2. 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) but
appear to the agents to be warranted by the historical path of the economy generated by the
combination 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 a
great deal about their environments with the assumption that they use rules of thumb and
try 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 of
the 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 region
of the parameter space in which the steady state is locally stable, in which case learning
tends to be complete, or into a region in which the steady state is locally unstable, in which
case learning may subsequently fail to converge and instead lead to persistent exchange rate
dynamics.
This mechanism is different from that proposed by Arifovic (1996) and Arifovic and
Gencay (2000), in which irregular exchange rate dynamics arise from the interaction of GA
learning and the nonlinearity of an overlapping generations model,2 as well as the models
of Brock and Hommes (1997, 1998), Chen et al. (2001) and Westerhoff (2003), in which
agents switch between a small number of belief types. The mechanism is closer in spirit to
those of Youssefmir and Huberman (1997) and LeBaron (2001b). However, its simplicity
allows 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 forecast
ofxt+1 heldbyagentiattimet,and ¯Ft[xt+1]asthearithmeticaverageofthesetimetforecasts
1 There is a rich and growing literature on learning and adaptation in economic contexts. For some examples
of 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 zur
Muehlen (2004), Vriend (2000) and Young (1998).
2 In their model, agents live for two periods and adopt behavioral rules of thumb for consumption and currency
demand based on the hypothetical performances of these rules of thumb in the previous period. This model can
generate returns that exhibit features of chaos (Arifovic and Gencay) and empirically realistic fat tail and volatility
clustering properties (Lux and Schornstein, 2005).
3. 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 two
currencies.3 In equilibrium the representative investor expects the rate of appreciation of
currency 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 rate
in country 1 is greater than that in country 2, so that z < 0, then the value of currency one
would fall toward zero at a constant rate just sufficient to offset its higher nominal rate of
return and thus to leave an agent with perfect foresight indifferent between holding the two
currencies.4
If agents do not have enough information about the structure of the environment to form
rational expectations of the rate of appreciation of currency 1, then they must formulate
forecasts of this rate of appreciation inductively. We suppose that agents use forecast rules
of the following form
Fi
t [xt+1] = ai
t + bi
txt + ci
tx2
t−1, (3)
where ai
t, bi
t, and ci
t are scalars that can vary across agents i and time t. In other words, in each
period, agents believe that the rate of appreciation next period will be well approximated
by a linear function of the current rate of appreciation and the square of last period’s
rate of appreciation.5 While these forecast rules are misspecified, we hypothesize that this
misspecificationmightappearreasonabletoindividualagentsinthemodeloutofsteadystate
equilibrium.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 country
i.
4 The initial level of the exchange rate is arbitrary, so there would be an infinite number of possible perfect
foresight equilibrium paths for the exchange rate. The rate of appreciation x of the exchange rate, on the other
hand, would be determinate as indicated above.
5 We assume that agents can condition their expectations on the current period’s exchange rate, which itself is
determined by the average expectation of the agents. Thus, we are assuming that the agents adjust their forecasts
as they witness the current period’s exchange rate emerge and that trade does not take place in each period until
this process is complete.
6 As noted above, Arifovic (1996) models the learning of behavioral rules rather than forecasts. Bullard and
Duffie (1999) call the former learning how to optimize and the latter learning how to forecast. For a discussion of
this 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).
4. C. Georges / J. of Economic Behavior & Org. 60 (2006) 70–84 73
Fig. 1. Consistent steady state expectations locus.
is characterized by7
xt =
1
¯bt
((z − ¯at) − ¯ctx2
t−1) ∀t, (4)
where ¯a, ¯b, and ¯c are the arithmetic averages of the forecast rule parameters across the
agents in the population. For arbitrary constant values of ¯a, ¯b, and ¯c, this system can display
a variety of different local and global dynamics. For example, holding ¯b = 1 and ¯a = 0
we get the perfect foresight equilibrium xt = z ∀t if ¯c = 0. However as ¯c is progressively
lowered, the steady state of (4) diverges from z, and we first have dampened oscillations
near 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 condition
is satisfied:
z = ¯a + ¯bz + ¯cz2
. (5)
This condition (5) is linear in the average forecast rule parameters ¯a, ¯b, and ¯c, so the values
of 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 equilibrium
x = 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 “perceived
law of motion.” In Grandmont’s terminology, (4) describes the “actual temporary equilibrium dynamics.”
5. 74 C. Georges / J. of Economic Behavior & Org. 60 (2006) 70–84
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 under
imperfectforesight.Further,eveniftheaverageforecastparametersareconsistentwith(5),if
the rate of appreciation x of the exchange rate starts out of steady state, the exchange rate will
change over time under (4),8 and these dynamics need not be locally asymptotically stable.
Indeed,variousoutofsteadystatedynamics(intheabsenceoflearning)includingdampened
cycles, limit cycles, and chaos are still supported if the average forecast parameters are
constrained to satisfy (5) (Fig. 2).
We are interested in this paper in exploring how agents’ forecast parameters evolve
relative to the consistent steady state expectations locus (5) under learning as well as how
x 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, it
is 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 either
individual expectations (3) or average expectations. Thus, of particular interest to us is the
way in which agents update their forecast rule parameters ai, bi, and ci over time and the
effect of this learning process on the equilibrium exchange rate under dynamics (4).
Herewemodeltheevolutionofforecastrulesasaprocessofsociallearningviaamodified
genetic algorithm.9 I follow Arifovic (1994) and Bullard and Duffie (1998) in modifying
the GA to serve as a closer metaphor for social learning, which is more Lamarckian than
genetically based evolution.10 Here, long lived agents are periodically able to compare their
forecast rule parameter values to those of other agents as well as to combinations of these
values and to randomly drawn values. Thus, agents are able to experiment with alternative
forecast rules on a limited basis in each period and adopt the rule among these alternatives
with the smallest mean forecast error in recent periods.
Below, we focus on two cases. The first is a benchmark case in which the quadratic term
in 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 the
consistent 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 to
using GAs. For example, Arthur et al. (1997) and LeBaron et al. (1999) use a classifier system in conjunction with
a GA to update individual agents’ forecast rules. LeBaron (2001b) and Yang (2003) specify forecasting rules as
artificial 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 decision
spaces in a way that provides a good balance between the benefits and costs of experimentation for on-line decision
problems. 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), Andreoni
and 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) further
modify the GA to allow long lived agents to retain their own chromosomes over time.
6. C. Georges / J. of Economic Behavior & Org. 60 (2006) 70–84 75
Fig. 2. The left hand panel shows the equilibrium mapping (4) expressed as xt = h(xt−1) for average parameter
values ¯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. A
bifurcation diagram is given in the right panel. This diagram illustrates the changing nature of the long run attractor
of the system (4) as ¯c and ¯b are varied along the consistent expectations locus (5) holding ¯a = 0.1.
variation in ai
t is suppressed and we consider learning on bi
t and ci
t. To summarize,
• Case 1: Agents can vary ai
t and bi
t.
• Case 2: Agents can vary bi
t and ci
t.
7. The GA
I use a standard binary coded GA with a few modifications. At time t the forecast rule
of each agent is coded as a binary string with three segments that code the three parameters
ai
t, bi
t, and ci
t.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 period
is based on its hypothetical forecasting accuracy in recent periods. Specifically, the fitness
of a rule being considered at time t is taken to be minus a weighted sum of the absolute
values 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, the
fitness 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 and
imitation. In any given period, some members of the population are allowed to update their
rules. Each of these agent selects at random two other agents from the population. Of those
two, the agent with the higher current fitness is retained as the comparison agent. The
rules 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 that
there is no best representation for evolutionary searches and that rather evolutionary algorithms should be tailored
to individual problems (e.g., Fogel, 2000; Michalewicz, 1996; Herrera et al., 1998). I use a binary coding in order
to easily allow agents to experiment globally in the space of forecast rule parameters.
12 The fitness fi
t of rule i being considered at time t is calculated as fi
t = − t−1
s=t−mem|Fi
t [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 we
use a tournament style selection process for reproduction, only the relative fitness values will be relevant.
7. 76 C. Georges / J. of Economic Behavior & Org. 60 (2006) 70–84
replacing segments of the agent’s original bit string with the corresponding segments of the
comparison agent’s bit string) and/or mutated (by flipping bits of the original or combined
string) 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 selects
the 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 retain
their most recent rules as candidates in each period and these rules are the basis for mutation
and 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 few
bits will be flipped (i.e., the rules will be close in Hamming distance). Note however that
the parameter values encoded in a new candidate chromosome may be quite different from
the agent’s original values, since flipping a single bit can correspond to a large change in a
parameter value.13 Thus, the genetic operators allow agents to search the parameter space
globally 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 is
suppressed (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 appreciation
of 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 rules
in an effort to reduce their forecast errors. The individual forecast rule parameters a and b
continue to vary substantially even as the rate of appreciation of the exchange rate grows
close to its equilibrium value. However, both individual and average values approach the
parameter subspace defined by (5) and then wander near that subspace for some time before
settling on an apparently arbitrary point on this subspace.
A sample run with a population of 40 agents and the memory in the fitness function set
to one period is given in Fig. 3.
We see in Fig. 3A that the rate of appreciation of the exchange rate fluctuates away from
the perfect foresight equilibrium early on, but is attracted to the perfect foresight value over
time, as agents update their forecast rules in an effort to reduce their forecast errors. We see
in Fig. 3B, that starting from a common set of values, the agents’ forecast rule parameters a
and b at first scatter and then approach the consistent expectation locus (5), and ultimately
converge 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 + 24¯a) of steady states for
the forecast rule parameters, even though the steady state value of the appreciation rate is
unique (x = z). Agents’ parameter values tend to wander toward this locus over time, but
do 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 the
second bit causes a change of only half that amount.
8. C. Georges / J. of Economic Behavior & Org. 60 (2006) 70–84 77
Fig. 3. (A) The equilibrium appreciation rate (xt) in a simulation over 250 rounds. The lighter line is the perfect
foresight 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 the
pseudo 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 model
will 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 assumed
to move toward the corresponding parameter values of the actual law of motion (4), as in the case of Evans and
Honkapohja, then we would have dynamics under which ¯b → 0 and ¯a → (1/¯b)(¯a − z) in continuous time. The
unique steady state of this learning process would be b = 0 and a = z (a particular point on the locus (5)), which is
not locally asymptotically stable. Thus, the expectational stability framework does not predict the general attraction
to the consistent steady state expectations locus that we find and that is illustrated in Fig. 3B and Fig. 4B.
9. 78 C. Georges / J. of Economic Behavior & Org. 60 (2006) 70–84
to the learning process (i.e., the updating of the forecast rules over time). In contrast to
this, the quadratic term in (3) adds a nonlinearity to the equilibrium dynamics (4) in an
otherwise linear model, and so, as noted above, can produce irregular dynamics even in
the absence of learning.15
Before proceeding, consider again Fig. 2. Given ¯a = 0.1, the local dynamics (near
x = z) under (4) for parameter values on the consistent expectations locus (5) are stable and
monotonic 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 the
system becomes globally unstable.16 Suppose then that learning were to drive the forecast
rule parameters to (5) as in Case 1. Then, a priori, this would cause the appreciation
rate x either to converge to the perfect foresight value z or to fluctuate around this
value.
We find that the learning system tends to converge, with xt → z over time, if it
finds itself in a region of the forecast rule parameter space for which the dynamics
under (4) are fairly well behaved. However, if the dynamics under (4) are sufficiently
volatile, then learning breaks down and contributes to the volatility of the system. This
can 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 the
system either to converge or to blow up, we will focus on cases of persistent dynamics
below.17
The following simulation is instructive. Forty agents start out with the same forecast
parameter 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 the
exchange rate to converge toward an appreciation rate of z. During this period, individual
agents’ rules converge toward the consistent steady state expectations locus (5) (which
in this example is given by ¯c = −81.6 + 24¯b). However, as the average value ¯c falls,
z becomes an unstable equilibrium under (4), and volatility increases. As x fluctuates
more wildly, learning breaks down and the population scatters in the parameter space
(i.e., forecast rules become more heterogeneous). However, this scattering causes the
average value of ¯c to rise, leading the system back into a stable regime, and individuals
begin to cluster again around the consistent steady state expectation locus. When the
simulation was run for an additional 500 rounds, the cycle of convergence and divergence
continued.
It is worth noting that the dynamics of both this case and that of the previous section are
different 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 switching
between forecast types with different costs of forecasting. In that model, when the economy is near the rational
expectations equilibrium, the costs of generating rational expectations outweigh the benefits to the agents, who
then 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.
10. C. Georges / J. of Economic Behavior & Org. 60 (2006) 70–84 79
Fig. 4. (A) The evolution of the appreciation rate x over 200 periods. (B) Evolution of the population’s parameter
values bi and ci. Same simulation as in (A).
a particular indeterminacy. At any equilibrium, the rates of return on the two currencies
are equal, so at the level of the individual investor, any portfolio composition appears to be
equally profitable. Thus, mutant portfolio behavior will survive the election operator and be
admitted into the population, kicking the market out of equilibrium and triggering feedback
between the rates of return of individual currencies and agents’ portfolio compositions.
Consequently, in Arifovic’s model, no steady state equilibrium is stable under the genetic
algorithm. In the present model, there is also an indeterminacy in that any combination of
forecast rule parameters satisfying (5) and constant rate of appreciation x = z constitutes a
steady 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 is
stable under the GA in the sense of Arifovic (1996). At any such equilibrium, there are no
forecast 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)
11. 80 C. Georges / J. of Economic Behavior & Org. 60 (2006) 70–84
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 and
depend 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 perpetually
scatter and churn along the locus (5). Consequently, for the small population sizes that I have
focused on (typically 40), the average parameter values fluctuate considerably over time.
Consequently, rather than converging, ¯c continues to wander and occasionally becomes
sufficiently 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 become
unstable eventually under learning. In terms of Grandmont’s insight, agents tend in this
model eventually to observe and extrapolate transitory nonlinear trends in destabilizing
ways. A sample run is shown in Fig. 5.
Not surprisingly, adding noise also makes the dynamics of the appreciation rate more
irregular, more closely mimicking the behavior of actual foreign exchange markets. The
simulation above displays some interesting features. First, there is clustered volatility in the
appreciation 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 forecast
rule parameter values, which in the absence of volatility appears to be absorbing under the
learning 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 initiate
out of equilibrium dynamics as in Arifovic’s model.
12. C. Georges / J. of Economic Behavior & Org. 60 (2006) 70–84 81
equilibrium. Second, heterogeneity (of the forecast rules) appears to increase in response
to 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 actual
financial market patterns with any accuracy. For example, the simulated appreciation rate
shown in Fig. 5 displays a high degree of first order serial correlation and close to normal
kurtosis. 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 a
finite number of past rounds and considered various weights (of the form (1/s)ρ, where s
is the number of periods in the past and ρ ≥ 0).22 Indeed, the tendency to either converge
or blow up appears to be heightened by increasing the memory in the fitness functions in
this way.
Similarly, increasing the population size (e.g., to 500 or 2000) does not appear to promote
greater stability. It also does not appear to have a systematic impact on the ability of the
model (with random shocks) to produce clustered volatility. This result contrasts with the
finding of Lux and Schornstein that the irregular fluctuations in the Arifovic (1996) model
converge to regular cycles as the population is increased.23
12. Conclusion
I have illustrated the exchange rate dynamics for a very simple artificial currency market
under 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 a
consistent expectations equilibrium under which agents’ expectations are consistent with the actual behavior of
the economy in terms of a limited number of linear sample statistics.
21 See for example, Baillie and Bollerslev (1990) and De Vries (1994).
22 Forexample,forρ = 0,pastforecasterrorsreceiveequalweights.Forρ > 0,pastforecasterrorsarediscounted.
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 as
the population size increases, the dynamics become more regular, and a periodic cycle emerges. The feedback
between the rate of return of the domestic currency and the average portfolio composition in the Karaken–Wallace
model produces cyclical tendencies. Idiosyncratic saving and portfolio behavior confounds these tendencies in
small populations but averages out in large populations. In the present model, increased population size does not
appear to favor the learning of more stable or periodic configurations in the aggregate.
13. 82 C. Georges / J. of Economic Behavior & Org. 60 (2006) 70–84
linearity suppressed, learning tends to be complete, with the rate of appreciation of the
exchange 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 from
the interaction of learning and the model.
These simulations are interesting as an illustration of how learning can fail to be complete
and produce interesting dynamics in a very simple market environment. However, while
the fluctuations that arise in some of the simulations do exhibit clustered volatility, they are
unrealistically regular. Given the set of forecast rules at their disposal, investors in the model
are unable to identify the regularity that persists in these fluctuations. Nevertheless, the
model illustrates the volatility that can be produced by a simple nonlinear misspecification
in the forecast rules used by boundedly rational agents in an otherwise very simple and
linear 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 Artificial
Economies, and an anonymous referee for their helpful comments. All errors are mine.
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under rational expectations (i.e., under deduction if the agents knew the true model) but
appear to the agents to be warranted by the historical path of the economy generated by the
combination 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 a
great deal about their environments with the assumption that they use rules of thumb and
try 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 of
the 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 region
of the parameter space in which the steady state is locally stable, in which case learning
tends to be complete, or into a region in which the steady state is locally unstable, in which
case learning may subsequently fail to converge and instead lead to persistent exchange rate
dynamics.
This mechanism is different from that proposed by Arifovic (1996) and Arifovic and
Gencay (2000), in which irregular exchange rate dynamics arise from the interaction of GA
learning and the nonlinearity of an overlapping generations model,2 as well as the models
of Brock and Hommes (1997, 1998), Chen et al. (2001) and Westerhoff (2003), in which
agents switch between a small number of belief types. The mechanism is closer in spirit to
those of Youssefmir and Huberman (1997) and LeBaron (2001b). However, its simplicity
allows 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 forecast
ofxt+1 heldbyagentiattimet,and ¯Ft[xt+1]asthearithmeticaverageofthesetimetforecasts
1 There is a rich and growing literature on learning and adaptation in economic contexts. For some examples
of 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 zur
Muehlen (2004), Vriend (2000) and Young (1998).
2 In their model, agents live for two periods and adopt behavioral rules of thumb for consumption and currency
demand based on the hypothetical performances of these rules of thumb in the previous period. This model can
generate returns that exhibit features of chaos (Arifovic and Gencay) and empirically realistic fat tail and volatility
clustering properties (Lux and Schornstein, 2005).](data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7)