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Learning under misspecification: a behavioral explanation of excess volatility in stock prices and persistence in inflation

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  • Shanghai Univeristy of Finance and Economics

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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.
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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. 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+ρyt1+εt,0ρ < 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
1ρ, the variance is σ2
y=σ2
ε
1ρ2and 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 xt1, xt2,···,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(xt1α) + 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(xt1α).(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(xt1α), 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)
Ext1[xtαβ(xt1α)] = E(xt1α)[xtαβ(xt1α)] = 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=Pt1
i=0(xiαt)(xi+1 αt)
Pt
i=0(xiαt)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
(xiαt)2,
then the SAC-learning is equivalent to the following recursive dynamical system (see
Appendix A).
αt=αt1+1
t+ 1(xtαt1),
βt=βt1+1
t+ 1R1
th(xtαt1)xt1+x0
t+ 1 t2+ 3t+ 1
(t+ 1)2αt11
(t+ 1)2xt
t
t+ 1βt1(xtαt1)2i,
Rt=Rt1+1
t+ 1ht
t+ 1(xtαt1)2Rt1i.
(2.8)
9
The actual law of motion under SAC-learning is therefore given by
xt=f(αt1+β2
t1(xt1αt1), 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 αt1+β2
t1(xt1αt1), 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 αt1+β2
t1(xt1αt1) 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)
e2,
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)
e2=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 ut0.
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
(R1)(Rρ)+ρ
Rρyt.(3.4)
In particular, if {yt}is i.i.d., i.e. a= ¯yand ρ= 0, then p
t=a
R1=¯y
R1for 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
(R1)(1 ρ)=¯y
R1,(3.5)
V ar(p
t) = E(p
tp)2=ρ2σ2
ε
(Rρ)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=
¯ye2zs
R1. 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
tp)(p
t1p) = ρ3σ2
ε
(Rρ)2(1 ρ2),
Corr(p
t, p
t1) = 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(pt1α) + δt.(3.8)
Given the perceived law of motion and knowledge of all prices observed up to period t1,
the 2-period ahead forecasting rule for pt+1 that minimizes the mean-squared forecasting
error is
pe
t+1 =α+β2(pt1α).(3.9)
By substituting (3.9) into (3.3), we obtain the implied actual law of motion for prices
pt=1
Rα+β2(pt1α) + a+ρyt,
yt=a+ρyt1+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
Rβ2.(3.11)
Imposing the first consistency requirement of a SCEE on the mean, i.e. ¯p=α, yields
α=¯y
R1=: 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, pt1) of the ALM satisfies
Corr(pt, pt1) = β2+
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+
ρ+R1 = (1ρ)(1R)
ρ+R=r(1ρ)
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(1ρ2)
(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 βis 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
(R1) =pand 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, εti.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
Rβ=β(βR)
R.
Since β1< 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, αis 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 βof 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 β0.997. 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
p2.5. Given the variance of fundamental prices
(3.6) and the variance of market prices (C.6),
σ2
p
σ2
p
=(β2ρ+R)(Rρ)2
(R2β4)(Rρβ2)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
pconverges 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
Rαt1+β2
t1(pt1αt1) + a+ρyt,
yt=a+ρyt1+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
(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
(d) Autocorrelation function in SCEE
Figure 1: (a) SCEE αis the intersection point of the mean ¯p=α(1β2)+¯y
Rβ2(bold curve)
with the perceived mean α(dotted line); (b) SCEE βis the intersection point of the
first-order autocorrelation coefficient F(β) = β2+
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, εti.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
(b) Ratio of variances in SCEE
Figure 2: (a) SCEE βwith 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 1β2
Rβ2in front of αin the unconditional mean in (3.11)
and by F(β) in (3.13). On one hand, since 0 1β2
Rβ2<1, it can be seen from (3.11) that
αis stable. On the other hand, the proof of uniqueness of βin Appendix D shows that
0< F (β)<1, and that therefore βis 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 αtα(1.0) under SAC-learning; (b) time series βtβ(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+ρyt1+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
1λρyt+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
tπ)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
tπ)(π
t1π) = γ2ρσ2
ε
(1 λρ)2(1 ρ2),
Corr(π
t, π
t1) = 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
t1) = 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(πt1α) + vt(4.5)
18
The implied actual law of motion then becomes
πt=λ[α+β2(πt1α)] + γyt+ut,
yt=a+ρyt1+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
1λβ2.(4.7)
Imposing the first consistency requirement on the mean, i.e. ¯π=λα(1β2)+γ¯y
1λβ2=α, we get
α=γ¯y
1λ=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, πt1) = 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+
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 0λ < 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
(1λ)(1ρ)=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(1λ2β4)
(λβ2ρ+1)+(1ρ2)(1λβ2ρ)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, εtN(0, σ2
ε), utN(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 βin 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 σuεbut not on their absolute values.
20
Tallman (2003). Furthermore, Figure 4d illustrates that inflation in the SCEE with high
βhas 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 βtogether
with the first-order autocorrelation coefficient of REE inflation as functions of ρ. For
0.84 < ρ < 0.918, two stable SCEE βexist separated by an unstable SCEE. The large
SCEE βis larger than the first-order autocorrelation coefficient of REE inflation, while
the small SCEE βis 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 βwith strong persistence.
If ρ > 0.918, there exists only one stable SCEE βwith 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(1λ2β4)
(λβ2ρ+1)+(1ρ2)(1λβ2ρ)1
γ2·σ2
u
σ2
ε
goes up. At γ= 0.05, a new SCEE β0.975
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 βwith the high stable βclose 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
(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)
(d) time series at (α, β) = (0.03,0.9961)
Figure 4: (a) SCEE αis the unique intersection point of the mean of inflation ¯π=
λα(1β2)+γ¯y
1λβ2(bold curve) with the perceived mean α(dotted line); (b) SCEE βis an
intersection point of the first-order autocorrelation of inflation γ2(λβ2+ρ)+λβ2(1ρ2)(1λβ2ρ)σ2
u
σ2
ε
γ2(λβ2ρ+1)+(1ρ2)(1λβ2ρ)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, εtN(0, σ2
ε), utN(0, σ2
u), a = 0.0004.
22
Figure 5: First-order autocorrelation coefficient of REE inflation (dotted real curve),
stable SCEE βwith 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[αt1+β2
t1(πt1αt1)] + γyt+ut,
yt=a+ρyt1+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(1β2)
1λβ2<1, it can be seen from (4.7) that αis 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
(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
(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
(f) γ= 0.1
Figure 6: SCEE βwith 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 λλβ2
1λβ2in 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) αt0.03
(b) βt0.3066
0 0.5 1 1.5 2
x 104
0.08
0.09
0.1
0.11
0.12
0.13
t
αt
(c) αt0.03
(d) βt0.9961
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)αt1+xtαt1]
=αt1+1
t+ 1[xtαt1].
Let
zt:= (x0αt)(x1αt) + ···+ (xt1αt)(xtαt)
= (x0αt11
t+ 1(xtαt1))(x1αt11
t+ 1(xtαt1)) +
···+ (xt1αt11
t+ 1(xtαt1))(xtαt11
t+ 1(xtαt1))
= (x0αt1)(x1αt1) + ··· + (xt2αt1)(xt1αt1)
+xtαt1
t+ 1 (2αt1x0x1+···+ 2αt1xt2xt1) + t1
(t+ 1)2(xtαt1)2
+t
t+ 1(xt1αt1)(xtαt1)t
(t+ 1)2(xtαt1)2
=zt1+1
t+ 1(xtαt1)[2(t1)αt1x02x1− · ·· − 2xt2xt1+t(xt1αt1)]
1
(t+ 1)2(xtαt1)2,
=zt1+1
t+ 1(xtαt1)[x0+ (t+ 1)xt1(t+ 2)αt1]1
(t+ 1)2(xtαt1)2
=zt1+ (xtαt1)hxt1+x0
t+ 1 t+ 2
t+ 1αt1+1
(t+ 1)2αt11
(t+ 1)2xti
=zt1+ (xtαt14,
where Φ4=xt1+x0
t+1 t2+3t+1
(t+1)2αt11
(t+1)2xt.
Write
nt:= (x0αt)2+ (x1αt)2+···+ (xtαt)2
= (x0αt11
t+ 1(xtαt1))2+···+ (xtαt11
t+ 1(xtαt1))2
= (x0αt1)2+ (x1αt1)2+···+ (xt1αt1)2+t+t2
(t+ 1)2(xtαt1)2
=nt1+t
t+ 1(xtαt1)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[nt1+t
t+ 1(xtαt1)2]
=t
t+ 1Rt1+t
(t+ 1)2(xtαt1)2
=Rt1+1
t+ 1ht
t+ 1(xtαt1)2Rt1i.
Furthermore,
βt=zt
nt
=βt1+1
ntnt1
[ztnt1zt1nt]
=βt1+1
ntnt1zt1+ (xtαt14nt1zt1nt1+t
t+ 1(xtαt1)2
=βt1+1
ntnt1(xtαt14nt1zt1
t
t+ 1(xtαt1)2
=βt1+1
nt(xtαt14βt1
t
t+ 1(xtαt1)2
=βt1+R1
t
t+ 1h(xtαt1)xt1+x0
t+ 1 t2+ 3t+ 1
(t+ 1)2αt1xt
(t+ 1)2t
t+ 1βt1(xtαt1)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 +···+ρn1a+ρnyt
Rn+···
=a+ρyt
R+a+ρa +ρ2yt
R2+···+
a(1ρn)
1ρ+ρnyt
Rn+···
=
X
n=1
a
1ρ1
Rnρ
Rn+
X
n=1 ρ
Rnyt
=a
1ρ1
R1ρ
Rρ+ρ
Rρyt
=aR
(R1)(Rρ)+ρ
Rρyt.(B.1)
C First-order autocorrelation coefficient of price
We rewrite (3.10) as
pt¯p=β2
R(pt1¯p) + ρ
Rεt+ρ2
Rεt1+···.(C.1)
Thus
E[(pt¯p)(pt1¯p)] = Ehβ2
R(pt1¯p)2+ρ
Rεt(pt1¯p) + ρ2
Rεt1(pt1¯p) + ···i
=β2
RE(pt1¯p)2+ 0 + ρ2
RE[εt1(pt1¯p)] + ···.(C.2)
E[(pt¯p)2]
=Ehβ2
R(pt1¯p)(pt¯p) + ρ
Rεt(pt¯p) + ρ2
Rεt1(pt¯p) + ···i
=β2
RE[(pt1¯p)(pt¯p)] + ρ
RE[εt(pt¯p)] + ρ2
RE[εt1(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[εt1(pt¯p)] + ···i
+ρ
RE[εt(pt¯p)] + ρ2
RE[εt1(pt¯p)] + ···
=β2
R2E[(pt¯p)2] + ρ(β2ρ+R)
R2E[εt(pt¯p)] + ρE[εt1(pt¯p)] + ···.
That is,
E[(pt¯p)2] = ρ(β2ρ+R)
R2β4E[εt(pt¯p)] + ρE[εt1(pt¯p)] + ···.(C.4)
In the following we will calculate E(εtkpt), k = 0,1,2,···.
E[εt(pt¯p)] = β2
RE[(pt1¯p)εt] + ρ
RE[ε2
t] + ρ2
RE[εt1εt] + ···
=ρ
Rσ2
ε.
E[εt1(pt¯p)] = β2
RE[(pt1¯p)εt1] + ρ
RE[εtεt1] + ρ2
RE[ε2
t1] + ···
=β2
RE[(pt¯p)εt] + ρ2
Rσ2
ε
=β2
R·ρ
Rσ2
ε+ρ2
Rσ2
ε.
E[εt2(pt¯p)] = β2
RE[(pt1¯p)εt2] + ρ
RE[εtεt2] + ρ2
RE[εt1εt2] + ρ3
RE[ε2
t2] + ···
=β2
RE[(pt¯p)εt1] + ρ3
Rσ2
ε
=β2
R2ρ
Rσ2
ε+β2
R·ρ2
Rσ2
ε+ρ3
Rσ2
ε.
E[εtk(pt¯p)] = β2
RE[(pt1¯p)εtk] + ρ
RE[εtεtk] + ρ2
RE[εt1εtk] + ···+ρk+1
RE[ε2
tk] + ···
=β2
RE[(pt¯p)εtk+1] + ρk+1
Rσ2
ε
=···
=ρ
Rσ2
ε·β2
Rk+ρ
Rσ2
ε·β2
Rk1·ρ+ρ
Rσ2
ε·β2
Rk2·ρ2+···
+ρ
Rσ2
ε·β2
R·ρk1+ρ
Rσ2
ε·ρk
=σ2
ε
R·
ρk+1 β2
Rk+1
1β2
ρR
.
32
Hence
E[εt(pt¯p)] + ρE[εt1(pt¯p)] + ···
=σ2
ε
ρR ·P
k=0 ρ2(k+1) P
k=0 β2ρ
Rk+1
1β2
ρR
=σ2
ε
ρR ·
ρ2
1ρ2β2ρ
Rρβ2
1β2
ρR
=ρσ2
ε
(1 ρ2)(Rρβ2).(C.5)
Substituting (C.5) into (C.4), we obtain
V ar(pt) = E[(pt¯p)2]
=ρ(β2ρ+R)
R2β4E[εt(pt¯p)] + ρE[εt1(pt¯p)] + ···
=ρ(β2ρ+R)
R2β4·ρσ2
ε
(1 ρ2)(Rρβ2)
=σ2
ερ2(β2ρ+R)
(R2β4)(1 ρ2)(Rρβ2).(C.6)
Furthermore, based on (C.2),
Corr(pt, pt1) = E[(pt¯p)(pt1¯p)]/V ar(pt)
=β2
R+ρ2
RV ar(pt)nE[εt(pt¯p)] + ρE[εt1(pt¯p)] + ···o
=β2
R+ρ2
R
ρσ2
ε
(1ρ2)(Rρβ2)
σ2
ερ2(ρβ2+R)
(R2β4)(1ρ2)(Rρβ2)
=β2
R+ρ(R2β4)
R(ρβ2+R)
=β2+
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)28ρβ2R(1 ρ2)
(ρβ2+R)3=2R(1 ρ2)(R3ρβ2)
(ρβ2+R)3.
Therefore, if ρR
3, then R3β2ρRβ2R > 0. Thus G′′(β) = F′′(β)>0.
Note that G(0) >0, G(0) = 1<0 and G(1) <0, G(1) = 2R(1ρ2)
(ρ+R)21. 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)21
<2pR/(3ρ)R(1 ρ2)
(R/3 + R)21
=33(1 ρ2)
81
<33(1 R2/9)
8pR2/31
=(R1)(R+ 9)
8R<0.
That is, G(β) is monotone. Therefore, in the case that 0 < ρ < 1, βis 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
Rαt1+β2
t1(pt1αt1) + a+ρyt,
yt=a+ρyt1+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(θt1)Xt1+B(θt1)Wt,
34
where θ
t= (αt, βt, Rt), X
t= (1, pt, pt1, yt) and W
t= (1, εt),
A(θ) =
0 0 0 0
a(1+ρ)+α(1β2)
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(ptα)2, σ2
pp1:= lim
t→∞ E(ptα)(pt1α)
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
= ¯p(α, β)α,
=σ2[σ2
pp1βσ2],
dR
=σ2R.
That is,
=α(1 β2) + ¯y
Rβ2α=+ ¯y
Rβ2,
=F(β)β=β2+
ρβ2+Rβ.
(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(ρk1a+···+ρa +a) +
X
k=0
γ(λρ)kyt+ut
=
X
k=1
γa
1ρ[λk(λρ)k] + γ
1λρyt+ut
=γλa
(1 λ)(1 λρ)+γ
1λρyt+ut.(F.1)
G First-order autocorrelation coefficient of inflation
We rewrite model (4.6) as
πt¯π=λβ2(πt1¯π) + γ(yt¯y) + ut,
yt¯y=ρ(yt1¯y) + εt.
(G.1)
That is,
πt¯π=λβ2(πt1¯π) + γρ(yt1¯y) + γεt+ut,
yt¯y=ρ(yt1¯y) + εt.
(G.2)
36
E[(πt¯π)(πt1¯π)]
=Ehλβ2(πt1¯π)2+γρ(πt1¯π)(yt1¯y) + γ(πt1¯π)εt+ (πt1¯π)uti
=λβ2V ar(πt) + γρE[(πt1¯π)(yt1¯y)] + γE[(πt1¯π)εt] + (πt1¯π)ut
=λβ2V ar(πt) + γρE[(πt1¯π)(yt1¯y)]
=λβ2V ar(πt) + γρE[(πt¯π)(yt¯y)].(G.3)
V ar(πt)
=E(πt¯π)2
=Ehλβ2(πt¯π)(πt1¯π) + γρ(πt¯π)(yt1¯y) + γ(πt¯π)εt+ (πt¯π)uti
=λβ2E[(πt¯π)(πt1¯π)] + γρE[(πt¯π)(yt1¯y)] + γ(πt¯π)εt+ (πt¯π)ut]
=λβ2E[(πt¯π)(πt1¯π)] + γρE[(πt¯π)(yt1¯y)] + γ2σ2
ε+σ2
u,(G.4)
where the last equation is based on the fact that E[(πt¯π)εt] = Ehλβ2(πt1¯π)εt+
γρ(yt1¯y)εt+γε2
t+utεti=γσ2
εand E[(πt¯π)ut] = Ehλβ2(πt1¯π)ut+γρ(yt1
¯y)ut+γεtut+u2
ti=σ2
u.
Based on (G.3) and (G.4),
V ar(πt) = λβ2E[(πt¯π)(πt1¯π)] + γρE[(πt¯π)(yt1¯y)] + γ2σ2
ε+σ2
u
=λβ2hλβ2V ar(πt) + γρE[(πt¯π)(yt¯y)]i+γρE[(πt¯π)(yt1¯y)] + γ2σ2
ε+σ2
u
=λ2β4V ar(πt) + λβ2γρE[(πt¯π)(yt¯y)] + γρE[(πt¯π)(yt1¯y)] + γ2σ2
ε+σ2
u.
That is,
V ar(πt) = λβ2γρE[(πt¯π)(yt¯y)] + γρE[(πt¯π)(yt1¯y)] + γ2σ2
ε+σ2
u
1λ2β4.(G.5)
Thus, in order to obtain E[(πt¯π)(πt1¯π)] and V ar(πt), we need calculate E[(πt
¯π)(yt¯y)] and E[(πt¯π)(yt1¯y)].
E[(πt¯π)(yt¯π)] = Ehλβ2(πt1¯π)(yt¯y) + γρ(yt1¯π)(yt¯y) + γεt(yt¯π) + ut(yt¯π)i
=λβ2E{(πt1¯π)[ρ(yt1¯y) + εt]}+γρE[(yt1¯π)(yt¯y)]
+γE{εt[ρ(yt1¯y) + εt]}+E[ut(yt¯π)]
=λβ2ρE[(πt1¯π)(yt1¯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)(yt1¯π)]
=Ehλβ2(πt1¯π)(yt1¯y) + γρ(yt1¯π)2+γεt(yt1¯π) + ut(yt1¯π)i
=λβ2E[(πt1¯π)(yt1¯y)] + γρE(yt1¯π)2+ 0 + 0
=λβ2·γσ2
ε
(1 ρ2)(1 λβ2ρ)+γρ ·σ2
ε
1ρ2
=γσ2
ε
(1 ρ2)λβ2
1λβ2ρ+ρ
=γσ2
ε
(1 ρ2)·λβ2(1 ρ2) + ρ
1λβ2ρ
=γσ2
ε
(1 λβ2ρ)hλβ2+ρ
1ρ2i.(G.7)
Therefore, based on (G.5), (G.6) and (G.7),
V ar(πt) = 1
1λ2β4nλβ2γρE[(πt¯π)(yt¯y)] + γρE[(πt¯π)(yt1¯y)] + γ2σ2
ε+σ2
uo
=1
1λ2β4nλβ2γ2ρσ2
ε
(1 ρ2)(1 λβ2ρ)+γ2ρσ2
ε
(1 λβ2ρ)hλβ2+ρ
1ρ2i+γ2σ2
ε+σ2
uo
=σ2
ε
1λ2β4nγ2ρλβ2(2 ρ2) + ρ
(1 ρ2)(1 λβ2ρ)+γ2+σ2
u
σ2
εo
=σ2
ε
1λ2β4nγ2(λβ2ρ+ 1)
(1 ρ2)(1 λβ2ρ)+σ2
u
σ2
εo.(G.8)
According to (G.3),
E[(πt¯π)(πt1¯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, πt1) satisfies
Corr(πt, πt1) = E[(πt¯π)(πt1¯π)]/V ar(πt)
=λβ2+
γ2ρσ2
ε
(1ρ2)(1λβ2ρ)
σ2
ε
1λ2β4nγ2(λβ2ρ+1)
(1ρ2)(1λβ2ρ)+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(θt1)Xt1+B(θt1)Wt,
where θ
t= (αt, βt, Rt), X
t= (1, πt, πt1, 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(πtα)2, σ2
ππ1:= lim
t→∞ E(πtα)(πt1α)
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
= ¯π(α, β)α,
=σ2[σ2
ππ1βσ2],
dR
=σ2R.
39
That is,
=λα(1 β2) + γ¯y
1λβ2α=α(λ1) + γ¯y
1λβ2,
=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 .
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43
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