Complex Systems Workshop Lecture III: Behavioral Asset Pricing Model with Heterogeneous Beliefs Cars Hommes CeNDEF, UvA CEF 2013, July 9, Vancouver Cars Hommes (CeNDEF, UvA) Complex Systems CEF 2013, Vancouver 1 / 52
Outline 1 Introduction 2 The model 2-type model: fundamentalists vs. trend followers 3-type model: fundamentalists vs. optimists and pessimists 4-type model: fundamentalists vs. trend and bias 3 Empirical Validation Cars Hommes (CeNDEF, UvA) Complex Systems CEF 2013, Vancouver 2 / 52
Introduction Literature Hommes, C.H., (2013), Behavioral Rationality and Heterogeneous Expectations in Complex Economic Systems, Cambridge. (Chapters 6 and 7) W.A. Brock and C.H. Hommes, A rational route to randomness, Econometrica 65, (1997), 1059-1095. W.A. Brock and C.H. Hommes, Heterogeneous beliefs and routes to chaos in a simple asset pricing model, Journal of Economic Dynamics and Control 22, (1998), 1235-1274. Hommes, C.H. (2006), Heterogeneous Agent Models In Economics and Finance, In: Handbook of Computational Economics, Volume 2: Agent-Based Computational Economics, Edited by L. Tesfatsion and K.L. Judd, Elsevier Science B.V., 2006, 1109-1186. Hommes, C.H. and in t Veld, D. (2013), Behavioral heterogeneity and the financial crisis, CeNDEF working paper, July 2013. Cars Hommes (CeNDEF, UvA) Complex Systems CEF 2013, Vancouver 3 / 52
Introduction Traditional Rational View representative agent, who is perfectly rational expectations are model consistent Friedman hypothesis: irrational agents will lose money and will be driven out the market by rational agents simple (linear), stable model, driven by exogenous random news about fundamentals prices reflect economic fundamentals (market efficiency) Lucas: macroeconomic policy should be based on rational expectations Cars Hommes (CeNDEF, UvA) Complex Systems CEF 2013, Vancouver 4 / 52
Introduction Heterogeneous, interacting agents approach heterogeneous agents, heterogeneous beliefs market psychology, herding behavior (Keynes (1936)) bounded rationality (Simon (1957)) markets as complex adaptive, nonlinear evolutionary systems interactions of agents create aggregate structure explaining stylized facts Cars Hommes (CeNDEF, UvA) Complex Systems CEF 2013, Vancouver 5 / 52
Introduction Some Problems Interacting Agents Approach wilderness of bounded rationality many degrees of freedom for heterogeneity what exactly causes the outcome in a (large) computational HAM Cars Hommes (CeNDEF, UvA) Complex Systems CEF 2013, Vancouver 6 / 52
Introduction How to Discipline Bounded Rationality? stylized agent-based models behavioral rationality behavioral consistency: simple heuristics that work reasonably well evolutionary selection ( survival of the fittest ) and reinforcement learning laboratory experiments to test individual decision rules and aggregate macro behavior Cars Hommes (CeNDEF, UvA) Complex Systems CEF 2013, Vancouver 7 / 52
The model Asset Pricing Model with Homogeneous Beliefs Agents choose between risk free and risky asset: R = 1 + r > 1: gross return on risk free asset p t : price (ex div.) per share of risky asset y t : IID dividend process for risky asset z t : number of shares purchased at date t End of period wealth: W t+1 = R(W t p t z t ) + (p t+1 + y t+1 )z t = RW t + (p t+1 + y t+1 Rp t )z t Myopic mean variance maximization: demand z t solves Max{E t W t+1 a 2 V tw t+1 }, so z t = E t[p t+1 + y t+1 Rp t ] av t [p t+1 + y t+1 Rp t ] = E t[p t+1 + y t+1 Rp t ] aσ 2 (where σ 2 is common beliefs on variance V ht ), Cars Hommes (CeNDEF, UvA) Complex Systems CEF 2013, Vancouver 8 / 52
Equilibrium Price The model Market equilibrium between supply and demand: equilibrium pricing equation: E t (p t+1 + y t+1 Rp t ) aσ 2 = z s Rp t = E ht (p t+1 + y t+1 ) aσ 2 z s special case: constant zero supply of outside shares z s = 0: Rp t = E ht (p t+1 + y t+1 ) Cars Hommes (CeNDEF, UvA) Complex Systems CEF 2013, Vancouver 9 / 52
REE fundamental solution The model Equilibrium pricing equation: (common beliefs on future dividends E t [y t+1 ]) Rp t = E t [p t+1 + y t+1 ] no bubble" condition implies unique bounded fundamental solution p t : (discounted sum expected future cash flow) p t = E t[y t+1 ] R + E t[y t+2 ] R 2 + For special case of IID dividends, with E t [y t+1 ] = ȳ: p = ȳ R 1 = ȳ r Cars Hommes (CeNDEF, UvA) Complex Systems CEF 2013, Vancouver 10 / 52
The model Model in deviations from fundamental deviation from fundamental Pricing equation in deviations: x t = p t p Rx t = E t x t+1 Notice: rational bubble solutions: x t = x 0 R t with self-fulfilling belief xt+1 e = gx t 1, with g = R 2. Equilibrium pricing equation with heterogeneous beliefs: (in deviations from RE-fundamental) H Rx t = n ht E ht x t+1 h=1 Cars Hommes (CeNDEF, UvA) Complex Systems CEF 2013, Vancouver 11 / 52
The model Asset pricing model with heterogeneous beliefs agents choose to invest in risk free or risky asset R = 1 + r > 1: gross return on risk free asset p t : y t : n ht price (ex div.) per share of risky asset IID dividend process for risky asset : fraction of agents of type h Myopic mean variance maximization of expected wealth demand for risky asset by type h: z ht = E ht[p t+1 + y t+1 Rp t ] av ht [p t+1 + y t+1 Rp t ] = E ht[p t+1 + y t+1 Rp t ] aσ 2 (common beliefs on variance V ht = σ 2 and a risk aversion parameter) Cars Hommes (CeNDEF, UvA) Complex Systems CEF 2013, Vancouver 12 / 52
Market equilibrium The model Equilibrium of supply and demand: H h=1 n ht E ht [p t+1 + y t+1 Rp t ] aσ 2 = z s equilibrium pricing equation: H Rp t = n ht E ht (p t+1 + y t+1 ) aσ 2 z s h=1 special case: constant zero supply of outside shares z s = 0: H Rp t = n ht E ht (p t+1 + y t+1 ) + ɛ t h=1 (where noise term ɛ t (e.g. random supply of shares) has been added) Cars Hommes (CeNDEF, UvA) Complex Systems CEF 2013, Vancouver 13 / 52
Heterogeneous Beliefs The model Assumptions about beliefs of trader type h: B1 same constant beliefs on variances for all types h: V ht [p t+1 +y t+1 Rp t ] = V t [p t+1 +y t+1 Rp t ] = σ 2 B2 common and correct beliefs on future dividends: E ht [y t+1 ] = E t [y t+1 ], for all types h special case of IID dividends: E ht [y t+1 ] = E t [y t+1 ] = ȳ. B3 heterogeneous beliefs on future prices of the form: E ht [p t+1 ] = E t [p t+1 ] + E ht[x t+1 ] = pt+1 + f h(x t 1, x t L ) special case of IID dividends: E ht [p t+1 ] = p + f h (x t 1, x t L ) Under assumptions B1-B3 equilibrium pricing equation in deviations x t = p t p from the fundamental: H H Rx t = n ht E ht [x t+1 ] = n ht f ht h=1 Cars Hommes (CeNDEF, UvA) Complex Systems CEF 2013, Vancouver 14 / 52 h=1
The model Forecasting rules belief of type h on future prices: or in deviations: E ht [p t+1 ] = p + f h (x t 1,..., x t L ) E ht [x t+1 ] = f h (x t 1,..., x t L ) Important special cases: rational expectations: "f (x t 1,..., x t L )" = x t+1 (also perfect foresight on other belief-fractions n ht ) fundamentalists: f 0 (no knowledge about other beliefs and fractions n ht ) pure trend chasers: f (x t 1,..., x t L ) = gx t 1 pure bias: f (x t 1,..., x t L ) = b. simple example: linear forecast with one lag f ht = g h x t 1 + b h trend extrapolator: f ht = x t 1 + g h (x t 1 x t 2 ) Question: Do rational agents and/or fundamentalists drive out trend chasers and biased Cars Hommes beliefs? (CeNDEF, UvA) Complex Systems CEF 2013, Vancouver 15 / 52
The model Evolutionary selection of strategies evolutionary selection or reinforcement learning: more successful strategies attract more followers fractions of belief types are updated in each period, according to (discrete choice model, BH 1997, 1998) n ht = eβu h,t 1 Z t 1 where Z t 1 is normalization factor and β is intensity of choice. β = 0: all types equal weight β = : neoclassical limit, i.e. all agents use best predictor Cars Hommes (CeNDEF, UvA) Complex Systems CEF 2013, Vancouver 16 / 52
The model Evolutionary selection of strategies realized profits in period t π ht = R t z h,t 1 = (p t + y t Rp t 1 ) E h,t 1[p t + y t Rp t 1 ] aσ 2 (with y t = ȳ + δ t ) (x t Rx t 1 + δ t ) E h,t 1[x t Rx t 1 ] aσ 2 Fitness function or performance measure (weighted sum of) realized profits U ht = π ht + wu h,t 1 C h where C h 0 are costs for predictor h, and w is memory strength (w = 1: infinite memory; fitness accumulated wealth w = 0: memory one lag; fitness most recently realized net profit) Cars Hommes (CeNDEF, UvA) Complex Systems CEF 2013, Vancouver 17 / 52
The model Asset Pricing Model with Heterogeneous Beliefs (in deviations from the RE fundamental) Rx t = H n ht f h (x t 1,..., x t L ) = h=1 n ht = e βu h,t 1 Hh=1 e βu h,t 1 H n ht f ht h=1 U h,t 1 = (x t 1 Rx t 2 ) f h,t 2 Rx t 2 aσ 2 C h Cars Hommes (CeNDEF, UvA) Complex Systems CEF 2013, Vancouver 18 / 52
The model Examples with Heterogeneous Linear Beliefs (in deviations from the RE fundamental) example with linear predictors f ht = g h x t 1 + b h : Rx t = H n ht (g h x t 1 + b h ) h=1 n ht = e βu h,t 1 Hh=1 e βu h,t 1 U h,t 1 = (x t 1 Rx t 2 ) (g hx t 3 + b h Rx t 2 ) aσ 2 C h Cars Hommes (CeNDEF, UvA) Complex Systems CEF 2013, Vancouver 19 / 52
The model 2-type model: fundamentalists vs. trend followers Two-type Example: Fundamentalists versus trend Two trader types, with forecasting rules f 1t = 0, fundamentalists at costs C f 2t = gx t 1, g > 0, trend followers Define difference in fractions: m t = n 1t n 2t Rx t = 1 mt 2 gx t 1 m t+1 = tanh( β 2 [ gx t 2 aσ 2 (x t Rx t 1 ) C]) Cars Hommes (CeNDEF, UvA) Complex Systems CEF 2013, Vancouver 20 / 52
The model 2-type model: fundamentalists vs. trend followers Two-type Example: Fundamentalists versus trend Theorem. (existence and stability steady states) Let x be positive solution of m eq = tanh( βc/2) m = 1 2R/g tanh( β 2 [ g aσ 2 (R 1)(x ) 2 C]) = m 1 0 < g < R: E 1 = (0, m eq ) globally stable steady state 2 very strong trend chaser, i.e g > 2R: three steady states E 1 = (0, m eq ), E 2 = (x, m ) and E 3 = ( x, m ) 3 R < g < 2R: if costs C > 0, then we have a pitchfork bifurcation for β = β, that is, a unique steady state for β < β and three steady states for β > β. Cars Hommes (CeNDEF, UvA) Complex Systems CEF 2013, Vancouver 21 / 52
The model 2-type model: fundamentalists vs. trend followers Two-type Example: Fundamentalists versus trend Moreover, Hopf bifurcation of non-fundamental steady states E 2 and E 3 : stable for β < β < β unstable for β > β Corollary: costly fundamentalists and cannot drive out trend chasers driven by short run profits. Cars Hommes (CeNDEF, UvA) Complex Systems CEF 2013, Vancouver 22 / 52
The model 2-type model: fundamentalists vs. trend followers Bifurcation diagram and Lyapunov exonent plot 2 0.1 1 0.05 0 0 1 0.05 2 2 2.5 3 3.5 4 0.1 2 2.2 2.4 2.6 2.8 3 3.2 3.4 3.6 3.8 4 Figure: Bifurcation diagram (left) and largest Lyapunov exponent plot (right) for 2-type model with costly fundamentalist versus trend followers. Cars Hommes (CeNDEF, UvA) Complex Systems CEF 2013, Vancouver 23 / 52
The model 2-type model: fundamentalists vs. trend followers Time series of prices and fractions and attractors 1.75 1.5 1.25 1 0.75 (a) (b) 2 1 0 0.5-1 0.25 100 200 300 400 500-2 100 200 300 400 500 1 (c) 1 (d) 0.8 0.8 0.6 0.6 0.4 0.4 0.2 0.2 100 200 300 400 500 100 200 300 400 500 1 (e) 1 (f) 0.8 0.8 0.6 0.6 0.4 0.4 0.2 0.2 0.25 0.5 0.75 1 1.25 1.5 1.75-2 -1 0 1 2 Figure: Time series of prices and fractions and attractors in the phase space for 2-type model with costly fundamentalist versus trend followers. Cars Hommes (CeNDEF, UvA) Complex Systems CEF 2013, Vancouver 24 / 52
The model 2-type model: fundamentalists vs. trend followers Homoclinic Orbit for β = + Two possibilities for the unstable manifold W u (E) (Brock and Hommes, 1998, Lemma 4, p.1251): 1 if g > (1 + r) 2, then unstable manifold W u (E) equals the (unbounded) unstable eigenvector; typical solutions are exploding, diverging to infinity with trend followers dominating the market; 2 if 1 + r < g < (1 + r) 2, then unstable manifold W u (E) is bounded; all time paths converge to the (locally) unstable saddle-point fundamental steady state; homoclinic orbits Cars Hommes (CeNDEF, UvA) Complex Systems CEF 2013, Vancouver 25 / 52
The model 3-type model: fundamentalists vs. optimists and pessimists Three Type Example Three types (zero costs) f 1t = 0 fundamentalists f 2t = b b > 0, positive bias (optimists) f 3t = b b < 0, negative bias (pessimists). Rx t = n 2,t b 2 + n 3,t b 3 n j,t+1 = exp( β aσ 2 (b j Rx t 1 )(x t Rx t 1 ))/Z t, j = 1, 2, 3 Cars Hommes (CeNDEF, UvA) Complex Systems CEF 2013, Vancouver 26 / 52
Three Type Example The model 3-type model: fundamentalists vs. optimists and pessimists Theorem. (existence and stability steady state) Assume opposite bias, i.e. b 2 > 0 > b 3, then (1) has unique steady state E, which equals the fundamental steady state when b 2 = b 3. E exhibits a Hopf bifurcation for β = β : E stable for 0 < β < β and E unstable for β > β. Corollary: Biased beliefs lead to a different route to complexity Cars Hommes (CeNDEF, UvA) Complex Systems CEF 2013, Vancouver 27 / 52
Three Type Example The model 3-type model: fundamentalists vs. optimists and pessimists Theorem (neoclassical limit, i.e. β = ) When biased beliefs are exactly opposite, i.e. b 2 = b 3 = b > 0, then (1) has globally stable 4-cycle. For all three types, average profit along this 4-cycle is b 2. Corrolary Fundemantalists with zero costs and infinite memory can not beat opposite biased beliefs! Cars Hommes (CeNDEF, UvA) Complex Systems CEF 2013, Vancouver 28 / 52
The model 3-type model: fundamentalists vs. optimists and pessimists Bifurcation diagram and largest Lyapunov exponent plot (a) 0.15 0.1 0.05 0-0.05-0.1-0.15 30 40 50 60 70 80 90 100 70 80 90 100 (b) 0-0.05-0.1-0.15-0.2-0.25-0.3 30 40 50 60 Figure: Bifurcation diagram (top panel) and largest Lyapunov exponent plot (bottom panel) for 3-type model with fundamentalists versus optimists and pessimists. Cars Hommes (CeNDEF, UvA) Complex Systems CEF 2013, Vancouver 29 / 52
Phase plot The model 3-type model: fundamentalists vs. optimists and pessimists 0.2 0.1 x t 0 0.1 0.2 0.2 0.1 0 0.1 0.2 x t 1 Figure: Phase plot for 3-type model with fundamentalists versus optimists and pessimists, as in Brock and Hommes, 1998, Figures 7 and 8. Cars Hommes (CeNDEF, UvA) Complex Systems CEF 2013, Vancouver 30 / 52
x Time series The model 3-type model: fundamentalists vs. optimists and pessimists 0.2 0.1 0 0.1 0.2 1 0.75 n 1 0.5 0.25 0 1 0.75 n 2 0.5 0.25 0 1 0.75 n 3 0.5 0.25 0 Figure: Time series for 3-type model with fundamentalists versus optimists and pessimists, as in Brock and Hommes, 1998, Figures 7 and 8. Cars Hommes (CeNDEF, UvA) Complex Systems CEF 2013, Vancouver 31 / 52
Four Belief Types The model 4-type model: fundamentalists vs. trend and bias (zero costs; memory one lag) example. g 1 = 0 b 1 = 0 fundamentalists g 2 = 0.9 b 2 = 0.2 trend + upward bias g 3 = 0.9 b 3 = 0.2 trend + downward bias g 4 = R = 1.01 b 4 = 0 trend chaser (1) Rx t = 4 n h,t (g h x t 1 + b j ) h=1 n h,t+1 = exp( β aσ 2 (g hx t 2 + b h Rx t 1 )(x t Rx t 1 ))/Z t, h = 1, 2, 3, 4 Cars Hommes (CeNDEF, UvA) Complex Systems CEF 2013, Vancouver 32 / 52
The model 4-type model: fundamentalists vs. trend and bias Four Belief Types Rational Route to Randomness: β < β : fundamental steady state globally stable β = β : Hopf bifurcation of steady state β < β < β : periodic and quasi-periodic price fluctuations on attracting invariant circle high values of β: strange attractors β = : convergence to (locally unstable) fundamental steady state Theoretical Question: Is the system close to homoclinic orbits and chaos, when the intensity of choice β is high? Cars Hommes (CeNDEF, UvA) Complex Systems CEF 2013, Vancouver 33 / 52
Figure: Bifurcation diagram (top panel) and largest Lyapunov exponent plot (bottom Cars Hommes panel) (CeNDEF, foruva) 4-type model. Complex Systems CEF 2013, Vancouver 34 / 52 The model 4-type model: fundamentalists vs. trend and bias Bifurcation diagram and largest Lyapunov exponent plot 0.2 0.15 0.1 0.05 0 0.05 0.1 0.15 40 50 60 70 80 90 100
The model 4-type model: fundamentalists vs. trend and bias Chaotic and noisy chaotic time series, and strange attractor (a) (b) 0.75 1 0.5 0.5 0.25 0 0-0.25-0.5-0.5-1 -0.75 100 200 300 400 500 (c) 100 0.5 300 400 500 (d) 0.1 0.75 200 0.05 0.25 0 0-0.25-0.05-0.5-0.75-0.75-0.5-0.25 0 0.25 0.5 0.75-0.05 0 0.05 0.1 Figure: Chaotic (top left) and noisy chaotic (top right) time series of asset prices in adaptive belief system with four trader types. Strange attractor (bottom left) and enlargement of strange attractor (bottom right). Cars Hommes (CeNDEF, UvA) Complex Systems CEF 2013, Vancouver 35 / 52
Figure: Forecasting errors for nearest neighbor method applied to chaotic returns series (lowest graph) as well as noisy chaotic returns series, for time horizons 1 20 and for different noise levels, in ABS with four trader types. Cars Hommes (CeNDEF, UvA) Complex Systems CEF 2013, Vancouver 36 / 52 Forecasting errors The model 4-type model: fundamentalists vs. trend and bias 1 0.8 Prediction Error 0.6 0.4 0.2 chaos 5% noise 10% 30% 40% 0 0 2 4 6 8 10 12 14 16 18 20 Prediction Horizon
Empirical Validation Empirical Validation: PE and PD ratios S&P500, 1871 2003 50 PY Ratio Fund. Ratio 100 90 PY Ratio Fund. Ratio 40 80 70 30 60 50 20 40 30 10 20 10 1880 1900 1920 1940 1960 1980 2000 1880 1900 1920 1940 1960 1980 2000 Price-to-Earnings Price-to-Dividends Cars Hommes (CeNDEF, UvA) Complex Systems CEF 2013, Vancouver 37 / 52
Empirical Validation S&P 500, 1950-2012 + benchmark fundamental p t = 1+g 1+r y t (g constant growth rate dividends) Cars Hommes (CeNDEF, UvA) Complex Systems CEF 2013, Vancouver 38 / 52
Empirical Validation BH-Model and risk premium market clearing (with zero net supply) H h=1 n h,t E h,t [p t+1 + y t+1 ] (1 + r)p t av t [R t+1 ] = 0 equilibrium pricing equation p t = 1 1 + r H n h,t E h,t (p t+1 +y t+1 ), or r = h=1 H h=1 n h,t E h,t [p t+1 + y t+1 p t ] p t, estimation: required rate of return r = risk free interest rate + risk premium Cars Hommes (CeNDEF, UvA) Complex Systems CEF 2013, Vancouver 39 / 52
Empirical Validation Stochastic cash flow with constant growth rate log y t Gaussian random walk with drift: log y t+1 = µ + log y t + υ t+1, υ t+1 i.i.d. N(0, σ 2 υ), This implies y t+1 y t = e µ+υ t+1 = e µ+ 1 2 σ2 υe υ t+1 1 2 σ2 υ = (1 + g)ε t+1, where g = e µ+ 1 2 σ2 υ 1 and ε t+1 = e υ t+1 1 2 σ2 υ, which implies E t (ε t+1 ) = 1. all types correct beliefs about cash flows E h,t [y t+1 ] = E t [y t+1 ] = (1 + g)y t E t [ε t+1 ] = (1 + g)y t. Cars Hommes (CeNDEF, UvA) Complex Systems CEF 2013, Vancouver 40 / 52
Empirical Validation RE fundamental benchmark for constant growth cash flow p t = 1 1 + r E t(p t+1 + y t+1 ) "no bubble" condition implies unique bounded RE fundamental price p t : (discounted sum of expected future dividends) pt = E t(y t+1 ) + E t(y t+2 ) 1 + r (1 + r) 2 +... = 1 + g 1 + r y (1 + g)2 t + (1 + r) 2 y t +... = 1 + r r g y t. fundamental price to cash flow ratio δ t = p t y t = 1 + r r g = m Cars Hommes (CeNDEF, UvA) Complex Systems CEF 2013, Vancouver 41 / 52
Empirical Validation Reformulation BH-model in terms of price to cash flows equilibrium pricing equation p t = 1 1 + r H n h,t E h,t (p t+1 + y t+1 ) h=1 in terms of price-to-cash flows δ t = p t /y t δ t = 1 H R {1 + n h,t E h,t [δ t+1 ]}, h=1 R = 1 + r 1 + g Cars Hommes (CeNDEF, UvA) Complex Systems CEF 2013, Vancouver 42 / 52
Empirical Validation Heterogeneous Beliefs in terms of price-to-cash flows deviation price-to-cash flow from fundamental x t = δ t m = δ t 1 + g r g belief of type h about price-to-cash flow: E ht [δ t+1 ] = E t [δ t ] + f h (x t 1,..., x t L ) = m + f h (x t 1,..., x t L ) pricing equation in deviations from fundamental H R x t = n ht f h (x t 1,..., x t L ), h=1 R = 1 + r 1 + g Cars Hommes (CeNDEF, UvA) Complex Systems CEF 2013, Vancouver 43 / 52
Empirical Validation Evolutionary Fitness Measure realized net profits in period t U ht = π ht = R t z h,t 1 = (p t + y t Rp t 1 ) E h,t 1[p t + y t Rp t 1 ] av t 1 [p t + y t Rp t 1 ] Assume (in analogy with BH) V t 1 [p t + y t Rp t 1 ] = V t 1 [p t + y t Rp t 1] = y 2 t 1η 2 fitness in deviations from fundamental U ht = π ht = (1 + g)2 aη 2 (x t R x t 1 )(E h,t 1 [x t R x t 1 ]) Cars Hommes (CeNDEF, UvA) Complex Systems CEF 2013, Vancouver 44 / 52
Empirical Validation Two-types: Fundamentalists versus trend Two trader types, with forecasting rules f 1t = φ 1 x t 1, 0 φ 1 < 1 fundamentalists f 2t = φ 2 x t 1, φ 2 > 1, trend extrapolators Cars Hommes (CeNDEF, UvA) Complex Systems CEF 2013, Vancouver 45 / 52
Empirical Validation Fractions of the Two Types fractions of belief types are updated in each period according to discrete choice model (BH 1997,1998) n h,t = exp[βπ h,t 1 ] Hk=1 exp[βπ k,t 1 ] = 1 1 + k h exp[ β πh,k t 1 ], where β > 0 is intensity of choice and π h,k t 1 = π h,t 1 π k,t 1 difference in realized profits types h and k In 2-type case, fraction of type 1: n t = 1 1 + exp { β [(φ 1 φ 2 )x t 3 (x t 1 R x t 2 )]} Cars Hommes (CeNDEF, UvA) Complex Systems CEF 2013, Vancouver 46 / 52
Empirical Validation Estimation of two type model in deviations from fundamental; synchronous updating; Boswijk et al., JEDC 2007 R x t = n t φ 1 x t 1 + (1 n t )φ 2 x t 1 + ɛ t R = 1 + r 1 + g 1.074 φ 1 = 0.762: fundamentalists, mean reversion φ 2 = 1.135 trend extrapolators β 10 φ t = n tφ 1 + (1 n t )φ 2 R market sentiment φ t < 1: mean reversion; φ t > 1: explosive, trend following Cars Hommes (CeNDEF, UvA) Complex Systems CEF 2013, Vancouver 47 / 52
Empirical Validation Fraction Fundamentalists & Market Sentiment Explanation: dot com bubble triggered by economic fundamentals and strongly amplified by trend following behavior Cars Hommes (CeNDEF, UvA) Complex Systems CEF 2013, Vancouver 48 / 52
Empirical Validation Average Response to Fundamental shock (2000 runs) 20 19 18 17 16 15 0 5 10 15 20 25 30 short term overreaction and long term mean reversion Cars Hommes (CeNDEF, UvA) Complex Systems CEF 2013, Vancouver 49 / 52
Empirical Validation Financial Crisis Extreme Event in Linear RE Model Quantiles of 2000 simulated predictions of the PE-ratio in deviations from fundamental Cars Hommes (CeNDEF, UvA) Complex Systems CEF 2013, Vancouver 50 / 52
Empirical Validation Financial Crisis not Extreme in Nonlinear Switching Model Quantiles of 2000 simulated predictions of the PE-ratio in deviations from fundamental Cars Hommes (CeNDEF, UvA) Complex Systems CEF 2013, Vancouver 51 / 52
Empirical Validation Conclusions Asset Pricing Model with Heterogeneous Beliefs rational route to randomness as β increases, with temporary bubbles and crashes weak correlation of beliefs: stable price behavior; strong coordination of beliefs: unstable price dynamics counter-examples to Friedman hypothesis: fundamentalists do not drive out irrational technical analysts, driven by short run profits empirical validation: explanation of bubbles and crashes consistent with learning-to-forecast laboratory experiments Cars Hommes (CeNDEF, UvA) Complex Systems CEF 2013, Vancouver 52 / 52