1 s2.0-s0167268105000983-main

Journal of Economic Behavior & OrganizationVol. 60 (2006) 70–84

Learning with misspecification inan artificial currency market

Christophre Georges∗

Department of Economics, Hamilton College, Clinton, NY 13323, USA

Received 18 March 2002; accepted 23 August 2004Available online 20 June 2005

Abstract

Agents evolve their forecast rules over time via a modified genetic algorithm in a simple artificialcurrency market. These forecast rules can be nonlinearly misspecified. When the misspecification issuppressed, learning tends to be complete. When it is not suppressed, learning can generate persistentexchange rate dynamics.© 2005 Elsevier B.V. All rights reserved.

JEL classification: D83; D84; E44

Keywords: Learning; Exchange rates; Volatility; Genetic algorithms

1. Introduction

This paper considers the behavior of the exchange rate in a very simple artificial currencymarket with two currencies and artificial agents who evolve their forecast rules over time viaa modified genetic algorithm. Persistent exchange rate dynamics arise due to the tendencyof agents to adopt misspecified forecast rules under learning. This provides an illustration ofGrandmont’s (1998) argument that local instability is likely to arise if agents are allowed toextrapolate a variety of nonlinear trends in the data. This behavior would not be warranted

∗ Tel.: +1 315 859 4472; fax: +1 315 859 4477.E-mail address: [email protected].

URL: http://academics.hamilton.edu/economics/cgeorges/.

0167-2681/$ – see front matter © 2005 Elsevier B.V. All rights reserved.doi:10.1016/j.jebo.2004.08.005

C. Georges / J. of Economic Behavior & Org. 60 (2006) 70–84 71

under rational expectations (i.e., under deduction if the agents knew the true model) butappear to the agents to be warranted by the historical path of the economy generated by thecombination of model fundamentals and expectations.

2. Background

The analysis in this paper follows the recent literature on learning in macroeconomics,finance, and game theory that replaces the assumption that agents are rational and know agreat deal about their environments with the assumption that they use rules of thumb andtry to improve upon these rules over time through experimentation and imitation.1

In the model presented below, there is a unique rational expectations equilibrium. How-ever, there is also a continuum of forecast rule parameter values that support this equilibrium(i.e., that yield correct expectations in steady state). Given a nonlinear misspecification ofthe forecast rules, the steady state equilibrium of the exchange rate dynamics is not every-where stable along this continuum. Learning may drive the parameter values into a regionof the parameter space in which the steady state is locally stable, in which case learningtends to be complete, or into a region in which the steady state is locally unstable, in whichcase learning may subsequently fail to converge and instead lead to persistent exchange ratedynamics.

This mechanism is different from that proposed by Arifovic (1996) and Arifovic andGencay (2000), in which irregular exchange rate dynamics arise from the interaction of GAlearning and the nonlinearity of an overlapping generations model,2 as well as the modelsof Brock and Hommes (1997, 1998), Chen et al. (2001) and Westerhoff (2003), in whichagents switch between a small number of belief types. The mechanism is closer in spirit tothose of Youssefmir and Huberman (1997) and LeBaron (2001b). However, its simplicityallows us to study the source of the volatility in some detail.

3. Structure of the market

There are two currencies, and xt is the rate of appreciation of currency one in period t.Equilibrium is characterized by uncovered interest parity. We denote Fi

t [xt+1] as the forecastofxt+1 held by agent i at time t, and Ft[xt+1] as the arithmetic average of these time t forecasts

1 There is a rich and growing literature on learning and adaptation in economic contexts. For some examplesof and perspectives on this literature see Arthur et al. (1997), Aumann (1997), Axtell (2000), Axtell et al. (2001),Blume and Easley (1992), Brock and Hommes (1997, 1998), Brock and deFontnouvelle (2000), Bullard (1994),Chakrabarti (2000), Chen and Yeh (2001, 2002), Cross (1983), Evans and Honkapohj (2001), Foster and Young(2001), Fudenberg and Levine (1998), Honkapohja and Mitra (2003), Levy et al. (1994), Lux (1995, 1998),Rubinstein (1998), Sargent (1993, 1999), Sandroni (2000), Simon (1955, 1978), Sobel (2000), Tetlow and von zurMuehlen (2004), Vriend (2000) and Young (1998).

2 In their model, agents live for two periods and adopt behavioral rules of thumb for consumption and currencydemand based on the hypothetical performances of these rules of thumb in the previous period. This model cangenerate returns that exhibit features of chaos (Arifovic and Gencay) and empirically realistic fat tail and volatilityclustering properties (Lux and Schornstein, 2005).

72 C. Georges / J. of Economic Behavior & Org. 60 (2006) 70–84

across the agents in the population. Then, we assume the following equilibrium condition:

Ft[xt+1] = z ∀t, (1)

where z reflects the nominal interest rate spread between assets denominated in the twocurrencies.3 In equilibrium the representative investor expects the rate of appreciation ofcurrency one to just offset the difference in the nominal returns of the two currencies.

4. Expectations

If all agents had perfect foresight, then the equilibrium would be

xt = z ∀t. (2)

The exchange rate would appreciate continuously at rate z. If for example the interest ratein country 1 is greater than that in country 2, so that z < 0, then the value of currency onewould fall toward zero at a constant rate just sufficient to offset its higher nominal rate ofreturn and thus to leave an agent with perfect foresight indifferent between holding the twocurrencies.4

If agents do not have enough information about the structure of the environment to formrational expectations of the rate of appreciation of currency 1, then they must formulateforecasts of this rate of appreciation inductively. We suppose that agents use forecast rulesof the following form

Fit [xt+1] = ai

t + bitxt + ci

tx2t−1, (3)

where ait , b

it , and ci

t are scalars that can vary across agents i and time t. In other words, in eachperiod, agents believe that the rate of appreciation next period will be well approximatedby a linear function of the current rate of appreciation and the square of last period’srate of appreciation.5 While these forecast rules are misspecified, we hypothesize that thismisspecification might appear reasonable to individual agents in the model out of steady stateequilibrium.6 Given these forecast rules (3) and the equilibrium condition (1), equilibrium

3 For the simulations below, we define z ≡ (1 + r2)/(1 + r1) − 1, where ri is the nominal interest rate in countryi.

4 The initial level of the exchange rate is arbitrary, so there would be an infinite number of possible perfectforesight equilibrium paths for the exchange rate. The rate of appreciation x of the exchange rate, on the otherhand, would be determinate as indicated above.

5 We assume that agents can condition their expectations on the current period’s exchange rate, which itself isdetermined by the average expectation of the agents. Thus, we are assuming that the agents adjust their forecastsas they witness the current period’s exchange rate emerge and that trade does not take place in each period untilthis process is complete.

6 As noted above, Arifovic (1996) models the learning of behavioral rules rather than forecasts. Bullard andDuffie (1999) call the former learning how to optimize and the latter learning how to forecast. For a discussion ofthis distinction and a variety of other issues that arise in agent based modeling in financial markets, see LeBaron(2001a). For evidence on the prevalence of technical analysis in actual foreign exchange markets, see for example,Taylor and Allen (1992), Menkhoff (1998) and Lui and Mole (1998).


Fig. 1. Consistent steady state expectations locus.

is characterized by7

xt = 1

bt

((z − at) − ctx2t−1) ∀t, (4)

where a, b, and c are the arithmetic averages of the forecast rule parameters across theagents in the population. For arbitrary constant values of a, b, and c, this system can displaya variety of different local and global dynamics. For example, holding b = 1 and a = 0we get the perfect foresight equilibrium xt = z ∀t if c = 0. However as c is progressivelylowered, the steady state of (4) diverges from z, and we first have dampened oscillationsnear this steady state and then pass through bifurcations into ranges with limit cycles, chaos,and explosive dynamics.

5. Consistent steady state expectations

Average expectations are correct in steady state in all periods if the following conditionis satisfied:

z = a + bz + cz2. (5)

This condition (5) is linear in the average forecast rule parameters a, b, and c, so the valuesof these parameters that satisfy it lie on a plane, a section of which is represented in Fig. 1.

If the average forecast rule parameters satisfy (5), then the perfect foresight equilibriumx = z is a steady state under (4), so that if the appreciation rate x is initially at this level, it will

7 In the terminology of Evans and Honkapohja, (4) is the “actual law of motion” and (3) represents the “perceivedlaw of motion.” In Grandmont’s terminology, (4) describes the “actual temporary equilibrium dynamics.”


remain there over time. Thus, while there is a unique perfect foresight equilibrium xt = z ∀t,there is a continuum of forecast rule parameter values that support that equilibrium underimperfect foresight. Further, even if the average forecast parameters are consistent with (5), ifthe rate of appreciation x of the exchange rate starts out of steady state, the exchange rate willchange over time under (4),8 and these dynamics need not be locally asymptotically stable.Indeed, various out of steady state dynamics (in the absence of learning) including dampenedcycles, limit cycles, and chaos are still supported if the average forecast parameters areconstrained to satisfy (5) (Fig. 2).

We are interested in this paper in exploring how agents’ forecast parameters evolverelative to the consistent steady state expectations locus (5) under learning as well as howx evolves relative to the perfect foresight equilibrium z.

6. Learning

If agents were to adopt arbitrary values of ai, bi, and ci that did not change over time, itis highly unlikely that (5) would be satisfied so that z would be a steady state for x under(4). Consequently, the actual appreciation rate would typically not be consistent with eitherindividual expectations (3) or average expectations. Thus, of particular interest to us is theway in which agents update their forecast rule parameters ai, bi, and ci over time and theeffect of this learning process on the equilibrium exchange rate under dynamics (4).

Here we model the evolution of forecast rules as a process of social learning via a modifiedgenetic algorithm.9 I follow Arifovic (1994) and Bullard and Duffie (1998) in modifyingthe GA to serve as a closer metaphor for social learning, which is more Lamarckian thangenetically based evolution.10 Here, long lived agents are periodically able to compare theirforecast rule parameter values to those of other agents as well as to combinations of thesevalues and to randomly drawn values. Thus, agents are able to experiment with alternativeforecast rules on a limited basis in each period and adopt the rule among these alternativeswith the smallest mean forecast error in recent periods.

Below, we focus on two cases. The first is a benchmark case in which the quadratic termin agents’ forecast rules is suppressed. Thus, in Case 1, ci

t = 0 for all i and t. In Case 2, the

8 Note that, the average forecast in t − 1 is correct if xt = z. However, this condition is not guaranteed by theconsistent steady state expectations condition (5) unless xt−1 is also equal to z.

9 There are numerous approaches in the literature to updating individual agent’s forecast rules in addition tousing GAs. For example, Arthur et al. (1997) and LeBaron et al. (1999) use a classifier system in conjunction witha GA to update individual agents’ forecast rules. LeBaron (2001b) and Yang (2003) specify forecasting rules asartificial neural nets. Chen and Yeh (2001, 2002) use genetic programming in conjunction with a “business school.”The GA was introduced by Holland (1975) who argued that it gives a method for searching complex decisionspaces in a way that provides a good balance between the benefits and costs of experimentation for on-line decisionproblems. For general treatments of this and related methods, see Goldberg (1989), Mitchell (1996), Michalewicz(1996), or Fogel (2000). For other applications in economics see for example Holland and Miller (1991), Andreoniand Miller (1995), Bullard and Duffie (1999) and Vriend (2000).10 Arifovic (1994) modifies the standard GA by including an election operator by which new chromosomes are

evaluated before being admitted into the population. As noted above, Arifovic (1996) and Arifovic and Gencay(2000) apply such a GA to behavioral rules in an artificial currency market. Bullard and Duffie (1998) furthermodify the GA to allow long lived agents to retain their own chromosomes over time.


Fig. 2. The left hand panel shows the equilibrium mapping (4) expressed as xt = h(xt−1) for average parametervalues a = 0.1, b = 2.2, and c = −28.8, and nominal interest rates r1 = 0.2 and r2 = 0.15 in the two countries.These average parameter values satisfy (5), and xt = z ≈ −0.042 is a steady state of (4) given these values. Abifurcation diagram is given in the right panel. This diagram illustrates the changing nature of the long run attractorof the system (4) as c and b are varied along the consistent expectations locus (5) holding a = 0.1.

variation in ait is suppressed and we consider learning on bi

t and cit . To summarize,

• Case 1: Agents can vary ait and bi

t .• Case 2: Agents can vary bit and ci

t .

7. The GA

I use a standard binary coded GA with a few modifications. At time t the forecast ruleof each agent is coded as a binary string with three segments that code the three parametersait , bi

t , and cit .

11 In each period the equilibrium rate of appreciation of the exchange rate xt

is determined according to (4). The fitness or performance of a forecast rule in any periodis based on its hypothetical forecasting accuracy in recent periods. Specifically, the fitnessof a rule being considered at time t is taken to be minus a weighted sum of the absolutevalues of the forecast errors that the rule would have generated in the past mem periods.12

For example, if the memory of the fitness function is set to one period, then in period t, thefitness of a rule depends on how well it would have forecasted xt−1 given xt−2 and xt−3.

Agents’ forecast rules are allowed to evolve through individual experimentation andimitation. In any given period, some members of the population are allowed to update theirrules. Each of these agent selects at random two other agents from the population. Of thosetwo, the agent with the higher current fitness is retained as the comparison agent. Therules of the agent and the comparison agent are then randomly combined (by crossover:

11 There has been a movement away from using binary coding in GAs as a number of authors have argued thatthere is no best representation for evolutionary searches and that rather evolutionary algorithms should be tailoredto individual problems (e.g., Fogel, 2000; Michalewicz, 1996; Herrera et al., 1998). I use a binary coding in orderto easily allow agents to experiment globally in the space of forecast rule parameters.12 The fitness f i

t of rule i being considered at time t is calculated as f it = −�t−1

s=t−mem|Fit [xs] − xs|(1/(t − s))ρ ,

where memory mem is the number of past rounds used to evaluate the rule and ρ ≥ 0 is a decay factor. Since weuse a tournament style selection process for reproduction, only the relative fitness values will be relevant.


replacing segments of the agent’s original bit string with the corresponding segments of thecomparison agent’s bit string) and/or mutated (by flipping bits of the original or combinedstring) to produce a third candidate rule. Of these three candidates (the agent’s original rule,the rule of the comparison agent, and the combined and/or mutated rule), the agent selectsthe one with the highest fitness (smallest recent forecast errors) to use in the current period.

This experimentation and social learning is incremental in the sense that agents retaintheir most recent rules as candidates in each period and these rules are the basis for mutationand crossover. Further, for low values of the mutation rate, mutation (pure experimentation)will tend to produce candidate rules that are ‘close’ to the old rules in the sense that fewbits will be flipped (i.e., the rules will be close in Hamming distance). Note however thatthe parameter values encoded in a new candidate chromosome may be quite different fromthe agent’s original values, since flipping a single bit can correspond to a large change in aparameter value.13 Thus, the genetic operators allow agents to search the parameter spaceglobally rather than just locally.

8. Case 1: quadratic term suppressed

As a benchmark, we consider the model when the quadratic term in the forecast rules issuppressed (i.e., in which ci

t = 0 for all i and t. Forecast rules are then simple AR(1) rules.The rate of appreciation x of currency one thus follows (4) with ct = 0 ∀t.

In this case, for a variety of runs, I find that learning is complete. The rate of appreciationof the exchange rate may fluctuate away from the perfect foresight equilibrium early on,but tends toward the perfect foresight value over time as agents update their forecast rulesin an effort to reduce their forecast errors. The individual forecast rule parameters a and bcontinue to vary substantially even as the rate of appreciation of the exchange rate growsclose to its equilibrium value. However, both individual and average values approach theparameter subspace defined by (5) and then wander near that subspace for some time beforesettling on an apparently arbitrary point on this subspace.

A sample run with a population of 40 agents and the memory in the fitness function setto one period is given in Fig. 3.

We see in Fig. 3A that the rate of appreciation of the exchange rate fluctuates away fromthe perfect foresight equilibrium early on, but is attracted to the perfect foresight value overtime, as agents update their forecast rules in an effort to reduce their forecast errors. We seein Fig. 3B, that starting from a common set of values, the agents’ forecast rule parameters aand b at first scatter and then approach the consistent expectation locus (5), and ultimatelyconverge to a single (arbitrary) point on that locus.

It is worth emphasizing that learning under the GA above apparently admits a continuum(characterized by (5), which in this example is given by b = 1 + 24a) of steady states forthe forecast rule parameters, even though the steady state value of the appreciation rate isunique (x = z). Agents’ parameter values tend to wander toward this locus over time, butdo not tend to converge to any particular point on the locus. This feature of our model is

13 For example, flipping the first bit of a chromosome causes ai to jump by half of its range, whereas flipping thesecond bit causes a change of only half that amount.


Fig. 3. (A) The equilibrium appreciation rate (xt) in a simulation over 250 rounds. The lighter line is the perfectforesight equilibrium appreciation rate. (B) The evolution of the population of parameter values ai and bi

also seen in the model of Youssefmir and Huberman and contrasts, for example, with thepseudo learning rule given by Evans and Honkapohja’s expectational stability test.14

9. Case 2: quadratic term active

With any linear autoregressive forecast rule, the (temporary) equilibrium of our modelwill be characterized by a linear AR process, so that any irregular dynamics will be due

14 If the average values of the forecast parameters (the parameters of the perceived law of motion) were assumedto move toward the corresponding parameter values of the actual law of motion (4), as in the case of Evans andHonkapohja, then we would have dynamics under which b → 0 and a → (1/b)(a − z) in continuous time. Theunique steady state of this learning process would be b = 0 and a = z (a particular point on the locus (5)), which isnot locally asymptotically stable. Thus, the expectational stability framework does not predict the general attractionto the consistent steady state expectations locus that we find and that is illustrated in Fig. 3B and Fig. 4B.


to the learning process (i.e., the updating of the forecast rules over time). In contrast tothis, the quadratic term in (3) adds a nonlinearity to the equilibrium dynamics (4) in anotherwise linear model, and so, as noted above, can produce irregular dynamics even inthe absence of learning.15

Before proceeding, consider again Fig. 2. Given a = 0.1, the local dynamics (nearx = z) under (4) for parameter values on the consistent expectations locus (5) are stable andmonotonic for c > 0 and cyclical for c < 0. As indicated in the right hand panel of Fig. 2,for c < 0 close to zero, cycles are damped, but as c becomes progressively more negative,the equilibrium passes through a series of bifurcations into limit cycles and chaos before thesystem becomes globally unstable.16 Suppose then that learning were to drive the forecastrule parameters to (5) as in Case 1. Then, a priori, this would cause the appreciationrate x either to converge to the perfect foresight value z or to fluctuate around thisvalue.

We find that the learning system tends to converge, with xt → z over time, if itfinds itself in a region of the forecast rule parameter space for which the dynamicsunder (4) are fairly well behaved. However, if the dynamics under (4) are sufficientlyvolatile, then learning breaks down and contributes to the volatility of the system. Thiscan lead the system to blow up, to wander into a more stable configuration, or to fluc-tuate irregularly and persistently. While many of the simulations that we ran led thesystem either to converge or to blow up, we will focus on cases of persistent dynamicsbelow.17

The following simulation is instructive. Forty agents start out with the same forecastparameter values under which, in the absence of learning, (4) would produce damped cycles.However, with learning, cycles start damped and then vary in amplitude.

Fig. 4 shows that in this particular run of the simulation learning initially causes theexchange rate to converge toward an appreciation rate of z. During this period, individualagents’ rules converge toward the consistent steady state expectations locus (5) (whichin this example is given by c = −81.6 + 24b). However, as the average value c falls,z becomes an unstable equilibrium under (4), and volatility increases. As x fluctuatesmore wildly, learning breaks down and the population scatters in the parameter space(i.e., forecast rules become more heterogeneous). However, this scattering causes theaverage value of c to rise, leading the system back into a stable regime, and individualsbegin to cluster again around the consistent steady state expectation locus. When thesimulation was run for an additional 500 rounds, the cycle of convergence and divergencecontinued.

It is worth noting that the dynamics of both this case and that of the previous section aredifferent from those of Arifovic (1996). The Karaken–Wallace model used there exhibits

15 In Arifovic (1996), nonlinearity is introduced not through forecast rules but rather through structure of the un-derlying OLG model. Persistent fluctuations in Brock and Hommes (1997, 1998) are driven by irregular switchingbetween forecast types with different costs of forecasting. In that model, when the economy is near the rationalexpectations equilibrium, the costs of generating rational expectations outweigh the benefits to the agents, whothen switch to less costly rules of thumb which cause local instability.16 By (4) we see that there is a negative and unstable root at x = z for c < −b/2|z|.17 The fairly high incidence of the system blowing up would be removed if we were to place reasonable bounds

on the forecasts.


Fig. 4. (A) The evolution of the appreciation rate x over 200 periods. (B) Evolution of the population’s parametervalues bi and ci. Same simulation as in (A).

a particular indeterminacy. At any equilibrium, the rates of return on the two currenciesare equal, so at the level of the individual investor, any portfolio composition appears to beequally profitable. Thus, mutant portfolio behavior will survive the election operator and beadmitted into the population, kicking the market out of equilibrium and triggering feedbackbetween the rates of return of individual currencies and agents’ portfolio compositions.Consequently, in Arifovic’s model, no steady state equilibrium is stable under the geneticalgorithm. In the present model, there is also an indeterminacy in that any combination offorecast rule parameters satisfying (5) and constant rate of appreciation x = z constitutes asteady state learning equilibrium. While many of these equilibria are not locally asymptot-ically stable, due to the feedback between actual and forecasted appreciation rates, each isstable under the GA in the sense of Arifovic (1996). At any such equilibrium, there are noforecast rules with greater fitness than any agent’s current rule (as with Arifovic’s model),but also, the set of rules with equal fitness has measure zero (in contrast to Arifovic’s model)


Fig. 5. The evolution of the appreciation rate x over 1500 periods with noise.

and so will never be discovered even though these rules would pass election if selected.18

Thus, all of the interesting dynamics of the present model occur out of steady state anddepend on the evolution of forecast rules under learning out of steady state.

10. Shocks

When random shocks are added to the equilibrium dynamics (added linearly to (4)),the qualitative results above are preserved except that the population appears to perpetuallyscatter and churn along the locus (5). Consequently, for the small population sizes that I havefocused on (typically 40), the average parameter values fluctuate considerably over time.Consequently, rather than converging, c continues to wander and occasionally becomessufficiently negative so as to produce instability and cause learning to break down. Thus,the presence of noise in the model increases the likelihood that the steady state will becomeunstable eventually under learning. In terms of Grandmont’s insight, agents tend in thismodel eventually to observe and extrapolate transitory nonlinear trends in destabilizingways. A sample run is shown in Fig. 5.

Not surprisingly, adding noise also makes the dynamics of the appreciation rate moreirregular, more closely mimicking the behavior of actual foreign exchange markets. Thesimulation above displays some interesting features. First, there is clustered volatility in theappreciation rate, with fluctuations centered broadly on the unique perfect foresight equi-librium x = z. While that equilibrium is unique, it is supported by a continuum of forecastrule parameter values, which in the absence of volatility appears to be absorbing under thelearning dynamics. Bursts of volatility are apparently set off by the average forecast param-eters drifting into the region of this continuum in which z is a locally unstable steady state

18 Further, even if selected, this would constitute a jump to a new steady state equilibrium and would not initiateout of equilibrium dynamics as in Arifovic’s model.


equilibrium. Second, heterogeneity (of the forecast rules) appears to increase in responseto these bursts of volatility, subsequently stabilizing the appreciation rate dynamics.19

Nevertheless, the model presented here is a toy model that is too simple to simulate actualfinancial market patterns with any accuracy. For example, the simulated appreciation rateshown in Fig. 5 displays a high degree of first order serial correlation and close to normalkurtosis. Agents are leaving even very basic linear structure in the time series unexploited,20

and the time series does not display the leptokurtosis (high incidence of extreme events)displayed in actual market data.21

11. Longer memories and larger populations

Increasing the memory in the fitness functions does not appear to promote greater sta-bility. As noted above, we considered weighted sums of the absolute forecast errors from afinite number of past rounds and considered various weights (of the form (1/s)ρ, where sis the number of periods in the past and ρ ≥ 0).22 Indeed, the tendency to either convergeor blow up appears to be heightened by increasing the memory in the fitness functions inthis way.

Similarly, increasing the population size (e.g., to 500 or 2000) does not appear to promotegreater stability. It also does not appear to have a systematic impact on the ability of themodel (with random shocks) to produce clustered volatility. This result contrasts with thefinding of Lux and Schornstein that the irregular fluctuations in the Arifovic (1996) modelconverge to regular cycles as the population is increased.23

12. Conclusion

I have illustrated the exchange rate dynamics for a very simple artificial currency marketunder a simple nonlinear forecast rule with learning. In the baseline case with the non-

19 This is a result similar to that of LeBaron (2001b) where homogeneity leads to low liquidity in the market—common expectations can make it difficult for agents to unwind their positions, leading to large price movements.Above, homogeneity does not lead to volatility per se, but volatility does cause homogeneity to disappear.20 How much systematic forecast error is acceptable? Hommes and Sorger (1998) argue that we should expect

agents to learn to uncover some but not all of the structure underlying the equilibrium dynamics. For example,nonlinear structure would be missed by agents using linear forecasting tools. They introduce the notion of aconsistent expectations equilibrium under which agents’ expectations are consistent with the actual behavior ofthe economy in terms of a limited number of linear sample statistics.21 See for example, Baillie and Bollerslev (1990) and De Vries (1994).22 For example, forρ = 0, past forecast errors receive equal weights. Forρ > 0, past forecast errors are discounted.23 In contrast to the present paper, Lux and Schornstein find that the Arifovic model can produce empirically

realistic fat tails and volatility clustering in returns for small GA populations. However, they also find that asthe population size increases, the dynamics become more regular, and a periodic cycle emerges. The feedbackbetween the rate of return of the domestic currency and the average portfolio composition in the Karaken–Wallacemodel produces cyclical tendencies. Idiosyncratic saving and portfolio behavior confounds these tendencies insmall populations but averages out in large populations. In the present model, increased population size does notappear to favor the learning of more stable or periodic configurations in the aggregate.


linearity suppressed, learning tends to be complete, with the rate of appreciation of theexchange rate tending to converge to its rational expectations equilibrium value. However,when the forecast rule is quadratic, persistent out of steady state fluctuations may arise fromthe interaction of learning and the model.

These simulations are interesting as an illustration of how learning can fail to be completeand produce interesting dynamics in a very simple market environment. However, whilethe fluctuations that arise in some of the simulations do exhibit clustered volatility, they areunrealistically regular. Given the set of forecast rules at their disposal, investors in the modelare unable to identify the regularity that persists in these fluctuations. Nevertheless, themodel illustrates the volatility that can be produced by a simple nonlinear misspecificationin the forecast rules used by boundedly rational agents in an otherwise very simple andlinear model.

Acknowledgements

I would like to thank Jim Bullard, Tom Michl, John Miller, Jeff Pliskin, William Brock,participants in the Seventh Annual Conference of the Society for Computational Eco-nomics and the Brookings Workshop on Multi-Agent Computation in Natural and ArtificialEconomies, and an anonymous referee for their helpful comments. All errors are mine.

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