Tài liệu DYNAMIC HIGHER ORDER EXPECTATIONS doc

This paper de-velops a tractable framework for solving and analyzing linear dynamic rational expectations models in which privately informed agents form higher order expectations.. But

KRISTOFFER P NIMARK

Abstract In models where privately informed agents interact, agents may need to form

higher order expectations, i.e expectations of other agents’ expectations This paper

de-velops a tractable framework for solving and analyzing linear dynamic rational expectations

models in which privately informed agents form higher order expectations The framework

is used to demonstrate that the well-known problem of the infinite regress of expectations

identified by Townsend (1983) can be approximated to an arbitrary accuracy with a finite

dimensional representation under quite general conditions The paper is constructive and

presents a fixed point algorithm for finding an accurate solution and provides weak

condi-tions that ensure that a fixed point exists To help intuition, Singleton’s (1987) asset pricing

model with disparately informed traders is used as a vehicle for the paper.

Keywords: Dynamic Higher Order Expectations, Private Information, Asset Pricing

1 IntroductionMany economic decisions involve predicting the actions of other agents For instance, firms

in oligopolistic markets may need to predict how much production capacity their competitorswill invest in and traders in financial markets may need to predict how much other traderswill be willing to pay for an asset at the next trading opportunity In settings where allagents are identical and share the same information, this becomes a trivial problem: Anindividual agent can predict the behavior of other agents by introspection, since all agentswill choose the same action in equilibrium The problem becomes more interesting if thecommon information assumption is relaxed because predicting the actions of others is thendistinct from predicting ones own’s actions But since other agents face a symmetric problem,

in order to predict the behavior of agents that form expectations about the actions of others,

an individual agent needs to predict other agents’ expectations about the actions of others,and so on, leading to the well-known infinite regress of expectations.1

This paper develops

a tractable framework for analyzing linear dynamic rational expectations models in whichprivately informed agents form higher order expectations The framework is then used to

Date: First version November 2006, this version March 21, 2011 The author thanks Francisco Barillas, Vasco Carvalho, Jesus Fernandez-Villaverde, Christian Matthes, Mirko Wiederholt, Thomas Sargent and seminar participants at New York University, Birkbeck College, Goethe University Frankfurt, the 2007 annual meeting of the Society for Economic Dynamics in Prague, Australian Workshop for Macro Dynamics, Institute for International Economic Studies at Stockholm University, University of Amsterdam, and the 2009 Econometric Society meeting in San Francisco for useful comments and suggestions Financial support from Ministerio de Ciencia e Innovacion (ECO2008-01665), Generalitat de Catalunya (2009SGR1157), Barcelona GSE Research Network and the Government of Catalonia is gratefully acknowledged.

Address: CREI, Universitat Pompeu Fabra, Ramon Trias Fargas 25-27, Barcelona 08005.

e-mail : knimark@crei.cat web page: www.kris-nimark.net

1 See for instance Townsend (1983) and Sargent (1991).

1

demonstrate that the infinite regress of expectations can be approximated to an arbitraryaccuracy with a finite dimensional representation under quite general conditions.

Conceptually, there are two distinct steps involved in deriving this result The first is

to put structure on higher order expectations by assuming that it is common knowledgethat agents form model consistent, or rational, expectations That is, all agents know thatall agents know, and so on, that all agents form model consistent expectations given theirinformation sets which gives enough structure to allow any order of expectation to be deter-mined recursively.2

The intuition is the following: Rationality of individual agents ensuresthat first order expectations are model consistent in exactly the same way that expectationsare model consistent in a standard common information rational expectations model Sincethis is common knowledge, the joint distribution of first order expectations and the truestate is also known to all agents Individual agents then form model consistent second orderexpectations by exploiting this knowledge This argument can be applied recursively to findany order of expectation, as in the static decision settings of Morris and Shin (2002) andWoodford (2002) Here, we show how common knowledge of model consistent expectationscan also be used to determine dynamic higher order expectations That is, expectationstoday of what other agents will expect tomorrow about an event the day after tomorrow,and so on This type of dynamic higher order expectations arise naturally in settings whereprivately informed agents optimize intertemporally

Deriving the dynamics of higher order expectations does not by itself solve the problem

of the infinite regress of expectations However, common knowledge of rational expectationsgives enough structure to the problem to allow us to prove the following two results: (i) Theimpact of expectations on the endogenous variables tends to zero as the order of expectationsincreases, and (ii) the variance of the approximation error introduced by only considering afinite number of orders of expectations converges to zero as the maximum order of expecta-tions considered increases.3

These are the main results of the paper and they can be shown

to hold under quite general conditions In the context of Singleton’s (1987) asset pricingmodel it is demonstrated that an accurate finite dimensional representation exists under thesame conditions that guarantee that a solution exists when agents are perfectly informed.Finite numbers can still be very large, and one may ask if these results are relevant

in practice First, the paper is constructive and provides a proof that an accurate finitedimensional representation exists as well as derive an algorithm for finding it Secondly, andagain in the context of Singleton’s (1987) asset pricing model, it is demonstrated numericallythat the equilibrium dynamics can be captured by a low number of orders of expectation,i.e by a vector of dimension in the single digits This latter result may be reassuring tothose who on grounds of human cognitive constraints doubt that economic agents form aninfinite hierarchy of higher order expectations

2 In the terminology established by Harsanyi (1967-8), there is a common prior about the true state of nature and the joint probability distribution of the true state of nature and the “types” Different “types” are distinguishable only by the realizations of the private signals that they have observed in the past The common prior then endows agents with sufficient knowledge to form model consistent expectations of the signals observed by other agents.

3 A result with similar implications for games with countable number of players and a compact action space can be found in Weinstein and Yildiz (2007).

Introducing information imperfections into macro economics and finance is not a new ideaand well-known early references include Phelps (1970), Lucas (1972, 1973, 1975), Townsend(1983), Singleton (1987) and Sargent (1991) However, recently, there has been a renewedinterest in the topic and several interesting results have emerged First, private informationabout quantities of common interest to all agents have been shown to introduce inertiaand sluggishness of endogenous variables in settings with strategic complementarities, e.g.Woodford (2002), Morris and Shin (2006), Nimark (2008), Mackowiak and Wiederholt (2009)and Angeletos and La’o (2009) Secondly, private information may also have normativepolicy implications as shown by Angeletos and Pavan (2007), Lorenzoni (2009) and Pacielloand Wiederholt (2011) Third, in financial markets, private information may introducespeculative behavior akin to the “beauty contest” metaphor of Keynes (1936), e.g Allen,Morris and Shin (2006), Bacchetta and van Wincoop (2006), Kasa, Walker and Whiteman(2006), Grisse (2009), Cespa and Vives (2007).

In spite of the renewed interest, no general solution methodology with known propertieshas emerged for solving this class of models This paper aims to help fill this gap and in order

to understand its contribution it is useful to put it into the context of alternative solutionmethods used by previous literature As a consequence of the infinite regress expectationsone could characterize most existing models of private information and strategic interaction

as efforts to avoid modeling higher order expectations explicitly, and instead find alternativerepresentations where higher order expectations do not occur as state variables.4

The mostcommon strategy for finding a finite dimensional representation in dynamic decision models

is to make private information short lived One way to achieve this is to assume that agentspool their information between periods as in Lucas (1975) or to analyze finite horizon models

as in Allen, Morris and Shin (2006) and Cespa and Vives (2007) Another way to makeprivate information short lived to assume that all shocks are observed perfectly by all agentswith a lag This assumption was first introduced by Townsend (1983) as a way to restrict thedimension of the relevant state for ‘forecasting the forecasts of others’ Optimal forecast ofany variable of interest can then be constructed using projections onto the perfectly revealedstate and a finite dimensional vector of signals

This paper demonstrates how higher order expectations can be modeled explicitly in adynamic setting without making additional assumptions to ensure that private information

is short lived The approach has at least two advantages First, the explicit modeling ofhigher order expectations helps intuition as it makes the link between private informationand the dynamics of endogenous variables more transparent Secondly, since relatively fewmodeling compromises are needed, the solution method is more suitable than some of thealternatives for empirical work As demonstrated by Nimark (2010) and Melosi (2011) thealgorithm presented here is both flexible enough and computationally fast enough to usefor likelihood based estimation of dynamic models with private information It thus makes

it feasible to empirically validate and to quantify the importance of the results from thetheoretical literature mentioned above

4 Notable exceptions are Woodford (2002), Morris and Shin (2002) and Adam (2007) who by restricting their attention to models of static decisions are able to analyze higher order expectations explicitly.

The framework presented here can also help us understand the properties of alternativeapproximation approaches Hellwig (2002) and Hellwig and Venkateswaran (2009) modifiesTownsend’s solution method by rewriting the equilibrium dynamics partly as an MA processand setting the lag T with which the state is revealed to be a very large number Intuitively,

it seems plausible to conjecture that in a stationary environment, the equilibrium dynamicsfound using this method should converge to some limit as T tends to infinity Here we showformally that there does indeed exists a finite dimensional representation of the form used

by Hellwig and Venkateswaran (2009), that as the lag T tend to infinity, converges to thetrue infinite dimensional solution

Finally, a novel approach to solve dynamic models with private information that is worthmentioning and that does not rely on restricting the dimension of the state has been proposed

by Kasa, Walker and Whiteman (2006) and further developed in Rondina and Walker (2010).These papers present methods that can be used to ensure that equilibrium outcomes are notperfectly revealing of the state in models where the number of signals is the same as thestochastic dimension of the model In this class of models, Rondina and Walker (2010) showthat endogenous variables can display waves of optimism and pessimism The approach isanalytically elegant and complementary to the methods proposed here, which are suitable forsettings where agents face a standard filtering problem with more shocks than observables

so that non-invertibility of the equilibrium process is guaranteed

The next section defines the relevant mathematical space for analyzing dynamic higherorder expectations and sets notation This is followed by a brief presentation of the model

of Singleton (1987) that will serve as a vehicle for the argument of the paper Section 4derives properties of higher order expectations that must hold in any equilibrium Section

5 introduces an average expectations operator and shows how it can be used to computeequilibrium outcomes Section 6 contains the main results of the paper It is here thatthe approximation results are presented, demonstrating that a finite number of orders ofexpectations are sufficient for an arbitrarily accurate representation of equilibrium Section

7 presents an algorithm to find the equilibrium and proves that an equilibrium exists underquite general conditions Section 8 presents properties of the solved model and shows that inpractise, only a low number of orders of expectations are necessary as equilibrium dynamicsconverge rapidly as the maximum order of expectation is increased Section 9 demonstratesthat the equilibrium dynamics can be recast in the form used by Hellwig and Venkateswaran(2009) and Section 10 concludes The Appendix contains some proofs left out of the maintext

2 PreliminariesBefore analyzing the dynamics of higher order expectations, it is necessary to invest alittle in notational machinery as well as to define exactly what is meant by a higher orderexpectation

2.1 The inner product space L2 In the model presented in the next section, the signalsthat traders observe and their expectations of fundamentals and endogenous variables areelements of the inner product space L2, which we now define

Definition 1 (The inner-product space L2.) The inner product space L2 is the collection ofall random variables X with finite variance

and with inner-product

Definition 2 Let Ω be a subspace of L2 An orthogonal projection of X onto Ω , denoted

PΩX, is the unique element in L2 satisfying

2.2 Defining higher order expectations There is a continuum of agents indexed by

j ∈ (0, 1) Agent j’s first order expectation of a variable θt ∈ L2 conditional on his period tinformation set Ωt(j) is denoted as

θ(1)t (j) ≡ E [θt | Ωt(j)] (2.5)The average first order expectation θ(1)t is obtained by taking averages of (2.5) across agents

θ(k)t ≡

ZE

a hierarchy of expectations are denoted

θ(0:k)t =h θt(0) θt(1) θt(k)

i0

(2.10)

2.2.1 Expectations about future expectations In later sections, it will prove useful to alsohave a notation for the average expectation held in period t of the average expectation held

in period t + 1 of the value of a variable in period t + 2, and so on For that purpose, wedefine the following notation The first order expectation in period t of θt+1 is defined as

a MA(2) process does not carry information that is useful for forecasts beyond a two periodhorizon Private information about an AR(1) process on the other hand is long lived Tosolve the second class of models, Singleton assumes that the innovations to the AR(1) processare perfectly and publicly observed with a two period lag This allows him to derive a finitedimensional state representation The rest of this paper uses the same set up as in Singleton’sModels 8-12 as a vehicle to show how dynamic models with private information can be solvedwithout assuming that the shocks to the hidden process ever become common knowledge.3.1 Model Set Up There is a continuum of competitive traders indexed by j ∈ (0, 1) who

at time t divide their wealth between a risky asset with price pt and coupon payment ct and

a risk free asset with return r The wealth of trader j then evolves according to

wt+1(j) = zt(j) [pt+1+ ct+1] − [zt(j)pt− wt(j)] (1 + r) (3.1)where zt(j) is the asset holdings of trader j who chooses his portfolio to maximize

E−e−γwt+1(j) | Ωt(j)

(3.2)and Ωt(j) is the information set of trader j at time t (defined below) The coupon paymentsfollow the known autoregressive process

ct= c + ψct−1+ ut : ut∼ N 0, σ2

u

(3.3)

Maximizing (3.2) subject to (3.1) yields agent j’s optimal demand for the risky asset

ztd(j) = (E [pt+1 | Ωt(j)] − (1 + r) pt) + (c + ψct)

where δ is the conditional variance of (pt+1+ ct+1) The supply of the asset at time t, zts,depends linearly on the price pt and additively on the persistent stochastic shock θt and thei.i.d disturbance t

For later reference, note that 0 < λ < (1 + r)−1

3.2 Traders’ Information Sets The basic structure of the model described above isidentical to Model 8-12 in Singleton (1987) Where this paper differ from Singleton’s is inthe assumption on what traders can observe In Singleton’s paper the information set ΩS

of trader j at time t is given by

ΩSt(j) = {st−T(j), pt−T, ct−T : T ≥ 0; vt−T, t−T : T ≥ 2} (3.10)where

st(j) = θt+ ηt(j) : ηt(j) ∼ N 0, ση2
∀ j (3.11)Each trader observes the price of the asset, pt, and the coupon payment, ct, perfectly Thepersistent component θt of the supply process is not perfectly revealed by the observation ofthe price due to the unobservable transitory supply shock t The transitory supply shock tthus serves the same purpose here as the noise traders do in Admati (1985) Trader j alsoobserves a private signal st(j) of the persistent supply process θt and it is due to the privatemeasurement error ηt(j) that the need to ’forecast the forecasts of others’ arises Singletonuses a similar method to overcome the infinite dimension of the state as Townsend (1983),i.e he assumes that the shocks to the supply process become known to all traders after afinite number of periods (which in Singleton’s case is after two periods) This allows for afinite dimensional time series representation of the model

While the assumption of public revelation of shocks with a lag is convenient from a ing perspective, it is not an assumption that is always realistic We want to solve the modelwithout imposing that all shocks are observed perfectly after a finite number of periods Theinformation set of our trader is therefore given by

model-Ωt(j) = {st−T(j), pt−T, ct−T : T ≥ 0} (3.12)

Traders thus form expectations about the future price of the asset by observing the privatesignal st(j), the commonly observable price pt and the coupon payment ct It is commonknowledge that all traders choose their portfolio to maximize (3.2) subject to the structuralequations (3.3) - (3.6).

3.3 The full information solution To solve the model we need to integrate out theaverage expectations term R E [pt+1 | Ωt(j)] dj from equation (3.8) Under full information,this could be done by iterating (3.8) forward

3.4 A complication With privately informed traders, we can still use forward substitution

of the Euler equation (3.8) This yields the equilibrium price as a function of higher orderexpectations of future values of the persistent supply process θt

where we used the notation for higher order expectations of future values of θt defined

in Section 2.2.1 The current price of the asset thus depends on the average expectation inperiod t of θt+1, the average expectation in period t of the average expectation in period t + 1

of θt+2 and so on As has been noted before, e.g Allen, Morris and Shin (2006), averagehigher order expectations, i.e expectations about other agent’s expectations generally differfrom average first order expectations and we cannot use the law of iterated expectations tointegrate out the higher order expectations in the price equation (3.15) To see why, notethat the law of iterated expectations can loosely speaking be attributed to the fact thatagents do not believe that they have ‘incorrect’ expectations so that they do not expect torevise their own expectations in a particular direction That is, first order expectations aremartingales The same is not true about expectations about other agents’ expectations Forinstance, an investor may believe that the average ‘market expectation’ of the fundamentalvalue of an asset is incorrect, but as more information becomes available to others over timethe ‘market expectation’ will be revised towards what the investor believes is the asset’s truevalue It is the fact that it can be rational to expect others to revise their expectations in

a certain direction that makes the law of iterated expectations inapplicable to higher orderexpectations It is also this fact that makes the dynamics of models with private informationinteresting

3.5 The strategy The rest of the paper is devoted to finding a finite dimensional sentation of the equilibrium price (3.15) of the form

repre-pk,t= λψ

1 − λψct− akθt(0:k)− δγλt (3.16)that is arbitrarily close to the solution to the equilibrium price (3.15) and where a0

k and θt(0:k)are finite dimensional vectors We will demonstrate that the discounted sum of higher orderexpectations of all future values of θt in (3.15) can be approximated by a linear function of

a finite number of orders of expectations of the current value of θt so that the variance ofapproximation error ∆k,t in

|λρ| < 1

4 Equilibrium properties of higher order expectations

It is possible to characterize some properties of higher order expectations using only that

it is common knowledge that agents form expectations rationally The properties derived inthis section will be important for the approximation results presented in Section 6 below, butthey also help develop intuition by making the link between common knowledge of rationalexpectations and the properties of higher order expectations explicit

4.1 First order expectations We start by establishing some properties of first orderexpectations This may seem pedantic, since properties of first order expectations are wellknown However, this will lay the groundwork for recursively deriving similar, but moreinteresting, properties of higher order expectations We start by defining a useful subspace

of L2

Definition 3 The (closed) subspace Ωt(j) ≡ sp {st−T(j), pt−T, ct−T : T ≥ 0} is the spacespanned by the history of variables observed by trader j at period t Projections onto Ωt(j)are denoted Pt,j.

From the projection theorem (e.g Brockwell and Davis (2006) ) we then know that thereexist an element θ(1)t (j) ∈ L2 such that

D

θt− θ(1)t (j), ωjE= 0 ∀ ωj ∈ Ωt(j) (4.1)that is, there exists a minimum variance expectation of θtconditional on trader j’s informa-tion set Given the linear structure of the model, past realizations of vt, t and ηt(j) form anorthogonal basis for the subspace Ωt(j) The conditional expectation E [θt | Ωt(j)] thus has

a representation of the form

θ(1)t (j) = A(L)vt+ B(L)t+ C(L)ηt(j) (4.2)where by the ex ante symmetry of traders, the (potentially infinite order) lag polynomialsA(L), B(L) and C(L) are common across traders Expectations will differ across tradersonly because of different realizations of the idiosyncratic noise shocks ηt(j)

4.2 The variance of first order expectations Here, the orthogonality property (4.1)and the representation (4.2) will be used to prove that the variance of average higher orderexpectations are bounded by the variance of lower order expectations This result will later

be used for the approximation results in Section 6 as well as for the existence results inSection 7 We start by showing that the variance of trader j’s first order expectations of θt

is bounded by the variance of the actual process θt

Lemma 1 The variance of trader j’s expectation of θt is bounded by the variance of θt, i.e

E [θt]2 ≥ Ehθt(1)(j)i

2

(4.3)Proof Define trader j’s first order expectation error ε(1)t (j) as

θt− θ(1)t (j) ≡ ε(1)t (j) (4.4)and rearrange

θt ≡ θ(1)t (j) + ε(1)t (j) (4.5)The variance of the l.h.s is E [θt]2 By (4.1), the error ε(1)t (j) is orthogonal to θ(1)t (j) ∈ Ωt(j)

so the variance of the r.h.s is simply the sum of the variances of the individual terms, whichgives the equality

According to the representation (4.2), trader j0s expectation has both a common andidiosyncratic component The fact that the idiosyncratic component is orthogonal to thecommon component allows us to prove our next result.

Lemma 2 The variance of the average expectation of θt is bounded by the variance of θt,i.e

E [θt]2 ≥ Ehθ(1)t i

2

(4.9)Proof The representation (4.2) implies that the variance of trader j’s first order expectations

is the sum of the variances of the terms in the MA representation

Ehθ(1)t (j)i

2

= E [A(L)vt]2+ E [B(L)t]2+ E [C(L)ηt(j)]2 (4.10)Since R ηt(j)dj = 0∀t the average first order expectation is simply

4.3 Variance bounds for higher order expectations

Proposition 1 The variance of higher order expectations of θt are bounded by the variance

of lower order expectations, i.e

It is straightforward to extend this result to higher order expectations of future values of

Using the same steps as in Lemma 1and Lemma 2, it can be established that

so that average first order expectation about future values of θ are also bounded since

E [θt+1]2 = E [θt]2 Defining a k order future expectation error

is common knowledge that allows us to derive the variance bounds above In the absence

of common knowledge of model consistent expectations, we would have to make alternativeassumptions about how traders in the model believe that other traders form expectations

in order to determine how traders form second order expectations Whether the variancebounds derived above would hold or not, would then depend on the properties of the secondorder beliefs about how other traders form expectations

4.4 Properties of the law of motion for higher order expectations In the solutionalgorithm proposed in Section 7, we conjecture (and verify) that the hierarchy of higherorder expectations of θt follows a vector autoregressive process of the form

We now prove that (4.24) must be a stable process

Proposition 3 If θt follows a stable process, i.e if |ρ| < 1, then common knowledge ofrational expectations implies that max |eig (M )| < 1

Proof The proof is by contradiction and is a direct corollary of Proposition 1 Consider thecase if max |eigM | = 1 This implies that at least one k 6= 0 order of expectation of θt has

a unit root and as a consequence that the variance of at least one k0 order of expectation of

θt is increasing without bound in t But from Proposition 1 we know that the variances ofhigher order expectations are bounded by the variance of θt and therefore (4.24) must be a

In this section, we have derived some properties of higher order expectations that must hold

in any equilibrium when it is common knowledge that traders form expectations rationally.Specifically, we showed that the variance of higher order expectations are bounded usingorthogonality properties of expectation errors While based on a simple insight, these resultswill later turn out to be very useful for both deriving an accurate finite dimensional solution

as well as for demonstrating that such a solution exists

5 The equilibrium priceThis section demonstrates how a simple matrix operator can be used to compute theequilibrium price for a given law of motion of the hierarchy of higher order expectations.The law of motion for the hierarchy of expectations is derived in the Section 7

5.1 An average higher order expectations operator To compute the higher orderexpectations we will use the linear operator H : R∞ → R∞ defined so that

That is, H applied to a hierarchy of expectations move the hierarchy one step up in order

of expectations If the state of the economy is given by θ(0:∞)t then the average expectations

of the true state is given by Hθ(0:∞)t and the operator H thus annihilates the first element

of a vector of higher order expectations The operator is given by the matrix

H ≡ 0 I∞ 

(5.2)where I∞ is the identity matrix.5

5.2 Equilibrium asset prices We can now derive an explicit expression for the rium price of the asset Given the conjectured law of motion (3.18) and the higher orderexpectations operator we can now compute the higher order expectations of the future values

equilib-of θt in the forward iteration (3.15) of the price Euler equation (3.8)

The one step ahead average expectation of θt is simply given by first applying H to thecomplete hierarchy of expectation to get the average expectation of the state and then apply

M to form the average expectation of the state in the next period The average expectation

in period t of the value of the persistent supply shock θt is then given by

be generated by a similar operator if θ was a non-persistent process.

where e1 is a vector with 1 in the first element and zeros elsewhere Using similar reasoning,the expectation in period t of the average expectation in period t + 1 of θt+2 is then givenby

where we used that θt = e01θ(0:∞)t The relationship between M and M H is thus analogous

to that of physical and risk neutral dynamics in the finance literature Of course, theinterpretation is different In the standard framework, fundamentals follows the physicalmeasure but assets are priced as if traders are risk neutral and fundamentals follow therisk neutral measure Here, fundamentals follow the process described by M which is thusanalogous to the physical measure, but the asset is priced as if traders observed the truestate and the fundamentals followed a VAR process with coefficient matrix M H

6 A Finite Dimensional Approximation

In the previous sections, several properties of higher order expectations were derived thatmust hold in equilibrium Though some of these properties may be interesting per se, here

we show how they can be used to prove a practical result: The equilibrium characterized by

an infinite number of orders of expectations can be approximated to an arbitrary accuracy

by a finite dimensional system That is, for practical purposes, we do not need to considerthe complete hierarchy of expectation, but instead we can find a maximum (and finite) order

of expectation that we need to consider, for any desired degree of accuracy We denote thismaximum order of expectation k

Two results are proved formally here First, we show that the weight on higher orderexpectations tend to zero as the order of expectation increases Secondly, we show that thevariance of the approximation error tends to zero as we increase the maximum number oforders of expectations k Both results are derived using a similar technique First, we define

an infinite series indexed by the number of orders of expectations We then show that theseries converges Since convergence of an infinite series implies that the individual elements

in the sequence tend to zero (while the converse is not generally true), convergence of a seriesindexed by the maximum order of expectation considered implies that the cumulative effect

of terms depending on orders of expectations higher than k tend to zero

6.1 The diminishing impact of higher order expectations The solved model willdeliver an expression for the equilibrium price pt as a function of the current hierarchy of

expectations about θt of the form

where ak is a row vector with elements defined as

ak ≡ a0 a1 · · · ak From the price equation (3.8) we already know that a0 = −δγλ The next propositionestablishes that as k → ∞ the coefficient ak tends to zero

Proposition 4 For |λρ| < 1,there exists a finite number k such that

|ak+ ak+1+ + ak+n−1+ ak+n| < ε (6.2)for any ε > 0 , for all n ≥ 1 and k > k

Proof The result is an immediate implication of the fact that {Σak}∞k=0 is a convergentseries, which we now establish To do so, we will use the fact that in the special case when

θt= θt(k)= 1∀k the equilibrium price equals the sum of the elements in the row vector a.First, note that (if by chance), all orders of expectations coincide so that θt= θ(k)t ∀ k thencommon knowledge of rational expectations implies that higher order expectations about thefuture values of θt must coincide with first order expectations That is, if there is agreementabout the current state, there must also be agreement about expected future states so that

e01(M H)k× 1∞= ρk : k = 0, 1, 2 (6.5)and substituting the right hand side of (6.5) into the price equation (5.6) gives

l.h.s of (6.7) equals the infinite sum of the elements of the row vector a∞, i.e.

which is finite Since the infinite series (6.11) converges, there exists a number k such that

|ak+ ak+1+ + ak+n−1+ ak+n| < ε for all n ≥ 1 and k > k (6.12)

Proposition 4 thus establishes that the coefficients akthat multiply the k order expectation

in the conjectured solution tend to zero as the order of expectation increases, and thatthis will hold under the same conditions that guarantee that a stable solution to the fullinformation model exists, i.e that |λρ| < 1

6.2 The variance of the approximation error Above, we demonstrated that the pact of expectations on the price tend to zero as the order of expectations increases Com-bined with the fact that the variance of higher order expectations are bounded, one mightconjecture that the variance of the contribution of the higher order expectation to the pricealso tend to zero as the order of expectation increases Here, we will now demonstrate thatthis is indeed the case but using a more direct approach that does not involve using theresult of Proposition 4 above Instead, we will define a particular convergent series (againindexed by k) so that the remainder of the sum corresponds to the variance of the approx-imation error Since the series converges, the remainder can be made arbitrarily small forlarge enough k To prove this result, we will need the following lemma

im-Lemma 3 The variance of the price pt is finite

Proof The proof uses that the higher order expectations about future expectations of futurevalues of θt in the price equation (3.15) are discounted by |ρ| < 1 and have finite variances

Definition 4 The approximation error ∆k,t associated with considering only k orders ofexpectations is defined as

so that

∆k,t = a − ak 0 
θt(0:∞) (6.16)Proposition 5 The variance of the approximation error ∆k,t tends to zero as k tends toinfinity

Proof First, define the sequence{zk}