Content uploaded by Cars Hommes

Author content

All content in this area was uploaded by Cars Hommes

Content may be subject to copyright.

Learning under misspeciﬁcation: a behavioral

explanation of excess volatility in stock prices and

persistence in inﬂation

Cars Hommes∗

, Mei Zhu

CeNDEF, School of Economics, University of Amsterdam

Roetersstraat 11, 1018 WB Amsterdam, Netherlands

May 17, 2011

Abstract

We propose a simple misspeciﬁcation equilibrium concept and a behavioral learning

process explaining excess volatility in stock prices and high persistence in inﬂation.

Boundedly rational agents use a simple univariate linear forecasting rule and in

equilibrium correctly forecast the unconditional sample mean and ﬁrst-order sam-

ple autocorrelation. In the long run, agents thus learn the best univariate linear

forecasting rule, without fully recognizing the structure of the economy. In a ﬁrst

application, an asset pricing model with AR(1) dividends, a unique stochastic con-

sistent expectations equilibrium (SCEE) exists characterized by high persistence and

excess volatility, and it is globally stable under learning. In a second application,

the New Keynesian Phillips curve, multiple SCEE arise and a low and a high per-

sistence misspeciﬁcation equilibrium co-exist. Learning exhibits path dependence

and inﬂation may switch between low and high persistence regimes.

Keywords: Bounded rationality; Stochastic consistent expectations equilibrium;

Adaptive learning; Excess volatility; Inﬂation persistence

JEL classiﬁcation: C62; D83; D84; E30

∗Corresponding author. Tel.: +31 20 525 4246; fax: +31 20 525 4349.

E-mail address: C.H.Hommes@uva.nl.

1

1 Introduction

Expectation feedback plays a crucial role in economics and ﬁnance. Since the intro-

duction by Muth (1961), and its application in macroeconomics by Lucas (1972), the

Rational Expectation Hypothesis (REH) has become the predominant paradigm. A Ra-

tional Expectation Equilibrium (REE) is in fact a ﬁxed point of an expectation feedback

system. Typically it is assumed that rational agents perfectly know not only the correctly

speciﬁed market equilibrium equations, but also their parameter values conditional upon

all available information.

Despite its popularity, the REH has been criticized for its highly demanding and

unrealistic information requirements. Adaptive learning models have been proposed as

an alternative to rational expectations; see, e.g. Sargent (1993, 1999) and Evans and

Honkapohja (2001) for extensive surveys. In contrast to rational expectations, adaptive

learning models assume that agents do not have perfect knowledge about market equilib-

rium equations, but agents are assumed to have some belief, the perceived law of motion,

about the actual law of motion; the corresponding parameters are not known, but are

estimated by adaptive learning based on available observations. The implied actual law

of motion under adaptive learning is thus a time-varying self referential system, depending

on the perceived law of motion. Under this framework, a rational expectations equilibrium

is simply a situation in which the implied law of motion exactly coincides with the per-

ceived law of motion, and adaptive learning may converge to such a rational expectations

equilibrium. In other words, convergence of adaptive learning to a rational expectations

equilibrium can occur when the perceived law of motion is correctly speciﬁed.

In general a perceived law of motion will be misspeciﬁed. White (1994) argues that

an economic model or a probability model is only a more or less crude approximation to

whatever might be the ”true” relationships among the observed data and consequently it

is necessary to view economic and/or probability models as misspeciﬁed to some greater

or lesser degree. Sargent (1991) ﬁrst develops a notion of equilibrium as a ﬁxed point

of an operator that maps the perceived law of motion (a vector ARMA process) into a

statistically optimal estimator of the actual law of motion. This may be viewed as an

early example of a Restricted Perceptions Equilibrium (RPE), as deﬁned by Evans and

Honkapohja (2001), formalizing the idea that agents have misspeciﬁed beliefs, but within

the context of their forecasting model they are unable to detect their misspeciﬁcation.

Branch (2006) gives an excellent survey and argues that the RPE is a natural alternative

2

to rational expectation equilibrium because it is to some extent consistent with Muth’s

original hypothesis of REE while allowing for bounded rationality by restricting the class

of the perceived law of motion.

The main contribution of our paper is to develop a behavioral equilibrium concept,

where agents try to learn a simple but misspeciﬁed forecasting rule. Our equilibrium

concept - Stochastic Consistent Expectations Equilibrium (SCEE) - may be viewed as

the simplest RPE and therefore it seems more likely that agents might coordinate their

expectations and learn such a simple behavioral equilibrium. The actual law of motion

(ALM) of the economy is a two (or higher) dimensional linear stochastic system. Agents

are forecasting one variable - say the price - of the economy using a simple univariate AR(1)

forecasting rule. In a SCEE the mean and the ﬁrst-order autocorrelation of realized prices

in the economy coincide with the corresponding mean and ﬁrst-order autocorrelation of

agents’ AR(1) perceived law of motion (PLM). In addition, a simple adaptive learning

scheme - Sample Autocorrelation Leaning (SAC-learning) - with an intuitive behavioral

interpretation, enforces convergence to the (stable) SCEE.

We illustrate our behavioral equilibrium concept in two standard applications. In the

ﬁrst - an asset pricing model with an exogenous stochastic dividend process - the SCEE

is unique and the SAC-learning scheme always converges to the SCEE. The SCEE is

characterized by excess volatility with asset prices much more volatile (with the variance

in asset prices more than doubled) than under REE. In the second application - a New

Keynesian Philips curve (NKPC) - with an exogenous AR(1) process for the output gap

and an independent and identically distributed (i.i.d.) stochastic shock to inﬂation -

multiple stable SCEE may co-exist. In particular, for empirically plausible parameter

values a SCEE with highly persistent inﬂation exists, matching the stylized facts of US-

inﬂation data.

Related literature

Our behavioral equilibrium is closely related to the Consistent Expectations Equilib-

rium (CEE) introduced by Hommes and Sorger (1998), where agents believe that prices

follow a linear AR(1) stochastic process, whereas the implied actual law of motion is a

deterministic chaotic nonlinear process. Along a CEE, price realizations have the same

sample mean and sample autocorrelation coeﬃcients as the AR(1) perceived law of mo-

tion. A CEE is another early example of a RPE and may be seen as an ”approximate

3

rational expectations equilibrium”, in which the misspeciﬁed perceived law of motion is

the best linear approximation within the class of perceived laws of motion of the actual

(unknown) nonlinear law of motion. Hommes and Rosser (2001) investigate CEE in an

optimal ﬁshery management model and used numerical simulations to study adaptive

learning of CEE in the presence of dynamic noise. The adaptive learning scheme used

here is SAC-learning, where the parameters of the AR(1) forecasting rule are updated

based on the observed sample average and ﬁrst-order sample autocorrelation. S¨ogner and

Mitl¨ohner (2002) apply the CEE concept to a standard asset pricing model with inde-

pendent and identically distributed (i.i.d.) dividends and showed that the unique CEE

coincides with the REE. As we will see in the current paper, introducing autocorrelations

in the stochastic dividend process will lead to learning equilibrium diﬀerent from REE.

Tuinstra (2003) analyzes ﬁrst-order consistent expectations equilibria numerically in a

deterministic overlapping generations (OLG) model. Hommes et al (2004) generalize the

notion of CEE to nonlinear stochastic dynamic economic models, introducing the concept

of stochastic consistent expectations equilibrium (SCEE). In a SCEE, agents’ perceptions

about endogenous variables are consistent with the actual realizations of these variables

in the sense that the unconditional mean and autocorrelations of the unknown nonlinear

stochastic process, which describes the actual behavior of the economy, coincide with the

unconditional mean and autocorrelations of the AR(1) process agents believe in. They

applied this concept to an OLG model and studied the existence of SCEE and its relation-

ship to sample autocorrelation learning (SAC-learning) based on numerical simulations.

Showing theoretically existence of SCEE and its relationship to adaptive learning has

proven to be technically diﬃcult, while convergence of SAC-learning has been studied only

by numerical simulations. The principle technical diﬃculty here is to calculate autocor-

relation coeﬃcients, prove existence of ﬁxed points in a nonlinear system and analyze the

relationship between SCEE and sample autocorrelation learning. Branch and McGough

(2005) obtain existence results on ﬁrst-order SCEE theoretically and analyze the stabil-

ity of SCEE under real-time learning numerically in a stochastic non-linear self-referential

model where expectations are based on an AR(1) process. Lansing (2009) considers a spe-

cial class of SCEE in the New Keynesian Philips curve, where the value of the Kalman

gain parameter in agents’ forecast rule is pinned down using the observed autocorrelation

of inﬂation changes. Lansing (2010) studies a Lucas-type asset pricing model and found

numerically a near-rational restricted perceptions equilibrium, for which the covariance of

an underparameterized (one parameter) PLM coincides with the covariance of an approx-

4

imate ALM. Bullard et al. (2008, 2010) add judgment into agents’ forecasts and use the

concept of SCEE to provide a related interesting concept of exuberance equilibria. They

study the resulting dynamics in the New Keynesian model and a standard asset pricing

model, respectively, where the driving variables are white noises (no autocorrelations).

The current paper studies the existence of SCEE and its stability under SAC-learning

in two standard applications: an asset pricing model and the New Keynesian Philips

curve. In both applications the driving variables (dividends or real marginal costs) are

assumed to follow AR(1) processes. More speciﬁcally, while the perceived law of motion

agents believe in is an AR(1) process with white noise, the true process of economy is not

an AR(1) process but a linear stochastic process driven by an exogenous autocorrelated

process. In addition to the conceptual contribution of introducing a behavioral learning

equilibrium, our paper makes two methodological contributions. First, we prove existence

of SCEE under general conditions in a misspeciﬁed framework, where prices (inﬂation)

have the same mean as REE. Second, we present the ﬁrst proof that the SAC-learning

converges to stable SCEE and provide simple and intuitive stability conditions. SCEE

thus represents a ﬁxed point of learning dynamics under misspeciﬁcation. Moreover,

we provide interesting results in our two applications. In the asset pricing model, we

show that the SCEE is unique and (globally) stable and characterized by market prices

ﬂuctuating around fundamental prices and exhibiting stronger serial autocorrelations and

higher volatility than the REE for plausible parameters. In the New Keynesian Philips

curve, we show that multiple SCEE may exist. In particular, for a large set of plausible

parameters a SCEE exists with highly persistent inﬂation. Coordination on a behavioral

learning equilibrium may thus explain high persistence in inﬂation (Milani, 2007).

Some other related literature, for example Timmermann (1993, 1996), Bullard and

Duﬀy (2001), Guidolin and Timmermann (2007) and Bullard et al. (2010), shows the

eﬀects of learning on asset returns from diﬀerent perspectives. Timmermann (1993, 1996)

shows that learning helps to explain excess volatility and predictability of stock returns

in the similar present value asset pricing model. In Timmermann (1993, 1996), the per-

ceived law of motion is correctly speciﬁed but the related parameters are estimated by

adaptive learning, and in the long run learning converges to REE. Bullard and Duﬀy

(2001) introduce adaptive learning into a general-equilibrium life-cycle economy with

capital accumulation and show that in contrast to perfect-foresight dynamics, the sys-

tem under least-squares learning possesses equilibria that are characterized by persistent

excess volatility in returns to capital. Guidolin and Timmermann (2007) characterize

5

equilibrium asset prices under adaptive, rational and Bayesian learning schemes in a

model where dividends evolve on a binomial lattice and ﬁnd that learning introduces se-

rial correlation and volatility clustering in stock returns. Bullard et al. (2010) construct

a simple asset pricing example with constant known dividends and i.i.d. asset supply

and ﬁnd that exuberance equilibria, when they exist, can be extremely volatile relative

to fundamental equilibria. An important conceptual diﬀerence with these references is

our behavioral interpretation of the SCEE as the simplest example of RPE. A behavioral

SCEE together with an intuitive SAC-learning scheme may explain agents’ coordination

on (almost) self-fulﬁlling equilibria.

The paper is organized as follows. Section 2 introduces the main concepts, i.e. ﬁrst-

order SCEE and sample autocorrelation learning in a general framework. Section 3 studies

existence and stability under SAC-learning theoretically as well as numerically in a stan-

dard asset pricing model. Section 4 presents a second application, the New Keynesian

Philips curve, and shows existence of multiple SCEE and the relationship to SAC-learning

theoretically and numerically. Finally, section 5 concludes.

2 Preliminary concepts

This section brieﬂy introduces the main concepts. Suppose that the law of motion of

an economic system is given by the stochastic diﬀerence equation

xt=f(xe

t+1, yt, ut),(2.1)

where xtis the state of the system (e.g. asset price or inﬂation) at date tand xe

t+1 is the

expected value of xat date t+ 1. This denotation highlights that expectations may not

be rational. Here fis a continuous function, {ut}is an i.i.d. noise process with mean zero

and ﬁnite absolute moments1, where the variance is denoted by σ2

u, and ytis a driving

variable (e.g. dividends or the output gap), assumed to follow an exogenous stochastic

AR(1) process

yt=a+ρyt−1+εt,0≤ρ < 1,(2.2)

where {εt}is another i.i.d. noise process with mean zero and ﬁnite absolute moments,

with variance σ2

ε, and uncorrelated with {ut}. The mean of the stationary process ytis

1The condition on ﬁnite absolute moments is required to obtain convergence results under SAC-

learning.

6

¯y=a

1−ρ, the variance is σ2

y=σ2

ε

1−ρ2and the kth-order autocorrelation coeﬃcient of ytis

ρk, see for example, Hamilton (1994).

Agents are boundedly rational and do not know the exact form of the actual law

of motion in (2.1). We assume that, in order to forecast xt+1, agents only use past

observations xt−1, xt−2,···,etc. Hence agents do not recognize that xtis driven by an

exogenous stochastic process yt. Instead agents believe that the economic variable xt

follows a simple linear stochastic process. More speciﬁcally, agents’ perceived law of

motion (PLM) is an AR(1) process, as in Hommes et al. (2004) and Branch and McGough

(2005), i.e.

xt=α+β(xt−1−α) + δt,(2.3)

where αand βare real numbers with β∈(−1,1) and {δt}is a white noise process; αis the

unconditional mean of xtwhile βis the ﬁrst-order autocorrelation coeﬃcient. Given the

perceived law of motion (2.3), the 2-period ahead forecasting rule for xt+1 that minimizes

the mean-squared forecasting error is

xe

t+1 =α+β2(xt−1−α).(2.4)

Combining the expectations (2.4) and the law of motion of the economy (2.1), we obtain

the implied actual law of motion (ALM)

xt=f(α+β2(xt−1−α), yt, ut),(2.5)

with ytan AR(1) process as in (2.2).

Stochastic consistent expectations equilibrium

We are now ready to recall the deﬁnition of stochastic consistent expectations equi-

librium (SCEE). Following Hommes et al. (2004)2, the concept of ﬁrst-order SCEE is

deﬁned as follows.

Deﬁnition 2.1 A triple (µ, α, β), where µis a probability measure and αand βare

real numbers with β∈(−1,1), is called a ﬁrst-order stochastic consistent expectations

equilibrium (SCEE) if the three conditions are satisﬁed:

S1 The probability measure µis a nondegenerate invariant measure for the stochastic

diﬀerence equation (2.5);

2In Hommes et al. (2004), the actual law of motion is xt=f(xe

t+1, ut),without the driving variable

yt. However, the deﬁnitions of SCEE and SAC-learning can still be applied here.

7

S2 The stationary stochastic process deﬁned by (2.5) with the invariant measure µhas

unconditional mean α, that is, Eµ(x) = Rx dµ(x) = α;

S3 The stationary stochastic process deﬁned by (2.5) with the invariant measure µhas

unconditional ﬁrst-order autocorrelation coeﬃcient β.

That is to say, a ﬁrst-order SCEE is characterized by the fact that both the uncondi-

tional mean and the unconditional ﬁrst-order autocorrelation coeﬃcient generated by the

actual (unknown) stochastic process (2.5) coincide with the corresponding statistics for

the perceived linear AR(1) process (2.3). This means that in a ﬁrst-order SCEE agents

correctly perceive the mean and the ﬁrst-order autocorrelation (persistence) of economic

variables although they do not correctly specify their model of the economy.

Our SCEE concept may be viewed as the simplest example of a RPE. It should be

stressed that the SCEE has an intuitive behavioral interpretation. In a SCEE agents use

a linear forecasting rule with two parameters, the mean αand the ﬁrst-order autocorre-

lation β. Both can be observed from past observations by inferring the average price (or

inﬂation level) and the (ﬁrst-order) persistence of the time series. For example, β= 0.5

means that, on average, prices mean revert toward their long-run mean by 50 percent.

These observations could be made approximately and simply by observing the time series

of aggregate variables. It is interesting to note that in learning-to-forecast laboratory

experiments with human subjects, for many subjects forecasting behavior can indeed be

described by simple rules, such as a simple AR(1) rule, see for example, Hommes et al.

(2005), Adam (2007), Heemeijer et al. (2005), Hommes (2011).

Finally, we note that in a ﬁrst-order SCEE, the orthogonality condition imposed by

Restricted Perceptions Equilibrium (RPE)

Ext−1[xt−α−β(xt−1−α)] = E(xt−1−α)[xt−α−β(xt−1−α)] = 0

is satisﬁed. The orthogonality condition shows that agents can not detect the correlation

between their forecasting errors and the agent’s perceived model, see Branch (2006). The

ﬁrst-order SCEE is a RPE where agents have their model incorrect; but within the context

of their forecasting model agents are unable to detect their misspeciﬁcation.

Sample autocorrelation learning

In the above deﬁnition of ﬁrst-order SCEE, agents’ beliefs are described by the linear

forecasting rule (2.4) with ﬁxed parameters αand β. However, the parameters αand

8

βare usually unknown. In the adaptive learning literature, it is common to assume

that agents behave like econometricians using time series observations to estimate the

parameters as additional observations become available. Following Hommes and Sorger

(1998), we assume that agents use sample autocorrelation learning (SAC-learning) to

learn the parameters αand β. That is, for any ﬁnite set of observations {x0, x1,···, xt},

the sample average is given by

αt=1

t+ 1

t

X

i=0

xi,(2.6)

and the ﬁrst-order sample autocorrelation coeﬃcient is given by

βt=Pt−1

i=0(xi−αt)(xi+1 −αt)

Pt

i=0(xi−αt)2.(2.7)

Hence αtand βtare updated over time as new information arrives.

Adaptive learning is sometimes referred to as statistical learning, because agents act as

statisticians or econometricians and use a statistical procedure such as OLS to estimate

and update parameters over time. SAC-learning may be viewed as another statistical

learning procedure. We would like to stress however that SAC-learning has a simple

behavioral interpretation that agents simply infer the sample average and persistence (i.e.

ﬁrst-order autocorrelation) from time series observations. We focus on the entire sample

average for αtin (2.6) and sample ﬁrst-order autocorrelation for βtin (2.7) over the entire

time-horizon, but one could also restrict the learning to the last Tobservations with

Trelatively small (e.g., T= 100 or even smaller). It is an easy and natural way for

agents, especially those without professional training, to estimate mean and ﬁrst-order

autocorrelation directly based on data instead of some complicated statistical techniques.

Deﬁne

Rt=1

t+ 1

t

X

i=0

(xi−αt)2,

then the SAC-learning is equivalent to the following recursive dynamical system (see

Appendix A).

αt=αt−1+1

t+ 1(xt−αt−1),

βt=βt−1+1

t+ 1R−1

th(xt−αt−1)xt−1+x0

t+ 1 −t2+ 3t+ 1

(t+ 1)2αt−1−1

(t+ 1)2xt

−t

t+ 1βt−1(xt−αt−1)2i,

Rt=Rt−1+1

t+ 1ht

t+ 1(xt−αt−1)2−Rt−1i.

(2.8)

9

The actual law of motion under SAC-learning is therefore given by

xt=f(αt−1+β2

t−1(xt−1−αt−1), yt, ut),(2.9)

with αt, βtas in (2.8) and ytas in (2.2).

In Hommes and Sorger (1998), the map fin (2.9) is a nonlinear deterministic function

depending only on αt−1+β2

t−1(xt−1−αt−1), without the driving variable ytand the noise

ut. Hommes et al. (2004) extend the CEE framework to SCEE, with fa nonlinear

stochastic process (but without exogenous driving variable yt). In this paper the map f

is a linear function, depending on not only αt−1+β2

t−1(xt−1−αt−1) and utbut also on

an exogenous AR(1) process yt. Hence, the true law of motion of the economy is a two

dimensional linear stochastic process, while agents try to forecast using a univariate linear

model. In the following we give two typical examples in economies and study existence

of ﬁrst-order SCEE and its relationship to SAC-learning in detail.

3 An asset pricing model with AR(1) dividends

A simple example of the general framework (2.1) is given by the standard present value

asset pricing model with stochastic dividends; see for example Brock and Hommes (1998).

Here we consider AR(1) dividends instead of independent and identically distributed

(i.i.d.) dividends.

Assume that agents can invest in a risk free asset or in 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 ytdenotes the (random) dividend process. Let e

Et,e

Vt

denote the subjective beliefs of a representative agent about the conditional expectation

and conditional variance of excess return pt+1 +yt+1 −Ryt. By the assumption that the

agent is a myopic mean-variance maximizer of tomorrow’s wealth, the demand ztfor the

risky asset by the representative agent is then given by

zt=e

Et(pt+1 +yt+1 −Rpt)

eae

Vt(pt+1 +yt+1 −Rpt)=e

Et(pt+1 +yt+1 −Rpt)

eaσ2,

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

variance of the excess return is assumed to be constant over time3, i.e. e

Vt(pt+1 +yt+1 −

Rpt)≡σ2.

3This assumption is consistent with the assumption that agents believe that prices follow an AR(1)

process and dividends follow a stochastic AR(1) process with ﬁnite variance.

10

Equilibrium of demand and supply implies

e

Et(pt+1 +yt+1 −Rpt)

eaσ2=zs,

where zsdenotes the supply of outside shares in the market, assumed to be constant over

time. Without loss of generality4, we assume zero supply of outside shares, i.e. zs= 0.

The market clearing price in the standard asset pricing model is then given by

pt=1

Rpe

t+1 +ye

t+1,(3.1)

where pe

t+1 is the conditional expectation of next period’s price pt+1 and ye

t+1 is the con-

ditional expectation of next period’s dividend yt+1.

Dividend {yt}is assumed to follow an AR(1) process (2.2). Suppose that the risky

asset (share) is traded, after payment of real dividends yt, at a competitively determined

price pt, so that ytis known by agents, and

ye

t+1 =a+ρyt.(3.2)

The market clearing price in the standard asset pricing model with AR(1) dividends is

then given by

pt=1

Rpe

t+1 +a+ρyt,(3.3)

where dividend ytfollows the AR(1) process (2.2). Compared with our general framework

(2.1), here the map fis a simple linear function and the noise ut≡0.

3.1 Rational expectations equilibrium with AR(1) dividends

Under the assumption that agents are rational, a straightforward computation (see

Appendix B) shows that the rational expectations equilibrium p∗

tsatisﬁes

p∗

t=aR

(R−1)(R−ρ)+ρ

R−ρyt.(3.4)

In particular, if {yt}is i.i.d., i.e. a= ¯yand ρ= 0, then p∗

t=a

R−1=¯y

R−1for any

t= 0,1,2,···.

Thus based on (3.4), the unconditional mean and the unconditional variance of the

rational expectation price p∗

tare given by, respectively,

p∗:= E(p∗

t) = a

(R−1)(1 −ρ)=¯y

R−1,(3.5)

V ar(p∗

t) = E(p∗

t−p∗)2=ρ2σ2

ε

(R−ρ)2(1 −ρ2).(3.6)

4In the case zs>0, the diﬀerence with the analysis below only lies in the mean of the SCEE α∗=

¯y−eaσ2zs

R−1. The analysis on autocorrelations and variances remains the same.

11

Furthermore, the ﬁrst-order autocovariance and autocorrelation coeﬃcient of the rational

expectation price p∗

tare given by, respectively,

E(p∗

t−p∗)(p∗

t−1−p∗) = ρ3σ2

ε

(R−ρ)2(1 −ρ2),

Corr(p∗

t, p∗

t−1) = ρ. (3.7)

3.2 Existence of ﬁrst-order SCEE

We now relax the rational expectation assumption and assume that agents are bound-

edly rational and believe that the price ptfollows a univariate AR(1) process

pt=α+β(pt−1−α) + δt.(3.8)

Given the perceived law of motion and knowledge of all prices observed up to period t−1,

the 2-period ahead forecasting rule for pt+1 that minimizes the mean-squared forecasting

error is

pe

t+1 =α+β2(pt−1−α).(3.9)

By substituting (3.9) into (3.3), we obtain the implied actual law of motion for prices

pt=1

Rα+β2(pt−1−α) + a+ρyt,

yt=a+ρyt−1+εt.

(3.10)

For the PLM (3.8) and the ALM (3.10), we ﬁrst study the existence and uniqueness of

ﬁrst-order SCEE.

Since 0 ≤β2

R<1 and 0 ≤ρ < 1, the price process (3.10) is stationary and ergodic.

Denote the unconditional expectation of ptby ¯p. Then ¯psatisﬁes

R¯p=α(1 −β2) + β2¯p+a+ρ¯y=α(1 −β2) + β2¯p+ ¯y.

Hence

¯p=α(1 −β2) + ¯y

R−β2.(3.11)

Imposing the ﬁrst consistency requirement of a SCEE on the mean, i.e. ¯p=α, yields

α=¯y

R−1=: α∗.(3.12)

Hence using (3.5), we conclude that in a SCEE the unconditional mean of market

prices coincides with the REE fundamental prices. That is to say, in a SCEE market

prices ﬂuctuate around the fundamental prices.

12

Next consider the second consistency requirement of a SCEE on the ﬁrst-order auto-

correlation coeﬃcient βof the PLM. A straightforward computation (see Appendix C)

shows that the ﬁrst-order autocorrelation coeﬃcient Corr(pt, pt−1) of the ALM satisﬁes

Corr(pt, pt−1) = β2+Rρ

ρβ2+R=: F(β).(3.13)

Deﬁne G(β) := F(β)−β. In the case that ρ > 05, since G(0) = ρ > 0 and G(1) =

1+Rρ

ρ+R−1 = (1−ρ)(1−R)

ρ+R=−r(1−ρ)

ρ+R<0, there exists at least one β∗∈(0,1), such that

G(β∗) = 0, i.e.

F(β∗) = β∗.

Furthermore, because F(0) = ρand F′(β) = 2βR(1−ρ2)

(ρβ2+R)2>0 for β∈(0,1), we have

F(β)> ρ for β∈(0,1). Hence

β∗> ρ.

It can be shown (see Appendix D) that β∗is unique. We thus have the following propo-

sition on ﬁrst-order stochastic consistent expectations equilibrium.

Proposition 1 In the case that 0< ρ < 1, there exists a unique nonzero ﬁrst-order

stochastic consistent expectations equilibrium (α∗, β∗)for the asset pricing model with

AR(1) dividends (3.10), which satisﬁes α∗=¯y

(R−1) =p∗and β∗> ρ.

This proposition states that in a SCEE self-fulﬁlling market prices have the same mean as

the fundamental prices, but a higher ﬁrst-order autocorrelation coeﬃcient than the funda-

mental prices. In other words, in a SCEE market prices ﬂuctuate around the fundamental

prices but with a higher persistence than under REE.

3.2.1 Numerical analysis

Now we illustrate the above results numerically. For example, consider R= 1.05, ρ =

0.9, a = 0.005, εt∼i.i.d. U (−0.01,0.01) (i.e. uniform distribution on [−0.01,0.01])6.

Figures 1a 7and 1b illustrate the existence of a unique stable ﬁrst-order SCEE, where

(α∗, β∗) = (1,0.997). The time series of fundamental prices and market prices with

5In the case that ρ= 0, F(β) = β2

R, which can be obtained from (3.13). Hence G(β) = β2

R−β=β(β−R)

R.

Since β≤1< R, the only equilibrium is β= 0. Therefore, in the case that ρ= 0, there is no nonzero

ﬁrst-order stochastic consistent expectations equilibrium (SCEE).

6As shown theoretically above, the numerical results are independent of selection of the parameter

values within plausible ranges, sample paths, initial values and distribution of noise.

7In Figure 1a, we take β= 0.9. However in fact, α∗is independent of β, as can be seen from (3.12).

13

(α, β) = (α∗, β∗) are shown in Figure 1c, which illustrates that the market price ﬂuctuates

around the fundamental price but has more persistence and exhibits excess volatility. In

fact, based on Proposition 1, in a SCEE the mean of the market prices is equal to that

of the fundamental prices and the ﬁrst-order autocorrelation coeﬃcient β∗of the market

prices is greater than that of the fundamental prices ρ, implying that the market prices

have higher persistence. In order to further illustrate this, the autocorrelation functions

of the market prices and the fundamental prices are shown in Figure 1d. It can be seen

from Figure 1d that autocorrelation coeﬃcients of the market prices are higher than those

of the fundamental prices and hence the market prices have higher persistence.

We now investigate how the ﬁrst-order SCEE and excess volatility of market prices

depend on the autoregressive coeﬃcient of dividends ρ, which is also the ﬁrst-order au-

tocorrelation of fundamental prices. Consistent with Proposition 1, Figure 2a illustrates

that the ﬁrst-order autocorrelation of market prices is higher than that of fundamental

prices, especially much higher as ρ > 0.4. In fact, based on empirical ﬁndings, e.g. Tim-

mermann (1996), the autoregressive coeﬃcient of dividends ρis about 0.9, where the

corresponding β∗≈0.997. In the case ρ > 0.4, correspondingly the variance of market

prices is larger than that of fundamental prices, as illustrated in Figure 2b. In the Figure

2b, the ratio of variance of market prices and variance of fundamental prices is greater

than 1 for 0.4< ρ < 1. For ρ= 0.9, σ2

p

σ2

p∗≈2.5. Given the variance of fundamental prices

(3.6) and the variance of market prices (C.6),

σ2

p

σ2

p∗

=(β2ρ+R)(R−ρ)2

(R2−β4)(R−ρβ2)β=β∗(ρ)

.

Proposition 1 demonstrates ρ < β∗(ρ)<1 for 0 < ρ < 1, and hence β∗(ρ) converges to 1

as ρtends to 1. Thus as ρtends to 1, σ2

p

σ2

p∗converges to 1, consistent with Figure 2b. So

for plausible parameter values of ρ, the variance of market prices is greater than that of

fundamental prices, indicating that market prices have excess volatility in the SCEE.

3.3 Stability under SAC-learning

In this subsection we study the stability of SCEE under SAC-learning in the asset

pricing model with AR(1) dividends. The asset pricing model with AR(1) dividends

under SAC-learning is given by

pt=1

Rαt−1+β2

t−1(pt−1−αt−1) + a+ρyt,

yt=a+ρyt−1+εt,

(3.14)

14

0 0.5 1 1.5 2

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

α

p

(a) SCEE for α∗

0 0.2 0.4 0.6 0.8 1

0

0.2

0.4

0.6

0.8

1

β

F(β)

(b) SCEE for β∗

1 1.02 1.04 1.06 1.08 1.1

x 104

0.7

0.8

0.9

1

1.1

1.2

1.3

1.4

1.5

1.6

t

pt

Fundamental price

Market price

(c) Time series of prices in SCEE

0 10 20 30 40 50

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

k

ρk

Fundamental price

Market price

(d) Autocorrelation function in SCEE

Figure 1: (a) SCEE α∗is the intersection point of the mean ¯p=α(1−β2)+¯y

R−β2(bold curve)

with the perceived mean α(dotted line); (b) SCEE β∗is the intersection point of the

ﬁrst-order autocorrelation coeﬃcient F(β) = β2+Rρ

ρβ2+R(bold curve) with the perceived ﬁrst-

order autocorrelation β(dotted line); (c) 1,000 observations of fundamental prices (dotted

curve) and market prices (bold curve) in the SCEE; (d) autocorrelation functions of 10,000

fundamental prices (lower dots) and market prices (higher stars) in the SCEE. Parameter

values are R= 1.05, ρ = 0.9, a = 0.005, εt∼i.i.d. U(−0.01,0.01).

15

0 0.2 0.4 0.6 0.8 1

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

ρ

β*

(a) 1-order autocorrelation in SCEE

0.4 0.5 0.6 0.7 0.8 0.9 1

0

1

2

3

4

ρ

σp

2/σp*

2

(b) Ratio of variances in SCEE

Figure 2: (a) SCEE β∗with respect to ρ; (b) ratio of variance of market prices and

variance of fundamental prices with respect to ρ, where R= 1.05.

with αt, βtas in (2.8). This is an expectations feedback system. Realized prices inﬂuence

the perceptions agents have about economic reality and these perceptions feed back into

the actual dynamics of the economy and determine future prices that will be realized.

In order to study the dynamical behavior of the model (3.14), we ﬁrst check the

stability of the unique SCEE (α∗, β∗) in Proposition 1. The stability of the SCEE is in

fact determined by the coeﬃcient 1−β2

R−β2in front of αin the unconditional mean in (3.11)

and by F(β) in (3.13). On one hand, since 0 ≤1−β2

R−β2<1, it can be seen from (3.11) that

α∗is stable. On the other hand, the proof of uniqueness of β∗in Appendix D shows that

0< F ′(β∗)<1, and that therefore β∗is stable. We thus have stability of the unique

SCEE under SAC-learning.

Proposition 2 The unique SCEE (α∗, β∗)in Proposition 1 is stable under SAC-learning,

that is, the SAC-learning process (αt, βt)converges to the unique SCEE (α∗, β ∗)as time

ttends to ∞.

Proof. See Appendix E.

This proposition shows that the SCEE describes the long-run behavior of SAC-learning

when agents use a simple AR(1) forecasting rule.

3.3.1 Numerical analysis

Figure 3 shows that SAC-learning (αt, βt) converges to the unique stable SCEE (α∗, β∗).

Figure 3a indicates that the mean of the market prices under SAC-learning αttends to

16

0 0.5 1 1.5 2 2.5 3

x 104

0

0.2

0.4

0.6

0.8

1

1.2

t

αt

(a)

0 500 1000 1500 2000

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

t

βt

(b)

1 1.02 1.04 1.06 1.08 1.1

x 104

0.7

0.8

0.9

1

1.1

1.2

1.3

1.4

1.5

1.6

t

pt

Fundamental price

Market price

(c)

Figure 3: (a) Time series αt→α∗(1.0) under SAC-learning; (b) time series βt→β∗(0.997)

under SAC-learning; (c) time series of market prices under SAC-learning and fundamental

prices. Initial values p0= 0.3, y0= 0.08.

the mean α∗= 1 in the SCEE, while Figure 3b shows that the ﬁrst-order autocorrelation

coeﬃcient of the market prices under SAC-learning βttends to the ﬁrst-order autocor-

relation coeﬃcient β∗= 0.997 in the SCEE. Therefore, given the same sample path of

noise, the time series of the market prices under SAC-learning is almost the same as

that in the SCEE, which can be seen by comparing Figure 3c to Figure 1c. That is,

the market prices under SAC-learning ﬂuctuate around the fundamental prices but have

excess volatility and stronger autocorrelation. Therefore, the self-referential SCEE and

learning oﬀer a possible explanation of bubbles within a stationary time series framework,

as suggested in Bullard et al. (2010).

4 The New Keynesian Philips curve with AR(1) driv-

ing variable

Now consider a second application of SCEE and learning in macroeconomics, the New

Keynesian Philips curve with an AR(1) driving variable as suggested by Lansing (2009).

Assume that the inﬂation and the output gap (real marginal cost) evolve according to

πt=λπe

t+1 +γyt+ut,

yt=a+ρyt−1+εt,

(4.1)

where πtis the inﬂation at time t, πe

t+1 is expected inﬂation at date t+ 1 and ytis the

output gap or real marginal cost. λ∈[0,1) is the representative agent’s subjective time

discount factor, γ > 0 is related to the degree of price stickiness in the economy and

17

ρ∈[0,1) describes the linear dependence of the output gap on its past value. utand

εtare i.i.d. stochastic disturbances with zero mean and ﬁnite absolute moments with

variances σ2

uand σ2

ε, respectively. The key diﬀerence with the standard asset pricing

model is that this model includes two stochastic disturbances, not only the noise εtof

the AR(1) driving variable, but also an additional noise utin the New Keynesian Philips

curve. We refer to utas a markup shock that is often motivated by the presence of a

variable tax rate and to εtas a demand shock that is uncorrelated with the markup shock.

4.1 Rational expectations equilibrium

If agents are rational, then a straightforward computation (see Appendix F) gives the

rational expectations equilibrium

π∗

t=γλa

(1 −λ)(1 −λρ)+γ

1−λρyt+ut.(4.2)

Hence the mean and variance of rational expectations equilibrium π∗

tare, respectively,

π∗:= E(π∗

t) = γa

(1 −λ)(1 −ρ),(4.3)

V ar(π∗

t) = E(π∗

t−π∗)2=γ2σ2

ε

(1 −λρ)2(1 −ρ2)+σ2

u.(4.4)

Furthermore, the ﬁrst-order autocovariance and autocorrelation of rational expectations

equilibrium π∗

tare, respectively,

E(π∗

t−π∗)(π∗

t−1−π∗) = γ2ρσ2

ε

(1 −λρ)2(1 −ρ2),

Corr(π∗

t, π∗

t−1) = ργ2

γ2+ (1 −λρ)2(1 −ρ2)σ2

u

σ2

ε

.

Note that in the special case σ2

u= 0, the above expression reduces to Corr(π∗

t, π∗

t−1) = ρ

as in Eq. (3.7). Moreover, the larger the noise level σ2

uin the markup shock, the smaller

the ﬁrst-order autocorrelation in the fundamental rational equilibrium inﬂation.

4.2 Existence of ﬁrst-order SCEE

Suppose now that agents are boundedly rational and that their perceived law of motion

for inﬂation is a univariate AR(1) process.

πt=α+β(πt−1−α) + vt(4.5)

18

The implied actual law of motion then becomes

πt=λ[α+β2(πt−1−α)] + γyt+ut,

yt=a+ρyt−1+εt.

(4.6)

Denote the unconditional expectation of πtby ¯πand the unconditional expectation of

ytby ¯y. Then ¯y=a/(1 −ρ) and ¯πsatisﬁes

¯π=λα(1 −β2) + λβ2¯π+γ¯y.

Hence

¯π=λα(1 −β2) + γ¯y

1−λβ2.(4.7)

Imposing the ﬁrst consistency requirement on the mean, i.e. ¯π=λα(1−β2)+γ¯y

1−λβ2=α, we get

α=γ¯y

1−λ=γa

(1 −λ)(1 −ρ)=: α∗.

Therefore using (4.3), in a SCEE the unconditional mean of inﬂation coincides with the

REE fundamental inﬂation.

After straightforward computations (see Appendix G), we obtain

Corr(πt, πt−1) = γ2(λβ2+ρ) + λβ2(1 −ρ2)(1 −λβ2ρ)σ2

u

σ2

ε

γ2(λβ2ρ+ 1) + (1 −ρ2)(1 −λβ2ρ)σ2

u

σ2

ε

=: F(β).(4.8)

Note that if we replace λby 1

R,γby ρ

Rand σuby 0, then the autocorrelation in (4.8) is

simpliﬁed to β2+Rρ

ρβ2+R, which coincides with the autocorrelation in the asset pricing model

in (3.13).

The second consistency requirement of ﬁrst-order autocorrelation coeﬃcient βyields,

F(β) = β.

Deﬁne G(β) := F(β)−β. Since 0 < ρ < 1 and 0 ≤λ < 1,

G(0) = γ2ρ

γ2+ (1 −ρ2)σ2

u

σ2

ε

>0

and

G(1) = γ2(λ+ρ) + λ(1 −ρ2)(1 −λρ)σ2

u

σ2

ε

γ2(λρ + 1) + (1 −ρ2)(1 −λρ)σ2

u

σ2

ε

−1

=−γ2(1 −λ)(1 −ρ)−(1 −λ)(1 −ρ2)(1 −λρ)σ2

u

σ2

ε

γ2(λρ + 1) + (1 −ρ2)(1 −λρ)σ2

u

σ2

ε

<0.

19

Therefore, there exists at least one β∗∈(0,1), such that G(β∗) = 0, i.e. F(β∗) = β∗.In

the special case without autocorrelation in the driving variable yt, i.e. ρ= 0, equation

(4.8) gives F(β) = λβ2. Hence the ﬁrst-order SCEE for ρ= 0 is β∗= 0 and coincides

with the REE.

Proposition 3 In the case that 0< ρ < 1and 0≤λ < 1, there exists at least one

nonzero ﬁrst-order stochastic consistent expectations equilibrium (SCEE) (α∗, β∗)for the

New Keynesian Philips curve (4.6) with α∗=γ a

(1−λ)(1−ρ)=π∗.

It turns out that in the NKPC multiple SCEE may co-exist. To see this, rewrite the

ﬁrst-order autocorrelation, F(β) = λβ2+ρ(1−λ2β4)

(λβ2ρ+1)+(1−ρ2)(1−λβ2ρ)1

γ2·σ2

u

σ2

ε

. It is easy to see that

if γor σ2

ε

σ2

uincreases, then F(β) increases, and therefore multiple SCEE may occur. The

simulations in the following subsection illustrate this point more clearly.

4.2.1 Numerical analysis

Now we illustrate the existence of SCEE and the eﬀects of ρ, γ and σ2

ε

σ2

unumerically.

Based on empirical ﬁndings, such as Lansing (2009), Gali et al. (2001) and Fuhrer (2006,

2009), we ﬁrst examine a plausible case8in which γ= 0.075, σu= 0.003162, σε= 0.01, ρ =

0.9, λ = 0.99, εt∼N(0, σ2

ε), ut∼N(0, σ2

u), a = 0.0004. Hence σ2

u

σ2

ε= 0.1. Figure 4a

illustrates existence of a unique (stable) α∗, where α∗= 0.03. Figure 4b shows that

there exist three β∗, where β∗= 0.3066,0.7417,0.9961. That is, there exist three ﬁrst-

order SCEE: two stable ones (α∗, β∗) = (0.03,0.3066),(0.03,0.9961) and an unstable

one (α∗, β∗) = (0.03,0.7417). Considering that the SAC-learning converges to a stable

SCEE (see the next subsection), the stable (learnable) SCEE are the most interesting.

Figures 4c and 4d illustrate the two time series for the two (stable) SCEE (α∗, β∗) =

(0.03,0.3066),(0.03,0.9961), suggesting that inﬂation has diﬀerent persistence at diﬀerent

SCEE. That is, the SCEE is an important factor in aﬀecting inﬂation persistence. In

fact, the time series of inﬂation in the SCEE with high β∗in Figure 4d has similar

persistence characteristics and amplitude of ﬂuctuation as in empirical inﬂation data in

8As shown in Lansing (2009), based on regressions using either the output gap or labor’s share of

income over the period 1949.Q1 to 2004.Q4, ρ= 0.9, σε= 0.01.Estimates of the NKPC parameters

λ, γ, σuare sensitive to the choice of the driving variable, the sample period, and the econometric model

etc. Later we also examine the eﬀects of some other parameters on SCEE. Furthermore, based on the

above theoretical results, ajust aﬀects the mean of inﬂation ¯πbut not the autocorrelation coeﬃcient

F(β). For σuand σε,F(β) only depends on their ratio σu/σεbut not on their absolute values.

20

Tallman (2003). Furthermore, Figure 4d illustrates that inﬂation in the SCEE with high

β∗has stronger persistence than the REE inﬂation, where the ﬁrst-order autocorrelation

coeﬃcient of REE inﬂation is 0.865 less than β∗= 0.9961.

In order to further study the eﬀects of ρ, Figure 5 illustrates SCEE β∗together

with the ﬁrst-order autocorrelation coeﬃcient of REE inﬂation as functions of ρ. For

0.84 < ρ < 0.918, two stable SCEE β∗exist separated by an unstable SCEE. The large

SCEE β∗is larger than the ﬁrst-order autocorrelation coeﬃcient of REE inﬂation, while

the small SCEE β∗is smaller than the ﬁrst-order autocorrelation coeﬃcient of REE in-

ﬂation. In the next subsection we will show that for a large range of initial values of

inﬂation the SAC-learning converges to the stable high SCEE β∗with strong persistence.

If ρ > 0.918, there exists only one stable SCEE β∗with stronger persistence than REE.

Therefore for plausible values of ρaround 0.9, inﬂation in a SCEE often generates high-

persistence as shown in Figure 4d. This result is consistent with the empirical ﬁnding

in Adam (2007) that the Restricted Receptions Equilibrium (RPE) describes subjects’

inﬂation expectations surprisingly well and provides a better explanation for the observed

persistence of inﬂation than REE.

Figure 6 illustrates how the number of SCEE depends on γ. The simulations show that

for plausible γthere exist at least one and at most three SCEE β∗. For suﬃciently small

γ(<0.05), there exists only one low β∗, as shown in Figure 6a. As γincreases, the graph

of F(β) = λβ2+ρ(1−λ2β4)

(λβ2ρ+1)+(1−ρ2)(1−λβ2ρ)1

γ2·σ2

u

σ2

ε

goes up. At γ= 0.05, a new SCEE β∗≈0.975

is created at a tangent bifurcation, see Figure 6b. Immediately after that there exist three

β∗, two stable equilibria and one unstable. That is, at γ= 0.05, a tangent bifurcation

occurs. Figures 6c and 6d illustrate the three β∗with the high stable β∗close to 1. As

γincreases, the two stable β∗-values grow. At γ= 0.084, another tangent bifurcation

occurs, where the lower β∗-values coincide, as shown in Figure 6e. For γ > 0.084, there is

only one large β∗, see Figure 6f, which corresponds to high-persistence SCEE of inﬂation.

Hence a larger γtends to lead to higher persistence of inﬂation. Intuitively with a larger

γ, it can be seen from (4.1) that the driving variable (output gap or real marginal cost)

has a larger impact on inﬂation. Hence when the driving variable is relatively important,

a high-persistence SCEE occurs. If on the other hand, σ2

uincreases, that is, the noise

to inﬂation increases, the ratio σ2

ε

σ2

udecreases and the strong reverses and low-persistence

SCEE become more likely. That is, intuitively clear, as more noise to inﬂation dominates

the driving variable, this leads to a low-persistence inﬂation equilibrium.

21

0.02 0.025 0.03 0.035 0.04

0.02

0.022

0.024

0.026

0.028

0.03

0.032

0.034

0.036

0.038

0.04

α

π

(a) SCEE for α∗

0 0.2 0.4 0.6 0.8 1

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

β

F(β)

(b) SCEE for β∗

0 50 100 150 200 250 300 350

0.01

0.015

0.02

0.025

0.03

0.035

0.04

0.045

0.05

t

πt

(c) time series at (α∗, β∗) = (0.03,0.3066)

0 50 100 150 200 250 300 350

0

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

t

πt

Inflation at REE

Inflation at SCEE

(d) time series at (α∗, β∗) = (0.03,0.9961)

Figure 4: (a) SCEE α∗is the unique intersection point of the mean of inﬂation ¯π=

λα(1−β2)+γ¯y

1−λβ2(bold curve) with the perceived mean α(dotted line); (b) SCEE β∗is an

intersection point of the ﬁrst-order autocorrelation of inﬂation γ2(λβ2+ρ)+λβ2(1−ρ2)(1−λβ2ρ)σ2

u

σ2

ε

γ2(λβ2ρ+1)+(1−ρ2)(1−λβ2ρ)σ2

u

σ2

ε

(bold curve) and the perceived ﬁrst-order autocorrelation β(dotted line); (c) time series

of inﬂation in stable low-persistence SCEE (α∗, β∗) = (0.03,0.3066); (d) times series of

inﬂation in stable high-persistence SCEE (α∗, β∗) = (0.03,0.9961) (bold curve) and time

series of REE inﬂation (dotted curve), where γ= 0.075, σu= 0.003162, σε= 0.01, ρ =

0.9, λ = 0.99, εt∼N(0, σ2

ε), ut∼N(0, σ2

u), a = 0.0004.

22

0.7 0.75 0.8 0.85 0.9 0.95 1

0

0.5

1

1.5

ρ

β*

1−order autocorrelation of REE

High stable β*

Low stable β*

Middle unstable β*

Figure 5: First-order autocorrelation coeﬃcient of REE inﬂation (dotted real curve),

stable SCEE β∗with respect to ρ(bold curves), unstable SCEE β∗(dotted curve), where

γ= 0.075, σu= 0.003162, σε= 0.01, λ = 0.99.

4.3 Stability under SAC-learning

The SAC-learning dynamics in the New Keynesian Philips curve with AR(1) driving

variable is given by

πt=λ[αt−1+β2

t−1(πt−1−αt−1)] + γyt+ut,

yt=a+ρyt−1+εt.

(4.9)

with αt, βtas in (2.8). This is another expectations feedback system with expectation

feedback from inﬂation forecasting. Realized inﬂations inﬂuence the beliefs agents have

about economic reality and these beliefs feed back into the actual dynamics of economy

and determine the future realized inﬂations together with an exogenous driving variable

output gap or real marginal costs.

We further check the relationship between stability of SCEE (α∗, β∗) and SAC-learning.

For α∗, since 0 ≤λ(1−β2)

1−λβ2<1, it can be seen from (4.7) that α∗is stable. For β∗, because

of the complexity of the ﬁrst-order autocorrelation F(β) in (4.8), it is diﬃcult to check

the stability of SCEE, or even the number of SCEE. We have the following relationship

between the SCEE and the SAC-learning.

Proposition 4 If (0 ≤)F′(β∗)<1, then the SCEE (α∗, β∗)is stable, that is, the SAC-

learning (αt, βt)converges to the SCEE (α∗, β∗)as time ttends to ∞.

The proof is given in Appendix H. If the stable SCEE is not unique, the convergence de-

pends on initial states of the system, as illustrated in the following numerical simulations.

23

0 0.2 0.4 0.6 0.8 1

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

β

F(β)

(a) γ= 0.01

0 0.2 0.4 0.6 0.8 1

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

β

F(β)

(b) γ= 0.05

0 0.2 0.4 0.6 0.8 1

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

β

F(β)

(c) γ= 0.065

0 0.2 0.4 0.6 0.8 1

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

β

F(β)

(d) γ= 0.075

0 0.2 0.4 0.6 0.8 1

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

β

F(β)

(e) γ= 0.084

0 0.2 0.4 0.6 0.8 1

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

β

F(β)

(f) γ= 0.1

Figure 6: SCEE β∗with γ= 0.01 (a); γ= 0.05 (b); γ= 0.065 (c); γ= 0.075 (d);

γ= 0.084 (e) and γ= 0.1 (f), where σ2

u

σ2

ε= 0.1, ρ = 0.9, λ = 0.99.

24

4.3.1 Numerical analysis

For (π0, y0) = (0.028,0.01), Figures 7a and 7b show that the SAC-learning dynamics

(αt, βt) converges to the stable low-persistence SCEE (α∗, β∗) = (0.03,0.3066). Figure

7a illustrates that the mean of inﬂation αttends to the mean α∗= 0.03. Figure 7b

illustrates that the ﬁrst-order autocorrelation coeﬃcient of inﬂation βtslowly tends to

the low-persistence stable ﬁrst-order autocorrelation coeﬃcient β∗= 0.3066. For the

diﬀerent initial value (π0, y0) = (0.1,0.15), our numerical simulation shows that the mean

of inﬂation αtunder SAC-learning still tends to the mean α∗, but slowly9(see Figure

7c), while Figure 7d indicates that the ﬁrst-order autocorrelation coeﬃcient of inﬂation

βtunder SAC-learning tends to the higher stable ﬁrst-order autocorrelation coeﬃcient

β∗= 0.996110. Correspondingly given the same sample path of noise, the time series of

inﬂation under SAC-learning can also replicate the time series of inﬂation in the SCEE

after long-term learning as shown in the preceding asset pricing model.

Numerous simulations show that as initial values of inﬂation are (relatively) higher

than the mean α∗= 0.03, the sample autocorrelation learning βtgenerally enters the

high-persistence region. In particular, a large shock to the inﬂation may easily cause a

jump of the SAC-learning process into the high-persistence region.

5 Conclusion

In this paper we have introduced a very simple type of misspeciﬁcation equilibrium

and a plausible corresponding behavioral learning process. Boundedly rational agents use

a univariate linear forecasting rule and in equilibrium correctly forecast the unconditional

sample mean and ﬁrst-order sample autocorrelation. Hence, to a ﬁrst order approxima-

tion the simple linear forecasting rule is consistent with observed market realizations.

Sample autocorrelation learning simply means that agents are slowly updating the two

coeﬃcients –sample mean and ﬁrst-order autocorrelation– of their linear rule. In the long

run, agents thus learn the best univariate linear forecasting rule, without fully recognizing

the structure of the economy.

We have applied our SCEE and SAC-learning concepts to a standard asset pricing

9The slow convergence is caused by the slope λ−λβ2

1−λβ2in the expression (4.7), which is very close to 1

for λ= 0.99, as shown in Figure 4a.

10As shown in Figure 4b, F′(β∗) is close to 1 and hence the convergence of SAC-learning is very slow.

25

0 500 1000 1500 2000 2500 3000

0.026

0.028

0.03

0.032

0.034

t

αt

(a) αt→0.03

0 2000 4000 6000 8000 10000

0

0.5

1

t

βt

(b) βt→0.3066

0 0.5 1 1.5 2

x 104

0.08

0.09

0.1

0.11

0.12

0.13

t

αt

(c) αt→0.03

0 0.5 1 1.5 2

x 104

0

0.5

1

t

βt

(d) βt→0.9961

Figure 7: Time series of αtand βtunder SAC-learning with diﬀerent initial values

(π0, y0) = (0.028,0.01)(a), (b) and (π0, y0) = (0.1,0.15) (c), (d).

26

model with AR(1) dividends and a New Keynesian Philips curve driven by an AR(1)

process for the output gap or marginal costs. In both applications, the law of motion

of the economy is linear, but it is driven by an exogenous stochastic AR(1) process.

Agents however are not fully aware of the exact linear structure of the economy, but

use a simple univariate forecasting rule, to predict asset prices or inﬂation. In the asset

pricing model a unique SCEE exists and it is globally stable under SAC-learning. An

important feature of the SCEE is that it is characterized by high-persistence and excess

volatility in asset prices, signiﬁcantly higher than under rational expectations. In the New

Keynesian model, multiple SCEE arise and a low and a high-persistence misspeciﬁcation

equilibrium co-exist. The SAC-learning exhibits path dependence and it depends on the

initial states whether the system converges to the low-persistence or the high-persistence

inﬂation regime. In particular, when there are shocks– e.g. oil shocks– temporarily causing

high inﬂation, SAC-learning may lock into the high-persistence inﬂation regime.

Are these simple misspeciﬁcation equilibria empirically relevant or would smart agents

recognize their (second order) mistakes and learn to be perfectly rational? This empirical

question should be addressed in more detail in future work, but we provide some argu-

ments for the empirical relevance of our equilibrium concept. Firstly, in our applications

the SCEE already explain some important stylized facts: (i) high persistence and excess

volatility in asset prices, (ii) high persistence in inﬂation and (iii) regime switching in

inﬂation dynamics, which could explain a long phase of high US inﬂation in the 1970s

and early 1980s as well as a long phase of low inﬂation in the 1990s and 2000s. Secondly,

we stress the behavioral interpretation of our misspeciﬁcation equilibrium and learning

process. The univariate AR(1) rule and the SAC-learning process are examples of simple

forecasting heuristics that can be used without any knowledge of statistical techniques,

simply by observing a time series and roughly ”guestimating” its sample average and

its ﬁrst-order persistence coeﬃcient. Coordination on a behavioral forecasting heuristic

that performs reasonably well to a ﬁrst-order approximation seems more likely than co-

ordination on more complicated learning or sunspot equilibria. Even though some smart

individual agents might be able to improve upon the best linear, univariate forecasting

rule, a majority of agents might still stick to their simple univariate rule. It therefore

seems relevant to describe aggregate phenomena by simple misspeciﬁcation equilibria and

behavioral learning processes. In fact, there is already some experimental evidence for

the relevance of misspeciﬁcation equilibria in Adam (2007). More recently Assenza et al.

(2011) and Pfajfar and Zakelj (2010) ran learning to forecasting experiments with human

27

subjects in a New Keynesian framework with expectations feedback from individual in-

ﬂation and output gap forecasts. Simple linear univariate models explain a substantial

part of individual inﬂation and output gap forecasting behavior.

In future work we plan to consider more general economic settings and study SCEE

and their relationship to SAC-learning. An obvious next step is to apply our SCEE and

SAC-learning framework to higher dimensional linear economic systems, with agents fore-

casting by univariate linear rules. In particular, the fully speciﬁed New Keynesian model

of inﬂation and output dynamics would be an interesting (two-dimensional) application.

Finally, it is interesting and challenging to study SCEE and misspeciﬁcation under het-

erogeneous expectations and allow for switching between diﬀerent rules. Branch (2004)

and Hommes (2011) provide some empirical and experimental evidence on heterogeneous

expectations, while Berardi (2007) and Branch and Evans (2006, 2007) have made some

related studies on heterogeneous expectations and learning in similar settings.

Acknowledgements

We would like to thank Kevin Lansing, Mikhail Anufriev, Florian Wagener, Cees Diks

and other participants in the KAFEE lunch seminar in University of Amsterdam, the 3rd

POLHIA workshop in Universita Politecnica delle Marche, and the 11th Workshop on

Optimal Control, Dynamic Games and Nonlinear Dynamics in University of Amsterdam

for helpful comments. We are grateful to the EU 7th framework collaborative project

”Monetary, Fiscal and Structural Policies with Heterogeneous Agents (POLHIA)”, grant

no. 225408, for ﬁnancial support. Mei Zhu also acknowledges ﬁnancial support from

NSFC(10871005).

28

Appendix

A Recursive dynamics of SAC-learning

The sample average is

αt=1

t+ 1[x0+x1+···+xt]

=1

t+ 1[(t+ 1)αt−1+xt−αt−1]

=αt−1+1

t+ 1[xt−αt−1].

Let

zt:= (x0−αt)(x1−αt) + ···+ (xt−1−αt)(xt−αt)

= (x0−αt−1−1

t+ 1(xt−αt−1))(x1−αt−1−1

t+ 1(xt−αt−1)) +

···+ (xt−1−αt−1−1

t+ 1(xt−αt−1))(xt−αt−1−1

t+ 1(xt−αt−1))

= (x0−αt−1)(x1−αt−1) + ··· + (xt−2−αt−1)(xt−1−αt−1)

+xt−αt−1

t+ 1 (2αt−1−x0−x1+···+ 2αt−1−xt−2−xt−1) + t−1

(t+ 1)2(xt−αt−1)2

+t

t+ 1(xt−1−αt−1)(xt−αt−1)−t

(t+ 1)2(xt−αt−1)2

=zt−1+1

t+ 1(xt−αt−1)[2(t−1)αt−1−x0−2x1− · ·· − 2xt−2−xt−1+t(xt−1−αt−1)]

−1

(t+ 1)2(xt−αt−1)2,

=zt−1+1

t+ 1(xt−αt−1)[x0+ (t+ 1)xt−1−(t+ 2)αt−1]−1

(t+ 1)2(xt−αt−1)2

=zt−1+ (xt−αt−1)hxt−1+x0

t+ 1 −t+ 2

t+ 1αt−1+1

(t+ 1)2αt−1−1

(t+ 1)2xti

=zt−1+ (xt−αt−1)Φ4,

where Φ4=xt−1+x0

t+1 −t2+3t+1

(t+1)2αt−1−1

(t+1)2xt.

Write

nt:= (x0−αt)2+ (x1−αt)2+···+ (xt−αt)2

= (x0−αt−1−1

t+ 1(xt−αt−1))2+···+ (xt−αt−1−1

t+ 1(xt−αt−1))2

= (x0−αt−1)2+ (x1−αt−1)2+···+ (xt−1−αt−1)2+t+t2

(t+ 1)2(xt−αt−1)2

=nt−1+t

t+ 1(xt−αt−1)2.

29

All these results are consistent with those in Appendix 1 of Hommes, Sorger & Wagener

(2004). Note that in our paper Rtis diﬀerent from ntin Hommes et al. (2004). In fact,

Rt=1

t+ 1nt

=1

t+ 1[nt−1+t

t+ 1(xt−αt−1)2]

=t

t+ 1Rt−1+t

(t+ 1)2(xt−αt−1)2

=Rt−1+1

t+ 1ht

t+ 1(xt−αt−1)2−Rt−1i.

Furthermore,

βt=zt

nt

=βt−1+1

ntnt−1

[ztnt−1−zt−1nt]

=βt−1+1

ntnt−1zt−1+ (xt−αt−1)Φ4nt−1−zt−1nt−1+t

t+ 1(xt−αt−1)2

=βt−1+1

ntnt−1(xt−αt−1)Φ4nt−1−zt−1

t

t+ 1(xt−αt−1)2

=βt−1+1

nt(xt−αt−1)Φ4−βt−1

t

t+ 1(xt−αt−1)2

=βt−1+R−1

t

t+ 1h(xt−αt−1)xt−1+x0

t+ 1 −t2+ 3t+ 1

(t+ 1)2αt−1−xt

(t+ 1)2−t

t+ 1βt−1(xt−αt−1)2i.

30

B Rational expectations equilibrium of prices

Under the assumption that the transversality condition lim

k→∞

Et(pt+k)

Rk= 0 holds, the

REE price can be computed as

p∗

t=1

REtp∗

t+1 +Etyt+1

=a+ρyt

R+1

REtp∗

t+1

=a+ρyt

R+1

REt1

REt+1p∗

t+2 +Et+1yt+2

=a+ρyt

R+Et(a+ρyt+1)

R2+1

R2Etp∗

t+2

=a+ρyt

R+a+ρa +ρ2yt

R2+1

R2Etp∗

t+2

=···

=a+ρyt

R+a+ρa +ρ2yt

R2+···+a+ρa +···+ρn−1a+ρnyt

Rn+···

=a+ρyt

R+a+ρa +ρ2yt

R2+···+

a(1−ρn)

1−ρ+ρnyt

Rn+···

=

∞

X

n=1

a

1−ρ1

Rn−ρ

Rn+

∞

X

n=1 ρ

Rnyt

=a

1−ρ1

R−1−ρ

R−ρ+ρ

R−ρyt

=aR

(R−1)(R−ρ)+ρ

R−ρyt.(B.1)

C First-order autocorrelation coeﬃcient of price

We rewrite (3.10) as

pt−¯p=β2

R(pt−1−¯p) + ρ

Rεt+ρ2

Rεt−1+···.(C.1)

Thus

E[(pt−¯p)(pt−1−¯p)] = Ehβ2

R(pt−1−¯p)2+ρ

Rεt(pt−1−¯p) + ρ2

Rεt−1(pt−1−¯p) + ···i

=β2

RE(pt−1−¯p)2+ 0 + ρ2

RE[εt−1(pt−1−¯p)] + ···.(C.2)

E[(pt−¯p)2]

=Ehβ2

R(pt−1−¯p)(pt−¯p) + ρ

Rεt(pt−¯p) + ρ2

Rεt−1(pt−¯p) + ···i

=β2

RE[(pt−1−¯p)(pt−¯p)] + ρ

RE[εt(pt−¯p)] + ρ2

RE[εt−1(pt−¯p)] + ···.(C.3)

31

Thus based on (C.2) and (C.3),

E[(pt−¯p)2] = β2

Rhβ2

RE[(pt−¯p)2] + ρ2

RE[εt(pt−¯p)] + ρ3

RE[εt−1(pt−¯p)] + ···i

+ρ

RE[εt(pt−¯p)] + ρ2

RE[εt−1(pt−¯p)] + ···

=β2

R2E[(pt−¯p)2] + ρ(β2ρ+R)

R2E[εt(pt−¯p)] + ρE[εt−1(pt−¯p)] + ···.

That is,

E[(pt−¯p)2] = ρ(β2ρ+R)

R2−β4E[εt(pt−¯p)] + ρE[εt−1(pt−¯p)] + ···.(C.4)

In the following we will calculate E(εt−kpt), k = 0,1,2,···.

E[εt(pt−¯p)] = β2

RE[(pt−1−¯p)εt] + ρ

RE[ε2

t] + ρ2

RE[εt−1εt] + ···

=ρ

Rσ2

ε.

E[εt−1(pt−¯p)] = β2

RE[(pt−1−¯p)εt−1] + ρ

RE[εtεt−1] + ρ2

RE[ε2

t−1] + ···

=β2

RE[(pt−¯p)εt] + ρ2

Rσ2

ε

=β2

R·ρ

Rσ2

ε+ρ2

Rσ2

ε.

E[εt−2(pt−¯p)] = β2

RE[(pt−1−¯p)εt−2] + ρ

RE[εtεt−2] + ρ2

RE[εt−1εt−2] + ρ3

RE[ε2

t−2] + ···

=β2

RE[(pt−¯p)εt−1] + ρ3

Rσ2

ε

=β2

R2ρ

Rσ2

ε+β2

R·ρ2

Rσ2

ε+ρ3

Rσ2

ε.

E[εt−k(pt−¯p)] = β2

RE[(pt−1−¯p)εt−k] + ρ

RE[εtεt−k] + ρ2

RE[εt−1εt−k] + ···+ρk+1

RE[ε2

t−k] + ···

=β2

RE[(pt−¯p)εt−k+1] + ρk+1

Rσ2

ε

=···

=ρ

Rσ2

ε·β2

Rk+ρ

Rσ2

ε·β2

Rk−1·ρ+ρ

Rσ2

ε·β2

Rk−2·ρ2+···

+ρ

Rσ2

ε·β2

R·ρk−1+ρ

Rσ2

ε·ρk

=σ2

ε

R·

ρk+1 −β2

Rk+1

1−β2

ρR

.

32

Hence

E[εt(pt−¯p)] + ρE[εt−1(pt−¯p)] + ···

=σ2

ε

ρR ·P∞

k=0 ρ2(k+1) −P∞

k=0 β2ρ

Rk+1

1−β2

ρR

=σ2

ε

ρR ·

ρ2

1−ρ2−β2ρ

R−ρβ2

1−β2

ρR

=ρσ2

ε

(1 −ρ2)(R−ρβ2).(C.5)

Substituting (C.5) into (C.4), we obtain

V ar(pt) = E[(pt−¯p)2]

=ρ(β2ρ+R)

R2−β4E[εt(pt−¯p)] + ρE[εt−1(pt−¯p)] + ···

=ρ(β2ρ+R)

R2−β4·ρσ2

ε

(1 −ρ2)(R−ρβ2)

=σ2

ερ2(β2ρ+R)

(R2−β4)(1 −ρ2)(R−ρβ2).(C.6)

Furthermore, based on (C.2),

Corr(pt, pt−1) = E[(pt−¯p)(pt−1−¯p)]/V ar(pt)

=β2

R+ρ2

RV ar(pt)nE[εt(pt−¯p)] + ρE[εt−1(pt−¯p)] + ···o

=β2

R+ρ2

R

ρσ2

ε

(1−ρ2)(R−ρβ2)

σ2

ερ2(ρβ2+R)

(R2−β4)(1−ρ2)(R−ρβ2)

=β2

R+ρ(R2−β4)

R(ρβ2+R)

=β2+Rρ

ρβ2+R.

D Proof of uniqueness of β∗(Proposition 1)

Using the ﬁrst-order autocorrelation F(β) in (3.13), it can be calculated that

F′′(β) = 2R(1 −ρ2)

(ρβ2+R)2−8ρβ2R(1 −ρ2)

(ρβ2+R)3=2R(1 −ρ2)(R−3ρβ2)

(ρβ2+R)3.

Therefore, if ρ≤R

3, then R−3β2ρ≥R−β2R > 0. Thus G′′(β) = F′′(β)>0.

Note that G(0) >0, G′(0) = −1<0 and G(1) <0, G′(1) = 2R(1−ρ2)

(ρ+R)2−1. Hence if

33

G′(1) ≤0, then G′(β∗)<0. If G′(1) >0, then there exists a minimal point β1such that

G′(β1) = 0. Moreover, since G(1) <0, then G(β1)<0 (otherwise, G(1) ≥G(β1)≥0,

which is contradictory to G(1) <0). Hence β∗(∈(0, β1)) is unique and G′(β∗)<0, hence

0< F ′(β∗)<1.

If ρ > R

3, then G′′(β)β=√R/(3ρ)=F′′(β)β=√R/(3ρ)= 0 and G′(β)β=√R/(3ρ)is maxi-

mal. Thus in the case that ρ > R

3,

G′(β) = F′(β)−1

=2βR(1 −ρ2)

(ρβ2+R)2−1

<2pR/(3ρ)R(1 −ρ2)

(R/3 + R)2−1

=3√3(1 −ρ2)

8√Rρ −1

<3√3(1 −R2/9)

8pR2/3−1

=−(R−1)(R+ 9)

8R<0.

That is, G(β) is monotone. Therefore, in the case that 0 < ρ < 1, β∗is unique and

G′(β∗)<0, hence 0 < F ′(β∗)<1.

E Proof of Proposition 2

Under the SAC-learning deﬁned in Section 2, the state dynamics equations become

pt=1

Rαt−1+β2

t−1(pt−1−αt−1) + a+ρyt,

yt=a+ρyt−1+εt.

(E.1)

Set γt= (1 + t)−1. Since all functions are smooth, the learning rule (2.8) satisﬁes the

conditions (A.1-A.3) of Section 6.2.1 in Evans and Honkapohja (2001, p.124).

In order to check the conditions (B.1-B.2) of Section 6.2.1 in Evans and Honkapohja

(2001, p.125), we rewrite the system in matrix form by

Xt=A(θt−1)Xt−1+B(θt−1)Wt,

34

where θ′

t= (αt, βt, Rt), X′

t= (1, pt, pt−1, yt) and W′

t= (1, εt),

A(θ) =

0 0 0 0

a(1+ρ)+α(1−β2)

R

β2

R0ρ2

R

0 1 0 0

a0 0 ρ

,

B(θ) =

1 0

0ρ

R

0 0

0 1

.

As shown in Evans and Honkapohja (2001, p.186), A(θ) and B(θ) satisfy the Lipschitz

conditions and Bis bounded. Since εtis assumed to have bounded moments, condition

(B.1) is satisﬁed. Furthermore, the eigenvalues of matrix A(θ) are 0 (double), ρand β2

R.

According to the assumption |β| ≤ 1 and 0 < ρ < 1, all eigenvalues of A(θ) are less than

1 in absolute value. Then it follows that there is a compact neighborhood including the

SCEE solution (α∗, β∗) on which the condition that |A(θ)|is bounded strictly below 1 is

satisﬁed.

Thus the technical conditions for Section 6.2.1 of Chapter 6 in Evans and Honkapohja

(2001) are satisﬁed. Moreover, since ptis stationary under the condition |β| ≤ 1 and

0< ρ < 1, the limits

σ2:= lim

t→∞ E(pt−α)2, σ2

pp−1:= lim

t→∞ E(pt−α)(pt−1−α)

exist and are ﬁnite. Hence according to Section 6.2.1 of Chapter 6 in Evans and Honkapo-

hja (2001, p.126), the associated ODE is

dα

dτ = ¯p(α, β)−α,

dβ

dτ =σ−2[σ2

pp−1−βσ2],

dR

dτ =σ2−R.

That is,

dα

dτ =α(1 −β2) + ¯y

R−β2−α=−rα + ¯y

R−β2,

dβ

dτ =F(β)−β=β2+Rρ

ρβ2+R−β.

(E.2)

35

Furthermore,

J F (α∗, β∗) =

−r

R−(β∗)20

0F′(β∗)−1

.

Based on the analysis in Appendix D, F′(β∗)−1<0. Therefore, the unique stable

SCEE (α∗, β∗) corresponds to the unique stable ﬁxed point of the ODE (E.2). Thus the

SAC learning (αt, βt) converges to the unique stable SCEE (α∗, β∗) as time ttends to ∞.

F Rational expectations equilibrium inﬂation

Under the assumption that the transversality condition limk→∞ λkEt(π∗

t+k) = 0 holds,

the REE inﬂation is computed as

π∗

t=λEtπ∗

t+1 +γyt+ut

=λEt[λEt+1π∗

t+2 +γyt+1 +ut+1 ] + γyt+ut

=λ2Etπ∗

t+2 +γλa +γλρyt+γyt+ut

=···

=

∞

X

k=1

γλk(ρk−1a+···+ρa +a) +

∞

X

k=0

γ(λρ)kyt+ut

=

∞

X

k=1

γa

1−ρ[λk−(λρ)k] + γ

1−λρyt+ut

=γλa

(1 −λ)(1 −λρ)+γ

1−λρyt+ut.(F.1)

G First-order autocorrelation coeﬃcient of inﬂation

We rewrite model (4.6) as

πt−¯π=λβ2(πt−1−¯π) + γ(yt−¯y) + ut,

yt−¯y=ρ(yt−1−¯y) + εt.

(G.1)

That is,

πt−¯π=λβ2(πt−1−¯π) + γρ(yt−1−¯y) + γεt+ut,

yt−¯y=ρ(yt−1−¯y) + εt.

(G.2)

36

E[(πt−¯π)(πt−1−¯π)]

=Ehλβ2(πt−1−¯π)2+γρ(πt−1−¯π)(yt−1−¯y) + γ(πt−1−¯π)εt+ (πt−1−¯π)uti

=λβ2V ar(πt) + γρE[(πt−1−¯π)(yt−1−¯y)] + γE[(πt−1−¯π)εt] + (πt−1−¯π)ut

=λβ2V ar(πt) + γρE[(πt−1−¯π)(yt−1−¯y)]

=λβ2V ar(πt) + γρE[(πt−¯π)(yt−¯y)].(G.3)

V ar(πt)

=E(πt−¯π)2

=Ehλβ2(πt−¯π)(πt−1−¯π) + γρ(πt−¯π)(yt−1−¯y) + γ(πt−¯π)εt+ (πt−¯π)uti

=λβ2E[(πt−¯π)(πt−1−¯π)] + γρE[(πt−¯π)(yt−1−¯y)] + γ(πt−¯π)εt+ (πt−¯π)ut]

=λβ2E[(πt−¯π)(πt−1−¯π)] + γρE[(πt−¯π)(yt−1−¯y)] + γ2σ2

ε+σ2

u,(G.4)

where the last equation is based on the fact that E[(πt−¯π)εt] = Ehλβ2(πt−1−¯π)εt+

γρ(yt−1−¯y)εt+γε2

t+utεti=γσ2

εand E[(πt−¯π)ut] = Ehλβ2(πt−1−¯π)ut+γρ(yt−1−

¯y)ut+γεtut+u2

ti=σ2

u.

Based on (G.3) and (G.4),

V ar(πt) = λβ2E[(πt−¯π)(πt−1−¯π)] + γρE[(πt−¯π)(yt−1−¯y)] + γ2σ2

ε+σ2

u

=λβ2hλβ2V ar(πt) + γρE[(πt−¯π)(yt−¯y)]i+γρE[(πt−¯π)(yt−1−¯y)] + γ2σ2

ε+σ2

u

=λ2β4V ar(πt) + λβ2γρE[(πt−¯π)(yt−¯y)] + γρE[(πt−¯π)(yt−1−¯y)] + γ2σ2

ε+σ2

u.

That is,

V ar(πt) = λβ2γρE[(πt−¯π)(yt−¯y)] + γρE[(πt−¯π)(yt−1−¯y)] + γ2σ2

ε+σ2

u

1−λ2β4.(G.5)

Thus, in order to obtain E[(πt−¯π)(πt−1−¯π)] and V ar(πt), we need calculate E[(πt−

¯π)(yt−¯y)] and E[(πt−¯π)(yt−1−¯y)].

E[(πt−¯π)(yt−¯π)] = Ehλβ2(πt−1−¯π)(yt−¯y) + γρ(yt−1−¯π)(yt−¯y) + γεt(yt−¯π) + ut(yt−¯π)i

=λβ2E{(πt−1−¯π)[ρ(yt−1−¯y) + εt]}+γρE[(yt−1−¯π)(yt−¯y)]

+γE{εt[ρ(yt−1−¯y) + εt]}+E[ut(yt−¯π)]

=λβ2ρE[(πt−1−¯π)(yt−1−¯π)] + 0 + γρ2σ2

ε

(1 −ρ2)+γσ2

ε+ 0.

37

Thus

E[(πt−¯π)(yt−¯π)] = γσ2

ε

(1 −ρ2)(1 −λβ2ρ).(G.6)

Hence based on (G.6),

E[(πt−¯π)(yt−1−¯π)]

=Ehλβ2(πt−1−¯π)(yt−1−¯y) + γρ(yt−1−¯π)2+γεt(yt−1−¯π) + ut(yt−1−¯π)i

=λβ2E[(πt−1−¯π)(yt−1−¯y)] + γρE(yt−1−¯π)2+ 0 + 0

=λβ2·γσ2

ε

(1 −ρ2)(1 −λβ2ρ)+γρ ·σ2

ε

1−ρ2

=γσ2

ε

(1 −ρ2)λβ2

1−λβ2ρ+ρ

=γσ2

ε

(1 −ρ2)·λβ2(1 −ρ2) + ρ

1−λβ2ρ

=γσ2

ε

(1 −λβ2ρ)hλβ2+ρ

1−ρ2i.(G.7)

Therefore, based on (G.5), (G.6) and (G.7),

V ar(πt) = 1

1−λ2β4nλβ2γρE[(πt−¯π)(yt−¯y)] + γρE[(πt−¯π)(yt−1−¯y)] + γ2σ2

ε+σ2

uo

=1

1−λ2β4nλβ2γ2ρσ2

ε

(1 −ρ2)(1 −λβ2ρ)+γ2ρσ2

ε

(1 −λβ2ρ)hλβ2+ρ

1−ρ2i+γ2σ2

ε+σ2

uo

=σ2

ε

1−λ2β4nγ2ρλβ2(2 −ρ2) + ρ

(1 −ρ2)(1 −λβ2ρ)+γ2+σ2

u

σ2

εo

=σ2

ε

1−λ2β4nγ2(λβ2ρ+ 1)

(1 −ρ2)(1 −λβ2ρ)+σ2

u

σ2

εo.(G.8)

According to (G.3),

E[(πt−¯π)(πt−1−¯π)] = λβ2V ar(πt) + γρE[(πt−¯π)(yt−¯y)]

=λβ2V ar(πt) + γ2ρσ2

ε

(1 −ρ2)(1 −λβ2ρ).(G.9)

Thus, the correlation coeﬃcient Corr(πt, πt−1) satisﬁes

Corr(πt, πt−1) = E[(πt−¯π)(πt−1−¯π)]/V ar(πt)

=λβ2+

γ2ρσ2

ε

(1−ρ2)(1−λβ2ρ)

σ2

ε

1−λ2β4nγ2(λβ2ρ+1)

(1−ρ2)(1−λβ2ρ)+σ2

u

σ2

εo

=λβ2+γ2ρ(1 −λ2β4)

γ2(λβ2ρ+ 1) + (1 −ρ2)(1 −λβ2ρ)σ2

u

σ2

ε

=γ2(λβ2+ρ) + λβ2(1 −ρ2)(1 −λβ2ρ)σ2

u

σ2

ε

γ2(λβ2ρ+ 1) + (1 −ρ2)(1 −λβ2ρ)σ2

u

σ2

ε

.

38

H Proof of Proposition 4

In order to check the conditions (B.1-B.2) of Section 6.2.1 in Evans and Honkapohja

(2001, p.125), we rewrite the system in matrix form by

Xt=A(θt−1)Xt−1+B(θt−1)Wt,

where θ′

t= (αt, βt, Rt), X′

t= (1, πt, πt−1, yt) and W′

t= (1, ut, εt),

A(θ) =

0 0 0 0

λα(1 −β2) + γa λβ20γρ

0 1 0 0

a0 0 ρ

,

B(θ) =

1 0 0

0 1 γ

0 0 0

0 0 1

.

As shown in Evans and Honkapohja (2001, p.186), A(θ) and B(θ) clearly satisfy the

Lipschitz conditions and Bis bounded. Since utand εtare assumed to have bounded

moments, condition (B.1) is satisﬁed. Furthermore, the eigenvalues of matrix A(θ) are 0

(double), ρand λβ2. According to the assumption |β| ≤ 1, 0 ≤λ < 1 and 0 < ρ < 1,

all eigenvalues of A(θ) are less than 1 in absolute value. Then it follows that there is a

compact neighborhood including the SCEE solution (α∗, β∗) on which the condition that

|A(θ)|is bounded strictly below 1 is satisﬁed.

Thus the technical conditions for Section 6.2.1 of Chapter 6 in Evans and Honkapohja

(2001) are satisﬁed. Moreover, since πtis stationary under the condition |β| ≤ 1, 0 ≤λ <

1 and 0 < ρ < 1, then the limits

σ2:= lim

t→∞ E(πt−α)2, σ2

ππ−1:= lim

t→∞ E(πt−α)(πt−1−α)

exist and are ﬁnite. Hence according to Section 6.2.1 of Chapter 6 in Evans and Honkapo-

hja (2001, p.126), the associated ODE is

dα

dτ = ¯π(α, β)−α,

dβ

dτ =σ−2[σ2

ππ−1−βσ2],

dR

dτ =σ2−R.

39

That is,

dα

dτ =λα(1 −β2) + γ¯y

1−λβ2−α=α(λ−1) + γ¯y

1−λβ2,

dβ

dτ =F(β)−β=γ2(λβ2+ρ) + λβ2(1 −ρ2)(1 −λβ2ρ)σ2

u

σ2

ε

γ2(λβ2ρ+ 1) + (1 −ρ2)(1 −λβ2ρ)σ2

u

σ2

ε

−β.

(H.1)

Hence a SCEE corresponds to a ﬁxed point of the ODE (H.1). Furthermore, the SAC

learning (αt, βt) converges to the stable SCEE (α∗, β∗) as time ttends to ∞.

References

[1] Adam, K., 2007. Experimental evidence on the persistence of output and inﬂation.

Economic Journal 117, 603-636.

[2] Assenza, T., Heemeijer, P., Hommes, C. and Massaro, D., 2011. Individual Expecta-

tions and Aggregate Macro Behavior. CeNDEF Working Paper, University of Amster-

dam.

[3] Berardi, M., 2007. Heterogeneity and misspeciﬁcation in learning. Journal of Economic

Dynamics & Control 31, 3203-3227.

[4] Branch, W.A., 2004. The theory of rationally heterogeneous expectations: evidence

from survey data on inﬂation expectations. Economic Journal 114, 592-621.

[5] Branch, W.A., 2006. Restricted perceptions equilibria and learning in macroeconomics,

in: Colander, D. (Ed.), Post Walrasian Macroeconomics: Beyond the Dynamic Stochas-

tic General Equilibrium Model. Cambridge University Press, New York, pp. 135-160.

[6] Branch, W.A., Evans, G.W., 2006. Intrinsic heterogeneity in expectation formation.

Journal of Economic Theory, 127, 264-295.

[7] Branch, W.A., Evans, G.W., 2007. Model uncertainty and endogenous volatility. Re-

view of Economic Dynamics 10, 207-237.

[8] Branch, W.A., McGough, B., 2005. Consistent expectations and misspeciﬁcation in

stochastic non-linear economies. Journal of Economic Dynamics & Control 29, 659-676.

[9] Brock, W., Hommes, C., 1998. Heterogeneous beliefs and routes to chaos in a simple

asset pricing model. Journal of Economic Dynamics & Control 22, 1235-1274.

40

[10] Bullard, J., Evans, G.W., Honkapohja, S., 2008. Monetary policy, judgment and

near-rational exuberance. American Economic Review 98, 1163-1177.

[11] Bullard, J., Evans, G.W., Honkapohja, S., 2010. A model of near-rational exuberance.

Macroeconomic Dynamics 14, 166-188.

[12] Bullard, J., Duﬀy, J., 2001. Learning and excess volatility. Macroeconomic Dynamics

5, 272-302.

[13] Evans, G.W., Honkapohja, S., 2001. Learning and Expectations in Macroeconomics.

Princeton University Press, Princeton.

[14] Fuhrer, J.C., 2006. Intrinsic and inherited inﬂation persistence. International Journal

of Central Banking 2, 49-86.

[15] Fuhrer, J.C., 2009. Inﬂation persistence. Working paper.

[16] Gali, J., Gertler, M., L´opez-Salido, J.D., 2001. European inﬂation dynamics. Euro-

pean Economic Review 45, 1237-1270.

[17] Guidolin, M., Timmermann, A., 2007. Properties of equilibrium asset prices under

alternative learning schemes. Journal of Economic Dynamics & Control 31, 161-217.

[18] Hamilton, J.D., 1994. Time Series Analysis. Princeton Univeristy Press.

[19] Heemeijer, P., Hommes, C.H., Sonnemans, J., Tuinstra, J., 2009. Price stability

and volatility in markets with positive and negative expectations feedback. Journal of

Economic Dynamics & Control 33, 1052-1072.

[20] Hommes, C.H., Sonnemans, J., Tuinstra, J., van de Velden, H., 2005. Coordination

of expectations in asset pricing experiments. Review of Financial Studies 18, 955-980.

[21] Hommes, C.H., 2011. The heterogeneous expectations hypothesis: some evidence for

the lab. Journal of Economic Dynamics & Control 35, 1-24.

[22] Hommes, C.H., Sorger, G., 1998. Consistent expectations equilibria. Macroeconomic

Dynamics 2, 287-321.

[23] Hommes, C.H., Rosser, J.B., 2001. Consistent expectations equilibria and complex

dynamics in renewable resource markets. Macroeconomic Dynamics 5, 180-203.

41

[24] Hommes, C.H., Sorger, G., Wagener, F., 2004. Consistency of linear forecasts in a

nonlinear stochastic economy. Unpublished working paper, University of Amsterdam.

[25] Lansing, K.J., 2009. Time-varing U.S. inﬂation dynamics and the new Keynesian

Phillips curve. Review of Economic Dynamics 12, 304-326.

[26] Lansing, K.J., 2010. Rational and near-rational bubbles without drift. Economic

Journal 120, 1149-1174.

[27] Lucas, R., 1972. Expectations and the neutrality of moeny. Journal of Economic

Theory 4, 103-124.

[28] Milani, F., 2007. Expectations, learning and macroeconomic persistence. Journal of

Monetary Economics 54, 2065-2082.

[29] Muth, J.E., 1961. Rational expectaions and the theory of price movements. Econo-

metrica 29, 315-335.

[30] Pfajfar, D., and B. Zakelj, 2010. Inﬂation Expectations and Monetary Policy Design:

Evidence from the Laboratory. Mimeo, Tilburg University.

[31] Sargent, T.J., 1991. Equilibrium with signal extraction from endogenous variables.

Journal of Economic Dynamics & Control 15, 245-273.

[32] Sargent, T.J., 1993. Bounded Rationality in Macroeconomics. Oxford University

Press Inc., New York.

[33] Sargent, T.J., 1999. The Conquest of American Inﬂation. Princeton University Press,

Princeton, NJ.

[34] S¨ogner, L., Mitl¨ohner, H., 2002. Consistent expectations equilibria and learning in a

stock market. Journal of Economic Dynamics & Control 26, 171-185.

[35] Tallman, E. W., 2003. Monetary policy and learning: some implications for policy

and research. Federal Reserve Bank of Atlanta, ECONOMIC REVIEW, Third Quater

2003.

[36] Timmermann, A., 1993. How learning in ﬁnancial markets generates excess volatility

and predictability in stock prices. Quarterly Journal of Economics 108, 1135-1145.

42

[37] Timmermann, A., 1996. Excess volatility and predictability in stock prices in autore-

gressive dividend models with learning. Review of Economic Studies 63, 523-557.

[38] Tuinstra, J., 2003. Beliefs equilibria in an overlapping generations model. Journal of

Economic Behavior & Organization 50, 145-164.

[39] White, H., 1994. Estimation, Inference and Speciﬁcation Analysis. Cambridge Uni-

versity Press, Cambridge.

43