heterogeneous agent models lecture 1 introduction rational

Heterogeneous Agent Models Lecture 1 Introduction Rational vs. Agent Based Modelling Heterogeneous Agent Modelling

Heterogeneous Agent ModelsLecture 1

IntroductionRational vs. Agent Based Modelling

Heterogeneous Agent Modelling

Mikhail Anufriev

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

July, 2013


Outline

1 Overview

2 Financial Market Model

3 Heterogeneous Agent Models


Overview

Economics as Expectation Feedback System

Expectations play utmost role in any human activitywhere and when to go to a vacation

choice of university degree and specific courses

when to buy a car, house, etc.

investment choice

Economics is an expectation feedback system

expectations affect people’s decisions

individual decisions are aggregated

aggregate variables affect expectations


Overview

Cobweb (“hog cycle”) Model

market for a non-storable consumption good

production lag: producers form price expectations one periodahead

temporary equilibrium: market clearing prices at each time step

What will be the price at this market?

dynamics: prices evolve over time forming a trajectory

how does this trajectory look like? Is it “simple” or not?

dynamics depend on functional form of demand and supply andon the way how expectations are formed


Overview

Naive Expectations in Cobweb Model

Naive expectationspe

t = pt−1


Overview

Price Dynamics

pet : producers’ price expectation for period t

pt : realized market equilibrium price

temporary equilibrium

qdt = D(pt) , demand

qst = S(pe

t ) supplyqd

t = qst market clearing

pet = H(pt−1, pt−2, . . . ) expectations

Price dynamics

pt = D−1(S(pet ))

= D−1(S(H(pt−1, pt−2, . . . ))).


Overview

Rational View

representative agent, who is perfectly rational

Rational Expectations: expectations are model consistent

Friedman argument: “irrational agents will lose money andwill be driven out the market by rational agents”

prices reflect economic fundamentals (market efficiency)


Overview

Heterogeneous, Interacting Agents Approach

heterogeneous agents, heterogeneous beliefs

bounded rationality (Simon, 1957)

market psychology (Keynes, 1936)herding behaviorinductive reasoning(imperfect) learning from mistakes

economy as complex adaptive, nonlinear, evolutionary systems

aggregation of interacting agents leads to non-trivialmacro-economic phenomena (agent-based modelling)


Overview

Some Problems of Interacting Agents Approach‘wilderness’of bounded rationality

many degrees of freedom for heterogeneity

would not irrational decision makers be driven out?

non-transparency: what exactly causes the outcome in a (large)computational ABM?

How to Discipline Bounded Rationality?

stylized agent-based models (Heterogeneous Agent Models)

behavioral consistency: simple heuristics that work reasonablywell

evolutionary selection (‘survival of the fittest’) andreinforcement learning

laboratory experiments to test individual decision rules andaggregate macro behavior


Overview

Overview

Lecture 1: Rational vs. agent-based vs. HAM approachesfinancial market model

Lecture 2: Bifurcation theory and Chaos theorychaos in economics, lessons of nonlinearity for economics

Lecture 3: Learning to Forecast Experimentsrole of market feedback

Lecture 4: Heuristic Switching Modelbehavioral model based on evolutionary switching betweenforecasting heuristics, which explain data very well


Financial Market Model

General setup

Market

Two alternatives for investors, two assets:

Risk-free asset gross riskless return on the asset R = 1 + r > 1

Risky asset dividend process yt

endogenous price pt per share

Demand zt for the risky asset is derived from myopicmean-variance utility maximization

Supply zs per investor is fixed (and assumed to be 0)



General setup

Mean-Variance DemandWealth evolution:

Wt+1 = R(Wt − ptzt) + (pt+1 + yt+1)zt

mWt+1 = RWt + (pt+1 + yt+1 − R pt)zt

Agent solves

maxzt

[Et Wt+1 −

a2

Vt Wt+1

],

where a > 0 is risk aversion.

Demand is

zt =Et[pt+1 + yt+1 − R pt]

a Vt[pt+1 + yt+1 − Rpt]



Academic view of financial markets

Efficient Market HypothesisEugene Fama (1965), Paul Samuelson (1965)

Market is defined as informationally efficient if

price pt of an asset reflects all available (relevant) information

...that is price is an unbiased estimation of the aggregate beliefsabout the future perspectives (fundamental value p∗t )

Fundamental price is

p∗t =

∞∑k=1

yt+k

(1 + r)k

where yt is the dividend at time t

Formally: pt = Et[p∗t ]




Efficient Market Hypothesis

Eugene Fama (1965), Paul Samuelson (1965)

pt = Et[p∗t ] = p∗t + εt

errors εt are unpredictable on the basis of information availableat time t

the returns are unpredictable:

Et[pt+1 + yt+1 − (1 + r)pt] = 0

Question: Why are the markets efficient?




Theoretical Underpinnings of Efficiency1st argument: Milton Friedman (1953)

arbitrage:people who argue that speculation is generally destabilizingseldom realize that this is largely equivalent to saying thatspeculators lose money, since speculation can be destabilizing ingeneral only if speculators on average sell when currency is lowin price and buy when it is high

Et[pt+1 + yt+1 − (1 + r)pt] = 0

m

1 + r =1pt

(Et[pt+1 + yt+1]) ⇔ pt =1

1 + rEt[pt+1 + yt+1]




Theoretical Underpinnings of Efficiency2nd argument: Robert Lucas (1978)

fully rational behavior

optimizing behavior of representative investorabsence of systematic mistakes about price distribution

assume demand functions:

zt,n =Et,n[pt+1 + yt+1]− pt(1 + r)

an Vt,n[pt+1 + yt+1]

thenpt =

11 + r

∑n

wt,n Et,n[pt+1 + yt+1]

where wt,n is the relative weight on the group “n” of agents




Theoretical Underpinnings of Efficiency2nd argument: Robert Lucas (1978)

Under Rational Expectations

pt =1

1 + r

∑n


m

pt =1

1 + r

∑n

wt,n Et[pt+1 + yt+1]

m

pt =1

1 + rEt[pt+1 + yt+1]



Empirical Consequences

Empirical Consequences“No-trade theorem”

Instead, we observe persistent trading volume:

0

1e+07

2e+07

3e+07

4e+07

5e+07

6e+07

7e+07

01/1980 01/1984 01/1988 01/1992 01/1996 01/2000 01/2004

volume of IBM shares (daily), 1980 - 2004





Both technical and fundamental analysis are useless (except by luck):

“blindfolded chimpanzee throwing darts at The Wall Street Journalcan select a portfolio that performs as well as those managed by theexperts”





No-Excess Volatility:

pt = Et[p∗t ] = p∗t + εt

thereforeσpt ≤ σp∗t




Empirical ConsequencesNo-Excess Volatility:

Instead, we observe excess volatility:

Shiller, “Do Stock Prices Move Too Much to be Justified by Subsequent Changes in

Dividends?”, American Economic Review, 71, pp.421–436, 1981




Summary of rational view on financial market

all investors possess rational expectations; they instantaneouslydiscount all available information

no opportunities for speculative profit

no place for market psychology or herding

bubbles and crushes are either non-existent or “rational”

trading volume is almost zero

no autocorrelations of returns



Implications of Heterogeneity

Heterogeneity of investorsassume that traders are not alike

pt =1

1 + r

∑n


BUT:

Et,n[pt+1] =1

1 + rEt,n

[∑m

wt+1,m Et,m[pt+2 + yt+2]

]there is no way to form expectations about others’ expectationsabout dividends and – even more so – about others’ expectationsof price

consequently, complete knowledge is impossible and agents areboundedly rational




Heterogeneity of investors

Brian Arthur (1995)

we can think of the economy ultimately as a vast collection of beliefsor hypotheses, constantly being formulated, acted upon, changed anddiscarded; all interacting and competing and evolving and coevolving;forming an ocean of ever-changing, predictive models-of-the-world

Santa-Fe Artificial Stock MarketArthur, Holland, LeBaron, Palmer and Tayler (1997)”Asset Pricing under Endogenous Expectations in an Artificial StockMarket”




Santa-Fe Artificial Stock Market: Setupeach agent is endowed with multiple forecasting models,M = 100forecasting model is a predictor

predictor checks the market condition and gets “active” ifcondition is satisfied

some of these conditions are “fundamental”some are “technical-trading”

predictor also provides two values a and b, which implyprediction E[pt+1 + yt+1] = a(pt + yt) + bfor each predictor the accuracy (squared forecasting error) istracked and updated, when the predictor is “active”

agent uses H most precise “active” predictors to combine themand get a forecaston a slower scale (on average every 250 or every 1000 periods)predictors get updated by Genetic Algorithm




Santa-Fe Artificial Stock Market: Results

Two regimes:slow-learning-rate regime

GA is evoked every 1, 000 periods in averageaccuracy of predictors are slowly updatedmarket converges rapidly and then stays in evolutionary stable REregime, where agents use similar rules, trading volume is low,technical trading does not emerge

fast-learning-rate regimeGA is evoked every 250 periods in averageaccuracy of predictors are updated relatively fastmarket is often close to the RE solution, but temporary bubblesand crashes emerge systematically, technical trading emerged inthe market


Heterogeneous Agent Models

Heterogeneous Agent Modelling

Brock and Hommes (1998, JEDC) propose a simple model which canreach a similar conclusion.

It also answers the following questions:

Can heterogeneous agents destabilize markets?

Can “less-rational” traders survive against “more-rational”?

Is the market with heterogeneous agents efficient?

Is it possible to mimic “stylized facts” with a simple(low-dimensional) model?



Model with Heterogeneous Agents

H = 2, 3, . . . types of traders

individual demand is

zh,t =Eh,t[pt+1 + yt+1 − R pt]

a Vh,t[pt+1 + yt+1 − Rpt]

Assume that agents have...

heterogeneous expectations about price

the same risk aversion

homogeneous expectations of the variance

complete knowledge of the dividend process (which is IID forsimplicity)



Fundamental SolutionEquilibrium pricing equation with homogeneous investors:

Rpt = Et[pt+1 + yt+1]

There exists an unique bounded fundamental solution p∗t (discountedsum expected future cash flow):

p∗t =Et[yt+1]

R+

Et[yt+2]

R2 + · · ·

For a special case of IID dividends, with Et[yt+1] = y:

p∗ =y

R− 1=

yr

Equilibrium pricing equation with heterogeneous investors:H∑

h=1

nh,tzh,t = 0 ⇔ pt =1

1 + r

H∑h=1

nh,t Eh,t[pt+1] +y

1 + r



Dynamics in Deviations

write the equation in the deviations from the rationalexpectations benchmark xt = pt − p∗

xt =1

1 + r

H∑h=1

nh,t Eh,t[xt+1]

there are two homogeneous, self-fulfilling solutions:if Eh,t[xt+1] = 0 then xt = 0 fundamental solution.if Eh,t[xt+1] = R2xt−1 then xt = Rxt−1 bubble solution.



Forecasting rulesAssume that belief of type h on future prices has form

Eh,t[pt+1] = p∗+fh(xt−1, ..., xt−L) ⇔ Eh,t[xt+1] = fh(xt−1, ..., xt−L)

Important special cases:

rational expectations ”f (xt−1, ..., xt−L)” = xt+1(assumes perfect foresight on all other belief-fractions nh,t+1!)

fundamentalists f ≡ 0

pure trend chasers f (xt−1, ..., xt−L) = g xt−1

strong trend chaser: g > Rcontrarian: g < 0strong contrarian: g < −R

pure bias: f (xt−1, ..., xt−L) = b.



Main QuestionDo rational agents and/or fundamentalists drive out trend chasers andbiased beliefs?

In BH paper the standard asset pricing model with two, three or fourtypes of agents is investigated

costly fundamentalists versus cheap trend followers

fundamentalists versus opposite biases

fundamentalists versus bias versus trend

Remark: All are described by linear forecast with one lag

fh,t = gh xt−1 + bh



Evolutionary selection of strategies

1 take realized excess return: Rt = pt + yt − Rpt−1

2 recall the demand by type h

zh,t−1 =Eh,t−1[pt + yt − Rpt−1]

aσ2

3 compute the realized profits in period t of type h

πh,t = Rt zh,t−1 = (pt + yt − Rpt−1)Eh,t−1[pt + yt − Rpt−1]

aσ2

= (xt − Rxt−1 + δt)Eh,t−1[xt − Rxt−1]

aσ2

with yt = y + δt.



Evolutionary selection of strategies4 define the performance measure as a (weighted sum of) realized

profitsUh,t = πh,t + wUh,t−1 − Ch

where Ch ≥ 0 are costs for predictor h, and w is memorystrength

w = 1: infinite memory; fitness ≡ accumulated wealthw = 0: memory is one lag; fitness ≡ most recently realized netprofit

5 update fractions of belief types according to the discrete choicemodel

nh,t =eβUh,t−1

Zt−1

where Zt−1 is normalization factor and β is intensity of choice.



Asset Pricing Model with Heterogeneous Beliefs

pricing equation

R xt =

H∑h=1

nh,tfh(xt−1, . . . , xt−L) =

H∑h=1

nh,tfh,t

fraction of different investors’ types

nh,t =eβUh,t−1∑H

k=1 eβUk,t−1

performance of different types

Uh,t−1 = (xt−1 − Rxt−2)fh,t−2 − Rxt−2

aσ2 − Ch



Example with linear predictors fh,t = ghxt−1 + bh

pricing equation

xt =

H∑h=1

nh,t(ghxt−1 + bh)

fraction of different investors’ types

nh,t =eβUh,t−1∑H

k=1 eβUk,t−1

performance of different types

Uh,t−1 = (xt−1 − Rxt−2)(ghxt−3 + bh − Rxt−2)

aσ2 − Ch



Two Types Example

Two-types example: Fundamentalists versus trend-followers

Two typesfundamentalists, f1,t = 0, at cost C

trend-followers, f2,t = gxt−1 at cost 0

Define difference in fractions: mt = n1,t − n2,t

Derive 3-dimensional system

Rxt = n2,tgxt−1

mt+1 = tanh(β2

[− gxt−2

aσ2 (xt − Rxt−1)− C])



Two Types Example

Fundamentalists versus trend-followersTheorem (Existence and stability of fixed points.)Let meq = tanh(−βC/2), m∗ = 1− 2R/g and x∗ be a positivesolution of

tanh(β

2[ g

aσ2 (R− 1)(x∗)2 − C])

= m∗

Then:

0 < g < R fundamental st-st E1 = (0,meq) is globally stable

g > 2R there are three st-st’s: E1 = (0,meq), E2 = (x∗,m∗) andE3 = (−x∗,m∗)

R < g < 2R E1 = (0,meq) is stable for β < β∗, E2 = (x∗,m∗) andE3 = (−x∗,m∗) are stable for β∗ < β < β∗∗.

system undergoes a pitchfork bifurcation for β = β∗

and Hopf bifurcation for β = β∗∗



Two Types Example

Rational Route to Randomness(C = 1, g = 1.2, r = 0.1, y = 10)

Corollary: fundamentalists cannot drive out trend chasers.



Two Types Example

Time series immediately after the secondary bifurcationβ = 2.81,C = 1, g = 1.2, r = 0.1, y = 10

98

99

100

101

102

103

0 100 200 300 400 500 600 700 800 900 1000

Pric

e

Time

initial price aboveinitial price below

noisy



Two Types Example

Time series far from the secondary bifurcationβ = 4,C = 1, g = 1.2, r = 0.1, y = 10

94

96

98

100

102

104

106

0 100 200 300 400 500 600 700 800 900 1000

Pric

e

Time

initial price above noisy



Other Examples

Fundamentalists versus two opposite biased beliefs

Three typesfundamentalists, f1,t = 0, at cost 0

positive bias, f2,t = b2 > 0 at cost 0

negative bias, f3,t = b3 < 0 at cost 0

Derive 3-dimensional system

R xt = n2,tb2 + n3,tb3

nj,t+1 = exp(

βaσ2 (bj − Rxt−1)(xt − Rxt−1)

)/Zt, j = 1, 2, 3



Other Examples


Theorem. (Existence and stability steady state.)

The system has unique fixed point E, which equals to thefundamental fixed point when b2 = −b3.

E exhibits a Hopf bifurcation for some β = β∗, so that E isstable for 0 < β < β∗ and E is unstable for β > β∗.



Other Examples


Theorem. (Neoclassical limit, i.e. β =∞)When biased beliefs are exactly opposite, i.e. whenb2 = −b3 = b > 0, then the system has globally stable 4-cycle.For all three types, average profit along this 4-cycle is b2.

Corollary. Fundamentalists with zero costs and infinite memory cannot beat opposite biased beliefs!

heterogeneous agent models lecture 1 introduction rational

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