Topics in Nonlinear Economic Dynamics: Bounded Rationality, Heterogeneous Expectations ... ·...

Introduction Cobweb Model Learning Experiments Heterogeneous Expectations Conclusions

Topics in Nonlinear Economic Dynamics:

Bounded Rationality, Heterogeneous Expectations

and Complex Adaptive Systems

Cars Hommes

CeNDEF, Amsterdam School of EconomicsUniversity of Amsterdam

PhD - Workshop Series in Advanced Quantitative Methods inEconomics & Finance, 12 November, 2010, St Andrews, UK

Cars Hommes CeNDEF, University of Amsterdam

PhD-Workshop Series in Advanced Quantitative Methods in Economics & Finance, 12 November, 2010, St Andrews, UK


Plan of Lectures:

I Lecture 1a: theory:cobweb model with heterogeneous expectations

I Lecture 1b: theory + some empirical testing:asset pricing model with heterogeneous expectations

I Lecture 2: laboratory testing:Learning-to-Forecast-Experiments




Main ingredients of Lecture 1a:The cobweb model with heterogeneous expectations

I what is a good theory of expectations, when markets arecomplex and agents are boundedly rational ?

I two “consistency” stories on bounded rationality in complexsystems: adaptive and evolutionary learning, and combinethem

I framework: classical cobweb or “hog cycle” modelI confront theory with laboratory experiments




Some Literature

I Hommes, C.H., Bounded Rationality and Learning in Complex Markets,(2009), In: Handbook of Research on Complexity, Edited by J. Barkley Rosser,Jr., Cheltenham: Edward Elgar, pp.87-123.

I Hommes, C.H., Heterogeneous Agent Models (HAM) in Economics andFinance, in Tesfatsion, L. and Judd, K.L., Handbook of ComputationalEconomics Volume 2: Agent-Based Computational Economics, Elsevier, 2006,pp.1109–1186.

I W.A. Brock and C.H. Hommes, A rational route to randomness, Econometrica65, (1997), 1059-1095.




Cobweb (‘hog cycle’) Model

I market for non-storable consumptions goodI production lag; producers form price expectations one period

aheadI partial equilibrium; market clearing pricesI key variables

I pet : producers’ price expectation for period t

I pt: realized market equilibrium price pt




Cobweb (‘hog cycle’) Model (continued)

D(pt) = a−dpt(+εt) a ∈ R, d ≥ 0 demand (1)

Sλ (pet ) = tanh(λ (pe

t −6))+1, λ > 0, supply (2)

D(pt) = Sλ (pet ) market clearing (3)

pet = H(pt−1, ...,pt−L), expectations (4)

Price dynamics: pt = D−1Sλ (H(pt−1, ...,pt−L))Expectations Feedback System:dynamical behavior depends upon expectations hypothesis;supply driven, negative feedback




Nonlinear S-shaped Supply Curves

λ = 0.22 λ = 2

2 4 6 8 10

0.5

1

1.5

2

2 4 6 8 10

0.5

1

1.5

2

stable (under naive) strongly unstable (under naive)




Adaptive Expectations (“error learning”), Nerlove 1958

pet = (1−w)pe

t−1 +wpt−1

= pet−1 +w(pt−1−pe

t−1

= wpt−1 +(1−w)wpt−2 + · · ·(1−w)j−1wpt−j + · · ·

1-D (expected) price dynamics: pet = wD−1S(pe

t−1)+(1−w)pet−1

I more stable in linear models, butI possibly low amplitude chaos in nonlinear models




Simple Benchmarks

naive expectations adaptive expectations (w = 0.2)pe

t = pt−1 pet = (1−w)pe

t−1 +wpt−1)

predictable ‘hog cycle’, with systematic forecasting errors




Rational Expectations (Muth, 1961)

agents compute expectations from market equilibrium equations

pet = Et[pt] or pe

t = pt or pet = p∗

implied price dynamicspt = p∗+δt

perfect foresight, no systematic forecasting errorsImportant Note: this is impossible in complex, heterogeneous world




Rational Expectations Benchmark (p∗ = 5.93)

Problem: need perfect knowledge of “law of motion”




Adaptive Learning

agents are boundedly rational, and adapt their behaviorbased upon time series observations(e.g. Sargent (1993), Grandmont (1998), Evans and Honkapohja(2001))

I perhaps heterogeneous agents can learn a REE?I or does a complex system converge to a

“boundedly rational learning equilibrium”(with excess volatility)?




Sample Auto-Correlation (SAC) Learninginitial states: p0, α0 and β0.sample average after t periods:

αt−1 =1t

t−1

∑i=0

pi , t ≥ 2 (5)

the sample autocorrelation coefficient at the first lag, after t periods:

βt−1 =∑t−2

i=0(pi−αt−1)(pi+1−αt−1)

∑t−1i=0(pi−αt−1)2

, t ≥ 2 (6)

perceived law of motion: simple AR1 forecasting rule

pet = αt−1 +βt−1(pt−1−αt−1) (7)




Adaptive Learning in Cobweb Model

learning by average SAC-learning

always quick convergence to REE




Computer Screen Lab Experiment




(unknown) "law of motion" of market price

pt =a−∑K

j=1 Sλ (pet )

b+ εt

I stable treatment: λ = 0.22I unstable treatment: λ = 0.5I strongly unstable treatment: λ = 2




Rational Expectations Benchmark (p∗ = 5.93)

Problem: need perfect knowledge of “law of motion”




Some Evidence from the Laboratory

convergence to REE excess volatility




Learning to Forecast Experiments & Heterogeneity

0

2

4

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10

5 10 15 20 25 30 35 40 45 50

PRICE_GROUP2

EXP21

EXP22

EXP23

EXP24

EXP25

EXP26

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PRICE_GROUP1

EXP11

EXP12

EXP13

EXP14

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EXP16

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PRICE_GROUP2

EXP21

EXP22

EXP23

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EXP26

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SIG

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SIG

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SIG




Stylized facts of laboratory cobweb experiments(Hommes et al. (2007), Macroeconomic Dynamics)

I stable cobweb converges to REI unstable cobweb:

I sample average very close to RE priceI excess volatility: sample variance higher than REI no linear predictability: no autocorrelations




Which theory of expectations fits laboratory experiments?

I naive/adaptive expectations: price dynamics too regularI rational expectations: no excess volatility in unstable caseI adaptive learning (e.g. by average or by SAC):

always converges to RE, even in the unstable treatment

Conclusion:heterogeneity is needed to explain all stylized facts simultaneously




Two “consistency” stories

I adaptive learningconsistent expectations equilibrium (Hommes and Sorger, MD1998)agents try to learn the best linear forecasting rule, in anunknown nonlinear environment

I evolutionary selection expectations (Brock and Hommes,1997, 1998)agents tend to follow (linear) rules that have performed better inthe recent past, according to past realized evolutionary fitness




Heterogeneous Beliefs and Evolutionary Learning(Brock and Hommes (1997))

I agents choose between two different types of forecasting rulesI sophisticated rule at information costs C > 0 (Simon (1957)

or a simple rule freely availableI agents evaluate the net past performance of all rules, and tend to

follow rules that have performed better in the recent pastevolutionary fitness measure ≡ past realized net profits




Evolutionary Selection of Expectations Rules

discrete choice model, with asynchronous updating:

nht = (1−δ )eβUh,t−1

Zt−1+δnh,t−1,

where Zt−1 = ∑eβUh,t−1 is normalization factor,Uh,t−1 past strategy performance, e.g. (weighted average) pastprofits

δ is probability of not updatingβ is the intensity of choice.β = 0: all types equal weight (in long run)β = ∞: fraction 1−δ switches to best predictor




Cobweb Model with Heterogeneous Beliefsmarket clearing

a−dpt = n1tspe1t +n2tspe

1t(+εt)

n1t and n2t = 1−n1t fractions of two typesforecasting rules:rational or fundamentalists or SAC-learning at cost C > 0versus free naive

pe1t = pt rational

= p∗ fundamentalist

= αt−1 +βt−1(pt−1−αt−1) SAC-learning

pe2t = pt−1 naive




Fundamentalists versus naive

0.2

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-2 -1.5 -1 -0.5 0 0.5 1 1.5 2

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0 10 20 30 40 50

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Fundamentalists versus naive (continued)

I chaotic price fluctuations (when intensity of choice large)I sample average of prices close to fundamental priceI strong negative first order autocorrelation in prices (βt →−0.85)

Question: will boundedly rational agents detect negative AC?Replace fundamentalists by SAC-learning




SAC-learning versus naive

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0 2 4 6 8 10

sa

c

t

agents learn to be contrarians, with first order AC βt →−0.62part of the (linear) structure has been “arbitraged away”




SAC-learning versus naive (with memory)

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weaker autocorrelation in prices, (βt →−0.48).




Concluding Remarks

I heterogeneity in expectations needed to explain experimentsI mixture of evolutionary selection and adaptive learning

broadly explains stylized facts in experiments:I sample mean close to REE p∗I excess volatility compared to REE, in unstable treatmentI little linear structure, because agents learn to be contrarians

Remark:Hommes and Lux (2010) fit a GA-learning model to match allstylized facts in the cobweb experiments



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