Heterogeneous Agent Models Lecture 3 Role of Expectations in Theory Learning to Forecast Experiments
Heterogeneous Agent Models
Lecture 3
Role of Expectations in Theory
Learning 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 pet )
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 ,
Sλ (pet )
=
D(pt ) =
pet
a ∈ R, b ≥ 0
arctan(λ pet ) ,
Sλ (pet )
demand
λ>0
supply
market clearing
= 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 <
S0 (p∗ )
<1
D0 (p∗ )
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 Expectations
Muth, 1961
the agents’ expectations are the same as generated by the
economic 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 equilibrium
equation
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 equivalent
to perfect foresight
pt = p∗
when small shock is added to the demand equation
pt = p∗ +
εt
b
so expectations are self-fulfilling and systematic forecasting
errors 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 equivalent
to perfect foresight
pt = p∗
when small shock is added to the demand equation
pt = p∗ +
εt
b
so expectations are self-fulfilling and systematic forecasting
errors 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
with noise:
Rational Expectations
constant price
small fluctuations
no 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” Model
Nerlove, 1958
agents correct previous errors
pet = pet−1 + w (pt−1 − pet−1 ) =
= (1 − w) pet−1 + w pt−1 =
= w pt−1 + (1 − w)w pt−2 + · · · + (1 − w)j−1 w pt−j + . . .
with weighted factor w ∈ (0, 1]
w = 1 : naive expectations
Solution:
1-D system in terms expected price dynamics:
pet = w D−1 S(pet−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 pet )
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
weight w = 0.4
convergence to the st-st
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 less
recognizable 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 Contras
Advantages
principle of RE can be applied to all dynamical problems:
different markets and systems
in the absence of RE there would be some
profitable opportunities for agents
RE is a benchmark to test deviations due to
biases, incomplete information, poor memory, etc.
Disadvantages
RE assume perfect knowledge about market equilibrium
equations, i.e. about the law of motion of the economy
RE assume perfect computational abilities of the agents
if 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 Contras
Advantages
principle of RE can be applied to all dynamical problems:
different markets and systems
in the absence of RE there would be some
profitable opportunities for agents
RE is a benchmark to test deviations due to
biases, incomplete information, poor memory, etc.
Disadvantages
RE assume perfect knowledge about market equilibrium
equations, i.e. about the law of motion of the economy
RE assume perfect computational abilities of the agents
if 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 large
consequences for the system
Type of bifurcation matters
Nonlinear systems are consistent with unpredictability as they
exhibit chaotic behaviour
Even simple nonlinear systems exhibit complex dynamics: how
realistic is the rational expectations assumption then?
knowledge of the actual laws of motion
learning of people from actual mistakes
“survival” of the fittest story
In a nonlinear world, simple heuristics that work reasonably well
may 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 large
consequences for the system
Type of bifurcation matters
Nonlinear systems are consistent with unpredictability as they
exhibit chaotic behaviour
Even simple nonlinear systems exhibit complex dynamics: how
realistic is the rational expectations assumption then?
knowledge of the actual laws of motion
learning of people from actual mistakes
“survival” of the fittest story
In a nonlinear world, simple heuristics that work reasonably well
may 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 large
consequences for the system
Type of bifurcation matters
Nonlinear systems are consistent with unpredictability as they
exhibit chaotic behaviour
Even simple nonlinear systems exhibit complex dynamics: how
realistic is the rational expectations assumption then?
knowledge of the actual laws of motion
learning of people from actual mistakes
“survival” of the fittest story
In a nonlinear world, simple heuristics that work reasonably well
may 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 large
consequences for the system
Type of bifurcation matters
Nonlinear systems are consistent with unpredictability as they
exhibit chaotic behaviour
Even simple nonlinear systems exhibit complex dynamics: how
realistic is the rational expectations assumption then?
knowledge of the actual laws of motion
learning of people from actual mistakes
“survival” of the fittest story
In a nonlinear world, simple heuristics that work reasonably well
may 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 expectations
Earlier experiments: indirect focus / expectations on exogenous time
series: 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 ȳ = 3
e
pt+1,1 +···+pet+1,6
1
1
pt = 1+r
p̄et+1 + ȳ + εt = 1+r
+
ȳ
+
ε
t
6
Idea of Experiment:
submit forecasts
6 human subjects
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 expectations
Earlier experiments: indirect focus / expectations on exogenous time
series: 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 ȳ = 3
e
pt+1,1 +···+pet+1,6
1
1
pt = 1+r
p̄et+1 + ȳ + εt = 1+r
+
ȳ
+
ε
t
6
Idea of Experiment:
submit forecasts
6 human subjects
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 Experiments
Subjects’ task and incentives
forecasting a price for 50 periods
better forecasts yield higher earnings
Subjects 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 see
past prices (up to pt−1 ), own past forecasts (up to pt,h ) and
own earnings (up to et−1,h )
Subjects do not know
exact 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 the
payoff 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
Number
100
90
80
70
60
50
40
30
20
10
0
1
prediction
real number
Round
6
11
16
21
26
Total
earnings:
Earnings
this period:
10357
1298
31
36
What is your prediction
this period?
Your prediction must
be between 0 and 100
earnings per period: et,h = max 1 −
41
Round
Prediction Real value
1
2
3
4
5
6
7
8
9
10
33,70
33,70
37,00
40,10
43,50
50,00
48,35
38,70
30,10
28,25
46
Remaining
time:
00
Prediction:
1
49 (pt
− pet,h )2 , 0 ×
1
2
euro
50,23
56,63
65,32
65,00
66,12
64,53
58,35
42,35
40,01
Heterogeneous Agent Models Lecture 3 Role of Expectations in Theory Learning to Forecast Experiments
Experiments
Rational Benchmark
If everybody predicts fundamental price pf =
70
ȳ
r
= 60, then pt = pf +
fundamental price
price under rational expectations
65
Price
60
55
1
0.5
50
0
-0.5
45
-1
0
10
20
30
40
50
40
0
10
20
30
Time
40
50
εt
1+r
Heterogeneous Agent Models Lecture 3 Role of Expectations in Theory Learning to Forecast Experiments
Experiments
1000
gr 1
gr 2
gr 3
gr 4
gr 5
gr 6
800
600
400
200
0
0
10
20
30
40
50
Heterogeneous Agent Models Lecture 3 Role of Expectations in Theory Learning to Forecast Experiments
Experiments
Experiment with stabilizing fundamentalists
pricing equation
pt =
1
1+r
(1 − nt )p̄et+1 + nt pf + ȳ + 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
Group 2
70
fundamental price
65
Group 5
70
experimental price
experimental price
60
Price
60
Price
fundamental price
65
55
50
55
50
45
45
40
40
0
10
20
30
40
50
0
10
20
Group 1
70
fundamental price
65
70
experimental price
40
50
experimental price
60
Price
Price
fundamental price
65
60
55
50
55
50
45
45
40
40
0
10
20
30
40
50
0
10
20
Group 4
90
80
70
60
50
40
30
20
10
fundamental price
30
40
50
Group 7
70
experimental price
fundamental price
65
experimental price
60
Price
Price
30
Group 6
55
50
45
40
0
10
20
30
40
50
0
10
20
30
40
50
Heterogeneous Agent Models Lecture 3 Role of Expectations in Theory Learning to Forecast Experiments
Experiments
2 Groups with (Almost) Monotonic Convergence
Group 2
55
55
45
65
Predictions
45
65
Predictions
Group 5
65
Price
Price
65
55
45
2
0
-2
35
0
55
45
2
0
-2
35
10
20
30
40
50
0
10
20
30
40
50
Heterogeneous Agent Models Lecture 3 Role of Expectations in Theory Learning to Forecast Experiments
Experiments
2 Groups with Constant Oscillations
Group 1
55
55
45
65
Predictions
45
65
Predictions
Group 6
65
Price
Price
65
55
45
5
0
-5
35
0
55
45
5
0
-5
35
10
20
30
40
50
0
10
20
30
40
50
Heterogeneous Agent Models Lecture 3 Role of Expectations in Theory Learning to Forecast Experiments
Experiments
2 Groups with Damping Oscillations
Group 4
90
Price
Price
50
65
55
10
45
90
70
50
30
10
75
65
55
45
Predictions
Predictions
30
30
0
-30
0
10
20
30
Group 7
75
70
40
50
10
0
-10
0
10
20
30
40
50
Heterogeneous Agent Models Lecture 3 Role of Expectations in Theory Learning to Forecast Experiments
Experiments
Price in Experiments. Groups 11–14
100
80
60
40
20
gr 11
gr 12
gr 13
gr 14
0
0
10
20
30
40
50
Heterogeneous Agent Models Lecture 3 Role of Expectations in Theory Learning to Forecast Experiments
Experiments
Price in Experiments. HSTV 2005, Groups 8-10
100
gr 8
gr 9
gr 10
80
60
40
20
0
0
10
20
30
40
50
Heterogeneous Agent Models Lecture 3 Role of Expectations in Theory Learning to Forecast Experiments
Experiments
Price in Experiments. HSTV 2005. Group 3
Experiment and simulation price for Group 3
70
65
Price
60
55
50
45
simulation
40
0
10
20
experiment
30
40
50
Heterogeneous Agent Models Lecture 3 Role of Expectations in Theory Learning to Forecast Experiments
Experiments
Estimation of individual prediction rules
OLS regression of predictions on the lagged prices and predictions
pei,t+1 = α +
5
X
βk pt−k +
k=1
5
X
γk pei,t−k + i,t
k=0
leaving insignificant coefficients out
adaptive expectations
pet+1,h = w pt−1 + (1 − w) pet,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 by
responding to a perturbation in the same direction.
A negative feedback system reverses a change in input and responds
to 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 a
group 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
Price
120
100
80
60
40
20
Price
120
100
80
60
40
20
20
40
60
pt = 60 −
20
21
80 100 120
Prediction
pet − 60 + εt
20
40
pt = 60 +
Prediction
60
80 100 120
20
21
pet − 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 −
1
49 (pt
− pet,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
80
Price
60
40
20
0
10
20
30
Time
40
50
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
80
Price
60
40
20
0
10
20
30
Time
40
50
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 different
environments
negative feedback: heavy fluctuation (during 5 periods) and then
fast convergence
positive feedback: no convergence, in some groups slow
oscillations
coordination of individual predictions
How do people behave (form expectations and learn) in the
expectations feedback system?