Topics in Nonlinear Economic Dynamics: Bounded Rationality, Heterogeneous Expectations and Complex Adaptive Systems

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1 Topics in Nonlinear Economic Dynamics: Bounded Rationality, Heterogeneous Expectations and Complex Adaptive Systems CeNDEF, Amsterdam School of Economics University of Amsterdam PhD - Workshop Series in Advanced Quantitative Methods in Economics & Finance, 12 November, 2010, St Andrews, UK

2 Plan of Lectures: Lecture 1a: theory: cobweb model with heterogeneous expectations Lecture 1b: theory + some empirical testing: asset pricing model with heterogeneous expectations Lecture 2: laboratory testing: Learning-to-Forecast-Experiments

3 Main ingredients of Lecture 1a: The cobweb model with heterogeneous expectations what is a good theory of expectations, when markets are complex and agents are boundedly rational? two consistency stories on bounded rationality in complex systems: adaptive and evolutionary learning, and combine them framework: classical cobweb or hog cycle model confront theory with laboratory experiments

4 Some Literature 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 Hommes, C.H., Heterogeneous Agent Models (HAM) in Economics and Finance, in Tesfatsion, L. and Judd, K.L., Handbook of Computational Economics Volume 2: Agent-Based Computational Economics, Elsevier, 2006, pp W.A. Brock and C.H. Hommes, A rational route to randomness, Econometrica 65, (1997),

5 Cobweb ( hog cycle ) Model market for non-storable consumptions good production lag; producers form price expectations one period ahead partial equilibrium; market clearing prices key variables p e t : producers price expectation for period t p t : realized market equilibrium price p t

6 Cobweb ( hog cycle ) Model (continued) D(p t ) = a dp t (+ε t ) a R, d 0 demand (1) S λ (p e t ) = tanh(λ(p e t 6)) + 1, λ > 0, supply (2) D(p t ) = S λ (p e t ) market clearing (3) p e t = H(p t 1,...,p t L ), expectations (4) Price dynamics: p t = D 1 S λ (H(p t 1,...,p t L )) Expectations Feedback System: dynamical behavior depends upon expectations hypothesis; supply driven, negative feedback

7 Nonlinear S-shaped Supply Curves λ = 0.22 λ = stable (under naive) strongly unstable (under naive)

8 Adaptive Expectations ( error learning ), Nerlove 1958 p e t = (1 w)p e t 1 + wp t 1 = p e t 1 + w(p t 1 p e t 1 = wp t 1 + (1 w)wp t 2 + (1 w) j 1 wp t j + 1-D (expected) price dynamics: p e t = wd 1 S(p e t 1 ) + (1 w)pe t 1 more stable in linear models, but possibly low amplitude chaos in nonlinear models

9 Simple Benchmarks naive expectations adaptive expectations (w = 0.2) p e t = p t 1 p e t = (1 w)p e t 1 + wp t 1) predictable hog cycle, with systematic forecasting errors

10 Rational Expectations (Muth, 1961) agents compute expectations from market equilibrium equations p e t = E t [p t ] or p e t = p t or p e t = p implied price dynamics p t = p + δ t perfect foresight, no systematic forecasting errors Important Note: this is impossible in complex, heterogeneous world

11 Rational Expectations Benchmark (p = 5.93) Problem: need perfect knowledge of law of motion

12 Adaptive Learning agents are boundedly rational, and adapt their behavior based upon time series observations (e.g. Sargent (1993), Grandmont (1998), Evans and Honkapohja (2001)) perhaps heterogeneous agents can learn a REE? or does a complex system converge to a boundedly rational learning equilibrium (with excess volatility)?

13 Sample Auto-Correlation (SAC) Learning initial states: p 0, α 0 and β 0. sample average after t periods: α t 1 = 1 t t 1 i=0 p i, t 2 (5) the sample autocorrelation coefficient at the first lag, after t periods: β t 1 = t 2 i=0 (p i α t 1 )(p i+1 α t 1 ) t 1 i=0 (p i α t 1 ) 2, t 2 (6) perceived law of motion: simple AR1 forecasting rule p e t = α t 1 + β t 1 (p t 1 α t 1 ) (7)

14 Adaptive Learning in Cobweb Model learning by average SAC-learning always quick convergence to REE

15 Computer Screen Lab Experiment

16 (unknown) "law of motion" of market price p t = a K j=1 S λ (p e t ) b + ε t stable treatment: λ = 0.22 unstable treatment: λ = 0.5 strongly unstable treatment: λ = 2

17 Rational Expectations Benchmark (p = 5.93) Problem: need perfect knowledge of law of motion

18 Some Evidence from the Laboratory convergence to REE excess volatility

19 Learning to Forecast Experiments & Heterogeneity PRICE_GROUP2 EXP21 EXP22 EXP23 EXP24 EXP25 EXP26 PRICE_GROUP1 EXP11 EXP12 EXP13 EXP14 EXP15 EXP16 PRICE_GROUP2 EXP21 EXP22 EXP23 EXP24 EXP25 EXP SIG SIG SIG

20 Stylized facts of laboratory cobweb experiments (Hommes et al. (2007), Macroeconomic Dynamics) stable cobweb converges to RE unstable cobweb: sample average very close to RE price excess volatility: sample variance higher than RE no linear predictability: no autocorrelations

21 Which theory of expectations fits laboratory experiments? naive/adaptive expectations: price dynamics too regular rational expectations: no excess volatility in unstable case 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

22 Two consistency stories adaptive learning consistent expectations equilibrium (Hommes and Sorger, MD 1998) agents try to learn the best linear forecasting rule, in an unknown nonlinear environment evolutionary selection expectations (Brock and Hommes, 1997, 1998) agents tend to follow (linear) rules that have performed better in the recent past, according to past realized evolutionary fitness

23 Heterogeneous Beliefs and Evolutionary Learning (Brock and Hommes (1997)) agents choose between two different types of forecasting rules sophisticated rule at information costs C > 0 (Simon (1957) or a simple rule freely available agents evaluate the net past performance of all rules, and tend to follow rules that have performed better in the recent past evolutionary fitness measure past realized net profits

24 Evolutionary Selection of Expectations Rules discrete choice model, with asynchronous updating: n ht = (1 δ) eβu h,t 1 Z t 1 + δn h,t 1, where Z t 1 = e βu h,t 1 is normalization factor, U h,t 1 past strategy performance, e.g. (weighted average) past profits δ is probability of not updating β is the intensity of choice. β = 0: all types equal weight (in long run) β = : fraction 1 δ switches to best predictor

25 Cobweb Model with Heterogeneous Beliefs market clearing a dp t = n 1t sp e 1t + n 2tsp e 1t (+ε t) n 1t and n 2t = 1 n 1t fractions of two types forecasting rules: rational or fundamentalists or SAC-learning at cost C > 0 versus free naive p e 1t = p t rational = p fundamentalist = α t 1 + β t 1 (p t 1 α t 1 ) SAC-learning p e 2t = p t 1 naive

26 Fundamentalists versus naive

27 Fundamentalists versus naive (continued) chaotic price fluctuations (when intensity of choice large) sample average of prices close to fundamental price strong negative first order autocorrelation in prices (β t 0.85) Question: will boundedly rational agents detect negative AC? Replace fundamentalists by SAC-learning

28 Introduction Cobweb Model Learning Experiments Heterogeneous Expectations Conclusions SAC-learning versus naive sac t agents learn to be contrarians, with first order AC βt 0.62 part of the (linear) structure has been arbitraged away

29 SAC-learning versus naive (with memory) weaker autocorrelation in prices, (β t 0.48).

30 Concluding Remarks heterogeneity in expectations needed to explain experiments mixture of evolutionary selection and adaptive learning broadly explains stylized facts in experiments: sample mean close to REE p excess volatility compared to REE, in unstable treatment little linear structure, because agents learn to be contrarians Remark: Hommes and Lux (2010) fit a GA-learning model to match all stylized facts in the cobweb experiments