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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
40
45
50
55
60
65
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?