The Heterogeneous Expectations Hypothesis:

Some Evidence from the Lab

Cars Hommesa

a CeNDEF, School of Economics, University of Amsterdam,

Roetersstraat 11, NL-1018 WB Amsterdam, Netherlands

January 11, 2010

Abstract

This paper surveys learning to forecast experiments (LtFEs) with human

subjects to test theories of expectations and learning. Subjects must repeatedly forecast a market price, whose realization is an aggregation of individual

forecasts. Emphasis is given to how individual forecasting rules interact at

the micro level and which structure they co-create at the aggregate, macro

level. In particular, we focus on the question wether the evidence from laboratory experiments is consistent with heterogeneous expectations.

JEL codes: E52, D83, D84, C62.

Keywords: Heterogeneous Expectations, bounded rationality, learning, heuristics switching.

Acknowledgment: This work was presented at a plenary lecture at the 15th International

Conference on Computing in Economics and Finance (CEF), Sydney, July 15–17, 2009. I would

like to thank the participants of the CEF-conference for many stimulating discussions. This survey is based on longlasting cooperation and many ideas from my co-authors Joep Sonnemans,

Jan Tuinstra, Henk van de Velden, Peter Heemeijer, Buz Brock and Tiziana Assenza and particularly benefitted from ideas, simulations and figures provided by Mikhail Anufriev, Te Bao,

Domenico Massaro and Thomas Lux. Financial support from the EU 7th framework collaborative project “Monetary, Fiscal and Structural Policies with Heterogeneous Agents (POLHIA)”,

grant no. 225408, is gratefully acknowledged. None of the above are responsible for errors in this

paper.

1
“Recent theoretical work is making it increasingly clear that the multiplicity of equilibria ... can arise in a wide variety of situations involving sequential trading, in

competitive as well as finite-agent games. All but a few of these equilibria are, I

believe, behaviorally uninteresting: They do not describe behavior that collections

of adaptively behaving people would ever hit on. I think an appropriate stability

theory can be useful in weeding out these uninteresting equilibria ... But to be useful, stability theory must be more than simply a fancy way of saying that one does

not want to think about certain equilibria. I prefer to view it as an experimentally

testable hypothesis, as a special instance of the adaptive laws that we believe govern

all human behavior” (Lucas, 1986, pp. S424-425).

1 Introduction

Individual expectations about future aggregate outcomes is the key feature that

distinguishes social sciences and economics from the natural sciences. Daily weather

forecasts, either by the public or by experts, do not affect the probability of rain.

In contrast, overly optimistic expectations about the economic outlook may have

exaggerated the strong rise in world wide financial markets in the late 1990s and,

more recently, the excessive growth in housing prices in 2000-2008, while an overly

pessimistic outlook by the public and by economists may have amplified the recent

financial crisis and deepened the current economic crisis.

Economic decisions today thus depend upon expectations about the future state

of the economy. A theory of individual expectations or market beliefs is therefore a

crucial part of economic theory. The market is an expectations feedback system and

any dynamic economic model depends crucially upon its underlying expectations

hypothesis. But how then should one model individuals who learn from the past

and adapt their behavior as more and more market realizations become available

over time?

Since the seminal works of Muth (1961) and Lucas (1972), the Rational Expectations Hypothesis (REH) has become the leading paradigm on expectation forma2
tion in economics, and rational expectations representative agent models have become the mainstream tool of analysis. In such a framework, all agents are the same

and forecast rationally, using all available information. Rational expectations are

by assumption model consistent and coincide on average with realizations, without

systematic forecasting errors. The rational expectations (RE) approach has important advantages: it is simple, elegant and puts strong discipline on individual

(forecasting) behavior minimizing the number of free parameters. But drawbacks

of the rational agent paradigm are also well known: it is unrealistic in assuming

perfect knowledge about the economy (typically it assumes knowledge of the law of

motion of the economy) and, even if such knowledge were known, RE requires extremely strong computing abilities of the agents to compute the equilibrium. Most

importantly, RE models are at odds with empirical observations and behavior in

laboratory experiments wit human subjects and, for example, the decline of worldwide financial markets by almost 50% between October 2008 and March 2009 is

hard to reconcile with rational behavior.

Economics, or at least a significant part of it, is currently witnessing a paradigm

shift to an alternative, behavioral view, where agents are boundedly rational. This

alternative view dates back to Simon (1957) and contains many elements from psychology, e.g. through the work of Tversky and Kahnemann (1974). The need for a

new paradigm in economics has recently been forcefully advocated by Akerlof and

Shiller (2009), Colander et al. (2009) and DeGrauwe (2009). Concerning expectations of boundedly rational agents, an alternative theory of adaptive learning has

been developed, see e.g. Sargent (1993) for an early and Evans and Honkapohja

(2001) for a more detailed overview. Boundedly rational agents do not know the law

of motion of the economy, but instead use time series observations to form expectations and adapt their behavior trying to learn model parameters of their perceived

law of motion as more observations become available. Adaptive learning sometimes

enforces convergence to REE, but it may also lead to non-RE equilibria, such as

the learning equilibria in Bullard (1994). Adaptive learning models are sometimes

“cautious modifications of rational expectations theories” (Sargent, 2008, p26) and

3
other times large deviations from rationality explaining excess volatility through

expectations driven fluctuations (Grandmont, 1998). Bounded rationality however

also has important drawbacks. In particular, the ‘wilderness” of bounded rationality (Sims, 1980) leads to (too) many degrees of freedom and too many free

parameters. There are simply too many ways of modeling non-rational behavior. This “wilderness” of bounded rationality, seems particularly relevant when

individuals have heterogeneous expectations.

A rough estimate indicates that in the past 20 years more than 1000 papers

on bounded rationality and learning have appeared. Among these, in the last

decade hundreds of (theoretical) papers on agent-based models populated with

boundedly rational agents employing heterogeneous strategies/expectations have

appeared, especially with applications in finance; see the comprehensive surveys

of LeBaron (2006) and Hommes (2006) in the Handbook of Computational Economics Volume 2: Agent-Based Computational Economics (Tesfatsion and Judd,

2006 and very recent surveys by Hommes and Wagener (2009) and Chiarella et al.

(2009) in the Handbook of Financial Markets: Dynamics and Evolution (Hens and

Schenk-Hopp ́e, 2009) as well as papers and references in the Handbook of Economic

Complexity (Rosser, 2009)1. Most of these papers present stylized, theoretical models or larger, agent-based simulation models, with the realistic feature that these

models can mimic many stylized facts in financial time series (Lux, 2009) and in

macro data (Delli-Gatti et al., 2008).

The empirical validation of heterogeneity is an important area of current research. For example, Baak (1999) and Chavas (2000) estimate heterogeneous agent

models (HAMs) on hog and beef market data, and found empirical evidence for

heterogeneity of expectations. For the beef market Chavas (2000) estimates that

about 47% of the beef producers behave naively (using only the last price in their

forecast), 18% of the beef producers behaves rationally, whereas 35% behaves quasirationally (i.e. use a univariate autoregressive time series model to forecast prices).

1There is also an extensive literature on heterogeneous belief models based on heterogeneous

information, see e.g. Williams (1977), Shefrin and Statmann (1994), Zapatero (1998), Basak

(2000) and Anderson et al. (2005).

4
A number of recent contributions have estimated heterogeneous agent models with

fundamentalists and chartist strategies on stock prices (e.g. Boswijk et al. (2007),

de Jong, Verschoor and Zwinkels (2009), exchange rates (e.g. Gilli and Winker

(2003), Westerhoff and Reitz (2003) and several commodities (e.g. gold prices

(Alfarano et al., 2005), and oil prices (ter Ellen and Zwinkels, 2009)). Most of

these studies find significant time-variation in the fractions of agents using a meanreverting fundamental versus a trend-following strategy. Empirical evidence for

heterogeneous trading strategies in the Spanish Stock Market Exchange has been

found in Vaglica et al. (2008); in particular, Lillo et al. (2008) show that the

investors can be classified in different groups, including trend followers and contrarians, whose change of inventory of the stock is positively respectively negatively

correlated with stock return.

A related, complementary branch of empirical literature uses survey data to

measure individual expectations; see Pesaran and Weale (2006) for a stimulating

overview. An advantage of survey data analysis is that it can focus exclusively on

the expectations-generating process, avoiding the dilemma of testing joint hypotheses. There is quite some evidence on forecasting heterogeneity in survey data. For

example, Frankel and Froot (1987, 1990), Allen and Taylor (1990) and Taylor and

Allen (1992) already found that financial experts use different forecasting strategies to predict exchange rates. They tend to use trend extrapolating rules at short

horizons (up to 3 months) and mean-reverting fundamentalists rules at longer horizons (6 months to 1 year) and, moreover, the weight given to different forecasting

techniques changes over time. Vissing-Jorgensen (2003) presents evidence of heterogeneous beliefs of individual investors about the prospect of the stock market,

while Shiller (2000) finds evidence that investor’s sentiment changes over time, with

both institutions and individual investors becoming more optimistic in response to

recent significant increases of the stock market. Evidence concerning heterogeneity

in survey data on exchange rate expectations can also be found in MacDonald and

Marsh (1996), Elliott and Ito (1999), Prat and Uctum (2000) and B ́enassy-Qu ́er ́e

et al. (2003). Dreger and Stadtmann (2008) show that for exchange rate forecasts

5
at a 6 months horizon, different expectations about macroeconomic fundamentals

is what drives heterogeneity. Mankiw et al. (2003) find evidence for heterogeneity

in inflation expectations in the Michigan Survey of Consumers and show that the

data are inconsistent with either rational or adaptive expectations, but may be

consistent with a sticky information model. Capistr ́an and Timmermann (2009)

show that heterogeneity of inflation expectations of professional forecasters varies

systematically over time, and depends on the level and the variance of current inflation. Pfajfar and Santoro (2009) measure the degree of heterogeneity in private

agents’ inflation forecasts by exploring time series of percentiles from the empirical

distribution of survey data. They show that heterogeneity in inflation expectations

is pervasive and identify three regions of the distribution corresponding to different

expectations formation mechanisms: a static or highly autoregressive region on the

left hand side of the median, a nearly rational region around the median and a

fraction of forecasts on the right hand side of the median consistent with adaptive

learning and sticky information. Branch (2004, 2007) estimates a simple switching model with heterogeneous expectations, along the lines of Brock and Hommes

(1997), and provides empirical evidence suggesting that models which allow the

degree of heterogeneity to change over time provide a better fit on exchange rate

survey data.

In this paper, we discuss laboratory experiments with human subjects that can

be used to validate expectations hypotheses and learning models. Lucas (1986)

already stressed the importance of laboratory experiments in studying adaptive

learning and its stability (see the quote at the beginning). In particular, we are

interested in the potential role of heterogeneity in expectations. We quote from

Muth (1961, p.321, emphasis added) on expectations heterogeneity and its aggregate effect: “Allowing for cross-sectional differences in expectations is a simple

matter, because their aggregate affect is negligible as long as the deviation from the

rational forecast for an individual firm is not strongly correlated with those of the

others. Modifications are necessary only if the correlation of the errors is large and

depends systematically on other explanatory variables”

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In this paper we discuss learning to forecast experiments (LtFEs) which provide

a controlled laboratory environment to study individual expectations as well as

aggregate outcomes, and investigate questions such as:

• How do individuals form expectations and how do they

bf learn and adapt their behaviour?

• How do individual forecasting rules interact at the micro level and which

aggregate outcome do they co-create at the macro level ?

• Will coordination occur, even when there is limited information, or will heterogeneity persist?

• When does learning enforce convergence to REE and when do “learning equilibria” arise?

The goals of this paper are twofold. Firstly, we summarize a number of LtFEs

in different market environments. Secondly, we fit a theory of heterogeneous expectations to the experimental data and discuss how well the model explains the

data across different market settings. This poses a particular challenge: is there a

general, perhaps even a universal theory of heterogeneous expectations, that is, can

one come up with one single expectations hypothesis explaining all LtFEs across

different market settings?

The paper is organized as follows. Section 2 summarizes related literature on

experiments on expectations. Section 3 discusses an experiment in a cobweb market

modeling framework, while Section 4 describes an asset pricing experiment. Section

5 presents a simple forecasting heuristics switching model, where agents switch

between different forecasting rules based upon their recent performance, and fits

the model to the asset pricing experiments. In Section 6 the same switching model

is fitted to experimental data in different market settings, where the only difference

comes from the type of expectations feedback, positive versus negative. Section 7

briefly discusses some recent experiments in a New Keynesian macro setting and

Section 8 concludes.

7
2 Learning-to-forecast experiments (LtFEs)

Laboratory experiments with human subjects, with full control over the market environment and economic fundamentals, form an ideal tool to study interactions at

the micro-level and how individual behavior affects aggregate market outcomes.

Duffy (2008ab) provides stimulating and up to date surveys of “experimental

macroeconomics”. Early work in this area focussed on market mechanisms, such as

double auctions, and the availability of information and futures markets, ensuring

that equilibrium will be reached (e.g. Smith (1962), Plott ad Sunder (1982), Sunder

(1995) and Plott and Smith (20xx)). More recently, unstable market environments

where equilibrium may not be reached, but instead bubbles and crashes may arise

have also been designed (e.g. Smith et al. (1988) and Lei et al. (2001).

In experimental work expectations often plays an indirect or implicit role. However, in order to avoid joint hypothesis testing there is an expanding literature on

exclusive experimental testing of the expectations hypothesis. An early example

is Schmalensee (1976), who presents subjects with historical data on wheat prices

and asks them to predict the mean wheat price for the next 5 periods. Williams

(1987) considers expectation formation in an experimental double auction market,

which varies from period to period by small shifts in the market clearing price.

Participants predict the mean contract price for 4 or 5 consecutive periods and

the participant with the lowest forecast error earns $1.00. In Dwyer et al. (1993)

and Hey (1994) subjects have to predict a time series generated by an (exogenous)

stochastic process such as a random walk or a simple linear first order autoregressive process. Kelley and Friedman (2002) consider learning in an Orange Juice

Futures price forecasting experiment, where prices are driven by a linear stochastic process with two exogenous variables (weather and competing supply). But in

these papers there is no expectations feedback, since market realizations are not

affected by individual forecasts.

Here, we focus on so-called learning-to-forecast experiments (LtFEs), where subject’s only task is to forecast the price of some commodity for a number, say 50-60,

8
periods, with the realized market price in each period determined by (average) individual expectations. In LtFEs subjects’ forecasting decisions are thus separated

from market-trading decisions. The subjects in the experiments do not participate themselves directly in other market activities, such as trading or producing,

but are forecasters (e.g. advisors to large producers or financial investors) whose

earnings increase when forecasting errors decrease. At the beginning of each period, individual forecasts are collected, which feed directly into (unknown) demand

and/or supply functions and computerized trading yields a market price, derived

from equilibrium between aggregate demand and supply, that becomes available to

the subjects at the end of the period. Demand and supply curves are derived from

maximization of expected utility, profit or wealth and thus consistent with rational

optimizing behavior.

These LtFEs were motivated by the bounded rationality literature, in order to

distinguish between different theories of expectations and learning. Sargent (1993),

for example, emphasizes two different requirements of rational expectations. The

first requirement imposes that individuals maximize an objective function (utility,

profit, wealth, etc.) subject to perceived constraints, while the second requirement

imposes mutual consistency of these perceptions. Marimon and Sunder (1994)

were the first to set up experiments testing individual rationality and mutual consistency either jointly or separately and used different experimental designs to distinguish between “learning-to-optimize” versus “learning-to-forecast” experiments

(Marimon and Sunder, 1994, p.134). The LtFEs focus exclusively on the role of

expectations, using computerized optimal individual demand and supply schedules

once these individual forecasts have been made.

In LtFEs, subjects typically only have qualitative information about the market.

They know that the price ptis an aggregation of individual forecasts, derived

from equilibrium between demand and supply and are able to infer the type of

expectations feedback, positive or negative. Positive (negative) feedback means

that an increase of (average) individual forecasts leads to a higher (lower) market

equilibrium price. Positive feedback is important in speculative asset markets,

9
Figure 1: Typical computerscreen in Learning to Forecast Experiment

where higher market expectations lead to an increase of speculative demand and

therefore an increase of the realized asset price. Negative feedback may be dominant

in supply driven commodity markets, where an increase in expected prices leads to

higher production and thus to a lower realized market price. Subjects in the LtFEs

know past prices and their own past forecasts and earnings, typically in table as

well as in graphic form, as illustrated by an example in Figure 1. They do however

not know the forecasts of other participants, the exact market equilibrium equation,

the exact demand and supply schedules and the exact number of other demanders

and/or suppliers in the market. The type of information in the experiment is thus

very similar to models of bounded rationality and adaptive learning, were agents

try to learn a perceived law of motion, based upon time series observations without

knowing the underlying actual law of motion of the market.

Quite a number of LtFEs have appeared in the literature. In a series of papers,

Marimon, Spear and Sunder studied expectation formation in inflationary overlapping generations economies. Marimon, Spear and Sunder (1993) find experimental

evidence for expectationally driven cycles and coordination of beliefs on a sunspot

2-cycle equilibrium, but only after agents have been exposed to exogenous shocks of

a similar kind. Marimon and Sunder (1995) present experimental evidence that a

“simple” rule, such as a constant growth of the money supply, can help coordinate

agents’ beliefs and help stabilize the economy. More recently, a number of LtFEs

within other macro economic frameworks have been performed. Adam (2007) uses

10
a simple model of sticky prices and shows that a restricted perception equilibrium

explains the experimental data better than the RE benchmark solution. Pfajfar

and Santoro (2009) and Assenza et al. (2009) run LtFEs in a new Keynesian framework. Pfajfar and Santoro (2009) find evidence for heterogeneity of expectations

in their experimental data and three different types of forecasting rules: simple

heuristics (e.g. trend following rules), adaptive learning and rational expectations.

The LtFEs of Assenza et al. (2009) will be briefly discussed in Section 7.

A learning to forecast experiment may be seen as a test bed for the expectations

hypothesis in a benchmark model, assuming that all other assumptions such as

rational, utility and profit maximizing behavior are satisfied. A learning to forecast

experiment thus provides clean data on individual expectations as well as aggregate

price behavior. Here we will discuss learning to forecast experiments, based on three

benchmark models: (1) the cobweb model, (2) a standard asset pricing model and

(3) a New Keynesian macro model. The underlying laws of motion are of the form

pt = F(p

e

1,t, · · · , pe

H,t) cobweb (2.1)

pt = F(p

e

1,t+1, · · · , peH,t+1) asset pricing (2.2)

(πt, yt) = F(π

e

1,t+1, · · · , πe

H,t+1; y

e

1,t+1, · · · , ye

H,t+1). New Keynesian (2.3)

In the cobweb LtFE experiments in Hommes al. (2007), the realized market price

ptin (2.1) is a (nonlinear) function of all individual one-period ahead forecasts p

e

h,t.

In the asset pricing LtFE in Hommes et al. (2005a, 2008) the realized market price

ptin (2.2) is a (nonlinear) function of all two-period ahead individual forecasts

p

e

h,t+1 of next periods price pt+1. There is another important difference between

the cobweb and the asset pricing LtFEs: negative versus positive expectations

feedback. Positive feedback means that the realized market price increases, when

an individual price forecast increases. This feature is characteristic of speculative

asset markets, where an increase of the price forecast leads to higher demand for

the asset and therefore to higher market prices; mathematically it means that the

map F in (2.2) is an increasing function of individual forecasts. Negative feedback

11
prevails in supply driven markets, where a higher expected price leads to increased

production and thus a lower realized market price; the map F in (2.1) underlying

the cobweb experiments is decreasing in individual forecasts. In Section 6 we

review the LtFE of Heemeijer et al. (2009), comparing positive versus negative

feedback systems. Despite the fact that the only difference is the sign (positive

versus negative) of the coefficient in the linear price-expectations feedback rule,

the aggregate price behaviors and individual expectations turn out to be rather

different.

Finally, in the New Keynesian macro model expectations on two different variables interact and realized inflation πt and realized output gap yt

in (2.3) simultaneously depend (linearly) on all two-period ahead individual forecasts of both

inflation and the output gap. In Section 7 we will briefly discuss the LtFE of Assenza et al. (2009), where the dynamics of inflation and the output gap is driven

simultaneously by individual expectations of both inflation and the output gap.

3 Cobweb Experiments

In this section we discuss LtFE in the classical cobweb framework. This is exactly

the same framework employed in the seminal paper of Muth (1961) introducing

rational expectations. These cobweb LtFEs may thus be seen as a direct test

of the REH in the cobweb model, assuming all other modeling assumptions (e.g.

producers’ profit maximization and consumers utility maximization) are satisfied.

Carlson (1967) already conducted hand-run experiments with subjects as cobweb

suppliers. Holt and Villamil (1986) and Hommes et al. (2000) conducted individual

cobweb experiments, where price fluctuations are induced by decisions of a single

individual. Wellford (1989) conducted several computerized cobweb experiments,

where market prices were determined by subjects quantity decisions.

Here we focus on the LtFEs in Hommes et al. (2007) with K = 6 participants

per session. The participants were asked to predict next period’s price of a commodity under limited information on the structural characteristics of the market.

12
Participants were only informed about the basic principles of the cobweb-type market. They were advisors to producers, whose only job is to accurately forecast the

price of the good for 50 subsequent periods. Pay-offs were defined as a quadratic

function of squared forecasting errors, truncated at 0:2

E = Max{1300 − 260(p

e

i,t − p

∗

t

)

2

, 0}. (3.1)

Participants were informed that the price would be determined by market clearing

and that it would have to be within the range [0, 10]. Furthermore, they knew that

there was (negative) feedback from individual price forecasts to realized market

price in the sense that if their forecast would increase, the supply would increase

and consequently the market clearing price would decrease. Subjects however did

not know how large these feedback effects would be, as they had no knowledge of

underlying market equilibrium equations. Subjects thus had qualitative information

about the market, but no quantitative details.

The realized price ptin the experiments was determined by the (unknown)

market equilibrium between demand and supply:

D(pt) = X

K

i=1

S(p

e

i,t), (3.2)

with p

e

i,t the price forecast of participant i at time t. Demand was exogenously

given by a simple linear schedule:

D(pt) = a − dpt + ηt (3.3)

with ηt a small stochastic shock drawn from a Normal distribution representing

small random demand fluctuations. Supply S(p

e

i,t) was determined by the nonlinear

schedule

S(p

e

i,t) = tanh(λ(p

e

i,t − 6)) + 1. (3.4)

21300 points corresponded to 0.5 Euro, so that maximum earnings were 25 Euro’s. Average

earnings ranged from 11.5 to 21 Euro (in about 1.5 hours), over the different treatments.

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This increasing, nonlinear supply schedule can be derived from producer’s expected

profit maximization with a convex cost function. Subjects in the experiment thus

do not participate themselves in production decisions, but supply is computed as

if each individual producer maximizes expected profit, given his/her individual

price forecast. The parameter λ tunes the nonlinearity of the supply curve and

the stability of the underlying cobweb model. The resulting equilibrium price is

obtained as:

pt = D

−1

(

X

K

i=1

S(p

e

i,t)) =

a −

PK

i=1 S(p

e

i,t)

d

+ 2t, (3.5)

where 2t = ηt/d. Given the parameters a, d and λ the aggregate realized price pt

depends on individual price expectations as well as the realization of the (small)

stochastic shocks. While the parameters of the demand function and the realizations of the noise component remained unchanged across all treatments at a = 13.8,

d = 1.5 and 2t =

ηt

d ∼ N(0, 0.5), the slope parameter of the supply function was

varied. Here we consider two treatments. A stable treatment had λ = 0.22, for

which under naive expectations the price converges quickly to the rational expectations equilibrium. In another strongly unstable treatment, with λ = 2, under

naive expectations the RE price is unstable and prices converge to a 2-cycle, as

illustrated in Fig. 2. Along the 2-cycle producers are “irrational” in the sense

that they make systematic forecasting errors, predicting a high (low) price when

realized market price will be low (high).

Under rational expectations, all individuals would predict the unique price p

∗

,

at which demand and supply intersect. Given that all individuals have rational

expectations, realized prices will be given by

pt = p

∗ + 2t

, (3.6)

that is, small random fluctuations around the RE steady state. Given the limited

market information one can not expect that all individuals have rational expectations at the outset, but one can hope that in such a simple, stationary environment

14
 0

 2

 4

 6

 8

 10

 0 10 20 30 40 50

 0

 2

 4

 6

 8

 10

 0 10 20 30 40 50

Figure 2: Cobweb dynamics in the strongly unstable treatment in two benchmark

simulations. Left Panel: convergence to a (noisy) 2-cycle under naive expectations. Right Panel: convergence to (noisy) RE equilibrium price under learning by

average.

individuals would learn to have rational expectations. For example, if price expectations are given by the sample average of past prices, convergence to the RE-price is

enforced, as illustrated in Fig. 2 (right panel). The LtFE has in fact been designed

to test whether individuals are able to learn from their systematic mistakes under

naive expectations and coordinate on a learning algorithm enforcing convergence

to the RE steady state.

Figure 3 shows time series of realized market prices together with the individual

forecasts (top panels) as well as the average forecast (middle panels) for two typical

experimental groups, one stable treatment (left panels) and one strongly unstable

treatment (right panels). An immediate observation is that in the stable treatment,

after a short learning phase of about 10 periods, the price volatility is low and individual forecasts as well as average forecasts are very close to the RE benchmark,

with price fluctuations entirely driven by the small random shocks in the experiments. Aggregate price behaviour and individual forecasts are very different in

the strongly unstable treatment. Realized prices exhibit large fluctuations, while

individual forecasts are very volatile, even towards the end of the experiment. The

bottom panel of figure 3 shows the degree of heterogeneity, as measured by the standard deviations of individual forecasts (6 individuals) averaged over the six groups,

in the stable respectively the strongly unstable treatments. In the stable treatment

heterogeneity quickly decreases over time, showing that individuals coordinate on

15
0

2

4

6

8

10

5 10 15 20 25 30 35 40 45 50

PRICE_GROUP2

EXP21

EXP22

EXP23

EXP24

EXP25

EXP26

1

2

3

4

5

6

7

8

9

10

5 10 15 20 25 30 35 40 45 50

PRICE_GROUP3

EXP31

EXP32

EXP33

EXP34

EXP35

EXP36

3

4

5

6

7

8

5 10 15 20 25 30 35 40 45 50

PRICE_GROUP2 AVEXP2

1

2

3

4

5

6

7

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5 10 15 20 25 30 35 40 45 50

PRICE_GROUP3 AVEXP3

0.0

0.4

0.8

1.2

1.6

2.0

5 10 15 20 25 30 35 40 45 50

SIG

0.0

0.4

0.8

1.2

1.6

2.0

5 10 15 20 25 30 35 40 45 50

SIG

Figure 3: Cobweb Learning to Forecast Experiments Top Panels: stable: coordination on RE unstable: persistent heterogeneity Middle Panels: Bottom Panels:

degree of heterogeneity measured by standard deviations of individual forecasts (6

individuals) averaged over the six groups in each treatment.

a forecast close to the RE-benchmark steady state. In the strongly unstable treatment heterogeneity decreases somewhat over time, but only slowly, and remains

at least 3 times as high as in the stable treatment. Hence, in the classical cobweb framework used in Muth (1961) to introduce rational expectations, our LtFEs

show that only in the stable cobweb case, the interaction of individual forecasting

rules enforces convergence to the RE-benchmark. In the unstable treatment, heterogeneity in forecasting is persistent and leads to an aggregate effect upon prices

characterized by excess volatility.

The behaviour in Figure 3 is typical for all cobweb experiments. Hommes et

16
al. (2007) summarize the stylized facts of the cobweb LtFE experiments as follows:

(1) the sample mean of realized prices is close to the RE benchmark p

∗

in all

treatments; (2) the sample variance of realized prices depends on the treatment: it

is close to the RE benchmark in the stable treatment, but significantly higher in

the unstable treatment; (3) realized market prices are irregular and do not exhibit

significant linear autocorrelations.

These stylized facts across different treatments appear hard to explain by standard learning mechanisms offered by the theoretical literature. For example, naive

expectations are inconsistent with the experiments, because in the unstable treatment it predicts too much regularity (convergence to a 2-cycle) in aggregate price

behavior. Average price expectations, which is just the simplest form of adaptive

learning obtained when regressing prices on a constant, also are inconsistent with

the experiments, because for both treatment it predicts convergence to the REbenchmark (see Fig. 2, right panel). Hommes (2009) discusses a number of other

homogeneous learning algorithms and concludes that heterogeneity in forecasting

rules is needed to explain the stylized facts of the cobweb experiments across different treatments. Apparently, the interaction of agents’ individual forecasting

rules washes out linear predictability in aggregate price behavior. In the stable

treatment, this interaction leads to coordination on the “correct” RE benchmark

steady state, but in the unstable treatment heterogeneity persists and prices are

excessively volatile.

Hommes and Lux (2009) present a model of heterogenous individual learning

via genetic algorithms (GAs) to explain the cobweb LtFEs3. Genetic algorithms

require a functional specification of the forecasting rule, whose fitness-maximizing

parameter values are searched for via the evolutionary algorithm. Hommes and

Lux (2009) use a simple first order autoregressive rule:

p

e

i,t+1 = αi + βi(pt − αi). (3.7)

3Colucci and Valori (2006) fit simple adaptive learning rules to the individual cobweb LtFE of

Hommes et al. (2000). Arifovic (1994) calibrates a GA-learning model to the cobweb experiments

of Wellford (1989).

17
Figure 4: Simulated prices and learning parameters.

Such a first order autoregressive (AR1) rule seems a natural forecasting scheme

as agents could implement it using the sample average as their estimate of αi and

the first order sample autocorrelation as the estimate of βi

4

. Moreover, the AR1

forecasting rule (3.7) has a simple behavioral interpretation, with αi representing an

anchor or observed average price level around which the market price fluctuates, and

βi representing the observed persistence or anti-persistence of price fluctuations5.

Hommes and Lux (2009) show that the interaction of individual GA-learning

rules simultaneously reproduces all stylized facts in aggregate price behaviour observed in the experiments across the different treatments. Figure ?? shows typical

price time series under GA-learning as well as time series of the two parameters

in the AR1 forecasting rule for the stable treatment (left panel) and the strongly

unstable treatment (right panel). In the stable treatment the parameters converge

to a neighborhood of the RE benchmark, consistent with the observed coordi4See Hommes and Sorger (1998), where the parameters of an AR1 rule are updated according

to sample autocorrelation learning.

5

In similar cobweb LtFE experiments Heemeijer et al. (2009) recently estimated individual

forecasting rules, and many individuals actually used forecasting rules of the simple AR1-form

(3.7).

18
nation of individual forecasts in the experiments, while in the strongly unstable

treatment parameters continue to fluctuate and prices keep moving away from the

RE-benchmark, consistent with the persistent heterogeneity in the strongly unstable treatment of the experiments (cf. Fig. 3).

4 Asset Pricing Experiment

Before discussing the asset pricing Learning-to-forecast experiments (LtFEs) in

Hommes et al. (2005), we briefly discuss the underlying benchmark model.

4.1 An asset pricing model with heterogeneous beliefs

This section discusses a standard one-period asset pricing model, extended to the

case with heterogeneous beliefs, as in Campbell et al. (1997) and Brock and

Hommes (1998). Agents can either invest in a risk free or in a risky asset. The risk

free asset is in perfect elastic supply and pays a fixed rate of return r; the risky

asset pays an uncertain dividend. Let pt be the price per share (ex-dividend) of

the risky asset at time t, and let yt be the stochastic dividend process of the risky

asset. Next period’s wealth is given by

Wt+1 = RWt + (pt+1 + yt+1 − Rpt)zt, (4.1)

where R = 1 + r is the gross rate of risk free return and zt denotes the number of

shares of the risky asset purchased at date t. Let Eht and Vht denote the ‘beliefs’ or

forecasts of trader type h about conditional expectation and conditional variance.

Agents are assumed to be myopic mean-variance maximizers so that the demand

zht of type h for the risky asset solves

Maxzt{Eht[Wt+1] −

a

2

Vht[Wt+1]}, (4.2)

19
where a is the risk aversion parameter. The demand zht for risky assets by trader

type h is then

zht =

Eht[pt+1 + yt+1 − Rpt]

aVht[pt+1 + yt+1 − Rpt]

=

Eht[pt+1 + yt+1 − Rpt]

aσ2

, (4.3)

where the conditional variance Vht = σ

2

is assumed to be equal for all types and

constant.6 Let z

s denote the supply of outside risky shares per investor, assumed

to be constant, and let nht denote the fraction of type h at date t. Equilibrium of

demand and supply yields

X

H

h=1

nht

Eht[pt+1 + yt+1 − Rpt]

aσ2

= z

s

, (4.4)

where H is the number of different trader types. The forecasts Eht[pt+1 + yt+1]

of tomorrows prices and dividends are made before the equilibrium price pt has

been revealed by the market and therefore will depend upon a publically available

information set It−1 = {pt−1, pt−2, . . . ; yt−1, yt−2, . . .} of past prices and dividends.

Solving the heterogeneous market clearing equation for the equilibrium price gives

pt =

1

1 + r

X

H

h=1

nhtEht[pt+1 + yt+1] − aσ2z

s

. (4.5)

The quantity aσ2z

s may be interpreted as a risk premium for traders to hold all

risky assets. I te experimets discussed below, z

s = 0 so tat (4.5) correspods to te

case of risk-eutral ivestors. Moreover, it will be assumed divideds are IID, wit mea

y ̄, ad all traders ave correct expectatios about divideds, Eht[yt+1] = Et[yt+1] =  ̄y,

so tat (4.5) simplifies to

pt =

1

1 + r

X

H

h=1

nhtEht[pt+1] +  ̄y. (4.6)

6Gaunersdorfer (2000) studies the case with time varying beliefs about variances and Chiarella

and He (2002,2003) investigate the model with heterogeneous risk aversion coefficients.

20
4.2 Experimental Design

In the asset pricing LtFEs six subjects are forecast advisors to large pension funds.

Subjects only task is to forecast the price of a risk asset for 50 periods and, based

on this forecast, the pension fund ten decides how much to invest in the risky asset

according to the mean-variance demand function (4.3). The realized asset price in

the experiment is given as

pt =

1

1 + r

3

(1 − nt)p

e ̄

t+1 + nt p

f +  ̄y + εt

 ́

where p

f =  ̄y/r is the fundamental price, p

e ̄

t+1 is the average two-period ahead

price forecast over six individuals and εt are small shocks, e.g. representing small

random fluctuations in asset demand7. Since the mean-dividend  ̄y and the interest

rate r are known to the subjects, they can in principle use these to compute the

fundamental price and use it in their forecast. The fraction ntin (4.7) is the share

of computerized fundamental robot traders, given by

nt = 1 − exp ¡−

1

200 |pt−1 − p

f

|

¢

. (4.7)

The fraction of robot traders increases as the price moves further away from the

fundamental benchmark. The fundamental trader thus acts as a “far from equilibrium” stabilizing force in the market, mimicking the feature that more traders in

the market expect the price to return in the direction of the fundamental when the

deviations becomes large8. Subjects’ earnings depend on forecasting performance

and are given by a quadratic scoring rule

ei,t =







1 −

3

pt−p

e

i,t

7

 ́2

if |pt − p

e

i,t| < 7 ,

0 otherwise ,

(4.8)

7Bottazzi et al. (2008) consider asset pricing LtFEs were, in addition to a price forecast

subjects must also forecast the variance of excess returns, which is then used in the mean-variance

demand function (4.3) to compute the market clearing price.

8

In the experiment nt never exceeds 0.25, while the weight of the other traders are equal to

(1 − nt)/6. Hommes et al. (2008) investigate price behavior in asset pricing LtFEs without robot

traders.

21
0 10 20 30 40 50

40

45

50

55

60

65

70

Period

Rational Expectation

simulated price

fundamental

0 10 20 30 40 50

40

45

50

55

60

65

70

Period

Naive Expectation

simulated price

fundamental

0 10 20 30 40 50

40

45

50

55

60

65

70

Period

Sample Average Expectation

simulated price

fundamental

0 10 20 30 40 50

40

45

50

55

60

65

70

Period

AR 2 Expectation

simulated price

fundamental

Figure 5: Simulation benchmarks: Top Panels RE (left) and naive (right); Bottom

Panels: average expectations (left) ad AR2 trend following rule p

e

t+1 = (60 +

pt−1)/2 + (pt−1 − pt−2).

so that forecasting errors exceeding 7 would result in no reward at a given period. At the end of the session the accumulated earnings of every participant were

converted to euros (1 point computed as in (4.8) corresponded to 50 cents).

4.3 Benchmark Simulations

Fig. 5 shows the simulation of realized prices, which would occur for a number of

homogeneous benchmark expectations rules. When all individuals use the rational,

fundamental forecasting rule, p

e

i,t+1 =  ̄y/r = p

f

, for all i and t, the realized price

pt = εt/(1 + r) randomly fluctuates around the fundamental level p

f = 60 with

small amplitude, due to the small socks. In the experiment, one can not expect

rational behavior at the outset, but aggregate prices might converge to their fundamental value through individual learning. Under naive expectations the price

slowly converges towards its fundamental value. The same is true under average

expectations, the simplest form of adaptive learning obtained when regressing the

price on a constant, but the convergence is extremely slow. If all subjects would

22
use the simple AR2 rule

p

e

t+1 =

60 + pt−1

2

+ (pt−1 − pt−2), (4.9)

price oscillations as illustrated in the bottom right panel of Fig. 5 arise. This is

an example of an anchoring and adjustment rule, which plays an important role in

psychology (Tversky ad Kahneman, 1974), because it extrapolates the last price

change from a reference point or anchor (p

f + pt−1)/2 describing the “long-run”

price level9.

4.4 Experimental Results

Fig. 6 shows time series of prices, individual predictions and forecasting errors

in six different markets of the experiment. A striking feature of aggregate price

behavior is that three different qualitative patterns emerge. The prices in groups

2 and 5 converge slowly and almost monotonically to the fundamental price level.

In groups 1 and 6 persistent oscillations are observed during the entire experiment.

In groups 4 and 7 prices are also fluctuating but their amplitude is decreasing.10

A second striking result concerns individual predictions. In all groups participants were able to coordinate their forecasting activity. The forecasts, as shown

in the lower parts of the panels in Fig. 7, are dispersed in the first periods but

then become very close to each other in all groups. The coordination of individual

forecasts has been achieved in the absence of any communication between subjects

and knowledge of past and present predictions of other participants.

Figure 7 (right panel) illustrates the degree of heterogeneity, as measured by

the standard deviation of individual forecasts, in three different groups. A first

9At this stage one could argue that the anchor of this rule, defined as the average between the

last observed price and the fundamental price, was unknown in the experiment, since subjects

were not provided explicitly with the fundamental price. It is remarkable however that for a

number of subjects the linear estimated forecasting rule was surprisingly close to the anchor and

adjustment rule (4.9)

10Price dynamics in group 3 (not shown, but see the concluding remarks) is more difficult to

classify. Similar to group 1 it started with moderate oscillations, then stabilized at a level below

the fundamental, suddenly falling in period t = 40, probably due to a typing error of one of the

participants.

23
[!t]

 35

 45

 55

 65

 0 10 20 30 40 50

Predictions

 45

 55

 65

Price

Group 2

-2

 0

 2

 35

 45

 55

 65

 0 10 20 30 40 50

Predictions

 45

 55

 65

Price

Group 5

-2

 0

 2

 35

 45

 55

 65

 0 10 20 30 40 50

Predictions

 45

 55

 65

Price

Group 1

-5

 0

 5

 35

 45

 55

 65

 0 10 20 30 40 50

Predictions

 45

 55

 65

Price

Group 6

-5

 0

 5

 10

 30

 50

 70

 90

 0 10 20 30 40 50

Predictions

 10

 30

 50

 70

 90

Price

Group 4

-30

 0

 30

 45

 55

 65

 75

 0 10 20 30 40 50

Predictions

 45

 55

 65

 75

Price

Group 7

-10

 0

 10

Figure 6: Asset Pricing Experiments: realized market prices, six individual predictions (middle part of each panel) and individual errors (bottom part of each

panel).

24
 35

 45

 55

 65

 0 10 20 30 40 50

Predictions

 45

 55

 65

Price

Group 2

-2

 0

 2

-0.5

 0

 0.5

 1

 1.5

 2

 2.5

 3

 3.5

 4

 0 10 20 30 40 50

Variance of Predictions for Group 2

experiment

simulation

 35

 45

 55

 65

 0 10 20 30 40 50

Predictions

 45

 55

 65

Price

Group 1

-5

 0

 5

 0

 2

 4

 6

 8

 10

 0 10 20 30 40 50

Variance of Predictions for Group 1

experiment

simulation

 10

 30

 50

 70

 90

 0 10 20 30 40 50

Predictions

 10

 30

 50

 70

 90

Price

Group 4

-30

 0

 30

 0

 50

 100

 150

 200

 250

 300

 350

 400

 0 10 20 30 40 50

Variance of Predictions for Group 4

experiment

simulation Figure 7: Left Panels: prices for laboratory experiments in three different groups.

Lower parts of left panels show individual predictions and forecasting errors (inner frames). Right Panels : Evolution of the degree of heterogeneity as measured

by the standard deviation of individual forecasts in corresponding groups.

25
observation is that for each of the three groups, there is considerable time variation in the degree of heterogeneity. In the converging group 2 heterogeneity quickly

decreases, to a level below 1 after period 15 and close to 0 after period 25. It should

be noted that despite the fact that coordination is quick, prices are not close to the

fundamental value. Hence, coordination on the “wrong”, non-fundamental price

occurs. In the oscillating group 1, heterogeneity fluctuates, with stronger coordination (i.e. a smaller degree of heterogeneity) during trends and weaker coordination

during trend reversals. The same features, but in a more extreme form, arise in the

dampened oscillation group 4. During the strong trend from periods 4 − 13 coordination is very strong, with the degree of heterogeneity falling from a initial level

above 150 to values less than 5. Thereafter, coordination weakens and the degree

of heterogeneity peaks at price trend reversals, becomes very high after period 23

(note the scale on the vertical axis) with an extremely high peak around period

36-37. Note that, as heterogeneity increases, the asset price stabilizes.

To summarize, in the asset pricing LtFEs we observe the following stylized facts:

1 participants were unable to learn the rational, fundamental price; only in

some cases individual predictions moved (slowly) in the direction of the fundamental price towards the end of the experiment;

2 alltough the sessions were designed in exactly the same, three different price

patterns were observed: (i) slow, (almost) monotonic convergence, (ii) persistent price oscillations with almost constant amplitude, and (iii) large initial

oscillations dampening slowly towards the end of the experiment;

3 already after a short transient, participants were able to coordinate their

forecasting activity, submitting similar forecasts in every period.

One would like to have a model explaining all these stylized facts simultaneously.

We have not bee able to come up wit a homogeneous expectations model fitting

all these experiments. The fact that qualitatively different aggregate outcomes can

arise suggests that path dependence and heterogeneous expectations play a key role.

26
5 A Heterogeneous Expectations Model

In the last 15 years a large literature on heterogenous agent models has developed,

as surveyed e.g. by Hommes (2006) ad LeBaron(2006). In particular, Brock ad

Hommes (1997) introduced a heterogeneous expectations model, where agents tend

to switch towards forecasting strategies that have performed better in the recent

past. Here we discuss a modified version, a heuristics switching model, which has

recently been fitted to the asset pricing LtFEs by Anufriev and Hommes (2009)11.

Agents can choose from a number of simple forecasting heuristics. To discipline

the wilderness of bounded rationality, the set of forecasting heuristics needs to be

carefully chosen. We will choose forecasting heuristics that are similar to those

obtained from estimations of linear models on individual forecasting data in the

LtFEs in Hommes et al. (2005) and Heemeijer et al. (2009). To further discipline the wilderness of bounded rationality two forms of individual learning are

introduced. First, for some heuristics adaptive learning takes place, tat is, some

parameters of the heuristics are updated over time. Second, evolutionar selection

or performance based reinforcement learning takes place, that is, agents evaluate

the performances of all heuristics, and tend to switch to more successfully rules.

Hence, the impact of each of the rules is evolving over time.

To keep the model as simple as possible, Anufriev and Hommes (2009) restricted

attention to only four forecasting heuristics:

ADA p

e

1,t+1 = 0.65 pt−1 + 0.35 p

e

1,t (5.1)

WTR p

e

2,t+1 = pt−1 + 0.4 (pt−1 − pt−2) (5.2)

STR p

e

3,t+1 = pt−1 + 1.3 (pt−1 − pt−2) (5.3)

LAA p

e

4,t+1 =

p

av

t−1 + pt−1

2

+ (pt−1 − pt−2), (5.4)

11The heuristics switching model is similar to other models of reinforcement learning, e.g. Erev

and Roth (1998) and Camerer and Ho (1999). An important difference however is that our

model is built in a market environment rather than the strategic environments usually studied

in standard game theory. Schunk (2009) introduces a dynamic model of behavioral heterogeneity

in search behavior and shows that his experimental data is well explained by a model assuming

dynamic updating of utility reference points.

27
were p

av

t−1 =

Pt−1

j=0 pj

is the sample average of past prices. The first adaptive expectations (ADA) rule predicts that the price is a weighted average of the last observed

price pt−1 and the last price forecast p

e

t

. This ADA rule was obtained as the estimated linear rule of a number of subjects in the converging groups 2 and 5. The

second and the third rules are both trend following rules, with a weak trend (WTR)

parameter 0.4 and a strong trend (STR) parameter 1.3 respectively12. These rules

were obtained as the estimated linear rules for quite a number of subjects in the

oscillatory markets 1, 4, 6 and 7, with 0.4 ad 1.3 obtained as the smallest and

largest trend extrapolating coefficients. Finally, the fourth rule is an anchor and

adjustment rule, obtained from the linear AR2 rule (4.9), discussed in subsection

4.3, by replacing the (unknown) fundamental price p

f by a proxy given by the

(observable) sample average of past prices p

av

t−1

. The weight coefficient of the ADA

rule and the trend parameters of trend following rules have been fixed and it appears that the simulations below are robust with respect small changes of these

parameters. The LAA rule exhibits a simple form of adaptive learning, since the

anchor of the rule is the average of the last observed price and the sample average

of all observed prices.

Subjects switch between the different forecasting rules based upon quadratic

forecasting errors, consistent with the earnings incentives in the experiments. The

fitness or performance measure of forecasting heuristic i is given by

Ui,t−1 = −

¡

pt−1 − p

e

i,t−1

¢2

+ η Ui,t−2 (5.5)

where the parameter η ∈ [0, 1] measure the strength of the agents’ memory. Switching is described by a discrete choice model with asynchronous updating

ni,t = δ ni,t−1 + (1 − δ)

exp(β Ui,t−1)

P4

i=1 exp(β Ui,t−1)

(5.6)

12Haruvy et al. (2007) also provide experimental evidence that individual price expectations

are significantly affected by past price trends. Trend following behavior is often associated with

technical trading strategies in real financial markets; see Menkhoff and Taylor for a comprehensive

overview of the importance of technical analysis.

28
In the special case δ = 0, (5.6) reduces to the the discrete choice model with

synchronous updating used in Brock and Hommes (1997). The more general case,

0 ≤ δ ≤ 1, gives some persistence or inertia in the impact of rule h, reflecting the

fact (consistent with the experimental data) that not all the participants update

their rule in every period or at the same time (see Hommes et al., 2005b and Diks

and van der Weide, 2005). Hence, δ may be interpreted as the average per period

fraction of individuals who stick to their previous strategy. In the extreme case

δ = 1, the initial impacts of the rules never change; if 0 < δ ≤ 1, in each period

a fraction 1 − δ of participants update their rule according to the discrete choice

model. The parameter β ≥ 0 represents the intensity of choice measuring how

sensitive individuals are to differences in strategy performance. The higher the

intensity of choice β, the faster individuals will switch to more successful rules.

In the extreme case β = 0, the impacts in (5.6) move to an equal distribution

independent of their past performance. At the other extreme β = ∞, all agents

who update their heuristic (i.e., a fraction 1 − δ) switch to the most successful

predictor.

In all simulations below, we fix the parameters at the benchmark values β =

0.4, η = 0.7, δ = 0.9

13 and the initial fractions of the four strategies are equal,

i.e. nht = 0.25. The simulations thus only differ in their initial prices, which

have been chosen exactly the same as in the first two periods in the corresponding

experimental group

Fig. 8 compares the experimental data with the one-step ahead predictions

made by the heuristics switching model, for one converging group (group 5), one

oscillating group (group 6) and one dampened oscillating group (group 7); the

other groups yield very similar results; see Anufriev and Hommes (2009). Fig. 8

suggests that the switching model with four heuristics fits the experimental data

quite nicely. The one-step ahead predictions of the nonlinear switching model in

Fig. 8 use past experimental price data to determine the forecasts and the fractions

13These values have been obtained in Anufriev and Hommes (2009) after some trial and error

simulations. The simulation results however are fairly robust with respect to small changes in

these parameter values.

29
 35

 45

 55

 65

 0 10 20 30 40 50

Predictions

ADA WTR STR LAA

 45

 55

 65

Price

Group 5

simulation experiment

-2

 0

 2

 0

 0.2

 0.4

 0.6

 0.8

 1

 0 10 20 30 40 50

Fractions of 4 rules in the simulation for Group 5

ADA WTR STR LAA

 35

 45

 55

 65

 0 10 20 30 40 50

Predictions

ADA WTR STR LAA

 45

 55

 65

Price

Group 6

simulation experiment

-5

 0

 5

 0

 0.2

 0.4

 0.6

 0.8

 1

 0 10 20 30 40 50

Fractions of 4 rules in the simulation for Group 2

ADA WTR STR LAA

 45

 55

 65

 75

 0 10 20 30 40 50

Predictions

ADA WTR STR LAA

 45

 55

 65

 75

Price

Group 7

simulation experiment

-10

 0

 10

 0

 0.2

 0.4

 0.6

 0.8

 1

 0 10 20 30 40 50

Fractions of 4 rules in the simulation for Group 7

ADA WTR STR LAA

Figure 8: Left Panels: prices for laboratory experiments in different groups (red)

with corresponding one-step ahead predictions of the evolutionary model (blue).

Lower parts of left panels show predictions and forecasting errors (inner frames)

of four heuristics. Right Panels : Evolution of heuristic impacts during the onestep ahead predictions of the model. Fractions of four forecasting heuristics: adaptive

expectations (ADA, purple), weak trend followers (WTR, black), strong trend followers

(STR, blue) and anchoring adjustment heuristic (LAA, red).

30
of the strategies at each period t, i.e., the model simulation uses exactly the same

information that was available to participants in the experiments. An immediate

observations by comparing these simulations is that the one-period ahead forecasts

can easily follow the different patterns in aggregate price behavior, slow monotonic

convergence, sustained oscillations as well as dampened oscillations.

The right panels in Fig. 8 show the corresponding fractions of the four strategies for each group. In different groups different heuristics are taking the lead

after starting from a uniform distribution. In the monotonically converging group,

the impact of the different rules stays more or less equal, although the impact of

adaptive expectations gradually increases and slightly dominates the other rules in

the last 25 periods. The oscillatory group yields similar results as, with the LAA

rule dominating the market early and its impact increasing to about 90% towards

the end of the experiment. Finally, for the group with the dampened oscillations,

one step ahead forecast produces a rich evolutionary selection dynamics (bottom

panel), with three different phases where the STR, the LAA and the ADA heuristics

subsequently dominate. The STR dominates during the initial phase of a strong

trend in prices, but starts declining after it misses the first turning point of the

trend. The LAA does a better job in predicting the trend reversal and its impact

starts increasing. The LAA takes the lead in the second phase of the experiment,

with oscillating prices. But the oscillations slowly dampen and therefore, after period 35, the impact of adaptive expectations, which has been the worst performing

rule until that point, starts increasing and adaptive expectations dominates the

groups in the last 9 periods.

6 Positive versus Negative Feedback Experiments

Aggregate price behavior in the cobweb and the asset pricing LtFEs are quite different. While in the cobweb framework the price fluctuates around its fundamental

value, with a sample average of realized prices very close to the RE price, in the asset pricing experiments persistent deviations from the fundamental price with long

31
lasting under- or over-valuations have been observed. A key difference between

the cobweb and asset pricing experiments is the type of feedback: the asset pricing

(cobweb) framework exhibits positive (negative) feedback, that is the realized price

depends positively (negatively) on the average price forecast. In the case of positive (negative) feedback, when an individual forecast increases, the realized market

price goes up (down). A natural question then is whether the type of feedback,

positive versus negative, explains these differences in aggregate behavior.

In most markets both types of feedback may play a role. Positive feedback

however, seems particularly relevant in speculative asset markets. If many agents

expect the price of an asset to rise they will start buying the asset, aggregate demand will increase and so, by the law of supply and demand, will the asset price.

High price expectations thus become self-confirming and lead to high realized asset

prices. In markets where the role of speculative demand is less important, e.g. in

markets for non-storable commodities, negative feedback may play a more prominent role. Consider e.g. a supply-driven commodity market. If many producers

expect future prices to be high they will increase production which, according to

the law of supply and demand, will lead to a low realized market price.

Heemeijer et al. (2009) investigate how the expectations feedback structure

affects individual forecasting behaviour and aggregate market outcomes by considering market environments that only differ in the sign of the expectations feedback,

but are equivalent along all other dimensions. In this section we discuss these experiments and apply the heterogeneous expectations model of Section 5 to see

whether it can explain the different aggregate outcomes.

The distinction between positive and negative expectation feedback is related

to the concepts of strategic complements versus strategic substitutes. Haltiwanger

and Waldman (1985) argue that when actions are strategic complements, agents

have an incentive to imitate other agents. This is the case in an asset market, where

predicting a price close to the predictions of the other participants turns out to be

most profitable. However, coordination of predictions enhances the impact of the

irrational participants upon realized prices and convergence to the rational equilib32
rium price becomes unlikely. When actions are strategic substitutes, agents have

an incentive to deviate from what other agents are doing. This is the case in negative feedback markets, where agents have an incentive to predict high (low) prices

when the majority predicts prices below (above) the equilibrium price. The impact

of irrational individuals will be limited and convergence to the equilibrium price is

more likely. Coordination of predictions will only take place after convergence.

In recent experiments Fehr and Tyran (2001, 2005, 2007) study the impact of

different strategic environments (strategic complementarity versus strategic substitutability) on individual rationality and aggregate outcomes. Strategic substitutability (complementarity) prevails if an increase in the action of individual i

generates an incentive for j to decrease (increase) his action. Fehr and Tyran

study the adjustment of nominal prices after an anticipated money shock in a price

setting game with positively (complements) or negatively sloped (substitutes) reaction curves, and find much faster convergence in the case of substitutes. Sutan and

Willinger (2009) investigate a new variant of beauty contest games (BCG) in which

players actions are strategic substitutes versus strategic complements and find that

chosen numbers are closer to rational play in the case of strategic substitutes.

In the LtFEs of Heemeijer et al. (2009), the (unknown) price generating rules

in the negative and positive feedback systems were respectively:

pt = 60 −

20

21

[

X

6

h=1

1

6

p

e

ht] − 60] + 2t

, negative feedback (6.1)

pt = 60 +

20

21

[

X

6

h=1

1

6

p

e

ht − 60] + 2t

, positive feedback (6.2)

where 2tis a random shock to the pricing rule. First we will consider positive and

negative feedback systems with small IID shocks 2t, 2t ∼ N(0, 0.25), and later on

with large permanent shocks.

A common feature of the positive and negative feedback systems (6.1) and (6.2)

is that both have the same RE equilibrium steady state p

∗

, with p

∗ = 60 when the

shocks 2t have mean 0. The only difference between (6.1) and (6.2) is therefore the

33
 0

 20

 40

 60

 80

 100

 0 10 20 30 40 50

Predictions

 20

 40

 60

 80

Price

NEGATIVE

-3

 0

 3

 0

 20

 40

 60

 80

 100

 0 10 20 30 40 50

Predictions

 20

 40

 60

 80

Price

POSITIVE

-3

 0

 3

Figure 9: Negative (left panel) vs. positive (right panel) feedback experiments with

small IID shocks; prices (top panels), individual predictions (bottom panels) and

forecast errors (small panels).

sign of the slope of the linear map, 20/21 ≈ +0.95 resp. −20/21 ≈ −0.9514.

Figure 9 shows realized market prices as well as individual predictions in two

typical groups. A striking feature is that aggregate price behavior is very different in the positive versus negative feedback cases. In the negative feedback case,

the price relatively quickly settles down to the RE steady state price 60, while

in the positive feedback case, the market price oscillates slowly around its fundamental value. Individual forecasting behavior is also different for the different

feedback treatments: in the case of positive feedback, coordination of individual

forecasts occurs extremely quickly, within 2-3 periods. The coordination however

is on a “wrong” non-fundamental price. In contrast, in the negative feedback case

coordination of individual forecasts is slower and takes about 10 periods. More

persistence in heterogeneity of individual forecasts however ensures that, after 10

periods, the realized market price is very close to the RE benchmark of 60.

Can the heterogeneous expectations model of Section 5 explain these different

outcomes in individual and aggregate behaviour? Figure 10 shows realized market

prices together with the simulated prices (left panels), and the corresponding evolution of the fractions of the four strategies (right panels) of the heuristics switching

14In both treatments, the absolute value of the slopes is 0.95, implying in both cases that the

feedback system is stable under naive expectations

34
 0

 20

 40

 60

 80

 100

 0 10 20 30 40 50

Price

simulation experiment

 0

 0.2

 0.4

 0.6

 0.8

 1

 0 10 20 30 40 50

Impacts of Heuristics

ADA WTR STR LAA

 0

 20

 40

 60

 80

 100

 0 10 20 30 40 50

Price

simulation experiment

 0

 0.2

 0.4

 0.6

 0.8

 1

 0 10 20 30 40 50

Impacts of Heuristics

ADA WTR STR LAA

Figure 10: Positive feedback (bottom panels) and negative feedback (top panels)

markets with small shocks. Realized and simulated prices (left panels) and corresponding evolution of fractions of 4 strategies in heuristics switching model.

model with the same benchmark parameters as before, i.e. β = 0.4, η = 0.7, δ = 0.9.

The model matches aggregate price behaviour in both the negative and positive

feedback treatment. Furthermore, the time series of the fractions of the different

forecasting heuristics (Figure 10, right panels) provide an intuitive explanation of

why aggregate behavior is different. In the negative feedback treatment, the adaptive expectations strategy performs best and within 20 periods it captures more

than 90% of the market, thus enforcing convergence towards the fundamental equilibrium price. In contrast, in the positive feedback treatment the impact of the

strong trend-following rule (STR) quickly increases and it captures more than 75%

of the market after 15 periods. Thereafter, the impact of the STR rule gradually

declines, while the fraction of weak trend-followers (WTR) gradually increases due

to the fact that the STR-rule makes (somewhat) larger mistakes (especially at the

turning points) than the WTR-rule.

The difference in aggregate behavior is thus explained by the fact that trend

following rules are successful in a positive feedback environment reinforcing price

oscillations and persistent deviations from the fundamental equilibrium benchmark

price, while the trend-following rules are driven out by adaptive expectations in

35
the case of negative feedback.

0 10 20 30 40 50 60 70

0

10

20

30

40

50

60

70

80

90

100

Period

Experiment

Simulation

0 10 20 30 40 50 60 70

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Period

ADA

WTR

STR

A&A

0 10 20 30 40 50 60 70

0

10

20

30

40

50

60

70

80

90

100

Period

Experiment

Simulation

0 10 20 30 40 50 60 70

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Period

ADA

WTR

STR

A&A

Figure 11: Positive feedback (bottom panels) and negative feedback (top panels)

markets with large permanent shocks to the equilibrium steady state level. Realized

and simulated prices (left panels) and corresponding evolution of fractions of the 4

strategies in heuristics switching model.

0 10 20 30 40 50 60

0

5

10

15

20

25

period

distance

The median of the distance between the market price and RE

negative feedback

positive feedback

0 10 20 30 40 50 60

0

5

10

15

period

distance

The median of the standard deviation

negative feedback

positive feedback

Figure 12: Positive/Negative Feedback; Large Shocks Coordination & Price Discovery; median absolute distance to RE fundamental price; median standard deviation

of individual predictions

Bao et al. (2010) recently ran similar LtFEs with large permanent shocks 2t to

the price generating mechanisms (6.1) and (6.2). These shocks have been chosen

such that, both in the negative and positive feedback treatments, the fundamental

36
equilibrium price p

∗

changes over time according to:

p

∗

t = 56 0 ≤ t ≤ 21

p

∗

t = 41 22 ≤ t ≤ 43

p

∗

t = 62 44 ≤ t ≤ 65.

(6.3)

The purpose of these experiments was to investigate how the type of expectations

feedback may affect the speed of learning of a new steady state equilibrium price.

Figure 11 shows realized market prices together with the simulated market prices

(left panels), together with the evolution of the fractions of the four strategies

of the heuristics switching model (right panels) for a typical group of the negative feedback (top panels) and the positive feedback treatment (bottom panels).

The heuristics switching model is exactly the same as in Anufriev and Hommes

(2009), in the case of the asset pricing experiments (see Section 5), with the same

benchmark parameters as before, i.e. β = 0.4, η = 0.7, δ = 0.9. As in the case

of small shocks, there is a striking difference between positive and negative feedback markets. In the negative feedback market, after each large shock the price

quickly (within 5 periods) settles down to the RE benchmark, while in the positive

feedback market the price slowly oscillates with persistent deviations from the RE

benchmark. The heuristics switching model matches both patterns quite nicely and

provides an intuitive, behavioral explanation why these different aggregate patterns

occur. In the negative feedback market, trend following strategies perform poorly

and the adaptive expectations strategy quickly dominates the market (more than

50% within 10 periods) enforcing quick convergence to the RE benchmark after

each large shock. In contrast, in the positive feedback treatment, trend following

strategies perform well, the weak trend following rule dominates in the first 20

periods, while the strong trend following rule starts dominating after the first large

shock in period 22.

Figure 12 reveals some other striking features of aggregate price behavior and

individual forecasts. The left panel shows the time variation of the median distance to the RE benchmark price over all groups in both treatments. For the

37
negative feedback treatment, after each large shock the distance spikes but converges quickly back (within 5-6 periods) to 0, while for the positive feedback treatment after each shock the distance to the RE benchmark shows a similar spike

but does not converge to 0 and only decreases slowly. The right panel shows how

the degree of heterogeneity, that is, the median standard deviation of individual

forecasts, changes over time. For the positive feedback treatment after each large

shock heterogeneity decreases very quickly and converges to (almost) 0 within 3-4

periods, while in the negative treatment heterogeneity is more persistent for about

10 periods after each large shock. One may summarize these results in saying that

in the positive feedback treatment individuals quickly coordinate on a common

prediction, but that coordination on the “wrong” non-fundamental price occurs.

On the other hand, in the negative feedback treatment coordination is slow, heterogeneity is more persistent but price convergence is quick. Stated differently,

positive feedback markets are characterized by quick coordination and slow price

discovery, while negative feedback markets are characterized by slow coordination,

more persistent heterogeneity and quick price discovery. See Bao et al. (2009) for

a more detailed discussion.

7 A New Keynesian Macro Model

As a final example, we briefly discuss the LtFEs experiments in a standard New

Keynesian macro model of Assenza et al. (2009); see also Pfajfar and Santoro

(2009) for closely related LtFEs. The New Keynesian model is described by the

following equations:

yt = y

e

t+1 − φ(it − π

e

t+1) + gt

, (7.1)

πt = λyt + βπe

t+1 + ut

, (7.2)

it = φπ(πt − π) + π , (7.3)

38
where yt and y

e

t+1 are respectively the actual and the expected output gap at time t,

itis the nominal interest rate, πt and π

e

t+1 are respectively the actual and expected

inflation rates, π is the target inflation rate, λ, σ, β and φ are positive coefficients,

and gt and ut are (small) white noise shocks. The coefficient φπ measures the

response of the nominal interest rate it to changes in the inflation rate πt. Equation

(7.1) is the IS curve in which the actual output gap yt depends on the expected

output gap y

e

t+1 and on the real interest rate it − π

e

t+1. Equation (7.2) is the

expectations-augmented New Keynesian Phillips curve according to which actual

inflation depends on the actual output gap and expected inflation. Finally, equation

(7.3) is the monetary policy rule implemented by the Central Bank in order to keep

inflation at its target level π.

In the LtFEs of Assenza et al. (2009), two different groups of 6 subjects have to

provide two-period ahead forecasts of the inflation rate respectively the output gap

for 50 periods. Realized inflation and realized output gap are determined by the

(average) individual expectations of two different groups of 6 individuals. Subjects

only obtain qualitative information about the macro-economy, but they do not

know the underlying law of motion (7.1-7.3).

Figure 13: Learning to forecast experiment in New Keynesian Macro framework.

Left panel: time series of realized inflation, individual inflation forecasts, interest

rate and realized output gap. Right panel: time series of realized inflation, interest

rate, realized output gap and individual forecasts of output gap. Parameter values:

φπ = 1.5, ...

Figure 13 shows time series of realized inflation, output gap, interest rate to39
0 10 20 30 40 50

−1.5

−1

−0.5

0

0.5

1

1.5

output

experiment

simulation

0 10 20 30 40 50

1

1.5

2

2.5

3

3.5

4

inflation

experiment

simulation

0 10 20 30 40 50

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

ADA

WTR

STR

LAA

0 10 20 30 40 50

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

ADA

WTR

STR

LAA

Figure 14: New Keynesian Macro Model: Simulations of heuristics switching model

gether with individual forecasts of inflation and output gap. Both inflation and

output gap exhibit dampened oscillations eventually converging to the RE benchmark steady state. An interesting feature of these experiments is that an aggressive

monetary policy described by a Taylor type interest rate rule that adjust the interest rate more than one point for one in response to inflation (with a coefficient

φπ = 1.5) is able to stabilize heterogeneous expectations.

Figure 14 shows simulated time series of inflation and output gap, together with

the fractions of the forecasting rules, of the benchmark heuristics switching model

of Anufriev and Hommes (2009). The same heuristics switching model is used for

both inflation and output gap forecasts and the model fits the experimental data

quite well. The patterns of the weight of the forecasting heuristics are quite similar

for inflation and output, as one would expect since both time series are qualitatively

similar. The Learning Anchor and Adjustment (LAA) rule dominates most of the

time, with a peak of about 80% after 25 periods and the adaptive expectations rule

dominating in the last 10 periods when the economy stabilizes. There is a slight

difference in the initial phase, with the strong trend rule STR dominating inflation

40
forecasting in periods 5-10, picking up the stronger trend in inflation.

8 Concluding Remarks

LtFEs can be used to test theories of expectations and learning in benchmark

model settings. LtFEs are Taylor made experiments to test exclusively the expectations hypothesis, computerizing all other model assumptions. Different outcomes

in aggregate behavior have been observed in different market settings. To our best

knowledge, no homogeneous expectations model fits all LtFEs across different settings. Quick convergence to the RE-benchmark only occurs in stable (i.e. stable

under naive expectations) cobweb/negative feedback markets.

Lab experiments suggest that heterogeneity is a crucial aspect of a theory of

expectations, because a heterogeneous expectations model can explain observed path

dependence in the same market environment as well as different aggregate outcomes

across different market settings.

A challenge to economic theory is to come up with a universal and plausible

theory of heterogeneous expectations. The fact that a simple heuristics switching

model fits different LtFEs gives some hope that a general heterogeneous expectations hypothesis may explain individual expectations and aggregate outcomes

across different market settings.

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