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 Economics University of Amsterdam PhD - Workshop Series in Advanced Quantitative Methods in Economics & 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
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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
Introduction Cobweb Model Learning Experiments Heterogeneous Expectations Conclusions
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
Cars Hommes CeNDEF, University of Amsterdam
PhD-Workshop Series in Advanced Quantitative Methods in Economics & Finance, 12 November, 2010, St Andrews, UK
Introduction Cobweb Model Learning Experiments Heterogeneous Expectations Conclusions
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
Cars Hommes CeNDEF, University of Amsterdam
PhD-Workshop Series in Advanced Quantitative Methods in Economics & Finance, 12 November, 2010, St Andrews, UK
Introduction Cobweb Model Learning Experiments Heterogeneous Expectations Conclusions
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.
Cars Hommes CeNDEF, University of Amsterdam
PhD-Workshop Series in Advanced Quantitative Methods in Economics & Finance, 12 November, 2010, St Andrews, UK
Introduction Cobweb Model Learning Experiments Heterogeneous Expectations Conclusions
Cobweb (‘hog cycle’) Model
I market for non-storable consumptions goodI production lag; producers form price expectations one period
I more stable in linear models, butI possibly low amplitude chaos in nonlinear models
Cars Hommes CeNDEF, University of Amsterdam
PhD-Workshop Series in Advanced Quantitative Methods in Economics & Finance, 12 November, 2010, St Andrews, UK
Introduction Cobweb Model Learning Experiments Heterogeneous Expectations Conclusions
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
Cars Hommes CeNDEF, University of Amsterdam
PhD-Workshop Series in Advanced Quantitative Methods in Economics & Finance, 12 November, 2010, St Andrews, UK
Introduction Cobweb Model Learning Experiments Heterogeneous Expectations Conclusions
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
Cars Hommes CeNDEF, University of Amsterdam
PhD-Workshop Series in Advanced Quantitative Methods in Economics & Finance, 12 November, 2010, St Andrews, UK
Introduction Cobweb Model Learning Experiments Heterogeneous Expectations Conclusions
Rational Expectations Benchmark (p∗ = 5.93)
Problem: need perfect knowledge of “law of motion”
Cars Hommes CeNDEF, University of Amsterdam
PhD-Workshop Series in Advanced Quantitative Methods in Economics & Finance, 12 November, 2010, St Andrews, UK
Introduction Cobweb Model Learning Experiments Heterogeneous Expectations Conclusions
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
PhD-Workshop Series in Advanced Quantitative Methods in Economics & Finance, 12 November, 2010, St Andrews, UK
Introduction Cobweb Model Learning Experiments Heterogeneous Expectations Conclusions
Rational Expectations Benchmark (p∗ = 5.93)
Problem: need perfect knowledge of “law of motion”
Cars Hommes CeNDEF, University of Amsterdam
PhD-Workshop Series in Advanced Quantitative Methods in Economics & Finance, 12 November, 2010, St Andrews, UK
Introduction Cobweb Model Learning Experiments Heterogeneous Expectations Conclusions
Some Evidence from the Laboratory
convergence to REE excess volatility
Cars Hommes CeNDEF, University of Amsterdam
PhD-Workshop Series in Advanced Quantitative Methods in Economics & Finance, 12 November, 2010, St Andrews, UK
Introduction Cobweb Model Learning Experiments Heterogeneous Expectations Conclusions
Learning to Forecast Experiments & Heterogeneity
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Cars Hommes CeNDEF, University of Amsterdam
PhD-Workshop Series in Advanced Quantitative Methods in Economics & Finance, 12 November, 2010, St Andrews, UK
Introduction Cobweb Model Learning Experiments Heterogeneous Expectations Conclusions
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
Cars Hommes CeNDEF, University of Amsterdam
PhD-Workshop Series in Advanced Quantitative Methods in Economics & Finance, 12 November, 2010, St Andrews, UK
Introduction Cobweb Model Learning Experiments Heterogeneous Expectations Conclusions
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
Cars Hommes CeNDEF, University of Amsterdam
PhD-Workshop Series in Advanced Quantitative Methods in Economics & Finance, 12 November, 2010, St Andrews, UK
Introduction Cobweb Model Learning Experiments Heterogeneous Expectations Conclusions
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
Cars Hommes CeNDEF, University of Amsterdam
PhD-Workshop Series in Advanced Quantitative Methods in Economics & Finance, 12 November, 2010, St Andrews, UK
Introduction Cobweb Model Learning Experiments Heterogeneous Expectations Conclusions
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
Cars Hommes CeNDEF, University of Amsterdam
PhD-Workshop Series in Advanced Quantitative Methods in Economics & Finance, 12 November, 2010, St Andrews, UK
Introduction Cobweb Model Learning Experiments Heterogeneous Expectations Conclusions
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
Cars Hommes CeNDEF, University of Amsterdam
PhD-Workshop Series in Advanced Quantitative Methods in Economics & Finance, 12 November, 2010, St Andrews, UK
Introduction Cobweb Model Learning Experiments Heterogeneous Expectations Conclusions
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
Cars Hommes CeNDEF, University of Amsterdam
PhD-Workshop Series in Advanced Quantitative Methods in Economics & Finance, 12 November, 2010, St Andrews, UK
Introduction Cobweb Model Learning Experiments Heterogeneous Expectations Conclusions
Fundamentalists versus naive
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Cars Hommes CeNDEF, University of Amsterdam
PhD-Workshop Series in Advanced Quantitative Methods in Economics & Finance, 12 November, 2010, St Andrews, UK
Introduction Cobweb Model Learning Experiments Heterogeneous Expectations Conclusions
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
Cars Hommes CeNDEF, University of Amsterdam
PhD-Workshop Series in Advanced Quantitative Methods in Economics & Finance, 12 November, 2010, St Andrews, UK
Introduction Cobweb Model Learning Experiments Heterogeneous Expectations Conclusions
SAC-learning versus naive
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agents learn to be contrarians, with first order AC βt →−0.62part of the (linear) structure has been “arbitraged away”
Cars Hommes CeNDEF, University of Amsterdam
PhD-Workshop Series in Advanced Quantitative Methods in Economics & Finance, 12 November, 2010, St Andrews, UK
Introduction Cobweb Model Learning Experiments Heterogeneous Expectations Conclusions
SAC-learning versus naive (with memory)
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weaker autocorrelation in prices, (βt →−0.48).
Cars Hommes CeNDEF, University of Amsterdam
PhD-Workshop Series in Advanced Quantitative Methods in Economics & Finance, 12 November, 2010, St Andrews, UK
Introduction Cobweb Model Learning Experiments Heterogeneous Expectations Conclusions
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
Cars Hommes CeNDEF, University of Amsterdam
PhD-Workshop Series in Advanced Quantitative Methods in Economics & Finance, 12 November, 2010, St Andrews, UK