the next iteration with a probability that

increases with their relative performance,which is dened in terms of the functional value

of the arguments.This so called procreation operator means that with each iteration,the

population of trial arguments is likely to have a higher functional value,i.e.be`tter'.On

the top of the procreation,GA uses three evolutionary operators that allow for an ecient

search through the problem space:mutation,crossover and election,where the last operator

was introduced by the economic literature (Arifovic,1995).

Mutation At each iteration,every entry in each chromosome has a small probability to mu-

tate,in which case it changes its value from zero to one and vice versa.The mutation

operator utilizes the binary representation of the arguments.A single change of one bit

at the end of the chromosome leads to a minor,numerically insignicant change of the

argument.But with the same probability a mutation of a bit at the beginning of the

chromosome can occur,which changes the argument drastically.With this experimenta-

tion,GA can easily search through the whole parameter space and have a good chance

of shifting from a local maximum towards the region containing the global one.

Crossover Pairs of arguments can,with a predened probability,exchange predened parts

of their respective binary strings.In practice,the crossover is set to exchanges subset

of the argument.For example,if the objective function has two arguments,crossover

would swap the rst argument between pairs of trial arguments.This allows for experi-

mentation in terms of dierent mixtures of arguments.

Election Election operator is meant to screen inecient outcomes of the experimentation

phase.This operator transmits the new chromosomes (selected from the old generation

and treated with mutation and crossover) into the new generation only if their functional

value is greater than that of the original`old'argument.This operator ensures that

once the routine nds the global solution,it will not diverge from it due to unnecessary

experimentation.

7

The procreation routine and the three evolutionary operators have a straightforward eco-

nomic interpretation for a situation,in which the agents want to optimize their behavioral

rules,e.g.price forecasting heuristics.The procreation means that { like in the case of HSM

{ people focus on better solutions (or heuristics).The mutation and crossover are experimen-

tation with the heuristics'specications,and nally the election ensures that the experimen-

tation does not lead to suboptimal heuristics.

For the sake of presentations,we give the specic formulation of our GA in Appendix A.

For the technical discussion and examples of GA applications see Haupt and Haupt (2004).

3.2 Model specication

Consider the price-expectation feedback economy,which was introduced in the previous section

and which is captured by the reduced-form law of motion (1).We consider a set of I agents,

which we will call GA agents.GA agents use a general forecasting rule which requires exact

parameter specication,and each agent is endowed with H such specications.For the whole

following analysis,we take H = 20.Following the estimations by HHST09,we deploy a

modied rst order rule (FOR) in order to give our model robust micro-foundations.

To be specic,agent i 2 f1;:::;Ig at time t focuses on H linear prediction rules to predict

price at that period p

t

given by

(6) p

e

i;h;t

=

i;h;t

p

t1

+(1

i;h;t

)p

e

i;t1

+

i;h;t

(p

t1

p

t2

);

where p

e

i;h;t

is the prediction of price p

t

,formulated by the agent i at time t conditional on

using the rule h,and p

e

i;t1

is the nal prediction by the agent of the price at time t 1,which

the agent submitted to the market.

Notice that this specication (6) is a special case of the general FOR from equation (5),

with the anchor or the fundamental price weight

3

= 0.We experimented with specications

with some possible anchors,but they could not properly account for the dynamics in the

positive feedback market.

4

We will refer to the

i;h;t

2 [0;1] parameter as the price weight,

in contrast to the past predictions:the higher it is,the less conservative is the agent in her

predictions.The trend extrapolation parameter

i;h;t

shows the extent to which the agent

want to follow the recent price change.For convenience we will drop the subscripts of these

two parameters whenever we refer to them in general.

Contrary to the price weight ,the choice of the allowed interval for the trend extrapolation

is not immediately clear.Experimentation has led us to take the upper bound of to be equal

to 1:1,as this specication seems to t well to the positive feedback.

5

Thus,there are two

equally intuitive choices for the lower bound.At rst glance,the interval should be symmetric

around zero,so that the agents can learn to contrast the trend to the same degree as they

4

To be specic,we looked on the average price so far,and the fundamental price itself,see Appendix C

for a discussion.Note also that (6) is a combination of the two heuristics (trend extrapolation and adaptive

expectations) used by Anufriev et al.(2012),who also had no need for an anchor for their HSM.

5

Please refer to Appendix C for a discussion.

8

can extrapolate it.We refer to the model with such specied 2 [1:1;1:1] as GA FORT+C

(trend and contrarian).On the other hand,it is not immediately clear that people perceive

trend contrarian and extrapolative rules in the same way.In fact HHST09 report only two

subjects to use contrarian strategies,so one may nd it more appropriate to focus on a

model with 2 [0;1:1],which we will refer to as GA FORT (trend only).For the HHST09

experiment,the two specications for all practical reasons behave in the same way,so for the

sake of brevity we will report results only for the model with unrestricted contrarian rules GA

FOTR+C.However,this will not be the case for other experimental data,to which we will

come back in the next section.

Dene H

i;t

as the set of such heuristics of agent i at time t.We emphasize that these

sets can be heterogeneous between the agents.Furthermore GA agents do not want to have a

static set of heuristics.Instead,they want to learn:to optimize the heuristics so that they can

adapt to the changing landscape of the economy.For example,in some circumstances it may

pay o to focus on the trend in the data and agents would like to nd the optimal degree of

trend following,by experimenting with dierent trend extrapolation coecients

i;h;t

.Agents

do so by updating the set of heuristics with GA.

To be specic,the agents evaluate their heuristics based on the hypothetical forecasting

performance of these heuristics in the previous period,where their objective is prediction's

mean squared error (MSE).Formally,at the beginning of each period t,they focus on the

following hypothetical performance measure:

(7) U

i;h;t

= exp

(p

e

h;i;t1

p

t1

)

2

:

Thus dene the normalized performance measure as:

(8)

i;h;t

=

U

i;h;t

P

H

k=1

U

i;k;t

;

which is a logit transformation of MSE.The normalized performance measure (8) can be

directly interpreted as the weight attached to each heuristic h by agent i at time i { the

probability with which the agent wants to use that heuristic.

6

Agents compute the weights U

i;h;t

twice each period.First time happens during the learning

phase.Once p

t1

becomes available to the subjects,heuristic parameters from H

i;t1

are

encoded as GA binary strings and undergo one iteration of GA operators to form H

i;t

,where

the U

i;h;t

is used as the GA objective function.The important thing is that the agents

run independent GA iterations,or more formally,the model consists of I independent GA

procedures.Intuitively,the agents cannot observe each others heuristic sets,in specic they

cannot exchange successful specications.The GA procedure utilizes procreation and three

6

Notice that (8) is dierent than the actual experimental payo received by the subjects,and which was

used by Hommes and Lux (2011) in their GA model.We decided to use this performance measure for two

reasons.First,our model becomes general:it can be directly applied to other experimental data sets without

a change to its core ingredient.Second,in this way we obtain a clear link to the literature on HSM models,

which often uses the logit transformation of MSE in a similar fashion.

9

Parameter Notation Value

Number of agents I 6

Number of heuristics per agent H 20

Number of parameters N 2

Allowed price weight [

L

;

H

] [0;1]

Allowed trend extrapolation

FORT+C [

L

;

H

] [1:1;1:1]

FORT [

L

;

H

] [0;1:1]

Number of bits per parameter fL

1

;L

2

g f20;20g

Mutation rate

m

0:01

Crossover rate

c

0:9

Lower crossover cuto point C

L

20

Higher crossover cuto point C

H

1 (none)

Performance measure U() exp(MSE())

Table 1:Parameter specication used by the Genetic Algorithms agents.

evolutionary operators,with each agent using the same specication of parameters shown in

Table 1.U

i;h;t

is taken directly as the objective function,crossover exchanges parameters

i;h;t

and

i;h;t

between heuristics and each parameter is encoded with 20 bits,meaning 40 bits per

each heuristic.

7

After the agents have learned,that is,after the agents have updated their heuristic sets,

they have to pick one specic heuristic in order to dene their predictions for the period t,p

e

i;t

.

For this task they sample one heuristic h 2 H

i;t

with normalized (7) performance measure as

probabilities,but after the set of heuristics was updated (because of the optimization step,

the set H

i;t

in principle can be dierent from H

i;t1

).This means that the actual probabilities

(8) have to be recalculated.

The timing of the model is the following.Until some moment the agents cannot learn:the

rule itself requires past prices and predictions.In those periods,the agents sample random

predictions from a predened distribution which we take as exogenous.The heuristics are

initialized at random from a`uniform'distribution:each agent i has 20 strings of 40 bits

representing H

i;t

(Table 1);and each bit is initially 0 or 1 with equal probability.

Once the initial periods are over and the agents can start to learn,each period consists of

three steps:

1.Agents independently update their heuristics using one GA iteration,where the GA

criterion function is U

i;h;t

;

2.Agents pick particular heuristics and generate their predictions,and the probability that

a heuristic h by the agent i is chosen is given by the recalculated tness probability

i;h;t

;

3.The market price is realized according to (1) and agents observe the new price.

7

Our model seems to be robust against reasonable parameter changes.

10

We would like to emphasize the way in which we have chosen the specication of the model.

Given its complexity,estimation of its parameters is for all practical reasons infeasible.As we

will discuss later,there is a way to obtain likelihood measures on the model with Auxiliary

Particle Filter (APF),but it remains computationally demanding,and so can be used for

calculating a small set of comparative statistics at most.

8

On the other hand,GA parameters have no direct economic interpretation and we are not

interested in identifying themwith an uncanny precision.To the contrary,we prefer to rely on

a predened specication that has well-known properties and is widely used in the literature.

To be specic,we use exactly the same set of parameters as Hommes and Lux (2011).

9

As

mentioned before,the variable that requires calibrating is the allowed trend extrapolation .

We used the experimental data reported by HHST09 to ne-tune our model in this respect.

10

Despite the model not being directly estimated,we nd it to be able to replicate the stylized

results of the HHST09 experimental data,both on the aggregate and the individual level.

3.3 50-period ahead simulations for HHST09

First test for tness of our model to the experimental data are 50-period ahead simulations

for the HHST09 experiment (as was the number of periods for all experimental groups).

11

We

take the feedback equations (4) and (3) for the negative and positive feedback respectively

and simulate our model for 50 periods in total,without any information from the experiment

after (and including) period 2,specically the realized prices and predictions.

12

Such a 50-period ahead simulation depends on the random numbers used by the model,

since GA is a stochastic procedure.To understand the 50-period ahead behavior of the model,

we resample it until we obtain a satisfactory distribution of its dynamics.We use this Monte

Carlo (MC) experiment to compare the model with the experimental data (with MC sample

equal to 1

0

000).We emphasize that this is a dicult test,since it requires the model to`stay

close'to the data for 50 periods.

13

.

For initialization,the model requires exogenous initial prediction.

14

In the experiment,the

initial predictions varied between and within the groups,whereas the group average seemed

to aect the later dynamics under the positive feedback treatment (Anufriev et al.,2012).

Therefore,we do not want to use the same initial predictions for all the 50-period ahead

8

For a dual-core Pentium with 2:7GHz clock and 3:21 GB RAM,a shot of APF for one experimental group

takes approximately 25 minutes for a relatively small number of 32 particles.

9

The exception is that we take the mutation rate

m

= 0:9,instead of 0:6 like Hommes and Lux (2011).

Here we follow suggestion of Haupt and Haupt (2004).This has no signicant in uence our results.

10

Please refer to Appendix C for a discussion.

11

All simulations,as well as the GAand APF libraries,were written in Ox matrix algebra language (Doornik,

2007) and are available at request.

12

We include the realizations of"

t

to the feedback equation.

13

In one of the positive feedback treatment groups,one of the subjects`out of the blue'predicted ten times

higher price than both his previous forecast and the realized market price.This destabilized the whole market

for a number of periods.In the following analysis,we follow Anufriev et al.(2012) and omit this group and

hence focus on six positive feedback and six negative feedback treatment groups.

14

Recall from the previous discussion that we initialize the heuristics with a`uniform'distribution.We do

not change that for the 50-period ahead simulations,leaving only the issue of the initial predictions.

11

simulations.We follow suggestion by Diks and Makarewicz (2013) that a simple bootstrap

of the experimental initial predictions may be inecient for a large MC.Instead we sample

these from a distribution calibrated by Diks and Makarewicz (2013).

Negative feedback

Positive feedback

(a) Experimental Group 1

(b) Experimental group 1

(c) Sample GA FORT+C

(d) Sample GA FORT+C

Figure 1:HHST09:typical results for the experimental groups and sample 50-period ahead simu-

lations of the GAFORT+C model.Black line represents the price and green dashed lines

are the individual predictions.

For a rst impression,consider Figure 1 with sample paths from the experiment and

simulations of GA FORT+C.Notice that each group or simulation has dierent initial predic-

tions.Despite that,for both type of feedback,the experimental data and the 50-period ahead

simulations are similar.Under the positive feedback,the GA agents coordinate in terms of

predictions in around three,four periods (Figure 1d).Moreover the distance between their

predictions is smaller in the second period than in the rst.Despite this coordination,the

agents oscillate around the fundamental and the price overshoots and undershoots the fun-

damental price p

f

= 60,in a similar vein to the experimental groups under this treatment

(Figure 1b).Under the negative feedback (Figure 1c),the price is pushed to the fundamental

outcome in around 5 periods.Only then the GA agents can actually converge as well.Before

that,their behavior is volatile,much like for the case of the experimental groups (Figure 1a).

These sample time paths are representative for the MC study for the GA FORT+C.

15

As the authors of the experiment,we focus on the distance of the realized price from the

fundamental (aggregate behavior) and the standard deviation of the individual six predictions

at each period (degree of coordination;individual behavior).

15

Results for GA FORT are comparable.

12

Negative feedback

Positive feedback

(a) Realized price

(b) Realized price

(c) Predictions standard deviation

(d) Predictions standard deviation

Figure 2:HHST09:Monte Carlo for the GA FORT+C 50-period ahead simulations.Realized

price and coordination (standard deviation of the individual predictions) over time.

Green dashed line represents the experimental median,black pluses are real observa-

tions;blue dotted lines are the 95% condence interval and red line is the median for the

GA FORT+C.Left column displays the negative feedback,right the positive feedback.

1

0

000 simulated markets for each feedback.

We report the results on Figure 2.As for the prices,Figure 2a shows the median and

the 95% Condence Intervals (CI) model prices across time for the negative feedback.For

95% of the simulations,the price is within [50;70] interval after roughly 5 periods,while after

10th period clearly converges to the fundamental,as did happen in the experiment.Dierent

pattern emerges for the positive feedback treatment (Figure 2b).Here,the distribution of

the prices does not collapse into a small region even after 50 periods,when 95% of the prices

stay in a wide [55;75] interval.Oscillations are a clear pattern,with the MC median (and

the 95% CI) price going up until around 20th period,then down for the next twenty periods

and up again.The fundamental price p

f

= 60 is not an unlikely outcome throughout the

last 35 periods,but the price can easily reach 80 from below and 45 from above.This is

a similar pattern to the experimental prices:the 95% CI of our model contain the bulk of

the experimental prices,almost all the groups until period 40 and still roughly half of the

observations afterward.

The GA FORT+C replicates subjects'coordination (standard deviation of six individual

predictions within each period) as well.For the positive feedback (Figure 2d),the six exper-

imental groups coordinated after at most 5 periods.The same holds for the model in which

the 95% CI are narrow and in the same 5 periods fall close to zero and remain there so.For

the negative feedback (Figure 2c),the experimental subjects can have varied predictions even

until the 15th period,the time required for the model 95% CI to converge as well.

13

Negative feedback

Positive feedback

(a) price weight

(b) price weight

(c) trend extrapolation

(d) trend extrapolation

Figure 3:HHST09:Monte Carlo for the GA FORT+C 50-period ahead simulations.Chosen co-

ecients of the price weight and the trend extrapolation.Blue dotted lines are 95%

condence interval,purple dashed are 90% CI and red line is the median for the GA

model.Left column displays the negative feedback,right the positive feedback.1

0

000

simulated markets for each feedback.

To sum up,50-period ahead simulations of our model GA FOTR+C replicate the stylized

behavior of prices and individual predictions.To be specic,we are able to capture with

our 95% CI 65% of the real prices and 81:(3)% of the predictions standard deviations for

the negative feedback treatment and 81% of the real prices and 71:(6)% of the predictions

standard deviations for the positive feedback.This means that we are able to explain roughly

75% of the data with the 50-period ahead predictions,a real achievement for any model.

In addition,we look at the heuristics which were chosen by our GA agents within the

50-periods.Figure 3 reports MC results for the chosen price weight and trend extrapolation

,with median and 95% and 90% CI,for the negative and positive feedback.The dynamics

of heuristics conrm results by HHST09 and Anufriev et al.(2012).In our simulations,under

the positive feedback both the price and the trend become much more important than under

the negative feedback,where the agents prefer adaptive expectations.

3.4 One-period ahead predictions for HHST09

A good measure of the model's tness is the precision of its one-period ahead predictions

(Anufriev et al.,2012).The question is what the model predicts (and how wrong it is) will

happen in period t + 1,conditional on the experimental data until period t.In the context

of the 2009 experiment,Anufriev et al.(2012) showed that HSM is similar to the Rational

Expectations (RE) model for the case of the negative feedback,but outperforms it for the

14

positive feedback.The specic performance measure was the mean squared error of the one-

period ahead price predictions for the last 47 periods,dened for each group as

(9) MSE

M

X

=

1

47

50

X

t=4

p

Gr X

t

p

M

t

2

;

where p

Gr X

t

denotes realized price at the period t in an experimental group X and p

M

t

is the

price p

t

predicted by the model M conditional on the behavior of the group X until period

t 1.For the Rational Expectations model,p

e;RE

i;t

= 60 and p

RE

t

= 60 +"

t

regardless of the

realized prices or predictions until t and the type of the feedback.

Similar exercise can be done for our GA model,but it requires more involved statistical

measures in comparison with the HSM.The only stochastic element in the HSM are the

shocks to the feedback equation (1).This makes HSM a deterministic model conditional

on the price-predictions feedback.The same holds for RE and many other homogeneous

expectations models,including simple naive expectations,trend extrapolation and adaptive

expectations.Therefore,the MSE (9) measure can be computed directly for these models.

This is not the case for our GA model.GA agents will update their heuristic sets con-

ditional on the past prices { and this learning is stochastic and highly nonlinear in nature,

based on a non-smooth period-to-period transition distribution.To address this issue,we will

use Sequential Importance Sampling (with Resampling) technique (SIRS).

16

We emphasize that our model is agent-based by design,which means that we can attempt

to trace the individual behavior in the experiment.This is in contrast with RE or HSM,

which serve as a stylized tool of aggregate market description.Therefore,unlike Anufriev

et al.(2012),we will focus on an alternative performance measure which is one period ahead

forecast of individual decisions.As a result,we obtain a statistical measure for how well our

agent-based model explains the data on the agent (or individual) level.We will also comment

on how to make a crude test of dierent models with our approach.

We introduce the following notation.Let a

t

denote the state of the model at time t,by

which we mean the H

t

set of the six sets of chromosomes that correspond to the six sets of

heuristics H

i;t

.Notice that in our model for the HHST09 experiment,a

t

changes 49 times from

period t = 2,when the chromosomes are randomly initialized,until the period t = 50,when the

chromosomes are updated for the last time conditional on the realized prices and predictions in

the period t = 49.We can observe the chromosomes only indirectly,through the realized prices

and predictions picked by the agents (observational variables).Both the state and observed

variables are evolving according to a distribution q().Denote also p

e

t

= fp

e

1;t

;:::;p

e

6;t

g as the

set of six individual predictions fromperiod t in an experimental group.Here and later t in the

superscript denotes history of the variable,so p

t1

= fp

1

;:::;p

t1

g and p

e;t1

= fp

e

1

;:::;p

e

t1

g.

Our problem is to dene the baseline distribution q(p

e

t

ja

t1

),that is,to evaluate the dis-

16

In order to avoid a technical discussion which is not important for our paper,in the following we as-

sume that the reader is familiar with SISR and sequential MC techniques.If that is not the case,a general

introduction can be found in Doucet et al.(2000).

15

tribution of the real predictions p

e

i;t

given the predictions from the period t 1 and what

they signal could have been the chromosomes H

t1

from the period t 1.This is a typical

state-space model problem.Essentially,q(p

e

t

ja

t1

) can be decomposed as

(10) q(p

e

t

ja

t1

) = q(p

e

t

ja

t

) q(a

t

ja

t1

):

Following Anufriev et al.(2012),we assume that conditional on the history until t 1 and

the chromosome set H

t

,the distribution of the six realized predictions is given by standard

normal distribution.

17

Given the information structure of the experiment we can assume that

in each period,the individual predictions are independent between the agents,and their joint

density is a simple product of the marginal densities of individual forecasts.To be specic,

we represent it as

(11) p

e

t

q(p

e

t

ja

t

;p

t1

;p

e;t1

) =

6

i=1

N(^p

e;GA

i;t

^p

e;GA

i;t

;1):

We simplify the notation by suppressing p

e

t

q(p

e

t

ja

t

).By ^p

e;GA

t

we mean the individual price

forecasts predicted by the GA model.This is based not on the price actually picked by each

agent,but rather on their expected pick (Anufriev et al.,2012).Dene

(12) ^p

e;GA

i;t

=

H

X

h=1

(

i;h;t

p

e;GA

i;h;t

)

(see formula (8)) and hence dene

(13) p

GA

t

= F(^p

e;GA

1;t

;:::;^p

e;GA

6;t

)

to be the price predicted by the GA model for the feedback structure (1).For a general case,

(11) density of the experimental predictions p

e

t

is just a product of normal standard densities

centered around the forecasts predicted by a model.

Unfortunately,q(a

t

ja

t1

) is not that simple to work with.As explained earlier,it is not

feasible to represent this problemanalytically or to linearize it.Nevertheless,it is fairly simple

to simulate a

t

conditional on a

t1

.Therefore,we focus on SISR technique known as Auxiliary

Particle Filter (APF) (Johansen and Doucet,2008).

18

Denote the importance distribution as g() and assume that g(a

t

ja

t1

) = q(a

t

ja

t1

) (which

is a standard assumption for APF).It follows that g(p

e

t

ja

t1

) can be decomposed in the same

manner as the baseline distribution in equation (10).For g(p

e

t

ja

t

) we use (a product of six)

Student-t with one degree of freedom.This density is again analytically straightforward and

compares p

e

t

with ^p

e;GA

t

price forecasts predicted by the GA model as in equation (12).

17

For APF,the choice of variance of the distributions is not important.We use standard normal for the

sake of computational eciency.

18

It is extremely dicult,also in conceptual terms,to dene a reverse distribution of the model at period

t conditional on period t +1,given the complexity of GA operators.As a result,we leave open the question

whether econometrically more ecient ltering-smoothing techniques can be used for the case of our model.

16

Negative feedback Positve feedback

MSE Prices Predictions Prices Predictions

Trend extr.21:101 35:648 0:926 4:196

Adaptive 2:3 14:912 2:999 6:482

Contrarian 2:249 14:856 3:864 7:436

Naive 3:09 15:782 1:822 5:184

RE 2:571 15:21 46:835 54:811

HSM 2:999 17:106 0:889 4:156

GA:FORT+C 3:623 16:913 1:496 6:943

(0:395) (1:496) (0:01) (0:219)

GA:FORT 3:134 17:951 0:881 6:252

(2:077) (8:055) (0:08) (0:183)

Table 2:Mean squared error (MSE) of the Trend Extrapolation,Adaptive,Contrarian,Naive

and Rational Expectations,Heuristic Switching Model and Genetic Algorithms models

FORT+C (with contrarian rules) and FORT (without contrarian rules) one period ahead

predictions of the experimental prices and predictions,averaged over six negative feedback

and six positive feedback groups.In parenthesis the standard deviation of the respective

measures of the GA models are provided.

The specic APF algorithm is discussed by Johansen and Doucet (2008).We use 1024

particles a

<b>

t

for b 2 f1;:::;1024g (1024 sets of six heuristic sets for six agents),with full

resampling.The core problem of our investigation lies with the g(p

e

t

ja

t1

) distribution,which

cannot be tracked analytically.We approximate it with a MC integral in the following fashion.

At the beginning of each iteration t > 1 of AFP,for each particle a

<b>

t1

(that is,for each set of

six agents and their chromosomes),we simulate 256 counter-factual prediction realizations of

the next period predictions p

e;<b;s>

t

for each particle b,where s 2 f1;:::;256g.To be specic,

for the particle b,for each simulation s given H

t1

we draw one prediction p

e;<b;s>

i;t1

for each

agent.This also generates the counter-factual price p

<b;s>

t1

,conditional on which the agents

use GA evolutionary operators to update H

t1

into the counter-factual H

<b;s>

t

.We use these

to compute price ^p

<b;s>

t

and ^p

e;<b;s>

t

individual price forecasts predicted by the model as in

equation (13).We therefore dene

(14) g(p

e

t

ja

<b>

t1

) =

1

256

256

X

s=1

6

i=1

T

1

^p

e;<s;k>

i;t

p

e

i;t

;

where T

1

denotes Student-t distribution with one degree of freedom.

We use the baseline (11) and the importance (14) distributions for the standard APF par-

ticle weighting and updating.

19

We run a separate APF for each of the twelve investigated

experimental groups.Like in the 50-period ahead simulations,the chromosomes (or the parti-

cles) are initialized at random from the`uniform'distribution dened above.Notice,however,

19

Both the baseline and importance densities are a product of six independent densities,which can take

very low values in the rst periods for some experimental groups.To ensure numerical stability,we multiply

both joint densities by 10

60

(or each of the six marginal distributions by 10

10

) and truncate them at 10

100

.

17

that in this case we do not have the problem of the initial predictions or prices,since the

APF works on the period-to-period basis,independently for each experimental group.Inter-

estingly,for deterministic models like RE or HSM or homogeneous heuristic models,the APF

eectively reduces to the procedure reported by Anufriev et al.(2012),since all the particles

would be the same and had the same weights.

Negative feedback { group 1

(a) Price

(b) Price forecasts of subject 1

Positive feedback { group 1

(c) Price

(d) Price forecasts of subject 1

Figure 4:Sample results for Auxiliary Particle Filter for HHST09 experiment:one-period ahead

predictions of the GA FORT+C model for prices and price forecasts of subject 1 from

sample groups from each treatment.Black line denotes the experimental variable and

red boxes display the APF one-period ahead predictions.

For each experimental group,we focus on fourteen variables in total,which we obtain

by using the APF weighting of the particles.For each of the last 47 periods in each group,

we look at the (mean) one-period ahead prediction of the price,as well as at the (mean)

one-period ahead predictions of individual price forecasts.Next,for the prices and the six

individual predictions from that period,we compute MSE against the original data.Notice

that we compute the expected APF MSE's,instead of MSE's for the expected prices or

predictions.We compute these variables for each group,and average them separately over

the two treatments to obtain the average MSE of the model's prediction of the prices and

individual forecasts.

In the same manner,we can also compute the mean variance of the MSE of predicted

prices and price forecasts for the two treatments.A crude measure of the 95% CI (99% CI) of

these four MSE's is simply the average MSE plus and minus twice (thrice) the square root of

its respective variance.It can be used as a basis of a simple test:if these CI coincide with a CI

of a dierent stochastic model,or if they include the (deterministic) MSE of a deterministic

18

model,the two models cannot be distinguished;otherwise the one with lower MSE is better.

Sample results of the one-period ahead forecasts for GA FOTR+C specication are pre-

sented on Figure 4.For both treatments,the model clearly follows both the prices and

individual predictions.We report the average MSE of the one-period ahead predictions of

the prices and the individual price forecasts in Table 2.We also compare our model with

benchmark rules:HSM,RE and a small number of homogeneous linear heuristics.

20

Most

models (with the exception of the trend extrapolation heuristic) are indistinguishable for the

case of the negative feedback.This is not surprising,since under this type of feedback,almost

any reasonable learning mechanism will quickly converge to the fundamental price,just like

the subjects in the experiment.

On the other hand,the positive treatment,with its oscillations,oers a real test for the

models.This proves a disaster for RE,but not for the GA model,which outperforms RE

by a factor of 10 in terms of individual forecasts.Interestingly,the specication GA FOTR

without contrarian rules seems to have a slightly better t to both types of feedback,dierence

signicant especially for the positive one.Moreover,both specications (and more so for GA

FOTR) are at least as good as any other model for each treatment.

Together with the 50-period ahead C exercise,this shows that our model captures both the

aggregate and individual behavior in the LtF experiment reported by HHST09,both in terms

of short and long-run dynamics.We emphasize again that the APF measure is an important

evidence,since it evaluates the agent-based structure of our model against the behavior of the

individuals in the experiment.Moreover,our methodology can be easily adapted for other

stochastic models,including GA models with dierent forecasting rules.

4 Evidence from other experiments

Results of our Genetic Algorithms model for the HHST09 are promising.Nevertheless,the

experiment is based on a simple linear feedback.We argue that our model can be used

to explain more complicated experimental settings.To be specic,we look at three other

experiments that oer a hierarchy of challenge for the GA model:

1.BHST12:linear feedback with large and unanticipated shocks to the fundamental price;

2.HSTV07;V01:nonlinear (cobweb) negative feedback economy investigated by Hommes

and Lux (2011);

3.HSTV05:non-linear positive feedback economy,with two-period ahead predictions;

4.1 Shocks to the fundamental price

BHST12 report LtF experiment with almost the same structure as HHST09:positive and neg-

ative feedback of linear structure given by (2).They use the same dumping factor jBj =

20

21

20

For the denition of the benchmark rules,please refer to Table 7,Appendix D.

19

for the two treatments (positive and negative feedback),but there are two large and unantic-

ipated shocks to the fundamental price A.Regardless of the feedback,the fundamental price

changes from p

f

= 56 to p

f

= 41 starting from period t = 21 and then to p

f

= 62 starting

from period t = 44 until the last period t = 65.

Negative feedback

Positive feedback

(a) Experiment group 1

(b) Experiment group 8

(c) Sample GA FORT+C

(d) Sample GA FORT+C

Figure 5:BHST12:typical results for the experimental groups and sample 50-period ahead simu-

lations of the GAFORT+C model.Black line represents the price and green dashed lines

are the individual predictions.

The results of this experiment are similar to the one by HHST09 and sample time paths

are shown on Figure 5.Under the negative feedback (Figure 5a),shock to the fundamental

distracts the subjects coordination and is followed by a small number of volatile forecasts,

which are pushed to the new fundamental only when the price itself has converged.Under the

positive feedback (Figure 5b),shocks do not brake the coordination,and the predictions and

prices move smoothly towards the new fundamental,eventually over- or undershooting it.

We run the same 65-period ahead Monte Carlo (MC) study as for the HHST09 experi-

ment.Each simulation is initialized with a`uniform'distribution of heuristics (see the previous

section).As for the initial forecasts,we follow Diks and Makarewicz (2013) by using their pro-

cedure to estimate the distribution of the initial predictions for BHST12 and sample directly

fromit.

21

We again look at the prices and the standard deviation of the individual predictions.

The two model specications GA FORT+C and GA FORT yield comparable results,so we

report only the ones for the unrestricted GA FORT+C.

Lower part of Figure 5 presents sample,representative simulations for GA FORT+C.

Notice that the two experimental groups and the two GA simulations reported on that gure

21

See Appendix B for the estimated distribution.

20

Negative feedback

Positive feedback

(a) Realized price

(b) Realized price

(c) Distance from the fundamental

(d) Distance from the fundamental

(e) Predictions standard deviation

(f) Predictions standard deviation

Figure 6:BHST12:Monte Carlo for the GAFORT+C 65-period ahead simulations.Realized price,

distance of the realized price from the fundamental and coordination (standard deviation

of the individual predictions) over time.Green dashed line is the experimental median,

black pluses are real observations;blue dotted lines are the 95% condence interval and

red line is the median of GA FORT+C.Left column displays the negative feedback,right

the positive feedback.1

0

000 simulated markets for each feedback.

have dierent initial predictions.Figure 5c displays a sample time path for the negative

feedback.As in the case of the real group for this feedback,the two shocks to the fundamental

cause a break in the coordination and volatile behavior.The individual predictions are pushed

to the fundamental price once the price itself has converged.The opposite happens in the

case of the positive feedback (Figure 5d).The agents quickly coordinate,and the breaks

in the fundamental are unable to spoil it.Instead,GA agents move smoothly and over- or

under-shoot the new fundamental.

Figure 6 reports the Monte Carlo (MC) experiment for GA FORT+C with 1

0

000 MC sam-

ple,conducted in the same way as for the HHST09 experiment.Under the negative feedback

(Figure 6a),the median price of the model with its 95% CI requires around 5 periods to con-

verge to the fundamental at the beginning of the experiment and twice after the breaks to the

21

Negative feedback

Positive feedback

(a) price weight

(b) price weight

(c) trend extrapolation

(d) trend extrapolation

Figure 7:BHST12:Monte Carlo for the GA FORT+C 65-period ahead simulations.Chosen co-

ecients of the price weight and the trend extrapolation.Blue dotted lines are 95%

condence interval,purple dashed are 90% CI and red line is the median for the GA

model.Left column displays the negative feedback,right the positive feedback.1

0

000

simulated markets for each feedback.

Negative feedback Positve feedback

MSE Prices Predictions Prices Predictions

Trend extr.114:061 121:329 1:183 2:165

Adaptive 3:689 10:332 3:776 4:618

Contrarian 5:92 12:534 4:737 5:559

Naive 9:979 16:81 2:411 3:286

RE 13:871 20:923 55:133 60:859

HSM 38:309 45:679 0:9996 2:024

GA:FORT+C 85:144 110:28 1:893 4:487

(0:172) (0:44) (0:102) (0:124)

GA:FORT 10:228 26:317 1:177 3:532

(0:071) (0:384) (0:113) (0:147)

Table 3:BHST12 experiment:Mean squared error (MSE) of the Trend Extrapolation,Adaptive,

Contrarian,Naive and Rational Expectations,Heuristic Switching Model and Genetic

Algorithms model one period ahead predictions of the experimental prices and predictions,

averaged over eight negative feedback and eight positive feedback groups.In parenthesis

the standard deviation of the respective measures of the GA model are provided.

fundamental.The shape of this convergence is similar to the one visible in the behavior of the

experimental groups under this treatment.Moreover,if the new fundamental is lower/higher

than the old one,the GA agents will under-/over-shoot it respectively,much like the subjects

22

in the experiment.This is seen on Figure 6c,which gives the distance of the price from the

fundamental,for experimental groups and the GA FORT+C.Under the positive feedback,the

prices in the GA model move smoother and there are no sharp breaks in market predictions

as the fundamental changes (Figure 6b).The realized prices move slowly towards the new

fundamental (Figure 6d),to eventually pass it (under- or over-shoots it).

Negative feedback { group 1

(a) Price

(b) Price forecasts of subject 1

Positive feedback { group 8

(c) Price

(d) Price forecasts of subject 1

Figure 8:Sample results for Auxiliary Particle Filter for BHST12 experiment:one-period ahead

predictions of the GA FORT+C model for prices and price forecasts of subject 1 from

sample groups from each treatment.Black line denotes the experimental variable and

red boxes display the APF one-period ahead predictions.

The patterns of the coordination between the experimental subjects and the GA agents

are seen at Figures 6e and 6f.Under the negative feedback,a break in the fundamental causes

both the model and the experimental groups to have more varied predictions for up to ten

periods.Under the positive feedback treatment,the predictions of both the real subjects and

the GA agents retain small standard deviation within each period,after no more than ve

periods and towards the end of the experiment,despite of the breaks in the fundamental.

We also look on the coecients used by our GA agents,see Figure 7.These are very

similar to those from the 2009 experiment:trend extrapolation heuristics with a high price

weight are much more important for the positive feedback market,where the whole 95% CI

for the trend extrapolation become signicantly positive towards the end of the experiment.

We also computed the MSE of the price and individual forecasts predicted by our model

one-period ahead.For this task we apply APF specied as in the previous section.Sample

time paths are shown for GA FORT+C and FORT on Figures 8 and 9 respectively.Both

23

Negative feedback { group 1

(a) Price

(b) Price forecasts of subject 1

Positive feedback { group 8

(c) Price

(d) Price forecasts of subject 1

Figure 9:Sample results for Auxiliary Particle Filter for BHST12 experiment:one-period ahead

predictions of the GA FORT model for prices and price forecasts of subject 1 fromsample

groups from each treatment.Black line denotes the experimental variable and red boxes

display the APF one-period ahead predictions.

model specications replicate the prices and individual predictions for the positive feedback

groups,including the turning points of the oscillations.Under the negative treatment,the

models one-period ahead price forecasts are in general consistent with the data.However,

GA FORT+C has problems in replicating data in one or two rst periods immediately after

the breaks in the fundamental.65-period ahead simulations show that for this treatment,the

prices and predictions quickly converge to the fundamental and so it does not matter which

heuristics the GA agents choose.This leaves space for contrarian rules,with conservative high

weight on the past predictions.With such rules,once the agents observe a signicant decrease

of the price (as happens after the rst brake of the fundamental),they would forecast the next

price to be close to or higher than the old fundamental,and so the model (with the negative

expectations-price feedback) predicts even further drop of the price (c.f.Figures 8a and 8b).

On the other hand,such an outcome is not possible once the contrarian rules are constrained

out,as for the case of GA FORT+C (Figure 9).

We argue that this explains the results for the average MSE of the predicted prices and

individual forecasts,for the two treatments,which are reported in Table 3.GA FORT outper-

forms GA FORT+C signicantly for the positive and dramatically for the negative feedback,

and beats HSM for the negative feedback.In either case,it does signicantly better than RE

and is the only reported model to have a decent t to both feedback treatments.Together

with the 65-period ahead simulations,this shows that the GA model is a good explanation of

24

Mean(p) Var(p) Mean(p

e

) Var(p

e

)

Stable

Experiments 5:64

y

0:36

y

5:56

y

0:087

GA:AR1 5:565 0:326 5:576 0:1

GA:FORT+C 5:628 0:372 5:571 0:082

95% CI [5:613;5:643] [0:359;0:389] [5:553;5:59] [0:065;0:101]

GA:FORT 5:649 0:353 5:548 0:0565

95% CI [5:631;5:667] [0:341;0:371] [5:527;5:57] [0:043;0:077]

Unstable

Experiments 5:85

y

0:63

y

5:67

y

0:101

y

GA:AR1 5:817 0:647 5:645 0:16

GA:FORT+C 5:792 0:598 5:705 0:103

95% CI [5:744;5:841] [0:525;0:746] [5:667;5:739] [0:067;0:171]

GA:FORT 5:825 0:557 5:694 0:079

95% CI [5:786;5:863] [0:487;0:658] [5:67;5:719] [0:052;0:122]

Strongly unstable

Experiments 5:93

y

2:62

5:73 0:429

GA:AR1 6:2 2:161 5:434 0:769

GA:FORC+T 5:809 2:172 5:832 0:345

95% CI [5:693;5:908] [1:626;2:875] [5:735;5:918] [0:182;0:598]

GA:FORT 5:962 1:487 5:807 0:206

95% CI [5:876;6:045] [1:188;1:834] [5:75;5:858] [0:113;0:347]

Strongly unstable,large group

Experiments 5:937

y

1:783

5:781

y

0:204

y

GA:AR1 6:183 1:571 5:515 0:5

GA:FORT+C 5:812 1:699 5:852 0:194

95% CI [5:731;5:892] [1:368;2:157] [5:779;5:918] [0:122;0:338]

GA:FORT 5:972 1:316 5:804 0:173

95% CI [5:918;6:026] [1:118;1:553] [5:768;5:843] [0:111;0:253]

Table 4:HSTV07 experiment under four treatments,stable,unstable and strongly unstable with

6 or 12 subjects.Average price and prediction,and their variances.Mean experimental

statistics;GA simulations with AR1 prediction rule for mutation rate

m

= 0:01 (mean

statistics);GA simulations with rst order rule with or without contrarian rules (FORT+C

or FORT) (median statistics with 95% condence intervals).Asterisk and dagger denote

experimental statistic which falls into 95% CI of GA FORT+C and FORT respectively.

Source for the experimental and AR1 entries:Hommes and Lux (2011).

the dynamics from BHST12,especially the GA FORT specication.

4.2 Cobweb economy

HSTV07;V01 report an LtF experiment in a Cobweb economy setting.As discussed,Hommes

and Lux (2011) investigate this data sets with a GA model based on AR1 forecasting rule.It

is thus important to check if our model,with the FOR (6) instead,can as well account for the

dicult,non-linear price-expectations feedback of the Cobweb economy.

25

Treatments Stable Unstable Strongly unstable

MSE Prices Predictions Prices Predictions Prices Predictions

Trend extr.1:176 1:997 2:122 3:719 5:856 14:39

Adaptive 0:108 0:328 0:434 0:549 2:784 2:863

Contrarian 0:102 0:318 0:414 0:497 2:929 2:729

Naive 0:196 0:448 0:577 0:788 3:095 3:731

RE 0:048 0:248 0:364 0:385 2:257 1:844

HSM 0:212 0:474 0:52 0:732 3:065 3:691

GA:FORT+C 0:247 0:585 0:828 0:801 4:558 2:909

(0:086) (0:35) (0:238) (0:392) (0:76) (0:53)

GA:FORT 0:109 0:42 0:516 0:659 3:872 2:755

(0:069) (0:303) (0:198) (0:31) (0:607) (0:405)

Table 5:HSTV07 experiment:Mean squared error (MSE) of the Trend Extrapolation,Adaptive,

Contrarian,Naive and Rational Expectations,Heuristic Switching Model and Genetic Al-

gorithms model one period ahead predictions of the experimental prices and predictions,

averaged over six groups for each treatment (stable,unstable,strongly unstable).In paren-

thesis the standard deviation of the respective measures of the GA model are provided.

Following Hommes and Lux (2011),we simulate our model in the setting of the cobweb

experiment (see HSTV07 for formal denition of the feedback structure,for which we also use

the experimental errors),with 6 independent agents and three parameter treatments:stable,

unstable (on the verge of stability) and (strongly) unstable.We also look at the strongly

unstable specication with 12 agents,experiment reported by V01.

As a rst test for our model,we conduct a MC exercise in the vein of Hommes and Lux

(2011).For each treatment,we compute (as was the number of groups in each treatment) six

50-period ahead simulations with dierent randomnumbers.

22

Next we compute the mean and

standard deviation of the realized prices p and the predictions p

e

.The variances correspond

to the volatility of the market,which in the experiment was higher for the unstable cases

(which cannot be explained by RE).To obtain a proper Monte Carlo distribution,we repeat

this procedure 1

0

000 times.This allows us to generate 95% condence intervals.

23

We report

the results in Table 4 for the two GA model specications.

Our 50-period ahead simulations explain well the experimental data,yield similar results

to the ones reported in Hommes and Lux (2011) and perform signicantly better than RE.

The CI of our model replicate 12 and 11 out of 16 experimental statistics for GA FORT+C

and GA FORT respectively.Furthermore,the unexplained statistics are usually missed by a

small error that can be attributed to rounding issues of the simulations and the experiment.

The next exercise is the one-period ahead forecasting of the model.Here we look only

at the 18 groups from HSTV07.We use APF exactly as specied in the previous section

and focus on the same set of variables:predicted prices and individual forecasts,as well as

22

As in the two previous cases,the initial heuristics are`uniform'.Following Diks and Makarewicz (2013),

we have also estimated the distribution of the initial predictions,see Appendix B.

23

Hommes and Lux (2011) look only on the expected outcomes of their simulations.

26

Stable treatment { group 3 prices

(a) GA FORT+C

(b) GA FORT

Unstable treatment { group 3 prices

(c) GA FORT+C

(d) GA FORT

Strongly unstable treatment { group 3 prices

(e) GA FORT+C

(f) GA FORT

Figure 10:Sample results for Auxiliary Particle Filter for HSTV07 experiment:one-period ahead

predictions for prices from the third groups of the treatments for GA FOTR+C and

GA FOTR.Black line denotes the experimental variable and red boxes display the APF

one-period ahead predictions.

MSE of these measures.Sample price time paths for both model specications for each of the

three treatments are shown on Figure 10.For the stable and unstable treatment,GA model

closely follows the experimental prices and replicates reversals of their volatile,short period

oscillations.The model has worse t for the strongly unstable treatment.

We report the average MSE of our model for the three treatments in Table 5.It seems that

GA FORT does better than GA FORT+C,but the dierences are on the edge of signicance.

The less stable the treatment,the worse t has any model.Moreover,all the models (with

the exception of trend extrapolation) have similar performance regardless of the treatment.

24

This is similar to the negative feedback treatments from the two previous experiments,but

24

Notice that the scale of the prices in this experiment is [0;10] in the contrast with the two previous settings,

where the prices belonged to [0;100] intervals.The highest possible MSE in the linear experiments is 100 times

higher than in the cobweb experiment.

27

we speculate that for a dierent the reason.Especially for the non stable treatments,the

subject's behavior can be close to chaotic and so detecting it one period ahead is problematic.

However,50-period ahead simulations show that our model replicates this behavior in terms

of its long run distribution.

4.3 Two-period ahead asset pricing

HSTV05 report an experiment in which the subjects participated in a non-linear positive

feedback economy (asset-pricing model with robotic fundamental traders),in which the current

price depends on the expectations about the price in the next period.This means that the

subjects had to predict prices two periods ahead.

There is no one denite way in which the basic FOR rule (6) together with the GA model

can be translated into the two-period ahead setting.Some experimentation led us to the

following (and indeed the simplest) specication.The agents at time t predict p

t+1

the next

price based on the rst order rule and the last available period.Dene the prediction of price

from period t +1,made at period t by the agent i and her rule h as

(15) p

e

h;i;t

=

h;i;t

p

t1

+(1

h;i;t

)p

e

i;t1

+

h;i;t

(p

t1

p

t2

):

Then,at the beginning of period t,each agent focuses on p

e

h;i;t

(prediction of p

t+1

).On the

other hand,once p

t

is realized,the agents can evaluate their rules and to do so,they look at

their hypothetical performance two periods ago.To be specic,they focus on (p

t

p

e

h;i;t1

)

2

.

(a) Convergence

(b) Unclear

(c) Oscillations

Figure 11:HSTV05:sample 50-period ahead simulations for GA FORT+C with dierent initial

predictions and learning.The green lines are individual predictions,the black line is

the realized price and the purple dashed line is the fundamental price.

Contrary to the HHST09 experiment,HSTV05 obtain results which cannot be easily clas-

sied into a clear-cut stylized facts.In the seven treatment groups with the fundamental price

p

f

= 60 they observe groups which have converged to this fundamental,as well as groups

with oscillations of dierent amplitude and period.For this reason we abstain from a MC

experiment for the 50-period ahead performance of the model.Instead,we report sample

simulations of the experimental economy (with the fundamental price p

f

= 60) with our GA

agents,in which the initial predictions for the two rst periods are draws fromthe distribution

calibrated by Diks and Makarewicz (2013).

28

(a) 500 periods

(b) 2000 periods

Figure 12:HSTV05:sample long run behavior of the GA FORTR+C model with fundamental

price p

f

= 60.2

0

000-period ahead simulation (b) and its rst 500 periods (a).The

green lines are individual predictions,the black line is the realized price and the purple

dashed line is the fundamental price.

MSE Prices Predictions

Trend extr.17:4527 55:0898

Adaptive 44:125 25:3157

Contrarian 59:3905 30:8646

Naive 31:6864 20:8416

RE 96:0328 145:998

GA:FORT+C 48:51 48:56

(0:266) (0:496)

GA:FORT 39:236 43:8

(0:211) (0:529)

Table 6:HSTV05 experiment:Mean squared error (MSE) of the Trend Extrapolation,Adaptive,

Contrarian,Naive and Rational Expectations,and Genetic Algorithms FORT+C and

FORT models one period ahead predictions of the experimental prices and predictions,

averaged over eight negative feedback and eight positive feedback groups.In parenthesis

the standard deviation of the respective measures of the GA models are provided.

Many 50-period ahead sample time paths look much alike the experimental ones.Figure 11

displays three typical time paths of the simulated markets (for dierent realizations of the

random number generator) for GA FOTR+C.We use the experimental errors to the price-

expectations law of motion,hence the dierences are purely due to dierent realizations of

the learning and the initial individual predictions.In terms of the simulated prices,both

convergence to the fundamental price (Figure 11a) and oscillations (Figure 11c) are common.

However,sometimes the agents seem to diverge from the fundamental after being close to it

for a few periods (Figure 11b).Model specication without the contrarian rules FOTR has

similar time paths.

To further stress the volatile behavior of this market structure,we report one long run

simulation for GA FOTR+C.Figure 12 displays its rst 500 (Figure 12a) and all 2

0

000 (Fig-

ure 12b) periods.Volatile,unruly oscillations are persistent and can also reappear even if the

29

Group 8:GA FOTR+C

(a) Price

(b) Price forecasts of subject 1

Group 8:GA FOTR

(c) Price

(d) Price forecasts of subject 1

Figure 13:Sample results for Auxiliary Particle Filter for HSTV05 experiment:one-period ahead

predictions of the GA FORT+C and GA FORT models for prices and price forecasts

of subject 1 from group 8.Black line denotes the experimental variable and red boxes

display the APF one-period ahead predictions.

market settles on the fundamental price for some time,as seen on Figure 12a around period

170.This means that in the system the fundamental price is a stable steady state,but is not

the unique attractor.This result is intuitive:individual agents cannot impose fundamental

price,but will rather try to follow the current trend,and this through the non-linear feedback

easily amplies the oscillations.That is the reason why the experimental groups { under the

same conditions { could converge to the fundamental,diverge from it or oscillate in a varied

fashion.

Table 6 reports the results of one-period ahead forecasting performance of our model.RE

are the worst model,both on aggregate and individual level.Interestingly,the best model

are naive expectations,whereas trend extrapolation has a good t only to the aggregate level.

We think that this result is because the oscillations between the groups were dierent,and

so one particular trend extrapolation specication can explain one or two groups,but not all

of them.Naive expectations seem to work ne mostly because the period to period changes

in predictions are often relatively small.This model would not be able to explain lasting

trend in the data.GA models work moderately well,again with GA FORT restricted model

outperforming GA FORT+C.The constrained specications is either very close to or slightly

better than most of the homogenous models.This,together with the dynamics present in

the long-run simulations show that our model is able to capture a good measure of these

experimental dynamics,although there is probably a space for improvement which we leave

30

for future research.

5 Conclusions

In this paper we discuss a model in which the agents independently use Genetic Algorithms

to optimize a simple prediction heuristic.Our investigation derives from the intuitions of

the Heuristic Switching Model (Anufriev and Hommes,2012;Anufriev et al.,2012) and the

Genetic Algorithmmodel introduced by Hommes and Lux (2011).Following the experimental

results by (Heemeijer et al.,2009),we model our agents as learning how to use a simple rst

order rule (a mixture of adaptive and trend extrapolation expectations) in dierent economic

environments.We argue that our model is able to replicate many experimental ndings

from dierent Learning-to-Forecast experiments.We show this by means of 50-period ahead

simulations.Furthermore we use Auxiliary Particle Filter technique to evaluate the model's

one-period ahead predicting power of the individual behavior,a novelty in the behavioral

literature.

In Learning-to-Forecast experiments,subjects are asked to predict prices,while the realized

price depends on their predictions.This mimics many well studied economic environments,

such as an asset pricing market or a cobweb economy.As a result,Learning-to-Forecast

experiments are a perfect controlled setting to study how the human subjects try to adapt to

the price-predictions feedback.Our model retains the basic intuition of the Heuristic Switching

Model:among dierent prediction heuristics,the agents focus on those that have relatively

good hypothetical past performance.On the other hand,following Hommes and Lux (2011)

we show how the agents' exibility can be enhanced by explicit learning with the individual use

of Genetic Algorithms.Therefore the heterogeneity of heuristics between the agents emerges

endogenously and resembles the one among the experimental subjects.

This contrast the dominating framework of the perfectly rational expectations.Tradition-

ally,the economists assumed that people use sophisticated concepts such as a fundamental

price or a long run equilibrium.They disregarded the fact that the market practice,the

agents face constraints on their rationality and may be forced to use second-best prediction

rules.As a result,rational expectation fail to describe experimental dynamics,unless these

are extremely simple.To counter this approach,we propose on a model where the subjects use

simple forecasting rules,but adapt themto the current environment with a smart optimization

procedure.This allows for a realistic description of the human behavior,which also explains

the experimental data to the degree that was unattainable for the traditional literature.

We use the simple linear setting of the experiment reported by Heemeijer et al.(2009) to

set up our Genetic Algorithms model.In a Monte Carlo experiment based on 50-period ahead

simulations,the model replicates the dynamics of the experiment,both on the aggregate

and individual level and for both treatments.We also replicate the major insights of the

Heuristic Switching Model model { that the trend extrapolation is relatively more important

for the positive feedback,in which it reinforces the oscillating behavior.Our results therefore

31

validate the stylized investigation by Anufriev et al.(2012).We also use Auxiliary Particle

Filter technique to conrm that the one-period ahead predictions of our model follow closely

the experimental individual predictions and prices.This is a novelty in the literature,since

we are able to conduct a direct test on how an agent based model ts to an experimental data

set on the individual level.

We further investigate three more complicated experimental settings with our Genetic Al-

gorithms model.The experiment reported by Bao et al.(2012) adds large and unanticipated

shocks to the basic linear structure of the original Heemeijer et al.(2009) experiment.Second

experiment,reported by van de Velden (2001) and Hommes et al.(2007) and investigated by

Hommes and Lux (2011),focuses on a cobweb economy.Finally,the asset pricing experi-

ment reported by Hommes et al.(2005) introduces two-periods ahead feedback between the

predictions and the realized prices.For all the three experiments,we use the Auxiliary Par-

ticle technique to demonstrate that our model can successfully predict the subjects behavior

one period ahead.Moreover,50-period ahead simulations of the model show that it is able

to replicate the long-run distribution of the individual predictions and prices for the three

experiments.

The strength of our model is its generality and agent-based structure.We emphasize

that it replicates the individual behavior from Learning-to-Forecast experiments,which were

based on very dierent experimental economies.Moreover,the model allows for realistic

heterogeneity and learning.We therefore argue that it can be used to investigate settings

with a more complicated interactions between individual agents.This can include economies

with heterogeneous preferences,unequal market power,information networks or decentralized

price setting.In any of these cases,heterogeneous price expectations may have important

consequences for market eciency or dynamics.Our model can be directly used to explore

these phenomena.

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34

Appendices

A Formal denition of Genetic Algorithms

In this appendix we present a formalized version of the Genetic Algorithms (GA) which we

have used in the simulations.Note that it closely follows the standard specication suggested

by Haupt and Haupt (2004) and used by Hommes and Lux (2011).Furthermore we give some

motivation for this specication.

A.1 Genetic Algorithms as optimization procedure

Consider a maximization problem where the target function F of N arguments

n

is such

that a straightforward analytical solution is unavailable.Instead,we need to use a numerical

optimization procedure.

Traditional maximization algorithms,like the Broyden-Fletcher-Goldfarb-Shanno (BFGS)

algorithm,iterate a candidate argument for the optimum of the target function F by (1)

estimating the curvature around the candidate and (2) using this curvature to nd the optimal

direction and length of the change to the candidate solution.This so called`hill-climbing'

algorithm is very ecient in its use of the shape of the target function.On the other hand,it

will fail if the target function is`ill-behaved':non-continuous or almost at around the optima,

has kinks or breaks.Here the curvature cannot be reliably estimated.Another problem is

that the BFGS may perform poorly for a problem of large dimensionality.

The Genetic Algorithms are based on fundamentally dierent approach and therefore can

be used for a wider class of problems.The basic idea is that we should focus on a population of

arguments which compete only in terms of their respective function value.This competition

is modeled in an evolutionary fashion:mutation operators allow for a blind-search experi-

mentation,but the probability that a particular candidate will survive over time is relative

to its functional value.As a result,the target function may be as general as necessary,while

the arguments can be of any kind,including real numbers,integers,probabilities or binary

variables.The only constraint is that each argument must fall into a predened bounded

interval a

n

;b

n

.

A.2 Binary strings

Genetic algorithm (GA) uses H chromosomes g

h;t

2 H which are binary strings divided into

N genes g

n

h;t

,each encoding one candidate parameter

n

h;t

for the argument

n

.A chromosome

h 2 f1;:::;Hg at time t 2 f1;:::;Tg is specied as

(16) g

h;t

= fg

1

h;t

;:::;g

N

h;t

g;

i

such that each gene n 2 f1;:::;Ng has its length equal to an integer L

n

and is a string of

binary entries (bites)

(17) g

n

h;t

= fg

n;1

h;t

;:::;g

n;L

n

h;t

g;g

n;l

h;t

2 f0;1g for each j 2 f1;:::;L

n

g:

The interpretation of (16) and (17) is straightforward.An integer

n

is simply encoded by

(17) with its binary notation.Consider now an argument

n

which is a probability.Notice

that

P

L

n

1

l=0

2

j

= 2

L

n

1.It follows that a particular gene g

n

h;t

can be decoded as a normalized

sum

(18)

n

h;t

=

L

n

X

l=1

g

n;l

h;t

2

l1

2

L

n

1

:

A gene of zeros only is therefore associated with

n

= 0,a gene of ones only { with

n

= 1,

while other possible binary strings cover the [0;1] interval with an

1

2

L

n

1

increment.Any

desired precision can be achieved with this representation.Since 2

10

10

3

,the precision

close to one over trillion (10

12

) is obtained by a mere of 40 bites.

A real variable

n

from an [a

n

;b

n

] interval can be encoded in a similar fashion,by a linear

transformation of a probability:

(19)

n

h;t

= a

n

+(b

n

a

n

)

L

n

X

l=1

g

n;l

h;t

2

l1

2

L

n

1

where the precision of this representation is given by

b

n

a

n

2

L

n

1

.Notice that one can approximate

an unbounded real number by reasonably large (in absolute value) a

n

or b

n

,since the loss of

precision is easily undone by a longer string.

Binary representation has two advantages.First,as will be discussed later,it allows for

an ecient search through the parameter space.Second,any type of a well-dened argument

can be translated into a string of logical values.

25

A.3 Evolutionary operators

The core of GA are evolutionary operators.GA iterates the population of chromosomes

for T periods,where T is either large and predened,or depends on some convergence cri-

terion.First,at each period t 2 f1;:::;Tg each chromosome has its tness equal to a

nondecreasing transformation of the function value F.This transformation is dened as

V (F(

h;t

)) V (h

k;t

)!R

+

\f0g.For example,a non-negative function can be used directly

as the tness.If the problem is to minimize a function,a popular choice is the exponential

transformation of the function values,similar to the one used in the logit specication of the

Heuristic Switching Model (Brock and Hommes,1997).

25

Nevertheless there are GA extensions with real-valued genes.See Haupt and Haupt (2004) for an intro-

ductory discussion.

ii

Recall that in the case of our model,the function value V () is dened with the equation

(7).Hence,the tness is the normalized function value given by (8) (see the procreation

operator).

Chromosomes at each period can undergo the following evolutionary operators:procre-

ation,mutation,crossover and election.These operators rst generate an ospring population

of chromosomes from the parent population t and therefore transform both populations into

a new generation of chromosomes t +1 (notice the division of the process).In the following

analysis we will use all four of them.

A.3.1 Procreation

For the population at time t,GA picks H subset of chromosomes and picks < of

them into a set K.The probability that the chromosome h 2 will be picked into the K as

its zth element (where z 2 f1;:::;g) is usually dened by the power function:

(20) Prob(g

z

= g

h;t

) =

V (g

h;t

)

P

j2

V (g

j;t

)

:

This procedure is repeated with dierently chosen 's until the number of chromosomes in all

such sets K's is equal to H.For instance,the roulette is procreation with = H and = 1:

GA picks randomly one chromosome from the whole population,where each chromosome has

probability of being picked equal to its function value relative to the function value of all other

chromosomes.This is repeated exactly H times.

So called tournaments are often used for the sake of computational eciency.Here, <<

H.For instance,GA could divide the chromosomes into pairs and sample two ospring from

each pair.We will use the full roulette operator.

Procreation is modeled on the basic natural selection mechanism.We consider subsets of

the original population (or maybe the whole population at once).Out of each such a subset,

we pick a small number of chromosomes,giving advantage to these which perform better.We

repeat this procedure until the ospring generation is as large as the old one.Thus the new

generation is likely to be`better'than the old one.

A.3.2 Mutation

For each generation t 2 f1;:::;Tg,after the procreation has taken the place,each binary

entry in each new chromosome has a predened

m

probability to be swapped:ones turned

into zeros and vice versa.In this way the chromosomes represent now dierent numbers

and may therefore attain better t.Moreover,the mutation operator is where the binary

representation becomes most useful.

If the bites,which are close to the end of the gene,mutate,the new argument will be

substantially dierent fromthe original one.On the other hand,small changes can be obtained

by mutating bites from the beginning of the gene.Both changes are equally likely!In this

iii

way,GA can easily evaluate arguments which are both far away from and close to what the

chromosomes are currently encoding.As a result,GA eciently converges to the maximum,

but also are likely not to xate on a local maximum.This requires no additional investigation

of the initial conditions,as is the case of BFGS.

A.3.3 Crossover

Let 0 6 C

L

;C

H

6

P

N

n=1

L

n

be two predened integers.The crossover operator divides the

population of chromosomes into pairs.If C

L

< C

H

,it exchanges the rst C

L

and the last

C

H

bites between chromosomes in each pair with a predened probability

c

.Otherwise,the

crossover operator exchanges either the rst C

L

bites (if C

H

is not specied or less than C

L

) or

the last C

H

bites (if C

L

is not specied or higher than C

H

) in each pair of chromosomes with

this predened probability

c

.This operator facilitates experimentation in a dierent way

than the mutation operator:here the chromosomes experiment with dierent compositions of

the individual arguments,which on their own are already successful.

A.3.4 Election

The experimentation done by the mutation and crossover operators does not need to lead to

ecient binary sequences.For instance,a chromosome which actually decodes the optimal

argument should not mutate at all.To counter this eect,it is customary to divide the creation

of a new generation into two stages.First,the chromosomes procreate and undergo mutation

and crossover in some predened order.Next,the resulting set of chromosomes is compared

in terms of tness with the parent population.Thus,ospring will be passed to the new

generation only if it strictly outperforms the parent chromosome.In this way each generation

will be at least as good as the previous one,what in many cases facilitates convergence.

A.4 Parametrization

Recall that we use the full roulette operator as the procreation operator.The full parametriza-

tion of the GA model is presented in Table 1.This parametrization is used for all our simu-

lations,unless stated otherwise.Notice that since we have only two genes,we suppress C

H

in

the crossover operator,which thus exchanges the price weight genes between the chromosomes

(since L

1

= 20 and C

L

= 20 and C

H

= 1).

B Initialization of the model

The model will have a good t to the data only if it is properly initialized.For instance,in the

Heemeijer et al.(2009) experimental environment under the positive feedback,one can imagine

that the price oscillations require the agents to start relatively far from the fundamental price

(c.f.Anufriev et al.(2012)).Diks and Makarewicz (2013) investigate this issue in a systematic

fashion for the case of the Heemeijer et al.(2009) experiment.They argue that,for the sake

iv

of tractability,the initial predictions can be regarded as sampled by each individual from a

common distribution.We use their methodology to calibrate the initial period of our model to

the experiments reported by Bao et al.(2012) and Hommes et al.(2007);van de Velden (2001).

For all the reported Monte Carlo (MC) experiments,we re-sample the initial predictions (from

the distribution calibrated to the respective experimental data) for each run of the market

simulations.

In the case of the asset-pricing environment reported by Hommes et al.(2005),we initialize

our estimations by the distribution from Diks and Makarewicz (2013).In the case of that

experiment,we are not interested in the proper Monte Carlo distribution of the result,hence

we decided to forfeit the problem of initialization of our simulations.

Finally,notice that at the period 2,the rule (6) requires p

1

as the past price,but also p

1

p

0

for the trend extrapolation.It is plausible that after the rst period (with only one realized

price!),the agents do not think that there is any actual trend in the data { yet.Moreover Diks

and Makarewicz (2013) notice that the initial predictions are close neither to the fundamental

price,nor to the focal point.Therefore,it seems that these two points were not used by the

agents as any sort of a natural reference point.Since we do not see any other possible reference

point (a natural estimator of p

0

given the information,which was provided to the subjects),

we argue that the agents disregarded the trend in the second period and set = 0.This is

eectively the same as behaving as if p

0

= p

1

.Therefore,the actual GA learning starts at

the end of the second period.At the beginning of the second period,the agents randomly

generate their sets of heuristics and pick one at random (with equal probabilities).

Heemeijer et al.(2009) experiment

For this experiment we use the estimated Winged Focal Point (WFP) reported by Diks and

Makarewicz (2013),which is given by

(21) p

e

i;1

=

8

>

>

>

<

>

>

>

:

"

1

i

U(9:546;50) with probability 0:45739;

50 with probability 0:30379;

"

2

i

U(50;62:793) with probability 0:23882:

Bao et al.(2012) experiment

We reestimate WFP model for the data reported by Bao et al.(2012) using the same method-

ology as reported by Diks and Makarewicz (2013).This leads to WFP specied as

(22) p

e

i;1

=

8

>

>

>

<

>

>

>

:

"

1

i

U(16:406;50) with probability 0:32296;

50 with probability 0:35159;

"

2

i

U(50;70:312) with probability 0:32296:

Hommes et al.(2007);van de Velden (2001) experiments

In the case of the cobweb economy experiment,the subjects were asked to predict prices in

the [0;10] interval.Interestingly,the initial predictions still have the WFP form,with a large

v

proportion equal to the midpoint 5 and the rest (not necessarily rounded to a full integer)

distributed around this new focal point.To account for that,we reestimate the WFP and

obtain

(23) p

e

i;1

=

8

>

>

>

<

>

>

>

:

"

1

i

U(1:875;5) with probability 0:17983;

5 with probability 0:36344;

"

2

i

U(5;7:5) with probability 0:45673:

Hommes et al.(2005) experiment

In this experiment,the predictions are two-period ahead,hence the second prediction is still

uninformed much like the rst one.Upon inspecting the data,we noticed that a majority

of the test subjects (around three quarters) have the same prediction in the rst period,i.e.

p

e

i;1

= p

e

i;2

.As discussed,we are not interested in the MC distribution of this experiment,

therefore for the sake of simplicity we sample the rst period predictions p

e

i;1

from (21) and

make all of our agents use them again in the second period.In this way,we omit the question

of the joint distribution of the initial predictions from the two rst periods.

C Parametrization of the forecasting heuristic

In Appendix A,we have discussed the parametrization of the Genetic Algorithms (GA) speci-

cation of our model.This specication is thus used by our GA agents to optimize First-Order-

Rule (6),which on its own requires to be specied.Notice that this is an independent issue

from the sole GA coecients:the same GA can be used to optimize a heuristic in which the

trend extrapolation is free and set to some specic [

L

;

H

],or even restricted to zero (which

gives pure adaptive expectations).Moreover,one can be interested how our GA agents behave

of optimizing a dierent heuristic,including AR1 or some generalized version of FOR.

In this appendix,we will address two of these issues.First,following Heemeijer et al.

(2009) we will look on the importance of the anchor.It is natural to think that the trend

extrapolation should be in a unit circle, 2 [1;1] (c.f.Heemeijer et al.(2009)) and so we

will discuss the issue of the anchor with this restriction.In the second part of the appendix,

we will restrict this assumption and discuss the proper degree of trend extrapolation.

C.1 Is anchor important?

Heemeijer et al.(2009) show that people can be described by First-Order prediction rule with

heterogeneous parameter specication:

(24) p

e

i;t

=

1

p

t1

+

2

p

e

i;t1

+

3

Anchor +(p

t1

p

t2

)

where the three

i

span a simplex and is the trend extrapolation coecient.Notice that

(6) is a special case of (24) with the restriction that

3

= 0,which implies that the anchor is

vi

Negative feedback

Positive feedback

(a) Distance from the fundamental

(b) Distance from the fundamental

(c) Predictions standard deviation

(d) Predictions standard deviation

Figure 14:Monte Carlo for the full First-Order-Rule with anchor:realized price and coordination

of the predictions over time.Green line is the experimental median,black pluses are

real observations;blue lines are 95% condence interval and red line is the median for

MC with the GA model.Left column displays the negative feedback,right the positive

feedback.Sample size is 1

0

000 simulated markets for each feedback.

not used by the agents.

Due to econometric issues the authors specify the anchor as Anchor = 60 = p

f

.It is more

realistic to think that Anchor = p

t

P

t

s=1

p

s

,the average price so far.

26

In a similar fashion

to the Monte Carlo (MC) experiment reported in section 3,we consider the MC simulation of

1000 GA experiments in comparison with the real data.We report signed distance from the

fundamental and standard deviation of individual predictions.The results for the GA model

based on (24) and the restricted (6) with 2 [1;1] are presented on Figure 14 and Figure 15

respectively.

The result for the negative feedback are similar to other models.This is not surprising.

In general almost any model seems to explain this feedback well.The anchor-based FOR rule

fails however to explain the positive feedback.Intuition suggests that the anchor here should

not play a signicant role,as the agents will prefer to focus on the trend.Nevertheless,GA

model based on FOR as in (24) does not predict oscillations at all,but rather a cumbersome,

sluggish convergence towards the fundamental.For this reason,we have decided to exclude

the anchor from our model.

26

Another possibility is to have Anchor = p

t

P

t

s=

p

s

,where = maxf0;t Tg and T stands for some

cut-o point or memory span.We also tried GA model,where the anchor is directly learned as one of the GA

variables.The results were not encouraging for either choice.

vii

Negative feedback

Positive feedback

(a) Distance from the fundamental

(b) Distance from the fundamental

(c) Predictions standard deviation

(d) Predictions standard deviation

Figure 15:Monte Carlo for First-Order-Rule with 2 [1;1]:realized price and coordination of

the predictions over time.Green line is the experimental median,black pluses are real

observations;blue lines are 95% condence interval and red line is the median for MC

with the GA model.Left column displays the negative feedback,right the positive

feedback.Sample size is 1

0

000 simulated markets for each feedback.

C.2 Degree of trend extrapolation

As mentioned in Section 3,we argue that the model performs well if we specify the (6) rule

to use trend extrapolation with the coecient 2 [1:1;1:1].We are not able to support

our claim with any proper statistical analysis or optimization procedure and so we have not

identied this interval in a edge-sharp manner.Nevertheless,we did some grid search and

argue that this specication has better properties than many other that seem to be intuitively

plausible.The specic reason is the amplitude of the price oscillations under the positive feed-

back,which is the larger the higher is the allowed maximum trend extrapolation parameter.

To see that,we conduct a series of Monte Carlo experiments similar to the ones reported in

the Section 3 and Section 4.

First,the model for 2 [1;1] seems to behave quite well,still the oscillations from the

second part of the experiment remain unexplained.See Figure 15.Allowing for non-stationary

trend extrapolation 2 [1:5;1:5] results in a model with huge possible oscillations and little

predictive power,see Figure 16.Finally,having 2 [0:5;0:5] results in too small oscillations,

see Figure 17.This shows the importance of [

L

;

H

] specication for explaining the positive

feedback.

27

Moreover this suggests that trend extrapolation should be chosen from an interval such

27

Negative feedback does not seem to be aected greatly by this issue,which is reasonable since in fact the

GA agents on average learn to omit the trend in this environment.

viii

Negative feedback

Positive feedback

(a) Distance from the fundamental

(b) Distance from the fundamental

(c) Predictions standard deviation

(d) Predictions standard deviation

Figure 16:Monte Carlo for First-Order-Rule with 2 [1:5;1:5]:realized price and coordination

of the predictions over time.Green line is the experimental median,black pluses are

real observations;blue lines are 95% condence interval and red line is the median for

MC with the GA model.Left column displays the negative feedback,right the positive

feedback.Sample size is 1

0

000 simulated markets for each feedback.

that

L

2 (1:5;1) and

H

2 [1;1:5).Moreover we restrict

L

=

H

to make this interval

symmetric around zero.

28

In total,some experimentation led us to the reported specication

with 2 [1:1;1:1],seen on Figure 2.It performs better than the specication with 2

[1;1],since it is better at explaining the amplitude of the oscillations of the positive feedback,

while retaining its explanatory power in comparison with specication 2 [1:5;1:5].

28

To our knowledge,there is no theoretical reason for anything else.Also,simulations suggest that this is

not an important restriction,as long as

L

6 0:5.

ix

Negative feedback

Positive feedback

(a) Distance from the fundamental

(b) Distance from the fundamental

(c) Predictions standard deviation

(d) Predictions standard deviation

Figure 17:Monte Carlo for First-Order-Rule with 2 [0:5;0:5]:realized price and coordination

of the predictions over time.Green line is the experimental median,black pluses are

real observations;blue lines are 95% condence interval and red line is the median for

MC with the GA model.Left column displays the negative feedback,right the positive

feedback.Sample size is 1

0

000 simulated markets for each feedback.

x

Rule Prediction

Homogeneous rules

Trend extr.p

e

t

= p

t1

+(p

t1

p

t2

)

Adaptive p

e

t

= 0:75p

t1

+0:25p

e

t1

Contrarian p

e

t

= p

t1

0:5(p

t1

p

t2

)

Naive p

e

t

= p

t1

RE p

e

t

= p

f

Heterogeneous rules

HSM as in Anufriev et al.(2012)

GA model p

e

i;t

=

i;t

p

t1

(1

i;t

)p

e

i;t1

+

i;t

(p

t1

p

t2

)

GA FORT+C

i;t

2 [0;1] and

i;t

2 [1:1;1:1]

GA FORT

i;t

2 [0;1] and

i;t

2 [0;1:1]

Table 7:Specication of the forecasting rules p

e

t

.

D Denition of forecasting rules

Table 7 gives the exact specication for all the prediciton rules used in the one-period ahead

forecasting exercises for the four experiments.Notice that the price predicted by each rule

is F(p

e

t

;:::;p

e

t

),including the random error and that these rules do not have to be model

consistent.

xi

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