Introduction Framework Testability Disagreement & Uncertainty GMM Theorem Proof
Uncertainty and Disagreement in EquilibriumModels
Nabil I. Al-Najjar & Eran ShmayaNorthwestern University Tel Aviv University
RUD, Warwick, June 2014
Forthcoming: Journal of Political Economy
Introduction Framework Testability Disagreement & Uncertainty GMM Theorem Proof
Motivation
Equilibrium concepts, such as Nash equilibrium andrational expectation equilibrium, explicitly assume thatagents know the “true” and “objective” stochastic processP governing their environment
Motivation:There is a “true” process PP is not metaphysical: any Q 6= P can be “proven wrong”In particular, =⇒no disagreement
Sargent 2008 describes (critically)A rational expectations equilibrium asserts that the samemodel is shared by (1) all of the agents within the model, (2)the econometrician estimating the model, and (3) nature,also known as the data generating mechanism.
This ought to be disturbing to a subjectivist
Introduction Framework Testability Disagreement & Uncertainty GMM Theorem Proof
Outline
Examine the idea that equilibrium probabilities are “objective,”
Operationalize the idea of “objectivity”
“objective” = “statistically testable”
Relate to ongoing debates in Macro-Finance-Econometrics
Formulate the theoretical questionuntied to a specific economic context.. in a way a (decision) theorist can relate to
Introduction Framework Testability Disagreement & Uncertainty GMM Theorem Proof
Philosophy
Two polar traditions:
1 Rational Expectations: beliefs are anchored in empiricalfrequencies
Econometrically tractable
..but rules out “uncertainty” and disagreement
2 Subjectivist Tradition: a belief is “a state of mind”Consistent with heterogeneity and disagreement
..but difficult to implement in practice (anything goes?)
The Hope:Merge the two traditions into a subjectivist equilibrium theory
This paper shows why this is very hard;better understanding of the sources of difficulty
Introduction Framework Testability Disagreement & Uncertainty GMM Theorem Proof
Dynamical Economic Systems
A dynamic model is a stochastic process P
Finite set of period outcomes SSequences of outcomes
H = S × S × · · ·
P is a distribution on H
Information: finite initial histories hn−1
Standard topology and measuresBorel sigma-algebra on products topologies
Finite S is technically convenient but not critical
Introduction Framework Testability Disagreement & Uncertainty GMM Theorem Proof
Stationarity
StationarityFor every integer m, the marginal distribution of P on
Sk+1 × · · ·Sk+m
does not vary with k = 1,2, . . .
Important in theoretical and econometric models
Later we relax it
P is the set of stationary distributions
Introduction Framework Testability Disagreement & Uncertainty GMM Theorem Proof
Example: Asset Pricing Models
Stochastic consumption process {cn} of the representativeagent
(Assume rich enough finite outcome space)
{cn} is exogenously givenAsset prices and returns are determined by the model
Time separable utility∑∞
t=1 δn u(cn)
Introduction Framework Testability Disagreement & Uncertainty GMM Theorem Proof
Example: Asset Pricing Models
The stochastic discount factor (SDF) is the process:
mn ≡ δu′(cn+1)
u′(cn)
Our primitive is a process P on {mn}.
The equilibrium return of any asset is determined via theEuler equation:
EP[mn+1 Rn+1 |hn] = 1
where {Rn} is an asset’s (stochastic) gross rate of return
Introduction Framework Testability Disagreement & Uncertainty GMM Theorem Proof
Example: Markov Perfect equilibrium
Exogenous variables evolve according to a Markoviantransition π
Notation too heavy; see the paper
Ingredients:Firms use Markovian strategies
Known π+ Markovian strategies =⇒Markovian P
P has “every right” to be called the true objective theory,shared by agents and outside observers
Real “sense” that probabilities are “objective”..hence no room for disagreementNo uncertainty about the long-runStandard econometrics “works”
Introduction Framework Testability Disagreement & Uncertainty GMM Theorem Proof
Tests: Idea
Say an outside observer (a modeler, econometrician)assumes that agents hold correct equilibrium beliefs
Verifying this requires an objective criterion that comparesbeliefs with an observed realization of the process
Idea to formalize:“If P is incorrect and the correct alternative Q ispresented, P can be proven wrong with some positiveprobability”
Introduction Framework Testability Disagreement & Uncertainty GMM Theorem Proof
Tests: Formal Definition
A statistical test is a function
T : P × H → {0,1}
that takes as input a distribution P and a history h and returns ayes/no answer
The set of all histories consistent with P:
TP ≡ {h : T (P,h) = 1}
These are the empirical predictions of P relative to test T :if P is correct, then the observed sequence of outcomesmust be in TP
Introduction Framework Testability Disagreement & Uncertainty GMM Theorem Proof
Type I Errors
The Type I error of a test T at P is the number:
1− P(TP)
In our asymptotic setting, it is natural to consider tests thatare free of Type I error:
P(TP) = 1, ∀P
With finite observations, small but positive Type I errorsmust be allowed
Asymptotic setting makes for shaper statements andinterpretation of the results
Introduction Framework Testability Disagreement & Uncertainty GMM Theorem Proof
Testability
DefinitionA stationary distribution P is testable if for every stationaryQ 6= P there is a test T ? such that
1 T ? is free of Type I error
2 Q(T ?
P)< 1
Q(T ?
P)
= probability that P is confirmed when false
Just the Type II error of T ?
Definition says: for any Q there is some test for which TypeII error is not 100% (i.e., has some power)
Introduction Framework Testability Disagreement & Uncertainty GMM Theorem Proof
Testability: Intuition
Imagine we propose P as an equilibrium model of assetprices
Suppose P is not testable
Then someone can suggest another model Q 6= P suchthat no matter what Type I error-free test we use, there isno way to prove P false
P has no testable implications against some alternative Q
We think of testability as a desirable property of a model
Introduction Framework Testability Disagreement & Uncertainty GMM Theorem Proof
Disagreement
There is a strong intuition that people hold different beliefsabout their environments
Differing beliefs perhaps reflect persistent imperfections inlearning
Empirical paradoxes and theoretical responses:Trade volume puzzlesSpeculative tradeOne response: models of heterogeneous priors
Introduction Framework Testability Disagreement & Uncertainty GMM Theorem Proof
Precluding Disagreement
Definition
Two beliefs Q1,Q2 are compatible if for every event B,Q1(B) = 1 if and only if Q2(B) = 1.
Mutual absolute continuity
DefinitionA stationary belief P precludes disagreement if for everystationary belief Q compatible with P, we have Q = P.
P accommodates disagreement if people can hold differentcompatible beliefs
Introduction Framework Testability Disagreement & Uncertainty GMM Theorem Proof
Uncertainty about Long-Run Fundamentals
What do we mean by long-run fundamentals?
Suppose P is a Markov process
There is uncertainty about next period outcome ..and theperiod next, and next..
But there is no uncertainty about the long-run distributionFormally, Markov processes are mixingThere is uncertainty about near horizon events but notabout the long-run
Value of information: additional observations will not helpme better predict the long-run
Implication: I already know it
Introduction Framework Testability Disagreement & Uncertainty GMM Theorem Proof
No-Long Run Uncertainty
For a sequence (xk , xk+1, . . . ) define the limiting average:
V (xk , xk+1, . . . ) = limn→∞
1n
n∑i=k
xi ,
whenever the limit exists
Introduction Framework Testability Disagreement & Uncertainty GMM Theorem Proof
No-Long Run Uncertainty
For a function f : Sk → Rm, with finite k , define:
fn ≡ f (sn−k+1, . . . , sn) ,n = k , k + 1, . . .
fk , fk+1, . . . is a payoff stream that depends on pastrealizations via the (stationary) formula f
f ≡ {fk , fk+1, . . .}
V (f ) is the random variable defined by:
V (f )(h) = limn→∞
1n
n∑i=k
fk
i.e., the limiting average of the stream fk , fk+1, . . .generated along h
Introduction Framework Testability Disagreement & Uncertainty GMM Theorem Proof
No-Long Run Uncertainty
DefinitionP displays no long-run structural uncertainty if for every f andhistory ht−1 with P(ht−1) > 0
EP(· | ht−1) V (f ) = EP V (f )
We may entertain doubts about the short-run outcomes,but not about the long-run behavior of the process
Introduction Framework Testability Disagreement & Uncertainty GMM Theorem Proof
Moment Conditions: Asset Pricing Revisited
Moment conditions are “finite-horizon features of P”Example: the Euler equation of asset pricing
EP[mn+1 Rn+1 |hn]− 1 = 0
Assume CRRA utility, with unknown parameter γFundamental parameters to be estimated:
Z = {δ, γ, moments of P . . .}
Empirical moment condition
1n
N∑n=1
[cγn
cγn+1Rn+1 | information
]− 1 = 0
Introduction Framework Testability Disagreement & Uncertainty GMM Theorem Proof
Moment estimation underlie many statistical techniquesused in econometric practice
Generalized method of moments (GMM) introduced inHansen (1982)
GMM itself covers many techniques such as linear andnon-linear regressions, VAR, . . . etc
Introduction Framework Testability Disagreement & Uncertainty GMM Theorem Proof
Moment Conditions and Statistical Identification
DefinitionA moment condition is a bounded continuous function
f : Z × Sk → Rq
k and q are positive integersZ ⊂ Rm is compact
f identifies P if there is a unique z = z(P, f ) such that
EP f (z, s1, . . . , sk ) = 0.
Introduction Framework Testability Disagreement & Uncertainty GMM Theorem Proof
Empirical Identification
In GMM, starting with a theoretical moment condition, oneestimates the true zP by the element
z ∈ Z
that minimizes the empirical moment condition
(Details are standard: see the paper or a time series textbook)
DefinitionA stationary distribution P can be empirically identified ifzn
P−→ z for every moment condition that identifies P.
Introduction Framework Testability Disagreement & Uncertainty GMM Theorem Proof
This says all of the implications of P can be recovered, viamoment conditions, from observing the evolution of theprocess.
Alternatively, if P ′ cannot be empirically identified, thenthere must be an implication of P ′
EP′ f (z, s1, . . . , sk ) = 0
such that z cannot be recovered, even asymptotically asdata grows without bound
Introduction Framework Testability Disagreement & Uncertainty GMM Theorem Proof
Main Result
Theorem
For any stationary process P, the following four statements areequivalent:
1 P is testable;
2 P precludes disagreement;
3 P precludes structural uncertainty;
4 P can be empirically identified.
Introduction Framework Testability Disagreement & Uncertainty GMM Theorem Proof
The Testable Case
Useful result:
PropositionThe conditions of the Theorem are satisfied for
Every irreducible memory k Markov processEvery hidden Markov process where the underlyingMarkov process is irreducible.
This case covers:
All MPE models in IO, PE, etc..Parametric consumption-based asset pricing models
Lucas-style asset pricing modelExogenous and endogenous variables are Markovian
By the Theorem, they preclude disagreement and uncertaintyabout long run fundamentals
Introduction Framework Testability Disagreement & Uncertainty GMM Theorem Proof
Simple intuition Skip slide
Two stationary Markov processes, indexed by transitions
{π1 6= π2}
Simplest possible example: i.i.d. processes
Each Pπi is testable (by the proposition)
Given Pπi , easy to see that:There can be no disagreement (any stationary P ′
compatible with Pπi must coincide with it)
Given Pπi there is no uncertainty about the long-run
GMM is consistent
Introduction Framework Testability Disagreement & Uncertainty GMM Theorem Proof
Uncertain beliefs
Introduce doubts about which of {π1, π2} is the true model
µ ∈ ∆ ≡ interior ∆{π1, π2}
Disagreement and uncertainty about the long-run are possible
However: Let µ, µ′ be any pair of beliefs in ∆ and T any Type Ierror-free test. Then
Pµ{T (µ′,h) = 1} = 1
An agent who believes the data is generated by µ also believesthat no zero-Type-I-error-test can disprove the wrong belief µ′
How should Rational Expectations be defined in this case?
Introduction Framework Testability Disagreement & Uncertainty GMM Theorem Proof
Proof Skip slide
Zero Type I error means:
1 = Pµ′{T (µ′,h) = 1}
= Pπ1{T (µ′,h) = 1}µ′(π1) + Pπ2{T (µ′,h) = 1}µ′(π2).
=⇒
Pπ1{T (µ′,h) = 1} = Pπ2{T (µ′,h) = 1} = 1
=⇒
Pµ{T (µ′,h) = 1} = Pπ1{T (µ′,h) = 1}µ(π1)+Pπ2{T (µ′,h) = 1}µ(π2)
= 1
The other equality is proved similarly.
Introduction Framework Testability Disagreement & Uncertainty GMM Theorem Proof
Concluding Observations
There is an emerging consensus that injecting subjectivityin equilibrium analysis is essential to explaining theoreticaland real-world puzzles
Much of economic theory is based on the idea that there isa ‘true’ objective process (deep down we are objectivists)
My view is that this commitment is like enter into a “bargainwith the devil,” eventually exacting a substantial price
My view also is that this has little to do with the axioms ofdecision theory
Question: is there a way to develop subjectivist statistics?
Introduction Framework Testability Disagreement & Uncertainty GMM Theorem Proof
Additional slidesnot shown in the talk.
Introduction Framework Testability Disagreement & Uncertainty GMM Theorem Proof
Non-Parametric Models
Empirical models that make no explicit assumptions aboutthe structure of P
Use GMM to estimate moments of P
These model assume that GMM is consistent
By the Theorem, these models assume that the underlyingprocess is testable and thus preclude disagreement andstructural uncertainty
Introduction Framework Testability Disagreement & Uncertainty GMM Theorem Proof
The Non-Testable Case
An implication of that theorem is that in stationarynon-testable environment there is always something tolearn about the long-run fundamentals.
As agents learn from observing the process
Changing beliefs can lead to non-stationary decisions,undermining the stationarity of the process
Goal: introduce limited form of non-stationarity so we can stillleverage the Theorem
Introduction Framework Testability Disagreement & Uncertainty GMM Theorem Proof
Learning, Stationarity, and Optimization
Write S = X × AAction profiles A at time nExogenous variables X not affected by these actions
Evolution of the systemstochastic process P on H = (X × A)∞
PX the marginal of P on X∞
PA the marginal on A∞
Require PX to be stationary, but not necessarily the entireprocess P
Makes it possible to apply the Theorem restricted to PX
Beliefs about the exogenous variables are stationary, butactions (and beliefs about actions) are not necessarily so
Introduction Framework Testability Disagreement & Uncertainty GMM Theorem Proof
Passive Learning
MPE setting except that agents are uncertain about thetransition π of exogenous variables
Pakes and Ericson (1998) study a model along these lines
Their goal is to empirically test for the implications ofagents’ passive learning on industry dynamics.
Agents have beliefs µi , i ∈ I, with a finite and commonsupport Π = {π1, . . . , πL},L ≥ 1
Agents learn, but learning is passiveagents’ actions do not influence exogenous variablesRules out agents’ active experimentationKeeps the theoretical and empirical analysis tractable
Formally, each πl ∈ Π takes the form πl(xn)
Introduction Framework Testability Disagreement & Uncertainty GMM Theorem Proof
Passive Learning
Let PX (πl) be the stationary distribution on X∞ induced bythe transitions πl ∈ Π
Agent i believes that the exogenous variables evolveaccording to the stationary distribution
PX (µi) = µi(π1)PX (π1) + · · ·+ µi(πl)PX (πl)
Agents believe none can influence this distribution
How about actions?
Introduction Framework Testability Disagreement & Uncertainty GMM Theorem Proof
Learning and Optimization
Consider three cases:
Case 1: L = 1, so agents know the true transition π.
All of P falls under the testable case
Case 2: L > 1 and agents share a common prior (µi = µ,for all i) with support {π1, . . . , πL}.
Case 3: L > 1 and agents have different priors (µi 6= µj , forsome i , j ) with common support {π1, . . . , πL}.
We show that under assumptions about payoffsthat PA may be inconsistent with any stationarybehavior
Introduction Framework Testability Disagreement & Uncertainty GMM Theorem Proof
The Econometrician’s Perspective
Pakes and Ericson (1998) exploit this to conduct tests forstationarity
An econometrician making the usual assumptions ofrational expectations (e.g. assume GMM or testability) willget very strange results looking at the data
This issue is endemic (was pointed out in asset pricingmodels, Lewellen-Shanken (2002), Brav-Heaton (2002),Weitzman (2005) among many others)
Introduction Framework Testability Disagreement & Uncertainty GMM Theorem Proof
Testing Beliefs vs. Testing Behavior
When beliefs disagree or display uncertainty aboutlong-run fundamentals, the theorem says that it is notpossible to devise an objective test to refute alternativebeliefs Q
It not mean that disagreement and structural uncertaintycan have observable implications on behavior and marketoutcomes
The Theorem does imply that in this case one would needtools that look very different form GMM
Introduction Framework Testability Disagreement & Uncertainty GMM Theorem Proof
The Econometrician’s Perspective Redux
Pakes and Ericson (1998) develop econometric techniques in aMarkovian setting to detect firms’ learning
Main issue is that beliefs are not directly observable
Need to look for indirect manifestations of subjectiveuncertainty
Idea: look for whether early observations have a persistentimpact on outcomes
Other example can be found in the context of asset pricing
But all this is very hard. GMM is so nice.. when it works!
Introduction Framework Testability Disagreement & Uncertainty GMM Theorem Proof
Bayesian Games
This part is informal; see the paper for detailsNature moves to select a stationary process forfundamentals
Industry shocks in an IO or trade modelPlayers strategies are part of an equilibrium
In particular, they best respond to their beliefs about theevolution of fundamentals and strategies of others
Unless the game is “weird” when players learn, theirbehavior is no longer stationary
Introduction Framework Testability Disagreement & Uncertainty GMM Theorem Proof
Idea of the Proof
The four properties above characterize ergodicdistributions
Given a stationary P and any (Borel) function g : SN → R, theergodic theorem states that the limit
g = limn→∞
1n
n−1∑k=0
g (sk , sk+1, . . . )
exists P-a.s. and that EP g = EPg.
If the limit g is constant (i.e., g = EPg almost surely) for every gthen P is called ergodic.
Introduction Framework Testability Disagreement & Uncertainty GMM Theorem Proof
Idea of the Proof
Ergodic Decomposition Theorem: Every stationaryprocess is a mixture of ergodic processes.For an ergodic process P, the decomposition is trivia
P has distinct support from any other ergodic processTestability
Empirical averages coincide with the theoreticalprobabilities
Moment conditionsIf P is a non-trivial mixture, then there is uncertainty aboutthe true ergodic component
There is uncertainty about long-run fundamental propertiesof the process