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Statistical Sunspots

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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 expectations 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 perceptions 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 developed in the context of a New Keynesian, an asset-pricing, and a pure monetary model.
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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
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
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 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
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-
2The introduction to White (1994) states explicitly “ economic or probability model is ...a
crude approximation to ...the ‘true’ relationships...Consequently, it is necessary to view models as
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.”
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).
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
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
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
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.
2 Restricted Perceptions and Endogenous Fluctu-
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,
Etyt+1 +γzt,(1)
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
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
Etyt+1 =b2yt1,(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.
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
Etyt+1 =b2yt1+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.
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
yt=αb2yt1+αd (b+φ)ηt+γzt.(5)
Notice that the perceived law of motion is misspecified: the actual law of motion (5)
depends on yt1, ηt,and zt, while the perceived law of motion depends only on yt1
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:
tΘ) = 0,(6)
where X0
t= (yt1, 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)yt1αb2ρyt2+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).
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)1EXtytT(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
T(Θ) = "1dση
0 (1 )σy
ση#corr (yt1, yt)
corr (yt1, 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 yt1as 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
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
The T-map for the fundamentals RPE can be identified by setting d= 0 and
solving the orthogonality condition (6) for b:
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(b+φ) (1 )
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,
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
d2ξ(b, α, ρ, φ)σ2
ξ(b, α, ρ, φ) = γ2{ρb[1 αb(1 )]}(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
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 Tmap.
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=1αφ
α(1φ2), 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.
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
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 (yt1, 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:
=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
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
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 band the fundamental
bholds: bˆ
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
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
b0.97 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 d6= 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
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
sufficiently small, the fundamentals RPE is
Figure 3: E-stability Regions
0.80 0.85 0.90 0.95 1.00
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, ddecreases 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
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
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 bis first decreasing and then increasing.
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 band 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 band decreases das 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.
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
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
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:
Etyt+1 +δyt1+γzt,0< α < 1
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
Leading the perceived law of motion forward one period and taking expectations,
leads to the actual law of motion for yt
1+b2+δyt1+αb1b2yt1=αd (b1+φ)ηt+zt(10)
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
αρb1b2yt3+αd(b1+φ) (ηtρηt1) + 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
Result 6 There exists a fundamentals RPE with 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,±d6= 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
Etyt+1 +γzt,(11)
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≤ −2b2or
2b2b1<1b2and 0 < b2<1⇒ −1 + b2< b1<1b2.
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 N2 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
Define the state vector X0
t= (yt, yt1, ..., ytp, θt, θt1, ..., θtq, zt, ηt). Such a state
vector will arise when agents form expectations from a perceived law of motion in the
general class:
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
assume throughout that ηtis a covariance stationary AR(1) process given by
where |φ|<1 and νtis white noise with variance σ2
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
tyt+1 =b2yt1+d(b+φ)ηt.(16)
The remaining 1 nof agents form expectations from the perceived law of motion
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.
Inserting expectations (16)-(17) into (13) leads to the actual law of motion
yt=αnb2yt1+ [α(1 n)+γ]zt+α[nd (b+φ) + (1 n)fφ]ηt.(18)
It is useful to define the state vector X0
t= (yt, yt1, ηt, zt). Then it is possible to
re-write (12), (15), and (18) into its standard VAR(1) form:
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= (yt1, ηt)0, x2
t= (zt, ηt)0. In an RPE, beliefs
satisfy the following least-squares orthogonality conditions:
t= 0,
t= 0.
Denote Ω EXtX0
t. It follows that
ξ1= Ω1,2:31
2:3,2:3 T1(b, c, d, f ),
ξ2= Ω1,3:42
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
dα(1 )(bdn + (f+dn fn)φ)
1b2n αφ ,
fα(bdn + (f+dn fn)φ)
1b2nαφ .
14The notation Ωi:j,k:lrefers to the (ij)×(kl) sub block of the matrix Ω.
Evidently, d=f= 0 is a solution to these pairs of equations. Then, the remaining
components of the T-map are:
1 + b2nαρ,(20)
cγ+c(1 n)αρ
1b2nαρ .(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 )αφ
(1 nαφ +bnα (φ21)) (1 )= 1,
which implies an RPE value
(1 φ2).(22)
Note that the RPE value of bis 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
1(1 n)αρ b2nαρ,
k[b(1 + a1ρ)a1ρ]
α2[φ+nb nφb(b+φ)]2[a1+φb(1 + a1φ)] αφ(1 )(1 a2
1)[φ+nb nφb(b+φ)],
d= (1 )f,
where e
(1a1ρ)(1ρ2),a1=αnb2and a2=α(1 n)+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+φφ2< α < 1
ii. 1αφ
α(1φ2)< n 1
iii. ˜ρ < ρ < 1,
where ˜ρ=(αφ)φ(1αφ)(1φ2)
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
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
Note that the first-order and second-order autocorrelations of the AR(2) model with-
out ηtare b1
1b2+b2. Hence their forecasts are, respectively,
E1,tyt+1 =b2yt1+ (b+φ)t,
E2,tyt+1 =b1E2,t yt+b2yt1+f φηt= (b2
1+b2)yt1+b1b2yt2+ (b1+φ)fηt.
The actual law of motion is
yt=α[nb2+ (1 n)(b2
1+b2)]yt1+ (1 n)αb1b2yt2+γzt
+α[n(b+φ)z+ (1 n)(b1+φ)f]ηt
where a1=α[nb2+ (1 n)(b2
1+b2)], a2= (1 n)αb1b2and a4=α[n(b+φ)d+ (1
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
Although sunspot RPE existence is analytically unavailable, the Appendix conjectures
that it is possible for their to exist multiple sunspot RPE with non-zero dand f,
and numerical examples confirm the conjecture. In particular, with α= 0.95, ρ = 0.9
then for all 0 n1 there is an open set of φconsistent with existence of sunspot
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:
tyt+1 =b2yt1+d(b+φ)ηt(24)
tyt+1 =ye
t+1 (25)
where the latter expectations are defined by the recursions
t+1 =t1+f(c+φ)ηtcye
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 θt1, ηt.
15Weill and Gregoir (2007) prove the existence of MA(q) restricted perceptions equilibria.
With these expectations, the actual law of motion for the economy is given by the
following recursive equations
yt=αnb2yt1+α(1 n)c(θt1ye
t) + α[nd (b+φ) + (1 n)f(c+φ)] ηt+zt
t+1 =c(θt1ye
t) + f(c+φ)ηt
As before, let X0
t=yt, yt1, ye
t+1, θt, ηt, zt, then with appropriate conformable ma-
trices B, C , we have
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
dα(1 ) [bdn + (f(1 n) + dn)φ+c(dnφ(b+φ)(1 n)(1 φ2))]
1b2nαφ c(α(1 n) + φ(b2nαφ 1)) (26)
fdnα (b+φ)
1αφ(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:
(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.
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:
2ntanh hω
2EU 1EU 2i+ 1o
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.
EU j=EytEj
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-
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 1EU 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:
Etπt+1 +κyt+ut(28)
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.”
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=ρut1+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
g0, 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
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
In order to parameterize deviations from rational expectations, we assume that a
fraction nof agents form their forecasts from
and the remaining 1 nagents forecast from
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
1 + κθ πt1+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
17The derivation of this equation under heterogeneous expectations was provided by Branch and
McGough (2009).
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 θdoes 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.
Figure 9: Optimal Monetary Policy.
0.04 0.06 0.08 0.10 0.12 0.14
n=0.78 n=0.25 n=0.85 REE
0 1 2 3 4 5
0.0 0.2 0.4 0.6 0.8
0.0 0.2 0.4 0.6 0.8 1.0
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
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
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
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=Sωθ.
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,
Figure 10: Optimal Monetary Policy and Misspecification Equilibria.
0.0 0.2 0.4 0.6 0.8 1.0
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
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
Vtdenote the subjective beliefs of agents about
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
t(pt+1 +dt+1 Rpt)
t(pt+1 +dt+1 Rpt)=b
t(pt+1 +dt+1 Rpt)
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
t(pt+1 +dt+1
Rpt)σ2. The dividend process dtis assumed to follow an AR(1) process
where ρ[0,1) and εtis an i.i.d. process with the standard deviation σε. Thus
t(pt+1) + ρdtRpt
The market clearing price is
tpt+1 +ρ
As in the previous sections, assume that a fraction nof agents form their forecasts
and the remaining 1 nagents forecast from
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.
The actual law of motion is given by
R[(1 n)+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
(1 a2
1)(1 a1ρ)(1 ρ2)+a2
3(1 + a1φ)σ2
(1 a2
1)(1 a1φ)(1 φ2).(31)
where a1=nb2
R>0, a2=(1n)+ρ
R, a3=nd(b+φ)+(1n)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 φ0 or φ1, 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=Rφ
n(1φ2)>0, c=ρ
Rnb2ρ(1n)ρ. It
follows that the comparative static effect of φon band cis 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, band care not affected by σ2
νwhile |f|and |b|decrease
as σ2
νgrows given other parameters constant, which can be seen from the expression of f2and
d2. From this, σ2
νseems to affect the variances. But further analysis indicates that at the non-
fundamental RPE,
ν[n2(b+φ)2(1 bφ)2+ 2n(1 n)φ(b+φ)(1 bφ) + (1 n)2φ2]
Therefore, the expression of f2implies that a2
νdoes not depend on σ2
ν. Thus σ2
νdoes not affect
the variances of market prices at the RPE.
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
Figure 11: Effects of φon the stock price variance
Fundamental RPE
Non-fundamental RPE
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 band 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 band 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.
Figure 12: Effects of non the variances of market prices and excess volatility
Fundmental RPE
Non-fundamental RPE
Varianceof p
Fundamental RPE
Non-fundamental RPE
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
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
and where the take-it-or-leave it offers determines the amount qttraded in the period-t
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)1αb1α
without a loss of generality set b0. As in the previous applications, agents will
differ by expectations-type. Money-demand of type jis given by
j,t =βφ1φe
Then letting ˆmt= log Mtand assuming ˆmtˆmt1zt, 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
=nE1π+ (1n)E2π. Now we assume that the (log) money-supply growth
process is driven by two stationary exogenous AR(1) processes:
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
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.
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-
tπt+1 =b2πt1+d(b+φ)ηt
and the remaining 1 nbuyers hold
tπt+1 =1zt+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
2tanh ωEU 1EU 2+ 1HωEU 1EU 2
Recalling that F(n) = EU 1EU 2, because an RPE exists it follows that there exists
a well-defined mapping Sω=HωF, 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
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.
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 45line 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.
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.
Online Appendix
Proof of Proposition 1.
A sunspot RPE exists if and only if |b|<1 and d2>0. It is straightforward to
see that |b|<1α > 1/(1 + φφ2). Similarly, d2>0ρ > ˜ρ, where
α2+α3φ3+ 3α2φ46α2φ2α2φ61+3αφ
In the corollary, fix 4/5< α < 1, and then finding conditions on φand ρconsistent
with b<1 and d2>0 produces the expressions in the text.
Proof of Lemma 3.
b(b) = αb2+ρ
1 + αˆ
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
b(b) =
1αhφ(2 αφ)ρ(1 φ2)2i
ρ+α[1 + φ42φ2ρφ (2 αφ)] >1αφ
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
2(1 ρ2)
(1 + b2αρ)2and α(b+φ) (1 )
1b2αφ .
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
fund < b
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,
however, saddle-path E-stable. Setting d= 0, i.e. restricting to the resting point of
the dcomponent of the E-stability o.d.e., leads to the following o.d.e.
1 + αb2ρb.
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
Proof of Proposition 5. We can write the actual law of motion as
yt=ξ1(b)yt1+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
(1 ρξ1(b
fund)) (1 ξ2
fund)) (1 ρ2),(33)
while for the non-fundamental RPE:
γ2(1 + ρξ1(b
(1 ρξ1(b
non-fund)) (1 ξ2
non-fund)) (1 ρ2)+ξ2
non-fund, d) (1 + φξ1(b
non-fund)) σ2
(1 φξ1(b
non-fund)) (1 ξ2
non-fund)) (1 φ2).
Based on Lemma 2, b
fund > b
non-fund, and hence ξ1(b
fund)> ξ1(b
non-fund). Thus
γ2(1 + ρξ1(b
(1 ρξ1(b
fund)) (1 ξ2
fund)) >γ2(1 + ρξ1(b
(1 ρξ1(b
non-fund)) (1 ξ2
non-fund)) .(35)
Moreover, for ρsufficiently large and/or σ2
sufficiently small, 2
ε(1φ2)tends to
0. Then we have for ρsufficiently large and/or σ2
sufficiently small,
γ2(1 + ρξ1(b
(1 ρξ1(b
fund)) (1 ξ2
fund)) >ξ2
non-fund, d) (1 + φξ1(b
(1 φξ1(b
non-fund)) (1 ξ2
+γ2(1 + ρξ1(b
(1 ρξ1(b
non-fund)) (1 ξ2
non-fund)) .
Proof of Proposition 7.
(1 φ2)
f=dnα (1 φ2)
+ (1 n)αφ2φ
Using the expression for cwe 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+(1 φ2)2
Then Tb(0) > bif 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.
t, γ1.
=E(ytyt1), γ2.
=E(ytyt2). Based on the orthogonality
Eyt1(ytbyt1t) = 0 (37)
Eyt1(ytb1yt1b2yt2f ηt) = 0 (38)
Eyt2(ytb1yt1b2yt2f ηt) = 0 (39)
t(ytbyt1t) = 0 (40)
t(ytb1yt1b2yt2f ηt)=0.(41)
That is,
γ10dφσ= 0 (42)
(1 b2)γ1b1γ0fφσyη = 0 (43)
γ2b1γ1b2γ0fφ2σyη = 0 (44)
(1 )σ2
η= 0 (45)
(1 b1φb2φ2)σf 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(ztyt1) + a4E(ηtyt1) (48)
γ2=a1γ1+a2γ0+γE(ztyt2) + a4E(ηtyt2).(49)
Note that
E(ztyt1) = E[(ρzt1+εt)yt1] = ρE(ztyt),
E(ztyt2) = E[(ρzt1+εt)yt2] = ρ2E(ztyt),
E(ztyt) = E[zt(a1yt1+a2yt2+γzt+a4ηt)]
=a1E(ztyt1) + a2E(ztyt2) + γEz2
= (a1ρ+a2ρ2)E(ztyt) + γEz2
That is,
E(ytzt) = γσ2
(1 a1ρa2ρ2)(1 ρ2)
=σyz .
E(ytηt) = a4σ2
(1 a1φa2φ2)(1 φ2)
=σ , E(yt2ηt) = φE(yt1ηt) = φ2E(ytηt).
1 γσy z
=γσy z
(1 a2)4a1(1 + a2)4a2(1 a2)4
a14(1 a2
2)4a1(1 + a2)4(1 a2a2
(1 a2)4a1(1 + a2)4a2(1 a2)4
a14(1 a2
2)4a1(1 + a2)4(1 a2a2
=γσy z 4
(1 a2) + a1(1 + a2)ρ+a2(1 a2)ρ2
a1+ (1 a2
2) + a1(1 + a2)ρ+ (1 a2a2
+a4σ 4
(1 a2) + a1(1 + a2)φ+a2(1 a2)φ2
a1+ (1 a2
2) + a1(1 + a2)φ+ (1 a2a2
where 4=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.
Based on (42)-(46),
= 0 (55)
(1 b2)γ0
(1 b2)γ0
= 0 (56)
+b2(b1+ (1 b2)φ)fφσyη
(1 b2)γ0
= 0 (57)
(1 )a4
(1 a1φa2φ2)d= 0 (58)
(1 b1φb2φ2)a4
(1 a1φa2φ2)f= 0 (59)
From the equations (58) and (59), it is easy to see
1d. (60)
Put (60) into (58) and then we can obtain
d= 0 or b=1αφ 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,
γ0b= 0 (62)
= 0 (63)
+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
1b2[1,1] and
1b2+b2[1,1], then b1=¯ρ1(1¯ρ2)
1and b2=¯ρ2¯ρ2
1. Based on the Brouwer fixed
point theorem, there is a fixed point (b,¯ρ
2) satisfying (62)-(64). That is, there is
a fixed point (b, b
1, b
2) satisfying (62)-(64), where b
1and b
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
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=1αφα(1n)(b1+b2φ)(1φ2)
αn(1φ2), based on the equations (42) and (60), it
is easy to see that
γσy z 41+a2
(1 a1φa2φ2)(1 φ2)42dφa4σ2
(1 a1φa2φ2)(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] = [n(b+φ)(1)+(1n)(b1+φ)(1b1φb2φ2)]
1. Therefore,
d2=γσy z 41(1 )2(1 b1φb2φ2)(1 φ2)
ν[φ(1 )43− 42
where 43=α[n(b+φ)(1 ) + (1 n)(b1+φ)(1 b1φb2φ2)].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
then based on (61) and (66), the non-fundamental RPE can be obtained.
Details on calculating the price variance
Since pt=a1pt1+a2dt+a3ηt, then
t+ 2a1a2E(pt1dt)+2a1a3E(pt1ηt).
Note that
E(pt1dt) = E[(a1pt2+a2dt1+a3ηt1)dt]
1E(dt1idt) + a3Σ
(1 a1ρ)(1 ρ2).(67)
E(ptdt) = E[(a1pt1+a2dt+a3ηt)dt]
(1 a1ρ)(1 ρ2),(68)
E(pt1ηt) = a3φ
(1 a1φ)(1 φ2),(69)
E(ptηt) = a3
(1 a1φ)(1 φ2).(70)
2+ 2a1a2
3+ 2a1a3
2(1 + a1ρ)σ2
(1 a2
1)(1 a1ρ)(1 ρ2)+a2
3(1 + a1φ)σ2
(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=EytEj
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 1EU 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.
Figure 14: Misspecification Equilibrium (ME) for the RPE
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