Chaos and Complexity in Economics€¦ · I markets as complex adaptive, nonlinear evolutionary...

Chaos and Complexity in Economics

Cars Hommes

CeNDEF, School of EconomicsUniversity of Amsterdam

18e Nationale Wiskunde DagenNoordwijkerhout, 3-4 Februari 2012

Cars Hommes University of Amsterdam


Main Themes

I nonlinear dynamics and chaos

I financial market model with heterogeneous traders

I estimation of nonlinear switching model

I laboratory experiments and nonlinear models



Why is economics so difficult?

Isaac Newton, about 1700“I can predict the motion of heavenly bodies,but not the madness of crowds”

irrationality is difficult to predict and to model

expectations are difficult to predict and to model



The Traditional Rational ViewExpectations are model-consistent

Milton Friedman, 1953: Irrational traders will be driven out of themarket by rational traders, who will earn higher profits.

Robert Lucas, 1971: economic policy should be based on rationalexpectations models in macro-economics



alternative, complexity, agent-based modeling approachin Economics

I bounded rationality and behavioral economics versus perfectrationality (Simon (1957) versus Lucas (1971)

I heterogeneous agents versus representative agentI market psychology, herding behavior (Keynes (1936)) versus

rationalityI markets as complex adaptive, nonlinear evolutionary systems

versus representative agent modelI computational versus analytical approach



Chaos in Nonlinear Systemsquadratic example: xt+1 = 4xt(1− xt)

20 40 60 80 100

0

0.2

0.4

0.6

0.8

1

13

t

x t

20 40 60 80 100

0

0.2

0.4

0.6

0.8

1 13

t

x t

x0 = 0.1 x0 = 0.1001

sensitive dependence on initial conditions



Period-doubling Bifurcation Route to Chaosfor Quadratic Map xt+1 = λxt(1− xt)

3 3.2 3.4 3.6 3.8 4

0.2

0.4

0.6

0.8

1

λ∞

λ→

fλ(xt)↑

fλ(x)=λx(1−x) and x0=0.5



Lyapunov Exponentmeasuring sensitive dependence

Lyapunov exponent ≡average rate of expansion/contraction along orbit

λ(x0) ≡ limn→∞

1n

n−1∑i=0

ln(| f ′(f i(x0)) |),

I λ(x0) < 0: stable periodic behaviourI λ(x0) > 0: chaos and sensitive dependence



Chaos in Quadratic Map xt+1 = λxt(1− xt)

3.4 3.5 3.6 3.7 3.8 3.9 4

−0.4

−0.2

0

0.2

0.4

0.6

λ→

λ(x0)↑

(a) f(x)=λx(1−x), x0=0.4 and n=1000

3.4 3.5 3.6 3.7 3.8 3.9 40

0.2

0.4

0.6

0.8

1

λ→

fλ(xt)↑

(b) f(x)=λx(1−x) and x0=0.4



Financial Market Model

Investors can choose between risky asset and a risk free assetI R = 1 + r > 1: gross return on risk free assetI risky asset pays stochastic dividends

I yt: stochastic dividend process for risky assetI pt: price (ex div.) per share of risky asset

I price of risky asset determined by market clearing:

Rpt =H∑

h=1

nhtEht(pt+1 + yt+1)

I nht: fraction of agents of type h



Rational Expectations (RE) fundamental benchmark

Rpt = Et(pt+1 + yt+1)

common beliefs on future earnings and pricesunique bounded RE fundamental price p∗t :

p∗t =Et(yt+1)

R+

Et(yt+2)R2 + ....

For special case of IID dividends, with E(yt+1) = y

p∗ =y

R− 1=

yr

pricing equation in deviations xt = pt − p∗t from fundamental:

Rxt = Etxt+1



Behavioral asset pricing model (Adaptive Belief Systems)standard asset pricing model with heterogeneous beliefsone risky asset, one risk free asset

I price of risky asset determined by market clearingI beliefs about future prices given by simple, linear ruleI forecasting strategies updated according to discrete choice

model with realized profitsequilibrium price of risky asset

Rpt =H∑

h=1

nhtEht(pt+1 + yt+1)

(in deviations xt = pt − p∗t from RE-fundamental)

Rxt =H∑

h=1

nhtEhtxt+1



Fractions of Strategy Type hfractions of belief types are updated in each period:discrete choice model (BH 1997,1998) with asynchronousupdating:

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



Example with 4 belief types(zero costs; memory one lag)xe

h,t+1 = ghxt−1 + bh

g1 = 0 b1 = 0 fundamentalistsg2 = 1.1 b2 = 0.2 trend + upward biasg3 = 0.9 b3 = −0.2 trend + downward biasg4 = 1.21 b4 = 0 trend chaser

Rxt =4∑

h=1nh,t(ghxt−1 + bh)

nh,t+1 =e

β

aσ2 (ghxt−2+bh−Rxt−1)(xt−Rxt−1)

Zt, h = 1, 2, 3, 4



Chaos in Financial Market Model withFundamentalists versus Chartists

-0.75 -0.5 -0.25 0 0.25 0.5 0.75

-0.75

-0.5

-0.25

0

0.25

0.5

0.75

(c)

-0.05 0 0.05 0.1

-0.05

0

0.05

0.1(d)

100 200 300 400 500

-0.75

-0.5

-0.25

0

0.25

0.5

0.75

(a)

100 200 300 400 500

-1

-0.5

0

0.5

1

(b)



Chaos in Financial Market Model withFundamentalists versus Chartists

40 50 60 70 80 90 100−0.15

−0.1

−0.05

0

0.05

0.1

0.15

0.2



Model very sensitive to noise



Nearest Neighbor Forecasting

0 2 4 6 8 10 12 14 16 18 200

0.2

0.4

0.6

0.8

1

Pre

dict

ion

Err

or

Prediction Horizon

chaos

5% noise

10%

30%

40%



S&P 500, 1871-2003 + benchmark fundamental p∗t = 1+g1+r yt

(g constant growth rate dividends)

1880 1900 1920 1940 1960 1980 2000

4.0

4.5

5.0

5.5

6.0

6.5

7.0

7.5 S&P 500 Fund. Value

1880 1900 1920 1940 1960 1980 2000

10

20

30

40

50

60

70

80

90

100PY Ratio Fund. Ratio

log S&P 500 and fundamental PD-ratio and PD-fundamental



Estimation 2-type model

R∗xt = nt{0.762 xt−1}+ (1 − nt){1.135 xt−1}+ εt(0.056) (0.036) (1)

nt = {1 + exp[−10.29(−0.373xt−3)(xt−1 − R∗xt−2)]}−1

(6.94)(2)

R2 = 0.82, AIC = 3.18, AICAR(1) = 3.24, σε = 4.77, QLB(4) = 0.44

1880 1900 1920 1940 1960 1980 2000

0.0

0.2

0.4

0.6

0.8

1.0

nt

1880 1900 1920 1940 1960 1980 2000

0.7

0.8

0.9

1.0

1.1

φt

fraction fundamentalists average market sentimentCars Hommes University of Amsterdam


How to model bounded rationality?

wilderness of bounded rationality:"There is only one way you can be right, but there are many ways youcan be wrong"

I use laboratory experiments with human subjects to test abehavioral theory of heterogeneous expectations

I fit simple complexity model to laboratory dataI test simple complexity model on real economic/financial data



Example: Laboratory Experiments wit Human subjectspsychology – behavioral economics – computer science

Computer Screen Learning to Forecast Experiment



Rational Expectations Benchmark

If everybody predictsrationally fundamentalprice,then

pt = pf +εt

1 + r

40

45

50

55

60

65

70

0 10 20 30 40 50

Pri

ce

Rational expectations



Naive and Trend-following Expectations Benchmarks

40

45

50

55

60

65

70

0 10 20 30 40 50

Pri

ce

Naive expectations

40

45

50

55

60

65

70

0 10 20 30 40 50

Pri

ce

AR(2) expectations

which expectations theory is correct?



Prices in the Experiment with Humans

40

45

50

55

60

65

70

0 10 20 30 40 50

Pric

e

Group 2

fundamental price experimental price

40

45

50

55

60

65

70

0 10 20 30 40 50

Pric

e

Group 5


40

45

50

55

60

65

70

0 10 20 30 40 50

Pric

e

Group 1


40

45

50

55

60

65

70

0 10 20 30 40 50

Pric

e

Group 6


10 20 30 40 50 60 70 80 90

0 10 20 30 40 50

Pric

e

Group 4


40

45

50

55

60

65

70

0 10 20 30 40 50

Pric

eGroup 7




Individual Forecasts in Experiments with Humans

10 5020

40

60

80group 1

10 5020

40

60

80group 2

10 5020

40

60

80group 3

10 500

50

100group 4

10 5020

40

60

80group 5

10 5020

40

60

80group 6

10 5020

40

60

80group 7

10 500

50

100group 8

10 500

50

100group 9

10 500

50

100group 10



Heterogeneous Expectations Hypothesis

Heuristics Switching Model

I there are a few simple heuristicsI adaptive expectations ADAI trend extrapolating rule STR, WTRI anchor and adjustment rule LAA

I impact of heuristics changes over timeI agents gradually switch to heuristics that have performed better

in the recent past



Group 5 (Convergence)

Parameters: β = 0.4, η = 0.7, δ = 0.9

40

45

50

55

60

65

70

0 10 20 30 40 50

simulation experiment

0

0.2

0.4

0.6

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1

0 10 20 30 40 50

ADA WTR STR LAA



Group 6 (Constant Oscillations)

Parameters: β = 0.4, η = 0.7, δ = 0.9

40

45

50

55

60

65

70

0 10 20 30 40 50


0

0.2

0.4

0.6

0.8

1

0 10 20 30 40 50

ADA WTR STR LAA



Group 7 (Dampened Oscillations)

Parameters: β = 0.4, η = 0.7, δ = 0.9

40

45

50

55

60

65

70

0 10 20 30 40 50


0

0.2

0.4

0.6

0.8

1

0 10 20 30 40 50

ADA WTR STR LAA



Concluding Remarks

I chaotic financial market model mimics bubble and crashdynamics

I simple 2-type model with fundamentalists versus chartists fitsUS stock market data

I nonlinear heuristic switching model with path dependence fitsexperimental data

I theory of evolutionary selection of heterogeneous expectationsfits experimental data



Chaos and Complexity in Economics€¦ · I markets as complex adaptive, nonlinear evolutionary...

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