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Heterogeneous Agent Models Lecture 3 Role of Expectations in Theory Learning to Forecast Experiments

Heterogeneous Agent ModelsLecture 3

Role of Expectations in TheoryLearning to Forecast Experiments

Mikhail Anufriev

EDG, Faculty of Business, University of Technology Sydney (UTS)

July, 2013

Heterogeneous Agent Models Lecture 3 Role of Expectations in Theory Learning to Forecast Experiments

Outline

1 Rational Expectations

2 Experiments

3 Learning to Forecast Experiment

Heterogeneous Agent Models Lecture 3 Role of Expectations in Theory Learning to Forecast Experiments

Rational Expectations

Cobweb Model

Demand:

D(pt) = 0.7− 0.25 pt

Supply:

Sλ(pet ) = arctan(4.8 pe

t )

Heterogeneous Agent Models Lecture 3 Role of Expectations in Theory Learning to Forecast Experiments

Rational Expectations

Different Expectation Schemes

Naive Expectations

D(pt) = a− b pt , a ∈ R , b ≥ 0 demandSλ(pe

t ) = arctan(λ pet ) , λ > 0 supply

D(pt) = Sλ(pet ) market clearing

pet = H(pt−1, ..., pt−L) expectations

naive expectations pet = pt−1

deterministic dynamics: pt = D−1(Sλ(pt−1)

)the steady-state p∗ is such that

D(p∗) = Sλ(p∗)

stability conditions

−1 <S′(p∗)D′(p∗)

< 1

Heterogeneous Agent Models Lecture 3 Role of Expectations in Theory Learning to Forecast Experiments

Rational Expectations

Different Expectation Schemes

Naive Expectations: Trajectories

without noise:cycle of period 2

with noise:quasi-cycle of period 2

predictable hog cycle with systematic forecasting errors

Heterogeneous Agent Models Lecture 3 Role of Expectations in Theory Learning to Forecast Experiments

Rational Expectations

Rational Expectations

Rational ExpectationsMuth, 1961

the agents’ expectations are the same as generated by theeconomic theory

the perceived law of motion

pet = H(pt−1, ..., pt−L)

is not systematically different from the actual law of motion

pt = D−1(

Sλ(H(pt−1, ..., pt−L)

))agents compute correct expectations from market equilibriumequation

pet = Et[pt]

Heterogeneous Agent Models Lecture 3 Role of Expectations in Theory Learning to Forecast Experiments

Rational Expectations

Rational Expectations

Rational Expectations in the Cobweb Model

in the model without shocks rational expectations are equivalentto perfect foresight

pt = p∗

when small shock is added to the demand equation

pt = p∗ +εt

b

so expectations are self-fulfilling and systematic forecastingerrors are impossible

no problem of stability as well (no dynamics)!

Heterogeneous Agent Models Lecture 3 Role of Expectations in Theory Learning to Forecast Experiments

Rational Expectations

Rational Expectations

Rational Expectations in the Cobweb Model

in the model without shocks rational expectations are equivalentto perfect foresight

pt = p∗

when small shock is added to the demand equation

pt = p∗ +εt

b

so expectations are self-fulfilling and systematic forecastingerrors are impossible

no problem of stability as well (no dynamics)!

Heterogeneous Agent Models Lecture 3 Role of Expectations in Theory Learning to Forecast Experiments

Rational Expectations

Rational Expectations

Rational Expectations: Trajectories

without noise:Perfect foresight

constant price

with noise:Rational Expectations

small fluctuationsno systematic forecasting errors

Heterogeneous Agent Models Lecture 3 Role of Expectations in Theory Learning to Forecast Experiments

Rational Expectations

Adaptive Expectations

Adaptive Expectations: “Error Learning” ModelNerlove, 1958

agents correct previous errors

pet = pe

t−1 + w (pt−1 − pet−1) =

= (1− w) pet−1 + w pt−1 =

= w pt−1 + (1− w)w pt−2 + · · ·+ (1− w)j−1w pt−j + . . .

with weighted factor w ∈ (0, 1]

w = 1 : naive expectationsSolution:

1-D system in terms expected price dynamics:

pet = w D−1(S(pe

t−1))

+ (1− w) pet−1

price dynamics is recovered:

pt = D−1(S(pet ))

Heterogeneous Agent Models Lecture 3 Role of Expectations in Theory Learning to Forecast Experiments

Rational Expectations

Adaptive Expectations

Adaptive Expectations: Trajectories

Demand:

D(pt) = 0.7− 0.25 pt

Supply:

Sλ(pet ) = arctan(4.8 pe

t )

Weight:

w = 0.15, 0.4

Heterogeneous Agent Models Lecture 3 Role of Expectations in Theory Learning to Forecast Experiments

Rational Expectations

Adaptive Expectations

Adaptive Expectations: Illustration

weight w = 0.15

convergence to the st-st

weight w = 0.4

randomly looking

Heterogeneous Agent Models Lecture 3 Role of Expectations in Theory Learning to Forecast Experiments

Rational Expectations

Adaptive Expectations

Cobweb with Adaptive Expectations

adaptive expectations brings stability

the stability region enlarges

amplitude of fluctuations decreases

adaptive expectations brings chaos with excess volatility

errors under adaptive expectations with chaos have lessrecognizable structure than without it

Heterogeneous Agent Models Lecture 3 Role of Expectations in Theory Learning to Forecast Experiments

Rational Expectations

Adaptive Expectations

Adaptive Expectations: Correlations of Forecasting Errors

weight w = 0.15 weight w = 0.4

Heterogeneous Agent Models Lecture 3 Role of Expectations in Theory Learning to Forecast Experiments

Rational Expectations

Adaptive Expectations

Rational Expectations: Pros and ContrasAdvantages

principle of RE can be applied to all dynamical problems:different markets and systems

in the absence of RE there would be someprofitable opportunities for agents

RE is a benchmark to test deviations due tobiases, incomplete information, poor memory, etc.

Disadvantages

RE assume perfect knowledge about market equilibriumequations, i.e. about the law of motion of the economyRE assume perfect computational abilities of the agentsif RE is only long-run phenomenon, then dynamics matter

especially if forecasting errors look not systematic

Heterogeneous Agent Models Lecture 3 Role of Expectations in Theory Learning to Forecast Experiments

Rational Expectations

Adaptive Expectations

Rational Expectations: Pros and ContrasAdvantages

principle of RE can be applied to all dynamical problems:different markets and systems

in the absence of RE there would be someprofitable opportunities for agents

RE is a benchmark to test deviations due tobiases, incomplete information, poor memory, etc.

Disadvantages

RE assume perfect knowledge about market equilibriumequations, i.e. about the law of motion of the economyRE assume perfect computational abilities of the agentsif RE is only long-run phenomenon, then dynamics matter

especially if forecasting errors look not systematic

Heterogeneous Agent Models Lecture 3 Role of Expectations in Theory Learning to Forecast Experiments

Rational Expectations

Adaptive Expectations

Implications of the Nonlinear Dynamics for Economics

Small changes in parameter values may lead to largeconsequences for the system

Type of bifurcation matters

Nonlinear systems are consistent with unpredictability as theyexhibit chaotic behaviour

Even simple nonlinear systems exhibit complex dynamics: howrealistic is the rational expectations assumption then?

knowledge of the actual laws of motionlearning of people from actual mistakes“survival” of the fittest story

In a nonlinear world, simple heuristics that work reasonably wellmay be the best what agents can achieve

Heterogeneous Agent Models Lecture 3 Role of Expectations in Theory Learning to Forecast Experiments

Rational Expectations

Adaptive Expectations

Implications of the Nonlinear Dynamics for Economics

Small changes in parameter values may lead to largeconsequences for the system

Type of bifurcation matters

Nonlinear systems are consistent with unpredictability as theyexhibit chaotic behaviour

Even simple nonlinear systems exhibit complex dynamics: howrealistic is the rational expectations assumption then?

knowledge of the actual laws of motionlearning of people from actual mistakes“survival” of the fittest story

In a nonlinear world, simple heuristics that work reasonably wellmay be the best what agents can achieve

Heterogeneous Agent Models Lecture 3 Role of Expectations in Theory Learning to Forecast Experiments

Rational Expectations

Adaptive Expectations

Implications of the Nonlinear Dynamics for Economics

Small changes in parameter values may lead to largeconsequences for the system

Type of bifurcation matters

Nonlinear systems are consistent with unpredictability as theyexhibit chaotic behaviour

Even simple nonlinear systems exhibit complex dynamics: howrealistic is the rational expectations assumption then?

knowledge of the actual laws of motionlearning of people from actual mistakes“survival” of the fittest story

In a nonlinear world, simple heuristics that work reasonably wellmay be the best what agents can achieve

Heterogeneous Agent Models Lecture 3 Role of Expectations in Theory Learning to Forecast Experiments

Rational Expectations

Adaptive Expectations

Implications of the Nonlinear Dynamics for Economics

Small changes in parameter values may lead to largeconsequences for the system

Type of bifurcation matters

Nonlinear systems are consistent with unpredictability as theyexhibit chaotic behaviour

Even simple nonlinear systems exhibit complex dynamics: howrealistic is the rational expectations assumption then?

knowledge of the actual laws of motionlearning of people from actual mistakes“survival” of the fittest story

In a nonlinear world, simple heuristics that work reasonably wellmay be the best what agents can achieve

Heterogeneous Agent Models Lecture 3 Role of Expectations in Theory Learning to Forecast Experiments

Experiments

Experimental Economics

possibility to address specific questions in a clean environment

reproducibility

Heterogeneous Agent Models Lecture 3 Role of Expectations in Theory Learning to Forecast Experiments

Experiments

Experiments about expectationsEarlier experiments: indirect focus / expectations on exogenous timeseries: Schmalensee (1976), Hey (1994), Marimon and Sunder (1994)

Learning-to-forecast experiments: Hommes et al (2005, RFS; 2008,JEBO), Adam (2009, EJ), Heemeijer et al (2009, JEDC)

Model of asset-pricing (Campbell, Lo and MacKinlay, 1997)

riskless asset with interest r = 0.05

risky asset with price pt and i.i.d. dividend yt with mean y = 3

pt = 11+r

(pe

t+1 + y + εt

)= 1

1+r

(pe

t+1,1+···+pet+1,6

6 + y + εt

)Idea of Experiment: 6 human subjects

submit forecasts pet+1,h and are paid according to the precision

computer generates price and reports it back to the participants

Heterogeneous Agent Models Lecture 3 Role of Expectations in Theory Learning to Forecast Experiments

Experiments

Experiments about expectationsEarlier experiments: indirect focus / expectations on exogenous timeseries: Schmalensee (1976), Hey (1994), Marimon and Sunder (1994)

Learning-to-forecast experiments: Hommes et al (2005, RFS; 2008,JEBO), Adam (2009, EJ), Heemeijer et al (2009, JEDC)

Model of asset-pricing (Campbell, Lo and MacKinlay, 1997)

riskless asset with interest r = 0.05

risky asset with price pt and i.i.d. dividend yt with mean y = 3

pt = 11+r

(pe

t+1 + y + εt

)= 1

1+r

(pe

t+1,1+···+pet+1,6

6 + y + εt

)Idea of Experiment: 6 human subjects

submit forecasts pet+1,h and are paid according to the precision

computer generates price and reports it back to the participants

Heterogeneous Agent Models Lecture 3 Role of Expectations in Theory Learning to Forecast Experiments

Experiments

Learning to Forecast ExperimentsSubjects’ task and incentives

forecasting a price for 50 periods

better forecasts yield higher earningsSubjects know

only qualitative information about the market

price pt derived from equilibrium between demand and supply

type of expectations feedback: positive(in this case) or negative

past information: at time t participant h can seepast prices (up to pt−1), own past forecasts (up to pt,h) andown earnings (up to et−1,h)

Subjects do not knowexact equilibrium equation

number and forecasts of other participants

Heterogeneous Agent Models Lecture 3 Role of Expectations in Theory Learning to Forecast Experiments

Experiments

Learning-to-Forecast Experiments

In a session six human subjects know only qualitative features. They:

start by submitting their first forecasts

observe realised price, which is determined by the computer

receive payoffs: the smaller the forecasting error is, the larger thepayoff is

submit new individual forecasts

and so on for 50 periods

Heterogeneous Agent Models Lecture 3 Role of Expectations in Theory Learning to Forecast Experiments

Experiments

Round Prediction Real value

1 33,70 50,232 33,70 56,633 37,00 65,324 40,10 65,005 43,50 66,126 50,00 64,537 48,35 58,358 38,70 42,359 30,10 40,01

10 28,25

Total Earnings Remainingearnings: this period: time:

10357 1298 00

What is your prediction Prediction:this period?

Your prediction mustbe between 0 and 100

0102030405060708090

100

1 6 11 16 21 26 31 36 41 46

prediction

real number

Round

Number

earnings per period: et,h = max(

1− 149 (pt − pe

t,h)2, 0)× 1

2 euro

Heterogeneous Agent Models Lecture 3 Role of Expectations in Theory Learning to Forecast Experiments

Experiments

Rational BenchmarkIf everybody predicts fundamental price pf = y

r = 60, then pt = pf + εt1+r

40

45

50

55

60

65

70

0 10 20 30 40 50

Pric

e

Time

fundamental priceprice under rational expectations

-1

-0.5

0

0.5

1

0 10 20 30 40 50

Heterogeneous Agent Models Lecture 3 Role of Expectations in Theory Learning to Forecast Experiments

Experiments

0

200

400

600

800

1000

0 10 20 30 40 50

gr 1gr 2gr 3gr 4gr 5gr 6

Heterogeneous Agent Models Lecture 3 Role of Expectations in Theory Learning to Forecast Experiments

Experiments

Experiment with stabilizing fundamentalists

pricing equation

pt = 11+r

((1− nt)pe

t+1 + nt pf + y + εt)

fraction of fundamental traders

nt = 1− exp(− 1

200 |pt−1 − pf |)

Heterogeneous Agent Models Lecture 3 Role of Expectations in Theory Learning to Forecast Experiments

Experiments

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

fundamental price experimental price

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45

50

55

60

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70

0 10 20 30 40 50

Pric

e

Group 1

fundamental price experimental price

40

45

50

55

60

65

70

0 10 20 30 40 50

Pric

e

Group 6

fundamental price experimental price

10 20 30 40 50 60 70 80 90

0 10 20 30 40 50

Pric

e

Group 4

fundamental price experimental price

40

45

50

55

60

65

70

0 10 20 30 40 50

Pric

eGroup 7

fundamental price experimental price

Heterogeneous Agent Models Lecture 3 Role of Expectations in Theory Learning to Forecast Experiments

Experiments

2 Groups with (Almost) Monotonic Convergence

35

45

55

65

0 10 20 30 40 50

Pred

ictio

ns

45

55

65

Pric

e

Group 2

-2 0 2

35

45

55

65

0 10 20 30 40 50

Pred

ictio

ns

45

55

65

Pric

e

Group 5

-2 0 2

Heterogeneous Agent Models Lecture 3 Role of Expectations in Theory Learning to Forecast Experiments

Experiments

2 Groups with Constant Oscillations

35

45

55

65

0 10 20 30 40 50

Pred

ictio

ns

45

55

65

Pric

e

Group 1

-5 0 5

35

45

55

65

0 10 20 30 40 50

Pred

ictio

ns

45

55

65

Pric

e

Group 6

-5 0 5

Heterogeneous Agent Models Lecture 3 Role of Expectations in Theory Learning to Forecast Experiments

Experiments

2 Groups with Damping Oscillations

10 30 50 70 90

0 10 20 30 40 50

Pred

ictio

ns

10

30

50

70

90

Pric

e

Group 4

-30 0

30 45 55 65 75

0 10 20 30 40 50

Pred

ictio

ns

45

55

65

75

Pric

e

Group 7

-10 0

10

Heterogeneous Agent Models Lecture 3 Role of Expectations in Theory Learning to Forecast Experiments

Experiments

Price in Experiments. Groups 11–14

0

20

40

60

80

100

0 10 20 30 40 50

gr 11gr 12gr 13gr 14

Heterogeneous Agent Models Lecture 3 Role of Expectations in Theory Learning to Forecast Experiments

Experiments

Price in Experiments. HSTV 2005, Groups 8-10

0

20

40

60

80

100

0 10 20 30 40 50

gr 8gr 9

gr 10

Heterogeneous Agent Models Lecture 3 Role of Expectations in Theory Learning to Forecast Experiments

Experiments

Price in Experiments. HSTV 2005. Group 3

40

45

50

55

60

65

70

0 10 20 30 40 50

Pric

e

Experiment and simulation price for Group 3

simulation experiment

Heterogeneous Agent Models Lecture 3 Role of Expectations in Theory Learning to Forecast Experiments

Experiments

Estimation of individual prediction rulesOLS regression of predictions on the lagged prices and predictions

pei,t+1 = α+

5∑k=1

βkpt−k +

5∑k=0

γkpei,t−k + εi,t

leaving insignificant coefficients outadaptive expectations

pet+1,h = w pt−1 + (1− w) pe

t,h

trend-extrapolating rules

pet+1,h = pt−1 + γ (pt−1 − pt−2)

anchoring and adjustment rule

pet+1,h = 1

2

(60 + pt−1

)+(pt−1 − pt−2

)

Heterogeneous Agent Models Lecture 3 Role of Expectations in Theory Learning to Forecast Experiments

Experiments

Learning-to-forecast experiments: Summary 1

“Stylized facts”

large bubbles in the absence of fundamentalists

qualitatively different patterns in the same environment(almost) monotonic convergence

constant oscillations

damping oscillations

coordination of individual predictions

forecasting rules with behavioral interpretation are used

Heterogeneous Agent Models Lecture 3 Role of Expectations in Theory Learning to Forecast Experiments

Experiments

Positive vs. Negative feedback

A positive feedback system reinforces a change in input byresponding to a perturbation in the same direction.

A negative feedback system reverses a change in input and respondsto a perturbation in the opposite direction.

Heterogeneous Agent Models Lecture 3 Role of Expectations in Theory Learning to Forecast Experiments

Experiments

Heterogeneous Agent Models Lecture 3 Role of Expectations in Theory Learning to Forecast Experiments

Experiments

Negative feedback in Economics

Heterogeneous Agent Models Lecture 3 Role of Expectations in Theory Learning to Forecast Experiments

Experiments

Negative feedback in Economics

Heterogeneous Agent Models Lecture 3 Role of Expectations in Theory Learning to Forecast Experiments

Experiments

Learning-to-Forecast Experiments

Question:How does the feedback affects the dynamical properties of price in agroup environment?

Another Learning-to-Forecast Experiment with two treatments.

Heterogeneous Agent Models Lecture 3 Role of Expectations in Theory Learning to Forecast Experiments

Experiments

Learning-to-Forecast Experiments

Negative feedback Positive feedback

20 40 60 80 100 120Prediction

20406080

100120

Price

20 40 60 80 100 120Prediction

20406080

100120

Price

pt = 60− 2021

(pe

t − 60)

+ εt pt = 60 + 2021

(pe

t − 60)

+ εt

Heterogeneous Agent Models Lecture 3 Role of Expectations in Theory Learning to Forecast Experiments

Experiments

Computer Screen

subjects’ payoff: et,h = max(

1− 149(pt − pe

t,h)2, 0)× 1

2 euro

Heterogeneous Agent Models Lecture 3 Role of Expectations in Theory Learning to Forecast Experiments

Experiments

Negative Feedback Experiment: Session 1

Heterogeneous Agent Models Lecture 3 Role of Expectations in Theory Learning to Forecast Experiments

Experiments

Negative Feedback Experiment: all sessions

20

40

60

80

0 10 20 30 40 50

Pri

ce

Time

Heterogeneous Agent Models Lecture 3 Role of Expectations in Theory Learning to Forecast Experiments

Experiments

Positive Feedback Experiment: Session 1

Heterogeneous Agent Models Lecture 3 Role of Expectations in Theory Learning to Forecast Experiments

Experiments

Positive Feedback Experiment: all sessions

20

40

60

80

0 10 20 30 40 50

Pri

ce

Time

Heterogeneous Agent Models Lecture 3 Role of Expectations in Theory Learning to Forecast Experiments

Experiments

Learning-to-forecast experiments: Summary 2

“Stylized facts” to explain:

qualitatively different aggregate patterns in differentenvironments

negative feedback: heavy fluctuation (during 5 periods) and thenfast convergence

positive feedback: no convergence, in some groups slowoscillations

coordination of individual predictions

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