Infinite Higher Order Beliefs and First Order Beliefs: 0 ...zhenhuo.weebly.com/uploads/2/6/4/8/26480716/slide1.pdf · Inﬁnite Higher Order Beliefs and First Order Beliefs: An Equivalence

Introduction Model Main Results Applications Extensions

Infinite Higher Order Beliefs and First Order Beliefs:

An Equivalence Result

Zhen HuoYale University

Marcelo Zouain PedroniUniversity of Amsterdam

Huo & Pedroni 1/51


Introduction

A large literature explores the role of dispersed information

A common feature of these models: infinite higher order beliefs

◦ important in accounting for observed dynamics in the data

◦ but make it hard to characterize the equilibrium

This paper: a novel characterization of widely used beauty contest models

◦ provides new insights for some classical problems

◦ allows for an easy way to solve for the equilibrium

Huo & Pedroni 2/51


This Paper: an Equivalence Result

Rational expectations equilibrium (REE)

yit = (1− α)Eit[θt] + αEit[yt]

yt =

∫yjt

◦ agents care about both exogenous fundamentals and others’ endogenous action

◦ fixed point problem with infinite higher order beliefs

Simple forecasting problem (SFP)

yit = Eit[θt]

◦ agents only care about exogenous fundamentals

◦ only first order beliefs

For REE, there exists an “equivalent” SFP with modified information Eit[·]

Huo & Pedroni 3/51


Main Results

1 Eit[·]: expectation conditional on α-modified signal process xit

◦ precisions of all idiosyncratic shocks in original signals xit discounted by α

2 Equivalence result at individual level

yit = (1− α)Eit[θt] + αEit[yt] = h(L)xit

Eit[θt] = h(L)xit

◦ the policy rule in equilibrium is the same as a simple forecasting problem

3 Equivalence result at aggregate level∫yit = (1− α)

∞∑k=0

αk Ek+1t [θt] = Et[θt]

◦ weighted sum of infinite higher order beliefs is the same as a first order belief

Huo & Pedroni 4/51


Implications of Main Results

Private signals are discounted in policy rule

◦ public signals help to coordinate, in addition to forecasting fundamentals

Dynamics of higher order beliefs v.s. first order beliefs

◦ higher order beliefs are very different from first order beliefs

◦ weighted sum of infinite higher order beliefs are similar to first order beliefs

Models with complicated dynamics can be solved and quantified

◦ apply this result to Woodford (2003) to identify REE v.s. SFP

Huo & Pedroni 5/51


Related Literature

Static beauty contest type models

◦ Morris and Shin (2002), Angeletos and Pavan (2007), and many others

Solving models with dynamics of higher order beliefs

1 Guess and verify: Woodford (2003), Angeletos and La’O (2009)

2 Time domain with truncation: Hellwig and Venkateswaran(2009), Nimark(2011)

3 Frequency domain: Rondina and Walker (2012), Huo and Nakayama (2015)

4 Time domain without truncation: this paper

Huo & Pedroni 6/51


Outline

1 General setup and main results

◦ static case

2 Applications with dynamic information

◦ inertia: long-lasting effect

◦ oscillation: overreaction and underreaction

◦ identification in Woodford (2003)

3 Extensions of basic equivalence results

◦ beauty contest models with more general best response

◦ beauty contest models with multiple actions

◦ beauty contest models in a network (joint with Laura Veldkamp)

Huo & Pedroni 7/51


A Beauty Contest Model: Best Response Function

A continuum of agents index by i ∈ [0, 1]

minyit

Eit[(1− α)(yit − θt)2 + α(yit − yt)2]

◦ θt: payoff relevant economic fundamental

◦ aggregate action

yt =

∫yit

◦ α: controls strategic complements or substitutes

Best response function


Huo & Pedroni 8/51


A Beauty Contest Model: Best Response Function

This type of best response function is common in macroeconomics

Business cycle fluctuations, Benhabib et.al (2015), Angeletos and La’O (2009)

qit = (1− α)Eit[zit] + αEit[qt]

Monopolistic firms’ pricing, Woodford (2003), Hellwig (2005)

pit = (1− α)Eit[mt] + αEit[pt]

Investment decision, Angeletos and Pavan (2007)

kit = (1− α)Eit[st] + αEit[kt]

Huo & Pedroni 9/51


A Beauty Contest Model: Information Process

Agents receive signals xit every period

xit can be any multivariate stationary process driven by Gaussian shocks

xit = M(L)

[εtuit

],

[εtuit

]∼ N

(0,

[Σε 00 Σu

])

◦ εt: aggregate shocks, affect all agents’ signal

◦ uit: idiosyncratic shocks, only affect agent i’s signal

◦ w.l.o.g, both Σε and Σu are diagonal matrices

Fundamental driven by aggregate shocks, will be relaxed later

θt = φ(L)εt

Huo & Pedroni 10/51


Infinite Higher Order Beliefs

yit =(1− α)Eit[θt] + αEit[yt]

=(1− α)Eit[θt] + αEit[∫

(1− α)Ejt[θt] + αEjt[yt]]

=(1− α)Eit[θt] + α(1− α)Eit[∫

Ejt[θt]]

+ α2Eit[∫

Ejt[yt]]

=(1− α)Eit[θt] + α(1− α)Eit[∫

Ejt[θt]]

+ α2(1− α)Eit[∫

Ejt[∫

Ejt[θt]]]

+α3Eit[∫

Ejt[∫

Ejt[yt]]]

...

Keynes (1936)’s beauty contest:

“It is not a case of choosing those that, to the best of one’s judgment, are reallythe prettiest, nor even those that average opinion genuinely thinks the prettiest.We have reached the third degree where we devote our intelligences to anticipatingwhat average opinion expects the average opinion to be. And there are some, Ibelieve, who practice the fourth, fifth and higher degrees.”

Huo & Pedroni 11/51





yjt

]=(1− α)Eit[θt] + αEit

[∫(1− α)Ejt[θt] + αEjt[yt]

]=(1− α)Eit[θt] + α(1− α)Eit

[∫Ejt[θt]

]+ α2Eit

[∫Ejt[yt]

]=(1− α)Eit[θt] + α(1− α)Eit

[∫Ejt[θt]

]+ α2(1− α)Eit

[∫Ejt

[∫Ejt[θt]

]]+α3Eit

[∫Ejt

[∫Ejt[yt]

]]...



Huo & Pedroni 11/51





yjt



]=(1− α)Eit[θt] + α(1− α)Eit

[∫Ejt[θt]

]+ α2Eit

[∫Ejt[yt]

]=(1− α)Eit[θt] + α(1− α)Eit

[∫Ejt[θt]

]+ α2(1− α)Eit

[∫Ejt

[∫Ejt[θt]

]]+α3Eit

[∫Ejt

[∫Ejt[yt]

]]...



Huo & Pedroni 11/51





yjt



]=(1− α)Eit[θt] + α(1− α)Eit

[∫Ejt[θt]

]+ α2Eit

[∫Ejt[yt]

]=(1− α)Eit[θt] + α(1− α)Eit

[∫Ejt[θt]

]+ α2(1− α)Eit

[∫Ejt

[∫Ekt[θt]

]]+ α3Eit

[∫Ejt

[∫Ekt[yt]

]]...



Huo & Pedroni 11/51



Define higher order beliefs recursively

◦ 1st order belief: Eit[E0t [θt]

]where E0

t [θt] ≡ θt

◦ 2nd order belief: Eit[E1t [θt]

]where E1

t [θt] ≡∫

Ejt[θt]

◦ 3rd order belief: Eit[E2t [θt]

]where E2

t [θt] ≡∫

Ejt[∫

Ejt[θt]]

...

◦ kth order belief: Eit[Ek−1t [θt]

]where Ekt [θt] ≡

∫Ejt

[Ek−1t [θt]

]

In equilibrium, it has to be that (given |α| < 1)

yit = (1− α)∞∑k=0

αk Eit[Ekt [θt]

]

Huo & Pedroni 12/51


Infinite Regress Problem

Eit[θt]: weighted average of current signals, and prior mean Eit−1[θt−1]

Eit[θt] = λ11Eit−1[θt−1] + g1xit

EitE1t [θt]: current signals, and prior mean Eit−1[θt−1],Eit−1E

1t−1[θt−1]

E2it[θt] = λ21Eit−1E

1t−1[θt−1] + λ22Eit−1[θt−1] + g2xit

EitE2t [θt]: current signals, and additional prior mean Eit−1E

2t−1[θt−1]

Forecasting infinite higher order beliefs requires infinite state variables

Equilibrium policy rule depends on all past signals

Huo & Pedroni 13/51


Equilibrium

All higher order beliefs are linear combinations of current and past signals

Definition

Given an information process xit, the rational expectations equilibrium is apolicy rule h(L) =

∑∞k=0 hkL

k, such that

yit = h(L)xit =

∞∑k=1

hkxit−k = (1− α)Eit[θt] + αEit[yt]

and

yt =

∫yit =

∫h(L)xit

Proposition

If α ∈ (−1, 1), then there exists a unique rational expectations equilibrium.

Huo & Pedroni 14/51


Equivalence Result

Huo & Pedroni 14/51


α-modified Information Process

Rational Expectations Equilibrium: yit = (1− α)Eit[θt] + αEit[yt]

xit = M(L)

[εtuit

],

[εtuit

]∼ N

(0,

[Σε 00 Σu

])

Simple Forecasting Problem: yit = Eit[θt]

xit = M(L)

[εtuit

],

[εtuit

]∼ N

(0,

[Σε 00 1

1−αΣu

])

Information transformation xit: α-modified signal process

◦ precisions of idiosyncratic shocks are discounted by α

Huo & Pedroni 15/51


Equivalence Result for Individual Actions

Theorem

Given a signal process xit, the equilibrium policy rule in REE is the same as inthe SFP with α-modified signal process xit:

yit = (1− α)Eit[θt] + αEit[yt] = h(L)xit,

Eit[θt] = h(L)xit.

Corollary

The forecasting rule of a weighted sum of infinite higher order beliefs is thesame as the first order belief with α-modified signal:

(1− α)∞∑k=0

αk Eit[Ekt [θt]

]= h(L)xit

Eit[θt] = h(L)xit

Huo & Pedroni 16/51


Equivalence Result for Aggregate Actions

Theorem

Given a signal process xit, the aggregate action in REE is the same as in theSFP with α-modified signal process xit:

yt =

∫Eit[θt] =

∫h(L)xit = h(L)M(L)

[εt0

]

Corollary

The forecasting rule of weighted sum of infinite higher order beliefs is the sameas first order belief with α-modified signal:

(1− α)∞∑k=0

αk Ek+1t [θt] = Et[θt]

Huo & Pedroni 17/51


Remarks

1 We prove the equivalence result via frequency domain method, which is notnecessary when applying the equivalence result

2 When θt and xit permit a finite state representation

◦ the policy rule h(L) can be obtained via Kalman filter

◦ the equilibrium also has a finite state representation

3 When θt or xit does not permit a finite state representation, the equivalenceresult still holds

◦ relevant for endogenous information

4 Conjecture: equivalence result also holds for non-stationary process

Huo & Pedroni 18/51


Static Case

Huo & Pedroni 18/51


Static Case: Morris and Shin (2002)

Assume θ is i.i.d and agents have improper prior about it

Agents receive a public and a private signal about θ

z = θ + ε, ε ∼ N (0, τ−1ε )

xi = θ + ui ui ∼ N (0, τ−1u )

The forecast about θ is

Ei[θ] =τε

τε + τuz +

τuτε + τu

xi

Huo & Pedroni 19/51


Static Case: Equilibrium

Equilibrium policy rule takes the following form

yi = (1− α)Ei[θ] + αEi[y] = h1z + h2xi

By method of undetermined coefficients

yi =τε

τε + (1− α)τuz +

(1− α)τuτε + (1− α)τu

xi

Recall that the forecast about θt is

Ei[θ] =τε

τε + τuz +

τuτε + τu

xi

The private signals are discounted by α in equilibrium

Huo & Pedroni 20/51


Where is the Discounting Coming From?

yi = (1− α)∞∑k=0

αk EiEk[θ]

Ei[θ] =τε

τε + τuz +

τu

τε + τuxi∫

Ei[θ] = E1[θ] =

τε

τε + τuz +

τu

τε + τuθ

EiE1[θ] =

τε

τε + τuz +

τu

τε + τu

(τε

τε + τuz +

τu

τε + τuxi

)...

EiEk[θ] =

[1−

(τu

τε + τu

)k+1]z +

(τu

τε + τu

)k+1

xi

As the order of higher order beliefs goes up, private signals are less important

Public signals are more useful in forecasting the aggregate action

Huo & Pedroni 21/51


REE and SFP in Static Case

Rational Expectations Equilibrium: yi = (1− α)Ei[θ] + αEi[y]

◦ signals: z = θ + ε, ε ∼ N (0, τ−1ε )

xi = θ + ui ui ∼ N (0, τ−1u )

Simple Forecasting Problem: yi = Ei[θ]

◦ signals: z = θ + ε, ε ∼ N (0, τ−1ε )

xi = θ + ui uit ∼ N (0, ((1− α)τu)−1)

α-modified signal: precision of private shock is discounted by α

Huo & Pedroni 22/51


Apply Equivalence Results in Static Model

Individual policy rule is the same in REE and SFP

yi =τε

τε + (1− α)τuz +

(1− α)τuτε + (1− α)τu

xi

Ei[θ] =τε

τε + (1− α)τuz +

(1− α)τuτε + (1− α)τu

xi

Aggregate allocation is the same in REE and SFP

y =

∫yi =

∫Ei[θ] =

τετε + (1− α)τu

z +(1− α)τu

τε + (1− α)τuθ

Sum of infinite higher order beliefs is the same as first order belief

(1− α)∞∑k=0

αk Ek+1[θ] = E[θ]

Huo & Pedroni 23/51


Applications

Huo & Pedroni 23/51


Application I: Inertia

Inertia: hump-shaped response

Noisy business cycles Angeletos and La’O (2009):

◦ Heterogeneity of information can contribute to significant inertia in theresponse of macroeconomic outcomes to such shocks

Effects of monetary policy shock Woodford (2003):

◦ Higher-order expectations adjust only sluggishly to a disturbance, even whenthe public’s average estimate of what has occurred adjusts fairly rapidly. Ifstrategic complementarities in price-setting are strong enough, the real effectsof a nominal disturbance may be both large and highly persistent.

Huo & Pedroni 24/51


Example: AR(1) Signal

The same best response function


Assume the fundamental θt follows an AR(1) process

θt = ρθt−1 + εt εt ∼ N (0, τ−1ε )

Agents receive a private signal about θt

xit = θt + uit uit ∼ N (0, τ−1u )

Huo & Pedroni 25/51


Understanding Inertia via Infinite Higher Order Beliefs

Recall that agent i’s best response is a weighted average of all higher order beliefs

yit = (1− α)∞∑k=0

αk Eit[Ekt [θt]

]

5 10 15 20 25 300

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

θt

Higher order beliefs display inertia, so does the aggregate action

Huo & Pedroni 26/51




yit = (1− α)∞∑k=0

αk Eit[Ekt [θt]

]

5 10 15 20 25 300

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

θt1st-order belief


Huo & Pedroni 26/51




yit = (1− α)∞∑k=0

αk Eit[Ekt [θt]

]

5 10 15 20 25 300

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

θt1st-order belief2nd-order belief


Huo & Pedroni 26/51




yit = (1− α)∞∑k=0

αk Eit[Ekt [θt]

]

5 10 15 20 25 300

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

θt1st-order belief2nd-order belief3rd-order belief4th-order belief

.

.

.


Huo & Pedroni 26/51




yit = (1− α)∞∑k=0

αk Eit[Ekt [θt]

]

5 10 15 20 25 300

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

yt: α = 0.5


.

.

.


Huo & Pedroni 26/51




yit = (1− α)∞∑k=0

αk Eit[Ekt [θt]

]

5 10 15 20 25 300

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

yt: α = 0.0yt: α = 0.5yt: α = 0.9θt1st-order belief2nd-order belief3rd-order belief4th-order belief

.

.

.


Huo & Pedroni 26/51


Apply Equivalence Result

In the AR(1) example, the corresponding α-modified signal is

xit = θt + uit uit ∼ N (0, ((1− α)τu)−1)

Solution to the corresponding simple forecasting problem

Eit[θt] =κ

κ+ (1− α)τuρEit−1[θt−1] +

(1− α)τu

κ+ (1− α)τuxit

κ =[ρ2 (κ+ (1− α)τu)−1 + τ−1

ε

]−1

The law of motion in rational expectations equilibrium

yit =κ

κ+ (1− α)τuρyit−1 +

(1− α)τu

κ+ (1− α)τuxit

Huo & Pedroni 27/51



In the AR(1) example, the corresponding α-modified signal is

xit = θt + uit uit ∼ N (0, ((1− α)τu)−1)


Eit[θt] =κ

κ+ (1− α)τu︸︷︷︸φ: Kalman gain

ρEit−1[θt−1] +(1− α)τu

κ+ (1− α)τuxit

κ =[ρ2 (κ+ (1− α)τu)−1 + τ−1

ε

]−1


yit =φ ρyit−1 + (1− φ) xit

yt =φ ρyt−1 + (1− φ) θt

Huo & Pedroni 28/51


Understanding the Inertia via the Equivalence Result

Impulse response of aggregation to a shock to θt at time 0

yt − yt−1 = ρt−1

[φt(1− φρ)− (1− ρ)

], for all t ≥ 0

whereφ =

κ

κ+ (1− α)τu

Necessary and sufficient condition for a hump-shaped response

φ >1

ρ− 1

◦ small τu: private signals not very precise

◦ large α: high strategic complementarity, discount the signal

Huo & Pedroni 29/51


Application II: Oscillation

Oscillation: overreaction and underreaction to fundamentals

Waves of optimisms and pessimisms, Rodina and Walker (2012)

Excess volatility of prices and returns, Kasa, et.al (2014)

Huo & Pedroni 30/51


Example: MA(1) Signal

The same best response function

yit = (1− α)Eit[εt] + αEit[yt]

Assume the fundamental εt follows an i.i.d process, εt ∼ N (0, 1)

Agents receive a private signal about εt

xit = εt + λεt−1 + uit uit ∼ N (0, τ−1u )

where λ ∈ (0, 1)

Huo & Pedroni 31/51




yit = (1− α)∞∑k=0

αk Eit[Ekt [εt]

]

1 2 3 4 5 6 7 8 9 10−0.2

0

0.2

0.4

0.6

0.8

1

1.2

ǫt

Signal: xit = εt + λεt−1 + uit

Huo & Pedroni 32/51




yit = (1− α)∞∑k=0

αk Eit[Ekt [εt]

]

1 2 3 4 5 6 7 8 9 10−0.2

0

0.2

0.4

0.6

0.8

1

1.2

ǫt

1st-order belief


Huo & Pedroni 32/51




yit = (1− α)∞∑k=0

αk Eit[Ekt [εt]

]

1 2 3 4 5 6 7 8 9 10−0.2

0

0.2

0.4

0.6

0.8

1

1.2

ǫt

1st-order belief2nd-order belief


Huo & Pedroni 32/51




yit = (1− α)∞∑k=0

αk Eit[Ekt [εt]

]

1 2 3 4 5 6 7 8 9 10−0.2

0

0.2

0.4

0.6

0.8

1

1.2

ǫt

1st-order belief2nd-order belief3rd-order belief4th-order belief

.

.

.


Huo & Pedroni 32/51




yit = (1− α)∞∑k=0

αk Eit[Ekt [εt]

]

1 2 3 4 5 6 7 8 9 10−0.2

0

0.2

0.4

0.6

0.8

1

1.2

yt: α = 0.5

ǫt


.

.

.


Huo & Pedroni 32/51




yit = (1− α)∞∑k=0

αk Eit[Ekt [εt]

]

1 2 3 4 5 6 7 8 9 10−0.2

0

0.2

0.4

0.6

0.8

1

1.2

yt: α = 0.0yt: α = 0.5yt: α = 0.9ǫt


.

.

.


Huo & Pedroni 32/51



In the MA(1) example, the corresponding α-modified signal is

xit = εt + λεt−1 + uit uit ∼ N (0, ((1− α)τu)−1)


Eit[εt] =ϑ

λ(1 + ϑL)xit

ϑ =1

2

λ+1

λ+

1

λ(1− α)τu−

√(λ+

1

λ+

1

λ(1− α)τu

)2

− 4


yit =ϑ

λ(1 + ϑL)xit

yt =ϑ

λ(1 + ϑL)(εt + λεt−1)

Huo & Pedroni 33/51


Understanding the Oscillation Behavior


The impulse response of Eit[εt] to εt shock

Eit[εt] =

ϑλ, if t = 0

− 1λ(λ− ϑ)(−ϑ)t, if t > 0

Underestimate εt, then overestimate εt+1

◦ underestimate ε0: Ei0[ε0] = ϑλ

< ε0 = 1

◦ overestimate ε1: Ei1[ε1] = ϑλ

(λ− ϑ) > ε1 = 0

◦ underestimate ε2: Ei2[ε2] = −ϑ2

λ(λ− ϑ) < ε2 = 0

Huo & Pedroni 34/51



Persistence of estimates ϑ monotonically increases in (1− α)τu

Size of undere(over)stimation does not monotonically increase in (1− α)τu

∣∣∣εt − Eit[εt]∣∣∣ =

1− ϑλ, if t = 0

1λ(λ− ϑ)ϑt, if t > 0

◦ if (1− α)τu high, εt can be accurately estimated, little undere(over)stimation

◦ if (1− α)τu low, most variations of xit are attributed to noise uit

Huo & Pedroni 35/51



Magnitude of oscillation does not change monotonically with α

Huo & Pedroni 36/51


Oscillation with Persistent Fundamental

Suppose agents best response is


whereθt = ρθt−1 + εt

1 2 3 4 5 6 7 8 9 100

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

yt: α = 0.5


.

.

.

Huo & Pedroni 37/51


Application III: Identification in Woodford (2003)

VAR evidence on the effects of monetary policy shock

◦ hump-shaped response of output and inflation

◦ inertia in dynamics

Woodford (2003): inertia can be the result of dispersed information

◦ inflation and output are driven by all higher order beliefs

◦ higher order beliefs are more persistent, harder to change

Can we tell whether the inertia is due to higher order or first order beliefs?

Huo & Pedroni 38/51


Setup in Woodford (2003)

Individual firm i’s pricing decision

pit = (1− α)Eit[qt] + αEit[pt]

Aggregate variables

◦ qt: exogenous aggregate nominal expenditure

◦ inflation rate: πt = pt − pt−1

◦ real output: yt = qt − pt

Signals

◦ growth rate of nominal expenditure is driven by monetary shock εt

∆qt = ρ∆qt−1 + εt, εt ∼ N (0, τ−1ε )

◦ firms observe a private signal

xit = qt + uit, uit ∼ N (0, τ−1u )

Huo & Pedroni 39/51



The corresponding α-modified signal is

xit = qt + uit uit ∼ N (0, ((1− α)τu)−1)


Eit[θt] =(φ− κ) (φ+ ρκ) (φ− ρ (1 + ρ)κL)

(φ+ ρκ)φ2 − (1 + ρ)2 κφ2L+ ρ (1 + ρ) (ρ2 (κ− φ)κ2 + ρκ3 + κφ2)L2xit

φ =κ+ (1− α) τu

κ =

((1 + ρ)2 φ2 + ρ2

(φ− ρ2κ

)(φ− κ)

φ (φ+ ρκ)2+ τ−1

ε

)−1

The policy rule in rational expectations equilibrium

pit =(φ− κ) (φ+ ρκ) (φ− ρ (1 + ρ)κL)

(φ+ ρκ)φ2 − (1 + ρ)2 κφ2L+ ρ (1 + ρ) (ρ2 (κ− φ)κ2 + ρκ3 + κφ2)L2xit

Huo & Pedroni 40/51


Impulse Response of Inflation and Output

Define τu = (1− α)τu

As τu decreases, the peak of impulse response delays

0 5 10 15 20 25 300

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

0.09

0.1

Inflation

τu = 2

τu = 0.5

τu = 0.01

0 5 10 15 20 25 30−0.02

0

0.02

0.04

0.06

0.08

0.1

0.12

0.14

Output

τu = 2

τu = 0.5

τu = 0.01

Huo & Pedroni 41/51


Can the Model Match the Impulse Response in the Data?

Choose ρ, τu, τε to match IRF of inflation and output in the first 15 periods

With relatively small τu, we can match the hump-shaped response

0 5 10 15−0.1

−0.05

0

0.05

0.1

0.15

0.2

0.25

Inflation

DataModel

0 5 10 15−0.4

−0.3

−0.2

−0.1

0

0.1

0.2

0.3

Output

DataModel

Source: Christiano et al. (2005)

By equivalence result, REE with τu = τu1−α yields the same impulse response

Huo & Pedroni 42/51


Identifying REE vs SPF using Survey Data

To obtain the same aggregate allocation

◦ REE displays smaller forecast dispersion and forecast error

Data on individual forecast errors help to distinguish REE from SFP

0 0.2 0.4 0.6 0.8 10

0.05

0.1

0.15

0.2

0.25

α

SFP

REE

Data

REE with α = 0.95 can match cross-sectional belief dispersion in the Surveyof Professional Forecasters

Huo & Pedroni 43/51


Extensions

Huo & Pedroni 43/51


Extension I: Arbitrary Fundamentals in Best Response Function

So far, the equivalence result works for


Action may depend on any shock

yit = γEit[ξit+k] + αEit[yt]

◦ ξit+k can depend on aggregate or idiosyncratic shocks

◦ ξit+k can depend on future or past shocks

A variation of the equivalence results applies for this case

Huo & Pedroni 44/51


Extension I: Arbitrary Fundamentals in Best Response Function

By Kalman filter, Eit[ξit+k] = g1(L)εt + g2(L)uit = ϕt + ωit

In equilibrium

yit = γ(ϕt + ωit) + αEit[yt]

= γ(ϕt + ωit) + αγ∞∑k=0

αk Eit[Ekt[ϕt]]

Recall we have shown that

(1− α)∞∑k=0

αk Eit[Ekt [ϕt]

]= h(L)xit

Eit[ϕt] = h(L)xit

Therefore, the individual policy rule is

yit = γ(ϕt + ωit) +αγ

1− αh(L)xit

Huo & Pedroni 45/51


Extension I: Example

Noisy business cycles Angeletos and La’O (2009):

yit = γ(ϕt + ωit) + αEit[yt]

◦ ωit, ϕt: idiosyncratic and aggregate components of TFP

◦ agents observe ϕt + ωit, but do not know ϕt perfectly

By our equivalence result,

(1− α)∞∑k=0

αk Eit[Ekt [ϕt]

]= h(L)xit

Eit[ϕt] = h(L)xit

The individual policy rule is

yit = γ(ϕt + ωit) +αγ

1− αh(L)xit

Huo & Pedroni 46/51


Extension II: Multiple Actions

Suppose there are more than one action that an agent needs to choose

yit = Eit[θt] + AEit [yt]

yit,1yit,2

...yit,n

=

Eit[θt,1]Eit[θt,2]

...Eit[θt,n]

+

a11 a12 . . . a1na21 a22 . . . a2n

......

. . ....

an1 an2 . . . ann

Eit[yt,1]Eit[yt,2]

...Eit[yt,n]

Huo & Pedroni 47/51


Extension II: Multiple Actions

If A is diagonalizable and all eigenvalues are within the unit circle

yit =

∞∑k=0

AkEit[Et[θt]

]

=

∞∑k=0

U

ω1

ω2

. . .

ωn

k

U−1Eit[Et[θt]

]

=n∑j=1

UEjV∞∑k=0

ωkj Ei,t[Ekt [θt]

]︸︷︷︸SFP with ωj -modified signal

yit is a weighted average of different first order beliefs

Huo & Pedroni 48/51


Extension III: Network

Suppose there are n agents in the economy.

Agent i’s best response function is

yit = Eit[θt] +n∑j=1

WijEit [yjt]

The fundamental is a stationary process driven by common shocks

θt = φ(L)ηt

Agent i receives a public signal zt and a private signal xit about θt

zt = θt + εt, εt ∼ N (0, τ−1ε )

xit = θt + uit, uit ∼ N (0, τ−1u )

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Extension III: Equivalence Result

k-th modified signal process

zt = θt + εt, εit ∼ N (0, τ−1ε )

xit = θt + uit, uit ∼ N (0, τ−1k )

where τ−1k is the k-th eigenvalue of τu(I−W)−1.

Simple forecasting problem with k-th modified signal process

Ekt[θt] = h1k(L)zt + h2

k(L)xit

Agent i’s policy rule is a weighted average of n different forecasting problems

yit =n∑k=1

UikU−1k 1n×1︸︷︷︸

weight on k’s SFP

[h1k(L)zt + h2

k(L)xit]

where Uk is the k-th eigenvector of τu(I−W)−1.

Huo & Pedroni 50/51


Conclusion

An equivalence result between REE and SFP with modified signals

◦ individual policy rule in REE is the same

◦ sum of infinite higher order beliefs is the same as a first order belief

Provide an easy solution and characterization for beauty contest models

◦ help understanding inertia and oscillation

◦ useful for quantitative exercise

This equivalence result also extends to models with network

Huo & Pedroni 51/51

Documents

Infinite Higher Order Beliefs and First Order Beliefs: 0 ...zhenhuo.weebly.com/uploads/2/6/4/8/26480716/slide1.pdf · Inﬁnite Higher Order Beliefs and First Order Beliefs: An Equivalence