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Statistical Sunspots∗

William Branch

University of California, Irvine

Bruce McGough

University of Oregon

Mei Zhu

Shanghai University of Finance and Economics

October 9, 2017

Abstract

This paper shows that belief-driven economic ﬂuctuations are a general

feature of many determinate macroeconomic models. Model misspeciﬁcation

can break the link between indeterminacy and sunspots by establishing the

existence of “statistical sunspots” in models that have a unique rational ex-

pectations equilibrium. Building on the insights of Marcet and Sargent (1989)

and Sargent (1991), with some state variables ‘hidden’ to a set of agents the

state vector expands to include agents’ expectations and, in a restricted per-

ceptions equilibrium, agents form beliefs by projecting the state vector onto

their set of observables. This set of observables can include serially correlated

non-fundamental factors (e.g. sunspots, judgment, expectations shocks, etc.).

Agents attribute, in a self-fulﬁlling way, some of the serial correlation observed

in data to extrinsic noise, i.e. statistical sunspots. This leads to sunspot equilib-

ria in models with a unique rational expectations equilibrium. Unlike rational

sunspots, these equilibria are stable under learning. Applications are devel-

oped in the context of a New Keynesian, an asset-pricing, and a pure monetary

model.

JEL Classiﬁcation: D82; D83; E40; E50

Keywords: adaptive learning, animal spirits, business cycles, optimal monetary

policy, heterogeneous beliefs.

∗This paper has beneﬁted from discussions with John Duﬀy, Stefano Eusepi, Cars Hommes, Blake

LeBaron, Guillaume Rocheteau, and John Williams. We also thank seminar participants at U.C.

Irvine, the 2017 Workshop on Expectations in Dynamic Macroeconomics at the St. Louis Fed, the

2017 Society of Computational Economics Meetings, and the Institute of Mathematical Behavioral

Science. Mei Zhu acknowledges ﬁnancial support from NSFC funding (grant no. 11401365) and

China Scholarship Council (ﬁle no. 201566485009).

1

1 Introduction

This paper illustrates a novel equilibrium phenomenon, which we call statistical

sunspots. Statistical sunspots are endogenous ﬂuctuations that arise in equilibrium

even in determinate models that feature a unique rational expectations equilibrium.

We consider economic environments where some state variables are hidden from a set

of agents, following a line of research that begins with Marcet and Sargent (1989).

With hidden state variables, the theory departs from strict rational expectations by

attributing to agents (potentially) misspeciﬁed forecasting models. In a restricted

perceptions equilibrium beliefs are formed by projecting the full state vector onto the

individuals’ restricted set of observables, thus preserving the cross-equation restric-

tions that are salient features of rational expectations models. The insight of this

paper is that this set of observables can include serially correlated extrinsic shocks,

statistical sunspots, which can be interpreted as judgment, sentiment, expectations

shocks, sunspots, etc. Because agents do not observe the full state vector, they at-

tribute, in a self-fulﬁlling way, some of the serial correlation observed in data to these

extrinsic factors. An important implication of breaking the link between indeter-

minacy and sunspots is that statistical sunspot equilibria are stable under learning,

unlike rational sunspots.

There is a long and venerable history in macroeconomics of proposing theories of

exogenous movements in expectations. For example, many forward-looking macroe-

conomic models can generate endogenous volatility through sunspot equilibria that

exhibit self-fulﬁlling dependence on extrinsic variables, i.e. “animal spirits.” Much

of the policy advice coming from New Keynesian models is predicated on ruling out

indeterminacy. In other settings, expectations can depend on “news” shocks, noisy

signals about future economic variables like productivity or future monetary/ﬁscal

policies. A third strand of literature assumes that a portion of expectations come

from statistical forecasting models (e.g. rational expectations or econometric learn-

ing rules) that are perturbed by expectational shocks or “add factors.”

While these approaches to endogenous volatility can have important empirical

implications for macroeconomic time series, there are theoretical drawbacks. In the

case of sunspot equilibria, their existence depends on an equilibrium indeterminacy

that raises the question of how individuals might coordinate on one of many possible

equilibria. Moreover, an extensive literature shows that these sunspot equilibria may

not be learnable when rational expectations are replaced with reasonable econometric

learning rules. A wide literature argues in favor of stability under econometric learning

as an important consistency and equilibrium selection criterion.1The news shocks

1See, for instance, Sargent (1993), Evans and Honkapohja (2001), Sargent (2008), and Woodford

2

and expectational shocks models are imposed by the modeler and, again, raise the

question of how agents might come to include them in their expectations. The issue of

coordinating on equilibria with endogenous volatility is of practical interest: without

a consistent theory of expectation formation it is not obvious that equilibria featuring

expectational shocks even add additional volatility.

We propose a model of expectation formation that is based on the observation

that statistical models are misspeciﬁed.2Forecasters may not observe all economic

variables, particularly exogenous shocks, or they may have a preference for parsimony

that leads them to under parameterize their models. For these reasons, there is a long

tradition by applied forecasters to estimate and forecast using parsimonious, vector

autoregressive (VAR) models. Motivated by these observations, we begin with a gen-

eral univariate, forward-looking model that depends on a serially correlated exogenous

process. An individual is said to have “restricted perceptions” when they use a mis-

speciﬁed (in some dimension) statistical model to formulate expectations. We ﬁrst

illustrate the main insights of the paper in the case where the fundamental shock is

hidden and all agents forecast with an AR(1) model plus a serially correlated extrinsic

shock (a “sunspot”). In a restricted perceptions equilibrium (RPE) each forecasting

model is optimal within the restricted class in the sense that their beliefs are the

least-squares projection of the true data generating process onto their restricted set

of regressors.3We show the existence of a fundamentals RPE and provide necessary

and suﬃcient conditions for the existence of multiple RPE that depend on a serially

correlated extrinsic variable, a “statistical sunspot.” Importantly, these sunspot equi-

libria exist even though we restrict attention to models that have a unique rational

expectations equilibrium. Rather than the continuum of sunspot equilibria that arise

in the corresponding rational expectations model, we show that at most three RPE

exist: the fundamental RPE and two symmetric RPE that are driven by sunspots.

The existence of these sunspot RPE depends on the strength of expectational feed-

back, the serial correlation of the hidden fundamental shock, the serial correlation of

the sunspot shock, and the signal to noise ratio of the shocks’ innovations.

Two features of these sunspot RPE are particularly interesting. Under certain

conditions, the sunspot RPE are stable under learning while the fundamental RPE is

unstable. In particular, we show that when the fundamental RPE is the unique RPE,

then it is strongly expectationally-stable (“E-stable”): agents endowed with a fore-

(2013).

2The introduction to White (1994) states explicitly “...an economic or probability model is ...a

crude approximation to ...the ‘true’ relationships...Consequently, it is necessary to view models as

misspeciﬁed.”

3Thus, in the spirit of Sargent (2008) we focus on environments where “agents inside a model

have views that can diverge from the truth in ways about which the data speak quietly and softly.”

3

casting model that depends on lagged endogenous state variables and a sunspot will

eventually converge to the fundamental RPE with a zero coeﬃcient on the sunspot.

The fundamental RPE is strongly E-stable in the sense that when agents’ statistical

model is over-parameterized to include the sunspot, their coeﬃcient estimate on the

sunspot converges to zero. If multiple RPE exist, however, then the fundamental RPE

is only weakly E-stable, agents who include the sunspot variable in their regression

will ﬁnd that their real time coeﬃcient estimates do not converge to the fundamental

RPE. Instead, we show that the sunspot RPE are E-stable, for suﬃciently serially

correlated sunspot shocks. Thus, the results of this paper break the link between

indeterminacy and the existence of expectationally stable sunspots.

We also show that, in the case of homogeneous beliefs, sunspot RPE are less

volatile than fundamental RPE, an unexpected but intuitive ﬁnding. The misspeci-

ﬁed AR(1) forecast model tracks the unobserved serial correlation of the fundamental

shock through two variables, the lagged endogenous variable and the serially corre-

lated sunspot shock. As a result, the AR(1) coeﬃcient in the fundamental RPE is

greater than it is in the sunspot RPE. If the sunspot shock becomes more serially

correlated and volatile, then the optimal forecasting equation puts more weight on

the sunspot but attributes a lower autocorrelation coeﬃcient to the endogenous state

variable. This makes the sunspot RPE have a variance that is bounded above by the

fundamental RPE.

The existence of sunspot RPE is robust across a range of extensions to the basic

framework. In extensions and generalizations of the benchmark result, a wide class

of “pseudo” ARMAX forecasting models are speciﬁed.4Extensions are considered

where there is hidden information to only a subset of the agents, where information

is hidden to all agents who are distributed across heterogeneous forecasting rules

that diﬀer in lag polynomials, the number of pseudo moving-average terms, and/or

whether the sunspot is included. Moreover, with heterogeneous expectations the self-

fulﬁlling nature of the sunspot is strengthened so that, depending on the diversity of

beliefs, the sunspot RPE can increase the equilibrium volatility.

These theoretical results have broader implications. Our ﬁrst application is to

re-consider a theme from studies into the design of monetary policy rules that, un-

der learning, policymakers face an improved stabilization trade-oﬀ via a systematic,

aggressive response whenever inﬂation deviates from its target rate.5The theoretical

underpinnings to these ﬁndings relates to the “anchoring” of private-sector expecta-

tions and whether policies that emphasize stabilizing inﬂation can improve outcomes

4Pseudo ARMA, also known as extended least-squares, is an alternative to maximum likelihood

to estimate moving average coeﬃcients.

5See, for example, Eusepi and Preston (2017).

4

by anchoring expectations that are susceptible to over-shooting and other self-fulﬁlling

feedback loops, a point made forcefully by Orphanides and Williams (2005). In our

application, the model environment is New Keynesian, where the central bank can

perfectly control aggregate demand and sets policy via a rule that adjusts aggregate

demand whenever inﬂation deviates from trend. Aggregate supply shocks lead to a

trade-oﬀ faced by policymakers who choose the reaction coeﬃcient in their policy

rule to minimize a weighted average of output and inﬂation volatility. When forming

optimal policy, the policymakers take the inﬂation expectations of agents, who are

distributed across two forecasting models, as given: the ﬁrst forecasting model nests

the unique rational expectations equilibrium as a linear function of the fundamental

(aggregate supply) shock; the remaining agents forecast using an AR(1) model. We

then study the eﬀect of restricted perceptions on the trade-oﬀs faced by policymak-

ers and the optimal design of policy rules. Heterogeneity and restricted perceptions

have a non-monotonic eﬀect on the trade-oﬀs faced by policymakers. Moving to a

setting with a relatively small degree of heterogeneity shifts in the policy frontier, im-

proving the trade-oﬀs faced by policymakers. Eventually as the proportion of agents

using the AR(1) model is large enough, the policy frontier shifts adversely and the

trade-oﬀ worsens. As the fraction using the AR(1) model increases eventually the

system bifurcates with the stable equilibria depending on sunspots and exhibiting

a lower variance. This bifurcation occurs for suﬃciently strong expectational feed-

back which, in the present environment, implies that policy is less “hawkish.” The

non-monotonic policy frontier and the possibility of a bifurcation imply that the op-

timal policy coeﬃcient is non-monotonic in departures from rational expectations.

At the bifurcation the optimal inﬂation reaction coeﬃcient drops precipitously as the

policymaker seeks to coordinate agents onto the sunspot RPE.

Because the trade-oﬀs and the optimal policy rule depend critically on the dis-

tribution of agents across forecasting models we then extend the benchmark model

to endogenize this distribution. Following Brock and Hommes (1997) and Branch

and Evans (2006), we extend the model expectation formation to include a discrete

choice between (potentially) misspeciﬁed forecasting models. An optimal policy mis-

speciﬁcation equilibrium is a symmetric Nash equilibrium where policymakers choose

their optimal rule taking beliefs as given and agents select the best performing model

taking the policy rule as given. We show that it is possible to have a multiplicity of

equilibria, with the rational expectations equilibrium existing always as an optimal

policy misspeciﬁcation equilibrium. However, there can also exist a sunspot RPE

equilibrium that features an adverse shift in the policy frontier and a more hawkish

policy response. Thus, in this particular policy experiment, it is possible for policy-

makers to be trapped in an ineﬃcient equilibrium while simultaneously having a more

hawkish policy stance. Two other applications are presented, a simple mean-variance

5

asset-pricing model and a pure monetary economy. The asset-pricing example shows

that excess volatility, a common ﬁnding in the learning literature since Timmermann

(1993), depends on the nature and distribution of expectations. The ﬁnal application

is a pure monetary model based on Lagos and Wright (2005) and Rocheteau and

Wright (2005). This environment naturally features negative expectational feedback.

It is possible for the equilibrium to feature “Intrinsic Heterogeneity” where agents

are distributed across multiple misspeciﬁed models that include sunspots, providing

an equilibrium explanation for heterogeneity observed in survey data on inﬂation

expectations.

The paper proceeds as follows. Section 2 develops the main theoretical results in

a simple environment with homogeneous expectations. Section 3 presents extensions

and generalizations including a model with lagged endogenous variables and a general

formulation with heterogeneous expectations. Section 4 presents the applications,

while Section 5 concludes. All proofs are contained in the Appendix.

1.1 Related literature

This paper is related to a literature that studies the equilibrium implication of econo-

metric model misspeciﬁcation. Marcet and Sargent (1989) ﬁrst introduced the idea

that in an environment with private information an adaptive learning process will

converge to, what they call, a limited information rational expectations equilibrium.

Subsequently, Sargent (1999) and Evans and Honkapohja (2001) extended the idea

to where the misspeciﬁcation never vanishes and learning converges to a restricted

perceptions, or misspeciﬁcation, equilibrium. Weill and Gregoir (2007) show the

possibility of misspeciﬁed moving-average beliefs being sustained in a restricted per-

ceptions equilibrium. Branch and Evans (2006) illustrate how the misspeciﬁcation

can arise endogenously by applying the Brock and Hommes (1997) mechanism to a

restricted perceptions environment. A very interesting result in Cho and Kasa (2017)

provides an equilibrium justiﬁcation for restricted perceptions as a sort of Gresham’s

law of Bayesian model averaging leads to correctly speciﬁed, within a rational expec-

tations equilibrium, models being driven out of the forecast model set. We are very

much in the spirit of Sargent’s (2008) essay on small deviations from the rational

expectations hypothesis that preserve beliefs being pinned down by cross-equation

restrictions and, yet, deliver an independent role for beliefs in economic ﬂuctuations.

The results here also relate to a very large literature on sunspot equilibria in

rational expectations models, e.g. Shell (1977), Cass and Shell (1983), Azariadis

(1981), Azariadis and Guesnerie (1986), Guesnerie (1986), and Guesnerie and Wood-

ford (1992). In the same spirit as this paper, Eusepi (2009) studies the connection

6

between expectations driven ﬂuctuations and indeterminacy in one and two-sector

business cycle models. Woodford (1990) was the ﬁrst to show that sunspot equilibria

could be stable under learning in overlapping generations models. Evans and Mc-

Gough (2005a) and Duﬀy and Xiao (2007) show that sunspot equilibria in applied

business cycle models like Benhabib and Farmer (1994) and Farmer and Guo (1994)

are unstable under learning. In New Keynesian models, sunspot equilibria are not

generally stable under learning, however, Evans and McGough (2005b) show that

if agents’ perceived law of motion represents the sunspot equilibrium as the mini-

mal state variable solution plus a serially correlated sunspot that satisﬁes a resonant

frequency condition, then sunspot equilibria can be stable under learning provided

there is suﬃcient negative expectational feedback. Related is also the diverse beliefs

that arise in a rational belief equilibrium pioneered by Kurz (1994) and with recent

applications by Kurz, Jin, and Motolese (2005). In the rational belief framework, be-

liefs are driven by extrinsic noise so long as subjective beliefs agree with the ergodic

empirical distribution.

A closely related paper is Angeletos and La’O (2013) who were ﬁrst to show

the existence of sunspot-like equilibria, that they call “sentiment shocks”, in models

that feature a unique rational expectations equilibrium. The key departure point

for Angeletos-La’O is a trading friction that limits communication between agents.

Decentralized trade and random matching imply that agents can hold diverse beliefs

about future prices (terms of trade) and, so, it is possible for their beliefs about the

future to be driven by a sentiment factor that leads to aggregate waves of optimism

and pessimism. In the framework here, heterogeneous beliefs is not essential for the

existence of sunspot RPE in determinate models. The key assumption is that there

is hidden information to a set of agents that lead them to form parsimonious, yet

misspeciﬁed, statistical models through which they form expectations.

Another closely related paper is Bullard, Evans, and Honkapohja (2008) who iden-

tify an exuberance equilibrium where agents hold a misspeciﬁed econometric model of

the economy, whose coeﬃcients guarantee that the perceived autocorrelation function

aligns with the autocorrelation function implied by the equilibrium data generating

process. Agents’ expectations come, in part, from the statistical model and then in-

clude add factors modeled as random shocks. An exuberance equilibrium is a Nash

equilibrium in the sense that a zero-mass agent’s best response would be to include the

add factors given all of the other agents’ behavior. In this sense, it is a self-fulﬁlling

equilibrium. Here we do not separate the forecasting into a statistical and judgment

component, instead asking whether if agents were to parameterize their model to

include exogenous shocks would they come to coordinate on such an equilibrium.

7

2 Restricted Perceptions and Endogenous Fluctu-

ations

Consider ﬁrst the minimal departure from rational expectations that leads to sta-

tistical sunspot equilibria that are stable under learning (“E-stable”) in determinate

(under rational expectations) models. A univariate model with homogeneous beliefs

feature E-stable statistical sunspot equilibria with distinct stochastic properties from

equilibria – under rational expectations or restricted perceptions – that do not depend

on extrinsic noise. It is shown that, even though the models under consideration have

a unique rational expectations equilibrium, learning dynamics will converge to the

sunspot equilibria so long as individuals’ beliefs place a prior on sunspot dependence.

2.1 Restricted perceptions equilibria with sunspots

We begin with a simple univariate model given by the pair of equations,

yt=αˆ

Etyt+1 +γzt,(1)

zt=ρzt−1+εt.(2)

Equation (1) is the expectational diﬀerence equation that determines the endogenous

state variable ytas a linear function of the time-tsubjective expectations ( ˆ

Et) of

yt+1 and a serially correlated process for “fundamental” shocks zt. The ztprocess is

assumed covariance stationary with 0 < ρ < 1 and εtis white noise with variance

σ2

ε. Subsequent sections provide examples of economies whose equilibrium conditions

lead to equations of the form (1)-(2). For now, assume that 0 < α < 1 so that the

model is determinate and features positive expectational feedback.6The model has

a unique rational expectations equilibrium of the form

yt= (1 −αρ)−1γzt.

The agents in the economy form subjective expectations using a linear forecasting

model, which are sometimes called “perceived laws of motion” or “PLM”. Hommes

and Zhu (2014), working in a similar framework, assign to agents forecasting models

of the AR(1) form

yt=byt−1+t⇒ˆ

Etyt+1 =b2yt−1,(3)

6The negative feedback case, where −1< α < 0, is considered later as an application to a

monetary model that naturally admits negative expectational feedback.

8

where tis a (perceived) white noise process.7In the AR(1) perceived law of mo-

tion the coeﬃcient bcorresponds to the ﬁrst-order autocorrelation coeﬃcient. Why

would individuals formulate and estimate an AR(1) forecasting model? It is a simple

and parsimonious econometric model that is appropriate in environments where the

fundamental shock ztis an unobservable, or hidden, state variable. Examples of hid-

den shocks depend on the precise model environment and include drifts in a central

bank’s inﬂation target, aggregate mark-up shocks, or asset ﬂoat. Hommes and Zhu

(2014) deﬁne a behavioral learning equilibrium as a stochastic process for ytsatisfying

(1), given that expectations are formed from (3), and with bequal to the ﬁrst-order

autocorrelation coeﬃcient of yt.

Similarly, we assume that the fundamental, i.e. payoﬀ relevant, shock zis un-

observable, or hidden, to agents when forming expectations. Agents formulate fore-

casting models that condition on observable endogenous state variables as well as ob-

servable exogenous variables that are extrinsic to the model, i.e. statistical sunspots.

Speciﬁcally, agents hold the perceived law of motion

yt=byt−1+dηt+t

ηt=φηt−1+νt)⇒ˆ

Etyt+1 =b2yt−1+d(b+φ)ηt.(4)

Expectations are formed from a forecast model that depends on lagged values of the

endogenous variable, y, and an extrinsic noise term, ηt, assumed to be a stationary

AR(1) with 0 < φ < 1.8For simplicity, ηtis uncorrelated with zt, i.e. Eεtνt= 0.9

The extrinsic noise, ηt, can be thought of as a statistical sunspot variable that prox-

ies for publicly announced consensus forecasts, waves of optimism/pessimism, senti-

ment shocks, judgment or add factors, political shocks, etc. We call them statistical

sunspots to distinguish them from rational sunspots that are typically martingale

diﬀerence sequences. A statistical sunspot, on the other hand, is a generically serially

correlated exogenous process that will impact agent beliefs only if there is a statisti-

cal relationship between the state yand η. Therefore, unlike the expectations and/or

news shocks models we hold that whether, and when, agents use ηin their forecast

7In Hommes and Zhu (2014), the PLM model in fact also includes a constant term. Since the

analysis of the mean is relatively trivial and is not the main point of this work, here we assume that

the means are zero, and known, without loss of generality. We also assume that the agents know

the stochastic process determining ηt. This does not impact the learning stability analysis since η

is exogenous, with a suﬃciently long history of η’s, the agents would precisely estimate φand σν.

Unlike η,yis determined via a self-referential system.

8The agents here respect the learning literature’s timing convention that exogenous variables are

contemporaneously observed while endogenous variables are observed with a lag. This breaks the

simultaneity of expectations and outcomes that are natural in rational expectations environments

but implausible under restricted perceptions.

9Relaxing this restriction leads, more generally, to stable sunspot equilibria.

9

model will arise as an equilibrium property: dis pinned down via cross-equation

restrictions. The following sections extend and generalize this benchmark case.

Is it reasonable that individuals would be able to observe ηtand not fundamental

variables such as zt? In our view, the answer is yes. The process ηtcould be any col-

lection of information that agents think is informative about the state of the market

or economy that does not have a direct, payoﬀ relevant, eﬀect except through agents’

beliefs. These could be subjective waves of optimism and pessimism in markets, judg-

ments included in statistical forecasts, conﬁdence indices, or any other collection of

observable information. The motivation for models with restricted perceptions is that

agents do not know the structural model that generates data. A good econometrician

would include all observable variables that help predict the state. We show that when

RPE exist that include dependence on ηtthese equilibria are stable under learning so

that eventually agents’ would come to believe that these non-fundamental variables

drive, in part, the endogenous state variable yt. Thus, these are self-fulﬁlling equilib-

ria. Surprisingly, these RPE with dependence on ηtarise in determinate models and

are stable under learning.

Given the perceived law of motion (4), the corresponding data generating process,

called the “actual law of motion” (ALM), can be found by plugging expectations into

(1):

yt=αb2yt−1+αd (b+φ)ηt+γzt.(5)

Notice that the perceived law of motion is misspeciﬁed: the actual law of motion (5)

depends on yt−1, ηt,and zt, while the perceived law of motion depends only on yt−1

and ηt. Thus, the PLM is underparameterized and so the equilibrium will not be a

rational expectations equilibrium but a restricted perceptions equilibrium (RPE).10 In

an RPE, agents’ beliefs, summarized by the coeﬃcients (b, d), are optimal within the

restricted class, i.e. they will satisfy the least-squares orthogonality condition:

EXt(yt−X0

tΘ) = 0,(6)

where X0

t= (yt−1, ηt) and Θ0

t= (b, d). In an RPE, agents are unable to detect their

misspeciﬁcation within the context of their forecasting model. A suﬃciently long

history of data will reveal the misspeciﬁcation to agents, so an RPE is appropriate

for settings where data are slow to reﬂect the serial correlation in the residuals of the

regression equations.

10Alternatively, one can write (5) as

yt= (αb2+ρ)yt−1−αb2ρyt−2+αd(b+φ)(1 −ρL)ηt+γεt

It is straightforward to see that if the forecast model was extended to be an AR(p) then the actual

law of motion is AR(p+ 1).

10

The set of restricted perceptions equilibria, and their stability under learning, are

characterized by studying the mappings from the PLM to the ALM whose ﬁxed points

are RPE, i.e. the “T-maps.” In particular, solving the orthogonality condition (6)

leads to

Θ = (EXtX0

t)−1EXtyt≡T(Θ) (7)

The T-map has the following interpretation. Given a value of Θ, the actual law of

motion is (5), and a regression of yon lagged yand the sunspot ηwould produce the

coeﬃcients T(Θ): the T-map is the least-squares projection of the ALM (5) onto the

PLM (4). An RPE is a ﬁxed point of the T-map (7). Straightforward calculations

produce

T(Θ) = "1−dση

σy

0 (1 −bφ)σy

ση#corr (yt−1, yt)

corr (yt−1, ηt)

where corr(x, w) is the correlation coeﬃcient between the variables x, w. The equilib-

rium coeﬃcients, (b, d), depend, in part, on the correlation between the endogenous

state variable ytand the lag variable yt−1as well as between ytand the sunspot ηt.

These correlation coeﬃcients, in turn, depend on the belief coeﬃcients (b, d). It is

this self-referential feature of the model that makes the set of RPE interesting to

characterize.

2.2 Existence

Afundamentals restricted perceptions equilibrium is an RPE in which b6= 0, d = 0,

since there is no dependence on the extrinsic variable. Conversely, if b, d 6= 0, then the

equilibrium is a sunspot RPE that features endogenous ﬂuctuations, i.e. a statistical

sunspot equilibrium. This section establishes existence, and characterizes the set of

RPE.

The T-map for the fundamentals RPE can be identiﬁed by setting d= 0 and

solving the orthogonality condition (6) for b:

b→αb2+ρ

1 + αb2ρ.(8)

A fundamentals RPE is a ﬁxed point, ˆ

b, of (8) and, it should be noted, is equivalent

to the behavioral learning equilibrium in Hommes and Zhu (2014).

The component of the T-map corresponding to dis given by

d→dα (b+φ) (1 −bφ)

1−αb2φ.

11

Evidently, d= 0 is a ﬁxed point of this mapping. There exists a fundamental RPE,

with the expression for ba complicated polynomial in α, ρ implied by (8).

When d6= 0,

b=b∗≡1−αφ

α(1 −φ2)

is also a ﬁxed point of the T-map. Given that value of b=b∗, the remainder of the

T-map can be solved for d, which after tedious calculations becomes

d2→ξ(b, α, ρ, φ)σ2

ε

σ2

ν

where

ξ(b, α, ρ, φ) = γ2{ρ−b[1 −αb(1 −bρ)]}(1 −αb2φ) (1 −φ2)

α(1 −αb2ρ)(1 −ρ2)(b+φ)φ(1 −αφ)

We have the following results.

Proposition 1 There exists a unique fundamentals restricted perceptions equilibrium

(b, d)=(ˆ

b, 0), where ˆ

bsolves (8). Moreover, a pair of symmetric sunspot RPE (b, d) =

(b∗,±d∗)exists if and only if

i. α > 1

1+φ(1−φ)≡˜α

ii. ˜ρ(α, φ)<ρ<1, for appropriately deﬁned ˜ρ.

Corollary 2 If 4/5< α < 1then sunspot RPE exist for suﬃciently large ρ.

Proposition 1 provides necessary and suﬃcient conditions under which a given

sunspot, parameterized by φ, is supported as a restricted perceptions equilibrium.

While Corollary 2 shows that for a given structural parameter α, there will exist

sunspot RPE provided the serial correlation of the fundamental shock is suﬃciently

strong.The existence of the statistical sunspot equilibria requires that αis suﬃciently

large, i.e. there is strong expectational feedback in the model.

Figure 1 illustrates Proposition 1 by plotting the ﬁxed points of the T−map.

The Tdcomponent of the T-map has a ﬁxed point at d= 0 for all values of b, this

corresponds to the fundamentals RPE. The ﬁxed point – if it exists – b=1−αφ

α(1−φ2), is

the vertical line in the ﬁgure. The Tbcomponent has a parabolic shape that intersects

the Tdcontour in three places, d= 0 and ±d∗. Where the Tband Tdcontours intersect

are restricted perceptions equilibria.

12

The ﬁgure also illustrates the comparative statics. As ρor αdecreases the Tb

component shifts left, lowering the RPE values for bin both types of equilibria, and

decreasing din the sunspot RPE. Similarly, α(and φ) shift the vertical segment of

the Td-map. As these parameter values decrease suﬃciently, eventually the Tbdoes

not intersect the Tdline, at which point there exists a unique RPE coinciding with

the fundamentals equilibrium.

Figure 1: Equilibrium Existence

Notice in Figure 1 that the autoregressive coeﬃcient bin the fundamentals RPE

is greater than the same coeﬃcient in the sunspot RPE. That is because when d6=

0 the agents’ model tracks the serial correlation in the model – arising from the

hidden shock zand the self-fulﬁlling serial correlation from agents’ beliefs – through

both the lagged endogenous variable and the extrinsic noise. Thus, it is not at all

obvious whether agents coordinating on the statistical sunspot equilibria will make

13

the resulting process for ytmore or less volatile than the equilibrium where they

condition on lagged yalone. Results on this question are presented below.

2.3 Expectational stability

An open issue in models that feature sunspots and other types of expectations shocks

is through what means agents can come to coordinate on these equilibria. A large

literature shows that indeterminate models featuring sunspot equilibria are generally

unstable under an econometric learning rule. For a sunspot equilibrium to be stable

typically requires that there be negative feedback from expectations onto the state

and that agents specify the equilibrium process in a particular manner, called the

“common factor” solutions by Evans and McGough (2005c). There is some debate

about the empirical relevance of many models with strong negative feedback. More-

over, a large literature on monetary policy rules makes an explicit case for policies

that ensure unique rational expectations equilibria.

On the other hand, Proposition 1 proves the existence of statistical sunspot equi-

libria in determinate models. We now further demonstrate that statistical sunspot

equilibria are stable under learning. The approach is an examination of the Ex-

pectational Stability (“E-stability”) properties of the equilibria following Evans and

Honkapohja (2001). In other models, expectations shocks are included as ad hoc

additions to the agents’ expectations or forecasting model leaving it unmodeled how

they might coordinate on the equilibrium.

To examine stability, step back from imposing that beliefs are optimal within the

restricted class. Recall that the T-maps deliver the optimal least-squares coeﬃcients

from a regression of yton the regressors (yt−1, ηt), given that ytis generated from an

actual law of motion implied by the perceived law of motion (parameterized by (b, d)).

The previous section focused on the ﬁxed-points of these maps. Here the focus is on

the local stability of the ﬁxed points.

It is well-established that, in a broad class of models, stability under reasonable

learning algorithms, such as recursive least-squares, are governed by “E-stability”

conditions found by computing the local stability of the rest points to the E-stability

ordinary diﬀerential equation:

dΘ

dτ =T(Θ) −Θ.(9)

Here τdenotes “notional” time, which can be linked to real time t. The E-stability

Principle states that Lyapunov stable rest points of the E-stability o.d.e. (9) are lo-

cally stable under least-squares, and other closely related learning algorithms. Notice

14

that both the fundamentals and sunspot RPE are rest points of the E-stability o.d.e.

since they are ﬁxed points of the T-map. That the E-stability principle governs sta-

bility of an equilibrium is intuitive since (9) dictates that the estimated coeﬃcients

Θ are adjusted in the direction of the best linear projection of the data generated by

the estimated coeﬃcients onto the class of statistical models deﬁned by the PLM (4).

Local stability of (9) thus answers the question of whether a small perturbation in the

perceived coeﬃcients Θ will tend to return to their restricted perceptions equilibrium

values.

The proofs to the E-stability results make use of the following Lemma.

Lemma 3 When ρ > ˜ρand α > ˜αso that there exist multiple restricted perceptions

equilibria, the following relationship between the sunspot RPE b∗and the fundamental

RPE ˆ

bholds: b∗≤ˆ

b.

Proposition 4 The E-stability properties of the AR(1) RPE are as follows:

1. when there exists a unique RPE it is E-stable;

2. when ρ > ˜ρand α > ˜α, the fundamental RPE is weakly E-stable. Moreover, the

two sunspot RPE are E-stable if and only if ˜

φL(α, ρ)<φ<1, for appropriately

deﬁned ˜

φL, depending on α, ρ.

As an illustration, consider the following numerical example. Set α= 0.95, ρ =

0.6, φ = 0.98, γ = 1, σ2

ε=σ2

ν= 1. Figure 2 plots the invariant manifolds of the T-map

along with the associated vector ﬁeld of the E-stability o.d.e. Their cross-points are

ﬁxed points of the T-map. That is, the ﬁgure summarizes the E-stability dynamics.

In the ﬁgure the solid lines are the contour plots giving the ﬁxed points of the T-

map, as in Figure 1. The vector ﬁeld and streams plot the direction of adjustment

according to the E-stability o.d.e..

The left plot (α= 0.90) corresponds to a parameterization with a unique (E-

stable) RPE. The RPE occurs with ˆ

b≈0.97 and ˆ

d= 0. It is also evident in the

ﬁgure that the unique fundamental RPE is E-stable. Contrast these results with the

right plot which holds the same parameter values except now α= 0.95. As in Figure

1 there are three RPE: the fundamentals RPE with ˆ

d= 0, and two sunspot RPE

with d∗6= 0. Interestingly, the sunspot RPE are E-stable with all initial values for

b, d > 0 converging to the equilibrium. The weak stability of the fundamental RPE

is apparent along the saddle ˙

d= 0.

Proposition 4 establishes conditions that depend on α, φ and ρ. Figure 3 illustrates

this relationship. The ﬁgure plots the sets of (α, φ) such that the sunspot RPE exists

15

Figure 2: E-stability Dynamics

and is E-stable, for various values of ρ. For suﬃciently large expectational feedback

α, the sunspot RPE is E-stable provided that it is suﬃciently serially correlated.

Elsewhere, there is a non-monotonic relationship between αand φthat deliver a

parameterization of the model consistent with E-stable sunspot RPE. As ρincreases

a given sunspot (i.e. ﬁx the value of φ) does not require as much feedback in order for

the sunspot RPE to exist. The existence of E-stable sunspot equilibria depends, non-

linearly, on the serial correlation properties of the omitted variable and the sunspot

as well as the strength of expectational feedback in the model.

2.4 Endogenous ﬂuctuations and economic volatility

Conventional models of sunspot equilibria are typically viewed as ineﬃcient since they

introduce serial correlation and volatility that would not exist without coordination

on the self-fulﬁlling equilibria. It is a natural question to ask whether the endogenous

ﬂuctuations that arise in a sunspot restricted perceptions equilibrium lead to more or

less economic volatility. It turns out that, with homogenous expectations, the sunspot

RPE exhibits lower volatility than the fundamentals RPE. This section explores this

question, ﬁrst, by establishing the following result.

Proposition 5 Let α > ˜αand ρ > ˜ρso that multiple restricted perceptions equilibria

exist. For ρsuﬃciently large and/or σ2

ν

σ2

ε

suﬃciently small, the fundamentals RPE is

16

Figure 3: E-stability Regions

0.80 0.85 0.90 0.95 1.00

0.0

0.2

0.4

0.6

0.8

1.0

α

ϕ

0.99

0.7

1

more volatile than the sunspot RPE.

Numerical analysis suggests that the result holds for a broad set of parameters.

The intuition for the result that the sunspot RPE are less volatile than the fundamen-

tals RPE is as follows. Imagine an increase in φ, i.e. making the statistical sunspot

more volatile. This has oﬀ-setting eﬀects for relatively large φ.11 On the one hand, it

increases b∗, which tends to push up the variance of the sunspot RPE. On the other

hand, d∗decreases as less weight is placed on the extrinsic shock ηsince its serial cor-

relation properties are further displaced from those of the fundamental shock z. This,

in turn, tends to push down the sunspot RPE’s variance. As ρbecomes large enough,

or the signal-noise ratio σ2

ν

σ2

ε

small enough, then the weight on the sunspot is suﬃciently

small so that the fundamentals RPE is more volatile. In general, the relative vari-

ances of the RPE depend on the elasticities of the RPE coeﬃcients b, d. Numerical

explorations suggest that the eﬀect of a lower self-fulﬁlling serial correlation via the

b-coeﬃcient outweighs the impact on the d-coeﬃcient.

This trade-oﬀ from changing the statistical properties of the sunspot variable can

be seen in the following numerical example. Set α= 0.95, ρ = 0.6, γ = 1, σ2

ε=σ2

ν= 1.

Figure 4 plots the excess volatility of the two RPE, i.e. the ratio of the RPE variance

to the variance in the rational expectations equilibrium, as a function of φ. The ﬁgure

clearly demonstrates the non-monotonic eﬀect of φon the variance of the sunspot

11In fact, the eﬀect of φon the b∗is ﬁrst decreasing and then increasing.

17

RPE. As φincreases from a relatively high level, the autocorrelation coeﬃcient b

increases and ddecreases, as the sunspot RPE is more similar to the the fundamentals

RPE. As φdecreases, making the sunspot less volatile, there is a non-monotonic eﬀect

on the RPE coeﬃcients. From a relatively high value of φ, the comparative static

eﬀect of decreasing φis to decrease b∗and increase d∗, as the sunspot better tracks the

serial correlation in the model. However, for some lower value of φ, the comparative

static eﬀect of decreasing φthen increases b∗and decreases d∗as the RPE becomes

closer to the fundamental RPE. Thus, in between these critical values there is a

non-monotonic eﬀect from φon economic volatility in the non-fundamentals RPE.

The statistical properties of the RPE sunspot equilibria are distinct from the

fundamentals RPE. Figure 5 illustrates that it is not just bthat is lower, but that the

ﬁrst-order autocorrelation of the sunspot equilibrium is always (weakly) lower than

for the fundamentals RPE. Moreover, inspection of the log spectral density (Figure

6) for the two processes shows that the fundamentals has the usual spectral shape of

a strongly autocorrelated process, while for the sunspot RPE a greater fraction of its

variance is explained by medium frequency ﬂuctuations.

Figure 4: Economic volatility in a restricted perceptions equilibrium.

18

Figure 5: First order autocorrelation in a restricted perceptions equilibrium.

Figure 6: Log spectral densities in a restricted perceptions equilibrium.

0.0 0.5 1.0 1.5 2.0 2.5 3.0

0.5

1

5

10

50

100

3 Extensions and Generalizations

This section presents extensions and generalizations that demonstrate the robustness

of the existence and stability results. In particular, we consider an extension of the

19

basic model to include a lagged endogenous variable in the data generating process

as well as a wide array of misspeciﬁed statistical models that agents might use to

form subjective expectations. We also allow for heterogeneity in forecast models,

considering both where some, or all, individuals include the sunspot variable in their

forecasts. Finally, while this paper develops its results with linear, univariate models,

we have also computed numerical examples of stable sunspot RPE in multivariate

and non-linear models. However, to focus on the basic result we omit these particular

extensions from the presentation.

3.1 Lagged endogenous variables

Statistical sunspot equilibria arise from restricted perceptions because, intuitively,

agents’ statistical model attributes, in a self-fulﬁlling way, the serial correlation ob-

served in data to come from autocorrelation of the endogenous variable and a serially

correlated extrinsic noise process. These results were developed in a model where

all of the serial correlation in the data generating process comes from the hidden

fundamental shock and aggregate subjective beliefs. A natural question is whether

the existence of sunspot RPE is robust to an environment where the data generating

process also depends on lags of the endogenous variable.

To answer this question, consider the following extension of the benchmark model:

yt=αˆ

Etyt+1 +δyt−1+γzt,0< α < 1

zt=ρzt−1+εt

If the agents’ statistical model remains an AR(1) plus sunspot, as in section 2, then

the results from the previous section extend in a natural way to this extended envi-

ronment. However, with the additional source of serial correlation arising from the

lagged y, it is plausible that a well-speciﬁed econometric model might include two lags

to better account for serial correlation in the residuals from an AR(1) regression. In

this case, with additional explanatory variables in the agents’ model it is not obvious

that sunspot RPE will continue to exist.

Speciﬁcally, assume that agents’ forecasts come from the statistical model

yt=b1yt−1+b2yt−2+dηt

ηt=φηt−1+νt

Leading the perceived law of motion forward one period and taking expectations,

leads to the actual law of motion for yt

yt=αb2

1+b2+δyt−1+αb1b2yt−1=αd (b1+φ)ηt+zt(10)

20

Because ztis hidden from agents, their model continues to be misspeciﬁed. To see

this, use the fact that zt= (1 −ρL)−2εtto re-write the law of motion (10) as

yt=αb2

1+b2+δyt−1+αb1b2−ρb2

1+b2−δρyt−2

−αρb1b2yt−3+αd(b1+φ) (ηt−ρηt−1) + εt

As agents forecast with an AR(2) plus sunspot the actual law of motion is an AR(3)

plus contemporaneous and lagged values of the sunspot. We can proceed as before,

compute the T-map from the least-square orthogonality condition and then ﬁnd the

set of RPE as ﬁxed points to the T-map. These expressions are particularly com-

plicated, though we can summarize the results of an extensive numerical analysis as

follows.

Result 6 There exists a fundamentals RPE with ˆ

b1≥0,ˆ

b26= 0,ˆ

d= 0. For αand

ρsuﬃciently large, and δsmall enough, there exists a pair of E-stable sunspot RPE

with b∗

16= 0, b∗

26= 0,±d∗6= 0.

As an example, set α= 0.95, δ = 0.25, γ = 1, ρ = 0.7, φ = 0.6, σε=σν= 0.1. The

T-map contours are now three-dimensional and so to illustrate the set of equilibria

Figure 7 plots in (b1, b2, d) space. The ﬁgure only plots those values consistent with

covariance stationary statistical models employed by agents.12 The T-map contour

surfaces are more complicated. To easily illustrate the equilibria the small white

spheres in the ﬁgure illustrate intersections of the T-map contour lines consistent

with E-stable RPE. That is, there are many possible RPE, however in our numerical

explorations we have always found, as in the previous section, a pair of symmetric

E-stable sunspot RPE. In the ﬁgure the sunspot RPE are d∗≈ ±3.

3.2 Generalizing beliefs

We now generalize the central insight to alternative forecasting models, but continue

again with the simple univariate model

yt=αˆ

Etyt+1 +γzt,(11)

zt=ρzt−1+εt.(12)

Let |α|<1 so that the model is determinate, can feature positive or negative feedback,

and possesses a unique rational expectations equilibrium of the form

yt= (1 −αρ)−1γzt.

12In particular, we restrict the space so that −1< b2<0⇒ −1 + b2< b1≤ −2√−b2or

2√−b2≤b1<1−b2and 0 < b2<1⇒ −1 + b2< b1<1−b2.

21

Figure 7: T-map in model with lags.

The generalization here is, rather than assuming homogeneous expectations, we as-

sume that individuals are distributed across N≥2 diﬀerent predictors, with each

predictor having a diﬀerent set of variables in the forecasting equation. Denoting nj

as the fraction using predictor j= 1,2, ..., N , the expectational diﬀerence equation

(11) can be re-written as

yt=α

N

X

j=1

njˆ

Ejyt+j+γzt.(13)

Deﬁne the state vector X0

t= (yt, yt−1, ..., yt−p, θt, θt−1, ..., θt−q, zt, ηt). Such a state

vector will arise when agents form expectations from a perceived law of motion in the

general class:

b(L)yt=c(L)θt+dzt+fηt(14)

where θtis a forecast error process deﬁned by the regression equation (details below).

Heterogeneity arises when agents specify a PLM with subsets of regressors, e.g. with

diﬀerent lag polynomials, diﬀerent moving averages of past forecast errors, and/or

including/excluding the fundamental and sunspot shocks. A regression equation like

(14) is the population equivalent of what Ljung and S¨oderstr¨om (1983) call a “pseudo-

linear regression” (or extended least-squares), which is essentially a pseudo ARMAX

model where θtis the diﬀerence between ytand the time-tconditionally predicted

value. Below, we consider diﬀerent natural speciﬁcations of (14), where we again

22

assume throughout that ηtis a covariance stationary AR(1) process given by

ηt=φηt−1+νt(15)

where |φ|<1 and νtis white noise with variance σ2

ν.

The idea throughout the paper is that there are “hidden states,” or certain vari-

ables in (14) that are unobservable to some sets of agents. A set of agents may use

forecasting models, for example, that omit the pseudo moving-average terms, while

to some agents the fundamental shocks ztmay be hidden information. Models may

diﬀer in the order of the autoregressive terms. Some groups of individuals may include

sunspot variables while others do not condition on sunspots. The central insight is

that the heterogeneity in regressors of the perceived law of motion implies that each

agents’ forecast model is misspeciﬁed. In a restricted perceptions equilibrium (RPE),

the perceived law of motion is the optimal predictor in the sense that the coeﬃcients

satisfy a least-squares projection of ytonto the space of observable variables for each

type of forecasting model.13

3.3 Fundamental vs. AR(1) beliefs

As a simple case, suppose that there are two types of agents, with the fundamental

shocks, z, hidden to a fraction n, who ﬁt to the data an AR(1) plus sunspot as in the

preceding section. That is, a fraction of agents, n, hold a perceived law of motion

yt=byt−1+dηt⇒ˆ

E1

tyt+1 =b2yt−1+d(b+φ)ηt.(16)

The remaining 1 −nof agents form expectations from the perceived law of motion

yt=czt+fηt⇒ˆ

E2

tyt+1 =cρzt+fφηt.(17)

These agents have information on the fundamentals as well as the observable sunspot

η. Below, we consider the case, also, where these “fundamentalist” expectations

depend on zalone and omit the sunspot, i.e. an RPE with d6= 0, f = 0. When f= 0,

the latter perceived law of motion is consistent with the unique rational expectations

equilibrium in the model. However, with a fraction of agents forming expectations

from the PLM (16) this perceived law of motion is also misspeciﬁed by omitting the

expectations of the other agents, or more speciﬁcally the lagged endogenous variable.

In this sense, the expectations of the other type of agents is a hidden state variable.

13In this section njis taken as given, i.e. “extrinsic heterogeneity.” In the applications below, the

njare determined as equilibrium objects.

23

Inserting expectations (16)-(17) into (13) leads to the actual law of motion

yt=αnb2yt−1+ [α(1 −n)cρ +γ]zt+α[nd (b+φ) + (1 −n)fφ]ηt.(18)

It is useful to deﬁne the state vector X0

t= (yt, yt−1, ηt, zt). Then it is possible to

re-write (12), (15), and (18) into its standard VAR(1) form:

Xt=BXt−1+Ct,(19)

where 0

t= (εt, νt). It is clear that the state vector Xtexpands the variables in (11)

to include the variables that are observable to agents and that enter as regressors

in their forecasting equations. Since the expectations of the other agents are not

observable, each agents’ PLM is underparameterized relative to the actual law of

motion for the state vector Xt. The appropriate equilibrium concept is, again, a

restricted perceptions equilibrium which requires that the agents’ underparameterized

forecasting model is optimal within its restricted class.

Deﬁne ξ1= (b, d)0, ξ2= (c, f )0, x1

t= (yt−1, ηt)0, x2

t= (zt, ηt)0. In an RPE, beliefs

satisfy the following least-squares orthogonality conditions:

Ex1

tyt−ξ0

1x1

t= 0,

Ex2

tyt−ξ0

2x2

t= 0.

Denote Ω ≡EXtX0

t. It follows that

ξ1= Ω1,2:3Ω−1

2:3,2:3 ≡T1(b, c, d, f ),

ξ2= Ω1,3:4Ω−2

3:4,3:4 ≡T2(b, c, d, f ),

map the perceived coeﬃcients into the values that project ytonto their respective

regressors. That is the coeﬃcients are estimated so that the regression error is orthog-

onal to the (restricted) regressors.14 The T-maps give the mapping from perceived to

actual coeﬃcients, where the actual coeﬃcients are the values one would obtain by

regressing yonto the regressors using a long history of data.

We now turn to characterizing the set of RPE. We ﬁrst show that there always

exists an RPE with d=f= 0, i.e. that does not feature self-fulﬁlling ﬂuctuations.

Tedious calculations show that the d, f components of the T-maps are

d→α(1 −bφ)(bdn + (f+dn −fn)φ)

1−b2n αφ ,

f→α(bdn + (f+dn −fn)φ)

1−b2nαφ .

14The notation Ωi:j,k:lrefers to the (i−j)×(k−l) sub block of the matrix Ω.

24

Evidently, d=f= 0 is a solution to these pairs of equations. Then, the remaining

components of the T-map are:

b→b2nα +ρ

1 + b2nαρ,(20)

c→γ+c(1 −n)αρ

1−b2nαρ .(21)

Hence, there exists an RPE where ytdepends on zand a single lag of y. Denote the

fundamental RPE ﬁxed points to (20)-(21) as ˆ

b, ˆc.

We now characterize the sunspot equilibria. When d, f 6= 0, the d, f components

of the T-maps, T1, T2combine to

(1 −n) (1 −bφ)αφ

(1 −nαφ +bnα (φ2−1)) (1 −bφ)= 1,

which implies an RPE value

b∗=1−αφ

nα (1 −φ2).(22)

Note that the RPE value of b∗is identical to the benchmark case when n= 1.

It is straightforward to verify that such an RPE value for bexists provided |α|>

1/(n(1 −φ2) + φ). Then the remaining RPE coeﬃcients are computed as

c=γ

1−(1 −n)αρ −b2nαρ,

f2=e

k[b(1 + a1ρ)−a1−ρ]

α2[φ+nb −nφb(b+φ)]2[a1+φ−b(1 + a1φ)] −αφ(1 −bφ)(1 −a2

1)[φ+nb −nφb(b+φ)],

d= (1 −bφ)f,

where e

k=σ2

ε

σ2

ν

a2

2(1−a1φ)(1−φ2)

(1−a1ρ)(1−ρ2),a1=αnb2and a2=α(1 −n)cρ +γ.

Proposition 7 There exists a unique restricted perceptions equilibrium that depends

only on fundamental state variables with coeﬃcients (b, c, d, f )=(ˆ

b, ˆc, 0,0). More-

over, there exists two sunspot RPE (b, c, d, f )=(b∗, c∗,±d∗,±f∗)if and only if

i. 1

1+φ−φ2< α < 1

ii. 1−αφ

α(1−φ2)< n ≤1

iii. ˜ρ < ρ < 1,

25

where ˜ρ=nα(α−φ)φ(1−αφ)(1−φ2)

(1−αφ)3+n2α2(1−φ2)3.

Proposition 7 is the generalization of the previous existence result to the case where

agents are distributed across two (potentially) misspeciﬁed forecasting models. The

main diﬀerence is that the critical values for αand ρdepend on both φand the

degree of heterogeneity n. The thresholds in Proposition 7 have a similar form as

Proposition 1. Moreover, for α, φ large enough (as dictated by the proposition), ˜ρis

decreasing in n, implying that heterogeneity loosens the bounds on the ρconsistent

with sunspot RPE. The corollary to this result shows that sunspot RPE can exist

when only the AR(1) model includes sunspot and the other type of agents are pure

fundamentalists.

Corollary 8 Consider the case of AR(1) with sunspots versus fundamentalist (f=

0). If the fraction using the AR(1) model n > 4/5, then ∃˜α(n, φ)and ˜ρ(α, φ, n)s.t.

sunspot RPE (b∗, c∗,±d∗)exists if and only if α > ˜α, ρ > ˜ρ.

3.4 AR(1) vs. AR(2) beliefs

This subsection and the next present two cases where the fundamental shocks z

are hidden to both groups of agents. In this section, the agents diﬀer by the lag

polynomials in their forecasting equations. This subsection considers a simple case

with one kind of agents forecasting with an AR(1) model and the remaining forecast

using an AR(2) model. Let ndenote the fraction of the agents using AR(1) model

plus sunspot. This is the heterogeneous expectations version of the model in Section

3.1 when δ= 0. The forecasting models are

yt=byt−1+dηt,

yt=b1yt−1+b2yt−2+fηt.

Note that the ﬁrst-order and second-order autocorrelations of the AR(2) model with-

out ηtare b1

1−b2,b2

1

1−b2+b2. Hence their forecasts are, respectively,

E1,tyt+1 =b2yt−1+ (b+φ)dηt,

E2,tyt+1 =b1E2,t yt+b2yt−1+f φηt= (b2

1+b2)yt−1+b1b2yt−2+ (b1+φ)fηt.

The actual law of motion is

yt=α[nb2+ (1 −n)(b2

1+b2)]yt−1+ (1 −n)αb1b2yt−2+γzt

+α[n(b+φ)z+ (1 −n)(b1+φ)f]ηt

=a1yt−1+a2yt−2+γzt+a4ηt,

(23)

26

where a1=α[nb2+ (1 −n)(b2

1+b2)], a2= (1 −n)αb1b2and a4=α[n(b+φ)d+ (1 −

n)(b1+φ)f].

Analytic results for the existence of sunspot RPE are unavailable. The following

result proves the existence of a fundamental RPE in this setting.

Proposition 9 There exists a restricted perceptions equilibrium that depends only on

fundamental state variables with coeﬃcients (b, b1, b2, d, f ) = (ˆ

b, ˆ

b1,ˆ

b2,0,0).

Although sunspot RPE existence is analytically unavailable, the Appendix conjectures

that it is possible for their to exist multiple sunspot RPE with non-zero d∗and f∗,

and numerical examples conﬁrm the conjecture. In particular, with α= 0.95, ρ = 0.9

then for all 0 ≤n≤1 there is an open set of φconsistent with existence of sunspot

RPE.

3.5 Pseudo moving-average vs. AR(1)

We continue to assume that the fundamental shocks ztare unobservable to all agents.

One group of agents still forecasts with the AR(1) plus sunspot, while the remain-

ing agents instead adjust expectations according to their lagged forecasting error, a

pseudo MA(1), plus a sunspot.

Assume that agents are distributed such that their forecasts come from one of two

perceived laws of motions:

yt=byt−1+dηt⇒ˆ

E1

tyt+1 =b2yt−1+d(b+φ)ηt(24)

yt=θt+cθt−1+fηt⇒ˆ

E2

tyt+1 =ye

t+1 (25)

where the latter expectations are deﬁned by the recursions

ye

t+1 =cθt−1+f(c+φ)ηt−cye

t

θt=yt−ye

t

This recursion arises under the consistent timing convention that yt, hence θt, is not

contemporaneously observable when forming expectations. As discussed in Marcet

and Sargent (1995), the second PLM (25) is a population equivalent of a pseudo linear

regression, or extended least-squares, a recursive alternative to maximum likelihood

for estimating the unobserved moving average terms in ARMAX models.15 It is a

pseudo MA(1) because θtis not restricted to be white noise innovations but instead

it satisﬁes the weaker condition that θtis orthogonal to the regressors θt−1, ηt.

15Weill and Gregoir (2007) prove the existence of MA(q) restricted perceptions equilibria.

27

With these expectations, the actual law of motion for the economy is given by the

following recursive equations

yt=αnb2yt−1+α(1 −n)c(θt−1−ye

t) + α[nd (b+φ) + (1 −n)f(c+φ)] ηt+zt

ye

t+1 =c(θt−1−ye

t) + f(c+φ)ηt

θt=yt−ye

t

zt=ρzt−1+εt

ηt=φηt−1+νt

As before, let X0

t=yt, yt−1, ye

t+1, θt, ηt, zt, then with appropriate conformable ma-

trices B, C , we have

Xt=BXt−1+Ct

Tedious calculations again lead to complicated polynomial expressions for the band

ccomponents of the T-map. For the sunspot coeﬃcients, the T-map components are

d→α(1 −bφ) [bdn + (f(1 −n) + dn)φ+c(dnφ(b+φ)−(1 −n)(1 −φ2))]

1−b2nαφ −c(α(1 −n) + φ(b2nαφ −1)) (26)

f→dnα (b+φ)

1−αφ(1 −n)−b2nαφ (27)

It is possible to characterize the RPE. As before, it is evident from (26)-(27) that

d=f= 0 is an RPE. Closed form solutions exist for the b, c subcomponents of the

T-map, however, the expressions are complicated and omitted for ease of exposition.

It is possible to show that the RPE values for b, c are solutions to a pair of cubic

polynomials with a unique real solution.

In the case that d, f 6= 0, the bcomponent of the T-map takes the same form as

in the previous case:

b→1−αφ

nα (1 −φ2)

Then, the RPE values for (c, d, f) can be found by solving (27) and the remaining

components of the T-map. Closed-form solutions are unavailable. Instead, we turn

to a numerical example.

Before presenting a speciﬁc example, a few brief comments about how the existence

results diﬀer with the pseudo MA(1) predictor:

1. For n suﬃciently small (including n= 0, the case of just a pseudo MA(1)) there

can exist multiple E-stable fundamental RPE.

28

2. For nsuﬃciently small (including n= 0) there can exist multiple sunspot RPE.

A subset of these sunspot RPE may be E-stable.

3. For nsuﬃciently large, there exists a unique fundamental RPE.

An exhaustive study of this case is beyond the scope of this paper and left for fu-

ture research. This section, instead, presents one numerical example, illustrating the

generality of sunspot RPE. To remain consistent with the previous results we set

n= 0.85, α = 0.99, ρ = 0.7, φ = 0.8. With these parameter values, there exists a

unique fundamental RPE.

Figure 8 plots various slices of the phase space. Since there are 4 parameters,

each plot graphs the two-dimensional phase space assuming that the remaining belief

parameters are ﬁxed at their RPE values.

Figure 8 demonstrates the existence of an E-stable sunspot RPE. This particular

ﬁgure is silent on the existence of other fundamental and sunspot RPE. That is

because these are plotting, essentially, slices of the full phase space locally around a

particular RPE.

4 Applications

We turn now to economic applications: a New Keynesian model with optimal policy;

an asset-pricing model; and, a pure monetary economy.

4.1 Misspeciﬁcation equilibria

Because of the prominent role played by the distribution of agents across models in the

applications, this section addresses the question of whether agents will coordinate on

a sunspot RPE by endogenizing the distribution of agents across forecasting models.

We follow Brock and Hommes (1997) and Branch and Evans (2006) in specifying

that the fraction of agents is determined via a multinomial logit (MNL) map. The

idea being that each agent makes a discrete choice between forecasting models in a

random utility setting. The likelihood that an agent will select a given forecasting

model is increasing in its forecast accuracy, as measured by mean-squared forecast

errors. In particular, we assume that nis determined by the following MNL map:

n=1

2ntanh hω

2EU 1−EU 2i+ 1o

29

Figure 8: E-stability Dynamics: pseudo MA(1) v. AR(1) beliefs. Top-left: (b, d)

plane. Top-right: (c, f ) plane. Bottom: (b, c) plane.

0.0

0.2

0.4

0.6

0.8

1.0

-1.0

-0.5

0.0

0.5

1.0

,

where

EU j=−Eyt−Ej

t−1yt2

where the unconditional expectation is with respect to the probability distribution

implied by the RPE. The parameter ω > 0 is the ‘intensity of choice.’ We focus

on the neoclassical case where ω→ ∞ so that agents only select the best perform-

30

ing forecasting model. The MNL map has a natural interpretation of introducing

randomness into forecasting which, like mixed actions in strategic games, provides a

means to remaining robust to forecast model uncertainty.

We construct a misspeciﬁcation equilibrium as follows. For each n, we calculate the

RPE. When there are multiple RPE, for a given n, we select the E-stable equilibria.

That is, we can deﬁne F: [0,1] →Ras

F(n) = EU 1−EU 2

Then, there is a mapping S: [0,1] →[0,1] where

Sω(n) = 1

2ntanh hω

2F(n)i+ 1o

Deﬁnition 10 A misspeciﬁcation equilibrium is a ﬁxed point n∗=Sω(n∗). Further-

more, a misspeciﬁcation equilibrium exhibits intrinsic heterogeneity if 0< n∗<1as

ω→ ∞.

4.2 Optimal monetary policy

A seminal result by Orphanides and Williams (2005) is that economies with non-

rational agents who update their forecasting models using an adaptive learning rule,

the optimal monetary policy rule involves a more aggressive response to inﬂation

deviations from target.16 The intuition behind this well-known result is that with

non-rational expectations the central bank seeks to minimize inﬂation volatility in or-

der to help anchor private-sector expectations. This section revisits Orphanides and

Williams (2005) with our theory of restricted perceptions and endogenous volatil-

ity. We ﬁnd that the size of the central bank’s reaction coeﬃcient responds non-

monotonically to departures from rational expectations.

We adapt Orphanides and Williams (2005) to the present environment:

πt=βˆ

Etπt+1 +κyt+ut(28)

yt=xt+gt(29)

where πtis the inﬂation rate, ytis the output gap, and ut, gtare aggregate supply and

aggregate demand shocks, respectively. Equation (28) is a standard New Keynesian

16For a general discussion of this robust ﬁnding, see the excellent survey by Eusepi and Preston

(2017), in particular “Result 5a.”

31

Phillips Curve that comes from the aggregate supply block.17 As before, we assume

that aggregate supply shocks follow a stationary AR(1) process, ut=ρut−1+εt.

Equation (29) is the aggregate demand equation and it relates the output gap to the

central bank’s policy variable, xt, up to noise gt. Without loss of generality, assume

σ2

g→0, so that the central bank can control aggregate demand perfectly. The latter

assumption is technically convenient and sidesteps issues about how policymakers can

exert such control.

The central bank has an optimal instrument rule of the form

xt=−θ(πt−¯π)

where ¯πis the long-run inﬂation target, which we set to ¯π= 0. The central bank

chooses the coeﬃcient in its policy rule in order to minimize a quadratic loss function

L= (1 −λ)Ey2

t+λEπ2

t

In order to parameterize deviations from rational expectations, we assume that a

fraction nof agents form their forecasts from

πt=bπt−1+dηt

and the remaining 1 −nagents forecast from

πt=cut+fηt

If n= 0, then the model is isomorphic to the model under rational expectations

and policymakers choose θwhile facing the usual trade-oﬀ between output and price

stability. With the speciﬁcations for expectations in hand, the actual law of motion

is given by

πt=βnb2

1 + κθ πt−1+β[n(b+φ)d+ (1 −n)fφ]

1 + κθ ηt+1 + (1 −n)βcρ

1 + κθ ut

The restricted perceptions equilibrium is pinned down as a function of the structural

parameters β, κ, ρ, φ, σ2

u, σ2

η, the policy parameter θ, and the distribution across pre-

dictors n. Denote π∗

t(θ) as the (E-stable) restricted perceptions equilibrium process

given θ. Then, the optimal policy coeﬃcient minimizes

(1 −λ)θ2+λE(π∗

t(θ))2

17The derivation of this equation under heterogeneous expectations was provided by Branch and

McGough (2009).

32

which is a standard quadratic objective function with a unique solution for θ∗. When

n= 0 there is a unique (closed-form) solution, see Orphanides and Williams (2005). In

general, a closed-form solution for θ∗does not exist, so we turn to numerical examples

by adopting the following parameterization: β= 0.99, κ = 0.1, ρ = 0.5, φ = 0.75,

and a baseline value for λ= 0.5. The main qualitative ﬁndings are robust to the

empirically plausible range of values for κ, and for larger values of ρ. These values

are consistent with the existence of an RPE with sunspots for suﬃciently large n.

4.2.1 Restricted perceptions and the policy frontier

How do restricted perceptions aﬀect optimal monetary policy and the trade-oﬀs faced

by policymakers? Figure 9 illustrates the answer. For now, taking the distribution

of agents across models ﬁxed and parameterized by n, we ﬁrst analyze how restricted

perceptions aﬀects the policy frontier, the so called “Taylor curve,” which illustrates

the trade-oﬀ between output volatility, σ2

y, and inﬂation volatility, σ2

π. The upper-

right panel plots the policy frontier for diﬀerent fractions of agents forecasting with

the AR(1) model.

Each point on a frontier gives the output and inﬂation volatility in a restricted

perceptions equilibrium, with optimally chosen θ∗, for diﬀerent weights on inﬂation

in the loss function λ. The slope of the frontier indicates the usual trade-oﬀ faced

by policymakers between stabilizing inﬂation and output volatility. The upper dot-

ted line, corresponding to n= 0, is the policy frontier in a rational expectations

equilibrium. The plot indicates what happens to the policymaker’s trade-oﬀs as the

economy moves away from the rational expectations equilibrium, with this shift pa-

rameterized by n. As nincreases from n= 0 the trade-oﬀ shifts the policy frontier

closer to the origin indicating a more favorable trade-oﬀ between inﬂation and out-

put volatility. Having some agents adopt an AR(1) forecasting model would lead to

lower output and inﬂation volatility. However, for suﬃciently large values of nthe

frontier again shifts out away from the origin, indicating that with a high fraction

of agents forecasting with the AR(1) model then both inﬂation and output volatility

will increase.

Why does the fraction of agents with restricted perceptions impact the policy

trade-oﬀ non-monotonically? The intuition can be found in the lower two panels. The

lower right panel plots inﬂation volatility within a restricted perceptions equilibrium

and ﬁxing λ= 0.5. For moderate values of n, the restricted perceptions equilibrium

exhibits less inﬂation volatility than the rational expectations equilibrium. This is

because aggregate expectations are a linear combination of expectations that are close

to rational expectations and less volatile restricted perceptions from the AR(1) model.

33

Figure 9: Optimal Monetary Policy.

0.04 0.06 0.08 0.10 0.12 0.14

3.0

3.5

4.0

4.5

5.0

σ2

y

σ2

π

n=0.78 n=0.25 n=0.85 REE

0 1 2 3 4 5

0

1

2

3

4

σ2

y

σ2π

0.0 0.2 0.4 0.6 0.8

0.00

0.05

0.10

0.15

0.20

0.25

n

θ*

0.0 0.2 0.4 0.6 0.8 1.0

3.0

3.5

4.0

4.5

n

Var π

However, as the fraction of agents using the AR(1) model increases then the perceived

serial correlation in the AR(1) model increases and reﬂects the (unobserved by these

agents) serial correlation from the supply shocks and self-fulﬁlling serial correlation

from beliefs. This, in turn, increases overall inﬂation volatility. At a critical value of

n, though, the equilibrium bifurcates and a sunspot RPE arises with lower volatility

than the fundamentals RPE.

The lower left panel illustrates how optimal monetary policy reacts to these chang-

ing policy trade-oﬀs. This panel plots the optimal value of θas a function of n, i.e.

θ∗(n). The most aggressive response in the policy rule occurs at the rational expecta-

tions equilibrium. As nincreases this lowers overall volatility and leads policymakers

to be less hawkish in responding to inﬂation. This provides the intuition for the

34

shift in the policy frontier: with lower inﬂation volatility and less hawkish policy

rules, output volatility decreases as well. However, after some critical value more

heterogeneity in expectations increases inﬂation volatility and policy becomes more

hawkish, leading to greater output volatility.

The lower left panel also illustrates how the bifurcation to a restricted perceptions

equilibrium with sunspots aﬀects how aggressively optimal policy should react to

inﬂation innovations. Recall before that the existence of these statistical sunspot

equilibria requires that the economy exhibits strong self-referential feedback, i.e. low

values of θ. The lower right panel shows that these equilibria are less volatile, so that

in this range of nthe policymakers want the private-sector to coordinate on a RPE

with sunspots. Thus, at a suﬃciently high value of nthe optimal policy response

drops sharply in order to induce coordination on the RPE with sunspots.

The upper left panel summarizes the eﬀects of heterogeneity on the policy trade-

oﬀs. Each point on the frontier is a restricted perceptions equilibrium value for output

and inﬂation volatility, ﬁxing the central bank’s preference for inﬂation stability at

λ= 0.5. Notice that there are three segments. As the fraction increases from n=

0, i.e. away from the rational expectations equilibrium, heterogeneity leads to less

inﬂation and output volatility as the trade-oﬀ favors for policymakers. After some

critical value, though, further increases in the degree of heterogeneity increases overall

volatility, worsening outcomes. The third segment corresponds with the bifurcation

to a RPE with sunspots, equilibria that exhibit monetary policy that is not as active,

higher inﬂation volatility and lower output volatility.

4.2.2 Optimal monetary policy and misspeciﬁcation equilibria

Given the eﬀects of heterogeneity on the trade-oﬀs faced by policymakers, a natural

question is whether it is possible for an economy to coordinate on a RPE with lower

inﬂation and output volatility. We extend the previous analysis by endogenizing the

distribution of agents across the two forecasting models within a misspeciﬁcation

equilibrium.

Optimal monetary policy seeks to minimize their loss function taking the distri-

bution of agents as given. Deﬁne θ∗(n) as follows:

θ∗(n) = arg min

θ(1 −λ)θ2+λE(π∗

t(θ, n))2

where the expectation is taking with respect to the distribution implied by the (E-

stable) RPE associated to n. Recalling the previous discussion, predictor choice takes

35

optimal policy as given, i.e. the MNL map is:

Sω(n, θ) = 1

2ntanh hω

2F(n, θ)i+ 1o

A misspeciﬁcation equilibrium is a ﬁxed point n∗=Sω(n∗, θ). Given n, optimal mon-

etary policy is chosen and given θthe agents make their discrete choice of forecasting

model. This is a simultaneous move game between private-sector agents selecting

their model and the central bank in choosing its policy rule optimally.

Deﬁnition 11 An Optimal Policy Misspeciﬁcation Equilibrium is a symmetric Nash

equilibrium deﬁned by the ﬁxed point n∗=Nω(n∗), where Nω=Sω◦θ∗.

Because Nω: [0,1] →[0,1], Nωis a continuous function, there exists at least one

Optimal Policy Misspeciﬁcation Equilibrium. However, because θ∗(n) and F(n, θ)

may be non-monotonic, there can also exist multiple Optimal Policy Misspeciﬁcation

Equilibria. It is not possible to establish conditions for uniqueness, however, we do

have the following existence result.

Proposition 12 There exists at least one Optimal Policy Misspeciﬁcation Equilib-

rium corresponding to n∗= 0.

Thus, the RPE that is isomorphic to the (unique) rational expectations equilib-

rium is always an Optimal Policy Misspeciﬁcation Equilibrium. However, there can

exist other Misspeciﬁcation Equilibria as well. The experiment is to see how policy

outcomes are aﬀected when policy and expectation formation are jointly determined

in equilibrium. Figure 10 demonstrates the results for a typical parameterization

that leads to multiple misspeciﬁcation equilibria. Here we adopt the same parame-

terization as the previous section except now we set ρ=.9, φ =.98. This particular

parameterization leads to RPE with sunspots for suﬃciently large values of n. The

right panel plots the set of (stable) misspeciﬁcation equilibria as a function of λ. Ev-

idently, for a range of λnot too large there exist equilibria at n∗= 0 and n∗= 1. A

central bank that places suﬃciently high weight on inﬂation stabilization will coordi-

nate the economy on the rational expectations equilibrium.

The left panel of Figure 10 plots the policy frontiers for each of the two equi-

libria. The frontier closest to the origin coincides with the n∗= 0 equilibrium, i.e.

the rational expectations equilibrium. For the range of λconsistent with multiple

equilibria, the RPE with sunspots leads to worse outcomes for the central bank: the

policy frontier for n∗= 1 lies to the northeast of the one for n∗= 0. However, for λ

suﬃciently large, there is a bifurcation and the frontiers coincide with low inﬂation,

36

Figure 10: Optimal Monetary Policy and Misspeciﬁcation Equilibria.

012345

0

1

2

3

4

σ2

y

σ2

π

0.0 0.2 0.4 0.6 0.8 1.0

0.0

0.2

0.4

0.6

0.8

1.0

λ

n

and high output, volatility. For lower values of λ, there exist multiple equilibria and

the central bank would prefer the economy coordinates on the n∗= 0 equilibrium.

However, it is possible that the central bank could ﬁnd itself trapped at the inferior

equilibrium.

How realistic is it that optimal policy will lead to coordination on a non-rational

expectations equilibrium? To a certain extent, this result is an implication of the

timing protocol between the central bank and the private-sector. If instead of deter-

mining n, θ simultaneously, the timing was changed to a Stackleberg game where the

central bank takes the expectation formation of agents into account when choosing θ.

In this case, the central bank would choose to be at the n= 0 equilibrium. However,

in order to coordinate the economy on the preferred outcome the central bank needs

to understand the expectation formation process, including the sets of forecasting

models and the MNL-map, an unrealistic assumption.

4.3 Mean-variance asset-pricing model

A simple example of the general framework (11)-(12) is the mean-variance asset pric-

ing model with stochastic AR(1) dividends; see for example Brock and Hommes

(1998), Branch and Evans (2010) and Hommes and Zhu (2014).

Assume that there exist two kinds of agents and agents can invest in a risk free

and a risky asset. The risk-free asset is perfectly elastically supplied at a gross return

R > 1. ptdenotes the price (ex dividend) of the risky asset and dtdenotes the

(random) dividend process. Let b

Et,b

Vtdenote the subjective beliefs of agents about

37

the conditional expectation and conditional variance of excess return pt+1 +dt+1 −Rpt.

Agents are assumed to be a myopic mean-variance maximizer of next period’s wealth.

Optimal demand hj

tfor the risky asset by a type-jagent is then given by

hj

t=b

Ej

t(pt+1 +dt+1 −Rpt)

ab

Vj

t(pt+1 +dt+1 −Rpt)=b

Ej

t(pt+1 +dt+1 −Rpt)

aσ2,

where a > 0 denotes the risk aversion coeﬃcient and the belief about the conditional

variance of the excess return is assumed to be constant over time, i.e. b

Vj

t(pt+1 +dt+1 −

Rpt)≡σ2. The dividend process dtis assumed to follow an AR(1) process

dt=ρdt−1+εt,(30)

where ρ∈[0,1) and εtis an i.i.d. process with the standard deviation σε. Thus

hj

t=b

Ej

t(pt+1) + ρdt−Rpt

aσ2.

The market clearing price is

pt=1

R

2

X

j=1

njb

Ej

tpt+1 +ρ

Rdt.

As in the previous sections, assume that a fraction nof agents form their forecasts

from

pt=bpt−1+dηt,

and the remaining 1 −nagents forecast from

pt=cdt+fηt.

In this context, ηtmight also be interpreted as news stories unrelated to fundamental

dividends in the same spirit as Shiller (2017).

4.3.1 Restricted perceptions equilibrium and price volatility

We now revisit the relative volatility of RPE with and without sunspots in this het-

erogeneous expectations framework. To gain intuition, we focus on extrinsic hetero-

geneity and consider misspeciﬁcation equilibria in the Appendix. With the diversity

in beliefs, but a common sunspot term, there is a further self-fulﬁlling channel at

work, which impacts the volatility of sunspot RPE.

38

The actual law of motion is given by

pt=1

Rnb2pt−1+1

R[(1 −n)cρ +ρ]dt+1

R[nd (b+φ) + (1 −n)fφ]ηt.

and the variance of ptis (see Appendix 5)

V ar(pt) = a2

2(1 + a1ρ)σ2

ε

(1 −a2

1)(1 −a1ρ)(1 −ρ2)+a2

3(1 + a1φ)σ2

ν

(1 −a2

1)(1 −a1φ)(1 −φ2).(31)

where a1=nb2

R>0, a2=(1−n)cρ+ρ

R, a3=nd(b+φ)+(1−n)fφ

R.Thus it is easy to compute

the comparative static eﬀects the structural parameters have on existence and the

excess volatility of RPE.

The persistence of the sunspot shock, φ, plays an important role in both the

existence and the excess volatility properties of the RPE18. Consider the comparative

static eﬀects of φ. As in Section 2, there are three RPE for a range of φthat consisting

of φthat are suﬃciently, but not too, large. When φincreases, the price volatility

of the fundamental RPE does not change, for obvious reasons, while the volatility at

the sunspot RPE is again ambiguous, ﬁrst decreasing in φand then increasing.

The intuition for the important role played by φin the price variance and excess

volatility is as follows. As φ→0 or φ→1, the sunspot shock is approximately white

noise or a random walk, respectively. In these cases, the forecasting power of the

sunspot is low, implying that sunspot RPE do not exist. For moderate values of φ,

on the other hand, sunspot RPE do exist and, consequently, φhas an eﬀect on the

equilibrium stock price volatility. Recall that b∗=R−φ

n(1−φ2)>0, c∗=ρ

R−nb∗2ρ−(1−n)ρ. It

follows that the comparative static eﬀect of φon b∗and c∗is ambiguous. Similarly,

for the sunspot coeﬃcients, d, f , where φhas a positive eﬀect for relatively small

values of φ, and then a smaller negative eﬀect for larger values of φ. It turns out

that for relatively small values for φ, the negative comparative eﬀect of φon b∗(i.e.

via a1and a2) plays a dominant role and the price volatility at ﬁrst decreases with

18We ﬁnd that the noise term of intrinsic shock (correspondingly σ2

ν) has no eﬀects on the RPE

and the variance at the corresponding RPE. It is easy to see that σ2

νhas no eﬀect on the fundamental

RPE. For the non-fundamental RPE, b∗and c∗are not aﬀected by σ2

νwhile |f∗|and |b∗|decrease

as σ2

νgrows given other parameters constant, which can be seen from the expression of f∗2and

d∗2. From this, σ2

νseems to aﬀect the variances. But further analysis indicates that at the non-

fundamental RPE,

a2

3σ2

ν=f∗2σ2

ν[n2(b∗+φ)2(1 −b∗φ)2+ 2n(1 −n)φ(b∗+φ)(1 −b∗φ) + (1 −n)2φ2]

R2.(32)

Therefore, the expression of f∗2implies that a2

3σ2

νdoes not depend on σ2

ν. Thus σ2

νdoes not aﬀect

the variances of market prices at the RPE.

39

small values of φ. As φbecomes large enough, a1and a2increase with respect to φ

and hence the price volatility eventually increases with the persistence of the sunspot

shock.

Figure 11: Eﬀects of φon the stock price variance

Fundamental RPE

Non-fundamental RPE

0.2

0.4

0.6

0.8

1.0

0

50

100

150

200

250

300

350

Φ

Varianceof p

Section 2 demonstrated that the volatility of the sunspot RPE is lower than the

fundamental RPE when all agents forecast with an AR(1) model. When there is

heterogeneous expectations, but all agents include the sunspot in their forecasts, there

is an additional self-reinforcing eﬀect. The strength of this eﬀect can be parameterized

by n. Accordingly, the comparative static eﬀect of non excess volatility is ambiguous.

It can be shown that ∂ b∗

∂n <0 and ∂c∗

∂n <0. That is, nhas a negative eﬀect on the

sunspot RPE coeﬃcients b∗and c∗. Conversely, for suﬃciently large n,|d∗|is an

increasing function of nand |f∗|ﬁrst increases and then decreases with n. It follows

that excess stock price volatility ﬁrst increases with nbut then decreases for nlarge

enough because of the relatively weaker impact on b∗and c∗: see the left panel of

Figure 12. The role played by heterogeneity in determining price volatility, however,

is diﬀerent for the fundamental RPE. Note that ˆ

bincreases as ngrows from 0 to 1 so

that as more agents use the AR(1) rule the persistence of dividend shocks increases.

Furthermore, numerical simulations indicate that ˆcﬁrst decreases ( n∈[0,0.4)) and

then increases ( n∈(0.4,1]): see the left panel of Figure 12. The excess volatility

of market prices at the fundamental RPE and non-fundamental RPE compared with

the rational expectations equilibrium price (i.e. when n= 0) is similar, as shown in

the right panel of Figure 12.

40

Figure 12: Eﬀects of non the variances of market prices and excess volatility

Fundmental RPE

Non-fundamental RPE

0.86

0.88

0.90

0.92

0.94

0.96

0.98

1.00

200

250

300

350

400

450

500

n

Varianceof p

,

Fundamental RPE

Non-fundamental RPE

0.86

0.88

0.90

0.92

0.94

0.96

0.98

1.00

1.5

2.0

2.5

3.0

n

Ratioof Variances

4.4 A pure monetary model

This section presents an economy that naturally admits intrinsic heterogeneity, i.e.

a misspeciﬁcation equilibrium with 0 < n∗<1 in the neoclassical limit ω→ ∞.

The economic environment is based on Rocheteau and Nosal (2017), who modify

the Lagos and Wright (2005) and Rocheteau and Wright (2005) model to include

monetary policy via interest paid on reserves with ex-ante buyers and sellers. This is

a perfect environment to explore heterogeneity because the model naturally admits

negative expectational feedback.

The inﬁnitely-lived economy consists of two subperiods at each date t. The ﬁrst

subperiod, called the decentralized market (DM), opens with bilateral matching of

buyers and sellers, who are anonymous and have scope to trade a perishable good but

because of a commitment problem all trade is quid pro quo with ﬁat money as the

payment instrument. In the second period, called the centralized market (CM), buy-

ers and sellers produce, with a linear technology, the numeraire good and re-balance

their portfolio, with a price φof money in terms of the numeraire. Additionally,

the central bank pays a gross interest on currency held at the beginning of the CM,

Rt=Rm, ﬁnanced via lump-sum taxes and assumed to be a constant peg. In addition,

the central bank engineers a growth rate of the money supply, Mt, via proportional

transfers to money holders. In Branch and McGough (2016) this basic environment

is extended to include buyers who diﬀer by expectations-type and the implications of

heterogeneous beliefs for trading and bargaining is studied. Here, we take a particu-

larly simple case in Branch and McGough (2016) where buyers make take-it-or-leave

41

it oﬀers and have dogmatic priors.19

Rocheteau and Nosal (2017) show that money-demand min this environment is

derived from the buyers’ ﬁrst-order condition:

βu0(qt+1 ) = φt

φe

t+1

1

Re

t+1

and where the take-it-or-leave it oﬀers determines the amount qttraded in the period-t

DM:

qt=φe

tRtmt

The buyers’ ﬁrst-order conditions balance the discounted marginal utility from con-

suming qagainst the expected real rate of return. Let u(q) = (q+b)1−α−b1−α

1−αand

without a loss of generality set b→0. As in the previous applications, agents will

diﬀer by expectations-type. Money-demand of type jis given by

mα

j,t =βφ−1φe

j,t+1Rm1−α

Then letting ˆmt= log Mtand assuming ˆmt−ˆmt−1≡zt, it is straightforward to

see that a log-linearization around the monetary steady-state leads to the following

expectational diﬀerence equation determining the (log) inﬂation rate:

πt= (1 −α)ˆ

Etπt+1 +αzt

where ˆ

Eπ =nE1π+ (1−n)E2π. Now we assume that the (log) money-supply growth

process is driven by two stationary exogenous AR(1) processes:

zt=ϑ1z1t+ϑ2z2t

and

z1t=ρ1z1t−1+ε1t

z2t=ρ2z2t−1+ε2t

with −1< ρi<1, i = 1,2. For simplicity, the constant interest rate peg and money-

supply as log deviations around a constant level imply a zero steady-state inﬂation

rate.

19Branch-McGough introduced the concept of Bayesian oﬀers as a bargaining solution where

buyers make take-it-or-leave-it oﬀers but sellers’ beliefs are not common knowledge within a match.

Thus, in order to make an oﬀer the buyer needs to place a prior over sellers’ beliefs. In the dogmatic

prior case, the buyer dogmatically believes the seller has the same beliefs.

42

Notice that when 1 < α < 2 the model has a unique rational expectations equi-

librium and exhibits negative feedback. This section shows the possibility of Intrinsic

Heterogeneity in a setting where z2tis unobservable to both groups of agents. As

before, one group of agents forecasts with an AR(1) plus sunspot while the other

forecasts with z1tplus sunspot. Unlike the previous examples, since both forecast

models omit z2tthey are misspeciﬁed models for any distribution n. Extending the

benchmark framework to the present environment leads to a fraction nwith expec-

tations

E1

tπt+1 =b2πt−1+d(b+φ)ηt

and the remaining 1 −nbuyers hold

E2

tπt+1 =cρ1zt+fφηt

As before, given na restricted perceptions equilibrium pins down the values of

b, c, d, f . Then one can compute the mapping that pins down the distribution across

models:

n=1

2tanh ωEU 1−EU 2+ 1≡HωEU 1−EU 2

Recalling that F(n) = EU 1−EU 2, because an RPE exists it follows that there exists

a well-deﬁned mapping Sω=Hω◦F, S : [0,1] →[0,1] and Sis continuous. Therefore,

there exists at least one misspeciﬁcation equilibrium n∗=Sω(n∗).

Although the form of the equilibrium condition and the set of forecasting functions

is very close to the benchmark example in Section 3, assuming zis bivariate makes

the restricted perceptions equilibrium and misspeciﬁcation equilibrium not easily pre-

sentable.20 Instead, we present an example of a case where Intrinsic Heterogeneity

exists, i.e. where F(0) >0 and F(1) <0, so that even in the neoclassical limit

ω→ ∞ agents have an incentive to deviate from a situation where everyone is us-

ing the same model. As an example, set α= 1.6, ρ1= 0.1, ρ2= 0.5, φ = 0.1, ϑ1=

0.5, ϑ2= 0.95, σ2

ε1=σ2

ε2=σ2

ν= 1, ω = 1000. Figure 13 plots the S-map and the net

predictor ﬁtness function F(n).

The bottom panel of Figure 13 plots the predictor ﬁtness diﬀerence as a function of

the distribution nof buyers who forecast with the AR(1) plus sunspot model. Under

the example parameterization F(0) >0 and F(1) <0 and, moreover, F(n) is a

monotonically decreasing function. Then the top panel plots the S-map. When n= 0

and F(0) >0 then the S-map dictates that Sω(0) = 1 for large ω. Similarly, n= 1

implies F(1) <0, Sω(1) = 0, ruling out misspeciﬁcation equilibria with homogeneous

expectations. Since we know from Brouwer’s theorem that a ﬁxed point to Sexists

20The Mathematica program that computes the analytic expressions and produces the ﬁgures is

available from the authors upon request.

43

Figure 13: Intrinsic Heterogeneity in a pure monetary economy.

it must be the case that the misspeciﬁcation equilibria are interior. In the top panel

this equilibrium occurs where the S-map crosses the 45◦line at ≈0.75. In this

particular numerical example the F(n) line is monotonically decreasing which implies

the existence of a unique misspeciﬁcation equilibrium with intrinsic heterogeneity.

However, this is not a general feature of F. For example, even with the negative

feedback there are many parameterizations that produce a monotonically increasing

Fline implying the existence of multiple misspeciﬁcation equilibria with sunspots.

Many parameterizations feature a unique equilibrium at either n= 0 or n= 1.

Finally, it is also possible that Fis a non-monotonic function of n. Because of the

broad set of possible equilibria and the complicated expressions for b, in particular,

preclude us from any analytical characterization of the equilibrium set beyond the

existence result discussed above.

44

5 Conclusion

The results in this paper show that sunspot equilibria can exist in models with a

unique rational expectations equilibrium. The minimal deviation from a general class

of macroeconomic models is that some state variables are unobserved, or hidden, to

a subset of agents. These restricted perceptions lead agents to extract information

about unobserved variables from the endogenous variables by specifying optimal par-

simonious forecast models that condition on observable state variables. In a restricted

perceptions equilibrium beliefs are optimal within the restricted class. The insight

in this paper is that while certain fundamental, i.e. payoﬀ relevant, variables may

be hidden to agents they may end up coordinating on an equilibrium that depends

on extrinsic variables that we call “statistical sunspots.” These statistical sunspots

overcome two limitations of sunspot theories based on rational expectations: they

exist in the empirically relevant range of models, i.e. within the determinacy region

–even in non-linear models – and, the sunspot equilibria can be stable under learning.

This paper focused on the theoretical properties of statistical sunspots with ap-

plications to optimal monetary policy, excess volatility in stock prices, and exis-

tence of heterogeneous expectations in monetary economies. The theory of statistical

sunspots, though, has broad practical interest for DSGE models. Under appropriate

conditions, statistical sunspots exist in standard formulations of real business cy-

cle models; that is, sunspot equilibria can exist without relying on non-convexities.

Statistical sunspots can also exist in New Keynesian models with optimal monetary

policy or Taylor-type rules that respect the “Taylor principle.” In particular, policy

advice to rule out expectations-driven cycles, i.e. unanchored expectations, is more

subtle than what one would conclude under strict rational expectations. Finally, since

statistical sunspots arise through the cross-equation restrictions of the restricted per-

ceptions equilibrium, they introduce additional over identifying restrictions that can

be used to test for sunspots. These questions are the subject of current research.

45

Online Appendix

Proof of Proposition 1.

A sunspot RPE exists if and only if |b∗|<1 and d∗2>0. It is straightforward to

see that |b∗|<1⇔α > 1/(1 + φ−φ2). Similarly, d∗2>0⇔ρ > ˜ρ, where

˜ρ=α2φ−αφ2−α3φ2+αφ4+α3φ4−α2φ5

α2+α3φ3+ 3α2φ4−6α2φ2−α2φ6−1+3αφ

In the corollary, ﬁx 4/5< α < 1, and then ﬁnding conditions on φand ρconsistent

with b∗<1 and d∗2>0 produces the expressions in the text.

Proof of Lemma 3.

Deﬁne

Tˆ

b(b) = αb2+ρ

1 + αˆ

bρ

which has the property that 0 < Tˆ

b(0) <1< Tˆ

b(1), and Tˆ

bis continuous. It suﬃces

to verify that Tˆ

b(b∗)> b∗. Tedious algebra leads to

Tˆ

b(b∗) =

1−αhφ(2 −αφ)−ρ(1 −φ2)2i

ρ+α[1 + φ4−2φ2−ρφ (2 −αφ)] >1−αφ

α(1 −φ2)=b∗

Proof of Proposition 4. E-stability is determined by looking at the eigenvalues of

the Jacobian matrix DT evaluated at the restricted perceptions equilibrium values.

In particular, E-stability requires that the eigenvalues of DT have real parts less than

one. Consider ﬁrst the E-stability of the fundamentals RPE, where d= 0. The

Jacobian is diagonal and the eigenvalues are

2bα (1 −ρ2)

(1 + b2αρ)2and α(b+φ) (1 −bφ)

1−b2αφ .

The ﬁrst eigenvalue is real and less than one, a result that follows from Hommes and

Zhu (2014). The second eigenvalue can be re-arranged to the condition

b∗

fund < b∗

non-fund.

When there is a unique RPE, coinciding with the fundamentals RPE, then b∗

non-fund >

1, implying that the fundamentals RPE is E-stable. However, when multiple RPE

exist, this condition violates, see Lemma 3. The fundamentals RPE in this case is,

46

however, saddle-path E-stable. Setting d= 0, i.e. restricting to the resting point of

the d−component of the E-stability o.d.e., leads to the following o.d.e.

db

dτ =αb2+ρ

1 + αb2ρ−b.

As shown in Hommes and Zhu (2014) the resting point b∗

fund is locally stable, hence

there is a stable saddle path with ˙

d= 0.

Now we turn to the E-stability properties of the non-fundamentals RPE. In this

case, the Jacobian is no longer diagonal and the resulting expressions for the eigen-

values are complicated. However, one can show that when α= ˜α, ρ = ˜ρthere is

an eigenvalue at 1 and another eigenvalue below one, for φsuﬃciently large. Then

it can be further shown that both eigenvalues have real parts less than one when

α > ˜α, ρ > ˜ρprovided that φ > ˜

φLfor some ˜

φLthat depends on αand ρ. The saddle

path is not surprising since it arises when we shut down coordination on the extrinsic

noise.

Proof of Proposition 5. We can write the actual law of motion as

yt=ξ1(b)yt−1+ξ2(b, d)ηt+γzt

where ξ1=αb2and ξ2=α(b+φ)d. It can be shown that for the fundamentals RPE

the variance is given by

γ2(1 + ρξ1(b∗

fund))σ2

ε

(1 −ρξ1(b∗

fund)) (1 −ξ2

1(b∗

fund)) (1 −ρ2),(33)

while for the non-fundamental RPE:

γ2(1 + ρξ1(b∗

non-fund))σ2

ε

(1 −ρξ1(b∗

non-fund)) (1 −ξ2

1(b∗

non-fund)) (1 −ρ2)+ξ2

2(b∗

non-fund, d∗) (1 + φξ1(b∗

non-fund)) σ2

ν

(1 −φξ1(b∗

non-fund)) (1 −ξ2

1(b∗

non-fund)) (1 −φ2).

(34)

Based on Lemma 2, b∗

fund > b∗

non-fund, and hence ξ1(b∗

fund)> ξ1(b∗

non-fund). Thus

γ2(1 + ρξ1(b∗

fund))

(1 −ρξ1(b∗

fund)) (1 −ξ2

1(b∗

fund)) >γ2(1 + ρξ1(b∗

non-fund))

(1 −ρξ1(b∗

non-fund)) (1 −ξ2

1(b∗

non-fund)) .(35)

Moreover, for ρsuﬃciently large and/or σ2

ν

σ2

ε

suﬃciently small, Eη 2

Ez2=σ2

ν(1−ρ2)

σ2

ε(1−φ2)tends to

0. Then we have for ρsuﬃciently large and/or σ2

ν

σ2

ε

suﬃciently small,

γ2(1 + ρξ1(b∗

fund))

(1 −ρξ1(b∗

fund)) (1 −ξ2

1(b∗

fund)) >ξ2

2(b∗

non-fund, d∗) (1 + φξ1(b∗

non-fund))

(1 −φξ1(b∗

non-fund)) (1 −ξ2

1(b∗

non-fund))

Eη2

Ez2

+γ2(1 + ρξ1(b∗

non-fund))

(1 −ρξ1(b∗

non-fund)) (1 −ξ2

1(b∗

non-fund)) .

(36)

47

Proof of Proposition 7.

Set

b∗=1−αφ

nα (1 −φ2)

f∗=dnα (1 −φ2)

nα + (1 −n)αφ2−φ

Using the expression for c∗we can deﬁne Tb(b∗, d, c∗, f∗)≡Tb(d). A sunspot RPE

exists if and only if b∗<1 and Tb(0) > b∗. Straightforward calculations show that

Tb(0) = (1 −αφ)2+nαρ (1 −φ2)2

ρ(1 −αφ)2+nα (1 −φ2)2

Then Tb(0) > b∗if and only if the conditions provided in the text. Corollary 8 follows

by following the same steps after deﬁning Tb(d) = Tb(b∗, d, c∗,0).

Proof of Proposition 9.

Denote γ0.

=Ey2

t, γ1.

=E(ytyt−1), γ2.

=E(ytyt−2). Based on the orthogonality

condition,

Eyt−1(yt−byt−1−dηt) = 0 (37)

Eyt−1(yt−b1yt−1−b2yt−2−f ηt) = 0 (38)

Eyt−2(yt−b1yt−1−b2yt−2−f ηt) = 0 (39)

Eηt(yt−byt−1−dηt) = 0 (40)

Eηt(yt−b1yt−1−b2yt−2−f ηt)=0.(41)

That is,

γ1−bγ0−dφσyη = 0 (42)

(1 −b2)γ1−b1γ0−fφσyη = 0 (43)

γ2−b1γ1−b2γ0−fφ2σyη = 0 (44)

(1 −bφ)σyη −dσ2

η= 0 (45)

(1 −b1φ−b2φ2)σyη −f σ2

η= 0.(46)

Based on the ALM (23), it is easy to get

γ0=a1γ1+a2γ2+γE(ztyt) + a4E(ηtyt) (47)

γ1=a1γ0+a2γ1+γE(ztyt−1) + a4E(ηtyt−1) (48)

γ2=a1γ1+a2γ0+γE(ztyt−2) + a4E(ηtyt−2).(49)

48

Note that

E(ztyt−1) = E[(ρzt−1+εt)yt−1] = ρE(ztyt),

E(ztyt−2) = E[(ρzt−1+εt)yt−2] = ρ2E(ztyt),

E(ztyt) = E[zt(a1yt−1+a2yt−2+γzt+a4ηt)]

=a1E(ztyt−1) + a2E(ztyt−2) + γEz2

t

= (a1ρ+a2ρ2)E(ztyt) + γEz2

t.

That is,

E(ytzt) = γσ2

ε

(1 −a1ρ−a2ρ2)(1 −ρ2)

.

=σyz .

Similarly,

E(ytηt) = a4σ2

ν

(1 −a1φ−a2φ2)(1 −φ2)

.

=σyη , E(yt−2ηt) = φE(yt−1ηt) = φ2E(ytηt).

Therefore,

γ0

γ1

γ2

=

1−a1−a2

−a11−a20

−a2−a11

−1 γσy z

1

ρ

ρ2

+a4σyη

1

φ

φ2

!(50)

=γσy z

(1 −a2)4a1(1 + a2)4a2(1 −a2)4

a14(1 −a2

2)4a1a24

(a2

1+a2−a2

2)4a1(1 + a2)4(1 −a2−a2

1)4

1

ρ

ρ2

(51)

+a4σyη

(1 −a2)4a1(1 + a2)4a2(1 −a2)4

a14(1 −a2

2)4a1a24

(a2

1+a2−a2

2)4a1(1 + a2)4(1 −a2−a2

1)4

1

φ

φ2

(52)

=γσy z 4

(1 −a2) + a1(1 + a2)ρ+a2(1 −a2)ρ2

a1+ (1 −a2

2)ρ+a1a2ρ2

(a2

1+a2−a2

2) + a1(1 + a2)ρ+ (1 −a2−a2

1)ρ2

(53)

+a4σyη 4

(1 −a2) + a1(1 + a2)φ+a2(1 −a2)φ2

a1+ (1 −a2

2)φ+a1a2φ2

(a2

1+a2−a2

2) + a1(1 + a2)φ+ (1 −a2−a2

1)φ2

,(54)

where 4=1

(1+a2)[(1−a2)2−a2

1].

Thus we can obtain the variance γ0, the ﬁrst-order covariance γ1and the second-

order covariance γ2. Correspondingly the ﬁrst-order and second-order autocorrela-

tions are ρ1=γ1

γ0and ρ2=γ2

γ0, respectively.

49

Based on (42)-(46),

ρ1−b−dφσyη

γ0

=γ1

γ0−b−dφσyη

γ0

= 0 (55)

ρ1−b1

1−b2−fφσyη

(1 −b2)γ0

=γ1

γ0−b1

1−b2−fφσyη

(1 −b2)γ0

= 0 (56)

ρ2−b2

1

1−b2

+b2−(b1+ (1 −b2)φ)fφσyη

(1 −b2)γ0

= 0 (57)

(1 −bφ)a4

(1 −a1φ−a2φ2)−d= 0 (58)

(1 −b1φ−b2φ2)a4

(1 −a1φ−a2φ2)−f= 0 (59)

From the equations (58) and (59), it is easy to see

f=1−b1φ−b2φ2

1−bφ d. (60)

Put (60) into (58) and then we can obtain

d= 0 or b=1−αφ −α(1 −n)(b1+b2φ)(1 −φ2)

αn(1 −φ2).(61)

In the case d∗= 0, then also f∗= 0 and the RPE is the fundamental equilibria

corresponding to the case without exogenous variable. In this case,

ρ1−b=γ1

γ0−b= 0 (62)

ρ1−b1

1−b2

=γ1

γ0−b1

1−b2

= 0 (63)

ρ2−b2

1

1−b2

+b2=γ2

γ0−b2

1

1−b2

+b2= 0.(64)

This means that at the RPE the ﬁrst-order autocorrelation of ALM are equal to those

of both of the two kinds of agents and the second-order autocorrelation of ALM are

equal to that of the AR(2) forecasting model. If we deﬁne ¯ρ1=b1

1−b2∈[−1,1] and

¯ρ2=b2

1

1−b2+b2∈[−1,1], then b1=¯ρ1(1−¯ρ2)

1−¯ρ2

1and b2=¯ρ2−¯ρ2

1

1−¯ρ2

1. Based on the Brouwer ﬁxed

point theorem, there is a ﬁxed point (b∗,¯ρ∗

1,¯ρ∗

2) satisfying (62)-(64). That is, there is

a ﬁxed point (b∗, b∗

1, b∗

2) satisfying (62)-(64), where b∗

1=¯ρ∗

1(1−¯ρ∗

2)

1−¯ρ∗2

1and b∗

2=¯ρ∗

2−¯ρ∗2

1

1−¯ρ∗2

1.

In the special case, if n= 1, i.e. all the agents use the AR(1) forecasting model,

then the RPE is the same as the BLE in Hommes & Zhu (2014). If n= 0, i.e. all the

50

agents use the AR(2) forecasting model, then the RPE corresponds to the equilibria

where the ﬁrst two autocorrelations of ALM are equal to those of PLM.

In the case b=1−αφ−α(1−n)(b1+b2φ)(1−φ2)

αn(1−φ2), based on the equations (42) and (60), it

is easy to see that

γσy z 41+a2

4σ2

ν

(1 −a1φ−a2φ2)(1 −φ2)42−dφa4σ2

ν

(1 −a1φ−a2φ2)(1 −φ2)= 0,(65)

where 41=4[(a1+ (1 −a2

2)ρ+a1a2ρ2)−b((1 −a2) + a1(1 + a2)ρ+a2(1 −a2)ρ2)],

42=4[(a1+ (1 −a2

2)φ+a1a2φ2)−b((1 −a2) + a1(1 + a2)φ+a2(1 −a2)φ2)] and

a4=α[n(b+φ)d+(1 −n)(b1+φ)f] = dα[n(b+φ)(1−bφ)+(1−n)(b1+φ)(1−b1φ−b2φ2)]

1−bφ . Therefore,

d2=γσy z 41(1 −bφ)2(1 −b1φ−b2φ2)(1 −φ2)

σ2

ν[φ(1 −bφ)43− 42

342],(66)

where 43=α[n(b+φ)(1 −bφ) + (1 −n)(b1+φ)(1 −b1φ−b2φ2)].Note that now b,

dand fare both the functions of b1and b2. From the theoretical point of view, put

the expressions of b,dand finto the equations (56) and (57) and then obtain the

two functions with respect to b1and b2. If there exists the equilibria for b∗

1and b∗

2,

then based on (61) and (66), the non-fundamental RPE can be obtained.

Details on calculating the price variance

Since pt=a1pt−1+a2dt+a3ηt, then

Ep2

t=E[a1pt−1+a2dt+a3ηt]2

=a2

1Ep2

t−1+a2

2Ed2

t+a2

3Eη2

t+ 2a1a2E(pt−1dt)+2a1a3E(pt−1ηt).

Note that

pt=a1pt−1+a2dt+a3ηt=a2Σ∞

i=0ai

1dt−i+a3Σ∞

i=0ai

1ηt−i.

Thus

E(pt−1dt) = E[(a1pt−2+a2dt−1+a3ηt−1)dt]

=a2Σ∞

i=0ai

1E(dt−1−idt) + a3Σ∞

i=0ai

1E(ηt−1−idt))

=a2Σ∞

i=0ai

1ρi+1Ed2

t

=a2ρ

1−a1ρEd2

t

=a2ρσ2

ε

(1 −a1ρ)(1 −ρ2).(67)

51

Similarly,

E(ptdt) = E[(a1pt−1+a2dt+a3ηt)dt]

=a2Σ∞

i=0ai

1ρiEd2

t

=a2σ2

ε

(1 −a1ρ)(1 −ρ2),(68)

E(pt−1ηt) = a3φ

1−a1φEη2

t=a3φσ2

ν

(1 −a1φ)(1 −φ2),(69)

E(ptηt) = a3

1−a1φEη2

t=a3σ2

ν

(1 −a1φ)(1 −φ2).(70)

Therefore,

Ep2

t=1

1−a2

1ha2

2+ 2a1a2

a2ρ

1−a1ρEd2

t+a2

3+ 2a1a3

a3φ

1−a1φEη2

ti

=a2

2(1 + a1ρ)σ2

ε

(1 −a2

1)(1 −a1ρ)(1 −ρ2)+a2

3(1 + a1φ)σ2

ν

(1 −a2

1)(1 −a1φ)(1 −φ2).

Misspeciﬁcation equilibria in the asset-pricing model

Given the important role played by nin determining excess volatility, we again

study whether sunspot RPE can emerge in a misspeciﬁcation equilibrium. As before,

we assume that

EU j=−Eyt−Ej

t−1yt2,

where the unconditional expectation is with respect to the probability distribution

implied by the RPE.

We construct a Misspeciﬁcation Equilibrium as follows. For each n, we calcu-

late the RPE. When there are multiple RPE, for a given n, we select the E-stable

equilibria. That is, we can deﬁne F: [0,1] →Ras

F(n) = EU 1−EU 2.

Then, there is a mapping S: [0,1] →[0,1] where

Sω(n) = 1

2ntanh hω

2F(n)i+ 1o.

Given the parameters above and ω= 0.1, the ME at the fundamental RPE (left

panel) and at one non-fundamental RPE (right panel) are illustrated in Figure 14

illustrates. Therefore given n, for E-stable equilibria, there are three ME, where

two are stable (one corresponding to fundamental RPE and one corresponding to

non-fundamental RPE) and another one is unstable.

52

Figure 14: Misspeciﬁcation Equilibrium (ME) for the RPE

0.0

0.2

0.4

0.6

0.8

1.0

0.0

0.2

0.4

0.6

0.8

1.0

n

SHnL

,

0.86

0.88

0.90

0.92

0.94

0.96

0.98

1.00

0.65

0.70

0.75

0.80

0.85

0.90

0.95

1.00

n

SHnL

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