Internal Rationality, Learning and Imperfect
Information∗
Szabolcs Deák
Paul Levine
Joseph Pearlman
University of Surrey
University of Surrey
City University, London
Bo Yang
Swansea University
January 25, 2017
Abstract
We construct, estimate and explore the monetary policy consequences of a New Keynesian (NK) behavioural model with bounded-rationality and heterogeneous agents. We
radically departs from most existing models of this genre in its treatment of bounded
rationality and learning. Instead of the Euler learning approach of most of the literature, we assume that agents are internally rational (IR) given their beliefs of aggregate
states and prices. We construct a heterogeneous agent NK model of fully rational (RE)
and IR agents where the latter use simple adaptive expectations rules to forecast aggregate variables exogenous to their micro-environment. They are capable of learning
from their forecasting errors by observing and comparing them with those of RE agents
making the composition of the two types endogenous. We find that IR results in an
NK model with more persistence and a smaller policy space for rule parameters that
induce stability and determinacy. In a Bayesian estimation of the model with fixed
proportions of RE and IR agents we find that a pure IR model fits the data better
than in any composite, including the pure RE case. However pure RE, with imperfect
rather than the standard perfect information assumption, outperforms the IR case
suggesting that Kalman-filtering learning with RE can match bounded-rationality in
matching persistence seen in the data.
JEL Classification: E03, E12, E32.
Keywords: New Keynesian Behavioural Model, Internal Rationality, Heterogeneous
Expectations, Reinforcement Learning, Imperfect Information
∗
Previous versions of this paper were presented at the CEF 2015 Conference, Taiwan, June 2015;
seminars at the Universities of Surrey and Leicester; workshops at the Bank of England, March 2015
and the Tinbergen Institute, October, 2015 and as a keynote lecture for the third annual workshop of
the Birkbeck Centre for Applied Macroeconomics, Birkbeck College, May 20, 2016. Comments from
participants are gratefully acknowledged, as are those from Cars Hommes. We acknowledge financial
support from the ESRC, grant reference ES/K005154/1.
Contents
1 Introduction
1
2 The Standard NK Behavioural Model
4
2.1
The Work-Horse NK Model . . . . . . . . . . . . . . . . . . . . . . . . . . .
4
2.2
The Brock-Hommes Behavioural NK Model . . . . . . . . . . . . . . . . . .
5
3 Euler Learning versus Internal Rationality
6
3.1
The Linearized RE Model . . . . . . . . . . . . . . . . . . . . . . . . . . . .
7
3.2
Euler Learning . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
9
3.3
Internal Rationality
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
4 Stability Analysis and Impulse Responses
15
4.1
Comparison of Euler Learning and Internal Rationality . . . . . . . . . . . . 16
4.2
Heterogeneous Expectations across Households and Firms . . . . . . . . . . 19
5 Bayesian Estimation
23
5.1
The Empirical Model, Data and Priors . . . . . . . . . . . . . . . . . . . . . 25
5.2
The Measurement Equations . . . . . . . . . . . . . . . . . . . . . . . . . . 26
5.3
Identification Diagnostics . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
5.4
Bayes Factor Comparison . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
5.5
Results for Rational Expectations Model . . . . . . . . . . . . . . . . . . . . 33
5.6
Results for Behavioural Models . . . . . . . . . . . . . . . . . . . . . . . . . 34
6 Imperfect Information and Informational Consistency
36
7 Model Validation
38
7.1
Matching Second Moments . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
7.2
Posterior Impulse Response Analysis
. . . . . . . . . . . . . . . . . . . . . 41
8 Conclusions
44
A
49
Foundations of the New Keynesian Model with Internal Rationality
A.1 The Rational Expectations Model
. . . . . . . . . . . . . . . . . . . . . . . 49
A.1.1 Households . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
A.1.2 Firms in the Wholesale . . . . . . . . . . . . . . . . . . . . . . . . . 51
A.1.3 Firms in the Retail Sector . . . . . . . . . . . . . . . . . . . . . . . . 51
A.1.4 Profits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
A.1.5 Flexi-Price Output . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
A.1.6 Closing the Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
A.1.7 Summary of Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56
A.1.8 Steady State . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
A.2 A Balanced-Non-Zero-Growth Steady State . . . . . . . . . . . . . . . . . . 58
A.3 Exogenous Point Expectations . . . . . . . . . . . . . . . . . . . . . . . . . 59
A.3.1 Households . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60
A.3.2 Retail Firms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
A.4 Stability with a Non-Zero Inflation Steady State . . . . . . . . . . . . . . . 63
A.5 Internal Rationality in the NK Model . . . . . . . . . . . . . . . . . . . . . 64
A.6 Heterogeneous Expectations . . . . . . . . . . . . . . . . . . . . . . . . . . . 65
A.7 Proof of Lemma . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66
A.8 Proof of Equation A.29
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67
B Priors and Posteriors
67
C Identification Strength at Priors and Posteriors
69
D Solution of Linearized Models under Imperfect Information
69
D.1 Perfect Information Case
. . . . . . . . . . . . . . . . . . . . . . . . . . . . 72
D.2 Imperfect Information Case . . . . . . . . . . . . . . . . . . . . . . . . . . . 73
D.3 When is the System Invertible? . . . . . . . . . . . . . . . . . . . . . . . . . 75
3
1
Introduction
This paper constructs, estimates and explores the monetary policy consequences of New
Keynesian (NK) ‘behavioural’ model with bounded-rationality and heterogeneous agents.
It departs from existing models of this genre in its approach to bounded rationality and
learning. There are broadly two choices made by the learning literature at this point:
Euler learning or internal rationality. The first, Euler learning , follows the pioneering
work of Evans and Honkapohja (2001) and assumes, in the case of households, that agents
forecast their own consumption decision decision next period. Furthermore they know
the minimum state variable (MSV) form of the equilibrium (equivalent to the saddle-path
under rational expectations) and use direct observations or VAR estimates of these states
to update their estimates each period using a discounted least-squares estimator. Then a
statistical learning equilibrium is one where this perceived law of motions and the actual
one coincide. For firms the same applies except the decision is on prices made by firms
who no longer locked into a contract.
Although this form of bounded rationality responds to what many regard as an extreme
assumption of model-consistent expectations, the departure is only a modest one in that
agents still need to know the MSV form of the equilibrium. The defining characteristic of
behavioural macro-models to limit the cognitive skills of at least a group of agents in the
model and this is achieved by introducing simple ‘heuristic’ learning rules. However this
raises the opposite concern regarding the bounds on bounded rationality: with heuristic
rules agents may fall considerably short of building rational expectations and such models
are particularly vulnerable to the Lucas critique when policy scenarios are studied. The
problem is that agents can depart from rationality in an infinite number of ways leading
into the ‘wilderness’ of Sims (1980).
In response the wilderness concern of Christopher Sims, the literature on behavioural
models adopts a basic general framework that is admirably set out in Young (2004).
To limit the departure from rationality and rule out stupid behaviour the approach of
reinforcement learning proposes that, although adaptation can be slow and there can be a
random component of choice, the higher the ‘payoff’ (defined appropriately) from taking
an action in the past, the more likely it will be taken in the future.
The alternative approach to learning adopted in this paper, first introduced by Eusepi
and Preston (2011) into an RBC model, assumes that agents are internally rational (IR)
given their beliefs of aggregate states and prices.1 As with the Euler equation approach,
agents cannot form model-consistent expectations and instead learn about these variables
using their knowledge of the MSV form of the equilibrium. The two approaches then differ
with respect to what agents learn about - their own decisions in the first approach, and
variables exogenous to the agents in the second approach.2
Proceeding to the linearization about a deterministic steady state, as is usual in the
literature, we show that EL of Evans and Honkapohja (2001) makes three explicit assumption: agents (1) know they are all identical; (2) know the structure of the saddle-path and
(3) observe the state vector including the shock processes. Our IR formulation, by contrast, make none of these assumptions and requires non-rational agents in the model to
forecast only macro-variables exogenous to their decision rules. Heuristic rules then can
be thought of as parsimonious forms of forecasting rules (as in Branch and Evans (2011))
which, for sophisticated agents, would take the form of high-order VARs. This we argue,
fits well the behavioural approach of assuming agents in the model with limited cognitive
skills.
The main contributions of this paper are as follows: first, we start with the full nonlinear formulation to provide rigorous foundations for NK behavioural models based on
internal rationality (IR) rather than Euler learning (EL); second, we examine empirically
the support for a composite RE-IR model of the Brock-Hommes variety by Bayesian
estimation; third in our comparisons of different composites including the pure RE and
IR cases, we impose what we term informational consistency where RE and IR agents in
the model share the same information as the econometrician estimating the model.
The nearest paper to ours is Massaro (2013) which presents a calibrated composite
1
See Adam and Marcet (2011) who apply the concept to asset-pricing, Spelta et al. (2012) for a housing
pricing model with internal rationality, Woodford (2013) who adopts a similar NK framework as in this
paper and Branch and McGough (2016) for a recent discussion of of IR, also referred to as the ‘infinite timehorizon approach to learning’. We adopt the general definition of internal rationality used in the first paper:
namely that “agents maximize utility under uncertainty, given their constraints and given a consistent set
of probability beliefs about payoff-relevant variables that are beyond their control or external ”. Then beliefs
can take the form of a well-defined probability measure over a stochastic process (the ‘fully Bayesian’ plan),
or they can adopt an ‘anticipated utility framework of Kreps (1998). Adam and Marcet (2011) adopt the
former approach whereas this paper and the other applications mentioned adopt the latter. Cogley and
Sargent (2008) compares the two and encouragingly find that anticipated utility can be seen as a good
approximation to fully Bayesain optimization.
2
See Graham (2011) for a discussion of this distinction.
2
heterogeneous expectations model of RE and IR-anticipated-utility agents. As in our
paper he emphasizes the need for policymakers to design robust rules that stabilize the
economy across different composites models; but here we we focus on the informational
assumptions made by the two sets of agents and we seek empirical support for the modelling
choices.3
The rest of the paper is structured as follows. Section 2 sets out the standard linear NK
RE model used in the literature and then proceeds to the Brock-Hommes composite model
of rational and boundedly rational agents. The latter assumes Euler learning whereas
in Section 3 we propose internal rationality as an alternative approach that makes far
less demands on the information assumptions of agents. Section 4 provides numerical
results on the dynamic properties of three possible models of expectations, rational (RE),
boundedly rational with Euler learning (EL) and boundedly but internally rational (IR).4
The Section first considers homogeneous expectations for which all agents (households and
firms) form either RE or IR or EL or expectations. Then we introduce heterogeneity in
a full Brock-Hommes NK model with a composite model of IR and RE agents. Section
5 estimates the latter, alongside the pure IR and RE models by Bayesian methods and
conducts a likelihood race. Section 6 first provides alternative estimation results without
estimating the proportions of rationality parameters. Then it adds an additional learning
mechanism assuming that RE agents do not observe all current state variables and only
have an imperfect information set. Section 7 examines the ability of the estimated model
to match the second moments in the data. Section 8 concludes the paper. The full nonlinear model that lies at the basis of the linearized forms is set out in a separate Appendix
which also contains details of the estimation results and the imperfect information solution
procedure.
3
IR also fits into the Agent-Based Modelling (ABM) framework: Sinitskaya and Tesfatsion (2014)
introduce forward-looking optimizing agents into an ABM model. They use essentially the IR concept
which they refer to as constructive rational decision-making. This results in a novel AB macro-model in
having internally rational optimizers: households maximize expected intertemporal utility over an infinite
time-horizon and firms do the same with their utility being taken as profit.
4
Jump and Levine (2017) provides analytical results for stability.
3
2
The Standard NK Behavioural Model
This section discusses the standard New Keynesian behavioural model framework used by
Jang and Sacht (2012), Jang and Sacht (2014), De Grauwe (2012a), De Grauwe (2012b),
Branch and McGough (2010), Massaro (2013), Cornea et al. (2014) and others.
2.1
The Work-Horse NK Model
We first set out the most basic three-equation linearized work-horse NK model with RE
yt = Et yt+1 − (rn,t − Et πt+1 ) + u1,t
(1)
πt = βEt πt+1 + λ(yt − ytF ) + u2,t
(2)
rn,t = ρr rn,t−1 + (1 − ρr )(θπ πt + θy yt ) + u3,t
or rn,t = ρr rn,t−1 + (1 − ρr )(θπ πt + θy (yt − ytF )) + u3,t
(3)
(4)
where yt , πt and rn,t are the output gap, the inflation rate and the nominal interest rate
respectively. All variables are expressed in log-deviation form about the steady state.
The shock processes ui,t , i = 1, 3 should be interpreted as exogenous shocks to demand
(or preferences), marginal costs and monetary policy respectively and are usually be AR1
processes.5 Expectations up to now are formed assuming rational expectations and perfect
information of the state vector (which includes the shock processes).
Equation (1) is the linearized Euler equation for consumption which is equated with
output in equilibrium (there is no government expenditure). (2) is the NK Phillips curve
with ytF the flexi-price output. In this simple model ytF = at , the technology shock process.
(3) is the nominal interest rate rule with the latter responding to output whereas in (4)
the feedback is from the output gap. We term these Taylor-type rules as ‘implementable’
and ‘conventional’ respectively.6
Before relaxing the RE assumption two points about this formulation need to be made.
First, there are no the lagged term in yt in the demand curve (1) nor a lagged terms in πt in
the Phillips Curve (2). These can enter through the introduction of external habit in the
consumers’ utility function and indexing respectively. But we choose to focus on learning
5
This feature of shock processes is criticized by De Grauwe (2012b) as it implies persistence is endogenously generated.
6
The implementable rule requires no observations of the output gap.
4
as a persistence mechanism, so both these features are omitted. Second, the linearization
even without these persistence terms is only correct about a zero-inflation steady state.
2.2
The Brock-Hommes Behavioural NK Model
In the Brock-Hommes framework which we later follow, the model become behavioural by
a departure from RE assumption and the introduction of two groups of agents. One group
is rational and the other forms expectations through simple ‘heuristic’ learning rules. RE
agents form model-consistent expectations fully aware of the existence or IR agents in the
composite model.
General adaptive learning rules7 that encompass those adopted by Brock and Hommes
(1997), Hommes (2013), Branch and McGough (2010), De Grauwe (2012b), and De Grauwe
(2012a) are
E∗t yt+1 = E∗t−1 yt + λy (yt−j − E∗t−1 yt ) ;
λy ∈ [0, 1], j = 0, 1
E∗t πt+1 = E∗t−1 πt + λπ (πt−j − E∗t−1 πt ) ;
λπ ∈ [0, 1], j = 0, 1
(5)
(6)
where we can in principle allow for both current and lagged observations of output and
inflation, j = 0, 1 respectively. Throughout the rest of the paper we make the following
information assumptions: for observations of aggregate output and inflation j = 1 which
is assumed in the EL approach. Later in the IR approach we need to model observations
of market-specific variables consisting of factor prices, profits and marginal costs. These
we assume can be observed without a lag and therefore j = 0.
The IS and NK equations now become
yt = ny,t Et yt+1 + (1 − ny,t )E∗t yt+1 − [rn,t − (nπ,t Et πt+1 (1 − nπ,t )E∗t πt+1 )] + u1,t (7)
πt = β(nπ,t Et πt+1 (1 − nπ,t)E∗t πt+1 ) + λ(yt − ytF ) + u2,t
(8)
where ny,t , nπ,t is the proportions of rational agents forecasting output and inflation respectively.
To complete the model we need expressions for the weights ny,t and nπ,t . These follow
7
Anufriev et al. (2015) provide lab-based support for such rules.
5
the reinforcement learning literature by choosing probabilities
nx,t =
exp(µx ΦRE
x,t ({xt })
AE
exp(µx ΦRE
x,t ({xt })) + exp(µx Φx,t ({xt }))
(9)
AE
where ΦRE
x,t ({xt )}) and Φx,t ({xt )}) are ‘fitness’ measures respectively of the forecast per-
formance of the rational and non-rational predictor of outcome {xt } = {yt }, {πt } given by
a MSE predictor:
RE
2
ΦRE
x,t ({xt })) = ρRE Φx,t−1 ({xt })) − (1 − ρRE )([xt − Et−1 xt ] + Cx )
(10)
AE
∗
2
ΦAE
x,t ({xt })) = ρAE Φx,t−1 ({xt })) − (1 − ρAE )[xt−j − Et−1−j xt−1 ] ; j = 0, 1 (11)
where Cx represents the relative costs of being rational in learning about variable xt . Thus
the proportion of rational agents in the steady state is given by
nx =
exp(−µx Cx )
exp(−µx Cx ) + 1
(12)
which is pinned down by the µx Cx (which can be positive or negative)
Equations (3) or (4) and (5) – (11) completes the linearized NK behavioural model.8
3
Euler Learning versus Internal Rationality
So far in the linearized model the justification for adaptive forecasts is not clear. In order
to address this we step back to the linearized form of the underlying non-linear model and
introduce the distinction between internal decisions and aggregate macro-variables. We
start with the linearization of the non-linear RE model set out in Appendix A about a
zero net inflation, zero growth deterministic steady state.
8
De Grauwe (2012b), and De Grauwe (2012a) construct a rather different composite model consisting
of ‘fundamentalist’ rather than rational agents alongside adaptive learners. For the former rational expectations E(·) are replaced with Ef yt+1 = ytF and Ef πt+1 = 0. Thus fundamentalists always believe next
periods output gap is zero and the net inflation rate will return to its steady-state value of zero. The same
author also assumes Cx = 0 in (10).
6
3.1
The Linearized RE Model
For household j the single-period utility is
Ut (j) = U (Ct (j), Ht (j)) = log(Ct (j)) − κ
Ht (j)1+φ
1+φ
(13)
In linearized form this leads to the Euler consumption equation
ct (j) = Et [ct+1 (j) − rt+1 ]
(14)
or alternatively after solving the household budget constrain forward in time, imposing
the Euler and transversality conditions we can utilize a consumption function of the form:
α1 ct (j) = α2 wt + α3 (ω1,t + rt ) + α4 ω2,t
ω1,t = α5 Et wt+1 − α6 Et rt+1 + βEt ω1,t+1
ω2,t = (1 − β)(γt (j) − rt ) − βrt + βEt ω2,t+1
α
cy
ct − (wt + ht )
γt =
γy
γy
where γt is exogenous profit per household (a function of aggregate consumption and
hours) and positive coefficients are given by
α1 ≡ 1 +
α
φcy
α2 ≡ (1 − β)(1 +
α3 ≡
α4 ≡
1 α
)
φ cy
γy
cy
βα
cy
α5 ≡ (1 − β)(1 +
α6 ≡ (1 +
1
)
φ
1
)
φ
Optimal household labour supply is given by
hst (j) =
1
(wt − ct (j))
φ
7
(15)
For optimizing firm m:
pot (m) − pt = ω4,t − ω3,t
ω3,t = ξβEt+1 [(ζ − 1)πt+1 + ω3,t+1 ] + (1 − βξ)(yt + uC,t )
ω4,t = ξβEt+1 [ζπt+1 + ω4,t+1 ] + yt + (1 − βξ)(uC,t + mct + mst )
Hence
pot (m) − pt = βEt [πt+1 + pot+1 (m) − pt+1 ] + (1 − βξ)(mct + mst )
(16)
For the wholesale sector:
yt = at + αhdt
mct = wt − yt + hdt
(17)
(18)
Aggregation of households and retail firms:
ct (j) = ct
(19)
hst (j) = hst
(20)
pot (m) = pot
(21)
ξπt = (1 − ξ)(pot − pt )
(22)
Equilibrium in output and labour markets:
yt = ct + dst
hst = hdt = ht
The model is completed with a Fischer equation
rt = rn,t−1 − πt
and policy rule as before giving 10 (or 12) equations for aggregates in ct (or ct , ω1,t , ω2,t ),
yt , hst , hdt , ht , wt , rt , rn,t , πt and mct given AR1 exogenous processes for technology,
demand and mark-up shock processes at , dst and mst .
8
Imposing the aggregation conditions (21) and (22) in (16) we arrive at the familiar
linearized Phillips curve
πt = βEt πt+1 +
(1 − ξ)(1 − βξ)
(mct + mst )
ξ
(23)
which on putting mct = wt − at + ht = (1 + φ)ht = (1 + φ)(yt − at )/α gives (2) with
λ=
3.2
(1−ξ)(1−βξ)(1+φ)
αξ
and u2,t = λmst . Finally in (1), u1,t = dst − Edst+1 .
Euler Learning
Our linearized form of the NK model can be expressed in a general linear state-space form:


zt+1
Et xt+1


 = A
zt
xt


 + Brn,t + 
Cǫt+1
0

;

ot = E 
zt
xt


(24)
where zt is a (n − m) × 1 vector of predetermined variables at time t with z0 given, xt , is
a m × 1 vector of non-predetermined variables and ot is a vector of other macro-variables
of interest (and can be augmented by components zt and xt ). All variables are expressed
as proportional deviations about a steady state. A, B, C and E are fixed matrices and
ǫt as a vector of random zero-mean shocks. Rational expectations are formed assuming a
full information set {zs , xs , ǫs }, s ≤ t, the model and the monetary rule.
Rules so far are of the form

rn,t = Dyt = D 
zt
xt

(25)

where D is constrained to be sparse in some specified way and zt must be augmented to
include the lag rn,t−1 . Substituting (25) into (55) gives the state space form:


zt+1
Et xt+1


 = F
zt
xt

 + Cǫt+1 ;

ot = E 
zt
xt


(26)
where F ≡ [A + BD] and, for our current model, assuming AR1 processes for u1,t , u2,t ,
9
ytF = at and that u3,t is i.i.d,
zt = [at
u1,t
xt = [yt
πt ]′
o = [ct
wt
xt = [ω1,t
o = [ct
u2,t
ht
ω2,t
yt
wt
rt
rn,t−1 ]′
mct ]′
or
πt ]′
ht
rt
mct ]′
We now proceed to the assumption that there are non-rational agents who are unable to
form model-consistent expectations. for such agents, in the learning literature pioneered
by Evans and Honkapohja (2001) learning rules are specified in terms of the minimum
state variable representation of the model-consistent solution to (26). If the number of
eigenvalues outside the unit circle is equal to the number of non-predetermined variables,
the system has a unique equilibrium which is also stable with saddle-path xt = −N zt
where N = N (D) and dependents on the rule. (See Blanchard and Kahn (1980); Currie
and Levine (1993)). Instability (indeterminacy) occurs when the number of eigenvalues
of F outside the unit circle is larger (smaller) than the number of non-predetermined
variables.
Partitioning F conformably with zt and xt , it follows that the RE solution takes the
form of a first-order VAR
zt = [A11 − A12 N ]zt−1 + Cǫt
(27)
xt = −N zt
(28)
Et xt+1 = −N Et zt+1 = −N [A11 − A12 N ]zt
(29)
In the learning literature, agents are the assumed to make their forecast (29) by using
(30) to estimate a first order VAR in zt . This is usually referred to as ‘Euler-learning’ or
by Ellison and Pearlman (2011) as ‘saddle-path learning’. For our model this implies that
agents (1) know they are all identical; (2) know the structure of the saddle-path and (3)
observe the state vector including the shock processes. The Brock-Hommes framework
shares assumptions (1) and (2) but reasonably assume that agents are using data on
10
inflation and output. We interpret the heuristic rules are parsimonious forecasting models
in which non-rational agents choose under-parameterized predictors (see Branch and Evans
(2011)).
We now express learning rules in terms of a subset of ot ≡ wt = [yt , πt , rn,t ]′ . Observing
these three time-series under RE enables agents (and the econometrician) to back out the
shocks and to express ot as an infinite VAR (Fernandez-Villaverde et al. (2007) and Levine
et al. (2012)). To show this write the RE solution as
zt = Ãzt−1 + B̃ǫt
(30)
wt = C̃zt−1 + D̃ǫt
(31)
and to ease the notation drop the tilde over the newly defined matrices. Because we have
three shocks and three observables, the matrix D is square. Assume now it is also nonsingular. Then ǫt = D −1 (wt − Czt−1 ) and substituting into (30) and denoting the lag
operator by L, we have
[(I − (A − BD −1 C)L]zt = BD −1 wt
(32)
Hence combining (30) – (32) we have
zt
∞
X
(A − BD −1 C)i BD −1 wt−i
=
i=0
∞
X
(A − BD −1 C)i BD −1 wt−i + Dǫt
wt = C
(33)
(34)
i=1
Convergence of the summations in (33) and (34) require that the matrix (A−BD −1 C) has
all eigenvalues within the unit circle. Then equation (34) is an infinite VAR for the three
observables wt = [yt , πt , rn,t ]′ which is estimatable from output, inflation and interest rate
data. It follows that the RE forecast is:
Et wt+1
∞
X
(A − BD −1 C)i BD −1 wt−i
= C
(35)
i=0
whereas the adaptive heuristic rules (5) and (6) are parsimonious representations of (35)
11
given by
E∗t yt+1 =
E∗t πt+1 =
∞
X
i=0
∞
X
λiy yt−i ;
λiπ πt−i ;
or
or
E∗t yt+1 =
E∗t πt+1 =
λiy yt−i
i=1
∞
X
λiπ πt−i
(36)
(37)
i=1
i=0
3.3
∞
X
Internal Rationality
With internal rationality and anticipated utility (also known as the ‘infinite horizon approach’), our model of learning is one in which agents are rational regarding their internal
decisions, but have no macroeconomic model to form expectations of aggregate variables.
We draw a clear distinction between aggregate and internal quantities so that identical
agents in our model are not aware of this equilibrium property (nor any others). We now
drop the three assumptions for Euler learning that agents (1) know they are all identical;
(2) know the structure of the saddle-path and (3) observe the state vector including the
shock processes.
We utilize the internal household and retail firm decision rules set out in section 3.1.
For household j we have
α1 ct (j) = α2 wt + α3 (ω1,t + rt ) + α4 ω2,t
ω1,t = α5 E∗t wt+1 − α6 E∗t rt+1 + βE∗t ω1,t+1
ω2,t = (1 − β)(γt − rt ) − βrt + βE∗t ω2,t+1
cy
α
γt =
ct − (wt + ht ))
γy
γy
1
hst (j) =
(wt − ct (j))
φ
where E∗t denotes bounded rationality which now takes the form of adaptive expectations
of factor prices and profits. With adaptive expectations
E∗t rt+1 = rn,t − E∗t πt+1
E∗t rt+i = rn,t+i−1 − E∗t πt+i ;
E∗t rn,t+i = E∗t rn,t+1 ;
i≥1
E∗h,t πt+i = E∗h,tπt+1 ;
i≥1
12
i≥2
E∗t wt+i = E∗t wt+1 ;
i≥1
E∗t γt+i = E∗t γt+1 ;
i≥1
Imposing the aggregation condition (19) we arrive at the IR consumption equation
α1 ct = α2 wt + α3 (ω1,t + rt ) + α4 ω2,t
(38)
1
ω1,t =
[α5 E∗t wt+1 − α6 (βE∗t rn,t+1 − E∗t πt+1 )] − α6 rn,t
1−β
β
((1 − β)(E∗t γt+1 − E∗t rt+1 ) − βE∗t rt+1
ω2,t = (1 − β)(γt − rt ) − βrt +
1−β
One-step ahead forecasts given by
E∗t xt+1 = E∗t xt + λx (xt−j − E∗t xt ) ;
x = w, rn , π, γ ;
j = 0, 1
(39)
Internally rational households make rational inter-temporal decisions for their consumption and hours supplied given adaptive expectations of the wage rate, the nominal interest
rate, inflation and profits. These macro-variables may in principle be observed with or
without a one-period lag (j = 1, 0). As stated earlier we assume j = 0 for market-specific
variables wt , γt , and j = 1 for aggregate inflation πt . However we assume the current
nominal interest rate, rn,t is announced and therefore also observed without a lag.
The payoffs for households is expressed on terms of a discounted sum of past weighted
forecast errors, Φh,t say, starting at t = 0 for with rational and non-rational households
respectively:
∗
2
∗
2
IR IR
ΦIR
h,t = ρh Φh,t−1 − ww (wt − Et−1 wt ) + wrn (rn,t − Et−1 rn,t ) +
+ wh,π (πt − E∗h,t−1 π)2 + wγ (γt − E∗t−1 γt )2 + Ch
RE RE
ΦRE
=
ρ
Φ
−
ww (wt − Et−1 wt )2 + wrn (rn,t − Et−1 rn,t )2
h,t
h
h,t−1
+ wh,π (πt − E∗h,t−1 π)2 + wγ (γt − E∗t−1 γt )2
and the proportion of rational households is given by
nh,t =
exp(µh ΦRE
h,t )
AE
exp(µh ΦRE
h,t ) + exp(µh Φh,t )
13
(40)
For retail firm m, we have
pot (m) − pt = βξE∗t [πt+1 + pot+1 (m) − pt+1 ] + (1 − βξ)(mct + mst )
(41)
Solving forward
pot (m)
− pt =
E∗t
∞
X
(βξ)i [βξπt+i+1 + (1 − βξ)(mct+i + mst+i )]
(42)
i=0
Note that we use (42) rather than (23) to form internally rational adaptive expectations
because we do not wish to assume price-setters know they are identical (which the aggregation equation (22) does) and are aware of the symmetric nature of the equilibrium. In
the latter case we would have
∞
(1 − ξ)(1 − βξ) ∗ X i
(1 − ξ)(1 − βξ)
πt =
β mct+i =
Et
ξ
ξ
i=0
mct +
β
∗
E (mct+1 + mst+1 )
1−β t
(43)
which omits learning about aggregate inflation. So with IR, retail firms so not know
the symmetric nature of the equilibrium and form expectations of aggregate inflation and
marginal costs whereas with EL they do know they are identical and only form expectations
of marginal costs.
With adaptive expectations
E∗t πt+i+1 = E∗t πt+1 ;
i≥0
E∗t (mct+i + mst+i ) = E∗t (mct+1 + mst+1 ) ;
i≥1
so that
pot (m) − pt =
β
βξ ∗
E πt+1 + (1 − βξ)(mct + mst ) +
E∗ (mct+1 + mst+1 )
1 − β f,t
1−β t
(44)
One-step ahead forecasts given by
E∗t xt+1 = E∗t xt + λx (xt−j − E∗t xt ) ;
x = πf , (mc + ms);
j = 0, 1
(45)
Internally rational retail firms make rational inter-temporal decisions for their price and
14
output given adaptive expectations of the aggregate inflation rate and their post-shock
real marginal shock wage rate. As before these variables may be observed with or without
a one-period lag (j = 1, 0). For aggregate inflation we assume j = 1 as for households, but
j = 0 for the market-specific variable mct . Note that we distinguish between household
and firms’ expectations of inflation and we do not impose the assumption that their rate
of learning is the same.
As for households, the payoffs for firms is expressed on terms of a discounted sum of
past weighted forecast errors, Φf,t say, starting at t = 0 for with rational and non-rational
firms respectively:
2
∗
2
∗
IR RE
π
)
+
w
(mc
+
ms
−
E
(mc
+
ms
))
+
C
−
w
(π
−
E
Φ
=
ρ
ΦIR
t
mc
t
t
t
t
π
t
f
t−1
f,t−1
f,t−1
f
f,t
f
2
2
RE
ΦRE
= ρRE
f Φf,t−1 − wπf (πt − Ef,t−1 πt ) + wmc (mct + mst − Et−1 (mct + mst ))
f,t
and the proportion of rational firms is given by
nf,t =
4
exp(µf ΦRE
f,t )
AE
exp(µf ΦRE
f,t ) + exp(µf Φf,t )
(46)
Stability Analysis and Impulse Responses
We now have three possible models of expectations, rational (i.e., model consistent),
boundedly rational with Euler learning and boundedly but internally rational. We denote these three cases by RE, EL and IR respectively. We first consider homogeneous
expectations for which all agents (households and firms) form either RE or IR or EL or
expectations. We then allow for the possibility that households and firms are heterogenous
across these groups (but retain intra-group homogeneity). Finally we allow for intra-group
heterogeneity in a full Brock-Hommes NK model with IR and RE agents. The decision
s IR ′
variables for the household, dh,t = [cRE
(hst )RE cIR
t
t (ht ) ] is a linear mapping from
′
∗
∗
∗
RE
∗
[wt γt rt rn,t Et cRE
t+1 Et rt+1 Et rn,t+1 Et wt+1 Eh,t πt+1 ] Et γt+1 ] where rt = rn,t−1 − πt . The
decision variables for the retail firm df,t = [(pot − pt )IR (pot − pt )IR ]′ is a linear mapping
′
′
∗ π
from [mct E∗t mct+1 , Et πt+1 , Ef,t
t+1 ] . In equilibrium [wt γt ct πt mct ] is a mapping from
[dh,t df,t ]′ .
15
4.1
Comparison of Euler Learning and Internal Rationality
To examine the stability properties put shock processes at = mst = mpst = 0 to give us
a perfect foresight model. From the household labour supply conditions
(hst )RE
=
(hst )IR =
1
(wt − cRE
t )
φ
1
(wt − cIR
t )
φ
and aggregating over RE and IR households we have that
hst = hdt = ht =
1
(wt − ct )
φ
(47)
Hence from wholesale production
α
(wt − yt ) ⇒
φ
(α + φ)
=
yt ⇒
α
(1 + φ)
= wt − yt + ht =
yt
α
1
(α + φ)
=
(yt − α(wt + ht )) = −
yt
γy
γy
yt = αht =
(48)
wt
(49)
mct
γt
(50)
(51)
Thus from (47)–(51) aggregate variables ht , wt , mct and γt are all linear functions of
aggregate output yt .
From the household decisions household aggregate demand is given by
yt = ct = nh,t−1 [Et yt+1 − (rn,t − Et πt+1 )] + (1 − nh,t−1 )cIR
t
(52)
where cIR
t is given by (38).
The supply side is given by
pot − pt = nf,t−1 (pot − pt )RE + (1 − nf,t−1 ) (pot − pt )IR
(1 − ξ) o
(pt − pt )
πt =
ξ
(53)
(54)
where (pot − pt )RE and (pot − pt )IR are given by (16) and (42) respectively with mst = 0.
16
Assuming λx = 1 in (45) (as in BM) and using the monetary rule, the linearized
deterministic component of the model given nh,t−1 and nf,t−1 can be expressed in a general
linear state-space form as
Hxt = F xt+1 + Gxt−1


y

 t


xt ≡  π t
 for ρr 6= 0


rn,t−1


yt
 for ρr = 0
xt ≡ 
πt
but matrices H, F and G are rather more complicated than in the EL case and ρr > 0
introduces higher-order dynamics.
In the numerical results below we fix parameters at their priors used later in the
Bayesian estimation apart from the adaptive learning parameter λx = 1 which we set at
unity. As stated above we make the following information assumptions: for observations
of aggregate output and inflation j = 1 which is assumed in the EL approach. Later in
the IR approach we need to model observations of market-specific variables consisting of
factor prices, profits and marginal costs. These we assume can be observed without a
lag and therefore j = 0. Note this only applies to the EL and IR agents but the RE
equilibrium for now assumes perfect information where agents observe all current values
of state variables. Later in section 6 we address this inconsistency and assume all agents
have the same imperfect information (II) set as for IR agents. However for rational agents
the stability conditions considered now can be derived from a perfect foresight equilibrium
and are independent of the information assumption.
Figures 1 and 2 compare the policy space across RE, EL and IR models for the parameters in the Taylor rule in (θy , θπ ) space with zero persistence in the rule (ρr = 0).
Figures 3 and 6 compare the models in (ρr , θπ ) space with θy = 0.3. Finally 5 sets ρr = 1
and compares EL and IR models in (αy , απ ) space having re-parameterized the rule as
rn,t = ρr rn,t−1 + απ πt + αy yt . Note that this rule reduces to a price-level rule when
αy = 0. The differences in the sizes of the policy spaces that result in a saddle-path
stable equilibrium are significant. Furthermore a clear ranking of the sizes of these spaces
17
emerges with RE ⊃ EL ⊃ IR. This means that unless the policy rule is designed for the
IR model, uncertainty as to which model of expectations is correct can lead to a rule that
is unstable or has infinite multiple equilibria (i.e., is indeterminate).
IR Model ρ =0
RE Model ρ =0
r
r
Determinacy
Indeterminacy
0.9
0.9
0.8
0.8
0.7
0.7
0.6
0.6
0.5
0.5
0
1
2
3
4
5
θ
6
7
8
Determinacy
Instability
1
θπ
θπ
1
0
9 10
1
2
3
4
5
θ
6
7
8
9 10
y
y
(a) RE
(b) EL
Figure 1: Comparison of Stability Properties of RE and EL Models. ρr = 0,
λx = 1.
RE Model ρr=0.
IR Model ρr=0
Determinacy
Indeterminacy
1
0.975
0.975
0.95
0.95
0.925
0.925
θπ
θπ
Determinacy
Indeterminacy
1
0.9
0.875
0.9
0.875
0.85
0.85
0.825
0.825
0.8
0.8
0
1
2
3
4
5
θy
6
7
8
9 10
0
(a) RE
1
2
3
4
5
θy
6
7
8
9 10
(b) IR
Figure 2: Comparison of Stability Properties of RE and IR Models. ρr = 0,
λx = 1
18
EL Model: θ =0.3, λ=1
RE Model θ =0.3
y
y
Determinacy
Indeterminacy
3
2.5
2.5
2
θπ
θπ
Determinacy
Instability
3
1.5
2
1.5
1
1
0.5
0.5
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
ρ
ρ
r
r
(a) RE
(b) EL
Figure 3: Comparison of Stability Properties of RE and EL Models. ρr > 0,
λx = 1.
4.2
Heterogeneous Expectations across Households and Firms
Now we come to the full Brock-Hommes NK model but with IR rather than EL boundedly
rational agents. Proportions of rational households and firms given by
nh,t =
nf,t =
exp(γΦRE
h,t )
exp(γΦh,t )RE + exp(γΦIR
h,t )
exp(γΦRE
f,t )
IR
exp(γΦRE
f,t ) + exp(γΦf,t )
where fitness for households given by
ΦRE
h,t
ΦIR
h,t
RE
−
weighted
sum
of
forecast
errors
+
C
Φ
= µRE
h
h,t−1
h
IR
= µIR
h Φh,t−1 − weighted sum of forecast errors
The composite RE-IR model then has an equilibrium (in the original non-linear form)
Htd = nh,t (Hts )RE + (1 − nh,t ) (Hts )IR
Ct = nh,t (Ct )RE + (1 − nh,t ) (Ct )IR = Yt
19
IR Model: θ =0.3, λ=1
EL Model: θ =0.3, λ=1
y
y
Determinacy
Instability
3
2.5
2.5
2
2
θπ
θπ
Determinacy
Instability
3
1.5
1.5
1
1
0.5
0.5
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
r
r
ρ
ρ
(a) EL
(b) IR
Figure 4: Comparison of Stability Properties of EL and IR Models. ρr > 0,
λx = 1.
EL Model : Integral Interest Rate Rule
IR Model: Integral Interest Rate Rule
Determinacy
Instability
1.2
1.1
1.1
1
1
0.9
0.9
0.8
0.8
0.7
0.7
θπ
θπ
Determinacy
Instability
1.2
0.6
0.6
0.5
0.5
0.4
0.4
0.3
0.3
0.2
0.2
0.1
0.1
0
0
0 0.25 0.5 0.75 1 1.25 1.5 1.75 2
0 0.25 0.5 0.75 1 1.25 1.5 1.75 2
αy
αy
(a) EL
(b) IR
Figure 5: Comparison of Stability Properties of EL and IR Models. ρr = 1,
λx = 1.
Pto
Pt
= nf,t
Pto
Pt
RE
+ (1 − nf,t )
Pto
Pt
IR
Note that rational agents in this model form model-consistent expectations taking into
account the presence of internally rational agents.
20
Local stability however depends only on the steady-state. The parameter Ch is a
fixed cost of being rational for households. Similarly for firms. Figure 6 provides a
stability analysis with a price level rule (ρr = 1, αy = 0 in the re-parameterized rule
rn,t = ρr rn,t−1 + απ πt + αy yt , Cf = Cf and nh = nf = n in the steady state as the costs
go from positive in which case n < 0, to negative giving n > 0. Fast learning (λx = 1)
results in a larger regions of instability (a smaller policy space) than the case of slower
learning (λx = 0.25).
So far we have confined the simulations to parameter regions of the model and policy
rule that result in saddle-path stability. If we enter a region of local instability, but global
boundedness, we see chaotic dynamics as highlighted generally in Hommes (2013) and for
an NK model with Euler learning in Branch and McGough (2010). Two points should be
made concerning this possible outcome. First, there is then enormous inflation volatility
under chaos so the model is one of hyper-inflation. Second, we have seen that this clearly
undesirable outcome can be avoided by an appropriate choice of monetary rule.
IR−RE Model; Price Level Rule; ρr=1, αy=0, λ=0.25
IR−RE Model: Price Level Rule; ρr=1, αy=0, λ=1
Determinacy
Instability
1.2
1.1
1.1
1
1
0.9
0.9
θπ
θπ
Determinacy
Instability
1.2
0.8
0.8
0.7
0.7
0.6
0.6
0.5
0.5
0.5 0.55 0.6 0.65 0.7 0.75 0.8 0.85 0.9 0.95
0.5 0.55 0.6 0.65 0.7 0.75 0.8 0.85 0.9 0.95
nh=nf
nh=nf
(a) fast learning.
(b) slow learning
Figure 6: Stability of IR-RE heterogeneous-agent model with price-level rule
under fast and slow learning.
Figures 11 to 13 plot the impulse response functions with standard parameters for the
rule for shocks to technology, monetary policy and the mark-up under fast and slow learning. Not surprisingly fast learning sees a irfs converge faster to the RE case, but in either
case IR introduces more persistence compared with RE. This suggests this feature should
21
−3
−3
Consumption
x 10
4
2
4
% dev from SS
6
−1
−2
−3
−4
−5
−3
5
−3
x 10
2
1
0
10
15
Quarters
Nominal interest rate
20
−1
−1.5
−2
5
−3
x 10
10
Quarters
Inflation
15
5
10
Quarters
Output Gap
15
20
5
10
Quarters
15
20
−3
x 10
−0.5
−1
% dev from SS
0
−2
−4
20
% dev from SS
% dev from SS
x 10
10
15
Quarters
Real interest rate
2
% dev from SS
5
Real wage
x 10
0
% dev from SS
% dev from SS
−3
Hours worked
x 10
8
3
2
1
0
20
5
10
Quarters
15
20
−0.5
−1
RE
IR slow learning
IR fast learning
−1.5
−2
−2.5
5
10
Quarters
15
20
Figure 7: RE versus IR-RE Composite Expectations with nh = nf = 0.5, λx =
0.25, 1.0; Taylor rule with ρr = 0.7, θπ = 1.5 and θy = 0.3. Technology Shock
0
−2
−4
−6
−3
Hours worked
x 10
−5
−10
5
−4
x 10
10
15
Quarters
Real interest rate
5
−4
x 10
−2
10
15
Quarters
Nominal interest rate
20
5
10
Quarters
Output Gap
15
20
5
10
Quarters
15
20
−4
x 10
4
2
0
−2
% dev from SS
0
% dev from SS
% dev from SS
0
−1
−3
20
6
−2
−4
−6
−4
5
−4
x 10
% dev from SS
Real wage
x 10
0
% dev from SS
% dev from SS
−4
Consumption
% dev from SS
−4
x 10
10
Quarters
Inflation
15
20
−2
5
10
Quarters
RE
15
20
IR slow learning
6
4
2
0
IR fast learning
−4
−6
5
10
Quarters
15
20
Figure 8: RE versus IR-RE Composite Expectations with nh = nf = 0.5, λx =
0.25, 1.0; Taylor rule with ρr = 0.7, θπ = 1.5 and θy = 0.3. Monetary Policy Shock
22
−1
−1.5
−3
Hours worked
x 10
% dev from SS
% dev from SS
−3
Consumption
Real wage
x 10
−0.5
% dev from SS
−3
x 10
−0.5
−1
−1.5
−2
−2.5
−2
−4
−6
−2
5
−4
15
20
5
−4
x 10
10
15
Quarters
Nominal interest rate
20
−2
−4
% dev from SS
0
5
10
Quarters
Output Gap
15
20
5
10
Quarters
15
20
−3
x 10
2
5
2
% dev from SS
% dev from SS
x 10
10
Quarters
Real interest rate
4
3
2
1
1.5
1
0.5
−6
5
−4
x 10
10
Quarters
Inflation
15
20
5
10
Quarters
15
20
% dev from SS
6
RE
IR slow learning
IR fast learning
4
2
5
10
Quarters
15
20
Figure 9: RE versus IR-RE Composite Expectations with nh = nf = 0.5, λx =
0.25, 1.0; Taylor rule with ρr = 0.7, θπ = 1.5 and θy = 0.3. Mark-up Shock
lead to a better fit of the data better without relying on other persistence mechanisms (
shocks, habit or indexing). This we examine in the estimation of our model.
Finally, using the estimated model of the next section, Table 1 provides a third order
perturbation solution of non-linear NK IR-RE Model. In the estimation the model is
linearized and the proportions nh,t and nf,t are fixed. Non-linear estimation is required to
pin down the parameters nh , nf in the steady state, and µh , µf and γ in the reinforcement
learning process. So here we impose them as reported in the table. We also scale the
estimated standard deviations of the shocks using a parameter σ = 1, 2. The results are
that our heterogenous agent model with IR alongside RE agents introduces high kurtosis
and skewness 9 in macro variables and learning results in the numbers of rational agents
increasing from the estimated deterministic steady state value of 0.093 and 0.099 to 0.13
and 0.22 for households and firms respectively in the stochastic steady state.
5
Bayesian Estimation
Bayesian estimation entails obtaining the posterior distribution of the model’s parameters,
say θ, conditional on the data. Using the Bayes’ theorem, the posterior distribution is
9
The absence of kurtosis in the standard NK model, often highlighted in the literature (see, for example,
De Grauwe (2012a) is in part simply the consequence of linearization and non-normality is a feature of
higher order approximations.
23
Variable
Ct
C
Ht
H
Wt
W
Πt
Π
Rn,t
Rn
ΦRE
h,t − Ch
ΦAE
h,t
ΦRE
f,t − Cf
ΦAE
f,t
Stochastic Mean
0.9993
1.0002
0.9996
0.9999
0.9999
-0.000065
-0.000084
-0.000011
-0.000069
Standard Deviation (%)
2.47
0.19
2.15
0.46
0.46
0.000020
0.000054
0.000009
0.000053
Skewness
0.2792
0.0192
0.2771
0.0159
0.0070
-0.7589
-1.8238
-0.7203
-2.2156
Kurtosis
0.0371
0.0327
0.0215
0.0645
0.0651
0.9487
5.7852
0.7834
8.8686
nh,t(γ = 1; σ = 1)
nf,t (γ = 1; σ = 1)
nh,t (γ = 100; σ = 1)
nf,t (γ = 100; σ = 1)
nh,t (γ = 1000; σ = 1)
nf,t (γ = 1000; σ = 1)
nh,t (γ = 1000; σ = 2)
nf,t (γ = 1000; σ = 2)
0.093301
0.098603
0.094221
0.101751
0.102506
0.130105
0.129993
0.224367
0.000004
0.000004
0.003634
0.004303
0.036343
0.043030
0.146939
0.174046
1.8039
2.2688
1.8039
2.2688
1.8039
2.2688
1.8403
2.3668
6.0897
9.2725
6.0897
9.2725
6.0897
9.2725
6.6096
10.5098
IR
Table 1: Third Order Solution of the Estimated NK IR-RE Model; µRE
h = µh =
IR
RE
µf = µf = 0.0; γ = 1, 100, 1000
obtained as:
p(θ|Y T ) = R
L(Y T |θ)p(θ)
L(Y T |θ)p(θ)dθ
(55)
where p(θ) denotes the prior density of the parameter vector θ, L(Y T |θ) is the likelihood of the sample Y T with T observations (evaluated with the Kalman filter) and
R
L(Y T |θ)p(θ)dθ is the marginal likelihood. Since there is no closed form analytical expression for the posterior, this must be simulated.
One of the main advantages of adopting a Bayesian approach is that it facilitates a formal comparison of different models through their posterior marginal likelihoods, computed
using the Geweke (1999) modified harmonic-mean estimator. With Maximum-Likelihood
Bayesian type approaches the data’s probabilistic assessments of different models are summarised by their likelihoods. For a given model mi ∈ M and common data set, the
marginal likelihood is obtained by integrating out vector θ,
T
L Y |mi =
Z
Θ
L Y T |θ, mi p (θ|mi ) dθ
24
(56)
where pi (θ|mi ) is the prior density for model mi , and L Y T |mi is the data density for
model mi given parameter vector θ. To compare models (say, mi and mj ) we calculate
the posterior odds ratio which is the ratio of their posterior model probabilities (or Bayes
Factor when the prior odds ratio,
P Oi,j
=
BFi,j =
p(mi )
p(mj ) ,
is set to unity):
p(mi |Y T )
L(Y T |mi )p(mi )
=
T
p(mj |Y )
L(Y T |mj )p(mj )
(57)
exp(LL(Y T |mi ))
L(Y T |mi )
=
L(Y T |mj )
exp(LL(Y T |mj ))
(58)
in terms of the log-likelihood. Components (57) and (58) provide a framework for comparing alternative and potentially misspecified models based on their marginal likelihood.
Such comparisons are important in the assessment of rival models, as the model which attains the highest odds outperforms its rivals and is therefore favoured. Given Bayes factors,
P
we can easily compute the model probabilities p1 , p2 , ···, pn for n models. Since ni=1 pi = 1
P
we have that p11 = ni=2 BFi,1 , from which p1 is obtained. Then pi = p1 BF (i, 1) gives the
remaining model probabilities.
5.1
The Empirical Model, Data and Priors
Our estimated empirical model differs from the linearized version used up to now in two
respects. First we introduce as further shock process – government spending – to the
model. Second, the steady state about which the perturbation solution is computed has
a non-zero growth and inflation. The estimation then is conducted to be consistent with
the long-term trend of output and inflation in the data used in the estimation.
We estimate all three models – the NK Rational Expectations model (RE Model), the
NK model with Individual Rationality (IR Model) and the Behavioural Composite model
with Heterogeneous Expectations (IR-RE Model) – by Bayesian methods using Dynare.
We use a subset of the observable set used in Smets and Wouters (2007) in first difference
at quarterly frequency but extend the sample length to the second quarter of 2008, before
the outbreak of the 2008 crisis. Namely, these observable variables are the log differences
of real GDP and the GDP deflator and the federal funds rate. We fit our NK models
to output, inflation and the nominal interest rate to empirically assess the ability of the
models to match the data. All series are seasonally adjusted. All data series are taken
25
from the FRED Database available through the Federal Reserve Bank of St.Louis and the
US Bureau of Labour Statistics. The sample period is 1984:1-2008:2.
5.2
The Measurement Equations
The corresponding measurement equations for the 3 observables are:10

D(log GDPt ) ∗ 100


 log(GDP DEFt /GDP DEFt−1 ) ∗ 100

F EDF U N DSt /4 ∗ 100


log




=




− log YYt−1 + trend + ǫy,t
t−1
Πt
log Π + consπ + ǫπ,t
R
log Rn,t
+ consr
n
Yt
Yt





(59)
where constants trend, consπ and consr are related to the steady state variables of our
model by
Π = consπ /100 + 1
Rn = consr /100 + 1
g = trend/100
From our non-zero-inflation-growth steady state this implies that we should impose the
restrictions
Π = consπ /100 + 1
(60)
g = trend/100
Π(1 + g)(1−̺)(σc −1)+1
Π
=
= consr /100 + 1
Rn =
βg
β
(61)
(62)
on β rather than calibrating it at the usual β = 0.99. This implies that β is calibrated as
β=
consπ + 100
consr + 100
(1 + g)(1−̺)(σc −1)+1
(63)
which is imposed in an external steady state that also calibrates ̺ to hit a target H =
Hss = 0.35 in the results below.
10
Yt = GDPt , Y t =trend and trend growth =log Y t − log Y t−1 = log(1 + g) ≈ g for small g.
26
We introduce measurement errors on two observables, output and inflation (ǫy,t and
ǫπ,t so in total there are 3 variables in the observations and 6 exogenous shocks (At , Gt ,
M St , ǫM,t plus measurement errors).
In the estimation procedure a number of the structural parameters are fixed, so as to
match their sample means or in accordance with previous studies or calibration for the
US economy. The fixed or calibrated parameters are collected into Θf so that:
Θf
=
=
h
h
β ζ c H α µRE
h
0.99 7.0 1/ζ
µRE
µIR
f
h
γ
µIR
f
i′
0.35 0.7 0.5 0.5 0.5 0.5 1.0
i′
These parameters are necessary to solve and linearize the models but are problematic for
estimation (e.g. identification). For example we impose steady state hours worked to
0.35 and α to 0.7 so that estimating φ pins down the calibrated value of κ (κ =
α
).
H 1+φ
IR RE IR ′
From Section 4 the parameters in the IR-RE model, [µRE
h µh µf µf ] , do not enter into
the first-order solution for the linearized model but only affect the second-order or higher
solutions. They cannot be identified in the first order solution that is used for estimation
so are imposed at their mid-point values as above. As in De Grauwe (2011) we fix γ
to unity so that allow for a moderate degree in the intensity of individual choice. The
remaining calibration values are standard choices in the DSGE literature.
For the remainder of parameters gamma and inverse gamma distributions are used as
priors when non-negativity constraints are necessary, and beta distributions for fractions
or probabilities. Normal distributions are used when more informative priors seem to be
necessary. The prior means and distributions of these parameters can be found in Table
2. The values of priors are in line with those in Smets and Wouters (2007). The Calvo
coefficient ξ is assumed to be beta distributed with prior mean of 0.5 and prior standard
deviation of 0.2, implying that prices are sticky for two quarters. We draw all the AR(1)
parameters ρA , ρM S and ρG , and the lagged interest rate ρr from the beta distribution in
order to restrict them to the open unit interval. The standard deviation on the priors for ξ
and ρA , ρM S , ρG and ρr gives larger intervals from which to draws these parameters. The
beta distribution we use on the adaptive expectations learning parameter λ also restricts
it to the open unit interval but we allow for an even larger standard deviation, i.e. less
27
informative priors, to account for the uncertainly about this parameter. For the beta
parameters, ξ, ρA , ρM S , ρG , ρr , and λ, we centre the prior density in the middle of the
unit interval. According to our model, the parameters for the proportions of rationality
nh and nf are obtained from Equation A.68 which are pinned down by the fixed cost
parameters (Ch , Cf ). To ensure a positive cost of being rational we draw our priors from
an interval nh , nf ∈ [0, 0.5]. Our prior mean nh , nf = 0.25 lies at the mid-point of this
range which is also beta distributed.
A common theme in papers that study empirical RBC/DSGE models is the difficulty
in pinning down the parameter of labour supply elasticity φ, inference on the inverse
Frisch elasticity of labour supply has been found susceptible to model specifications, and
exhibiting wide posterior probability intervals. So we assume a normal distribution with
mean 2.0 and standard deviation of 0.5 for the parameter which is well within the range of
point estimates reported in the RBC and labour literature. For these parameter priors for
which we believe that non-negativity constraints are necessary gamma and inverse gamma
distributions are used. For the Taylor rule parameter on inflation the prior is set to obey
the Taylor principle that the interest rate changes more than the increase in inflation and
response is centred at the value suggested by Taylor. With regard to the output gap the
response of interest rate is smaller but we do not rule out non-zero responses for both
parameters. Finally the priors on the standard deviations of the exogenous shocks and
measurement errors are assumed to have inverse gamma distributions. The uncertainty
held about these elements motivates an open interval for their priors that excludes zero
and is unbounded.
5.3
Identification Diagnostics
It is necessary to confront the question of parameter identifiability in our DSGE models
before taking them to the data, as model or parameter identification is a prerequisite for the
informativeness of different estimators, and their effectiveness when one uses the models
to address policy questions. In this section we focus on detecting parameter identification
difficulties that are inherent in the structure of the models. As mentioned we fix some
parameters before estimation because of their non-identification in the model solution (at
first order). The aim of this section is to scan the parameters we choose to estimate
28
Parameter
Notation
Calvo prices
ξ
Labour supply elasticity
φ
Speed of learning
λ
Proportion of rationality
Proportion of rational households nh
Proportion of rational firms
nf
Interest rate rule
Inflation
θπ
Output
θy
Interest rate smoothing
ρr
AR(1) coefficient
Technology
ρA
Price mark-up
ρM S
Government spending
ρG
Standard deviation of shocks
Technology trend
sd(ǫA,trend )
Technology
sd(ǫA )
Government spending
sd(ǫG )
Price mark-up
sd(ǫM S )
Monetary policy
sd(ǫM P S )
Standard deviation of measurement errors
Observation error (output)
sd(ǫy )
Observation error (inflation)
sd(ǫπ )
Prior
Density
B
N
B
distribution
Mean S.D/df
0.50
0.10
2.00
0.50
0.50
0.28
B
B
0.25
0.25
0.18
0.18
N
N
B
1.50
0.125
0.75
0.50
0.05
0.10
B
B
B
0.50
0.50
0.50
0.20
0.20
0.20
IG
IG
IG
IG
IG
0.10
0.10
0.10
0.10
0.10
2.00
2.00
2.00
2.00
2.00
IG
IG
0.10
0.10
Inf
Inf
Table 2: Prior Distributions
in terms of their identification in our models. Among the authors who have made the
most recent contributions to addressing the identification issues in DSGE models are
Iskrev (2008) and Iskrev (2010b), Canova and Sala (2009), and Komunjer and Ng (2011).
We use Iskrev (2010b)’s computational toolbox to perform formal identification checks
on the reduced form parameters and structural parameters. This approach is based on
evaluating analytically the information matrix of the reduced-form model and checking
for rank deficiency of gradient matrix (the Jacobian). Our checks are performed in terms
of a local analysis that is based on the identification evaluation at the point values of the
prior means in Table 2 and a ‘global’ prior exploration of point identification properties
by taking a Monte Carlo samples from the prior space. The identifiability of each draw
including the mean prior is established by studying the ranks of Jacobian of the model
and given the set of observable variables and the sample size (the sample moments).
29
We take our models to the identification toolbox that computes the Jacobian numerically of the model (the solution) and the moments for rank evaluations prior to estimating
them. To completely rule out a flat likelihood at the local point we also check collinearity
between the effects of different parameters on the likelihood. If there exists an exact linear dependence between a pair and among all possible combinations their effects on the
moments are not distinct and the violation of this condition must indicate a flat likelihood
and lack of identification. We find that the Jacobian matrix has full rank and that the
models can be identified locally within the prior space. This includes our key parameters
for behaviourial heterogeneity λ, nh and nf .
A further output of Iskrev (2010b)’s identification routine is the analysis of identification strength, i.e., focusing on weak identification, summarized in Figures 13 - 16 in
Appendix B. The procedures are based on either the asymptotic or a moment information
matrix. The first can be obtained given a sample of size T , whereas the second can be
computed based on Monte Carlo simulations for samples of size T , from which sample
moments of the observed variables are computed, forming a sample of N replicas of simulated moments. The corresponding information matrix is then obtained as IT (θ|mT )
= HT ΣmT HT , where ΣmT is the covariance matrix of simulated moments and HT the
derivatives of the vector collecting all the reduced-form coefficients. We now examine
more carefully all our parameters at the means of the prior and posterior distributions
and using the prior uncertainty. We focus this analysis on the two behaviourial models
and report the sensitivity measure and collinearity results for all parameters evaluated at
the prior mean, relative to the prior standard deviation and at the posterior mean obtained using the estimated models in the next section. Note that, in Appendix B, all their
parameters are sensitive in affecting the likelihood through their effects on the moments
of the observed variables. Table 3 highlights the effects of λ, nh and nf on the likelihood
which are very strong. Although some similarities in terms of pair-wise collinearity are
detected it is important to confirm that no linear dependance (non-identification) is found
across the estimated parameters in these models.
30
θi
λ
nh
nf
Prior
∆i θ i
25.0845
47.6276
186.7911
Sensitivity
Mean
Posterior Mean
∆prior
θ
∆i θ i
i
i
15.2038
19.5426
0.5148
27.7389
37.1511
5.3974
Prior
̺i
0.6532
0.9253
0.8390
Collinearity
Mean
Posterior
θj
̺i
θπ
0.5649
ǫA,trend 0.8685
θy
0.9927
Mean
θj
ǫπ
θπ
ξ
Table 3: Identification at Priors and Posteriors (Parameters λ, nh and nf )
5.4
Bayes Factor Comparison
We now turn to the estimation of the models using the Bayesian methods described above.
The joint posterior distribution of the estimated parameters is obtained in two steps. First,
the posterior mode and the Hessian matrix are obtained via standard numerical optimization routines. The Hessian matrix is then used in the Metropolis-Hastings (MH) algorithm
to generate a sample from the posterior distribution. Two parallel chains are used in the
Monte-Carlo Markov Chain Metropolis-Hastings (MCMC-MH) algorithm. Thus, 100,000
random draws (though the first 25% ‘burn-in’ observations are discarded to remove any
dependance from the initial conditions) from the posterior density are obtained via the
MCMC-MH algorithm, with the variance-covariance matrix of the perturbation term in
the algorithm being adjusted in order to obtain reasonable acceptance rates (between 20%30%). We run an iterative process of MCMC simulations in order to calibrate the scaling
factor to achieve the desired rate of acceptance which is key for the speed of convergence
of the MCMC-MH chains, which are also sensitive to the number of MCMC iterations.
The former ensures that more of the parameter region is searched more regularly, but at
the expense of reducing the acceptance ratio. In this estimation the number of draws we
choose is sufficient to allow for convergence. To formally test and to check the convergence,
besides calibrating the acceptance rate, we use the convergence indicators recommended
by Brooks and Gelman (1998b) and Gelman et al. (2004) which use the standard Analysis
of Variance considerations and compare the variances of the elements within the sequence
of the drawn parameter space and to the variance across several sequences produced by
the MH-MCMC simulator given different initial conditions.
Table 4 contains summary statistics of the posterior distributions of three NK models.
We report posterior means of the parameters of interest and 95% probability intervals as
well as the posterior model odds. First look at the last row of Table 4 that reports the
31
Parameter
Model RE
Model IR
ξ
0.83 [0.79:0.87] 0.43 [0.30:0.56]
φ
2.06 [1.22:2.99] 1.96 [1.17:2.77]
Adaptive learning
λ
0.91 [0.91:1.00]
Proportion of rationality
nh
1.00
0.00
nf
1.00
0.00
Interest rate rule
θπ
2.64 [1.98:3.29] 1.08 [1.04:1.13]
θy
0.08 [0.02:0.15] -0.00 [-0.01:0.01]
ρr
0.80 [0.75:0.87] 0.41 [0.26:0.46]
AR(1) coefficient
ρA
0.60 [0.47:0.75] 0.96 [0.94:0.99]
ρG
0.50 [0.19:0.86] 0.50 [0.16:0.80]
ρM S
0.47 [0.17:0.83] 0.50 [0.18:0.83]
Standard deviation of shocks
sd(ǫA,trend )
0.61 [0.53:0.70] 0.09 [0.06:0.11]
sd(ǫA )
1.52 [0.66:2.24] 0.55 [0.49:0.62]
sd(ǫG )
0.08 [0.03:0.13] 0.10 [0.03:0.21]
sd(ǫM S )
0.07 [0.03:0.12] 0.11 [0.02:0.27]
sd(ǫM P S )
0.17 [0.14:0.21] 0.04 [0.02:0.05]
Standard deviation of measurement errors
sd(ǫy )
0.07 [0.02:0.14] 0.06 [0.02:0.11]
sd(ǫπ )
0.07 [0.02:0.11] 0.52 [0.46:0.58]
Price contract length
1
5.88
1.75
1−ξ
LL and posterior model odd
Marginal likelihood
-125.24
-117.26
Prob.
0.0002
0.5473
Model IR-RE
0.44 [0.29:0.58]
1.92 [1.06:2.75]
0.90 [0.75:1.00]
0.09 [0.02:0.16]
0.10 [0.02:0.17]
1.08 [1.03:1.12]
0.00 [-0.01:0.01]
0.41 [0.36:0.46]
0.96 [0.93:0.99]
0.50 [0.16:0.83]
0.50 [0.17:0.83]
0.08
0.56
0.16
0.09
0.04
[0.06:0.11]
[0.49:0.63]
[0.02:0.41]
[0.03:0.17]
[0.02:0.05]
0.06 [0.03:0.09]
0.53 [0.45:0.58]
1.79
-117.45
0.4526
Table 4: Bayesian Posterior Distributions: Perfect Information Assumption for RE
Agents. For IR and IR-RE observations with a lag. α = 0.7 imposed. The trend or mean
of the data variables are calculated directly from the data and not estimated with the rest
of the model. The steady state is consistent with these values.
32
posterior model odds, revealing that Models IR and IR-RE substantially outperform their
RE counterpart with posterior probabilities of 55% and 45%. Both IR and IR-RE clearly
win the likelihood ranking and the rational extreme RE model is firmly rejected by the
data. As noted we employ the Bayes Factor (BF) from the model marginal likelihoods to
gauge the relative merits across the three models. Formally using the Bayesian statistical
language and according to Kass and Raftery (1995), a BF of 3-10 is “slight evidence”
in favour of model i over j. This corresponds to a marginal likelihood difference in the
range [ln3, ln10]= [1.10, 2.30]. A BF of 10-100 or a marginal likelihood range of [2.30,
4.61] is “strong to very strong evidence”; a BF over 100 (marginal likelihood difference
over 4.61) offers “decisive evidence”. The posterior BF of the marginal likelihoods of
RE and IR is e7.98 ≈ 2921.93 which offers decisive support to the presence of pure IR
and composite behaviours from the US data set we observe. However the BF difference
between the two non-rational models is much smaller (e0.19 ≈ 1.21) which in Kass and
Raftery (1995)’s language suggests that the data’s preference for IR over IR-RE is “barely
worth mentioning”. Although IR is supported by the data with a probability of 55%, the
evidence is favour of this in terms of the likelihood comparison, is not sufficient for us to
ignore IR-RE for conducting policy analysis.
5.5
Results for Rational Expectations Model
First, we briefly discuss the posterior estimates obtained for the US economy using the
standard canonical NK-rational model. It is interesting to note that the price stickiness
parameter, ξ, is estimated to be much larger than assumed in the prior distribution. This
implies that there is a considerable degree of price stickiness. We find that the implied
average contract duration is about 5.88 quarters from this model, suggesting that firms
change prices as infrequently as once every 5.88 quarters on average, which seems to be
somewhat counterintuitive by comparison with the studies in other developed economy
literature. In addition, the posteriors of this model also indicate a Frisch labour supply
elasticity, φ−1 = 0.49, a strong response to inflation that satisfies the Taylor principle, θπ =
2.46, and evidence for the monetary authority’s smoothing of interest rate (ρr = 0.80). In
terms of the persistence of the exogenous shocks, the estimates of the AR(1) coefficients
show that the technology shock is inertial. According to the estimated standard deviation,
33
the technology shock stands out as being the most volatile shock in this economy. As
expected the interest rate policy shock is less volatile and is less important in driving
inflation, consumption and output. The variations in measurement errors of output and
inflation are relatively moderate in this model.
Overall although many of these estimates are in the range often found in the existing
literature, this simple model with extreme rationality seems to be less able to deliver
plausible estimates of some parameters for an econometrician with the priors specified in
Table 2. Namely, ξ implies implausibly infrequent price changing horizons and the wide
confidence bands suggest that φ is not tightly estimated and is poorly determined. We
now move on to focusing on the preferred models that introduce individual rationality and
expectations heterogeneity in the next section.
5.6
Results for Behavioural Models
The first ‘behavioural’ heterogeneity model IR introduces individual rationality in the NK
Model as the agents’ information processing mechanism, departing from the above extreme
rational assumption from the RE NK model. In this framework individual household and
firms form exogenous adaptive point expectations on their consumption and pricing decisions, respectively. The IR solution equilibrium we propose departs from the standard RE
solutions and allows a process of adaptive learning driven by the speed of learning parameter λ ∈ [0, 1] for the household and firms respectively. The closer λ is to zero slower the
learning process is, which is the key mechanism of this setup because this introduces more
dynamics into the model, offering the possibility of capturing the observable data better.
The second behavioural model IR-RE assumes heterogeneous expectations, encompasses
the first and allows a proportion of agents to form expectations rationally. It not only emphasises individual rationality but formulates its payoffs (costs) for households and firms
which are now estimated in our procedure (i.e. the steady state proportions of rational
agents nh and nf which can derived from the fixed costs of rationality).
Now also in Table 4 we find that the point estimates under the both assumptions are
tight and plausible. In particular, focusing on the parameters characterizing the degree
of price stickiness ξ, the mean estimates report an average price contract duration of
around 1.75 and 1.79 quarters for IR and IR-RE. Their estimated 95% intervals imply
34
that price contacts change in the ranges of ∈ (1.43, 2.27) and ∈ (1.41, 2.38) suggesting now
that the firms of IR and IR-RE economies change prices as frequently as once about 1.4
quarters up to semi-annually or slightly longer. Another notable difference comes from the
estimates of φ−1 ∈ (0.36, 0.85) and φ−1 ∈ (0.36, 0.94) which are large Frisch labour supply
elasticities for IR and IR-RE models and are significantly estimated. The mean estimate
of φ and the value of α determine the point estimates for κ which is then found to be
fairly strong and in accordance with the values found in the literature. These estimates
provide the evidence of plausible explanation on the elements of the behavioural models
that interact endogenously to capture fluctuations around the data. The estimates of the
AR(1) coefficients show that the technology shock is now significantly inertial. With IR in
the model, the exogenous technology shock volatility contributes the most to the variation
in the data and the monetary policy volatility mattes much less for this aspect of the fit.
The price mark-up shock is larger than that of the RE model because the expectation
heterogeneity in the model increases inflation volatility and acts as a persistent force in
this behavioural economy in the inflation fluctuations (this is also evident in the following
section when we examine the implied model moments). The measurement error on inflation
also has some sizeable contribution.
As noted both behavioural models decisively dominate the purely rational model and
this is confirmed by the estimates of the 3 behavioural parameters associated with adaptive
learning and exogenous proportions of rationality which are statistically different from zero
according to the 95% intervals suggesting that they are playing important roles in this
economy that generate a degree of endogenous inertia in the NK models. The estimated
nh and nf suggest that around 9% households and 10% firms are fully rational when
making expected consumption and pricing decisions respectively. Note that this also
implies that the estimated fixed costs for being rational for households and firms are
nf
nh
=
1.0048
and
C
=
−
log
= 0.9542. Again the 95% intervals
Ch = − log 1−n
f
1−nf
h
combined with the large differences in likelihoods clearly reject the pure RE assumption
and indicate a degree of bounded individual rationality on economic dynamics. For those
who are internally rational and are capable of learning using simple adaptive rules from
their rational counterparts learning is fast for the households learning about their expected
future consumption and for the firms about the future movements in inflation with λ = 0.9.
35
In terms of the policy rule implications, we find that for both models the parameter
estimate for the degree of interest rate smoothing indicates that there is a low degree of
persistence in the nominal interest rate which is much lower than observed in the literature.
The response of output (θy ) is very low, nearly non-existent, while the feedback to inflation
(θπ ) is strong relatively, implying a stronger concern from the monetary authorities in the
inflation variability, relative to the moments in output, which is caused by the varying
forecast behaviours from agents’ heterogenous expectations.
Furthermore Figure 15 plots the prior and posterior distributions for Model IR-RE,
the most general behavioural model which also attains the better probabilistic assessment
from the data. The location and the shape of the posterior distributions are largely
independent of the priors we have selected since most priors are broadly less informative
for the parameters that are well-identified in the model structure. Most of the posterior
distributions seem to be roughly symmetric implying that the mean and median coincide.
Interesting to note that assuming very diffuse priors on the learning parameter λ, we find
that the data is very informative about this parameter, strengthening the strong empirical
support from the observations for the learning processes in the IR and IR-RE economies.
This observation also applies to the estimated proportions of rational or non-rational
agents. Indeed, from Figure 15 the posterior curves have updated significantly from the
prior ones11 .
6
Imperfect Information and Informational Consistency
Before we turn to the main subject of this section (informational consistency), we first
recall the identification analysis in Section 5.3, in Table 3, that the sensitivity effect of nf
at its posterior mean point is relatively weak (0.5148). Also at their estimated values, the
pair-wise collinearity between nf and ξ is very close to an exact linear dependence between
the pair (0.9927). From high correlations to near-exact collinearity one may suspect some
weak identification (due to partial identification). Figure 17 in the Appendix shows the
identification strength and sensitivity component in the moments using the composite
RE-IR estimation results and shows again the sensitive strength in the moments of nh
11
The identification analysis of the parameters at the mean of the posterior distribution in Section
5.3 and Appendix B also report the significant elasticity of the likelihood with respect to the estimated
parameters in the two IR models.
36
is very weak. Therefore in this section we compare the cases without estimating nf and
nh so the proportions nh = nf = n are fixed to the values we used in the simulation
and stability section earlier in the paper. Table 5 below compares all together 5 models
including the pure RE, pure IR models and two hybrid RE-IR models with n = 0.5 and
n = 0.1, respectively, along with a RE counterpart when the extreme perfect information
assumption is relaxed – this perfect (asymmetric) information (PI) assumption is described
in the following section.
The main findings in this section so far are consistent with the previous analysis, that
is, adding exogenously proportions of rationality to Pure RE model helps improve the
model’s empirical performance. The likelihood values increase significantly towards Pure
IR model which eventually decisively dominates the purely rational model. The parameterizations further indicate that a degree of bounded internal rationality on economic
dynamics is decisively supported by the data we observe. Again focusing on the parameters characterizing the degree of price stickiness, ξ, the mean estimates report an average
price contract duration of around 1.7 quarters suggesting that the firms that are internally rational change prices slightly over semi-annually, whereas the RE result reports an
estimated price stickiness of 1.4 years, which again implies an implausibly infrequent price
changing horizon.
In our estimation procedure up to now we have followed the standard practice of assuming agents forming rational expectations have perfect information (PI) of all states,
include shock processes, whereas the econometrician only observes the data variables.
Based on the methodology developed in Levine et al. (2012), we now relax this extreme
information assumption and instead assume that both private agents and the econometrician have the same imperfect information (II) and both use the Kalman Filter, the former
to form expectations of unobserved state variables and the latter to calculate the likelihood
in the Bayesian estimation. Although perfect information on idiosyncratic shocks may be
available to economic agents, it is implausible to assume that they have full information
on economy-wide shocks. Furthermore, as shown in Levine et al. (2012), as long as the
number of shocks exceed the number of observables (as is the case in our set-up), the
solution under II and PI differ and the latter provides an alternative source of endogenous
persistence that offers the potential for capturing the dynamics observed in the data. We
37
therefore empirically address these alternative information assumptions to assess whether
parameter estimates are consistent across them and which of these choices leads to a better
model fit.
Indeed, restricting II to the fully rational DSGE model yields the highest model performance in terms of their likelihood race. A LL of -126.60 attains the highest posterior
probability of 97% among all the competing models. Overall, the parameter estimates are
reasonably robust across information specifications, despite the fact that the II alternative
leads to a much better model fit based on the corresponding posterior marginal likelihood.
It is interesting to note that the point estimates of almost every single parameter under II
are tighter and more strongly determined compared with the case under the standard PI
assumption, i.e., the confident intervals are more tightly estimated with II, so this clearly
explains its superior performance in the likelihood race. The estimated contract length is,
however, higher than the IR cases, but is found to be shorter than the standard RE model.
Evidently, using formal Bayes Factor comparisons, we show that the departures of conventional DSGE modelling from both the extreme rationality assumption and assuming a
perfect information set when solving and estimating the model offer decisive improvement
in terms of the overall model fit. In the next section, we extend our empirical investigation
to check the absolute model fit to data, that is, how these alternatives to RE(PI) perform
in terms of capturing the observed population moments.12
7
Model Validation
In this section we examine the model summary statistics, such as first and second moments, which have been standard for researchers to assess models in the literature on
DSGE models, especially in the RBC tradition. The question of matching the moments
is whether the models estimated by a Bayesian-likelihood method correctly predict population moments, such as the variables’ volatility or their correlation. We consider second
moments and autocorrelations in turn. The conditional second moments implied by the
estimated models in Sections 6 and 7 such as impulse responses also focus on the Pure RE
(PI) and IR models and Pure RE(II).
12
Our composite RE-IR model is actually a RE(PI)-IR one. Software limitations for now prevent us
examining RE(II)-IR composites.
38
Parameter Pure RE (PI) RE-IR (n=0.5) RE-IR (n=0.1)
Pure IR
Pure RE (II)
ξ
0.80 [0.75:0.86] 0.43 [0.29:0.57] 0.42 [0.29:0.56] 0.42 [0.28:0.56] 0.78 [0.74:0.81]
φ
1.80 [0.82:2.73] 1.96 [1.16:2.80] 1.90 [1.07:2.75] 1.99 [1.06:2.80] 1.64 [0.81:2.46]
Adaptive learning
λ
0.93 [0.85:1.00] 0.91 [0.80:1.00] 0.91 [0.79:1.00]
Proportion of rationality
nh
1.00
0.50
0.10
0.00
1.00
nf
1.00
0.50
0.10
0.00
1.00
Interest rate rule
θπ
3.21 [2.62:3.86] 1.07 [1.02:1.11] 1.07 [1.03:1.11] 1.07 [1.03:1.11] 3.41 [2.84:3.94]
θy
0.04 [-0.05:0.13] -0.01 [-0.01:0.00] -0.00 [-0.01:0.01] 0.00 [-0.01:0.01] 0.04 [-0.03:0.10]
ρr
0.71 [0.61:0.83] 0.41 [0.35:0.46] 0.41 [0.35:0.46] 0.41 [0.36:0.46] 0.65 [0.58:0.73]
AR(1) coefficient
ρA
0.71 [0.60:0.81] 0.96 [0.93:0.99] 0.96 [0.93:0.99] 0.96 [0.93:0.99] 0.77 [0.73:0.82]
ρG
0.51 [0.22:0.86] 0.49 [0.17:0.83] 0.49 [0.17:0.83] 0.50 [0.16:0.81] 0.51 [0.19:0.84]
ρM S
0.51 [0.19:0.85] 0.51 [0.19:0.85] 0.49 [0.17:0.85] 0.50 [0.17:0.82] 0.58 [0.33:0.82]
Standard deviation of shocks
sd(ǫA,trend ) 0.66 [0.56:0.75] 0.10 [0.07:0.12] 0.08 [0.05:0.12] 0.09 [0.06:0.12] 0.70 [0.61:0.79]
sd(ǫA )
1.04 [0.64:1.57] 0.59 [0.52:0.66] 0.60 [0.53:0.68] 0.59 [0.51:0.65] 0.82 [0.67:0.98]
sd(ǫG )
0.06 [0.03:0.11] 0.08 [0.12:0.15] 0.10 [0.02:0.22] 0.10 [0.02:0.19] 0.08 [0.02:0.15]
sd(ǫM S )
0.06 [0.03:0.10] 0.08 [0.02:0.15] 0.07 [0.03:0.12] 0.04 [0.02:0.06] 0.09 [0.02:0.17]
sd(ǫM P S ) 0.09 [0.03:0.19] 0.04 [0.02:0.05] 0.04 [0.02:0.06] 0.06 [0.03:0.10] 0.05 [0.02:0.06]
Standard deviation of measurement errors
sd(ǫy )
0.08 [0.03:0.13] 0.06 [0.02:0.10] 0.07 [0.02:0.12] 0.06 [0.03:0.10] 0.07 [0.02:0.11]
sd(ǫπ )
0.18 [0.05:0.27] 0.54 [0.48:0.61] 0.54 [0.48:0.60] 0.53 [0.47:0.59] 0.24 [0.21:0.27]
Price contract length
1
5.00
1.75
1.72
1.72
4.55
1−ξ
Marginal likelihood and posterior model odd
LL
-143.15
-134.23
-131.30
-130.64
-126.60
Prob
0.0000
0.0005
0.0089
0.0171
0.9735
Table 5: Bayesian Posterior Distributions for RE (PI), IR, Hybrid RE-IR and
RE(II): Perfect Information and Imperfect Information (II) Assumptions for RE Agents.
For IR and RE-IR and RE(II) observations with a lag. The information sets for lag 0
and lag 1 cases at time t are It = {Yt , Πt , Rn,t } and It = {Yt−1 , Πt−1 , Rn,t } respectively.
α = 0.7, n = nh = nf = 0.5, 0.1 imposed in this estimation. The trend or mean of the
data variables are calculated directly from the data and not estimated with the rest of the
model. The steady state is consistent with these values.
39
7.1
Matching Second Moments
In terms of matching volatility Pure IR is able to generate output standard deviation
very close to the data and perform better at getting closer to the interest rate data,
whereas the pure RE model performs very poorly at capturing output volatility. The
behavioural models’ ability of matching inflation moments is distorted, generating much
volatility in inflation than the data and the RE models with II and PI, and as noted
this can be explained by the role played by the more volatile pricing shock found in the
estimated models which gives rise to the amplification effects on inflation dynamics caused
by the expectation heterogeneity in the behavioural economy. Table 6 also reports the
cross-correlations of the 3 observable variables vis-a-vis output. Model RE does well and
predicts the correct sign for the output inflation cross-correlation whereas the IR model
performs poorly and indeed have the wrong sign. However in terms of the output-interest
rate correlations, the II assumption improves in this dimension, capturing the correct sign
and getting much closer to the data.
Data
Pure RE
Pure IR
RE(II)
Data
Pure RE (PI)
Pure IR
Pure RE (II)
Standard Deviation
Output Inflation Interest rate
0.58
0.24
0.61
1.35
0.26
0.33
0.57
0.72
0.50
1.09
0.28
0.30
Cross-correlation with Output
1.00
-0.12
0.22
1.00
-0.04
0.03
1.00
0.03
0.03
1.00
0.09
0.12
Table 6: Selected Second Moments
If we look at the autocorrelations up to 10 lags in Figure 7.1, the picture is also somewhat mixed. Overall it shows fairly good goodness-of-fit of Pure IR model to data in
terms of capturing the autocorrelations up to many lags – in any case, almost all of the
moments are inside the 95% confidence intervals of the empirical moments of autocorrelations, which leads to some confidence in all the estimated models. Model RE with and
without PI is problematic in reproducing the output autocorrelations at the first two and
three orders, ACF lying well-outside of the lower interval and having the wrong sign. The
40
IR model is capable of generating more persistence in inflation and interest rate up to
many lags and the reason for this lies in the estimated fast learning mechanism of the
adaptive expectations scheme (λ = 0.9) in, for example, forecasting inflation movements.
Finally switching the information set from PI to II for the RE model produces a little
more persistence, captured by the implied correlograms of output and interest rate.
The findings in this section are generally in line with those in Jang and Sacht (2014),
who conduct an empirical investigation on moment matching using a bounded rationality
behavioural model à la De Grauwe (2011) estimated by the Simulated Method of Moments
for the Euro Area. They find that their results can mimic the real data well, slightly
outperforming the linear RE counterpart in some of the moments, or are at least as good
as the RE model in terms of providing fits for auto- and cross-covariances of the data.
Perhaps the main message to emerge from this RBC type of model validity exercise is
that it can be misleading to assess model fit using a selective choice of second moment
comparisons as there are trade-offs in terms of fitting some second moments well, at the
expense of others. As pointed out the most comprehensive form of assessment of competing
models is via likelihood comparisons. In this moments analysis there is some evidence that
shows a good fit of both RE(II) and Pure IR, in particular how the IR model captures
the autocorrelation dynamics and output volatility, to some dimensions of the data but
needs to analysed with some caution and the probabilistic assessment using the marginal
likelihoods provides the most decisive support.
7.2
Posterior Impulse Response Analysis
As shown from the estimated models and the moment analysis above, both the learning
mechanisms through agents IR and having imperfect information can introduce more
persistence into the standard RE solutions. As a result, the empirical models incorporating
either form of endogenous learning can outperform the standard RE model. The empirical
impulse response functions estimated in this section confirm the previous simulation. In
Figures 11 - 13 it introduces more persistence compared with RE, generating more humpshaped trajectories after the system is shocked. This is especially clear in the case of
assuming II, suggesting this feature always leads to a better fit of the data better without
relying on other persistence mechanisms.
41
Output
0.6
Inflation
1
0.8
0.4
ACF
ACF
0.6
0.2
0
0.4
0.2
0
-0.2
-0.2
-0.4
-0.4
1
2
3
4
5
6
7
8
9
10
1
2
3
Lag
4
5
6
7
8
9
10
Lag
Interest rate
1
0.8
ACF
0.6
0.4
0.2
0
-0.2
-0.4
1
2
3
4
5
6
7
8
9
10
Lag
Data
Pure RE
Pure IR
RE(II)
Figure 10: Autocorrelations of Observables in the Actual Data and in the Estimated
Modelsz
z
The approximate 95 percent confidence bands are constructed using the large-lag standard
errors (see Anderson (1976)).
42
Output
Hours worked
-0.1
0.4
0.3
0.2
0.4
-0.2
% dev from SS
% dev from SS
% dev from SS
0.5
Real wage
0.6
0.6
-0.3
-0.4
-0.5
0.2
0
-0.2
-0.6
-0.4
0.1
-0.7
5
10
15
20
5
10
Quarters
Real interest rate
20
5
0
-0.05
15
20
15
20
Inflation
-0.02
-0.02
-0.04
-0.04
% dev from SS
% dev from SS
0.05
10
Quarters
Nominal interest rate
0.1
% dev from SS
15
Quarters
-0.06
-0.08
-0.1
-0.12
-0.06
-0.08
-0.1
-0.12
-0.14
-0.1
-0.14
5
10
15
20
5
Quarters
10
Pure RE
15
Quarters
Pure IR
20
5
10
Quarters
RE(II)
Figure 11: Estimated Impulse Responses – Technology Shock
Output
Hours worked
0
0.02
Real wage
0
-0.04
-0.06
-0.08
-0.05
% dev from SS
% dev from SS
% dev from SS
0
-0.02
-0.1
-0.1
-0.2
-0.3
-0.15
-0.4
5
10
15
20
5
10
Quarters
15
20
5
Quarters
Real interest rate
Nominal interest rate
15
20
15
20
Inflation
0
0.05
0.08
0.06
0.04
0.02
% dev from SS
-0.02
% dev from SS
% dev from SS
10
Quarters
0
-0.05
-0.04
-0.06
-0.08
0
-0.1
-0.1
-0.02
5
10
15
20
5
Quarters
10
Pure RE
15
Quarters
Pure IR
20
5
10
Quarters
RE(II)
Figure 12: Estimated Impulse Responses – Monetary Policy Shock
Output
0
Hours worked
Real wage
0
-0.005
-0.02
-0.015
% dev from SS
% dev from SS
-0.01
-0.01
-0.015
-0.02
-0.04
-0.06
-0.025
-0.08
-0.02
-0.03
5
10
15
20
5
10
Quarters
× 10-3
15
20
5
Quarters
× 10-3
Real interest rate
10
15
20
15
20
Quarters
× 10-3
Nominal interest rate
3
Inflation
2.5
2
1
0
-1
% dev from SS
% dev from SS
2.5
% dev from SS
% dev from SS
-0.005
2
1.5
1
0
0
5
10
Quarters
15
20
1
0.5
0.5
-2
2
1.5
5
Pure RE
10
Quarters
15
Pure IR
20
RE(II)
5
10
Quarters
43
Figure 13: Estimated Impulse Responses – Mark-up Shock
8
Conclusions
We have argued that NK behavioural models with Euler Learning (EL) provide inadequate micro-foundations for the learning process. EL implicitly assumes that learning
rules are correctly specified so that agents are learning about the equilibrium saddle-path
solution. But the misspecifed rules typically assumed in the NK behavioural literature are
parsimonious and therefore incorrectly specified. By contrast Internal Rationality (IR) is
consistent with misspecified learning rules. An additional feature of IR is that this approach makes less informational demands in the sense that it does not assume that agents
know they are identical. This assumption is in fact used in the linearization that is the
starting point of the EL approach.
The paper has examined the policy space of feedback parameters in a Taylor-type rule
with interest rate persistence. We find that IR has a smaller policy space than ER which
in turn is smaller than RE, making it more prone to local instability and the possibility
of chaos. The differences in the sizes of the policy spaces that result in a saddle-path
stable equilibrium are significant. Furthermore a clear ranking of the sizes of these spaces
emerges with RE ⊃ EL ⊃ IR. This means that unless the policy rule is designed for the
IR model, uncertainty as to which model of expectations is correct can lead to a rule that
is unstable or indeterminate.
Finally in a Bayesian estimation of the RE-IR composite model we find in a likelihood
race that the RE model with imperfect information (II) outperforms the IR model which
in turn outperforms RE with perfect information (PI). Second moment comparisons with
the data are mixed, but RE with II and IR capture more dimensions than RE. These
results suggest that persistence can be injected into the NK model to improve data fit
in two contrasting ways: bounded-rationality with learning, and retaining RE, but with
imperfect information Kalman-filtering learning.
Our results for a very simple NK model suggest a new agenda for constructing empirical
medium-sized NK models. Future work will embed the RE-IR composite model into a
richer NK model along the lines of Smets and Wouters (2007), use non-linear estimation
methods to identify a number of parameters involving reinforcement learning that are not
identified using linear Bayesian estimation and examine optimal monetary policy. Future
work on the policy aspect will follow Hall and Mitchell (2007), Geweke and Amisano
44
(2012) and Deak et al. (2017) and estimate an optimal pool of RE and IR-RE composites
to design a robust rule across such model variants.
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Sims, C. (1980). Macroeconomics and Reality. Econometrica, 48, 1–48.
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A
Foundations of the New Keynesian Model with Internal
Rationality
We now make explicit the micro-foundations of the NK model and the consumption and
price-setting behaviour in particular by deriving the non-linear decision rules. We proceed
from rational expectations to internal rationality in stages.
A.1
A.1.1
The Rational Expectations Model
Households
Household j choose between work and leisure and therefore how much labour they supply.
Let Ct (j) be consumption and Ht (j) be the proportion of this available for work or leisure
spent at the former. The single-period utility is
Ut (j) = U (Ct (j), Ht (j)) = log(Ct (j)) − κ
49
Ht (j)1+φ
1+φ
(A.1)
In a stochastic environment, the value function of the representative household at time
t is given by
Vt (j) = Vt (Bt−1 (j)) = Et
"
∞
X
s
#
β U (Ct+s (j), Ht+s (j))
s=0
(A.2)
For the household’s problem at time t is to choose paths for consumption {Ct (j)}, labour
supply {Ht (j)} and holdings of financial savings to maximize Vt (j) given by (A.2) given
its budget constraint in period t
Bt (j) = Rt Bt−1 (j) + Wt Ht (j) + Γt − Ct (j)
(A.3)
where Bt (j) is the given net stock of financial assets at the end of period t, Wt is the wage
rate and Rt is the ex post real interest rate paid on assets held at the beginning of period
t given by
Rt =
Rn,t−1
Πt
(A.4)
where Rn,t and Πt are the nominal interest and inflation rates respectively and Γt are profits from wholesale and retail firms owned by households. Wt , Rt and Γt are all exogenous
to household j. As usual all variables are expressed in real terms relative to the price of
final output.
The first order conditions are
UC,t (j) = βEt [Rt+1 UC,t+1 (j)]
UL,t (j)
= Wt
UC,t (j)
(A.5)
(A.6)
An equivalent representation of the Euler consumption equation (A.5) is
1 = Et [Λt,t+1 (j)Rt+1 ]
where Λt,t+1 (j) ≡ β
UC,t+1 (j)
UC,t (j)
(A.7)
is the real stochastic discount factor for household j, over the
interval [t, t + 1].
50
1
Ct (j)
= βEt
φ
and UH,t = −κHtφ so these become
1
Ct
For our choice of utility function UC,t =
Rt+1
Ct+1 (j)
= Wt ⇒ Ht (j) =
κCt (j)Ht (j)
(A.8)
Wt
κCt (j)
1
φ
(A.9)
In a symmetric equilibrium of identical households Ct (j) = Ct , aggregate per household
consumption and Ht (j) = Ht , average hours worked,
A.1.2
Firms in the Wholesale
Wholesale firms employ a decreasing returns to scale production function in labour to
produce a homogeneous output
YtW = F (At , Ht ) = At Htα
(A.10)
where At is total factor productivity. Profit-maximizing demand for labour results in the
first order condition
Wt =
P W YtW
PtW
FH,t = α t
Pt
Pt Ht
(A.11)
Hence from (A.9) and (A.11) we have
Ht =
A.1.3

PW
α Ptt

κ

1
1+φ
(A.12)

Firms in the Retail Sector
The retail sector costlessly converts a homogeneous wholesale good into a basket of differentiated goods for aggregate consumption
Ct =
Z
1
(ζ−1)/ζ
Ct (m)
0
dm
ζ/(ζ−1)
(A.13)
where ζ is the elasticity of substitution. For each m, the consumer chooses Ct (m) at a
R1
price Pt (m) to maximize (A.13) given total expenditure 0 Pt (m)Ct (m)dm. This results
in a set of consumption demand equations for each differentiated good m with price Pt (m)
51
of the form
Ct (m) =
where Pt =
hR
1
1−ζ dm
0 Pt (m)
i
Pt (m)
Pt
1
1−ζ
−ζ
Ct ⇒ Yt (m) =
Pt (m)
Pt
−ζ
Yt
(A.14)
. Pt is the aggregate price index. Ct and Pt are Dixit-
Stigliz aggregates – see Dixit and Stiglitz (1977).
Following Calvo (1983), we now assume that there is a probability of 1 − ξ at each
period that the price of each retail good m is set optimally to Pt0 (m). If the price is not
re-optimized, then it is held fixed.13 For each retail producer m, given its real marginal
cost M Ct , the objective is at time t to choose {Pt0 (m)} to maximize discounted profits
Et
∞
X
k=0
ξ k Λt,t+k Yt+k (m) Pt0 (m) − Pt+k M Ct+k
subject to (A.14), where Λt,t+k ≡ β k
UC,t+k /Pt+k
UC,t /Pt
(A.15)
is now the nominal stochastic discount
factor over the interval [t, t + k]. The solution to this is
Et
∞
X
k
ξ Λt,t+k Yt+k (m)
k=0
Pt0 (m)
1
Pt+k M Ct+k = 0
−
(1 − 1/ζ)
(A.16)
and by the law of large numbers the evolution of the price index is given by
1−ζ
0
Pt+1
= ξPt1−ζ + (1 − ξ)(Pt+1
)1−ζ
(A.17)
In order to set up the model in non-linear form as a set of difference equations, required
for software packages such a Dynare, we need to represent the price dynamics as difference
equations. First define k periods ahead inflation as
Πt,t+k ≡
Pt+k
Pt+1 Pt+2 Pt+k−1
= Πt+1 Πt+2 · ·Πt+k
=
··
Pt
Pt Pt+1
Pt+k
Note that Πt,t+1 = Πt+1 and Πt,t = 1.
Next using (A.14) with Pt+k (m) = P0 (m), the price set at time t which survives with
13
Thus we can interpret
1
1−ξ
as the average duration for which prices are left unchanged.
52
probability ξ, we have that
Λt,t+k Yt+k (m) = β
k UC,t+k
UC,t
Hence cancelling out
Et
∞
X
k
(ξβ)
Pt
Pt+k
P0 (m) −ζ
Pt
P0 (m)
Pt+k
−ζ
Yt+k = β
and multiplying by
UC,t+k Πζ−1
t,t+k Yt+k
k=0
k UC,t+k
UC,t
UC,t
Pt
Πζ−1
t,t+k
P0 (m)
Pt
−ζ
Yt+k
we can write (A.16) as
Pt0 (m)
− Πt+k M Ct+k M St+k = 0
Pt
(A.18)
We seek a symmetric equilibrium where firms who are either re-setting their prices or
are locked into a contract are identical. In such an equilibrium, the price dynamics can
be written as difference equations as follows (see Appendix A):
Jt
Pt0
=
Pt
JJt
i
h
ζ−1
JJt − ξEt Πt+1 JJt+1 Λt,t+1 = Yt
i
h
Jt − ξEt Πζt+1 Jt+1 Λt,t+1 =
(A.19)
(A.20)
1
1−
1
ζ
!
Y t M Ct M S t
(A.21)
Jt 1−ζ
1 =
+ (1 − ξ)
JJt
ζ
ζ
α
J
t
∆t = ξΠtα ∆t−1 + (1 − ξ)
JJt
ξΠζ−1
t
M Ct =
PtW
Pt
=
Wt
FH,t
(A.22)
(A.23)
(A.24)
(A.25)
where (A.40) uses (A.11). We have introduced a mark-up shock M St . Notice that the
real marginal cost, M Ct , is no longer fixed as it in the flexi-price case model.
Price dispersion lowers aggregate output as follows. Market clearing in the labour
market gives
Ht =
YtW
At
α1
=
n
X
m=1
1
Ht (m) =
n X
Yt (m) α
m=1
At
53
=
Yt
At
1 X
ζ
n α
Pt (m) − α
m=1
Pt
(A.26)
using (A.10) and (A.14). Hence rearranging we have
Yt =
YtW
∆αt
(A.27)
where price dispersion is defined by
ζ
∆t ≡
n X
Pt (m) − α
Pt
m=1
!
(A.28)
Price dispersion is linked to inflation as follows. Assuming as before that the number
of firms is large we obtain the following dynamic relationship:
ζ
α
∆t = ξΠt ∆t−1 + (1 − ξ)
Jt
JJt
− ζ
α
(A.29)
Proof
See Appendix B.
A.1.4
Profits
To close the model with internal rationality we require total profits from retail and wholesale firms. Γt , remitted to households. This is given in real terms by
PW
PW
PW
Γt = Yt − t YtW + t YtW − Wt Ht = Yt − α t YtW
P
P
Pt
|
{zt
} | t
{z
}
retail
Wholesale
(A.30)
using the first-order condition (A.11).
A.1.5
Flexi-Price Output
In order to evaluate the output gap we need the flexi-price limit as ξ → 0 (but retaining
monopolistic competition with c > 0 and ζ < ∞ and the mark-up shock M St ). As for the
sticky-price model labour supply and demand conditions are
Wt = Ct κHtφ = Yt κHtφ = At κHtα+φ
PtM αYtW
= αM Ct At Htα−1
Wt =
Pt Ht
54
(A.31)
(A.32)
Hence Ht =
αM Ct
κ
1
1+φ
(A.33)
As ξ ⇒ 0 we have
M Ct M S t
Pt0
=1=
Pt
1 − 1ζ
(A.34)
It follow that
HtF
YtF
A.1.6
 1
 1+φ
1
α 1 − 1ζ
1+φ
α


=
=
M St κ
κM St
(A.35)
= At (HtF )α
(A.36)
Closing the Model
The model is closed with a resource constraint
Yt = Ct + Gt
(A.37)
where Gt is an exogenous demand process and a monetary policy rule for the nominal
interest rate is given by the following Taylor-type rule
log
Rn,t
Rn
Rn,t
Rn
= ρr log
Rn,t−1
Rn
= ρr log
Rn,t−1
Rn
+ (1 − ρr ) θθ log
Πt
Π
+ θy log
Yt
Y
+ ǫM P S,t, (A.38)
or
log
+ (1 − ρr )θπ log
Πt
Π
+ (1 − ρr )θy log
Yt
YtF
+ ǫM P S,t,
(A.39)
where YtF is the flexi-price level of output and ǫM P S,t is a monetary policy i.i.d shock
14
and finally there are two exogenous AR1 shock processes to technology and marginal cost
(the latter being interpreted as a mark-up shock):
log At − log A = ρA (log At−1 − log A) + ǫA,t
log Gt − log G = ρG (log Gt−1 − log G) + ǫG,t
log M St − log M S = ρM S (log M St−1 − log M S) + ǫM S,t
14
(A.40) is an ‘implementable’ form of the Taylor rule which stabilizes output about its steady state.
55
and ǫM,t is an i.i.d. shock to monetary policy.
A.1.7
Summary of Model
Households:
Ut = U (Ct , Ht ) = log Ct − κ
Ht1+φ
1+φ
UC,t = βEt [Rt+1 UC,t+1 ]
or Et [Λt,t+1 Rt+1 ] = 1 where Λt−1,t =
βUC,t
UC,t−1
Rn,t−1
Πt
1
=
Ct
= −κHtφ
Rt =
UC,t
UH,t
UL,t
= Wt
UC,t
Firms:
YtW
Yt
PtW
FH,t
Pt
Pt0
Pt
JJt
= F (At , Ht ) = At Htα
YtW
=
∆αt
PtW αYtW
=
= Wt
Pt Ht
Jt
=
JJt
i
h
JJ
Λ
= ξEt Πζ−1
t+1 t,t+1 + Yt
t+1
h
i
Jt = ξEt Πζt+1 Jt+1 Λt,t+1 +
1
1−
Jt 1−ζ
1 =
+ (1 − ξ)
JJt
ζ
ζ
Jt − α
α
∆t = ξΠt ∆t−1 + (1 − ξ)
JJt
ξΠζ−1
t
M Ct =
Wt
PtW
=
Pt
FH,t
Γt = Yt − αM Ct YtW
56
1
ζ
!
Y t M Ct M S t
Closure:
Yt =
Rn,t
log
=
Rn
Rn,t
=
or log
Rn
log At − log A =
Ct + Gt
Rn,t−1
Πt
Yt
ρr log
+ (1 − ρr ) θθ log
+ θy log
+ ǫM,t
Rn
Π
Y
Rn,t−1
Πt
Yt
ρr log
+ (1 − ρr )θπ log
+ (1 − ρr )θy log
+ ǫM,t
Rn
Π
YtF
ρA (log At−1 − log A) + ǫA,t
log Gt − log G = ρG (log Gt−1 − log G) + ǫG,t
log M St − log M S = ρM S (log M St−1 − log M S) + ǫM S,t
A.1.8
Steady State
In recursive form the zero-growth zero-inflation (Π = 1) steady state of can be written
1
β
Λ = β
R =
MC =
PW
P
C
Y
= 1−
= 1 − gy
H =
YW
Y
W
J
JJ
Hence with Π = 1, J
1
ζ
α∆α M C
κ(1 − gy )
= (AH)α
YW
=
∆α
PW Y W
= α
P H
Y M CUC
=
(1 − 1ζ )(1 − ξβΠζ )
Y UC
(1 − ξβΠζ−1 )
= JJ
=
∆ = 1
Γ = Y − αM CY W
For a particular steady state inflation rate Π > 1 the NK features of the steady state
57
become
1
1 − ξΠζ−1 1−ζ
=
1−ξ
1
J(1 − βξΠζ )
=
1−
ζ JJ(1 − βξΠζ−1 )
J −ζ
(1 − ξ)α JJ
∆ =
1 − ξΠζ
J
JJ
PW
MC =
P
Then P W Y W /P Y = M C∆/ 6= 1.
A.2
A Balanced-Non-Zero-Growth Steady State
We can easily set up the model with a balanced-exogenous-growth steady state. Now the
process for At is replaced with
At = Āt Act
Āt = (1 + g)Āt−1 exp(ǫA,t ) ⇒ log Āt = µ + log Āt−1 + ǫtrend,t
log Act − log Ac = ρA (log Act−1 − log Ac ) + ǫA,t
where µ = log(1 + g) ≈ g and At is a labour-augmenting technical progress parameter
which we decompose into a cyclical component, Act , modelled as a temporary AR1 process,
a stochastic trend, a random walk with drift, Āt . Thus the balanced growth deterministic
steady state path (bgp) is driven by labour-augmenting technical change growing at a net
rate g. If we put g = ǫtrend,t = 0 and Āt = 1, we arrive at our previous formulation with
Act = At .
Now stationarize variables by defining cyclical and stationary components:
(YtW )c ≡
Ctc ≡
Wtc ≡
YtW
= Act Htα
Āt
Ct
Āt
Wt
Āt
Utc ≡ log Ctc − κ
c
UC,t
≡
1
Ctc
58
Ht1+φ
1+φ
c
Λt,t+1 = β
UC,t+1
UC,t+1
= βg c
UC,t
UC,t
for all non-stationary variables where
gt ≡
βg ≡
(Āt − Āt−1 )
= (1 + g) exp(ǫA,t ) − 1
Āt
β
1 + gt
is the stochastic steady state growth rate and the Euler equation is still
Et [Λt,t+1 Rt+1 ]
and
ζ−1
c
c
c
c c
c
JJtc − ξEt [Π̃ζ−1
t+1 JJt+1 βg,t+1 (1 + gt+1 )] = JJt − ξβEt [Π̃t+1 JJt+1 ] = Yt UC,t UC,t
c
c
c
M Ct M S t
]Ytc UC,t
βg,t+1 (1 + gt+1 )] = Jtc − ξβEt [Π̃ζt+1 Jt+1
Jtc − ξEt [Π̃ζt+1 Jt+1
The steady state for the rest of the system is the same as the zero-growth one except
for the following relationships:
R =
1
Rn
=
βg
Π
(A.40)
where R and Rn are the real and nominal steady state interest rates and Π is inflation.
A.3
Exogenous Point Expectations
As a first step towards internal rationality we now formulate the consumption and pricing
decision of the household and firms respectively in terms of current and expected future
aggregate variables exogenous these agents.
59
A.3.1
Households
For households, solving (A.3) forward in time and imposing the transversality condition
on debt we can write
Bt−1 (j) = PVt (Ct (j)) − PVt (Wt Ht (j)) − PVt (Γt )
(A.41)
where the present (expected) value of a series {Xt+i }∞
i=0 at time t is defined by
PVt (Xt ) ≡ Et
∞
X
Xt+i
Xt
1
=
+
PVt+1 (Xt+1 )
Rt,t+i
Rt
Rt
(A.42)
i=0
where Rt,t+1 ≡ Rt Rt+1 Rt+2 · · · Rt+i is the real interest rate over the interval [t, t + i].
The forward-looking budget constraint (A.41) holds for the representative household.
In aggregate there is no net debt so Bt−1 = 0. Then in a symmetric equilibrium, substituting for Ht from (A.9) we have
PVt (Ct ) =
1
κ
1
φ
PVt

1+ φ1
W
 t
1
φ
Ct

(A.43)
(A.44)
 + PVt (Γt )
Solving (A.8) forward in time we have for i ≥ 1
1
Ct
i
= β Et
Rt+1,t+i
Ct+i
; i≥1
The internally rational solution to the household optimization problem seeks a solution
to its decision functions for Ct and Ht that are functions of non-rational point expectations
∞
∞
∗
∗
{E∗t Wt+i }∞
i=1 , {Et Rt+1,t+i }i=1 and {Et Γt+i }i=0 treated as exogenous processes given at
time t as opposed to rational model-consistent expectations {Et Wt+i }∞
i=1 etc.
15
With
point expectations we use (A.44) to obtain
E∗t Ct+i = Ct β i E∗t Rt+1,t+i ; i ≥ 1
E∗t (Wt+i Ht+i ) =
(A.45)
1
1+ φ
1 (E∗t Wt+i )
1
1
(A.46)
κ φ (E∗t Ct+i ) φ
15
With point expectations agents treat Et∗ (·) as certain, although the environment is stochastic (see
Evans and Honkapohja (2001), page 61). Since Et f (Xt ) ≈ f (Et (Xt )) and Et f (Xt Yt )) ≈ f (Et (Xt Yt ))
up to a first-order Taylor-series expansion, assuming point expectations is equivalent to using a linear
approximation of (A.43) and (A.44) as is usually done in the literature.
60
Substituting (A.45) and (A.46) into the forward-looking household budget constraint and
P
1
i
using ∞
i=0 β = 1−β , we arrive at
Ct
Rt (1 − β)
1
=
1+ 1
Wt φ
1
Rt (κCt ) φ
Ht =
Wt
κCt
1
∗
1+
∞
X
φ
1
Et Wt+i
−i
φ
(β )
+
E∗t Rt+1,t+i
i=1
!
∞
X
E∗t Γt+i
+
E∗t Rt,t+i
i=0
(A.47)
1
φ
(A.48)
Writing E∗t Rt,t+i = Rt E∗t Rt+1,t+i for i ≥ 1 (A.47) becomes
Ct
(1 − β)
1
=
1+ 1
Wt φ
1
(κCt ) φ
1
1+
∗
∞
X
φ
1
E
W
t+i
t
−i
(β φ )
+
∗
Et Rt+1,t+i
i=1
!
+ Γt +
∞
X
i=1
E∗t Γt+i
E∗t Rt+1,t+i
(A.49)
This can be written as
Ct
(1 − β)
Ω1,t
1
=
1+ 1
Wt φ
+ Ω1,t + Γt + Ω2,t
(A.50)
1
(κCt ) φ
∗
∗
1+ 1
1+ 1
∞
X
φ
φ
1
1
1
Et Wt+i
Et Wt+1
−i
−1
φ
φ
φ )−1 Ω
=
(β
)
+
(β
(β )
≡
1,t+1
E∗t Rt+1,t+i
E∗t Rt+1,t+1
i=1
Ω2,t ≡
∞
X
i=1
E∗t Γt+1
E∗t Γt+i
=
+ Ω2,t+1
E∗t Rt+1,t+i
E∗t Rt+1,t+1
With a non-zero balanced growth steady state β is simply replaced with βg in the expressions for Ct and Ω1,t above.
(A.48) and (A.50) and constitute the consumption and hours decision rules given point
∗
∞
∗
∞
expectations of {E∗t Wt+i }∞
i=1 , {Et Rt+1,t+i }i=1 and {Et Γt+i }i=0 .
A.3.2
Retail Firms
Turning next to price-setting by retail firms, write (A.20) and (A.21) as
Jt =
1
1−
1
ζ
JJt = Yt + Et
!
Y t M Ct M S t + E t
∞
X
∞
X
ξ k Λt,t+k Πζt,t+k Yt+k M Ct+k M St+k (A.51)
k=1
ξ k Λt,t+k Πζ−1
t,t+k Yt+k
k=1
61
(A.52)
Then assuming point expectations (as for households)
Jt =
=
1
1−
1
ζ
!
1
1−
JJt = Yt +
!
1
ζ
∞
X
k=1
!
(Yt M Ct M St + Ω3,t )
(A.53)
Y t M Ct M S t +
∞
X
ξ k E∗t Λt,t+k (E∗t Πt,t+k )ζ E∗t Yt+k E∗t M Ct+k E∗t M St+k
ξ k E∗t Λt,t+k (E∗t Πt,t+k )ζ−1 E∗t Yt+k
k=1
= Yt + Ω4,t
(A.54)
where noting that E∗t Λt,t+1 =
Ω3,t
1
E∗t Rt+1
and Πt,t+1 = Πt+1 we have
E∗t Πζt+1
(E∗t Πt+1 )ζ E∗t Yt+1 E∗t M Ct+1 E∗t M St+1
= ξ
+ξ ∗
Ω3,t+1
E∗t Rt+1
Et Rt+1
(A.55)
Ω4,t = ξ(
E∗t Πζ−1
E∗t Πt+1 )ζ−1 E∗t Yt+1
t+1
+
ξ
Ω4,t+1
E∗t Rt+1
E∗t Rt+1
Recalling that the optimal price re-setting decision rule is given by
(A.56)
PtO
Pt
=
Jt
JJt
,
(A.59) and (A.60) now give us the this rule given exogenous expectations of {E∗t Πt+i }∞
i=0 ,
∗
∞
∗
∞
∗
∞
{E∗t Rt,t+i }∞
i=0 , {Et Yt+i }i=0 , {Et M Ct+i }i=0 and {Et M St+i }i=0
With a non-zero-growth bgp this becomes
Jt
Jtc ≡
Āt
JJtc ≡
JJt
Āt
1
1−
=
1
ζ
!
Yt
+ Et
Āt
=
∞
X
Yt
Yt+k
M Ct M S t + E t
(1 + gt )k M Ct+k M
(A.57)
St+k
ξ k Λt,t+k Πζt,t+k
Āt
Āt+k
∞
X
k=1
ξ k Λt,t+k Πζ−1
t,t+k
k=1
Yt+k
(1 + gt )k
Āt+k
(A.58)
Then assuming point expectations (as for households) in stationarized form we have
Jtc =
=
1
1−
1
1−
JJt = Ytc +
1
ζ
!
!
1
ζ
∞
X
k=1
!
(Ytc M Ct M St + Ω3,t )
(A.59)
∞
X
c
c
(ξ(1 + gt ))k E∗t Λt,t+k (E∗t Πt,t+k )ζ E∗t Yt+k
E∗t M Ct+k E∗t M St+k
Y t M Ct M S t +
ξ k E∗t Λt,t+k (E∗t Πt,t+k )ζ−1 E∗t Yt+k
k=1
= Ytc + Ω4,t
(A.60)
62
where noting that E∗t Λt,t+1 =
Ω3,t
1
E∗t Rt+1
and Πt,t+1 = Πt+1 we have
E∗t Πζt+1
(E∗t Πt+1 )ζ E∗t Yt+1 E∗t M Ct+1 E∗t M St+1
= ξ(1 + gt )
+ ξ(1 + gt ) ∗
Ω3,t+1
E∗t Rt+1
Et Rt+1
(A.61)
Ω4,t = ξ(1 + gt )(
E∗t Πζ−1
E∗t Πt+1 )ζ−1 E∗t Yt+1
t+1
+
ξ(1
+
g
)
Ω4,t+1
t
E∗t Rt+1
E∗t Rt+1
(A.62)
Recalling that the optimal price re-setting decision rule is given by
PtO
Pt
=
Jt
JJt
,
(A.59) and (A.60) now give us the this rule given exogenous expectations of {E∗t Πt+i }∞
i=0 ,
∗
∞
∗
∞
∗
∞
{E∗t Rt,t+i }∞
i=0 , {Et Yt+i }i=0 , {Et M Ct+i }i=0 and {Et M St+i }i=0
A.4
Stability with a Non-Zero Inflation Steady State
Indeterminacy NK
Determinacy
Indeterminacy
Instability
3
2.5
θπ
2
1.5
1
0.5
1
1.001
1.002
1.003
1.004
1.005
1.006
1.007
1.008
1.009
1.01
PIEss
Figure 14: Stability and Indeterminacy in (Π, θπ ) space in RE Model.
Figure 14 shows that a non-zero inflation steady state induces indeterminacy and
instability which is removed by a more aggressive monetary feedback rule.
63
A.5
Internal Rationality in the NK Model
The final step to complete the IR equilibrium is to choose the learning rule for {E∗t Wt+i }∞
i=0 ,
∗
∞
∗
∞
∗
∞
∗
∞
{E∗t Rt,t+i }∞
i=0 and {Et Γt+i }i=0 for households and {Et Πt+i }i=0 , {Et Rt,t+i }i=0 , {Et Yt+i }i=0 ,
∞
∗
{E∗t M Ct+i }∞
i=0 and {Et M St+i }i=0 for retail firms.
We assume general adaptive expectations rules so that
E∗t [Wt+i ] = E∗t [Wt+1 ] for i ≥ 1
(A.63)
∞
∞
∗
∞
∗
∞
∗
∗
and similarly for {E∗t Γt+i }∞
i=0 , {Et Πt+i }i=0 , {Et Yt+i }i=0 , {Et M Ct+i }i=0 and {Et M St+i }i=0
whilst with point expectations
E∗t Rt+1,t+i
=
=
Rn,t Rn,t+1
Rn,t+i−1
···
Πt+1 Πt+2
Πt+i
∗
i
Et Rn,t+1
Rn,t
for i ≥ 1
∗
Et Rn,t+1
E∗t Πt+1
E∗t
(A.64)
which takes into account the observation of Rn,t at time t.
One-period ahead forecasts are given by
E∗t [Wt+1 ] = E∗t−1 [Wt ] + λW Wt − E∗t−1 [Wt ] ; λW ∈ [0, 1]
E∗t [Γt+1 ] = E∗t−1 [Γt ] + λΓ Γt − E∗t−1 [Γt ] ; λΓ ∈ [0, 1]
E∗t [Rn,t+1 ] = E∗t−1 [Rn,t ] + λRn Rn,t − E∗t−1 [Rn,t ] ; (households)
E∗h,t [Πt+1 ] = E∗t−1 [Πt ] + λh,Π Πt − E∗t−1 [Πt ] ; λh,Π ∈ [0, 1] (households)
E∗f,t [Πt+1 ] = E∗t−1 [Πt ] + λf,Π Πt − E∗t−1 [Πt ] ; λf,Π ∈ [0, 1] (firms)
E∗t [Yt+1 ] = E∗t−1 [Yt ] + λY Yt − E∗t−1 [Yt ] ; λY ∈ [0, 1]
E∗t [M˜C t+1 ] = E∗t−1 [M˜C t ] + λM C M˜C t − E∗t−1 [M˜C t ] ; λM C ∈ [0, 1]
where M˜C t ≡ M Ct M St
noting that we assume that M˜C t is observed by the firm, but it cannot disentangle M Ct
and M St .
64
With adaptive point expectations (A.49) now becomes
Ct
(1 − β)
=
1
(κCt )
1
φ

1
1+ φ

Wt
+
E∗t Rn,t+1
Rn,t
1
φ
E∗t Wt+1
1
1+ φ
ex )
β (E∗t Rt+1
1+ 1 
φ
−1

 + Γt +
E∗t Rn,t+1
E∗t Γt+1
Rn,t
ex − 1
E∗t Rt+1
where we have defined the ex post real interest rate as
Rtex ≡
Rn,t
Πt
(A.65)
whilst (A.61) and (A.62) now become
Ω3,t =
Ω4,t =
ξ(E∗t Πt+1 )ζ E∗t Yt+1 E∗t M Ct+1 E∗t M St+1
E∗t Rt+1 − ξ(Πt+1 )ζ
ξ(E∗t Πt+1 )ζ−1 E∗t Yt+1
E∗t Rt+1 − ξ(E∗t Πt+1 )ζ−1
This completes the internally equilibrium with point adaptive expectations.
A.6
Heterogeneous Expectations
We consider two formulations of the payoffs for households and firms. The first is expressed
on terms of a discounted sum of past weighted forecast errors, Φh,t say, starting at t = 0
for with rational and non-rational households respectively:
ΦRE
h,t
ΦIR
h,t
RE
= µRE
Φ
−
wW ((Wt − Eh,t−1 Wt )/W )2 + wh,Π ((Πt − Eh,t−1 Π)/Π)2
h
h,t−1
+ wΓ ((Γt − Eh,t−1 Γt )/Γ)2 + wR ((Rn,t − Et−1 Rn,t )/Rn )2 + Ch
IR
2
2
∗
∗
= µIR
h Φh,t−1 − wW ((Wt − Eh,t−1 Wt )/W ) + wh,Π ((Πt − Eh,t−1 Π)/Π)
∗
+ wΓ ((Γt − Eh,t−1
Γt )/Γ)2 + wR ((Rn,t − Et−1 Rn,t )/Rn )2
The parameter Ch is a fixed cost of being rational for households. For firms this becomes
ΦRE
f,t
ΦIR
f,t
2
2
RE
= µRE
f Φf,t−1 − wY ((Yt − Ef,t−1 Yt )/Y ) + wf,Π ((Πt − Ef,t−1 Π)/Π)
+ wM C ((M˜C t − Ef,t−1 M˜C t )/M C)2 + Cf
IR
∗
∗
= µIR
Φ
−
wY ((Yt − Ef,t−1
Yt )/Y )2 + wf,Π ((Πt − Ef,t−1
Π)/Π)2
f
f,t−1
∗
+ wM C ((M˜C t − Ef,t−1
M˜C t )/M C)2
65
where the parameter Cf is a fixed cost of being rational for firms and we allow for the
possibility that Ch 6= Cf .
Then the proportions of rational households and firms is given by
nh,t =
nf,t =
exp(γΦRE
h,t )
exp(γΦh,t )RE + exp(γΦIR
h,t )
exp(γΦRE
f,t )
IR
exp(γΦRE
f,t ) + exp(γΦf,t )
(A.66)
(A.67)
Thus the proportion of rational agents in the steady state is given by
exp(−γCh )
exp(−γCh ) + 1
exp(−γCf )
exp(−γCf ) + 1
nh =
nf
=
(A.68)
(A.69)
which is pinned down by the cost parameters (Ch , Cf ) (which can be positive or negative).
A.7
Proof of Lemma
In the first order conditions for Calvo contracts and expressions for value functions we are
confronted with expected discounted sums of the general form
Ωt = Et
"
∞
X
k
β Xt,t+k Yt+k
k=0
#
(A.70)
where Xt,t+k has the property Xt,t+k = Xt,t+1 Xt+1,t+k and Xt,t = 1 (for example an
inflation, interest or discount rate over the interval [t, t + k]).
Lemma
Ωt can be expressed as
Ωt = Yt + βEt [Xt,t+1 Ωt+1 ]
Proof
Ωt = Xt,t Yt + Et
"
∞
X
β k Xt,t+k Yt+k
k=1
66
#
(A.71)
= Yt + E t
"
= Yt + βEt
∞
X
β
k ′ =0
" ∞
X
k ′ +1
Xt,t+k′ +1 Yt+k′ +1
#
k′
β Xt,t+1 Xt+1,t+k′ +1 Yt+k′ +1
k ′ =0
= Yt + βEt [Xt,t+1 Ωt+1 ]
A.8
#
Proof of Equation A.29
In the next period, ξ of these firms will keep their old prices, and (1 − ξ) will change their
O . By the law of large numbers, we assume that the distribution of prices
prices to Pt+1
among those firms that do not change their prices is the same as the overall distribution
in period t. It follows that we may write
∆t+1 = ξ
X
jno change
Pt (j)
Pt+1
−ζ
+ (1 − ξ)
Jt+1
JJt+1
−ζ
Jt+1 −ζ
Pt (j) −ζ
+ (1 − ξ)
Pt
JJt+1
jno change
Pt −ζ X Pt (j) −ζ
Jt+1 −ζ
= ξ
+ (1 − ξ)
Pt+1
Pt
JJt+1
j
Jt+1 −ζ
ζ
= ξΠt+1 ∆t + (1 − ξ)
JJt+1
= ξ
B
Pt
Pt+1
−ζ
X
Priors and Posteriors
67
SE_epsAtrend
SE_epsA
SE_epsG
15
15
10
10
5
5
20
10
0
0
0.2
0
0.4
0.2
SE_epsMS
0.4
0.6
0.8
0
0
0.2
SE_epsMPS
15
0.4
SE_mes_y
40
15
30
10
10
20
5
5
10
0
0
0.2
0
0.4
0
0.2
SE_mes_pi
0
0.4
0
0.2
rhoA
15
30
10
20
5
10
0.4
rhoG
1.5
1
0
0.2
0.4
0
0.6
0.5
0.2
0.4
rhoMS
0.6
0.8
1
0
0
rho_r
1.5
0.5
1
theta_pie
15
10
10
1
5
5
0.5
0
0
0.5
0
1
0.4
0.6
theta_y
0
0.8
0.8
4
60
0.6
3
40
0.4
2
20
0.2
1
0
0.1
1
2
0
0.2
0
2
conspie
3
xi
80
0
0
phi
0
4
0.2
trend
0.4
0.6
0.8
consr
6
10
4
4
5
2
0
0.2
2
0.4
0.6
0.8
1
0
0.4
lam
0.6
0.8
0
1
nhobs
1.2
1.4
1.6
nfobs
6
4
2
2
1
4
2
0
0.2 0.4 0.6 0.8
1
0
0
0.2
0.4
0.6
0.8
0
0
0.5
1
Figure 15: Priors and Posteriors Distributions (NK Model IR-RE)
68
C
Identification Strength at Priors and Posteriors
We follow Iskrev (2010a)’s procedures and measure the identification strength, based on
the information matrix IT (θ), as sensitivity of the information derived from the likelihood to the parameters and collinearity between the effects of different parameters on
the likelihood. The ‘strength’ of identification can be decomposed into a ‘sensitivity’ and
‘correlation’ component. The first referring to the case when weak identification arises
when the moments do not change with θi and the second when collinearity dampens the
effect of θi . The former is defined as
∆i =
q
θi2 · IT (θ)(i,i)
(C.1)
which can also be normalised relative to the prior standard deviation for θi : σ(θi ), weighting the information matrix using the prior uncertainty:
= σ(θi ) ·
∆prior
i
q
IT (θ)(i,i)
(C.2)
It is possible to show the standard error of a parameter:
s.e.(θi ) =
1
1
q
∆ i 1 − ̺2
(C.3)
i
where ̺i denotes collinearity between the effects of different parameters so that lack of
identification and a flat likelihood may be due to either ∆i = 0 or ̺i = 1.
D
Solution of Linearized Models under Imperfect Information
We write a rational expectations (RE) model in the general non-linear form:
Et [f (yt , yt+1 , yt−1 , ǫt )] = 0
(D.1)
with Et now referring to expectations subject to an information set that may be imperfect.
Either analytically, or numerically using the methods of Levine and Pearlman (2011), a
69
Identification strength with asymptotic Information matrix (log−scale)
6
4
relative to param value
relative to prior std
2
0
−2
−4
−6
rho_r
nfobs
xi
xi
rho_r
epsMPS
epsMPS
Sensitivity component with asymptotic Information matrix (log−scale)
8
nfobs
theta_pie
theta_pie
lam
nhobs
lam
nhobs
mes_pi
epsAtrend
mes_pi
epsAtrend
phi
mes_y
rhoMS
theta_y
rhoA
epsA
epsG
epsMS
rhoG
−8
relative to param value
relative to prior std
6
4
2
0
−2
phi
mes_y
rhoMS
theta_y
rhoA
epsA
epsG
epsMS
rhoG
−4
Figure 16: Identification Strength at Prior Means in Model IR-RE
Identification strength with asymptotic Information matrix (log−scale)
5
0
relative to param value
relative to prior std
−5
mes_pi
theta_pie
mes_pi
theta_pie
rhoA
lam
lam
rho_r
epsAtrend
epsAtrend
theta_y
epsMPS
epsA
mes_y
xi
phi
nfobs
nhobs
rhoMS
epsMS
epsG
−15
rhoG
−10
Sensitivity component with asymptotic Information matrix (log−scale)
10
relative to param value
relative to prior std
5
0
rhoA
rho_r
theta_y
epsMPS
epsA
mes_y
xi
phi
nfobs
nhobs
rhoMS
epsMS
epsG
−10
rhoG
−5
Figure 17: Identification Strength at Posterior Means in Model IR-RE
70
Identification strength with asymptotic Information matrix (log−scale)
6
relative to param value
relative to prior std
4
2
0
−2
−4
epsAtrend
lam
epsMPS
xi
phi
lam
epsMPS
xi
phi
rho_r
rho_r
epsAtrend
theta_pie
theta_pie
mes_pi
mes_y
mes_y
mes_pi
theta_y
relative to param value
relative to prior std
theta_y
rhoA
rhoA
epsA
rhoMS
rhoMS
rhoG
Sensitivity component with asymptotic Information matrix (log−scale)
epsG
epsMS
6
rhoG
8
epsMS
−6
4
2
0
−2
epsA
epsG
−4
Figure 18: Identification Strength at Prior Means in Model IR
Identification strength with asymptotic Information matrix (log−scale)
5
0
relative to param value
relative to prior std
−5
rhoA
theta_pie
rhoA
theta_pie
mes_pi
rho_r
lam
lam
theta_y
epsAtrend
epsAtrend
epsMPS
xi
epsA
mes_y
phi
epsMS
rhoMS
epsG
−15
rhoG
−10
Sensitivity component with asymptotic Information matrix (log−scale)
6
4
relative to param value
relative to prior std
2
0
−2
−4
−6
mes_pi
rho_r
theta_y
epsMPS
xi
epsA
mes_y
phi
epsMS
rhoMS
epsG
rhoG
−8
Figure 19: Identification Strength at Posterior Means in Model IR
71
log-linearized form state-space representation can be obtained as


zt+1
Et xt+1


 = G
zt
xt


+H
Et zt
E t xt


+
B
0

 ǫt+1
(D.2)
where zt , xt are vectors of backward and forward-looking
variables, respectively, and ǫt is

G11 G12
, with H similarly defined. The
a vector of shock variables. We define G = 
G21 G22
reason for transforming the equations of the model from the linearized version of (D.1) is
that the corresponding solution method of Sims (2002) does not extend easily to imperfect
information. In addition we assume that agents all make the same observations at time t,
which are given, in non-linear and subsequently linearized forms respectively, by
Mtobs = m(yt )
mt =
h
M1 M2
i


zt
xt

+
h
L1 L2
i


Et zt
E t xt


(D.3)
Note that the expressions involving Et zt , Et xt arise from rewriting the model in BlanchardKahn form (D.2). The presence of these terms is what distinguishes our results on invertibility from those of Baxter et al. (2011), and in addition we do not make the assumption
that agents have full current information on all variables for which forward expectations
are present in the model.
Thus the information set at time t for all agents is {ms : s ≤ t}. For ease of notation
we assume that if any variables are observed with measurement error, then these variables
are included in the state space, and the measurement errors are then part of the vector ǫt .
Given the fact that expectations of forward-looking variables depend on the information
set, it is hardly surprising that the absence of perfect information will impact on the
path of the system. A full derivation of the solution for the general linear setup above is
provided in Pearlman et al. (1986), but is outlined below.
D.1
Perfect Information Case
We first consider the solution for (D.2) and (D.3) under perfect information; in this case
we assume that all stocks dated t−1 and other variables dated t in (D.2) are fully observed
72
during the course of period t. These would include beginning-of-period capital stock kt−1 ,
beginning-of-period net worth nt−1 , all flows such as output, consumption, investment,
all output and factor prices, inflation over the period and all end-of-period realizations of
exogenous stochastic processes such as at , gt etc.
For this perfect information case (where Et zt = zt , Et xt = xt ) there is a saddle path
satisfying:
xt + N zt = 0
h
where
N
I
i
(G + H) = ΛU
h
N
I
i
(D.4)
where ΛU is a matrix with unstable eigenvalues. If the number of unstable eigenvalues
of (G + H) is the same as the dimension of xt , then the system will be determinate.16
We then ask whether observations by the econometrician of the form (D.3) will lead to
invertibility.
From the saddle path relationship (D.4), it is clear that the reduced-form representation
of the model is now
zt = (G11 + H11 − (G12 + H12 )N )zt−1 + Bǫt
mt = (M1 + L1 − (M2 + L2 )N )zt (D.5)
Expressing mt in terms of zt−1 and ǫt , from Fernandez-Villaverde et al. (2007) we deduce
that a necessary and sufficient condition for invertibility (see (31)) is that D̃ = EB is
invertible, where E = M1 + L1 − (M2 + L2 )N .
D.2
Imperfect Information Case
We first briefly outline how the imperfect information setup is solved, and the provide the
conditions for invertibility. Following Pearlman et al. (1986), we use the Kalman filter
updating given by


zt,t
xt,t


=
zt,t−1
xt,t−1


 + J  mt −
h
M1 M2
i


zt,t−1
xt,t−1

−
h
L1 L2
i


zt,t
xt,t


where we denote zt,t ≡ Et [zt ] etc. The Kalman filter was developed in the context of
backward-looking models, but extends as we see here to forward-looking models. The basic
16
Note that in general the dimension of xt will not match the number of expectational variables in
(D.1). The algorithm in Levine and Pearlman (2011) will eliminate linear dependency among expectational
variables and will also convert the system at = ρat−1 + εt , bt = Et at+1 into at = ρat−1 + εt , bt = ρat .
73
idea behind it is that the best estimate of the states {zt , xt } based on current information
is a weighted average of the best estimate using last period’s information and the new
information mt . Thus the best estimator of the state vector at time t − 1 is updated by
multiple J of the error in the predicted value of the measurement as above, where J is
given by

J =
P D′
−N P D ′

 Γ−1


and D ≡ M1 − M2 G−1
22 G21 , M ≡ [M1 M2 ] is partitioned conformably with

zt
,
xt
Γ ≡ EP D ′ where E ≡ M1 + L1 − (M2 + L2 )N and P satisfies the Riccati equation (D.8)
below.
Using the Kalman filter, the solution as derived by Pearlman et al. (1986)
17
is given
by the following processes describing the pre-determined and non-predetermined variables
zt = z̃t + zt,t−1 and xt , and a process describing the innovations z̃t ≡ zt − zt,t−1 :
Predetermined :
zt+1,t = Czt,t−1 + CP D ′ (DP D ′ )−1 Dz̃t
Non-predetermined :
Innovations :
(D.6)
−1
′
′ −1
xt = −N zt,t−1 − G−1
22 G21 z̃t − (N − G22 G21 )P D (DP D ) Dz̃t
z̃t+1 = Az̃t − AP D ′ (DP D ′ )−1 Dz̃t + Bǫt+1
(D.7)
where
C ≡ G11 + H11 − (G12 + H12 )N,
A ≡ G11 − G12 G−1
22 G21 ,
D ≡ M1 − M2 G−1
22 G21
and P is the solution of the Riccati equation given by
P = AP A′ − AP D ′ (DP D ′ )−1 DP A′ + BB ′
(D.8)
where we assume that the shocks are normalized such that their covariance matrix is given
by the identity matrix.
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A less general solution procedure for linear models with imperfect information is in Lungu et al. (2008)
with an application to a small open economy model, which they also extend to a non-linear version.
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The measurement mt , as shown by Pearlman et al. (1986), can now be expressed as
mt = Ezt,t−1 + EP D ′ (DP D ′ )−1 Dz̃t
(D.9)
We can see that the solution procedure above is a generalization of the Blanchard-Kahn
solution for perfect information and that the determinacy of the system is independent of
the information set.
D.3
When is the System Invertible?
We now pose the question: given the econometrician’s information set, under what conditions do the RE solutions under agents’ different information sets actually differ? When
can the econometrician infer the full state vector, including shocks?
Fernandez-Villaverde et al. (2007) do not attempt to answer this question. Their focus
is on the complete reduced form of the solution from the perspective of the econometrician;
the source of this reduced form i.e its dependence on the information set of the agents is
not discussed at all. As we have seen above, the reduced form under any information set
is of the standard state space type investigated by Fernandez-Villaverde et al. (2007), but
the invertibility properties depend on the information set.
Levine et al. (2017) show the following: If EB is of full rank (i.e. number of observables
= number of shocks) but D is not of full row rank, then imperfect information is not
equivalent to full information, and the system is then not invertible. This is a new result
in the literature, which says that if a limited information set under perfect information
is invertible, it does not follow that the same limited information set under imperfect
information is also invertible.
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