Evolution of Market Heuristics (An Explanation of an Asset-Pricing - - PowerPoint PPT Presentation

Learning Experiment Homogeneous Expectations Individual Predictions Model Other Experiments Conclusion

Evolution of Market Heuristics

(An Explanation of an Asset-Pricing Experiment)

Mikhail Anufriev Cars Hommes

CeNDEF, Faculty of Economics and Business University of Amsterdam

Seminar at the Paris School of Economics 22 June 2009

Mikhail Anufriev, Cars Hommes CeNDEF, University of Amsterdam Evolution of Market Heuristics

Learning Experiment Homogeneous Expectations Individual Predictions Model Other Experiments Conclusion

Question

How do people behave (form expectations and learn) in the expectations feedback system?

◮ expectations are shaped given the market history ◮ expectations affect the outcome (e.g. price)

Mikhail Anufriev, Cars Hommes CeNDEF, University of Amsterdam Evolution of Market Heuristics

Learning Experiment Homogeneous Expectations Individual Predictions Model Other Experiments Conclusion

Possible Answers

◮ Fully rational (Rational Expectations) ◮ Belief-based and econometric learning

◮ Jordan (GEB, 1991), Bray and Savin (E, 1986), Kalai and Lehrer

(E, 1993), Cheung and Friedman (GEB, 1997), ...

◮ Evans and Honkapohja (2001)

◮ Reinforcement learning

◮ Arthur (AER, 1991), Arthur (JEE, 1993), Roth and Erev (GEB,

1995), Erev and Roth (AER, 1998), Camerer and Ho (E, 1999)

Mikhail Anufriev, Cars Hommes CeNDEF, University of Amsterdam Evolution of Market Heuristics

Learning Experiment Homogeneous Expectations Individual Predictions Model Other Experiments Conclusion

Here we present...

◮ a descriptive model of “reinforcement” learning in a

non-game-theoretic setting with limited information about environment...

◮ ...explaining the results of a recent experiment where

◮ subject predicted future price ◮ price process depended on the expectations

in forecasting agents rely on simple heuristics in learning agents update their “active” heuristics on the basis of heuristics’ performances

Mikhail Anufriev, Cars Hommes CeNDEF, University of Amsterdam Evolution of Market Heuristics

Learning Experiment Homogeneous Expectations Individual Predictions Model Other Experiments Conclusion

Experiment

Hommes, Sonnemans, Tuinstra, van de Velden (2005, RFS) participants forecast the next realization of an endogenous price process and are rewarded for a precision of their forecasts

◮ two assets in a market

◮ riskless with interest r ◮ risky with price pt and dividend yt whose mean is ¯

y

◮ price pt is derived from equilibrium between demand and supply ◮ positive relation between their forecast and demand ◮ in the beginning of time t every participant h knows

the past prices (up to pt−1), own past forecasts (up to pt,h) and

• wn earnings (up to et−1,h)

Mikhail Anufriev, Cars Hommes CeNDEF, University of Amsterdam Evolution of Market Heuristics

Learning Experiment Homogeneous Expectations Individual Predictions Model Other Experiments Conclusion

Experiment

Hommes, Sonnemans, Tuinstra, van de Velden (2005, RFS) participants forecast the next realization of an endogenous price process and are rewarded for a precision of their forecasts

◮ two assets in a market

◮ riskless with interest r ◮ risky with price pt and dividend yt whose mean is ¯

y

◮ price pt is derived from equilibrium between demand and supply ◮ positive relation between their forecast and demand ◮ in the beginning of time t every participant h knows

the past prices (up to pt−1), own past forecasts (up to pt,h) and

• wn earnings (up to et−1,h)

Mikhail Anufriev, Cars Hommes CeNDEF, University of Amsterdam Evolution of Market Heuristics

Learning Experiment Homogeneous Expectations Individual Predictions Model Other Experiments Conclusion

Formal Presentation of the Experiment

◮ fundamental price of the risky asset pf = ¯ y r = 60 ◮ 6 participants in the beginning of period t submit forecasts pe t+1,h ◮ fraction nt of “robot” traders predict pf

nt = 1 − exp

• −

1 200|pt−1 − pf |

• ◮ realized price depends on the next-period price forecasts

pt =

1 1+r

• (1 − nt)

pe

t+1,1+···+pe t+1,6

6

+ nt pf + ¯ y + εt

• with noise εt ∼ N(0, 0.25) for periods t = 0, . . . , 50

◮ subjects are paid according to the precision of their forecast

et,h = max

• 1 − 1

49(pt − pe t,h)2, 0

• × 1

2euro

Mikhail Anufriev, Cars Hommes CeNDEF, University of Amsterdam Evolution of Market Heuristics

Learning Experiment Homogeneous Expectations Individual Predictions Model Other Experiments Conclusion

Formal Presentation of the Experiment

◮ fundamental price of the risky asset pf = ¯ y r = 60 ◮ 6 participants in the beginning of period t submit forecasts pe t+1,h ◮ fraction nt of “robot” traders predict pf

nt = 1 − exp

• −

1 200|pt−1 − pf |

• ◮ realized price depends on the next-period price forecasts

pt =

1 1+r

• (1 − nt)

pe

t+1,1+···+pe t+1,6

6

+ nt pf + ¯ y + εt

• with noise εt ∼ N(0, 0.25) for periods t = 0, . . . , 50

◮ subjects are paid according to the precision of their forecast

et,h = max

• 1 − 1

49(pt − pe t,h)2, 0

• × 1

2euro

Mikhail Anufriev, Cars Hommes CeNDEF, University of Amsterdam Evolution of Market Heuristics

Learning Experiment Homogeneous Expectations Individual Predictions Model Other Experiments Conclusion

Formal Presentation of the Experiment

◮ fundamental price of the risky asset pf = ¯ y r = 60 ◮ 6 participants in the beginning of period t submit forecasts pe t+1,h ◮ fraction nt of “robot” traders predict pf

nt = 1 − exp

• −

1 200|pt−1 − pf |

• ◮ realized price depends on the next-period price forecasts

pt =

1 1+r

• (1 − nt)

pe

t+1,1+···+pe t+1,6

6

+ nt pf + ¯ y + εt

• with noise εt ∼ N(0, 0.25) for periods t = 0, . . . , 50

◮ subjects are paid according to the precision of their forecast

et,h = max

• 1 − 1

49(pt − pe t,h)2, 0

• × 1

2euro

Mikhail Anufriev, Cars Hommes CeNDEF, University of Amsterdam Evolution of Market Heuristics

Learning Experiment Homogeneous Expectations Individual Predictions Model Other Experiments Conclusion

Formal Presentation of the Experiment

◮ fundamental price of the risky asset pf = ¯ y r = 60 ◮ 6 participants in the beginning of period t submit forecasts pe t+1,h ◮ fraction nt of “robot” traders predict pf

nt = 1 − exp

• −

1 200|pt−1 − pf |

• ◮ realized price depends on the next-period price forecasts

pt =

1 1+r

• (1 − nt)

pe

t+1,1+···+pe t+1,6

6

+ nt pf + ¯ y + εt

• with noise εt ∼ N(0, 0.25) for periods t = 0, . . . , 50

◮ subjects are paid according to the precision of their forecast

et,h = max

• 1 − 1

49(pt − pe t,h)2, 0

• × 1

2euro

Mikhail Anufriev, Cars Hommes CeNDEF, University of Amsterdam Evolution of Market Heuristics

Learning Experiment Homogeneous Expectations Individual Predictions Model Other Experiments Conclusion

Information

Subjects know

◮ the environment (interest rate r and mean dividend ¯

y)

◮ some qualitative information (positive relation between the

forecasts and demand, higher demand should imply higher price)

◮ past prices and own forecasts

Subjects do not know

◮ exact equilibrium equation ◮ exact demand schedule of themselves and others ◮ number and identity of other participants

Mikhail Anufriev, Cars Hommes CeNDEF, University of Amsterdam Evolution of Market Heuristics

Learning Experiment Homogeneous Expectations Individual Predictions Model Other Experiments Conclusion

Rational Benchmark

If everybody predicts fundamental price, then pt = pf + εt 1 + r

40 45 50 55 60 65 70 10 20 30 40 50 Price Time fundamental price price under rational expectations

• 1
• 0.5

0.5 1 10 20 30 40 50

Mikhail Anufriev, Cars Hommes CeNDEF, University of Amsterdam Evolution of Market Heuristics

Learning Experiment Homogeneous Expectations Individual Predictions Model Other Experiments Conclusion

Price in the Experiment

40 45 50 55 60 65 70 10 20 30 40 50 Price Group 2 fundamental price experimental price 40 45 50 55 60 65 70 10 20 30 40 50 Price Group 5 fundamental price experimental price 40 45 50 55 60 65 70 10 20 30 40 50 Price Group 1 fundamental price experimental price 40 45 50 55 60 65 70 10 20 30 40 50 Price Group 6 fundamental price experimental price 10 20 30 40 50 60 70 80 90 10 20 30 40 50 Price Group 4 fundamental price experimental price 40 45 50 55 60 65 70 10 20 30 40 50 Price Group 7 fundamental price experimental price

Mikhail Anufriev, Cars Hommes CeNDEF, University of Amsterdam Evolution of Market Heuristics

Learning Experiment Homogeneous Expectations Individual Predictions Model Other Experiments Conclusion

2 Groups with (Almost) Monotonic Convergence

35 45 55 65 10 20 30 40 50 Predictions 45 55 65 Price

Group 2

• 2

2 35 45 55 65 10 20 30 40 50 Predictions 45 55 65 Price

Group 5

• 2

2

Mikhail Anufriev, Cars Hommes CeNDEF, University of Amsterdam Evolution of Market Heuristics

Learning Experiment Homogeneous Expectations Individual Predictions Model Other Experiments Conclusion

2 Groups with Constant Oscillations

35 45 55 65 10 20 30 40 50 Predictions 45 55 65 Price

Group 1

• 5

5 35 45 55 65 10 20 30 40 50 Predictions 45 55 65 Price

Group 6

• 5

5

Mikhail Anufriev, Cars Hommes CeNDEF, University of Amsterdam Evolution of Market Heuristics

Learning Experiment Homogeneous Expectations Individual Predictions Model Other Experiments Conclusion

2 Groups with Damping Oscillations

10 30 50 70 90 10 20 30 40 50 Predictions 10 30 50 70 90 Price

Group 4

• 30

30 45 55 65 75 10 20 30 40 50 Predictions 45 55 65 75 Price

Group 7

• 10

10

Mikhail Anufriev, Cars Hommes CeNDEF, University of Amsterdam Evolution of Market Heuristics

Learning Experiment Homogeneous Expectations Individual Predictions Model Other Experiments Conclusion

Estimation of Individual Predictions

...for the past 40 periods

◮ in converging groups agents use adaptive expectations

pe

t+1 = w pt−1 + (1 − w) pe t ◮ often agents used simple linear rules

pe

t+1 = α + β1 pt−1 + β2 pt−2

in particular trend-extrapolating rules pe

t+1 = pt−1 + γ (pt−1 − pt−2)

Mikhail Anufriev, Cars Hommes CeNDEF, University of Amsterdam Evolution of Market Heuristics

Learning Experiment Homogeneous Expectations Individual Predictions Model Other Experiments Conclusion

Summary of the Results of the Experiment

Results are inconsistent with fundamental forecasting. One would like to explain:

◮ three qualitatively different patters

◮ (almost) monotonic convergence ◮ constant oscillations ◮ damping oscillations

◮ coordination of agents in their predictions ◮ agents “used” simple prediction rules with behavioral

interpretation

Mikhail Anufriev, Cars Hommes CeNDEF, University of Amsterdam Evolution of Market Heuristics

Learning Experiment Homogeneous Expectations Individual Predictions Model Other Experiments Conclusion

Dynamics under Homogeneous Expectations

◮ all the agents use the same rule (adaptive or linear)

           pe

t+1 = f(pt−1, pt−2, pe t )

nt = 1 − exp

• −

1 200

• pt−1 − pf
• pt =

1 1 + r

• (1 − nt)pe

t+1 + nt pf + ¯

y + εt

• ◮ εt = 0: deterministic skeleton

◮ simulations with εt from the experiment

Mikhail Anufriev, Cars Hommes CeNDEF, University of Amsterdam Evolution of Market Heuristics

Learning Experiment Homogeneous Expectations Individual Predictions Model Other Experiments Conclusion

Dynamics under Homogeneous Expectations

◮ all the agents use the same rule (adaptive or linear)

           pe

t+1 = f(pt−1, pt−2, pe t )

nt = 1 − exp

• −

1 200

• pt−1 − pf
• pt =

1 1 + r

• (1 − nt)pe

t+1 + nt pf + ¯

y + εt

• ◮ εt = 0: deterministic skeleton

◮ simulations with εt from the experiment

Mikhail Anufriev, Cars Hommes CeNDEF, University of Amsterdam Evolution of Market Heuristics

Learning Experiment Homogeneous Expectations Individual Predictions Model Other Experiments Conclusion

Adaptive Expectations: pe

t+1 = w pt−1 + (1 − w) pe t

Dynamics globally converge to fundamental price.

45 50 55 60 65 70 10 20 30 40 50 Price under adaptive heuristic w = 0.25 w = 0.65

Mikhail Anufriev, Cars Hommes CeNDEF, University of Amsterdam Evolution of Market Heuristics

Learning Experiment Homogeneous Expectations Individual Predictions Model Other Experiments Conclusion

Extrapolative Expectations: pe

t+1 = α + β1 pt−1 + β2 pt−2

Special cases:

◮ trend-following heuristic

pe

t+1 = pt−1 + γ (pt−1 − pt−2) ◮ anchoring and adjustment heuristic

pe

t+1 = 0.5 (pf + pt−1) + (pt−1 − pt−2)

Definition

The extrapolative rule is called consistent in the steady-state with price p∗, if it predicts p∗ in this steady-state.

Mikhail Anufriev, Cars Hommes CeNDEF, University of Amsterdam Evolution of Market Heuristics

Learning Experiment Homogeneous Expectations Individual Predictions Model Other Experiments Conclusion

Extrapolative Expectations: pe

t+1 = α + β1 pt−1 + β2 pt−2

Special cases:

◮ trend-following heuristic

pe

t+1 = pt−1 + γ (pt−1 − pt−2) ◮ anchoring and adjustment heuristic

pe

t+1 = 0.5 (pf + pt−1) + (pt−1 − pt−2)

Definition

The extrapolative rule is called consistent in the steady-state with price p∗, if it predicts p∗ in this steady-state.

Mikhail Anufriev, Cars Hommes CeNDEF, University of Amsterdam Evolution of Market Heuristics

Learning Experiment Homogeneous Expectations Individual Predictions Model Other Experiments Conclusion

Extrapolative Expectations: pe

t+1 = α + β1 pt−1 + β2 pt−2

◮ The unique steady-state where the rule is consistent has p∗ = pf ◮ For the trend-following heuristic this is the only steady-state ◮ Stability conditions

• 1.5
• 1
• 0.5

0.5 1 1.5

• 2
• 1

1 2 β2 β1 Neimark-Sacker p i t c h

• f
• r

k period-doubling γ = 0.4 γ = 1.3 AA

Mikhail Anufriev, Cars Hommes CeNDEF, University of Amsterdam Evolution of Market Heuristics

Learning Experiment Homogeneous Expectations Individual Predictions Model Other Experiments Conclusion

Weak-Trend Extrapolation: pe

t+1 = pt−1 + γ (pt−1 − pt−2)

Dynamics converge to fundamental price.

45 50 55 60 65 70 10 20 30 40 50 Price under weak trend following heuristic γ = 0.4 γ = 0.99

Mikhail Anufriev, Cars Hommes CeNDEF, University of Amsterdam Evolution of Market Heuristics

Learning Experiment Homogeneous Expectations Individual Predictions Model Other Experiments Conclusion

Strong-Trend Extrapolation: pe

t+1 = pt−1 + γ (pt−1 − pt−2)

Dynamics diverge from fundamental price...

20 40 60 80 100 10 20 30 40 50 Price under strong trend following heuristic γ = 1.1 γ = 1.3

Mikhail Anufriev, Cars Hommes CeNDEF, University of Amsterdam Evolution of Market Heuristics

Learning Experiment Homogeneous Expectations Individual Predictions Model Other Experiments Conclusion

Strong-Trend Extrapolation: pe

t+1 = pt−1 + γ (pt−1 − pt−2)

...and settles on the quasi-periodic attractor.

20 40 60 80 100 20 40 60 80 100 Attractor under strong trend following heuristic γ = 1.1 γ = 1.3

200 points after 500 transitory steps

Mikhail Anufriev, Cars Hommes CeNDEF, University of Amsterdam Evolution of Market Heuristics

Learning Experiment Homogeneous Expectations Individual Predictions Model Other Experiments Conclusion

Anchoring and Adjustment: pe

t+1 = pf +pt−1 2

+ (pt−1 − pt−2)

45 50 55 60 65 70 10 20 30 40 50 Price under anchoring and adjustment heuristic fixed anchor learned anchor

with learning of anchor: pe

t+1 = pav

t−1+pt−1

2

+ (pt−1 − pt−2)

Mikhail Anufriev, Cars Hommes CeNDEF, University of Amsterdam Evolution of Market Heuristics

Learning Experiment Homogeneous Expectations Individual Predictions Model Other Experiments Conclusion

Model with Homogeneous Expectations

◮ pattern of monotonic convergence can be easily reproduced

adaptive rule, weak trend extrapolation

◮ pattern of constant oscillations can be reproduced

anchoring and adjustment rule without learning

◮ pattern of damping oscillations is reproduced (very imperfectly)

strong-trend extrapolations

Mikhail Anufriev, Cars Hommes CeNDEF, University of Amsterdam Evolution of Market Heuristics

Learning Experiment Homogeneous Expectations Individual Predictions Model Other Experiments Conclusion

Dynamics for Individual Rules: Converging Groups

• 1.5
• 1
• 0.5

0.5 1 1.5

• 2
• 1

1 2 Stability region and group 2

• 1.5
• 1
• 0.5

0.5 1 1.5

• 2
• 1

1 2 Stability region and group 5

Mikhail Anufriev, Cars Hommes CeNDEF, University of Amsterdam Evolution of Market Heuristics

Learning Experiment Homogeneous Expectations Individual Predictions Model Other Experiments Conclusion

Dynamics for Individual Rules: Oscillating Groups

• 1.5
• 1
• 0.5

0.5 1 1.5

• 2
• 1

1 2 Stability region and group 1

• 1.5
• 1
• 0.5

0.5 1 1.5

• 2
• 1

1 2 Stability region and group 6

Mikhail Anufriev, Cars Hommes CeNDEF, University of Amsterdam Evolution of Market Heuristics

Learning Experiment Homogeneous Expectations Individual Predictions Model Other Experiments Conclusion

Dynamics for Individual Rules: Damping Groups

• 1.5
• 1
• 0.5

0.5 1 1.5

• 2
• 1

1 2 Stability region and group 4

• 1.5
• 1
• 0.5

0.5 1 1.5

• 2
• 1

1 2 Stability region and group 7

Mikhail Anufriev, Cars Hommes CeNDEF, University of Amsterdam Evolution of Market Heuristics

Learning Experiment Homogeneous Expectations Individual Predictions Model Other Experiments Conclusion

Evolution of Individual Predictions

48 50 52 54 56 58 60 10 20 30 40 50 Group 2, participants 1 and 5 prediction 5 prediction 1 price

Naive Rule: pe

t+1 = pt−1

Adaptive Rule: pe

t+1 = 0.25 pt−1 + 0.75 pe t

Mikhail Anufriev, Cars Hommes CeNDEF, University of Amsterdam Evolution of Market Heuristics

Learning Experiment Homogeneous Expectations Individual Predictions Model Other Experiments Conclusion

Evolution of Individual Predictions

52 54 56 58 60 62 64 66 10 20 30 40 50 Group 6, participant 1 prediction price

Weak trend extrapolation: pe

t+1 = pt−1 + γ (pt−1 − pt−2) , γ ≃ 0.4

Mikhail Anufriev, Cars Hommes CeNDEF, University of Amsterdam Evolution of Market Heuristics

Learning Experiment Homogeneous Expectations Individual Predictions Model Other Experiments Conclusion

Evolution of Individual Predictions

48 50 52 54 56 58 60 62 64 66 10 20 30 40 50 Time Group 1, participant 3 prediction price

Anchoring adjustment rule: pe

t+1 = 0.5(pf + pt−1) + (pt−1 − pt−2)

Mikhail Anufriev, Cars Hommes CeNDEF, University of Amsterdam Evolution of Market Heuristics

Learning Experiment Homogeneous Expectations Individual Predictions Model Other Experiments Conclusion

Evolution of Individual Predictions

40 45 50 55 60 65 70 75 10 20 30 40 50 Time Group 7, participant 3 prediction price

Strong trend extrapolation: pe

t+1 = pt−1 + γ (pt−1 − pt−2) , γ ≃ 1.3

Mikhail Anufriev, Cars Hommes CeNDEF, University of Amsterdam Evolution of Market Heuristics

Learning Experiment Homogeneous Expectations Individual Predictions Model Other Experiments Conclusion

Summary

◮ participants tend to base their predictions on past observations,

following simple prediction rules (heuristics)

◮ learning of people has a form of switching from one heuristic to

another

◮ in every group some heterogeneity among participants remains

despite relatively close predictions

◮ it is some combination of forecasting rules which leads to

different outcomes

Mikhail Anufriev, Cars Hommes CeNDEF, University of Amsterdam Evolution of Market Heuristics

Learning Experiment Homogeneous Expectations Individual Predictions Model Other Experiments Conclusion

Informal Description of the Model

◮ there exist a number of simple heuristics (rules mapping the

information to the price prediction)

◮ heuristics are used by agents unevenly, so that every heuristic

has own impact on price determination

◮ agents evaluate the performances of all heuristics, so that

impacts are evolving

◮ agents tend to switch on the more successful heuristics ◮ initialization:

◮ prices in the first periods ◮ initial impacts Mikhail Anufriev, Cars Hommes CeNDEF, University of Amsterdam Evolution of Market Heuristics

Learning Experiment Homogeneous Expectations Individual Predictions Model Other Experiments Conclusion

Informal Description of the Model

◮ there exist a number of simple heuristics (rules mapping the

information to the price prediction)

◮ heuristics are used by agents unevenly, so that every heuristic

has own impact on price determination

◮ agents evaluate the performances of all heuristics, so that

impacts are evolving

◮ agents tend to switch on the more successful heuristics ◮ initialization:

◮ prices in the first periods ◮ initial impacts Mikhail Anufriev, Cars Hommes CeNDEF, University of Amsterdam Evolution of Market Heuristics

Learning Experiment Homogeneous Expectations Individual Predictions Model Other Experiments Conclusion

Informal Description of the Model

◮ there exist a number of simple heuristics (rules mapping the

information to the price prediction)

◮ heuristics are used by agents unevenly, so that every heuristic

has own impact on price determination

◮ agents evaluate the performances of all heuristics, so that

impacts are evolving

◮ agents tend to switch on the more successful heuristics ◮ initialization:

◮ prices in the first periods ◮ initial impacts Mikhail Anufriev, Cars Hommes CeNDEF, University of Amsterdam Evolution of Market Heuristics

Learning Experiment Homogeneous Expectations Individual Predictions Model Other Experiments Conclusion

Four forecasting heuristics

◮ adaptive rule

ADA pe

1,t+1 = 0.65 pt−1 + 0.35 pe 1,t ◮ weak trend-following rule

WTR pe

2,t+1 = pt−1 + 0.4 (pt−1 − pt−2) ◮ strong trend-following rule

STR pe

3,t+1 = pt−1 + 1.3 (pt−1 − pt−2) ◮ anchoring and adjustment heuristics with learnable anchor

LAA pe

4,t+1 = 0.5 pav t−1 + 0.5 pt−1 + (pt−1 − pt−2)

Mikhail Anufriev, Cars Hommes CeNDEF, University of Amsterdam Evolution of Market Heuristics

Learning Experiment Homogeneous Expectations Individual Predictions Model Other Experiments Conclusion

Stability of four heuristics

• 1.5
• 1
• 0.5

0.5 1 1.5

• 2
• 1

1 2 β2 β1 Neimark-Sacker p i t c h

• f
• r

k period-doubling γ = 0.4 γ = 1.3 AA

Mikhail Anufriev, Cars Hommes CeNDEF, University of Amsterdam Evolution of Market Heuristics

Learning Experiment Homogeneous Expectations Individual Predictions Model Other Experiments Conclusion

Price dynamics

◮ price dynamics

pt = 1 1 + r

• n1,tpe

1,t+1 + n2,tpe 2,t+1 + n3,tpe 3,t+1 + n4,tpe 4,t+1

• ×

× (1 − nt) + pf nt + ¯ y + εt

• ◮ fraction nt of robot traders is evolving as

nt = 1 − exp

• −

1 200|pt−1 − pf |

• ◮ impacts of heuristics ni,t are evolving as in discrete choice model

with asynchronous updating

Mikhail Anufriev, Cars Hommes CeNDEF, University of Amsterdam Evolution of Market Heuristics

Learning Experiment Homogeneous Expectations Individual Predictions Model Other Experiments Conclusion

Evolutionary switching

◮ performance measure of heuristic i is

Ui,t−1 = −

• pt−1 − pe

i,t−1

2 + η Ui,t−2 parameter η ∈ [0, 1] – the strength of the agents’ memory

◮ discrete choice model with asynchronous updating

ni,t = δ ni,t−1 + (1 − δ) exp(β Ui,t−1) 4

i=1 exp(β Ui,t−1)

parameter δ ∈ [0, 1] – the inertia of the traders parameter β ≥ 0 – the intensity of choice

Mikhail Anufriev, Cars Hommes CeNDEF, University of Amsterdam Evolution of Market Heuristics

Learning Experiment Homogeneous Expectations Individual Predictions Model Other Experiments Conclusion

Deterministic path

◮ Learning parameters are fixed: β = 0.4, η = 0.7, δ = 0.9 ◮ Model is initialized with

◮ two initial price p0 and p1 ◮ initial impacts of different heuristics n1,t, n2,t, n3,t, n4,t

Experiment Simulations Heuristics’ Initial Impacts p0 p1 p0 p1 ADA WTR STR LAA Group 2 48.94 51.21 49 50.5 0.25 0.35 0.15 0.25 Group 5 53.78 53.61 54 53.5 0.25 0.35 0.15 0.25 Group 1 53.05 56.45 51 54 0.15 0.35 0.35 0.15 Group 6 56.54 58.38 56 58 0.1 0.3 0.4 0.2 Group 4 43.72 47.33 42 47 0.1 0.9 Group 7 44.81 49.71 44 48 0.17 0.66 0.17

◮ Model is simulated for 49 periods with the same noise process as

in the experiment

Mikhail Anufriev, Cars Hommes CeNDEF, University of Amsterdam Evolution of Market Heuristics

Learning Experiment Homogeneous Expectations Individual Predictions Model Other Experiments Conclusion

Group 5 (Convergence)

Parameters: β = 0.4, η = 0.7, δ = 0.9

35 45 55 65 10 20 30 40 50 Predictions 45 55 65 Price

Group 5

• 2

2

35 45 55 65 10 20 30 40 50 Predictions ADA WTR STR LAA 45 55 65 Price

Group 5

simulation experiment

• 2

2

Mikhail Anufriev, Cars Hommes CeNDEF, University of Amsterdam Evolution of Market Heuristics

Learning Experiment Homogeneous Expectations Individual Predictions Model Other Experiments Conclusion

Group 1 (Constant Oscillations)

Parameters: β = 0.4, η = 0.7, δ = 0.9

35 45 55 65 10 20 30 40 50 Predictions 45 55 65 Price

Group 1

• 5

5

35 45 55 65 10 20 30 40 50 Predictions ADA WTR STR LAA 45 55 65 Price

Group 1

simulation experiment

• 5

5

Mikhail Anufriev, Cars Hommes CeNDEF, University of Amsterdam Evolution of Market Heuristics

Learning Experiment Homogeneous Expectations Individual Predictions Model Other Experiments Conclusion

Group 7 (Damping Oscillations)

Parameters: β = 0.4, η = 0.7, δ = 0.9

45 55 65 75 10 20 30 40 50 Predictions 45 55 65 75 Price

Group 7

• 10

10

45 55 65 75 10 20 30 40 50 Predictions ADA WTR STR LAA 45 55 65 75 Price

Group 7

simulation experiment

• 10

10

Mikhail Anufriev, Cars Hommes CeNDEF, University of Amsterdam Evolution of Market Heuristics

Learning Experiment Homogeneous Expectations Individual Predictions Model Other Experiments Conclusion

Comparison with Homogeneous Expectations: MSE

Fit of the experiment with parameters β = 0.4, η = 0.7, δ = 0.9

Specification Group 2 Group 5 Group 1 Group 6 Group 4 Group 7 Fundamental Prediction 18.037 11.797 15.226 8.959 291.376 22.047 ADA – exp prices 0.841 0.200 7.676 8.401 330.101 51.526 WTR – exp prices 4.419 1.983 8.868 6.252 308.549 30.298 STR – exp prices 585.789 478.525 638.344 509.266 1231.064 698.361 AA – exp prices 39.308 21.760 17.933 17.345 289.134 87.878 LAA – exp prices 5.475 3.534 5.405 14.404 307.605 69.749 ADA – fitted prices 0.514 0.199 6.832 7.431 312.564 36.436 WTR – fitted prices 4.222 1.844 8.670 6.228 292.150 19.764 STR – fitted prices 413.435 42.488 182.284 29.200 580.543 579.141 AA– fitted prices 26.507 13.228 11.117 13.981 258.010 63.777 LAA – fitted prices 2.055 1.859 4.236 13.433 284.880 45.153 4 heuristics (plots) 0.449 0.302 8.627 14.755 526.417 29.520 4 heuristics (fitted) 0.313 0.245 7.227 7.679 235.900 18.662

Mikhail Anufriev, Cars Hommes CeNDEF, University of Amsterdam Evolution of Market Heuristics

Learning Experiment Homogeneous Expectations Individual Predictions Model Other Experiments Conclusion

Comparison with Homogeneous Expectations: AR2

“Indirect fit” with parameters β = 0.4, η = 0.7, δ = 0.9

Specification Group 2 Group 5 Group 1 Group 6 Group 4 Group 7 Fundamental Prediction 0.946 0.671 2.673 3.610 2.311 2.002 ADA – exp prices 0.239 0.006 2.182 2.898 1.691 1.494 WTR – exp prices 0.066 0.529 0.383 0.627 0.203 0.165 STR – exp prices 1.494 2.583 0.112 0.020 0.240 0.342 AA – exp prices 1.095 1.848 0.010 0.038 0.045 0.094 LAA – exp prices 0.747 1.544 0.003 0.050 0.003 0.013 ADA – fitted prices 0.100 0.000 1.584 2.159 1.385 1.157 WTR – fitted prices 0.068 0.343 0.262 0.435 0.174 0.139 STR – fitted prices 1.358 2.192 0.078 0.001 0.147 0.242 AA– fitted prices 1.036 1.755 0.005 0.029 0.038 0.083 LAA – fitted prices 0.640 1.277 0.000 0.033 0.000 0.004 4 heuristics (plots) 0.383 0.744 0.011 0.008 0.157 0.239 4 heuristics (fitted) 0.144 0.499 0.009 0.003 0.121 0.048

Mikhail Anufriev, Cars Hommes CeNDEF, University of Amsterdam Evolution of Market Heuristics

Learning Experiment Homogeneous Expectations Individual Predictions Model Other Experiments Conclusion

One-period ahead prediction

◮ at any moment t take the experimental time series until time t ◮ use them to produce

◮ predictions of heuristics ◮ impacts of heuristics Mikhail Anufriev, Cars Hommes CeNDEF, University of Amsterdam Evolution of Market Heuristics

Learning Experiment Homogeneous Expectations Individual Predictions Model Other Experiments Conclusion

Group 5 (Convergence)

Parameters: β = 0.4, η = 0.7, δ = 0.9

35 45 55 65 10 20 30 40 50 Predictions ADA WTR STR LAA 45 55 65 Price

Group 5

simulation experiment

• 2

2 0.2 0.4 0.6 0.8 1 10 20 30 40 50 Fractions of 4 rules in the simulation for Group 5 ADA WTR STR LAA

Mikhail Anufriev, Cars Hommes CeNDEF, University of Amsterdam Evolution of Market Heuristics

Learning Experiment Homogeneous Expectations Individual Predictions Model Other Experiments Conclusion

Group 6 (Constant Oscillations)

Parameters: β = 0.4, η = 0.7, δ = 0.9

35 45 55 65 10 20 30 40 50 Predictions ADA WTR STR LAA 45 55 65 Price

Group 6

simulation experiment

• 5

5 0.2 0.4 0.6 0.8 1 10 20 30 40 50 Fractions of 4 rules in the simulation for Group 6 ADA WTR STR LAA

Mikhail Anufriev, Cars Hommes CeNDEF, University of Amsterdam Evolution of Market Heuristics

Learning Experiment Homogeneous Expectations Individual Predictions Model Other Experiments Conclusion

Group 1 (Constant Oscillations)

Parameters: β = 0.4, η = 0.7, δ = 0.9

35 45 55 65 10 20 30 40 50 Predictions ADA WTR STR LAA 45 55 65 Price

Group 1

simulation experiment

• 5

5 0.2 0.4 0.6 0.8 1 10 20 30 40 50 Fractions of 4 rules in the simulation for Group 1 ADA WTR STR LAA

Mikhail Anufriev, Cars Hommes CeNDEF, University of Amsterdam Evolution of Market Heuristics

Learning Experiment Homogeneous Expectations Individual Predictions Model Other Experiments Conclusion

Group 4 (Damping Oscillations)

Parameters: β = 0.4, η = 0.7, δ = 0.9

10 30 50 70 90 10 20 30 40 50 Predictions ADA WTR STR LAA 10 30 50 70 90 Price

Group 4

simulation experiment

• 50
• 25

25 50 0.2 0.4 0.6 0.8 1 10 20 30 40 50 Fractions of 4 rules in the simulation for Group 4 ADA WTR STR LAA

Mikhail Anufriev, Cars Hommes CeNDEF, University of Amsterdam Evolution of Market Heuristics

Learning Experiment Homogeneous Expectations Individual Predictions Model Other Experiments Conclusion

Group 7 (Damping Oscillations)

Parameters: β = 0.4, η = 0.7, δ = 0.9

45 55 65 75 10 20 30 40 50 Predictions ADA WTR STR LAA 45 55 65 75 Price

Group 7

simulation experiment

• 10

10 0.2 0.4 0.6 0.8 1 10 20 30 40 50 Fractions of 4 rules in the simulation for Group 7 ADA WTR STR LAA

Mikhail Anufriev, Cars Hommes CeNDEF, University of Amsterdam Evolution of Market Heuristics

Learning Experiment Homogeneous Expectations Individual Predictions Model Other Experiments Conclusion

MSE over 47 periods of the one-step ahead forecast

Specification Group 2 Group 5 Group 1 Group 6 Group 4 Group 7 Fundamental Prediction 16.6231 10.8238 15.7581 9.3245 300.9936 21.9123 naive 0.0388 0.0514 3.5415 2.4494 141.0558 13.2453 AA 5.1259 3.4323 2.9309 0.888 65.2296 5.0594 ADA 0.0712 0.0378 5.6734 4.6095 210.3313 19.5158 WTR 0.0862 0.1419 2.0905 1.1339 92.2163 9.2932 STR 0.5001 0.6605 2.9071 0.8131 124.3494 14.7224 LAA 0.4588 0.4756 0.456 0.6591 66.2637 5.8635 4 heuristics (δ = 1) 0.0814 0.1698 1.2417 0.6618 70.8516 7.0956 4 heuristics (Figures) 0.0646 0.1108 0.4672 0.2917 47.2492 4.3154 4 heuristics (best fit) 0.0493 0.0353 0.4423 0.1655 34.4932 2.9358 β ∈ [0, 10] 10 10 0.1 10 3 0.2 η ∈ [0, 1] 0.4 0.9 1 0.1 0.8 0.5 δ ∈ [0, 1] 0.9 0.6 0.5 0.7 0.6 0.4

Mikhail Anufriev, Cars Hommes CeNDEF, University of Amsterdam Evolution of Market Heuristics

Learning Experiment Homogeneous Expectations Individual Predictions Model Other Experiments Conclusion

Out-of-sample performance

The heuristic learning model vs. AR(2) model

Group 2 Group 5 Group 1 Group 6 Group 4 Group 7 average MSE40 0.0422 0.0383 0.4519 0.1702 42.6314 3.6141 1 p ahead 0.0122 0.0321 0.479 0.1921 15.0395 0.7857 2 p ahead 0.0122 0.0901 1.8599 1.0792 57.5144 1.5543 average MSE40 0.0641 0.1037 0.5242 0.2869 58.2751 5.0055 1 p ahead 0.0417 0.1456 0.4186 0.4097 7.694 0.7922 2 p ahead 0.0801 0.2304 1.5871 1.9035 16.2213 1.9338 1 p ahead 0.3732 0.4431 0.9981 0.5568 13.4616 0.6682 2 p ahead 0.5052 0.4045 3.5823 1.4944 44.6453 2.0098

Mikhail Anufriev, Cars Hommes CeNDEF, University of Amsterdam Evolution of Market Heuristics

Learning Experiment Homogeneous Expectations Individual Predictions Model Other Experiments Conclusion

The same experiment

45 55 65 75 10 20 30 40 50 Predictions 45 55 65 75 Price

Group 3

• 10

10

Mikhail Anufriev, Cars Hommes CeNDEF, University of Amsterdam Evolution of Market Heuristics

Learning Experiment Homogeneous Expectations Individual Predictions Model Other Experiments Conclusion

Group 3

Parameters: β = 0.4, η = 0.7, δ = 0.9

35 45 55 65 10 20 30 40 50 Predictions ADA WTR STR LAA 45 55 65 Price

Group 3

simulation experiment

• 10

10 0.2 0.4 0.6 0.8 1 10 20 30 40 50 Fractions of 4 rules in the simulation for Group 3 ADA WTR STR LAA

Mikhail Anufriev, Cars Hommes CeNDEF, University of Amsterdam Evolution of Market Heuristics

Learning Experiment Homogeneous Expectations Individual Predictions Model Other Experiments Conclusion

Smaller fundamental price, pf = 40

20 40 60 10 20 30 40 50 Predictions 20 40 60 Price

Group 8

• 30

30 20 40 60 80 10 20 30 40 50 Predictions 20 40 60 80 Price

Group 10

• 30

30

Mikhail Anufriev, Cars Hommes CeNDEF, University of Amsterdam Evolution of Market Heuristics

Learning Experiment Homogeneous Expectations Individual Predictions Model Other Experiments Conclusion

Group 8

Parameters: β = 0.4, η = 0.7, δ = 0.9

20 40 60 10 20 30 40 50 Predictions ADA WTR STR LAA 20 40 60 Price

Group 8

simulation experiment

• 30

30 0.2 0.4 0.6 0.8 1 10 20 30 40 50 Fractions of 4 rules in the simulation for Group 8 ADA WTR STR LAA

Mikhail Anufriev, Cars Hommes CeNDEF, University of Amsterdam Evolution of Market Heuristics

Learning Experiment Homogeneous Expectations Individual Predictions Model Other Experiments Conclusion

Group 10

Parameters: β = 0.4, η = 0.7, δ = 0.9

20 40 60 80 10 20 30 40 50 Predictions ADA WTR STR LAA 20 40 60 80 Price

Group 10

simulation experiment

• 30

30 0.2 0.4 0.6 0.8 1 10 20 30 40 50 Fractions of 4 rules in the simulation for Group 10 ADA WTR STR LAA

Mikhail Anufriev, Cars Hommes CeNDEF, University of Amsterdam Evolution of Market Heuristics

Learning Experiment Homogeneous Expectations Individual Predictions Model Other Experiments Conclusion

Results

◮ the model with evolutionary switching is built ◮ three qualitatively different patterns of the experiment have been

reproduced

◮ dynamics of model is path-depended ◮ Anufriev, M. and Hommes, C. (2007) “Evolution of Market

Heuristics”, CeNDEF Working paper 07-06 University of Amsterdam

◮ Anufriev, M. and Hommes, C. (2007) “An Evolutionary

Explanation of an Asset-Pricing Experiment”

◮ Code Evexex: http://www.cafed.eu/evexex ◮ Software Package E&F Chaos

Mikhail Anufriev, Cars Hommes CeNDEF, University of Amsterdam Evolution of Market Heuristics

Learning Experiment Homogeneous Expectations Individual Predictions Model Other Experiments Conclusion

Results

◮ the model with evolutionary switching is built ◮ three qualitatively different patterns of the experiment have been

reproduced

◮ dynamics of model is path-depended ◮ Anufriev, M. and Hommes, C. (2007) “Evolution of Market

Heuristics”, CeNDEF Working paper 07-06 University of Amsterdam

◮ Anufriev, M. and Hommes, C. (2007) “An Evolutionary

Explanation of an Asset-Pricing Experiment”

◮ Code Evexex: http://www.cafed.eu/evexex ◮ Software Package E&F Chaos

Mikhail Anufriev, Cars Hommes CeNDEF, University of Amsterdam Evolution of Market Heuristics

Learning Experiment Homogeneous Expectations Individual Predictions Model Other Experiments Conclusion

Future Directions

◮ application of the model to other forecasting experiments, in

particular to the negative feedback system

Mikhail Anufriev, Cars Hommes CeNDEF, University of Amsterdam Evolution of Market Heuristics

Learning Experiment Homogeneous Expectations Individual Predictions Model Other Experiments Conclusion

Measure of coordination

◮ average prediction error =

average dispersion + common prediction error

◮ in the experiment

1 6 6

i=1

• pe

i,t − pt

2 = 1 6 6

i=1

• pe

i,t − ¯

pe

t

2 +

• ¯

pe

t − pt

2 , where ¯ pe

t = 1 6

6

i=1 pe i,t ◮ in simulations

4

h=1 nh,t−1

• pe

h,t−pt

2 = 4

h=1 nh,t−1

• pe

h,t−¯

pe

t

2+

• ¯

pe

t −pt

2 , where ¯ pe

t = 4 h=1 nh,t−1pe h,t

Mikhail Anufriev, Cars Hommes CeNDEF, University of Amsterdam Evolution of Market Heuristics

Learning Experiment Homogeneous Expectations Individual Predictions Model Other Experiments Conclusion

Measure of coordination

◮ average prediction error =

average dispersion + common prediction error

◮ in the experiment

1 6 6

i=1

• pe

i,t − pt

2 = 1 6 6

i=1

• pe

i,t − ¯

pe

t

2 +

• ¯

pe

t − pt

2 , where ¯ pe

t = 1 6

6

i=1 pe i,t ◮ in simulations

4

h=1 nh,t−1

• pe

h,t−pt

2 = 4

h=1 nh,t−1

• pe

h,t−¯

pe

t

2+

• ¯

pe

t −pt

2 , where ¯ pe

t = 4 h=1 nh,t−1pe h,t

Mikhail Anufriev, Cars Hommes CeNDEF, University of Amsterdam Evolution of Market Heuristics

Learning Experiment Homogeneous Expectations Individual Predictions Model Other Experiments Conclusion

Measure of coordination

◮ average prediction error =

average dispersion + common prediction error

◮ in the experiment

1 6 6

i=1

• pe

i,t − pt

2 = 1 6 6

i=1

• pe

i,t − ¯

pe

t

2 +

• ¯

pe

t − pt

2 , where ¯ pe

t = 1 6

6

i=1 pe i,t ◮ in simulations

4

h=1 nh,t−1

• pe

h,t−pt

2 = 4

h=1 nh,t−1

• pe

h,t−¯

pe

t

2+

• ¯

pe

t −pt

2 , where ¯ pe

t = 4 h=1 nh,t−1pe h,t

Mikhail Anufriev, Cars Hommes CeNDEF, University of Amsterdam Evolution of Market Heuristics

Learning Experiment Homogeneous Expectations Individual Predictions Model Other Experiments Conclusion

Coordination

Table: Coordination in the experiment and over the simulations. Group 5 Group 1 Group 7 period exp sim exp sim exp sim 3-10 44.91 94.23 79.16 71.04 61.30 79.43 11-20 81.54 84.42 78.03 91.28 72.72 90.35 21-30 71.75 80.50 78.62 89.48 66.14 94.27 31-40 81.83 84.14 81.98 97.98 67.01 94.74 41-50 84.85 86.51 94.90 93.84 21.95 74.38 Group 2 Group 6 Group 4 period exp sim exp sim exp sim 3-10 40.88 69.98 58.59 77.00 76.44 87.45 11-20 67.25 86.48 78.94 80.96 90.41 88.44 21-30 75.73 80.53 76.16 79.69 83.41 96.70 31-40 83.88 80.51 79.33 88.72 48.47 88.03 41-50 91.44 84.66 72.09 92.26 31.74 65.78

Mikhail Anufriev, Cars Hommes CeNDEF, University of Amsterdam Evolution of Market Heuristics

Learning Experiment Homogeneous Expectations Individual Predictions Model Other Experiments Conclusion

Analysis of the Evolutionary Model

◮ deterministic skeleton

                   pe

i,t+1 = fi(pt−1, pt−2, pe i,t)

pt − pf = 1 1 + r exp

• −

1 200

• pt−1 − pf
• 4

i=1 ni,t (pe i,t+1 − pf )

ni,t = δ ni,t−1 + (1 − δ) exp(β Ui,t−1) 4

i=1 exp(β Ui,t−1)

Ui,t−1 = −

• pt−1 − pe

i,t−1

2 + η Ui,t−2

◮ parameters

◮ β ≥ 0 – the intensity of choice ◮ η ∈ [0, 1] – the strength of the agents’ memory ◮ δ ∈ [0, 1] – the inertia of the traders Mikhail Anufriev, Cars Hommes CeNDEF, University of Amsterdam Evolution of Market Heuristics

Learning Experiment Homogeneous Expectations Individual Predictions Model Other Experiments Conclusion

Stability for the Model with Fixed Impacts

0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 0.2 0.4 0.6 0.8 1 n3, fraction of STR Stability region for model with fixed fractions A n1, fraction of ADA n2, fraction of WTR n3, fraction of STR

Mikhail Anufriev, Cars Hommes CeNDEF, University of Amsterdam Evolution of Market Heuristics