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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 Model Main Results Applications Extensions
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
Introduction Model Main Results Applications Extensions
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
Introduction Model Main Results Applications Extensions
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
Introduction Model Main Results Applications Extensions
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
Introduction Model Main Results Applications Extensions
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
Introduction Model Main Results Applications Extensions
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
Introduction Model Main Results Applications Extensions
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
yit = (1− α)Eit[θt] + αEit[yt]
Huo & Pedroni 8/51
Introduction Model Main Results Applications Extensions
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
Introduction Model Main Results Applications Extensions
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
Introduction Model Main Results Applications Extensions
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
Introduction Model Main Results Applications Extensions
Infinite Higher Order Beliefs
yit =(1− α)Eit[θt] + αEit[yt]
=(1− α)Eit[θt] + αEit[∫
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]
]]...
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
Introduction Model Main Results Applications Extensions
Infinite Higher Order Beliefs
yit =(1− α)Eit[θt] + αEit[yt]
=(1− α)Eit[θt] + αEit[∫
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]
]]...
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
Introduction Model Main Results Applications Extensions
Infinite Higher Order Beliefs
yit =(1− α)Eit[θt] + αEit[yt]
=(1− α)Eit[θt] + αEit[∫
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
[∫Ekt[θt]
]]+ α3Eit
[∫Ejt
[∫Ekt[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
Introduction Model Main Results Applications Extensions
Infinite Higher Order Beliefs
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
Introduction Model Main Results Applications Extensions
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
Introduction Model Main Results Applications Extensions
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
Introduction Model Main Results Applications Extensions
Equivalence Result
Huo & Pedroni 14/51
Introduction Model Main Results Applications Extensions
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
Introduction Model Main Results Applications Extensions
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
Introduction Model Main Results Applications Extensions
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
Introduction Model Main Results Applications Extensions
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
Introduction Model Main Results Applications Extensions
Static Case
Huo & Pedroni 18/51
Introduction Model Main Results Applications Extensions
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
Introduction Model Main Results Applications Extensions
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
Introduction Model Main Results Applications Extensions
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
Introduction Model Main Results Applications Extensions
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
Introduction Model Main Results Applications Extensions
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
Introduction Model Main Results Applications Extensions
Applications
Huo & Pedroni 23/51
Introduction Model Main Results Applications Extensions
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
Introduction Model Main Results Applications Extensions
Example: AR(1) Signal
The same best response function
yit = (1− α)Eit[θt] + αEit[yt]
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
Introduction Model Main Results Applications Extensions
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
Introduction Model Main Results Applications Extensions
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
θt1st-order belief
Higher order beliefs display inertia, so does the aggregate action
Huo & Pedroni 26/51
Introduction Model Main Results Applications Extensions
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
θt1st-order belief2nd-order belief
Higher order beliefs display inertia, so does the aggregate action
Huo & Pedroni 26/51
Introduction Model Main Results Applications Extensions
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
θt1st-order belief2nd-order belief3rd-order belief4th-order belief
.
.
.
Higher order beliefs display inertia, so does the aggregate action
Huo & Pedroni 26/51
Introduction Model Main Results Applications Extensions
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
yt: α = 0.5
θt1st-order belief2nd-order belief3rd-order belief4th-order belief
.
.
.
Higher order beliefs display inertia, so does the aggregate action
Huo & Pedroni 26/51
Introduction Model Main Results Applications Extensions
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
yt: α = 0.0yt: α = 0.5yt: α = 0.9θt1st-order belief2nd-order belief3rd-order belief4th-order belief
.
.
.
Higher order beliefs display inertia, so does the aggregate action
Huo & Pedroni 26/51
Introduction Model Main Results Applications Extensions
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
Introduction Model Main Results Applications Extensions
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)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
The law of motion in rational expectations equilibrium
yit =φ ρyit−1 + (1− φ) xit
yt =φ ρyt−1 + (1− φ) θt
Huo & Pedroni 28/51
Introduction Model Main Results Applications Extensions
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
Introduction Model Main Results Applications Extensions
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
Introduction Model Main Results Applications Extensions
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
Introduction Model Main Results Applications Extensions
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 [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
Introduction Model Main Results Applications Extensions
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 [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
Signal: xit = εt + λεt−1 + uit
Huo & Pedroni 32/51
Introduction Model Main Results Applications Extensions
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 [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
Signal: xit = εt + λεt−1 + uit
Huo & Pedroni 32/51
Introduction Model Main Results Applications Extensions
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 [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
.
.
.
Signal: xit = εt + λεt−1 + uit
Huo & Pedroni 32/51
Introduction Model Main Results Applications Extensions
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 [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
1st-order belief2nd-order belief3rd-order belief4th-order belief
.
.
.
Signal: xit = εt + λεt−1 + uit
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Introduction Model Main Results Applications Extensions
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 [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
1st-order belief2nd-order belief3rd-order belief4th-order belief
.
.
.
Signal: xit = εt + λεt−1 + uit
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Introduction Model Main Results Applications Extensions
Apply Equivalence Result
In the MA(1) example, the corresponding α-modified signal is
xit = εt + λεt−1 + uit uit ∼ N (0, ((1− α)τu)−1)
Solution to the corresponding simple forecasting problem
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
The law of motion in rational expectations equilibrium
yit =ϑ
λ(1 + ϑL)xit
yt =ϑ
λ(1 + ϑL)(εt + λεt−1)
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Introduction Model Main Results Applications Extensions
Understanding the Oscillation Behavior
Signal: xit = εt + λεt−1 + uit
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
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Introduction Model Main Results Applications Extensions
Understanding the Oscillation Behavior
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
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Introduction Model Main Results Applications Extensions
Understanding the Oscillation Behavior
Magnitude of oscillation does not change monotonically with α
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Introduction Model Main Results Applications Extensions
Oscillation with Persistent Fundamental
Suppose agents best response is
yit = (1− α)Eit[θt] + αEit[yt]
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
θt1st-order belief2nd-order belief3rd-order belief4th-order belief
.
.
.
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Introduction Model Main Results Applications Extensions
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?
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Introduction Model Main Results Applications Extensions
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 )
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Introduction Model Main Results Applications Extensions
Apply Equivalence Result
The corresponding α-modified signal is
xit = qt + uit uit ∼ N (0, ((1− α)τu)−1)
Solution to the corresponding simple forecasting problem
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
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Introduction Model Main Results Applications Extensions
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
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Introduction Model Main Results Applications Extensions
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
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Introduction Model Main Results Applications Extensions
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
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Introduction Model Main Results Applications Extensions
Extensions
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Introduction Model Main Results Applications Extensions
Extension I: Arbitrary Fundamentals in Best Response Function
So far, the equivalence result works for
yit = (1− α)Eit[θt] + αEit[yt]
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
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Introduction Model Main Results Applications Extensions
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
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Introduction Model Main Results Applications Extensions
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
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Introduction Model Main Results Applications Extensions
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]
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Introduction Model Main Results Applications Extensions
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
Introduction Model Main Results Applications Extensions
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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Introduction Model Main Results Applications Extensions
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
Introduction Model Main Results Applications Extensions
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
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