Behavioral heterogeneity and the New Keynesian Phillips curve
Thesis MSc. Econometrics by Lisanne Cock
April 14, 2014
Supervisor: Prof. Dr. C.H. Hommes Second marker: Dr. D. Massaro
Acknowledgements
I would like to thank my supervisors Cars Hommes and Domenico Massaro for their useful feedback on my thesis. Especially, I would like to thank Domenico for his suggestions and comments, and for his patience with my MATLAB programming skills.
Furthermore I would like to thank my friends and family for being there for me and supporting for all these years. Last but not least, I want to thank Gerdie Knijp for her support during this project, and even more so for laughing and crying together during our five years as econometrics students.
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Introduction
The rational expectation hypothesis has been a widely accepted expectations formation sys-tem for modeling prices and inflation since the earliest models by Lucas (1972) and Sargent and Wallace (1975). In a Rational Expectations Equilibrium (REE) agents are assumed to know the market equilibrium and the parameter values (Evans and Honkapohja, 2001). In recent research, the REE is criticized for this demanding assumption, and less demanding alternatives have been proposed. One alternative for the REE is the behavioral approach of the Restricted Perceptions Equilibrium (RPE), where agents are assumed to have misspecified beliefs, but are unable to detect this misspecification within the context of their forecasting model (Evans and Honkapohja, 2001).
An example of this RPE is the Stochastic Consistent Expectations Equilibrium (SCEE) as introduced in Hommes et al. (2013). Agents form misspecified beliefs about the actual law of motion (ALM), but the unconditional sample mean and autocovariances of agent’s perceived law of motion coincide with the ALM. Hommes and Zhu (2014) discuss the simplest example of the SCEE where agents use a univariate linear forecasting rule. Agents do not fully recognize the true structure of the economy, but learn the best univariate forecasting rule in the long run. As a result the unconditional sample mean and the first order sample autocorrelation of the true and perceived law of motion coincide, agents are unable to distinguish between the true and perceived process and therefore have no reason to alter their behavior (Hommes et al.,
2013).1 Consequently, the forecasting errors are uncorrelated and consistent with the linear
model. These features are often mentioned in favor of the REE, but are also characteristics of this SCEE.
Hommes and Zhu (2014) use the SCEE and the simple univariate forecast rule to examine an asset pricing model and the New Keynesian Phillips Curve. For the latter application they find that for a large range of initial values of inflation, the boundedly rational agents converge to a stable high persistence SCEE. This finding is in line with research that shows that inflation tends to be persistent (exhibiting periods of near unit root) and is subject
1We assume that agents do not perform higher order analysis, in which case they would be able to detect
to regime shifting (periods of high and low persistence in inflation). Moreover, the result is consistent with a previous empirical finding by Adam (2007), who shows that a RPE provides a better explanation than a REE for the observed persistence of inflation, and Fuhrer (2009) and Cornea et al. (2012), who show that realized high persistence in inflation is not well described by the rational expectation hypothesis. In this thesis we shall built on the obtained results by Hommes and Zhu (2014) by using their framework and its application to the New Keynesian Phillips Curve (NKPC henceforth).
The NKPC relates expected one-period ahead inflation to current inflation and the output gap. Instead of assuming that all agents have the expectations as in Hommes and Zhu (2014), we will assume that the group is heterogeneous. Heterogeneity in individual expectations for inflation analysis is supported by various papers, such as Cornea et al. (2012), Branch (2004), Mankiw et al. (2003) and Pfajfar and Santoro (2010). Moreover, it seems quite intuitive to say that a combination of two forecast rules is likely to outperform any of the individual forecasts (Deutsch et al., 1994). In this paper two heuristics will be considered. The first rule is the simple linear rule as introduced in Hommes and Zhu (2014). It is assumed that agents behave like econometricians when they estimate the parameters of this rule. More specific, this means that they learn the parameter values in the long run using sample autocorrelation learning (SAC-learning). For a second forecast rule, fundamental expectations are considered, as proposed in a behavioral learning setting by Brock and Hommes (1997) and Cornea et al. (2012). As with rational expectations, fundamentalists recognize the structure of the process - they believe that future inflation depends on the discounted sum of expected future marginal
costs2 - but, unlike agents using rational expectations, they fail to acknowledge the presence
of other types of agents. This means that in case of homogeneous expectations, fundamental and rational expectations coincide.
Fuster et al. (2010) use a similar combination of heuristics - an “intuitive” linear rule and rational expectations - in a macroeconomic model. They use the weighted average of intuitive and rational expectations and define this to be natural expectations. They assume that each agent uses a weighted average of each heuristic with fixed proportions, and do not allow this
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fraction to vary over time. They find that natural expectations fail to account for the ending of a period of high persistence in inflation.
Recent research by Cornea et al. (2012) shows that the assumption by Fuster et al. (2010), i.e. agents use a fixed proportion of each heuristic, is overly restrictive. They find statistical evidence for both heterogeneous expectations and evolutionary switching in U.S. inflation data. In an evolutionary switching mechanism the past performance of a heuristic determines the fraction of agents using this rule in each period. Cornea et al. (2012) put forward several papers in which evidence is provided supporting usage of past forecast errors for an updating
process of the proportions of heterogeneous forecasters.3 Also, endogenous modeling of
het-erogeneous expectations is suggested by for example Brock and Hommes (1997). In addition to the past forecast errors, it is common in the literature to assume a small positive cost for heuristics requiring knowledge about the model. For the demanding fundamental heuristic we will therefore take into account the possibility of introducing this cost. This means that predictor choice will be based upon past performance as well as information costs.
The dynamics of the heuristics considered in this paper have been examined individually, but not yet in a heterogeneous framework. This means that we do know what happens in two extreme cases; all agents using either the simple linear rule or the fundamental heuristic. For the first case Hommes and Zhu (2014) find convergence to high persistence SCEE using SAC-learning for a large range of initial values of inflation. Secondly, Adam (2007) shows that under fundamental expectations low persistence steady states are obtained. The aim of this paper is to see what happens if we introduce an evolutionary switching system that allows agents to switch between these heuristics. Therefore the key question is:
Does an evolutionary switching mechanism between a simple univariate linear rule and a fundamental rule lead to coordination of agents on a high inflation persistence regime?
The rest of this paper is structured as follows. In Section 2 the model will be introduced. In the next section the dynamics of this model will be examined for two cases; (i) with fixed coefficients for the linear forecasting rule and (ii) with time varying coefficients for the linear forecasting rule, updated according to SAC-learning. For the first case we will investigate
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the influence of the intensity of choice and costs parameters on the dynamics. In the second case we will examine if SAC-learners and fundamentalists can coordinate towards a high persistence regime. Finally, in Section 4, some concluding remarks will be drawn.
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The model
2.1 New Keynesian Phillips curve
Hommes and Zhu (2014) apply the concept of SCEE and SAC-learning to the New Keynesian Phillips curve (NKPC). This curve relates inflation at time t to the expected one-period ahead inflation the output gap or real marginal costs. Underlying this framework is the assumption of imperfect competition in the goods market, since different prices are set by each firm as they produce differentiated goods. Furthermore, it is assumed that only a fraction of firms can reset their prices in a given period (Gal´ı, 2008). The implied inflation dynamics as derived by Calvo (1983), look as follows,
πt= δ Etπt+1+ κϕt+ ut, (2.1)
where πtis inflation at time t, Etπt+1 the expected one period ahead inflation, ϕtthe output
gap or the firm’s real marginal costs, δ represents the time discount factor (0 ≤ δ < 1),
and κ the degree of price stickiness.4 It is assumed that inflation is driven by an exogenous
AR(1) process, representing output gap or real marginal costs. Lansing (2009) argues that the theoretical results do not depend on the interpretation of the exogenous driving process, only in empirical applications this choice should be substantiated. Using the relationship
ϕt= kyt in Eq. (1.1) gives us the following system of equations
πt= δπet+1+ γyt+ ut,
yt= a + ρyt−1+ t,
(2.2)
where πet+1is the one period ahead subjective expected inflation, ytis the output gap, driven
4The derivation of Eq. (1) can be found in Hommes and Zhu (2014) as well as the justification of using the
by the exogenous AR(1) process, and γ measures the strength of the dependence of inflation on this exogenous driving process. The persistence of the AR(1) process is described by
ρ ∈ [0, 1). Both ut and t are i.i.d. stochastic disturbances with mean zero and variances σ2u
and σ2 respectively, with finite absolute moments.
2.2 Heuristics
Hommes and Zhu (2014) assume that the one period ahead inflation expectations are formed by boundedly rational agents. These agents do not know the underlying model and the pa-rameters driving inflation, but behave like econometricians and use a simple linear forecasting rule. Literature suggests heterogeneity in inflation analysis and fundamental expectations in a behavioral learning setting. In addition to the agents using the linear forecasting rule, we will therefore introduce another group of boundedly rational agents using a fundamental forecast.
Linear forecasting rule
The first group of boundedly rational agents behave like econometricians, meaning that they apply statistical rules to forecast inflation. They use a simple univariate forecasting rule a as in Hommes and Zhu (2014). For this group the expected one period ahead inflation forecast at time t for time t + 1 is given by
Eatπt+1= α + β2(πt−1− α), (2.3)
where α is the unconditional mean and β ∈ (−1, 1) the first-order autocorrelation coefficient. Hommes and Zhu (2014) take these parameters fixed, but show numerically that the param-eters converge to these fixed values under sample autocorrelation learning (SAC-learning). SAC-learning is a statistical learning procedure that assumes agents update the parameters of their forecast rule over time, as new information becomes available (Hommes and Sorger, 1998). By using this learning mechanism, agents learn the optimal misspecified model in the long run (Hommes and Zhu, 2014). The mean and first-order sample autocorrelation of the true and misspecified model will coincide, disabling agents to detect their misspecification.
Initially, these parameters will be taken fixed in this paper, but in a second analysis this assumption will be relaxed and they will be allowed to vary over time.
In order to obtain the SAC-learning mechanism, the sample average and first-order sample autocorrelation are specified as
αt= 1 t + 1 t X i=0 πi, and βt= Pt−1 i=0(πi− αt)(πi+1− αt) Pt i=0(πi− αt)2 ,
respectively. Now define
Rt= 1 t + 1 t X i=0 (πi− αt)2,
then the following recursive dynamical system is obtained5
αt= αt−1+ 1 t + 1(πt− αt−1), βt= βt−1+ 1 t + 1R −1 t (πt− αt−1)(πt−1+ π0 t + 1 − t2+ 3t + 1 (t + 1)2 αt−1− 1 (t + 1)2πt) − t t + 1βt−1(πt− αt−1) 2 , (2.4) Rt= Rt−1+ 1 t + 1 t t + 1(πt− αt−1) 2− R t−1 . Fundamentalists
Agents using a fundamental forecast rule assume that the forcing variable yt is driven by an
exogenous process
yt= a + ρyt−1+ t,
and they know the corresponding parameter values, but fail to recognize the presence of other
agents in the economy. The one-period ahead expectations are given by6
Eft πt+1=
γa
(1 − δ)(1 − δρ) + γ(1 − δρ)
−1ρy
t. (2.5)
5The derivation of this system can be found in Appendix A of Hommes and Zhu (2014). 6
2.3 Evolutionary switching
Instead of assuming that a fixed fraction of agents uses a certain heuristic, we will assume that agents can switch between the heuristics. The past performance of the heuristics determines the decisions of agents. In order to determine the past performance of each heuristic it is assumed that agents calculate the past forecast error. Defining the squared forecast error of heuristic h as
F Eth= (Eht−1πt− πt)2, (2.6)
the evolutionary fitness measure of heuristic h at time t is defined as
Uh,t = − F Eth PH j F E j t − Ch. (2.7)
The constant Chrepresents the cost associated with heuristic h, incorporating cognitive effort
and information gathering costs (Massaro, 2012).
Assumption 1. Cf ≥ 0
This first assumption is made since fundamentalists are assumed to recognize that the model is driven by an exogenous driving process and know the corresponding parameter values, which are quite demanding assumptions. Also, in literature it is common to assume small positive costs for sophisticated predictors such as rational expectations (Brock and Hommes, 1997).
Assumption 2. Ca= 0
The rational for the second assumption is that agents applying a linear rule do not require a lot of information about the true model, as they simply estimate an AR(1) model to past observations.
As in Brock and Hommes (1997) the mechanism that describes switching between
heuris-tics is a discrete choice model, where a fraction of nh,t agents use heuristic h at time t. This
fraction uses Eq (2.7), and looks as follows
nh,t =
exp(λUh,t−1)
PH
j=1exp(λUj,t−1)
where the parameter λ ∈ [0, ∞) represents the intensity of choice. This parameter measures the sensitivity of agents towards the past performance of the heuristics. If λ = 0, agents cannot
switch between heuristics and each strategy is adopted with probability nh,t = 1/H. In our
case this would imply that nh,t = 1/2, meaning that half of the agents use the fundamental
rule and half of the agents use the linear rule to forecast inflation. If λ → ∞ then in each period all agents switch to the best performing predictor of inflation.
2.4 Actual law of motion
We now have a New Keynesian framework in which two agents compete based on evolutionary switching. The dynamics of the framework are determined by the forecasts of agents, the exogenous driving process and a random term. The ALM can be obtained by combining the equations as introduced in this section:
πt= δ[na,tEatπt+1+ (1 − na,t) Eft πt+1] + γyt+ ut,
yt= a + ρyt−1+ t, Eat πt+1= α + β2(πt−1− α), Eft πt+1= γa (1 − δ)(1 − δρ) + γ(1 − δρ) −1ρy t, na,t= exp(λUa,t−1) PH h=1exp(λUh,t−1) , Uh,t−1= − (Eht−2πt−1− πt−1)2 PH j (E j t−2πt−1− πt−1)2 − Ch.
2.5 Sample mean and first-order autocorrelation
We know that in equilibrium the sample mean of the system in (2.2) with only SAC-learners is equal to the unconditional mean of the REE, and given by
α∗ = γa
(1 − δ)(1 − ρ). (2.9)
Since for both SAC-learners as well as for fundamentalists the equilibrium value of the un-conditional sample mean is given by Eq. (2.9), any combination of these heuristic will yield
the same unconditional sample mean.
In order to derive an analytical expression of the first-order ACF for our heterogeneous
framework we will treat na as a parameter, instead of including the switching mechanism.
Then, the ACF is given by7
Fn(β) = nδβ2+ γ2ρ(1 − nδρ)2(1 − n2δ2β4) γ2(1 − nδρ)2(nδρβ2+ 1) + (1 − δρ)2(1 − nδρβ2)(1 − ρ2)σu2 σ2 , (2.10)
where n = na is used for simplicity.
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Numerical Analysis
In this section we will analyze our framework, as introduced in the previous section, by simulating the time series for several initial values. We need to specify the difference between high and low persistence inflation regimes, before we can examine these regimes for both fixed coefficients as well as time varying coefficients.
3.1 High and low persistence regimes
In order to compare our dynamics with those found in Hommes and Zhu (2014) we will fix
the parameters analogously to δ = 0.99, γ = 0.075, a = 0.0004, ρ = 0.9, σ = 0.01 and
σu = 0.003162. This means we have t ∼ N (0, σ2) and ut ∼ N (0, σu2) and
σ2 u
σ2
= 0.1. With
these parameters, Hommes and Zhu (2014) find two stable SCEEs. They find
• convergence of the system to a low persistence stable steady state (α∗, βL∗) = (0.03, 0.3066)
for the initial conditions (π0, y0) = (0.028, 0.01);
• and convergence of the system to a high persistence stable steady state
(α∗, βH∗) = (0.03, 0.9961) for the initial conditions (π0, y0) = (0.1, 0.15).
A third SCEE with β∗ = 0.7417 is shown to be unstable. Let us define the autocorrelation
implied by homogeneous fundamental expectations, as βF. Given the parameter values, we
know from (2.10) (with n = 0) that βF∗ = 0.8653 for all t. Hommes and Zhu (2014) talk
about a low persistence steady state when they refer to βL∗, but this persistence is lower than
the persistence implied by the REE, corresponding to βF. In order to be able to make a
clear distinction between the two levels of persistence in following analysis, from now on, we
refer to βF if we talk about a low persistence inflation regime, and talk of an extremely low
persistence regime for βL∗.
3.2 Fixed coefficients
3.2.1 High persistence
Let us first consider the case with initial conditions for which Hommes and Zhu (2014) find convergence to the high persistence SCEE. The costs for the fundamental forecast are assumed
to be zero, Cf = 0 . Figure 3.1 shows realized inflation for increasing values of the intensity
of choice. From these figures it can be observed that, as we increase the intensity of choice more agents adopt the linear forecast strategy, and as a result the persistence of the time series increases. For λ = 1, we find that the first-order autocorrelation coefficient of the inflation process is 0.9597, and as we increase the intensity of choice to λ = 10, we find that
the autocorrelation gets close to βH∗ , the high persistence steady state from Hommes and Zhu
0 50 100 150 200 250 300 −0.02 0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 t πt 0 50 100 150 200 250 300 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 t n a
HE with fixed coefficients REE (a) λ = 1, AC = 0.9597 0 50 100 150 200 250 300 −0.05 0 0.05 0.1 0.15 0.2 t π t
HE with fixed coefficients REE 0 50 100 150 200 250 300 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 t n a (b) λ = 5, AC = 0.9909 0 50 100 150 200 250 300 −0.1 −0.05 0 0.05 0.1 0.15 0.2 0.25 t πt
HE with fixed coefficients REE 0 50 100 150 200 250 300 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 t n a (c) λ = 10, AC = 0.9925
Figure 3.1: Inflation and evolution of fractions for β = 0.9961 for increasing values of the intensity of choice parameter λ.
3.2.2 Extremely low persistence
Now let β correspond to the extremely low persistence steady state, i.e. β = β∗L. We expect
the persistence of the time series to be between the extremely low steady state βL∗ and the
low persistence corresponding to βF. Figure 3.2 shows that this indeed holds true, since
persistence in the HE is slightly lower than persistence implied by the REE. If we increase the intensity of choice λ, the number of agents using a linear heuristic increases, and as a consequence the first-order autocorrelation of the HE decreases.
0 50 100 150 200 250 300 −0.02 −0.01 0 0.01 0.02 0.03 0.04 0.05 0.06 t πt
HE with fixed coefficients REE 0 50 100 150 200 250 300 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 t n a (a) λ = 1, AC = 0.8510 0 50 100 150 200 250 300 −0.02 −0.01 0 0.01 0.02 0.03 0.04 0.05 0.06 t πt
HE with fixed coefficients REE 0 50 100 150 200 250 300 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 t na (b) λ = 10, AC = 0.8489
Figure 3.2: Inflation and evolution of fractions for β = 0.3066 for increasing values of the intensity of choice parameter λ.
3.3 Time varying coefficients
We have shown, that if we fix β = βH∗, the heterogeneous framework exhibits periods of high
persistence in inflation. Now, let us assume that agents do not fix α and β but learn the best possible forecast rule in the long run by updating these coefficients each period.
3.3.1 Convergence of coefficients
Let us first examine the case with initial values, π0 and y0, equal to the values for which
Hommes and Zhu (2014) find convergence towards a high persistence inflation regime. Figure
3.3 shows that if we fix the initial values of ytand πt and increase the intensity of choice, the
system seems to convergence towards a high persistence steady state. However, if we increase
our time frame, we observe that after a period in the high persistence regime, βtslowly starts
to decrease.
This behavior can be explained if we analyse realisations of first-order ACF for different
values of na. Figure 3.4 shows there are many adjacent steady states between the low
persis-tence steady state βF, implied by fundamentalists, and the high persistence steady state βH∗,
implied by the SAC-learners. The system does not lock into the high persistence steady state, as the error terms of the stochastic equations cause the system to jump into another basin of attraction corresponding to a different steady state. As a result, the SAC-learners will update their parameter values corresponding to this lower persistence steady state, causing the system to converge away from the high persistence steady state in the long run, to a
0 1 2 3 4 5 x 105 0 0.05 0.1 0.15 t αt 0 1.25 2.5 3.75 5 x 105 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 t βt
(a) αtand βt for λ = 1
0 1 2 3 4 5 x 105 0 0.05 0.1 0.15 t α t 0 1.25 2.5 3.75 5 x 105 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 t β t (b) αtand βtfor λ = 10
Figure 3.3: Convergence of αt and βt depends on the intensity of choice. For both (a) and
(b), π0= 0.1 and y0 = 0.15. The first-order autocorrelation coefficients corresponding to the
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 β F n(β) n=0 n=0.1 n=0.2 n=0.3 n=0.4 n=0.5 n=0.6 n=0.7 n=0.8 n=0.9 n=1
Figure 3.4: The first-order autocorrelation for several values of n = na and the perceived
first-order autocorrelation (black line). For n = 1 we can see the ACF for the SAC-learners and for n = 0 the ACF coincides with the first-order autocorrelation in a REE.
Numerous simulations (not shown) show that when dynamics fail to converge to a steady
state between βF and β∗H, αt does not converge to α∗ in the long run. An example of this
behavior was shown in Figure 3.5b, where βtstays between βL∗ and βF. Correspondingly, the
time series exhibits lower persistence in inflation than persistence in the REE.
Figure 3.5 shows that convergence of the system is influenced by the initial values π0 and
y0. For the same values of λ, the time series can temporarily stay in a high persistence steady
state and after this period slowly converge towards a lower persistence steady state, as seen in Figure 3.3b, or it can converge to a low persistence steady state immediately as in Figure
3.5a, if we take a low initial value for the exogenous driving process, y0. Multiple simulations
(again, not shown) show that decreasing the initial value of y0, from 0.15 to zero (and fixing
π0 = 0.1), causes the system to converge to a steady state between βF and βH∗ . Moreover,
persistence lower than the persistence corresponding to a REE. It turns out that in this case, the unconditional sample mean fails to converge to the equilibrium value in the long run, and remains high around 0.75.
0 1 2 3 4 5 x 105 0 0.05 0.1 0.15 t αt 0 1.25 2.5 3.75 5 x 105 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 t βt
(a) αt and βtfor π0= 0.1 and y0= 0.01
0 1 2 3 4 5 x 105 0 0.05 0.1 0.15 t αt 0 1.25 2.5 3.75 5 x 105 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 t βt
(b) αt and βt for π0= 0.028 and y0= 0.15
Figure 3.5: Convergence of αt and βt depends on initial values π0 and y0. For both (a) and
(b), λ = 10. The first-order autocorrelation coefficients corresponding to the time series are (a) AC = 0.8831 and (b) AC = 0.6855.
3.3.2 Introducing a fundamental cost
So far, we have assumed that the cost for the fundamental predictor was equal to zero. Since
we assumed that Cf ≥ 0, we will examine what happens if we introduce a small cost, i.e. let
Cf > 0. First, we introduce a cost into a framework with initial values corresponding to the
initial values in Figure 3.5b, so we let π0= 0.028 and y0 = 0.15. For these initial values,
case. In case of heterogeneous expectations β stays between βL∗ and βF. Introducing a cost
for the fundamental predictor, leads to more SAC-learners, and consequently, the system will
still converge towards a low persistence regime between βL∗ and βF.
0 1.25 2.5 3.75 5 x 106 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 t βt (a) Cf = 0 0 1.25 2.5 3.75 5 x 106 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 t βt (b) Cf = 0.25
Figure 3.6: Convergence of βt for π0 = 0.1, y0 = 0.15 and λ = 10. The first-order
autocor-relation coefficients corresponding to the inflation time series are (a) AC = 0.9664 and (b) AC = 0.9862.
Now let us re-examine the dynamics observed in Figure 3.3b, where π0= 0.1 and y0 = 0.15.
There, inflation stayed in the high persistence region temporarily, but slowly converged away
from β∗H. Figure 3.6b shows that if we introduce a small cost, βtremains in the high persistence
region for a longer period of time, but will ultimately converge to a lower persistence regime as well. Introducing a cost merely slows down convergence towards a steady state between
βF and βH∗. This means that ultimately, the system does not lock into the high persistence
steady state.
4
Conclusion
In this paper the research from Hommes and Zhu (2014), on the application of behavioral learning equilibria to the New Keynesian Phillips curve, is extended with the introduction heterogeneous expectations. The NKPC relates current inflation to one-period ahead inflation and an exogenous driving process. Hommes and Zhu (2014) find that agents can coordinate towards a high persistence inflation regime, using a simple linear rule to do the one-period
ahead forecast. In this paper it was examined whether a system with two types of boundedly rational agents who are allowed to switch between heuristics, can also converge towards a high persistence inflation regime. Agents estimating a simple linear model using past realized inflation, so-called SAC-learners, are assumed to update the parameters of their model each period according to a sample autocorrelation learning mechanism (SAC-learning). In addition to these SAC-learners, we considered fundamental agents who recognize that inflation is driven by an exogenous driving process, but fail to realize that there are other agents who do not recognize this. Agents were allowed to switch between heuristics and based their choice of heuristic on its past performance.
We examined the two stable steady states found by Hommes and Zhu (2014), a high and low persistence steady state, in a framework with fixed coefficients. After fixing the perceived autocorrelation coefficient of the simple linear forecast to the high persistence steady state, an increase in the intensity of choice led to an increase in the persistence of the HE. The dynamics of the HE were similar to the dynamics in the homogeneous case with only SAC-learners.
When SAC-learning was introduced in the framework, dynamics temporarily stayed in a high persistence inflation regime. In the neighborhood of the high persistence steady state there are many steady states corresponding to lower persistence in inflation. The ACF showed that, for each ratio of fundamentalists and SAC-learners, there is a steady state between the low persistence steady state implied by fundamentalists and the high persistence steady state implied by SAC-learners. Due to the stochastic nature of the system, it is therefore inevitable that at some point the system jumps into the basin of attraction of one of the lower persistence steady states, that lie between the high and low persistence steady state. By introducing a cost for the fundamental heuristic, convergence towards a lower persistence steady state can be slowed down, yielding inflation to exhibit long periods of high persistence. Nevertheless, agents ultimately fail to coordinate towards a high persistence inflation regime in the long run.
Future research
In future research it would be interesting to apply the framework to real inflation data. Some initial research, using U.S. inflation data, shows that the SAC-learners outperform the
fun-damentalists, and consequently agents only use the SAC forecasting rule. This can easily be explained by the structure imposed by the model we used, where we assume the exoge-nous driving process to be a simple AR(1) process. If we take data on output gap or labour share of income, which serves as a proxy for marginal costs, and estimate an AR(1) model, the optimal fundamental forecast turns out to be zero. Taking the parameters as used in our framework, results in a poor fundamental forecast and consequently all agents use the SAC-learning heuristic after a few periods. Therefore, it would be interesting to follow the approach taken by Cornea et al. (2012), who estimate an optimal VAR model to various possible exogenous driving variables. They use this VAR model for the fundamental forecast in a heterogeneous framework with naive expectations. Additional to using SAC-learners instead of naive agents, it would be interesting to introduce dynamic lag selection, so that an optimal VAR model is estimated each period. This framework would then no longer be in line with the framework of this paper, but it would be an interesting way to compare the heterogeneous framework with SAC-learners and fundamentalists to the homogeneous case with only SAC-learners.
In line with the previous remark of estimating an optimal VAR each period to variables driving inflation, it would be interesting to extend this to the SAC forecasting rule. One could introduce “less” boundedly rational agents by assuming that each period they estimate the optimal AR(p) model to the data. In a homogeneous setting, it would be interesting to see whether agents are able to learn higher order terms in the long run.
Appendix
A
Derivation of fundamental expectations
In order to find the one-period ahead fundamental forecast, expectations of the one-period ahead rational expectations equilibrium can be taken. The rational expectations equilibrium is given by πt∗ = γδa (1 − δ)(1 − δρ)+ γ 1 − δρyt+ ut. 8 (A.1)
At time t + 1, Eq. (A.1), is given by πt+1= γδa (1 − δ)(1 − δρ)+ γ 1 − δρyt+1+ ut+1. (A.2)
Taking expectations we obtain
Eftπt+1= γδa (1 − δ)(1 − δρ)+ γ 1 − δρEtyt+1, = γδa (1 − δ)(1 − δρ)+ γ 1 − δρ(a + ρyt), = γa (1 − δ)(1 − δρ)+ γ(1 − δρ) −1ρy t. (A.3)
B
Derivation of first-order autocorrelation function
We substitute equations (2.3) and (2.5) into Eq. (2.2) and rewrite it as
πt− ¯π = nδβ2(πt−1− ¯π) + δ(1 − n)
γρ
1 − δρ(yt− ¯y) + γ(yt− ¯y) + ut,
yt− ¯y = ρ(yt−1− ¯y) + t,
(B.1)
where we let n = na for convenience. We can rewrite the first expression from Eq. (B.1) by
plugging in the second expression
πt− ¯π = nδβ2(πt−1− ¯π) + δ(1 − n)γρ 1 − δρ ρ(yt−1− ¯y) + δ(1 − n)γρ 1 − δρ t+ γρ(yt−1− ¯y) + γt+ ut, = nδβ2(πt−1− ¯π) + δ(1 − n)γρ2 1 − δρ + γρ (yt−1− ¯y) + δ(1 − n)γρ 1 − δρ + γ t+ ut, = nδβ2(πt−1− ¯π) + γρ(1 − nδρ) 1 − δρ (yt−1− ¯y) + γ(1 − nδρ) 1 − δρ t+ ut. (B.2)
We know that the first-order autocorrelation of πt is given by
Corr(πt, πt−1) =E[(πt
− ¯π)(πt−1− ¯π)]
V ar(πt)
First, let us derive an expression for the numerator E[(πt− ¯π)(πt−1− ¯π)] = E nδβ2(πt−1− ¯π)(πt−1− ¯π) + γρ(1 − nδρ) 1 − δρ (yt−1− ¯y)(πt−1− ¯π) +γρ(1 − nδρ) 1 − δρ t(πt−1− ¯π) + ut(πt−1− ¯π) , = nδβ2V ar(πt) + γρ(1 − nδρ) 1 − δρ E[(yt−1− ¯y)(πt−1− ¯π)] +γρ(1 − nδρ) 1 − δρ E[t(πt−1− ¯π)] + E[ut(πt−1− ¯π)], = nδβ2V ar(πt) + γρ(1 − nδρ) 1 − δρ E[(yt−1− ¯y)(πt−1− ¯π)]. (B.4)
To obtain this expression we need (i) V ar(πt), and (ii) E[(πt− ¯π)(yt− ¯y)]. First
V ar(πt) = E(πt− ¯π)2, = E nδβ2(πt−1− ¯π)(πt− ¯π) + γρ(1 − nδρ) 1 − δρ (yt−1− ¯y)(πt− ¯π) +γ(1 − nδρ) 1 − δρ t(πt− ¯π) + ut(πt− ¯π) , = nδβ2E[(πt−1− ¯π)(πt− ¯π)] + γρ(1 − nδρ) 1 − δρ E[(yt−1− ¯y)(πt− ¯π)] +γ(1 − nδρ) 1 − δρ E[t(πt− ¯π)] + E[ut(πt− ¯π)], = nδβ2E[(πt−1− ¯π)(πt− ¯π)] + γρ(1 − nδρ) 1 − δρ E[(yt−1− ¯y)(πt− ¯π)] +γ(1 − nδρ) 1 − δρ γ(1 − nδρ)σ2 1 − δρ + σ2u, (B.5)
Now substitute Eq. (B.2) into (B.5) to obtain V ar(πt) = nδβ2 nδβ2V ar(πt) + γρ(1 − nδρ) 1 − δρ E[(yt−1− ¯y)(πt−1− ¯π)] + γ(1 − nδρ) 1 − δρ E[t(πt−1− ¯π)] + E[ut(πt−1− ¯π)] + γρ(1 − nδρ) 1 − δρ E[(yt−1− ¯y)(πt− ¯π)] + γ(1 − nδρ) 1 − δρ · γ(1 − nδρ)σ2 1 − δρ + σ 2 u, = n2δ2β4V ar(πt) + nδβ2 γρ(1 − nδρ) 1 − δρ E[(yt− ¯y)(πt− ¯π)] + γρ(1 − nδρ) 1 − δρ E[(yt−1− ¯y)(πt− ¯π)] + γ2(1 − nδρ)2σ2 (1 − δρ)2 + σ 2 u. (B.6) So, V ar(πt) = 1 1 − n2δ2β4 nδβ2γρ(1 − nδρ)
1 − δρ E[(πt− ¯π)(yt− ¯y)]
+γρ(1 − nδρ)
1 − δρ E[(πt− ¯π)(yt−1− ¯y)] +
γ2(1 − nδρ)2σ2 (1 − δρ)2 + σ 2 u . (B.7)
Again, we need (ii) E[(πt− ¯π)(yt− ¯y)], and additionally (iii) E[(πt− ¯π)(yt−1− ¯y)].
E[(πt− ¯π)(yt− ¯y)] = E
nδβ2(πt−1− ¯π)(yt− ¯y) + γρ(1 − nδρ) 1 − δρ (yt−1− ¯y)(yt− ¯y) +γ(1 − nδρ) 1 − δρ t(yt− ¯y) + ut(yt− ¯y) , = nδβ2E[(πt−1− ¯π))(ρ(yt−1− ¯y) + t)] + γρ(1 − nδρ)
1 − δρ E[(yt−1− ¯y)(yt− ¯y)]
+γ(1 − nδρ)
1 − δρ E[t(ρ(yt−1− ¯y) + t)] + E[ut(yt− ¯y)],
= nδρβ2E[(πt−1− ¯π))(yt−1− ¯y)] +
γρ(1 − nδρ) 1 − δρ · ρσ2 1 − ρ2 + γ(1 − nδρ) 1 − δρ σ 2 , (B.8)
since E[(yt−1− ¯y)(yt− ¯y)] = ρσ2/(1 − ρ2). Rearranging gives
E[(πt− ¯π)(yt− ¯y)] =
1 1 − nδρβ2 γρ(1 − nδρ) 1 − δρ · ρσ2 1 − ρ2 + γ(1 − nδρ) 1 − δρ σ 2 , = γσ 2 (1 − nδρ) (1 − nδρβ2)(1 − ρ2)(1 − δρ). (B.9)
Next, we will calculate (iii)
E[(πt− ¯π)(yt−1− ¯y)] = E
nδβ2(πt−1− ¯π)(yt−1− ¯y) + γρ(1 − nδρ) 1 − δρ (yt−1− ¯y) 2 +γ(1 − nδρ) 1 − δρ t(yt−1− ¯y) + ut(yt−1− ¯y) ,
= nδβ2E[(πt−1− ¯π)(yt−1− ¯y)] +
γρ(1 − nδρ)
1 − δρ E(yt−1− ¯y)
2
+γ(1 − nδρ)
1 − δρ E[t(yt−1− ¯y)] + E[ut(yt−1− ¯y)],
= nδβ2E[(πt−1− ¯π)(yt−1− ¯y)] +
γρ(1 − nδρ)
1 − δρ E(yt−1− ¯y)
2.
(B.10)
Now substitute the result from Eq. (B.9) and E(yt−1− ¯y)2= σ2/(1 − ρ2) into (B.10)
E[(πt− ¯π)(yt−1− ¯y)] = nδβ2·
γσ2(1 − nδρ) (1 − nδρβ2)(1 − ρ2)(1 − δρ)+ γρ(1 − nδρ) 1 − δρ · σ2 1 − ρ2, = γσ 2 (1 − nδρ) (1 − nδρβ2)(1 − δρ) nδβ2+ ρ 1 − ρ2 . (B.11)
Now we substitute equations (B.9) and (B.11) into (B.7)
V ar(πt) = 1 1 − n2δ2β4 nδβ2γρ(1 − nδρ) 1 − δρ γσ2(1 − nδρ) (1 − nδρβ2)(1 − ρ2)(1 − δρ) +γρ(1 − nδρ) 1 − δρ · γσ2(1 − nδρ) (1 − nδρβ2)(1 − δρ) nδβ2+ ρ 1 − ρ2 +γ 2(1 − nδρ)2σ2 (1 − δρ)2 + σ 2 u , = 1 1 − n2δ2β4 nδβ2γ2ρ(1 − nδρ)2σ2 (1 − nδρβ2)(1 − ρ2)(1 − δρ)2 + γ 2ρσ2 (1 − nδρ)2 (1 − nδρβ2)(1 − δρ)2 nδβ2+ ρ 1 − ρ2 +γ 2(1 − nδρ)2σ2 (1 − δρ)2 + σ 2 u , = σ 2 1 − n2δ2β4 1 − nδρ 1 − δρ 2 nδβ2γ2ρ (1 − nδρβ2)(1 − ρ2)+ γ2ρ 1 − nδρβ2 nδβ2+ ρ 1 − ρ2 + γ2 + σ 2 u σ2 , = σ 2 1 − n2δ2β4 1 − nδρ 1 − δρ 2 γ2ρ(nδβ2(2 − ρ2) + ρ) (1 − nδρβ2)(1 − ρ2) + γ 2 +σ 2 u σ2 , = σ 2 1 − n2δ2β4 1 − nδρ 1 − δρ 2 · γ 2(nδρβ2+ 1) (1 − nδρβ2)(1 − ρ2)+ σ2u σ2 . (B.12)
Substitute the result from Eq. (B.9) into (B.4) E[(πt− ¯π)(πt−1− ¯π)] = nδβ2V ar(πt) + γρ(1 − nδρ) 1 − δρ · γσ2(1 − nδρ) (1 − nδρβ2)(1 − ρ2)(1 − δρ), = nδβ2V ar(πt) + γ2ρσ2(1 − nδρ)2 (1 − nδρβ2)(1 − ρ2)(1 − δρ)2. (B.13)
Finally, substitute Eq. (B.13) and (B.12) into (B.3)
Corr(πt, πt−1) = E[(πt − ¯π)(πt−1− ¯π)] V ar(πt) , = nδβ2+ γ2ρσ2 (1−nδρ)2 (1−nδρβ2)(1−ρ2)(1−δρ)2 σ2 1−n2δ2β4 1−nδρ 1−δρ 2 ·(1−nδρβγ2(nδρβ2)(1−ρ2+1)2) + σ2 u σ2 , = nδβ2+ γ 2ρ(1 − nδρ)2(1 − n2δ2β4) γ2(1 − nδρ)2(nδρβ2+ 1) + (1 − δρ)2(1 − nδρβ2)(1 − ρ2)σ2u σ2 . (B.14)
References
Adam, K. (2007). Experimental Evidence on the Persistence of Output and Inflation. The Economic Journal 117 (520), 603–636.
Branch, W.A. (2004). The theory of rationally heterogeneous expectations: Evidence from survey data on inflation expectations. The Economic Journal 114, 592–621.
Brock, W.A. and C.H. Hommes (1997). A Rational Route to Randomness. Econometrica 65, 1059–1095.
Calvo, G.A. (1983). Staggered prices in a utility-maximizing framework. Journal of Monetary Economics 12, 383–398.
Cornea, A., C.H. Hommes, and D. Massaro (2012). Behavioral Heterogeneity in U.S. Inflation. CeNDEF Working Papers 12-03, University of Amsterdam.
Deutsch, M., C.W.J. Granger, and T. Ter¨asvirta (1994). The Combination of Forecasts Using
Evans, G.W. and S. Honkapohja (2001). Learning and Expectations in Macroeconomics. Princeton University Press.
Fuhrer, J.C. (2009). Inflation Persistence. Federal Reserve Bank of Boston.
Fuster, A., D. Laibson, and B. Mendel (2010). Natural expectations and macroeconomic fluctuations. Journal of Economic Perspectives 24, 67–84.
Gal´ı, J. (2008). Monetary Policy, Inflation, and the Business Cycle: An Introduction to the New Keynesian Framework. Princeton University Press.
Hommes, C.H. and G. Sorger (1998). Consistent expectations equilibria. Macroeconomic Dynamics 2, 287–321.
Hommes, C.H., G. Sorger, and F. Wagener (2013). Global Analysis of Dynamic Models in Economic and Finance: essays in Honour of Laura Gardini, Chapter Consistency of Linear Forecasts in a Nonlinear Stochastic Economy. Springer.
Hommes, C.H. and M. Zhu (2014). Behavioral Learning Equilibria. Journal of Economic Theory 150, 778–814.
Lansing, K.J. (2009). Time-varying U.S. inflation dynamics and the New Keynesian Phillips curve. Review of Economic Dynamics 12, 304–326.
Lucas, R.E. (1972). Expectations and the Neutrality of Money. Journal of economic theory 4, 103–124.
Mankiw, N., R. Reis, and J. Wolfers (2003). Disagreement about inflation expectations. Massaro, D. (2012). Bounded rationality and heterogeneous expectations in macroeconomics.
Ph. D. thesis, University of Amsterdam.
Pfajfar, D. and E. Santoro (2010). Heterogeneity, learning and information stickiness in inflation expectations. Journal of Economic Behavior & Organization 75, 426–444. Sargent, T.J. and N. Wallace (1975). Rational Expectations, the Optimal Monetary
