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
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
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
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.87-123. 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.1109 1186. W.A. Brock and C.H. Hommes, A rational route to randomness, Econometrica 65, (1997), 1059-1095.
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
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
Nonlinear S-shaped Supply Curves λ = 0.22 λ = 2 2 2 1.5 1.5 1 1 0.5 0.5 2 4 6 8 10 stable (under naive) 2 4 6 8 10 strongly unstable (under naive)
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
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
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
Rational Expectations Benchmark (p = 5.93) Problem: need perfect knowledge of law of motion
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)?
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)
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 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
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 10 10 10 8 8 8 6 6 6 4 4 4 2 2 2 0 5 10 15 20 25 30 35 40 45 50 0 5 10 15 20 25 30 35 40 45 50 0 5 10 15 20 25 30 35 40 45 50 PRICE_GROUP2 EXP21 EXP22 EXP23 EXP24 EXP25 EXP26 PRICE_GROUP1 EXP11 EXP12 EXP13 EXP14 EXP15 EXP16 PRICE_GROUP2 EXP21 EXP22 EXP23 EXP24 EXP25 EXP26 2.0 2.0 2.0 1.6 1.6 1.6 1.2 1.2 1.2 0.8 0.8 0.8 0.4 0.4 0.4 0.0 5 10 15 20 25 30 35 40 45 50 0.0 5 10 15 20 25 30 35 40 45 50 0.0 5 10 15 20 25 30 35 40 45 50 SIG SIG SIG
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
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
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
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
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
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
Fundamentalists versus naive 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2-2 -1.5-1 -0.5 0 0.5 1 1.5 2 2 1.5 1 0.5 0-0.5-1 -1.5 0 10 20 30 40 50 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 10 20 30 40 5 0.25 0.2 0.15 0.1 0.05 0-0.05-0.1-0.15-0.2 0-0.2-0.4-0.6-0.8 0 20 40 60 80 100 0 20 40 60 80 100
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
Introduction Cobweb Model Learning Experiments Heterogeneous Expectations Conclusions SAC-learning versus naive 0 0.7-0.1 0.6-0.2 0.5-0.3 0.4-0.4 0.3-0.5 0.2-0.6 0.1-0.7-1 -0.5 0 0.5 1 0 100 200 300 400 500 3 2.5 2 sac 1.5 1 0.5 0-0.5-1 0 2 4 6 8 10 t agents learn to be contrarians, with first order AC βt 0.62 part of the (linear) structure has been arbitraged away
SAC-learning versus naive (with memory) 0.9 3 0.9 0.8 0.7 0.6 0.5 2 1 0 0.8 0.7 0.6 0.5 0.4-1 0.4 0.3 0.2 0.1-3 -2-1 0 1 2 3-2 -3 0 20 40 60 80 100 120 140 0.3 0.2 0.1 0 20 40 60 80 100 120 140 1 0 0.8-0.1 0.6-0.2 0.4 0.2-0.3 0-0.4-0.2-0.5-0.4-0.6-0.6-0.8-0.7 0 1000 2000 3000 4000 5000 0 1000 2000 3000 4000 5000 weaker autocorrelation in prices, (β t 0.48).
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