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Federal Reserve Bank of Chicago
Modeling the Evolution of
Expectations and Uncertainty
in General Equilibrium
Francesco Bianchi and Leonardo Melosi
September 2013
WP 2013-12
Modeling the Evolution of Expectations
and Uncertainty in General Equilibrium
Francesco Bianchi
Duke University
University of Pennsylvania
Leonardo Melosi
Federal Reserve Bank of Chicago
This draft: September 2013
First draft: January 2012
Abstract
We develop methods to solve general equilibrium models in which forward-looking agents are subject
to waves of pessimism, optimism, and uncertainty that turn out to critically a¤ect macroeconomic
outcomes. Agents in the model are fully rational, conduct Bayesian learning, and they know that
they do not know. Therefore, agents take into account that their beliefs will evolve according to what
they will observe. This framework accommodates both gradual and abrupt changes in beliefs and
allows for an analytical characterization of uncertainty. Shocks to beliefs a¤ect economic dynamics
and uncertainty. We use a prototypical Real Business Cycle to illustrate the methods.
JEL Codes: D83, C11, E22.
Keywords: Markov switching, general equilibrium models, uncertainty, Bayesian learning, rational
expectations, downside risk, rare disasters.
We thank Fernando Alvarez, Gadi Barlevy, Je¤ Campbell, Martin Eichenbaum, Eric Leeper, Monika Piazzesi, Martin
Schneider, Dan Waggoner, and Tao Zha for very useful comments and discussions. We wish to thank participants at Macro
seminar of the Federal Reserve Bank of Chicago and at the SED 2013 in South Korea. Todd Messer provided excellent
research assistance. Francesco Bianchi gratefully acknowledges …nancial support from the National Science Foundation
through grant SES-1227397. Francesco Bianchi, Department of Economics, Duke University, 213 Social Sciences Building,
Durham, NC, 27708-0089, USA. Leonardo Melosi, Federal Reserve Bank of Chicago, 230 South La Salle street, Chicago, IL
60604-1413, USA. Email: lmelosi@frbchi.edu. The views in this paper are solely the responsibility of the authors and should
not be interpreted as re‡ecting the views of the Federal Reserve Bank of Chicago or any other person associated with the
Federal Reserve System.
1
1
Introduction
A centerpiece of the rational expectations revolution is that economic outcomes critically depend on
agents’ beliefs about future events. Most general equilibrium models are solved assuming that agents
have perfect knowledge about the stochastic properties of all the realized events. These are certainly
strong restrictions imposed upon the dynamics of beliefs. For instance, the private sector is likely to have
limited information about the future path of policymakers’decisions, the dynamics of dividend payments,
or the likely duration of observed changes in the returns to labor and capital. These assumptions, in
turn, in‡uence the expectations formation mechanism and hence the predictions we draw from rational
expectations models.
In this paper we develop methods to study dynamic general equilibrium models in which forwardlooking and fully rational agents learn about the stochastic properties of realized events. This modeling
framework captures waves of pessimism, optimism, and uncertainty that turn out to critically a¤ect
macroeconomic outcomes. Such outbursts of pessimism, optimism, and uncertainty may happen abruptly
or may gradually unfold over a long period of time in response to the behavior of other agents or to
the realizations of economic outcomes. Furthermore, this framework is well-suited to study the e¤ects
of shocks to beliefs and agents’uncertainty in Dynamic Stochastic General Equilibrium (DSGE) models.
All results are derived within a modeling framework suitable for structural estimation that will allow
researchers to bring the models to the data.
The evolution of agents’beliefs is modelled assuming the existence of di¤erent states of the world that
di¤er according to the statistical properties of the exogenous shocks or based on the behavior of some
of the agents in the model. Such regimes follow a Markov-switching process, which may be correlated
with other aspects of the model. For example, the government could be more likely to in‡ate debt away
when the level of spending is high. Agents are assumed to observe economic outcomes, but not the
regimes themselves. Agents will then adopt Bayesian learning to infer which regime is in place. This will
determine the evolution of agents’beliefs about future economic outcomes.
Our modeling framework goes beyond the assumption of anticipated utility that is often used in
models characterized by a learning process. Such an assumption implies that agents forecast future
events assuming that their beliefs will never change in the future. Instead, agents in our models know
that they do not know. Therefore, when forming expectations, they take into account that their beliefs
will evolve according to what they observe in the future. In our context, it is possible to go beyond the
anticipated utility assumption because there are only a …nite number of relevant beliefs and they are
strictly linked to observable outcomes through the learning mechanism in a way that we can keep track
of their evolution. It should also be noted that the proposed approach is based on agents being fully
rational and hence their beliefs always being consistent in equilibrium. Rationality in our approach is
essential in that it puts discipline on beliefs so as to make it possible to draw precise predictions from
economic models.
The proposed model framework is ‡exible enough to encompass both abrupt and gradual changes in
beliefs. For example, augmenting the modeling framework with signals about the regime in place allows
one to capture the sharp e¤ect of news on the evolution of the economy or to study the macroeconomic
2
implications of changes in animal spirits about future events. At the same time, through the learning
process, we can model situations in which agents’beliefs gradually change in response to the behavior of
other agents or the realizations of stochastic events. This sluggish adjustment of public expectations is
hard to reproduce through rational expectations models in which the functioning of the whole economy is
common knowledge among agents. Furthermore, the methods introduced in this paper can be combined
with the techniques developed by Bianchi (2012) to obtain an analytical characterization of the evolution
of uncertainty in response to changes in agents’beliefs.
We show how to apply these methods using a prototypical Real Business Cycle (RBC) model. In the
model, total factor productivity (TFP) growth can assume two values: high or low. For each value of
TFP growth, we allow for a long-lasting and a short-lasting regime. Therefore, while agents can observe
the current TFP growth rate, they are uncertain about its future values, because they do not know if the
current value is likely to last for a short time or for a long time. We consider a wide range of speci…cations,
allowing for smooth transitions or abrupt changes in agents’optimism about future realizations of TFP
growth. Each of these di¤erent speci…cations can be easily captured with the appropriate transition
matrix governing the evolution of TFP growth. This has the important implication that the dynamics of
pessimism, optimism, and uncertainty are consistent in equilibrium. Whenever a short-lasting regime is
in fact realized, with the bene…t of hindsight, agents’beliefs turn out to overreact to the regime change
because agents always take into account the possibility that the economy entered a long-lasting regime.
However, if, in fact, the regime is long-lasting, it takes time for agents’beliefs to line up with the actual
realization. This implies that although agents are fully rational, their beliefs are generally misaligned
with respect to the actual state of the economy. Such a misalignment is found to substantially in‡uence
consumption and capital allocation in the RBC model.
Finally, we expand our analysis of the RBC model to study the case in which agents receive signals
about the likely duration of the current regime. In this environment, signals work as shocks to agents’
beliefs that have …rst-order and second-order e¤ects. Uncertainty about macroeconomic events evolves
over time as agents’ beliefs drift, creating interesting comovements between volatility and real activity.
This feature might shed further light on the link between uncertainty and macroeconomic outcomes with
respect to the seminal work by Bloom (2009).
The methods developed in this paper are based on the idea of expanding the number of regimes to
take into account the learning mechanism. The central insight consists of recognizing that the evolution
of agents’beliefs can be captured by de…ning an expanded set of regimes indexed with respect to agents’
beliefs themselves. Once this structure has been imposed, the model can be recast as a Markov-switching
dynamic stochastic general equilibrium (MS-DSGE) model with perfect information. If regime changes
enter additively the model can be solved with standard solution methods such as gensys (Sims, 2002) and
Blanchard and Kahn (1980), following the approach described in Schorfheide (2005) and Liu, Waggoner,
and Zha (2011). If instead regime changes enter multiplicatively the model can be solved with any of the
methods developed for solving MS-DSGE models, such as Davig and Leeper (2007), Farmer, Waggoner,
and Zha (2009), Cho (2012), and Foerster, Rubio-Ramirez, Waggoner, and Zha (2011).
In both cases, the resulting solution is suitable for likelihood-based estimation. This is because
even if the …nal number of regimes is very large, there is a tight link between observable outcomes and
3
the evolution of agents’ beliefs. In other words, the transition matrix governing the joint dynamics of
the economy and agents’ beliefs is highly restricted. For example, Bianchi and Melosi (2012) apply
these methods and Bayesian techniques to estimate a model in which agents are uncertain about the
future stance of monetary policy. This paper is therefore related to a growing literature that models
parameter instability to capture changes in the evolution of the macroeconomy. This consists of two
branches: Schorfheide (2005), Justiniano and Primiceri (2008), Bianchi (2013), Davig and Doh (2008),
and Fernandez-Villaverde and Rubio-Ramirez (2008) introduce parameter instability in DSGE models,
while Sims and Zha (2006), Primiceri (2005), and Cogley and Sargent (2005) work with structural VARs.
Finally, to the extent that we can model situations in which agents’beliefs evolve in response to policymakers’behavior, our work is also linked to papers that study how in‡ation expectations respond to policy
decisions, such as Mankiw, Reis, and Wolfers (2004), Nimark (2008), Del Negro and Eusepi (2010), and
Melosi (2013, Forthcoming).
Schorfheide (2005) pioneers a method to estimate general equilibrium models in which agents learn
the realization of a discrete Markov-switching process that a¤ects the constants of the model-implied laws
of motion. In another pathbreaking contribution, Andolfatto and Gomme (2003) develop and calibrate a
DSGE model in which agents face a signal extraction problem to learn about the money growth regime.
Our work di¤ers from each of these two papers in one or more of the following dimensions. First, our
framework can accommodate situations in which agents learn about regime changes that do not only
a¤ect the constant terms of the model, but also its autoregressive component. For example, Bianchi
and Melosi (2012) use the proposed framework to estimate a model in which agents have to learn about
future policymakers’behavior. Second, in our framework agents have always enough information to infer
what the current state of the economy is or what other agents are doing: High or low growth, Hawkish or
Dovish monetary policy, etc. Nevertheless, agents face uncertainty about the statistical properties of what
they are observing. For example, agents could be uncertain about the persistence and the destination
of a particular state. As we shall show, in a model in which agents are forward looking these sources of
uncertainty have pervasive e¤ects on the law of motion of the economy. Third, our approach mitigates a
tension that often arises in the learning literature when agents are assumed to understand the structure
of the economy. On the one hand, it would be desirable to have regimes that are very di¤erent in order
to induce signi…cant changes on economic dynamics. On the other hand, this would make the learning
process relatively fast. Instead, in our approach even small di¤erences in persistence have very large
e¤ects on agents’expectations because regimes may be still very di¤erent in terms of the probability of
moving to alternative regimes. Therefore, our framework allows for both smooth or abrupt changes in
agents’ beliefs and for the possibility of signals that play the role of shocks to beliefs, a¤ecting agents’
expectations and uncertainty. Finally, our framework is suitable for likelihood estimation.
The remainder of the paper is organized as follows. Section 2 introduces the class of models and
derives the main results. In Section 3, we discuss the main advantages of the proposed methodology.
Section 4 applies the methods to an RBC model. Section 5 concludes.
4
2
The Model Framework
In this section, we introduce the modeling environment to which our methods are applicable. The class
of models we focus on has three salient features:
1. A model that can be expressed or approximated in the following form:1
0 ( t ) St
=
c ( t)
+
1 ( t ) St 1
+
( t ) "t +
(1)
t
where St is a vector containing all variables of the model known at time t (including conditional
expectations formed at time t),
t
is a vector containing the endogenous expectation errors, and
the random vector "t contains the familiar Gaussian shocks. The hidden variable
parameter values in place at time,
( t ) ; assumes discrete values
according to a Markov-switching process with transition matrix P.
t
t
controls the
2 f1; : : : ; ng, and evolves
2. Agents have to forecast the dynamics of the endogenous variables St+1 on the basis of Model (1)
and their information set at time t, It . This includes the history of model variables and shocks, but
not the history of regimes,
t
S t ; "t :
: It
3. Some regimes are assumed to bring about the same model parameters,
regimes into m blocks bj = f
t
2 f1; : : : ; ng : ( t ) =
bi g,
( t ). Let us group the
for j 2 f1; :::; mg.
Given that agents know the structure of the model (sub 1 ) and can observe the endogenous variables
and the shocks (sub 2 ), they can also determine which set of parameters is in place at each point in time.
However, while this is enough for agents to establish the history of blocks, agents cannot exactly infer the
realized regime
t;
because the regimes within each block share the same parameter values (sub 3 ). It is
very important to emphasize that regimes that belong to the same block are not identical in all respects,
as they can di¤er in their stochastic properties such as average persistence and the probability of switching
to other regimes. These properties are known to agents that will use them to learn about the regime in
place today and to form expectations about the future. Therefore, points 1-3 describe a model in which
agents learn about the latent variable
t.
As will be shown below, such a learning process a¤ects the
equilibrium law of motion of the economy. However, agents cannot extract any additional information
about the underlying regime from observing the history of the endogenous variables S t because this
re‡ects their own beliefs.
Henceforth, we will consider a benchmark case in which there are two blocks (m = 2) and two regimes
within each block. This choice is made in order to keep notation simple. The extension to the case
in which m > 2 is straightforward. The probabilities of moving across regimes are summarized by the
1
The assumption of model linearity is not essential and can be actually relaxed as discussed in Subsection 3.3. We focus
on linear models because linearization is still the most popular approach to solve and estimate Markov-switching DSGE
models.
5
transition matrix:
2
p11
6
6 p21
P =6
6 p
4 31
p41
p12
p13
p22
p23
p32
p33
p42
p43
p14
3
7
p24 7
7
p34 7
5
p44
(2)
in which the probability of switching to regime j given that we are in regime i is denoted by pij . Without
loss of generality, we assume that regimes
and
t
t
= 1 and
t
= 2 belong to Block 1, while regimes
t
=3
= 4 belong to Block 2. We consider only non-trivial blocks that satisfy p11 + p12 + p21 + p22 6= 0
and p33 + p34 + p43 + p44 6= 0. The excluded cases are trivial as both blocks would last only one
period. Furthermore, we require that the two regimes that belong to the same block di¤er either in their
persistence or in the probability of moving from one another; that is, we require that either p11 6= p22 or
p12 6= p21 and either p33 6= p44 or p34 6= p43 . This condition makes the within-block Bayesian learning
non-trivial. Finally, we will impose that p11 + p22 > 0 and p33 + p44 > 0. This last assumption guarantees
that within a block at least one of the two regimes can last more than one period. Summarizing, for each
block, we will maintain the following benchmark assumptions throughout the paper:
A1 Non-triviality assumption: p11 + p12 + p21 + p22 6= 0 and p33 + p34 + p43 + p44 6= 0:
A2 Non-trivial-learning assumption: Either p11 6= p22 or p12 6= p21 and either p33 6= p44 or p34 6= p43 :
A3 Non-jumping assumption: p11 + p22 > 0 and p33 + p44 > 0:
We will now proceed in two steps. First, in Subsection 2.1 we will characterize the evolution of agents’
beliefs within a block for given prior beliefs. Second, in Subsection 2.2 we will explain how agents’beliefs
are pinned down once the economy moves across blocks. The statistical properties of the economy, as
captured by the transition matrix, will allow us to distinguish two cases: Static and Dynamic priors. It is
worth emphasizing that in both cases the assumption of rationality will be maintained and all results will
be based on the Bayes’theorem. Finally, for each of these cases, we will describe how to recast the model
with information frictions as a perfect information rational expectations model obtained by expanding
the number of regimes to keep track of agents’beliefs.
2.1
Evolution of Beliefs Within a Block
In what follows, we will derive the law of motion of agents’beliefs conditional on being in a speci…c block.
The formulas derived below will provide a recursive law of motion for agents’ beliefs based on Bayes’
theorem. Such recursion applies for any starting values for agents’beliefs. These will be determined by
agents’beliefs at the moment the system enters the new block. We will characterize these initial beliefs
in the next subsection.
As we have noticed in the previous section, agents can infer the history of the blocks. Therefore, at
each point in time, agents know the number of consecutive periods spent in the current block since the
last switch. Let us denote the number of consecutive realizations of Block i at time t as
To …x ideas, suppose that the system is in Block 1 at time t, implying that
1
t
> 0 and
i, i 2
t
2 = 0.
t
f1; 2g.
Then,
there are only two possible outcomes for the next period. The economy can spend an additional period
6
in Block 1, implying that
2
t+1
1
t+1
=
1
t
+ 1 and
2
t+1
1
t+1
= 0; or it can move to Block 2, implying
= 0 and
= 1: In this subsection, we restrict our attention to the …rst case.
1
1
t 1 = 2j t 1 = 1 prob
t 1 = 1j t 1 ; the probabilobserved 1t consecutive realizations of Block 1, prob t = 1j 1t ;
Using Bayes’theorem and the fact that prob
ity of being in Regime 1 given that we have
is given by:2
prob
where
1
t
=
1
t 1
t
1
t
= 1j
+ 1 and for
=
1
t
prob
t
prob
1 = 1j
1
p21 ) + p21
t 1 = 1j t 1 (p11
1
p21 p22 ) + p21
t 1 (p11 + p12
> 1: Notice that for
1
t
= 1; prob
t
= 1j
1
t
(3)
+ p22
denotes the initial beliefs
that will be discussed in Subsection 2.2. Equation (3) is a rational …rst-order di¤erence equation that
allows us to recursively characterize the evolution of agents’ beliefs about being in Regime 1 while the
system is in Block 1. The probability of being in Regime 3 given that we have observed
realizations of Block 2, prob
prob
where
2
t
=
2
t 1
t
= 3j
t
2
t
+ 1 and for
= 3j
=
2
t
2
t
prob
2
t
consecutive
; can be analogously derived:
t
prob
1 = 3j
2
p43 ) + p43
t 1 = 3j t 1 (p33
2
p43 p44 ) + p43
t 1 (p33 + p34
+ p44
:
(4)
> 1:
The recursive equations (3) and (4) characterize the dynamics of agents’beliefs in both blocks for a
given set of prior beliefs. The following proposition states that under the benchmark assumptions A1-A3,
these recursive equations converge. This convergence result will be key to being able to recast Model
(1)-(2) in terms of a …nite dimensional set of regimes indexed with respect to agents’beliefs.
Proposition 1 Convergence. Under the benchmark assumptions A1-A3, for any " > 0 there exists a
1
2 N and
2
2 N such that:
prob (
t
= 1j 1 )
prob (
t
= 1j
1
+ 1) < "
prob (
t
= 3j 2 )
prob (
t
= 3j
2
+ 1) < "
Proof. See Appendix B.
In what follows, we denote the converging probabilities for prob (
~ b and ~ b , respectively.
1
2
2.2
t
= 1j 1 ) and prob (
t
= 3j 2 ) as
Evolution of Beliefs Across Blocks
In the previous subsection, we characterized the evolution of agents’ beliefs conditional on being in a
speci…c block. The formulas derived above apply to any set of initial beliefs. In this subsection, we will
pin down agents’ beliefs at the moment the economy moves across blocks. These beliefs will serve as
starting points for the recursions (3) and (4) governing the evolution of beliefs within a block.
Suppose for a moment that before switching to the new block, agents could observe the regime that
was in place in the old block. Notice that in this case the transition matrix conveys all the information
2
A detailed derivation of equation (3) is provided in Appendix A.
7
necessary to pin down agents’prior beliefs about the regime in place within the new block. Speci…cally,
we have that if the economy moves from Block 2 to Block 1, the probability of being in Regime 1 is given
by
prob
t
= 1j
t 1
= 3;
1
t
=1 =
p31
;
p31 + p32
if the economy was under Regime 3 in the previous period, or by
prob
t
= 1j
t 1
= 4;
1
t
=1 =
p41
p41 + p42
if the economy was under Regime 4 in the previous period. Symmetrically, the probability of being in
Regime 3 given that the economy just moved to Block 2 is given by
prob
t
= 3j
t 1
= 1;
2
t
=1 =
p13
;
p13 + p14
if the economy was under Regime 1 in the previous period, or by
prob
t
= 3j
t 1
= 2;
2
t
=1 =
p23
p23 + p24
if the economy was previously under Regime 2.
However, in the model, agents never observe the regime that is in place. Therefore, their beliefs at the
moment the economy moves from one block to the other will be a weighted average of the probabilities
outlined above. The weights, in turn, will depend on agents’ beliefs at the moment of the switch. In
what follows we will focus on three cases:
1. Static prior beliefs. In this case, the transition matrix P is such that every time the economy enters
a new block, agents’ beliefs about which regime has been realized do not depend on their beliefs
right before the switch. Thus, what has been observed in the past block does not help rational
agents to form expectations in the new block. Such a transition matrix has the virtue of delivering
a nice closed-form analytical characterization for the dynamics of beliefs.3
2. Dynamic prior beliefs. In this case, the transition matrix P is such that beliefs about which regime
is prevailing within a block a¤ect prior beliefs the moment the economy moves to the new block.
3. Signals. Exogenous signals $t about the current regime are also observed by agents. Signals are
assumed to be distributed according to p ($t j t ).
It is worth clarifying that nothing prevents the researcher from combining the three cases described
above. For example, static prior beliefs could characterize one block but not another or agents could
receive a signal every time the economy enters a new block.
3
A simpli…ed version of this problem with three regimes is studied by Barlevy (1998) within a partial equilibrium
framework.
8
2.2.1
The Case of Static Prior Beliefs
In the case of static prior beliefs, the transition matrix P implies that every time the system enters a
new block, rational agents’beliefs are the same regardless of the history of past beliefs. It is immediate
to show that necessary and su¢ cient conditions for this to happen are:
prob
t
= 1j
t 1
= 3;
1
t
=1
=
prob
t
= 3j
t 1
= 1;
2
t
=1
=
p41
p31
=
= prob
p31 + p32
p41 + p42
p13
p23
=
= prob
p13 + p14
p23 + p24
t
= 1j
t 1
= 4;
1
t
=1
(5)
t
= 3j
t 1
= 2;
2
t
=1
(6)
In other words, the transition matrix P is such that when the economy leaves a block, the relative
probability of the two regimes in the new block is not a¤ected by the regime that was in place before.
Agents’beliefs are uniquely pinned down by (5) and (6) because agents are fully rational and know the
transition matrix governing the evolution of regimes.
The recursive equations (3) and (4) combined with the initial conditions (5) and (6) uniquely characterize the dynamics of agents’beliefs in each block. To see this, notice that for each block, there is a
unique path for the evolution of agents’beliefs, given that (5) and (6) make agents’beliefs before entering
the block irrelevant. Furthermore, Proposition 1 guarantees that there exists a
1
2 N and
2
2 N such
that agents’beliefs converge for an arbitrary level of accuracy. Therefore, in the case of static priors the
number of consecutive periods spent in a block ( it ) is a su¢ cient statistic to pin down the dynamics
of beliefs in both blocks. Equipped with this important result, we can re-cast Model (1)-(2) in terms
of a new set of regimes indexed with respect to the number of consecutive periods spent in a block
i,
t
i 2 f1; 2g:
0 ( t ) St
where "t
N (0;
errors, and the
")
1
+
=
c ( t)
+
+
is a vector of exogenous Gaussian shocks,
2
regimes
1; 2
t
t
t
e11 and P
e12 are given by
where the matrices P
2
6
6
6
6
6
6
6
4
0 prob
0
..
.
0
0
1
t+1
= 2j
0
..
.
0
0
1
t
( t ) "t +
t
(7)
t
is a vector of endogenous expectation
evolve according to the transition matrix
e=
P
e11
P
1 ( t ) St 1
"
=1
e11 P
e12
P
e21 P
e22
P
#
;
::: 0
0
::: 0
..
. 0
0
0
: : : 0 prob
::: 0
9
3
prob
1
t
=
1
t+1
j
>
1
t
=
j
1
t
1
=
7
7
7
7
7
7
7
5
e12
P
2
6
6
4
1
1
t+1
prob
= 2j
..
.
1
prob
1
t+1
>
j
1
t
=1
1
t
=
01
(
3
1)
7
7
5
..
.
01
(
1)
with the elements of the matrices given by
prob
i
t+1
=
i
t
+ 1j
i
t
= prob
t
= 1j
1
t
(p11 + p12 ) + 1
prob
t
= 1j
1
t
(p21 + p22 )
(8)
where prob t = 1j 1t can be obtained from the recursive equation (3) and equation (5). The matrices
e21 and P
e22 can be analogously derived.
P
Notice that the newly de…ned set of regimes keeps track of both the parameters in place at each
point in time and the evolution of agents’beliefs. Since Model (7) is a Markov-switching DSGE model
with perfect information, it can be solved using the techniques developed by Schorfheide (2005), Liu,
Waggoner, and Zha (2011), Davig and Leeper (2007), Farmer, Waggoner, and Zha (2009), Cho (2012),
and Foerster, Rubio-Ramirez, Waggoner, and Zha (2011). The result is an MS-VAR in the DSGE state
vector St :
St = c
e +T
e St
t; P
t; P
1
+R
e "t
t; P
(9)
where the law of motion of the economy depends on agents’beliefs as captured by
t.
of Proposition 1 at hand, the solution of Model (7) with a truncated number of regimes
With the results
t
approximates
the solution of the original model (1) with learning. Notice that the accuracy of this approximation can
be made arbitrarily precise simply by increasing the number of regimes
. Furthermore, it is worth
pointing out that in the case of static priors the approximation error stems only from truncating agents’
learning process. For all regimes such that
i
t
<
i
agents’ beliefs exactly coincide with the analytical
values derived using (3) and (4) and conditions (5) and (6).
It is important to notice that the case of static prior belief does not constitute a deviation from
rationality or from the Bayes theorem. In fact, it directly stems from the application of the Bayes
theorem. Given condition (5), agents would be irrational to have di¤erent beliefs when the system enters
Block 1 in di¤erent periods. Condition (6) leads to this exact implication all the times the system enters
Block 2. We single out the case of static prior because when conditions (5)-(6) hold, the approximation
error from truncating agents’ learning process is negligible. This is for two reasons. First, Proposition
1 ensures that the change in agents’ beliefs is negligible once the system has spent a su¢ ciently long
period of time in a block. Second, the initial beliefs can be pinned down analytically thanks to conditions
(5)-(6).
2.2.2
The Case of Dynamic Prior Beliefs
When conditions (5) and (6) do not hold, past beliefs always in‡uence current beliefs. In this case, the
number of consecutive periods
t
spent in a block is no longer a su¢ cient statistic for agents’ beliefs.
However, as pointed out before, the recursive equations (3) and (4) hold for any prior beliefs. Therefore,
these equations still capture the dynamics of beliefs while the system stays in a block. Furthermore, it
follows that the su¢ cient conditions for convergence derived in Subsection 2.1 still apply. Nevertheless,
10
the initial conditions are now di¤erent from (5) and (6) as they will depend on beliefs in the past block.
Speci…cally, agents’starting beliefs upon the shift from Block 2 to Block 1 are given by
prob f
t
= 1jIt g =
prob
prob t 1 = 3jIt 1 p31 + 1
t 1 = 3jIt 1 (p31 + p32 ) + 1
prob
prob
t 1
t 1
= 3jIt
= 3jIt
1
= 1jIt
= 1jIt
1
1
p41
(p41 + p42 )
(10)
p23
(p23 + p24 )
(11)
while if the system just entered Block 2, starting beliefs read
prob f
t
= 3jIt g =
prob
prob t 1 = 1jIt 1 p13 + 1
t 1 = 1jIt 1 (p13 + p14 ) + 1
prob
prob
t 1
t 1
1
Notice that, using their information set It ; agents can keep track of both the number of consecutive
deviations and their starting beliefs. Therefore, in the case of dynamic prior beliefs two variables pin
down the dynamics of beliefs over time: how many consecutive periods the system has spent in the
current block and the initial beliefs agents had when the system entered the current block. We then
tackle the problem of solving Model (1)-(2) when prior beliefs are dynamic by making a grid for agents’
beliefs. Denote the grid for beliefs prob f
as Gb2 = fGg1 +1 ; :::; Gg1 +g2 g where 0
t
Gi
= 1jIt g as Gb1 = fG1 ; :::; Gg1 g and for beliefs prob f
1, all 1
i
t
= 3jIt g
g = g1 + g2 . Furthermore, we denote the
whole grid as G = G b1 [ Gb2 . Endowed with such a grid, we can recast the original model in terms of a
new set of regimes
2 f1; :::; g1 + g2 g, any t. The new regime
t
best approximates agents’beliefs; that is, in our notation prob f
1 and prob f
t
t
t
captures the knot of the grid G that
= 1jIt g when the system is in Block
= 3jIt g when the system is in Block 2. The transition probability matrix for these new
regimes can be pinned down using the recursions (3) and (4) and the initial conditions (10) and (11).
The algorithm below illustrates how exactly to perform this task.
b for the new regimes
Algorithm Initialize the transition matrix P
t,
b = 0g
setting P
g:
Step 1 For each of the two blocks, do the following steps (without loss of generality we describe the
steps for Block 1):
Step 1.1 For any grid point Gi 2 Gb1 ; 1
b (i; j) = prob
P
where prob
prob f
t
t 1
t 1
= 1jIt
1
= 1jIt
1
i
g1 , compute
(p11 + p12 ) + 1
= Gi and j
prob
t 1
= 1jIt
g1 is set so as to min jprob f
1
t
(p21 + p22 )
= 1jIt g
= 1jIt g is computed using the recursive equation (3) by approximating prob
Gj j, where
t 1
= 1jIt
Gi . To ensure the convergence of beliefs, we correct j as follows: if j = i and Gi 6= eb1 , then
set j = min (j + 1; g1 ) if Gi < eb1 or j = max (1; j 1) if Gi > eb1 .
Step 1.2 For any grid point Gi 2 Gb1 ; 1
i
satisfying
min
prob
b (i; l) = 1
g1 ; compute P
prob t 1 = 1jIt 1 p13 + 1
t 1 = 1jIt 1 (p13 + p14 ) + 1
11
prob
prob
t 1
t 1
= 1jIt
= 1jIt
1
1
b (i; j) with l > g1
P
p23
(p23 + p24 )
Gl
1
=
where prob
t 1
= 1jIt
= Gi .
1
b has all zero elements, stop. Otherwise, go to Step 3.
Step 2 If no column of P
P
b (i; j) = 0 set
Step 3 Construct the matrix T as follows. Set j = 1 and l = 1. While j
g, if gi=1 P
Pg b
j = j + 1. Otherwise, if i=1 P (i; j) 6= 0: (1) set T (j; l) = 1, (2) set T (j; v) = 0 for any 1 v g
and v 6= l, (3) set l = l + 1 and j = j + 1.
bR = T P
b T 0 . If no column of P
bR has all zero elements, set
Step 4 Write the transition equation as P
b=P
bR and stop. Otherwise, go to step 3.
P
Step 1.1 determines the regime j the system will go to if it stays in Block 1 next period and …lls up the
b with the probability of moving to Regime j. Step
appropriate element (i; j) of the transition matrix P
1.2 computes the regime l the system will go to if it leaves Block 1 and …lls up the appropriate element
b Steps 2-4 are not necessary but help to keep the dimension of the grid small, getting
(i; l) of matrix P.
rid of regimes that will never be reached. For computational convenience, we always add the convergence
points for the two blocks (i.e., eb in the case of Block 1) to the grid G. On many occasions, it is a good
1
idea to make the grid near the convergence knot very …ne to improve the precision of the approximation.
b for the new set of regimes is characterized, the original Model (1) can
Once the transition matrix P
be recast in terms of the new set of regimes
0 ( t ) St
where
t
=
c ( t)
t:
+
1 ( t ) St 1
+
( t ) "t +
(12)
t
2 f1; :::; g1 + g2 g. Therefore, up to an approximation error that can be made arbitrarily small,
the task of solving the model with learning in (1)-(2) boils down to solving the perfect-information model
(12) using solution algorithms for MS-DSGE models.4 The resulting law of motion is once again an
MS-VAR:
St = c
2.2.3
Signals
b +T
t; P
b St
t; P
1
+R
b "t
t; P
(13)
Let us assume that agents observe signals about the realized regime. To …x notation, denote the signal
as $t and, for simplicity, assume that it can have only two values, 1 or 2. We denote the probability that the signal is equal to q 2 f1; 2g, conditional on the regime being equal to h 2 f1; 2; 3; 4g as
prob f$t = qj
t
= hg. The model with signals can be solved by introducing a new system of regimes
which indexes the grid points corresponding to the probabilities prob
t
= 1jIt
; $t
and prob
t
t,
= 3jIt ; $t ,
and following the same logic used in the previous subsection. As we shall show, signals will allows us
to introduce shocks to beliefs; that is, exogenously driven changes in beliefs that are not associated with
changes in the economy’s state
t.
4
In Subsection 4.2.1 we present an example for which both the approximation error and the time required to solve the
model turn out to be very small.
12
b for the new set of regimes, one can implement the algorithm detailed
To …ll up the transition matrix P
in Subsection 2.2.2 with only the little tweak of updating beliefs using the information contained in the
observed signal. For instance, we compute the ex-post-probability prob
prob
where prob
t
= 1jIt ; $t
t
1
= 1jIt ; $t
t
= 1jIt ; $t
prob ($t = qj t = 1) prob t = 1jIt ; $t 1
; $t = q = P2
t
i=1 prob ($ t = qj t = i) prob ( t = ijIt ; $
1
1)
; q 2 f1; 2g
(14)
is computed using the recursive equation (3) for a given initial point in the
grid G that approximates prob
t 1
= 1jIt
1; $
t 1
. We use the probability computed in equation (14) to
determine the appropriate destination points of the grid G, which we denote as jq , q 2 f1; 2g. Note that for
any given initial belief prob
t 1
= 1jIt
1; $
t 1
2 G, the (ex-post) belief prob
t
= 1jIt ; $t
1; $
t
=q
now pins down the grid points, depending on the realization of the signal $t . Once these two destination
points in the grid are determined, we can …ll up the transition probability as follows:
b (i; jq ) = P2 prob
P
v=1
where
prob
t
= vjIt
t
1; $
= vjIt
t 1
=
1; $
t 1
P2
prob f$t = qj
u=1 prob
t 1
= ujIt
t
= vg ; q 2 f1; 2g
1; $
t 1
puv
(15)
(16)
and we approximate prob t 1 = 1jIt 1 ; $t 1 2 G. Note that in the case of binary signals, each row of
b has up to four non-zero elements. This completes the derivation of the submatrix
the transition matrix P
b11 , which governs the evolution of beliefs within Block 1. How to obtain the other submatrices P
b12 ,
P
b21 , and P
b22 is detailed in Appendix C.
P
3
Discussion
Summarizing, the methods outlined above show that one can recast the Markov-switching DSGE model
with learning as a Markov-switching rational expectations system in which the regimes are indexed with
respect to agents’beliefs. In the case of static priors, the number of consecutive realizations of a block
represents a su¢ cient statistic to index agents’ beliefs. In the case of dynamic priors, agents’ beliefs
are mapped into a grid. In both cases, a new transition matrix that characterizes the joint evolution of
agents’beliefs and model parameters is derived.
Section 3.1 highlights the main di¤erences of our approach from the tradition learning literature.
In Section 3.2, we discuss about the tractability of the proposed method and assess its suitability for
econometric applications. In Section 3.3, we deal with the applicability of our method to nonlinear
models.
3.1
Agents Know That They Do Not Know
It is worth emphasizing that this way of recasting the learning process allows us to easily model economies
in which agents know that they do not know. In other words, agents form expectations taking into account
that their beliefs will change in the future according to what they will observe in the economy. This is
13
why the laws of motion (9) and (13) characterizing the behavior of the model depend on the current
beliefs and the expanded transition matrix de…ning the joint evolution of agents’ beliefs and model
parameters. This represents a substantial di¤erence with the anticipated utility approach in which agents
form expectations without taking into account that their beliefs about the economy will change over time
(e.g., Evans and Honkapohja, 2001; Cogley, Matthes, and Sbordone, 2011). Furthermore, the approach
described above di¤ers from the one traditionally used in the learning literature in which agents form
expectations according to a reduced-form law of motion that is updated recursively using the discounted
least-squares estimator (Eusepi and Preston, 2011). The advantage of adaptive learning is the extreme
‡exibility given that, at least in principle, no restrictions need to be imposed on the type of parameter
instability characterizing the model. However, such ‡exibility does not come without a cost, given that
agents are not really aware of the model they live in, but only of the implied law of motion. Instead, in
this paper agents fully understand the model, they are uncertain about the future, and they are aware
of the fact that their beliefs will evolve over time based on what they observe.
3.2
Likelihood Estimation
It is also important to emphasize the extreme tractability of the approach taken in this paper. The
solutions (9) and (13) can be easily combined with an observation equation and used in an estimation
algorithm. For example, Bianchi and Melosi (2012) estimate a prototypical New-Keynesian DSGE model,
in which agents form beliefs about the likely duration of deviations from active in‡ation stabilization
policies. The estimation of this new class of models is possible for three main reasons. First, even if the
…nal number of regimes can be extremely high, the model imposes very speci…c restrictions on the allowed
regime paths and on the link between observable outcomes and agents’beliefs. This implies that when
evaluating the likelihood, a relatively small number of regime paths has to be taken into account. Second,
the statistical properties of the di¤erent regimes can vary substantially and depend on the probability of
moving across regimes. Therefore, identi…cation of the transition matrix is not only given by the frequency
with which the di¤erent regimes occur, but also by the laws of motion characterizing the di¤erent regimes.
Finally, the number of extra parameters with respect to a model with perfect information is very low, if
not zero, while the resulting dynamics can be substantially enriched. For example, Bianchi and Melosi
(forthcoming) show that a period of …scal distress can lead to a run-up in in‡ation that lasts for decades.
From a computational point of view, there might be a concern about the time required to solve the
model when the …nal number of regimes becomes very large. This turns out not to be a problem. If
regime changes enter in an additive way, a¤ecting only the matrix
c,
the model can be solved with
standard solution algorithms such as gensys (Sims, 2002) or Blanchard and Kahn (1980) and the high
dimensionality of the transition matrix is not found to give rise to computational hurdles. However, in
many situations we might want to model regime changes that enter in a multiplicative way. For example,
we might want to allow for changes in the Taylor rule parameters. In this case, the matrices
0
and
1
are also a¤ected and we need to rely on solution methods developed to solve MS-DSGE models. However,
according to our experience based on the use of the approach proposed by Farmer, Waggoner, and Zha
(2009), even in this case a solution can be obtained in a matter of seconds because the transition matrix
governing the evolution of the regimes is very sparse. Therefore, the methods described in this paper
14
provide a promising tools for modeling information frictions, animal spirits, and shocks to agents’beliefs
in a general equilibrium framework suitable for structural estimation.
3.3
Extension to Nonlinear Models
To illustrate our methods, we have referred to linear models of the form (1) because this is currently the
most popular modeling framework when it comes to solving and estimating general equilibrium models
subject to parameter instability. However, this restriction can be easily relaxed. In fact, it can be shown
that all the results about the dynamics of agents’beliefs in Section 2 apply to the nonlinear case as well.
To see this, one should notice that we maintain the assumption that agents fully understand the model
and can observe the endogenous variables and the shocks. Therefore, at each point in time they are able
to infer the block that is in place. Once the history of the realizations of the two blocks is known, the
dynamics of agents’beliefs are pinned down by the properties of the transition matrix P that is known
to agents. Therefore, the evolution of beliefs does not depend on the model under consideration or the
order of approximation.
It is worthwhile emphasizing that application of our method to nonlinear models sets a promising
research agenda aimed to investigate the macroeconomic e¤ects of swings in uncertainty due to changes
in fundamentals or policy-makers’behavior. In light of this, the progresses made in e¢ ciently estimating
DSGE model through perturbation methods (e.g., Fernandez-Villaverde and Rubio-Ramirez, 2006 and
Foerster, Rubio-Ramirez, Waggoner, and Zha, 2011) have to be regarded as important complementary
to this line of studies.
4
Applications
In this section, we introduce a prototypical RBC model to illustrate the properties of the methods detailed
above. Central to our discussion will be the evolution of optimism and pessimism and the implications
thereof for consumption and saving decisions. The representative household chooses the sequence of
consumption ct and capital kt :
e0 P1
max E
t=0
ct ;kt
subject to the resource constraint ct + kt = zt kt
1
t
ln ct
+ (1
) kt
1
with
< 1 and 0 <
et ( )
< 1. Let E
denote the expectation operator conditional on households’ information set at time t. We assume that
total factor productivity (TFP) zt follows an exogenous process, such that
ln zt =
iid
where "t v N (0; 1) and
t
( t ) + ln zt
1
+
z "t
denotes a discrete Markov process a¤ecting the drift of TFP. This process
evolves according to the transition probability matrix P. We assume that
t
2 f1; 2; 3; 4g. These values map into values for the TFP drift
H
and
t
2 f3; 4g =)
(17)
t ( t)
=
L,
where
L
<
H.
t
can take four values; that is,
( t ) as follows
t
2 f1; 2g =)
t ( t)
=
In Block 1, Regimes 1 and 2 di¤er in their likely
persistence: p11 < p22 . The same applies to Regimes 3 and 4 in Block 2: p33 < p44 . We call Regimes 1
15
and 2 high-growth regimes and Regimes 3 and 4 low-growth regimes. Households are assumed to observe
the history of the model variables (kt , ct , and zt ) and that of the TFP shocks ("t ). Therefore, households
can establish whether the economy is in the high-growth block or in the low-growth block.
1=(1
) e
1=(1
We introduce the stationary variables t
ln (zt =zt 1 ), e
ct
ct =zt
, kt
kt =zt
)
and,
following Schorfheide (2005) and Liu, Waggoner, and Zha (2011), we de…ne the steady state as the
stationary equilibrium in which all shocks are shut down, including the regime shocks to the growth rate
of TFP. We then derive a log-linear approximation to the equilibrium equations around the steady-state
equilibrium for these stationary variables. The log-linearized Euler equation reads:5
where M
1
et b
b
ct = E
ct+1
(
exp ( ),
1) 1 + (
(p1 + p2 )
1) M
1
1
+ M
1
(
et bt+1
1) + 1 E
(18)
being the ergodic mean of the log growth rate of the
economy, and pi stands for the ergodic probability of being in regime i, bt ; b
ct and b
kt denote log-deviations
H
+ (p3 + p4 )
1
b
kt
1
L
of the stationary TFP growth, consumption, and capital, respectively, from their steady-state value, and
b ( t)
t ( t)
css b
ct + kss b
kt =
is the log-deviation of TFP drift from its ergodic mean . The resource constraint is
M
1
kss
1
+
1
1
1
M
1
kss bt + M
1
kss + (1
)
Finally, the log-deviation of the growth rate of TFP from its ergodic level follows
bt = b ( t ) +
z "t
1
1
kss b
kt
1
(19)
(20)
As is standard for any RBC model, households adjust capital so as to smooth consumption intertemporally. The occurrence of TFP shocks and the succession of low-growth and high-growth regimes challenge
households’ability to smooth consumption over time. When the economy is in the high-growth regime,
households expect that, with some probability, the economy will enter into the low-growth regime in the
future, making it harder to raise future consumption. Therefore, ceteris paribus agents raise capital today
et b
so as to raise future expected consumption E
ct+1 vis-a-vis current consumption b
ct . When the economy
is in the low-growth regime, agents expect that, with some probability, the economy will enter into the
high-growth regime in the future, making it easier to raise future consumption. Therefore, ceteris paribus
agents reduce capital today so as to raise current consumption b
ct vis-a-vis expected future consumption
et b
E
ct+1 .
Clearly, the persistence of the regime in place critically a¤ects consumption and capital decisions.
When the current regime is expected to be short lasting, households generally adjust capital more aggressively than when it is expected to be long lasting, because they deem that a switch in the next period
is more likely. In contrast, households do not adjust capital so aggressively if they expect the regime to
be very long lasting. When households expect that low growth or high growth has become a structural
characteristic of the environment, they understand that consumption cannot be e¤ectively smoothed out
over time by adjusting capital. Thus, very persistent regimes are mostly characterized by structural
5
A detailed derivation of the steady-state equilibrium for the stationary variables and the log-linearized equations is
provided in Appendix D.
16
changes in the level of consumption.
Given that households have limited information, the log-linearized Model (18)-(20) cannot be solved
using the existing techniques that are used to solve Markov-switching models with perfect information.
However, we proceed as described in the previous sections, by introducing a new set of regimes that
capture the evolution of the representative household’s beliefs over time. It is important to notice that
in the RBC model described above, regime changes enter additively. In other words, they only a¤ect
the vector of constants
c(
) in the canonical forms (7) or (12). In this case, the state space can be
augmented with a series of dummy variables as in Schorfheide (2005), Liu, Waggoner, and Zha (2011),
and Bianchi, Ilut, and Schneider (2012) and the models under imperfect information can be easily solved
using standard solution methods for DSGE model, such as gensys (Sims, 2002) and Blanchard and Kahn
(1980). When regime changes enter multiplicatively, the matrices
0
and
1
are also a¤ected. In this
case, the model can be solved with any of the solution methods that have been developed for MS-DSGE
models. Bianchi and Melosi (2012a, Forthcoming) consider these cases and solve the model using the
algorithm developed by Farmer, Waggoner, and Zha (2009).
In what follows, we adopt a standard calibration of the RBC model. We set capital’s share parameter
to equal 0:33. The discount factor
is equal to 0:9976 and the parameter for the physical depreciation
of capital is set to equal 0:0250. The standard deviation of the TFP shock
is set to 0:007. We set
the growth rate of TFP in the high-growth state to equal the annualized rate of 4%:
assume that under low-growth, the growth rate of TFP is simply zero:
L
H
= :01. We
= 0. Furthermore, we consider
several parameterizations of the transition matrix P, allowing us to illustrate a number of di¤erent model
economies that can be potentially used to address a large set of empirical issues. In actual applications,
the parameterization of the transition will depend on the data and the associated empirical moments the
researcher is interested to match. In this paper, we do not address any speci…c empirical issue as our
objective is to show the scope of application of the methodology we propose.
Section 4.1 studies an economy in which agents have to learn about the likely persistence of the
observed TFP growth. In Section 4.2, we consider an economy that goes through two types of phases
over time: a high-growth phase that is mostly characterized by long-lasting high-growth periods with
rare short-lasting low-growth periods and a low-growth phase that is mostly characterized by persistent
periods of low-growth and high-growth periods of rather short duration. In Section 4.3, we study the
e¤ects of shocks to beliefs on aggregate dynamics and uncertainty.
4.1
Learning the Persistence of TFP Growth
In what follows, we consider an RBC model as described by equations (18)-(20) in which the economy
‡uctuates repeatedly between short-lasting periods of high growth and short-lasting periods of low growth.
However, once in a while a prolonged phase of high growth or low growth may occur. Agents are able to
infer what the growth rate is today, but they are uncertain about its likely duration. Whenever a change
in the level of growth occurs, the statistical properties of the macroeconomy, which are captured by the
transition matrix P, imply that rational agents’ pessimism and optimism are invariant with respect to
previous realizations of TFP growth. Notice that this corresponds to the case of static prior. We will …rst
assume that periods of high growth and low growth are symmetric. We will then consider the asymmetric
17
case.
4.1.1
The symmetric case
Let us assume that the persistence of the short-lasting regimes is the same in the two blocks: p11 = p33 =
0:5. Analogously, we set the probabilities of staying in the long-lasting regimes so that p22 = p44 = 0:95.
For simplicity we assume that regimes belonging to the same block do not communicate with each other;
that is, p12 = p21 = p34 = p43 = 0. We will be more general later on. Furthermore, the transition matrix
implies that once a switch to a new block occurs, agents always attach a 95% probability to being in the
short-lasting regime:
p31
p31 + p32
p13
p13 + p14
=
=
p41
= 0:95
p41 + p42
p23
= 0:95
p23 + p24
(21)
(22)
Notice that conditions (21)-(22) imply static prior beliefs: agents always enter the high-growth block and
the low-growth block with the same beliefs. In summary, we work with the following transition matrix:
2
0:50
6
6
0
P =6
6 0:475
4
0:0475
0
0:475
0:95
0:0475
0:025
0:50
0:0025
0
0:025
3
7
0:0025 7
7
7
0
5
0:95
To illustrate the consequences of ‡uctuations in agents’beliefs, we simulate the economy assuming a
typical path for the regimes and setting all Gaussian shocks "t to zero. We assume that consumption and
capital are initialized at its steady-state value. The results are reported in Figure 1. In each panel, the
gray and white areas correspond to periods of low and high growth, respectively. Short-lasting regimes
last for their typical duration of 2 quarters. Long-lasting regimes last for their typical duration of 20
quarters. The two right graphs report the evolution of consumption and capital in the model with learning
compared to the model with perfect information in which agents can observe the current regime. The
panel in the upper-left corner shows the evolution of agents’beliefs about being in the long-lasting highgrowth regime and in the long-lasting low-growth regime. The panel in the lower-left corner reports the
evolution of expected average TFP growth at 4-, 8-, 20-, and 40- quarter horizons. Notice that this is a
convenient measure of agents’optimism/pessimism that takes into account uncertainty about the regime
in place today and the possibility of regime changes.
Three features of Figure 1 deserve to be emphasized. First, right after a switch to a new block, agents
believe that this switch is most likely to be short lasting. This can be seen in the top left graph when
switches to new blocks occur. The reason is that agents are rational and hence are aware that regardless
of whether the past regime was short lasting or long lasting, the probability of switching to the shortlasting regime in the new block is always as high as 95%. This stems from the restrictions in (21)-(22),
which imply static prior beliefs. Second, whenever a short-lasting regime is in fact realized, with the
bene…t of hindsight, agents’ beliefs turn out to be slightly misaligned with the truth in the upper left
18
0.0375
Pr.LL-HG
Pr.LL-LG
Beliefs
0.8
0.4
0.2
0.6
1
50
100
150
Consumption Gap
1
0
-0.0125
-0.025
50
100
150
200
50
100
150
200
0.0125
4Q
8Q
20Q
40Q
Ergodic
0.01
0.0075
0.005
0.0025
50
100
150
Capital Gap
E(Avg TFP Growth)
0.0125
-0.0375
200
0.0125
0
0.025
0.0075
0.0025
-0.0025
-0.0075
-0.0125
200
Figure 1: Beliefs and Allocations. Top left graph: Evolution of beliefs of being in the long-lasting high-growth regime (red
solid line) and in the long-lasting low-growth regime (blue dashed line). Top right graph: Log-deviations of consumption from
the perfect-information benchmark. Bottom left graph: Expected average growth rate of technology (annualized percentage)
at various horizons. Bottom right graph: Log-deviations of capital from the perfect-information benchmark. In all graphs,
gray areas denote periods of low growth.
graph because agents rationally attach a non-zero probability to being in the long-lasting regime. Third,
the probability of being in the long-lasting regime smoothly increases as more realizations of the same
block are observed. The top left graph shows that the probability of being in the long-lasting regime rises
monotonically with the number of consecutive realizations of a particular growth rate. For instance, from
t = 117 to t = 136, the economy is in a long-lasting low-growth regime. While agents initially attach a
small probability to being in the long-lasting regime, they become fully convinced after 12 consecutive
periods of low TFP growth.
Furthermore, Figure 1 shows the evolution of optimism and pessimism and the associated dynamics
of the consumption gap and the capital gap, which are de…ned as the log-deviation of consumption and
capital from their corresponding levels under perfect information. When the economy enters the longlasting low-growth period imperfectly informed agents are not very pessimistic about the duration of the
low-growth regime. This is re‡ected in their expectations about the average growth rate of TFP that
barely moves in the bottom left graph. Given that they expect that the low-growth period will be short
lasting, they decide to slow down capital accumulation so as to smooth consumption. In contrast, if
agents knew the actual realization of the low-growth regime, they would have adjusted their stock of
capital less aggressively and consequently consumption would have fallen more dramatically. This is why
in Figure 1 we observe a positive consumption gap and a negative capital gap when the economy enters
a period of long-lasting low growth.
As the period of low-growth consolidates, imperfectly informed agents update their beliefs until they
eventually become convinced that they are in the long-lasting regime. This happens in roughly 12 quarters
19
Figure 2: Evolution of Uncertainty about Consumption, Capital, and TFP Growth. The horizon ranges from one quarter
ahead (i.e., lightest blue line) through …ve years ahead (i.e., darkest red line). The darker the color of a line, the longer the
h is measured as the standard deviation of the variable of interest in
t + h conditional on agents’information set at time t, It .
horizon of the uncertainty. Uncertainty at an horizon
period
after the switch. As illustrated in the bottom left graph, such slow-moving beliefs cause the expected
average growth rate of TFP over the next few years to also adjust sluggishly. This eventually determines
an adjustment in the path for consumption and the consumption gap slowly fades away. Interestingly,
at the end of the long-lasting low-growth period, the consumption gap becomes negative. The reason
is that the sluggish evolution of pessimism prompted households to decumulate capital rapidly at the
beginning of the period of low growth. The relatively small capital stock depresses consumption as
households become pessimistic, leading to a negative consumption gap. A specular pattern characterizes
the economy the moment it enters the long-lasting high-growth period at the beginning of the simulation.
As pointed out before, even when the economy repeatedly alternates between short-lasting periods,
agents’beliefs are slightly misaligned with the truth. Let us focus on the …rst 16 quarters during which
a sequence of short-lasting regimes are realized. While the economy is in the short-lasting high-growth
regime, imperfectly informed households consume more and accumulate less capital than in the case
of perfect information. The reason is that imperfectly informed agents attach some non-negligible albeit small - probability to being in the long-lasting regime. By the same token, when the economy is
going through a short-lasting period of low growth, imperfectly informed households consume less and
accumulate more capital than under perfect information.
Figure 2 shows the evolution of uncertainty about consumption, capital, and TFP growth rate. The
horizon ranges from one quarter ahead, light blue line, through …ve years ahead, dark red line. The darker
the color of a line, the longer the horizon of the uncertainty. Uncertainty at an horizon h is measured
using the standard deviation of the variable of interest at time t+h conditional on agents’information set
20
at time t, It . It is worth emphasizing that this measure of uncertainty is computed taking into account
the possibility of regime changes and the evolution of agents’ beliefs, using the methods described in
Bianchi (2012). Let us focus …rst on the evolution of uncertainty about future TFP growth during the
long-lasting period of high growth, captured by the large white area on the lowest panel. It should be
observed that when agents are mostly convinced to be in the short-lasting high-growth regime (i.e., in
the narrow white areas or at the beginning of the broad white areas), uncertainty is generally higher and
remarkably similar at all horizons. Furthermore, uncertainty about future TFP growth falls at all horizons
as agents become more convinced to be in the long-lasting regime. These two …ndings are not surprising,
since expecting a shorter (longer) duration of the current block raises (reduces) uncertainty about future
TFP developments. Furthermore, when long-lasting regimes occur, long-horizon uncertainty falls less
dramatically than short-horizon uncertainty. The reason is that as agents become more convinced to be
in a long-lasting regime, they deem a switch to a short-lasting regime as relatively more likely at longer
horizons.6
As far as the dynamics of uncertainty about consumption and capital (the highest and middle panels),
it is important to notice that the main source of uncertainty about future allocations is due to the need of
establishing whether and when big adjustments in allocations will occur. As shown in the right panel of
Figure 1, such large adjustments are observed when long-lasting regimes occur because beliefs adjust only
sluggishly to the truth. Long-horizon uncertainty about future allocations appears to be always higher
than short-horizon uncertainty, because agents know that such large adjustments are more likely to
happen as the considered horizon gets longer and longer. Quite interestingly, we observe that uncertainty
follows a hump-shaped pattern during a typical long-lasting regime. On the one hand, uncertainty rapidly
rises as agents gets more and more convinced of being in a long-lasting regime that is associated with
a large adjustment in allocations. On the other hand, as agents become more convinced to be in the
long-lasting regime, their uncertainty about future allocations falls because they become less uncertain
about future TFP growth. See the lowest graph of Figure 2. Finally, note that the hump-shaped becomes
more pronounced as the horizon grows large.
4.1.2
An Asymmetric Case
So far we have analyzed the case in which the stochastic properties of the short-lasting and the longlasting regimes are exactly the same across blocks. Now we study a situation in which the persistence of
the short-lasting regime is di¤erent across the two blocks. More speci…cally, we calibrate the probability
of staying in the short-lasting high-growth regime to be p11 = 0:75 > 0:5. The probability of staying
in the long-lasting high-growth regime is unchanged (p22 = 0:95).
We shall show that optimism and
pessimism now evolve at di¤erential speed in this model. The speed of learning within a block is a¤ected
by the relative persistence of the corresponding two regimes. As we shall show, if the persistences of the
two regimes become more similar, it takes longer for rational agents to …gure out which regime is in place.
Figure 3 shows the dynamic of beliefs, the expected average growth rate of TFP at various horizons
(4, 8, 20, and 40 quarters), the consumption gap, and the capital gap when the economy goes through
6
Recall that the parameterization of the transition matrix P implies that the long-lasting regimes of both blocks are
always followed by short-lasting regimes, which are associated with relatively higher uncertainty.
21
Pr.LL-HG
Pr.LL-LG
Beliefs
0.8
0.4
0.2
0.6
50
100
150
200
250
300
0.0375
0.025
0.0125
0
-0.0125
-0.025
-0.0375
0.0125
0.0125
0.01
0.0075
Capital Gap
E(Avg TFP Growth)
1
Consumption Gap
1
0.0075
0.005
4Q
8Q
20Q
40Q
Ergodic
0.0025
0
50
100
150
200
250
100
150
200
250
300
50
100
150
200
250
300
0.0025
-0.0025
-0.0075
-0.0125
300
50
Figure 3: Beliefs and Allocations. Top left graph: Evolution of beliefs of being in the long-lasting high-growth regime (red
solid line) and in the long-lasting low-growth regime (blue dashed line). Top right graph: Log-deviations of consumption from
the perfect-information benchmark. Bottom left graph: Expected average growth rate of technology (annualized percentage)
at various horizons. Bottom right graph: Log-deviations of capital from the perfect-information benchmark. In all graphs,
gray areas denote periods of low growth.
the same sequence of regimes as that in Figure 1 with the only di¤erence that now the typical duration of
the short-lasting high-growth regime is longer: 4 quarters instead of 2. The typical duration of all other
regimes is the same as that in the example of the previous subsection. The crucial point to notice is that
in Figure 3 the typical realization of 20 quarters of high growth is not enough for agents to …gure out
that the realized regime is of the long-lasting type. Agents attach only 80% probability of being in the
long-lasting regime after having observed 20 consecutive periods of high growth. In contrast, when the
economy is going through a period of long-lasting low growth, it takes roughly 12 quarters for households
to be fully convinced that they are in the long-lasting regime, exactly as in Figure 1.
Di¤erential speeds of learning have an impact on the dynamics of consumption and capital. During
the long-lasting high-growth regime, the misalignment of agents’beliefs is more persistent than in the case
of the low-growth regime. This implies a more persistent negative consumption gap because agents raise
capital more aggressively when they are quite convinced to be in the short-lasting high-growth regime.
Moreover, the consumption gap is less pronounced than that under a symmetric speed of learning because
the expected duration of the short-lasting high-growth regime is now more similar to that of the longlasting high-growth regime compared to the previous subsection.
Figure 4 shows the evolution of uncertainty about consumption, capital, and TFP growth rate. Two
facts deserve to be emphasized in the lower graph depicting the evolution of uncertainty about future
TFP growth. First, uncertainty about future TFP initially increases at all horizons during periods of
low-growth (gray areas). This outcome was not observed in the symmetric case depicted in Figure 2.
The reason is that now the two high-growth regimes are more similar in terms of their likely duration.
22
Figure 4: Evolution of Uncertainty about Consumption, Capital, and TFP Growth. The horizon ranges from one quarter
ahead (lightest blue line) through …ve years ahead (darkest red line). The darker the color of a line, the longer the horizon
of the uncertainty.
Therefore, agents are relatively less uncertain about the likely duration of TFP growth during high-growth
periods. It follows that when the system switches from high growth to low growth, agents’uncertainty
soars to re‡ect the higher uncertainty inherent in the low-growth block. Second, as agents become more
and more convinced to be in the long-lasting regimes, uncertainty declines, especially at shorter horizons.
This happened in the symmetric case too. These two facts explain why agent’s uncertainty about future
TFP follows a hump-shaped pattern during long-lasting low-growth regimes.
Finally note that uncertainty about future consumption and capital does not substantially change as
the economy experiences a long-lasting period of fast TFP growth (large white area). The reason is that
in this asymmetric case the persistence of the short-lasting high-growth regime is fairly similar to that
of the relative longer lasting regime. Thus even though agents are uncertain about which high-growth
regime is actually in place, this turns out to have relatively smaller practical implications for allocations
as it emerges from comparing the rights graphs of Figures 1 and 3.
4.2
A Two-Phase RBC Model
We will now model an economy that goes through two types of phases over time: a high-growth phase that
is mostly characterized by long-lasting high-growth periods with rare short-lasting low-growth periods
and a low-growth phase that is mostly characterized by persistent periods of low-growth and high-growth
periods of rather short duration. In such an economy waves of optimism and pessimism will spur from
the past realizations of TFP growth. Therefore, we need to keep track of agents beliefs before a change in
growth. This corresponds to the case of dynamic priors. In what follows, we study three RBC economies
featuring low- and high-growth phases. In Subsection 4.2.1, switches to a new phase are always preceded
23
Figure 5: Expected Growth Rate of Technology at Various Horizons as a Function of Beliefs. Lighter blue areas capture
expected rates that are lower than the ergodic rate. Darker red areas capture expected rates that are higher than the ergodic
rate. The horizontal axes report beliefs about being in the long-lasting high-growth regime (LL-HG) and beliefs about being
in the long-lasting low-growth regime (LL-LG).
by a changes in TFP growth.7 In Subsection 4.2.2, we study an economy in which switches to the lowgrowth phase are not necessarily marked by a changes in TFP growth. In Subsection 4.2.3, we consider
the case of an economy in which switches to both phases may occur with no apparent changes in TFP
growth. Note that whether changes of phases are preceded or not by change in TFP growth is important
because agents perfectly observe changes in TFP growth.
4.2.1
Changes of Phase Preceded by a Change in Growth
In order to model the two phases of the business cycle, we introduce the following restrictions on the
parameters of the transition matrix P:
p31
p31 + p32
p13
p13 + p14
p41
= 0:95
p41 + p42
p23
= 0:05 <
= 0:95
p23 + p24
= 0:05 <
(23)
(24)
Furthermore, we assume that the probability of staying in the short-lasting regimes is p11 = p33 = 0:75.
We set the probabilities of staying in the long-lasting regimes so that p22 = p44 = 0:95. We also assume
that the regimes belonging to the same block do not communicate with each other: p12 = p21 = p34 =
p43 = 0. This has the important implication that a change of phase is always preceded by an observable
change in growth. We will relax this restriction in the next two subsections. To sum up, the transition
7
Of course, this does not mean that every time that a change in growth occurs, agents immediately conclude that the
phase changed. They still have to learn about the nature of the observed change in growth.
24
matrix reads:
2
0:75
6
6
0
P =6
6 0:0125
4
0:0475
0
0:0125
0:95
0:0475
0:2375
0:75
0:0025
0
0:2375
3
7
0:0025 7
7
7
0
5
0:95
It is important to emphasize that, in this model, the fact that the economy is currently in the highgrowth or low-growth regime plays a minor role in a¤ecting agents’beliefs. Most of the action stems from
whether agents believe that the economy has been going through a high-growth phase or a low-growth
phase. Figure 5 shows agents’ expectations about the growth rate of TFP
ergodic level
t
in deviations from its
at di¤erent horizons and for various initial levels of probability of being in the long-lasting
high-growth regime (left plot) and low-growth regime (right plot). Notice that when agents expect to be
in the long-lasting high-growth (low-growth) regime, the expected growth rate of technology di¤ers from
that in the short-lasting low-growth (high-growth) regime only at very short horizons. This is because
agents are rational and are aware of conditions (23)-(24), implying that short-lasting regimes are more
likely to be followed by the long-lasting regime of the opposing block.
It should be noted that conditions (23)-(24) imply that agents’ beliefs are dynamic in this model.
Suppose that agents mostly expect to be in the long-lasting high growth period. Then if in the next
period the economy moves to the low-growth block, agents will mainly expect to be in the short-lasting
low-growth regime. In contrast if agents believe to be in the short-lasting high growth today, then a
switch to the low-growth block tomorrow will lead them to believe that the low-growth period is most
likely to be long-lasting.
The upper left graph of Figure 6 reports the evolution of agents’beliefs, consumption, and capital for
the case of dynamic prior beliefs. We simulate a typical path for the regimes where a low graph phase is
followed by a high-growth phase. Furthermore, we initialize agents’beliefs so that agents are con…dent
of being in a high-growth phase.8 As agents observe 4 quarters of high growth, followed by 20 quarters
of low growth, agents start to fear that the economy has switched to the low-growth phase. As a result,
households are less optimistic when the economy returns to the high-growth regime. When the second
realization of the long-lasting low-growth regime occurs, households become immediately convinced that
the long-lasting low-growth regime is in place. Symmetrically, when the economy returns to the shortlasting high-growth regime for the third time, households believe that the high-growth regime will be
long-lasting with only a 6% probability. Afterwards, the economy enters the high-growth phase by going
through a short-lasting low-growth regime. Households are initially very pessimistic about the persistence
of this regime expecting the low-growth regime to be long lasting. It takes two realizations of the longlasting high-growth regimes to make them fully con…dent that the economy has shifted to the high-growth
phase.
The lower left graph of Figure 6 provides further evidence that households slowly learn about changes
in the two paths. Observe that when the economy enters the …rst long-lasting low-growth period, households mostly believe that they are still in the high-growth phase and expect an average growth rate
8
This can easily happen if the economy went through a typical high-growth phase in the past and agents have …nally
learned about this phase.
25
Beliefs
0.8
0.4
0.2
0.6
25
50
75
100
0.0625
0.025
-0.0125
-0.05
-0.0875
0.0125
0.01
0.0075
0.005
0.0025
0
25
50
75
100
25
50
75
100
0.05
4Q
8Q
20Q
40Q
Ergodic
Capital Gap
E(Avg TFP Growth)
1
Pr.LL-HG
Pr.LL-LG
Consumption Gap
1
25
50
75
0.025
0
-0.025
-0.05
100
Figure 6: Beliefs and Allocations. Top left graph: Evolution of beliefs of being in the long-lasting high-growth regime (red
solid line) and in the long-lasting low-growth regime (blue dashed line). Top right graph: Log-deviations of consumption from
the perfect-information benchmark. Bottom left graph: Expected average growth rate of technology (annualized percentage)
at various horizons. Bottom right graph: Log-deviations of capital from the perfect-information benchmark. In all graphs,
gray areas denote periods of low growth.
of TFP over the next 20 or 40 quarters that is above the ergodic level. The same sluggishness in the
expected average growth rate of TFP can be observed as the economy enters the …rst long-lasting highgrowth period. Furthermore, the sluggish dynamics of optimism and pessimism are con…rmed by a quick
comparison of the expected average growth rate of TFP across short-lasting periods.
The behavior of consumption and capital during the low-growth and the high-growth phase is analyzed
in the right graphs of Figure 6. We observe that at the beginning of the …rst short-lasting high-growth
regime, which is associated with high optimism, the consumption gap is positive. The reason is that
imperfectly informed households expect this regime to be much longer lasting than what it actually turns
out to be. This implies that imperfectly informed households do not raise capital as aggressively as they
would if they knew that the high-growth regime is, in fact, short lasting. This leads to a negative capital
gap and a positive consumption gap. When the economy enters the long-lasting low-growth regime for the
…rst time, households mainly expect a short-lasting regime at …rst. As a result, households decide to cut
capital fairly aggressively to sustain current consumption. Households would do otherwise, if they knew
that the economy just entered the long-lasting low-growth regime, leading to a positive consumption gap
and a negative capital gap.
During the …rst long-lasting low-growth spell households update their beliefs until they realize that
this regime is most likely long lasting, signifying that the economy must have switched to the low-growth
phase. This change in agents’beliefs causes consumption and capital (the latter with some sluggishness)
to become similar to the perfect-information benchmark. Interestingly, the consumption gap changes
sign and becomes negative at the end of the …rst long-lasting low-growth spell and throughout the second
26
Figure 7: Evolution of Uncertainty about Consumption, Capital, and TFP Growth. The horizon ranges from one quarter
ahead (lightest blue line) through …ve years ahead (darkest red line). The darker the color of a line, the longer the horizon
of the uncertainty.
low-growth period. This is due to the fact that capital adjusts sluggishly to its perfect-information level.
When the second short-lasting high-growth regime occurs, agents are more convinced to have entered a
low-growth phase and then optimism is smaller than in the previous high-growth period, resulting in a
more contained hike in the consumption gap. The dynamics of the consumption gap and the capital gap
are clearly reversed during the high-growth phase.
Figure 7 shows the evolution of uncertainty about consumption, capital, and TFP growth rate during
the typical simulation. At time t = 1, agents mostly expect to be in a high-growth phase that is is
typically characterized by long-lasting high-growth regimes. Since agents expect to be in a long-lasting
regime, their uncertainty about future TFP growth is on the low side. As the economy switches to a
low-growth period at time t = 5, agents initially expect it to be short lasting because they are still quite
convinced to live in a high-growth phase in which the typical duration of low-growth periods is relatively
short. This conviction prompts agents to expect that the TFP growth is likely to change shortly. Thus,
their uncertainty sharply increases as the system enters the …rst low-growth spell, as shown in the lower
panel of Figure 7. At the end of the …rst long-lasting low-growth period, agents have mostly learned by
now that the system is going through a low-growth phase and therefore the current low-growth regime
is likely to last for a fairly long period. Consequently, uncertainty about future TFP at all horizons fall
down at the end of this long-lasting low-growth period. The initial rise in uncertainty and its subsequent
fall during the …rst long-lasting low growth regime explain the hump-shaped pattern that can be observed
in the bottom panel of Figure 7.
As the system switches to the second high-growth period, uncertainty suddenly soars because by then
agents have learned to be in the low-growth phase, implying that the realized high-growth regime is most
27
Absolute Forecast Error
x 10
=1|B =B )
-4
t
prob(ξ
t+h
2
0
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
x 10
2
0
-4
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
x 10
=3|B =B )
prob(ξ
4
t+h
4
-4
L
H
=3|B =B )
6
t
6
t
4
t+h
4
6
t
h=1
h=2
h=3
h=4
6
prob(ξ
t+h
prob(ξ
-4
L
H
=1|B =B )
x 10
2
0
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
prob( ξ =1|B )
t
H,t
2
0
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
prob( ξ =3|B )
t
L,t
Figure 8: Absolute Forecast Errors. Left panel: Absolute forecast error about the probability of being in the short-lasting
regimes prob
t+h = 1jIt (upper graphs) and prob
t+h = 3jIt (lower graphs), in h = 1; 2; 3; and 4 quarters
from period t conditional on being in the high-growth block with initial beliefs prob f t = 1jIt g, which are reported
on the horizontal axes. Right panel: Absolute forecast error about the probability of being in the short-lasting regimes
prob t+h = 1jIt (upper graph) and prob t+h = 3jIt (lower graph), in h = 1; 2; 3; and 4 quarters from period
t conditional on being in the low-growth block with initial beliefs prob f t = 3jIt g, which are reported on the horizontal
axes. Di¤erent color bars denote di¤erent horizons h of forecast. Bt denotes the block in place at time t, which can be the
high-growth one, BH , or the low-growth one, BL .
likely to be short lasting. This is the opposite of what happened during the …rst high-growth period
where agents were still convinced to be in the high-growth phase. When the system switches to the low
growth regime for the second time, agents are now fairly sure to be in the low-growth phase and hence
the likely duration of the current low growth in TFP is long. Consequently, uncertainty plummets at all
horizons. Uncertainty keeps falling as agents gets more and more convinced to be in the long-lasting low
growth regime.
Furthermore, it is interesting to notice the evolution of the short-horizon uncertainty about TFP
growth relatively to the long-horizon one. When agents expect an observed regime to be long-lasting
(short-lasting), short-horizon uncertainty is generally lower (higher) than long-horizon uncertainty. If the
current regime is long lasting, a switch to the short-lasting regime, which would sharply raise uncertainty
about future TFP growth, is likely to occur only at long horizons. Therefore, long-horizon uncertainty
becomes higher than the short-horizon one when agents expect the current regime to be long lasting. In
particular, note that during the …rst long-lasting regime of the two phases uncertainty is initially higher
at shorter horizon. After a few periods of these long-lasting regimes, high-horizon uncertainty turns
out to be higher than the short-horizon one. Again, this is explained by the sharp swings in agents’
beliefs occurring during these two long-lasting regimes. Agents start expecting a short-lasting regime but
28
eventually end up updating their beliefs in favor of a long-lasting regime.
Uncertainty about future consumption and capital follow a hump-shaped patter, when the economy
goes through a long-lasting regime that agents initially believed to be short-lasting. The reason is similar
to that we discussed in Section 4.1. On the one hand, uncertainty rapidly rose as agents gets more and
more convinced to be in a long-lasting regime that would prompt a large adjustment in allocations. On
the other hand, as agents become more convinced to be in the long-lasting regime, their uncertainty
about future allocations falls because they become less uncertainty about future TFP growth.
While the approximation error can be shown to be extremely tiny in the model with static prior
studied in Section 4.1, it is not obvious that the approximation error made in this application in which
agents’priors are dynamic is small. When agents’priors is dynamic, the approximation error is tightly
related to how …ne is the grid G that approximates agents’ beliefs, prob ( t jIt ). We initially set 100
equally spaced knots in our grid for each block. Furthermore, we add 194 knots to make the grid …ner
for beliefs near the convergence points for prob
t
= 1j
1
t
and prob
t
= 3j
1
t
, which are zero for both
blocks. After the re…nement of the grid of beliefs introduced in steps 3-4 of Section 2.2.2, we are left with
213 grid points per block. Even if the number of regimes seems enormous, solving the model takes 5.23
seconds in Matlab on a 64-bit desktop endowed with an Intel core processor i7-2600 CPU at 3.40 GHz.9 .
Figure 8 reports the absolute forecast error, which is computed by taking the absolute di¤erence between
b and the true probability that can be easily worked out
prob t+h jIt approximated using the matrix P
using equations (3) and (4). The forecast errors are computed using various initial beliefs of being in
the short-lasting regime of a given block, prob (
t
= 1jIt ) and prob (
t
= 3jIt ), which are reported on the
horizontal axes of the plots in Figure 8. The left panel refers to the situation in which the economy is
initially in the high-growth block, while the right panel shows the approximation errors when the initial
state of the economy is low growth. Approximation error appears to be very small at all horizons.10
Finally note that the plots are symmetric (i.e., the upper left one is identical to the lower right one
and the lower left one is identical to the upper right one) because the primitive transition matrix P is
symmetric and the initial grids for the beliefs in the two blocks are chosen to be identical.
4.2.2
Low-Growth Phases Not Necessarily Preceded by a Change in Growth
We have considered so far transitions between high-growth and low-growth phases that are always marked
by an observable change in TFP growth. This feature is due to the fact that so far we have assumed
that the probability of switching between regimes belonging to the same block is zero. In this section we
relax this assumption.
Let us use the baseline calibration and the same values for the transition matrix P as those used
in Subsection 4.2.1, with the only exception that now the probability of switching to the short-lasting
high-growth regime conditional on being in the long-lasting high-growth regime is nonzero. Speci…cally,
we set p21 = 0:04. The probabilities p23 and p24 are re-scaled so that (24) is satis…ed. In this context,
a switch from the high-growth phase to the low-growth phase may happen without the agents observing
b
In the case with static prior beliefs, which was analyzed in Section 4.1, it takes 0.10 second to compute the matrix P
and to solve the model with gensys.
10
We also checked stability of the law of motion as the number of grid points increases. The approximation turns out to
be accurate even using this criterion. We report results for the absolute forecast errors because these are not model speci…c.
9
29
Pr.LL-HG
Pr.LL-LG
0.8
Beliefs
Consumption Gap
1
0.4
0.2
0.6
25
50
75
100
0.025
-0.0125
-0.05
-0.0875
0.0125
0.01
0.0075
0.005
0.0025
0
25
50
75
100
25
50
75
100
0.05
4Q
8Q
20Q
40Q
Ergodic
Capital Gap
E(Avg TFP Growth)
1
0.0625
25
50
75
0.025
0
-0.025
-0.05
100
Figure 9: Beliefs and Allocations. Top left graph: Evolution of beliefs of being in the long-lasting high-growth regime (red
solid line) and in the long-lasting low-growth regime (blue dashed line). Top right graph: Log-deviations of consumption from
the perfect-information benchmark. Bottom left graph: Expected average growth rate of technology (annualized percentage)
at various horizons. Bottom right graph: Log-deviations of capital from the perfect-information benchmark. In all graphs,
gray areas denote periods of low growth.
any change in growth: The economy simply moves from the long-lasting high-growth regime to the shortlasting high-growth regime. Although the probability that such an unobserved switch would happen is
quite small (p21 = 0:04), such a possibility turns out to deeply in‡uence the dynamics of agents’beliefs
and allocations.
Figure 9 reports the evolution of beliefs, average expected growth rate, consumption gap, and capital
gap conditional on the same simulated path of regimes as that analyzed in the previous example.11 Let
us focus on the second half of the simulation when the economy enters the high-growth phase. The top
left graph of Figure 9 shows that agents’beliefs about being in the long-lasting high-growth regime do
not converge to unity even when a large number of high-growth periods occur. This is di¤erent from
what we observe in Figure 6. Thus, an important implication of introducing unobserved switches to the
low-growth phase is that agents will never become fully convinced to be in the high-growth phase.12
Furthermore, as short-lasting low-growth regimes occur, agents are relatively more concerned about the
possibility of having entered a long-lasting low-growth period. The reason is that agents are aware that
an unobserved switch to the low-growth phase may have occurred during the last spell of high-growth.
The right panels of Figure 9 show the consumption and capital gaps with respect to the perfectinformation benchmark. As we allow for the possibility of unobserved switches, the high-growth phase is
11
To ease the comparison with the previous case with no unobservable switches, the scale of the y-axes is set to be the
same as that in Figure 6.
12
Note that this is di¤erent from the case of slow learning analyzed in Subsection 4.1.2. In that case, agents can become
fully convinced to be in the long-lasting high-growth regime provided that they observe a su¢ ciently long spell in the
high-growth block.
30
Figure 10: Evolution of Uncertainty about Consumption, Capital, and TFP Growth. The horizon ranges from one quarter
ahead (lightest blue line) through …ve years ahead (darkest red line). The darker the color of a line, the longer the horizon
of the uncertainty. The gray areas denote periods of low TFP growth.
characterized by recurrent negative consumption gaps as the economy is going through short-lasting lowgrowth regimes. This is di¤erent from the case of observed switches in Figure 6 in which we observe only
one large negative consumption gap that fades away as the economy remains in the high-growth phase.
The reason is that the possibility of unobserved switches to the low-growth phase prompts households to
persistently interpret short-lasting low-growth regimes as long lasting. As a result, imperfectly informed
agents adjust their capital stock less aggressively than what they would have done if they knew that
the economy is going through a short-lasting low-growth regime. When long-lasting high-growth periods
occur, agents are initially not very optimistic, expecting a quite short-lasting period of high TFP growth.
As a result, they speed up capital accumulation. Quite interestingly and unlike the example in Figure
6, high pessimism during short-lasting low-growth periods causes the capital gap to not exhibit mean
reversion during a typical high-growth phase. In other words, the possibility of unobserved switches
induces households to hoard capital during high-growth phases. Capital hoarding during high-growth
phases is due to households’ inability to fully learn when the economy is in the high-growth phase
because of the possibility of unobserved switches to the low-growth phase. Finally, in the low-growth
phase, households learn faster that the economy is on a low-growth path than in Figure 6. The reason
is that agents take into account that an unobserved switch may have occurred during one of the shortlasting periods of high TFP growth. This results in beliefs that are less misaligned with the truth and
consequently smaller departures of consumption and capital allocations from the perfect-information
benchmark.
Figure 10 shows the evolution of uncertainty about consumption, capital, and the TFP growth rate
during the typical simulation. As usual, the gray areas denote periods of low TFP growth. Three impor-
31
tant results regarding the evolution of uncertainty about future TFP growth stand out. First, uncertainty
is generally higher in the high-growth phase because the possibility of hidden switches to the low-growth
phase makes the learning about the duration of the regime in place harder. Second, periods of low-growth
are generally characterized by monotonically decreasing uncertainty, with long-horizon uncertainty higher
than short-horizon uncertainty. Instead, periods of high-growth are associated with short-horizon uncertainty that is initially higher than long-horizon uncertainty. These patterns are explained by the overall
rising in pessimism due to the possibility of hidden switches to the low-growth phase. As discussed earlier,
when agents expect an observed regime to be long-lasting (short-lasting), short-horizon uncertainty is
generally lower (higher) than long-horizon uncertainty. Since low-growth regimes are suddenly interpreted
as long-lasting by agents, who become more convinced about that as the system stays in this regime,
short-horizon uncertainty will be quite low and declining over time. In contrast, high-growth regimes are
initially interpreted as short-lasting, implying higher uncertainty especially at shorter horizons. Third,
long-lasting high-growth regimes are characterized by hump-shaped dynamics of short-horizon uncertainty
and a monotonically-increasing evolution of long-horizon uncertainty, which ends up being higher than
short-horizon uncertainty. Unlike in Figure 7, long-horizon uncertainty never falls during the high-growth
regimes because the possibility of hidden switches cause agents to never get fully convinced to be in the
long-lasting regime. The hump-shaped dynamics of short-horizon uncertainty during the long-lasting
high-growth regimes is due to the swing in agents’beliefs. Agents initially believe to be in a short-lasting
regime and eventually become convinced to be most likely in the long-lasting regime. When this happens,
short-horizon uncertainty falls while long-horizon uncertainty plateaus.
4.2.3
Changes of Phases Not Necessarily Preceded by a Change in Growth
Let us now consider a two-phase RBC economy in which switches to both the low-growth phase and the
high-growth phase may be hidden. Speci…cally, consider the transition probability matrix:
2
0:75
6
6 0:04
P =6
6 0:0125
4
0:0475
0
0:0125
0:95
0:0475
0:2375
0:75
0:0025
0:15
0:2375
3
7
0:0025 7
7:
7
0
5
0:80
It should be noted that this parameterization is exactly the same as that in Section 4.2.2 apart from
two features. First, the long-lasting low-growth regime is relatively shorter lasting than in Section 4.2.2.
Second, there is a non-zero probability that a switch from the long-lasting low-growth regime to the
short-lasting low-growth regime occurs, since p43 = 0:15 > 0.
The matrix P captures an economy that alternates a high-growth phase (Regime 2 and Regime 3) to a
low-growth phase (Regime 1 and Regime 4) that is characterized by high-growth periods and low-growth
periods of roughly the same typical duration. Furthermore, switches from one phase to the other may
happen even though agents do not observe any changes in the TFP growth rate.
Consider the same sequence of regimes as that of the previous subsection; that is, a typical lowgrowth phase, in which the economy stagnates, followed by a typical high-growth phase. As in the
32
Pr.LL-HG
Pr.LL-LG
Beliefs
0.8
0.4
0.2
0.6
25
50
75
0.0625
0.025
-0.0125
-0.05
-0.0875
0.0125
25
50
75
25
50
75
0.05
4Q
8Q
20Q
0.01
40Q
Capital Gap
E(Avg TFP Growth)
1
Consumption Gap
1
Ergodic
0.0075
0.005
0.0025
0
25
50
0.025
0
-0.025
-0.05
75
Figure 11: Beliefs and Allocations. Top left graph: Evolution of beliefs of being in the long-lasting high-growth regime (red
solid line) and in the long-lasting low-growth regime (blue dashed line). Top right graph: Log-deviations of consumption from
the perfect-information benchmark. Bottom left graph: Expected average growth rate of technology (annualized percentage)
at various horizons. Bottom right graph: Log-deviations of capital from the perfect-information benchmark. In all graphs,
gray areas denote periods of low growth.
previous subsection, agents mostly expect to be in a high-growth phase at time 0, perhaps re‡ecting an
economy that went through a high-growth phase in the past. Figure 11 reports the evolution of beliefs,
average expected growth rate, consumption gap, and capital gap in this economy conditional on this
typical path of regimes. Note that the third gray areas starting from left coincides with the beginning of
the high-growth phase.
A quick comparison of Figures 9 and 11 reveals that the possibility of hidden switches to the highgrowth phase has important implication for the dynamics of beliefs. First, compared to the case depicted
in Figure 9, agents react more optimistically to the economy entering a high-growth regime, expecting
that this regime will last for longer. Second, the dynamics of beliefs in the simulation seem to be less
sensitive to the phase the economy is going through, suggesting that the presence of unobservable switches
make it harder for agents to learn about the phase and the type of regime that are currently in place.
Third, pessimism, captured by the blue dashed line in the upper left graph, overshoots as the system
enters a low-growth regime and then gradually falls as the economy remains in that regime. Note that
this pattern is not observed in the economy we have studied so far. Importantly, the direction of learning
now goes from the long-lasting regime to the short-lasting regime as the system stays in the low-growth
regime. This is opposite to what is observed in all the previous examples, in which pessimism always
increases as the system stays in the low-growth regime.
To understand why the direction of learning is opposite in this example, one should realize that
agents largely expect a long-lasting low-growth regime as they observe low TFP growth. Furthermore,
the probability about being in the long-lasting regime does not converge to unity as the spell in the
33
low-growth block goes to in…nity. This happens because agents are rational and are aware that the
probability of a hidden switch gets higher as the number of consecutive periods in the block grow large.
It follows that the possibility of a hidden switch to the short-lasting low-growth regime reduces agents’
ability to learn about the likely duration of low-growth regimes from observing the number of consecutive
periods spent in that block. These dynamics of beliefs during the low-growth regimes crucially a¤ect the
evolution of the average growth rate (lower left plot of Figure 11). Precisely, unlike the previous example
shown in Figure 9, agents raises their expected growth rate of TFP as the economy stays longer and
longer in a low-growth regime.
Comparing the right panels of Figure 9 and 11, consumption and capital allocations are generally less
far o¤ from their perfect-information level in the latter graph. The reason is that now the two low-growth
regimes are quite similar in terms of their relative persistence. Furthermore, unlike the previous example
with no hidden switch to the high-growth phase, capital gap steadily decreases during the low-growth
phase. The capital gap starts increasing only when the economy enters the high-growth phase; that is,
at the beginning of the third gray area in our simulation. The reason why the capital gap widens during
the low-growth phase is the high optimism about the likely duration of the high-growth regime during
the low-growth phase. In other words, agents overestimate the likely duration of the high-growth regime
during the low-growth phase, leading agents to adjust their capital stock less aggressively than what they
would have done if they knew that the realized high-growth regimes are actually short lasting. Recall that
the possibility of unobserved switches from the low-growth phase to the high-growth phase is the culprit
for such a higher optimism about the duration of high-growth regimes during the low-growth phases.
Figure 12 reports the evolution of uncertainty about future allocations and future TFP growth at
di¤erent horizons. We observe that low-growth regimes are associated with higher short-horizon uncertainty (light blue lines). Since the two regimes are characterized by very similar persistence, this
result may seem surprising at …rst. However, this outcome re‡ects the possibility of hidden switches to
a high-growth phase that is characterized by a growth rate that is substantially higher than that in the
stagnating phase. At longer horizon (dark red lines), uncertainty seems remarkably insensitive to regime
switches and quite anchored to its ergodic level even when the low-growth regimes occur. We observe
this pattern because hidden switches make agents’ task of predicting the growth rate of TFP at long
horizon harder. Uncertainty about allocations is fairly stable across all horizons, re‡ecting the fact that
the presence of hidden switches makes it harder for agents to learn about the likely duration of realized
regimes.
4.3
Disasters and Shocks to Beliefs
We want to model an economy in which there exists a fairly rare chance that a disaster occurs. The goal
is to study the e¤ects of the possibility of such a disaster on the dynamics of beliefs and, in turn, on
the aggregate variables. We expand the RBC model introduced in Section 4 so as to include a disaster
regime that causes the economy to contract sharply. We have a total of four regimes: Regime 1 and 2
are high-growth regimes,
H
= 0:01, and Regime 3 and 4 are low-growth regimes,
34
L
=
0:05. Let us
Figure 12: Evolution of Uncertainty about Consumption, Capital, and TFP Growth. The horizon ranges from one quarter
ahead (lightest blue line) through …ve years ahead (darkest red line). The darker the color of a line, the longer the horizon
of the uncertainty.
consider the following transition matrix:
2
0:85
6
6 0:05
P =6
6 0
4
0:20
0:1
0:045
0:005
0:9
0:05
0
0:99
0:01
0
0
0
0:80
3
7
7
7:
7
5
While the high-growth regimes exhibit similar persistence, the low-growth regimes have markedly di¤erent
persistence. There is a very small probability that, once the system is in Regime 3, it will stay in the
low-growth block next period; most likely it will switch to the long-lasting high-growth regime. Regime
4 is more persistent, leading to a severe contraction which is expected to last for four quarters. We
dubbed Regime 4 as the disaster regime. Also, note that the probability of staying in the high-growth
block is 0.95 for both the short-lasting and long-lasting high-growth regimes. However, Regime 1 has a
larger downside risk with a non-zero probability of moving to the disaster regime in the next period. The
disaster regime is quite unlikely to occur, with an ergodic probability of 0.96%.
When under the high-growth block, households receive a public signal $t about the regime in
place. The signal can take two values: 1 or 2. We assume that prob f$t = 1j
prob f$t = 1j
t
t
= 1g = 0:80 and
= 2g = 0:20, implying that receiving a signal $t = 1 is more likely when the econ-
omy is the short-lasting high-growth Regime 1. Conversely, receiving a signal $t = 2 is more likely when
the economy is in the long-lasting high-growth Regime 2. We study the evolution of allocations and
beliefs when Regime 2 is in place for its typical duration of 10 quarters. Households always receive the
same signal $t = 2 during the period except at time t = 3 and t = 6, when they receive $t = 1. Figure
35
Figure 13: Beliefs, Allocations, and Uncertainty. Top graph: Evolution of beliefs of being in the long-lasting high-growth
regime in the case of shocks to beliefs at time t=3 and t=6 (black dashed line) and in the case of no shock to beliefs (solid
blue line). Middle left graph: Expected average growth rate of technology (annualized percentage points) at various horizons
(4 quarters, 8 quarters, 20 quarters, and 40 quarters) for the case of shocks to beliefs at t=3 and t=6. From top to bottom:
the solid blue line denotes the horizon of 4 quarters, the solid black line denotes the horizon 40 quarters. Middle center
graph: Log deviations of consumption from the case of no shock to beliefs. Middle right graph: Log-deviations of capital
from the case of no shock to beliefs. Bottom graphs: Uncertainty about future TFP growth, consumption, capital at horizons
from 4 quarters to 20 quarters. Lighter blue areas denote shorter horizons. Darker red areas denote longer horizons.
13 shows the dynamics of beliefs and allocations (black dashed line) and compare them with those of an
economy in which households always receive $t = 2 at any time (solid blue line).
Receiving signals $3 = 1 and $6 = 1 in‡uences agents’ beliefs by reducing their optimism. Note
that nothing is really changed in the economy’s fundamentals as the economy remains in Regime 2 at
all times. Hence, the signals $3 = 1 and $6 = 1 play the role of shocks to beliefs with the e¤ect of
reducing optimism. If agents did not receive these two signals, their beliefs would have not changed (see
the black dashed line). Such shocks to beliefs change consumption and capital allocations. The …rst
shock to beliefs reduces consumption by 0:22% in quarterly rates and the second one by 0:25% three
quarters later. Furthermore, Figure 13 shows that shocks to beliefs have delayed e¤ects on consumption
by prompting agents to accumulate more capital. The higher accumulated capital pushes consumption
up at the end of the simulated periods, when the e¤ects of the signals $3 = 1 and $6 = 1 on agents’
beliefs fade away.
Importantly, the two shocks to beliefs have interesting second-order e¤ects. In this simulation, bad
news ($3 = 1 and $6 = 1) raise the downside risk and uncertainty. The bottom middle panel and the
bottom right panel show that the increase in downside risk translates into a spike in uncertainty about
36
future consumption and future capital at all horizons. Upon the arrival of the two bad pieces of news,
short-horizon uncertainty (light blue lines) about TFP growth increases whereas the long-horizon one
(dark red lines) is substantially una¤ected. The reason is that the disaster regime is not very persistent,
lasting only four quarters on average. Note that such changes in uncertainty are not supported by any
changes in the economy’s fundamentals, but rather are due to signals that change the perceived probability
of entering the disaster regime in the near future.
While this is not the …rst paper to use signals as shocks to beliefs (e.g., Lorenzoni, 2009, Angeletos
and La’O, 2010 and Forthcoming) the approach proposed in this paper has the important advantage
of keeping the model very tractable. This feature makes our methods potentially suitable for studying
shocks to beliefs in likelihood-based estimated large-scale DSGE models (e.g., Christiano, Eichenbaum,
and Evans, 2005 and Smets and Wouters, 2007).
5
Concluding Remarks
This paper has developed methods to solve general equilibrium models in which agents are subject to
waves of optimism, pessimism, and uncertainty. Agents in the model are fully rational, understand the
structure of the economy, and know that they do not know. Therefore, when forming expectations they
take into account that their beliefs will evolve in response to realized observable economic outcomes, the
behavior of other agents in the model, or both. The central insight consists of creating an expanded
number of regimes indexed with respect to agents’ beliefs. The resulting law of motion re‡ects agents’
uncertainty and can be expressed in state space form. Therefore, the framework proposed in this paper
is suitable for structural estimation. Another important contribution of the paper is to show the large
role played by changes in agents’beliefs in shaping the predictions we draw from rational expectations
models. The applications studied in this paper aim to show the scope of applicability of the proposed
methods. Finally, note that we have con…ned our analysis to the case of a primitive transition matrix of
four regimes. Nothing prevents us from constructing a larger transition matrix with a potentially richer
scope for learning. The choice of the transition matrix depends on the feature of the stochastic process
the researcher wishes to estimate. The results of this paper can be easily extended to more articulated
transition matrices.
As scholars develop methods to e¢ ciently solve and estimate DSGE models through high-order perturbation (e.g., Fernandez-Villaverde and Rubio-Ramirez, 2006 and Foerster, Rubio-Ramirez, Waggoner,
and Zha, 2011), the methods developed in this paper lay down a convenient framework for investigating
the e¤ects of changes in economic fundamentals or animal spirits on uncertainty and the feedback e¤ects
of such swings in uncertainty on the economic dynamics.
37
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40
Appendices
The appendices are organized as follows. Appendix A works out the recursions (3) and (4) that pin down
the dynamics of beliefs within blocks. Appendix B proves Proposition 1 that ensures the convergence
of the di¤erence equations (3)-(4). Appendix C details the algorithm to construct the transition matrix
b when agents receive signals. Appendix D characterizes the steady-state equilibrium for stationary
P
variables in the RBC model and obtains the log-linearized equations of this model.
Note that the convergence results, which are proven in Appendices B, could be derived by working
on the submatrices of each block. However, we have decided to work with the solution of the di¤erence
equations (3) and (4) because this approach is familiar to a wider audience.
A
Deriving the Law of Motion for Beliefs
In this appendix, we want to show two propositions.
Proposition 2 The rational di¤ erence equations (3) and (4) hold true
Proof. Recall that equation (3) describes the dynamics of beliefs within Block 1. Consequently, this equation
holds when 1t > 1. The Bayes’theorem can be applied to characterize the probability of being in Regime 1 given
that the system is in Block 1 ( 1t > 1):
prob
t
1
t
= 1j
p
= P4
1
t
i=1
But if
1
t
=
1
t 1
=
1
t
p
1
t 1
+ 1j
=
1
t 1
t
=1 p
+ 1j
t
t
=i p
= 1j
t
=
1
t 1
ij 1t 1
+ 1, then the likelihood is such that
p
1
t
=
1
t 1
+ 1j
t
=1 =p
1
t
=
1
t 1
+ 1j
t
=2 >0
p
1
t
=
1
t 1
+ 1j
t
=3 =p
1
t
=
1
t 1
+ 1j
t
=4 =0
and
The equality in the …rst expression re‡ects the fact that agents cannot distinguish regimes belonging to the same
block. The inequality sign in the …rst expression and the equality sign in the second expression are due to the fact
that the system is in Block 1 at time t, ruling out the possibility that either Regime 3 or Regime 4 is realized.
These results allow us to write:
p
= 1j 1t 1
prob t = 1j 1t = P2 t
1
t = ij t 1
i=1 p
P2
Since p t = ij 1t 1 = j=1 p t 1 = jj 1t 1 pji , then
prob
t
= 1j
1
t
P2
1
t 1 = jj t 1 pj1
j=1 p
P2
1
t 1 = jj t 1 pji
i=1
j=1 p
= P2
Furthermore, note that p t 1 = 2j 1t 1 = 1 p t 1 = 1j
equation (3). Equation (4) can be proved analogously.
B
1
t 1
and after straightforward manipulations leads to
Proof of Proposition 1
We will characterize the convergence of prob t = 1j 1t as the number of consecutive periods spent in Block 1, 1t ,
grows large. We will denote 1lim prob t = 1j 1t = x using prob t = 1j 1t ! x and the characteristic roots of
t !1
41
equation (3) with :
e1
e2
q
(p11
2p21
2
p22 ) + 4p21 p12
p11
p22
p11
2 (p11 + p12 p21 p22 )
q
2
p22 2p21 + (p11 p22 ) + 4p21 p12
2 (p11 + p12
p21
p22 )
(25)
(26)
The following propositions provide the conditions under which the di¤erence equation (3) converges to the stable
root e2 . An analogous pair of roots, e3 and e4 , with e4 being the stable root, can be derived for Block 2. Similarly,
all results that follow will also apply to Block 2.
Proposition 3 If (i) p11 + p12 p21 p22 6= 0, (ii) p11 p22 6= p21 p12 , (iii) p11 6= p22 or both p12 6= 0 and p21 6= 0,
and the initial probability is such that prob t = 1j 1t = 1 6= e1 ; then prob t = 1j 1t ! e2 2 [0; 1]. If conditions
(i), (ii), and (iii) hold and the initial probability is such that prob t = 1j 1t = 1 = e1 , then prob t = 1j 1t = e1
for any 1t .
Proof. The di¤erence equation (3) can be expressed as
prob
t
= 1j
1
t
=
a prob
c prob
t 1
t 1
= 1j
= 1j
1
t 1
1
t 1
+b
+d
(27)
where
a
p11
p21 ; b
c
p11 + p12
p21
p21
p22 ; d
p21 + p22
Condition (i) ensures that the di¤erence equation of interest is rational because it implies c > 0. We will deal with
the case of c = 0 later on. We then proceed as follows. Denote prob t = 1j 1t + dc as xt and re-write the di¤erence
equation above as
xt =
(28)
xt
1
where
p11 + p22
p11 + p12 p21 p22
p11 p22 p21 p12
(p11 + p12
p21
p22 )
2
Condition (ii) ensures that 6= 0. The case of = 0 will be studied later. The above equation can be reduced to
a homogeneous linear di¤erence equation by de…ning xt = 't ='t 1 where:
't
If 1 and
(29) is
2
't
1
+ 't
=0
2
t
1
=
C1
+ C2
't
=
(C1 + C2 t)
t
2;
t
1;
if
1
2
1
2
are the solutions of the characteristic equation, namely
't
(29)
1
if
1
6=
=
2
2
p
2
4 , then the general solution of
(30)
(31)
The general solution of (28) is then:
xt =
t
1
C1 t1 1
C1
42
+ C2
+ C2
t
2
t 1
2
(32)
when C2 = 0, xt =
1
for all t. When C1 = 0, xt =
2
for all t. When neither C1 nor C2 is zero, then
t+1
+C
1
2
xt =
2
+C
1
2
Note that
2
; C 6= 0
t
4 is required for the characteristic roots
1
and
2
p11 + p22
p11 + p12 p21
4
p22
2
(33)
to be real. This condition is
p11 p22
p21 p12
(p11 + p12
2
p21
p22 )
and after simplifying
p211 + p222 + 2p11 p22
4p11 p22
4p21 p12
Some straightforward manipulation leads us to
2
(p11
p22 )
4p21 p12
(34)
From condition (iii ), the inequality above is strict and the characteristic roots are unequal. The case in which
t
the characteristic roots are identical is tackled by the next proposition. Let j 2 j > j 1 j then j 1 = 2 j ! 0
and (33) impliesp that xt ! 2 as long as x1 6= 1 . The root with highest absolute value can be seen to be
p
+p
+
(p
p
)2 +4p21 p12
.
p22 )
22
11
22
always 11 2(p
11 +p12 p21
manipulations we obtain:
prob
t
= 1j
Recall that xt
1
t
! e2 =
p11
prob
p22
t
= 1j
2p21 +
+ dc . After some straightforward algebraic
1
t
q
(p11
2 (p11 + p12
2
p22 ) + 4p21 p12
p21
p22 )
where e2 is the stable root for the variable of interest prob t = 1j 1t . The unstable root for prob
be easily seen to be:
q
2
p11 p22 2p21
(p11 p22 ) + 4p21 p12
e1 =
2 (p11 + p12 p21 p22 )
We only need to show that e2 2 [0; 1]. We want to show that
q
2
p11 p22 2p21 + (p11 p22 ) + 4p21 p12
2 (p11 + p12
p21
t
= 1j
1
t
can
0
p22 )
If p11 +p12 p21 p22 > 0 and p11 p22 2p21 0, then the statement is clearly true. When p11 +p12 p21 p22 > 0
and p11 p22 2p21 < 0, then
q
2
(p11 p22 ) + 4p21 p12
(p11 p22 2p21 )
Since the right-hand side is positive we can square both sides of this equation:
(p11
2
p22 ) + 4p21 p12
(p11
p22
4p21 p12
4p221
4 (p11
2
2p21 )
p22 ) p21
If p21 = 0, the statement is true. If p21 > 0
p12
which is true. If p11 + p12
p21
p21 + (p11
p22 < 0, then p11
p11
p22
2p21
p22 )
0
p22 2p21 < 0. We need to show that
q
2
(p11 p22 ) + 4p21 p12
43
Since both sides of the inequality are negative, then
(p11
2
p22
2p21 )
2
(p11
p22 ) + 4p21 p12
and after manipulating:
p22 ) p21 + 4p221
4 (p11
4p21 p12
If p21 = 0, the inequality is obviously veri…ed. If p21 > 0, then
0
(p11
p22 ) + p12
p21
which is true. We want to show that
p11
p22
2p21 +
q
(p11
2 (p11 + p12
If p11 + p12
p21
2
p22 ) + 4p21 p12
p21
p22 > 0, then after some manipulations
q
2
(p11 p22 ) + 4p21 p12
Note that p11 + 2p12
yields:
p22 > p11 + p12
(p11
p21
1
p22 )
p11 + 2p12
p22
p22 > 0. Hence, taking the square on both sides of the inequality
2
p22 ) + 4p21 p12
(p11 + 2p12
p22 )
2
and …nally
4p21 p12
4p212 + 4 (p11
p22 ) p12
If p12 = 0, this is true. If p12 > 0, then
p21
which is true. If p11 + p12
If p11 + 2p12
p21
p12 + (p11
p22 )
p22 < 0, then after some manipulations
q
2
(p11 p22 ) + 4p21 p12 p11 + 2p12 p22
p22 < 0, this inequality is obviously true. If p11 + 2p12
(p11
2
p22 ) + 4p21 p12
p22
(p11 + 2p12
p22 )
0, then
2
and then
4p21 p12
4p212 + 4 (p11
p22 ) p12
If p12 = 0, this is true. If p12 > 0, then
p21
p12 + p11
p22
which is true.
The next proposition relaxes condition (iii ) of the above proposition.
Proposition 4 If (i) p11 + p12 p21 p22 6= 0, (ii) p11 p22 6= p21 p12 , (iii) p11 = p22 and either p12 = 0 or p21 = 0,
then prob t = 1j 1t ! e1 = e2 and the roots are either equal to zero (if p21 = 0) or one (if p12 = 0).
Proof. We want to show that if (i) p11 + p12 p21 p22 6= 0, (ii) p11 p22 6= p21 p12 , (iii) p11 = p22 and either p12 = 0
or p21 = 0, then prob t = 1j 1t ! e1 = e2 and the roots are either equal to zero (if p21 = 0) or one (if p12 = 0).
This result follows from observing that condition (iii) implies that condition (34) delivers coincident characteristic
roots e1 and e2 ; that is,
p21
e1 = e2 =
p21 p12
If p12 = 0, then prob t = 1j 1t ! e1 = e2 = 1. If p21 = 0, then prob t = 1j 1t ! e1 = e2 = 0.
If the two regimes have the same persistence (p11 = p22 ) and the system has remained in Block 1 for su¢ ciently
long, then agents will eventually believe they are in the regime that is an absorbing state (conditional on staying
44
in the block). The next proposition relaxes condition (ii ) of the previous propositions.
Proposition 5 If (i) p11 + p12
p21
p22 6= 0, (ii) p11 p22 = p21 p12 , then prob
t
= 1j
1
t
=
p11 p21
p11 +p12 p21 p22 :
Proof. We want to show that if (i) p11 + p12 p21 p22 6= 0, (ii) p11 p22 = p21 p12 , then prob t = 1j 1t =
p11 p21
= 0 in equation (28) and hence (using the notation introduced above)
p11 +p12 p21 p22 . Condition (ii) implies
p11 + p22
p11 + p12 p21
xt =
Recall that xt = prob
t
= 1j
1
t
p22
+ d=c, then it follows that
prob
t
1
t
= 1j
=
p11
p11 + p12
p21
p21
p22
.
11 p21
Note that if conditions (i ) and (ii ) are satis…ed, prob t = 1j 1t suddenly converges by jumping to p11 +pp12
p21 p22
as the system enters Block 1. The recursion (3) can be shown to become a linear di¤erence equation. The solution
of this equation is characterized in the following two propositions.
Proposition 6 If (i) p11 + p12
p21
p22 p11 +2p21 2 [0; 1] :
p21
p22 = 0 and (ii) p11 6= p21 , then prob
t
= 1j
1
t
!
p21
p22 p11 +2p21 ,
with
Proof. We want to show that if (i) p11 +p12 p21 p22 = 0 and (ii) p11 6= p21 , then prob t = 1j 1t ! p22 pp1121+2p21 ,
with p22 pp1121+2p21 2 [0; 1] : If p11 + p12 p21 p22 = 0, then c = 0 in the di¤erence equation (27), which hence boils
down to the …rst-order linear di¤erence equation below:
prob
where a = p11
t
= 1j
1
t
=
a
prob
d
t 1
= 1j
p21 ; b = p21 ; d = p21 + p22 . Stability is ensured by
1
t 1
+
a
d
=
b
d
(35)
p11 p21
p21 +p22
< 1. First note that the
benchmark assumption A1 combined with condition (i) implies that d 6= 0 and hence the ratio ad is well-de…ned.
Condition (ii) rules out the possibility that the ratio ad is zero. We will tackle this case in the next proposition.
p21
p21
as pp11
.
The condition p11 + p12 p21 p22 = 0 allows us to re-write the stability condition ad = pp11
21 +p22
11 +p12
Hence, showing that p12 + p21 > 0 implies stability. Recall that the benchmark assumption A2 requires that either
p11 6= p22 or p12 6= p21 . If the latter condition is satis…ed, then p12 + p21 > 0 trivially follows. If the latter condition
is not satis…ed, then it must be that p11 6= p22 , which, combined with condition (i), implies that p12 + p21 > 0. It
1
is easy to see that the di¤erence equation (35) implies that prob t = 1j 1t ! db 1 ad
, that is,
prob
t
1
t
= 1j
!
p21
p21 + p22
1
p11 p21
p21 + p22
1
After easy algebraic manipulations
prob
t
= 1j
Note that
0
To see that, recall that in this case, p11 + p12
this result into the inequalities above yields
1
t
!
p22
p21
:
p11 + 2p21
p22
p21
p11 + 2p21
p21
p22 = 0, implying that p22
0
p21
p12 + p21
which is clearly veri…ed.
45
1
1
p11 = p12
p21 . Substituting
Proposition 7 If (i) p11 + p12
p21
p22 = 0, (ii) p11 = p21 , then prob
t
= 1j
1
t
=
p21
p22 +p21 :
21
Proof. We want to show that if (i) p11 + p12 p21 p22 = 0, (ii) p11 = p21 , then prob t = 1j 1t = p22p+p
.
21
Condition (i) implies that c = 0 in the di¤erence equation (27), which hence boils down to the …rst-order linear
di¤erence equation below:
a
b
prob t 1 = 1j 1t 1 +
(36)
prob t = 1j 1t =
d
d
where a = p11 p21 ; b = p21 ; d = p21 + p22 . Condition (ii) implies that a = 0 and hence prob t = 1j 1t = b=d =
p21 = (p21 + p22 ).
21
It should be noted that when p11 = p21 , beliefs prob t = 1j 1t suddenly jump to p22p+p
for any 1t
1 (as
21
the system enters Block 1).
To sum up, given the benchmark assumptions A1-A3, we have shown that equation (3) always converges. Note
that Proposition 2 implies that beliefs do not converge to e2 , if the starting beliefs prob t = 1j 1t = 1 = e1 . The
next two propositions show that either e1 0 or e1 1, implying that the only admissible values for probabilities
are either zero or one. Therefore, there are only a few limiting cases in which equation (3) does not converge to
e2 . It can be shown that it is su¢ cient to set the probability ratios 0 < pi3 = (pi3 + pi4 ) < 1 for any i 2 f1; 2g to
rule out these cases that are not very relevant in practice.
Recall that
q
2
(p11 p22 ) + 4p21 p12
p11 p22 2p21
e1
2 (p11 + p12 p21 p22 )
We want to show that 0
e1
Proposition 8 If p11 + p12
1. This claim is implied by the following two propositions.
p22 > 0, then e1
p21
Proof. We want to show that
p11
p22
2p21
0.
q
(p11
2 (p11 + p12
If p11 + p12
p21
2
p22 ) + 4p21 p12
p21
p22 )
0
p22 > 0, then the above implies
p11
p22
2p21
q
(p11
p22 ) + 4p21 p12
(p11
p22 ) + 4p21 p12
2
Note that the benchmark assumption A3 excludes that p11 p22 2p21 = 0. Hence there are two possible cases
left: (a) if p11 p22 2p21 < 0, then the above is true; (b) if p11 p22 2p21 > 0, then we can take the square on
both sides of the above equation to get
(p11
2
2
p22
2p21 )
p221
p11 p21 + p22 p21
Straightforward manipulations lead to
p21 p12
If p21 = 0, then the above is true. Otherwise, we can divide both sides of the above inequality by p21 to get
p11 + p12
that is obviously true because p11 + p12
Proposition 9 If p11 + p12
p21
Proof. We want to show that
p11
p21
p22
0
p22 > 0.
p22 < 0, then e1
p22
p21
2p21
q
2 (p11 + p12
1:
(p11
p21
46
2
p22 ) + 4p21 p12
p22 )
1
Since p11 + p12
p21
p22 < 0, the above implies
q
2
p11 p22 2p21
(p11 p22 ) + 4p21 p12
and after simplifying
q
2
(p11
p22 ) + 4p21 p12
2 (p11 + p12
p11
p21
p22 )
p22 + 2p12
Note that the benchmark assumption A3 excludes that p11 p22 + 2p21 = 0. If p11
obviously true. If p11 p22 + 2p12 < 0, then taking the square on both sides
2
(p11
p22 ) + 4p21 p12
(p11
p22 + 2p12 )
p22 + 2p12 > 0, the above is
2
After some manipulations:
p12 + p11
that is obviously true because p11 + p12
C
p21
p12
p22
0
p22 < 0.
Algorithm for the Case with Signals
b = 0g
Algorithm Set i = 1 and initialize the matrix P
Step 1 Find j1
g1 and j2
prob
t
= 1jIt ; $t
g1 so as to min prob
1
g
t
= 1jIt ; $t
1
; $t = q
Gjq with q 2 f1; 2g where
prob ($t = qj t = 1) prob t = 1jIt ; $t 1
; $t = q = P2
t
j=1 prob ($ t = qj t = j) prob ( t = jjIt ; $
1)
; q 2 f1; 2g
(37)
and agents’beliefs about being in Regime 1 before observing the signal read:
prob
t
= 1jIt ; $t
1
=
prob
t
prob t
1 = 1jIt
= 1jIt 1 ; $t 1 (p11 p21 ) + p21
t 1 (p
p21 p22 ) + p21 + p22
1; $
11 + p12
1
(38)
using the approximation prob t 1 = 1jIt 1 ; $t 1 = Gi . To ensure convergence of beliefs, we correct j1
and j2 as follows. If jq = i and Gi =
6 e2 (q 2 f1; 2g), then set jq = jq + 1 if Gi < e2 and jq = max (1; jq 1)
e
if Gi > 2 .
Step 2 Setting prob
t 1
= 1jIt
1; $
t 1
= Gi , the (ex-ante) transition probability can be computed as:
b (i; jq ) = P2 prob
P
v=1
where
prob
t
= vjIt
t
1; $
= vjIt
t 1
=
Step 3 Find j1 > g1 and j2 > g1 so as to min prob
prob
t
= 3jIt ; $t
1
1; $
t 1
P2
u=1
t
prob f$t = qj
prob
t 1
= 3jIt ; $t
1
= ujIt
; $t = q
t
= vg ; q 2 f1; 2g
1; $
t 1
(39)
puv
(40)
Gjq with q 2 f1; 2g, where
prob ($t = qj t = 3) prob t = 3jIt ; $t 1
; $t = q = P4
t
j=3 prob ($ t = qj t = j) prob ( t = jjIt ; $
1)
; q 2 f1; 2g
and the beliefs about being in Regime 3 upon the shift to Block 2 (before having observed the signal $t ) are
given by:
P
t 1
pj3
t 1 = jjIt 1 ; $
j2b1 prob
t 1
P
prob t = 3jIt ; $
=P
t
1 p
ji
t 1 = jjIt 1 ; $
i2b2
j2b1 prob
=
prob
prob t 1 = 1jIt 1 ; $t 1 p13 + 1
t 1 (p
13 + p14 ) + 1
t 1 = 1jIt 1 ; $
using the approximation that prob
t 1
= 1jIt
1; $
47
t 1
prob
prob
t 1
t 1
= 1jIt
= 1jIt
= Gi . Setting prob
1; $
1
t 1
; $t 1
t 1
= 1jIt
p23
(p23 + p24 )
1; $
t 1
= Gi , the
(ex-ante) transition probabilities as
b (i; jq ) = P
b (i; jq ) +
P
4
2
X
X
v=3
prob
t 1
= ujIt
1; $
t 1
puv
u=1
!
prob f$t = qj
t
= vg ; q 2 f1; 2g
(41)
Step 4 If i = g1 then set i = i + 1 and go to step 6; otherwise, set i = i + 1 and go to step 1.
Step 5 Find j1 > g1 and j2 > g1 so as to min prob
prob
t
= 3jIt ; $t
1
t
= 3jIt ; $t
1
; $t = q
Gjq with q 2 f1; 2g where
prob ($t = qj t = 3) prob t = 3jIt ; $t 1
; $t = q = P4
t
j=3 prob ($ t = qj t = j) prob ( t = jjIt ; $
1)
; q 2 f1; 2g
and agents’beliefs about being in Regime 3 before observing the signal read:
prob
t
= 3jIt ; $t
1
=
prob
t
prob t
1 = 3jIt
= 3jIt 1 ; $t 1 (p33 p43 ) + p43
t 1 (p
p43 p44 ) + p43 + p44
1; $
33 + p34
1
(42)
using the approximation prob t 1 = 3jIt 1 ; $t 1 = Gi . To ensure convergence of beliefs, we correct j1
and j2 as follows. If jq = i and Gi =
6 e4 (q 2 f1; 2g), then set jq = min (jq + 1; g) if Gi < e4 and jq = jq 1
e
if Gi > 4 .
Step 6 Setting prob
t 1
= 3jIt
b (i; jq ) = P
b (i; jq ) +
P
Step 7 Find j1
g1 and j2
prob
t
1; $
t 1
4
4
X
X
v=3
= Gi , the (ex-ante) transition probability can be computed as:
prob
t 1
= ujIt
1
t 1
puv
u=3
g1 so as to min prob
= 1jIt ; $t
1; $
t
= 1jIt ; $t
1
!
prob f$t = qj
; $t = q
t
= vg ; q 2 f1; 2g
(43)
Gjq with q 2 f1; 2g, where
prob ($t = qj t = 1) prob t = 1jIt ; $t 1
; $t = q = P2
t
j=1 prob ($ t = qj t = j) prob ( t = jjIt ; $
1)
; q 2 f1; 2g
and the beliefs about being in Regime 1 upon the shift to Block 1 (before having observed the signal $t ) are
given by:
P
t 1
pj1
t 1 = jjIt 1 ; $
j2b2 prob
t 1
P
prob t = 1jIt ; $
=P
t
1 p
ji
t 1 = jjIt 1 ; $
i2b1
j2b2 prob
=
prob
prob t 1 = 3jIt 1 ; $t 1 p31 + 1
t 1 (p
31 + p32 ) + 1
t 1 = 3jIt 1 ; $
prob
prob
t 1
t 1
= 3jIt
= 3jIt
1; $
1
; $t 1
using the approximation that prob t 1 = 3jIt 1 ; $t 1 = Gg1 +i . Setting prob
the (ex-ante) transition probability can be computed as:
!
2
4
X
X
t
1
b (i; jq ) = P
b (i; jq ) +
P
prob t 1 = ujIt 1 ; $
puv prob f$t = qj
v=1
u=3
t 1
t 1
= Gi ,
= vg ; q 2 f1; 2g
(44)
t 1
t
p41
(p41 + p42 )
= 3jIt
1; $
Step 8 If i = g, then go to step 9; otherwise, set i = i + 1 and go to step 5.
b has all zero elements, then stop. Otherwise, go to step 10.
Step 9 If no column of P
Pg b
Step 10 Construct the matrix T as follows. Set j = 1 and l = 1. While j g, if i=1 P
(i; j) 6= 0 then do three
things: (1) set T (j; l) = 1, (2) set T (j; v) = 0 for any 1
v
g and v 6= l, (3) set l = l + 1 and (4) set
Pg b
j = j + 1; otherwise (i.e., if i=1 P
(i; j) = 0), set j = j + 1.
bR = T P
b T 0 . If no column of P
bR has all zero elements, set P
b=P
bR
Step 11 Write the transition equation as P
and stop. Otherwise, go to step 10.
48
D
Log-Linearization of the RBC Model
Solving the problem of the representative household in Section 4 leads to:
ct
1
et c 1
E
t+1
=
ct + k t
= zt kt
1
1
zt+1 kt
+ (1
) kt
+1
(45)
(46)
1
The stochastic process of TFP (17) and equations (45)-(46) imply that consumption and capital are non-stationary.
(1
) 1 e
(1
) 1
Denote the stationary variables e
ct
ct =zt
, kt
kt =zt
, t
ln (zt =zt 1 ), and Mt
zt =zt 1 as the
gross growth rate of TFP. The stationary version of the model reads:
h
i
1
1
et e
kt 1 + 1
(47)
e
ct 1 =
E
ct+1
Mt+11 Mt+1 e
e
ct + e
kt
1
Me
kt
=
1
+ (1
) Mt
1
e
kt
(48)
1
Following Schorfheide (2005) and Liu, Waggoner, and Zha (2011), we de…ne a steady-state equilibrium for the
stationary consumption e
ct and capital e
kt when "t = 0 all t and the growth rate of TFP is at its ergodic value .
The steady-state equilibrium level of consumption css and capital kss is:
"
kss
=
css
= M
1
M1
1
M
1
1+
h
kss + (1
!#
1
)M
1
1
1
(49)
i
1 kss
(50)
where M
exp ( ),
(p1 + p2 ) H + (p3 + p4 ) L is the ergodic mean of the log growth rate of the economy,
and pi stands for the ergodic probability of being in Regime i.
Taking the log-linear approximation of equations (47)-(48) around the steady-state equilibrium (49)-(50) leads
to
1
1
1
et b
et bt+1
b
ct = E
ct+1 (
1) 1 + (
1) M 1 b
kt
+ M 1(
1) + 1 E
1
1
where we use the fact that M 1 M kss 1 + 1
= 1 from equation (49) and bt
is the log-deviation
t
ct and b
kt denote log-deviations of the stationary consumption
of the growth rate of TFP from its ergodic mean . b
is the log-deviation of the TFP
and capital, respectively, from their steady-state value, and b ( t )
t ( t)
drift from its ergodic mean . The resource constraint is
css b
ct + kss b
kt =
M
1
kss
1
+
1
1
1
M
1
1
kss bt + M
1
kss + (1
and the log-deviations of the growth rate of TFP from its ergodic level follows
bt = bt ( t ) +
49
z "t :
)M
1
kss b
kt
1
(51)
Working Paper Series
A series of research studies on regional economic issues relating to the Seventh Federal
Reserve District, and on financial and economic topics.
Comment on “Letting Different Views about Business Cycles Compete”
Jonas D.M. Fisher
WP-10-01
Macroeconomic Implications of Agglomeration
Morris A. Davis, Jonas D.M. Fisher and Toni M. Whited
WP-10-02
Accounting for non-annuitization
Svetlana Pashchenko
WP-10-03
Robustness and Macroeconomic Policy
Gadi Barlevy
WP-10-04
Benefits of Relationship Banking: Evidence from Consumer Credit Markets
Sumit Agarwal, Souphala Chomsisengphet, Chunlin Liu, and Nicholas S. Souleles
WP-10-05
The Effect of Sales Tax Holidays on Household Consumption Patterns
Nathan Marwell and Leslie McGranahan
WP-10-06
Gathering Insights on the Forest from the Trees: A New Metric for Financial Conditions
Scott Brave and R. Andrew Butters
WP-10-07
Identification of Models of the Labor Market
Eric French and Christopher Taber
WP-10-08
Public Pensions and Labor Supply Over the Life Cycle
Eric French and John Jones
WP-10-09
Explaining Asset Pricing Puzzles Associated with the 1987 Market Crash
Luca Benzoni, Pierre Collin-Dufresne, and Robert S. Goldstein
WP-10-10
Prenatal Sex Selection and Girls’ Well‐Being: Evidence from India
Luojia Hu and Analía Schlosser
WP-10-11
Mortgage Choices and Housing Speculation
Gadi Barlevy and Jonas D.M. Fisher
WP-10-12
Did Adhering to the Gold Standard Reduce the Cost of Capital?
Ron Alquist and Benjamin Chabot
WP-10-13
Introduction to the Macroeconomic Dynamics:
Special issues on money, credit, and liquidity
Ed Nosal, Christopher Waller, and Randall Wright
WP-10-14
Summer Workshop on Money, Banking, Payments and Finance: An Overview
Ed Nosal and Randall Wright
WP-10-15
Cognitive Abilities and Household Financial Decision Making
Sumit Agarwal and Bhashkar Mazumder
WP-10-16
1
Working Paper Series (continued)
Complex Mortgages
Gene Amromin, Jennifer Huang, Clemens Sialm, and Edward Zhong
WP-10-17
The Role of Housing in Labor Reallocation
Morris Davis, Jonas Fisher, and Marcelo Veracierto
WP-10-18
Why Do Banks Reward their Customers to Use their Credit Cards?
Sumit Agarwal, Sujit Chakravorti, and Anna Lunn
WP-10-19
The impact of the originate-to-distribute model on banks
before and during the financial crisis
Richard J. Rosen
WP-10-20
Simple Markov-Perfect Industry Dynamics
Jaap H. Abbring, Jeffrey R. Campbell, and Nan Yang
WP-10-21
Commodity Money with Frequent Search
Ezra Oberfield and Nicholas Trachter
WP-10-22
Corporate Average Fuel Economy Standards and the Market for New Vehicles
Thomas Klier and Joshua Linn
WP-11-01
The Role of Securitization in Mortgage Renegotiation
Sumit Agarwal, Gene Amromin, Itzhak Ben-David, Souphala Chomsisengphet,
and Douglas D. Evanoff
WP-11-02
Market-Based Loss Mitigation Practices for Troubled Mortgages
Following the Financial Crisis
Sumit Agarwal, Gene Amromin, Itzhak Ben-David, Souphala Chomsisengphet,
and Douglas D. Evanoff
WP-11-03
Federal Reserve Policies and Financial Market Conditions During the Crisis
Scott A. Brave and Hesna Genay
WP-11-04
The Financial Labor Supply Accelerator
Jeffrey R. Campbell and Zvi Hercowitz
WP-11-05
Survival and long-run dynamics with heterogeneous beliefs under recursive preferences
Jaroslav Borovička
WP-11-06
A Leverage-based Model of Speculative Bubbles (Revised)
Gadi Barlevy
WP-11-07
Estimation of Panel Data Regression Models with Two-Sided Censoring or Truncation
Sule Alan, Bo E. Honoré, Luojia Hu, and Søren Leth–Petersen
WP-11-08
Fertility Transitions Along the Extensive and Intensive Margins
Daniel Aaronson, Fabian Lange, and Bhashkar Mazumder
WP-11-09
Black-White Differences in Intergenerational Economic Mobility in the US
Bhashkar Mazumder
WP-11-10
2
Working Paper Series (continued)
Can Standard Preferences Explain the Prices of Out-of-the-Money S&P 500 Put Options?
Luca Benzoni, Pierre Collin-Dufresne, and Robert S. Goldstein
Business Networks, Production Chains, and Productivity:
A Theory of Input-Output Architecture
Ezra Oberfield
WP-11-11
WP-11-12
Equilibrium Bank Runs Revisited
Ed Nosal
WP-11-13
Are Covered Bonds a Substitute for Mortgage-Backed Securities?
Santiago Carbó-Valverde, Richard J. Rosen, and Francisco Rodríguez-Fernández
WP-11-14
The Cost of Banking Panics in an Age before “Too Big to Fail”
Benjamin Chabot
WP-11-15
Import Protection, Business Cycles, and Exchange Rates:
Evidence from the Great Recession
Chad P. Bown and Meredith A. Crowley
WP-11-16
Examining Macroeconomic Models through the Lens of Asset Pricing
Jaroslav Borovička and Lars Peter Hansen
WP-12-01
The Chicago Fed DSGE Model
Scott A. Brave, Jeffrey R. Campbell, Jonas D.M. Fisher, and Alejandro Justiniano
WP-12-02
Macroeconomic Effects of Federal Reserve Forward Guidance
Jeffrey R. Campbell, Charles L. Evans, Jonas D.M. Fisher, and Alejandro Justiniano
WP-12-03
Modeling Credit Contagion via the Updating of Fragile Beliefs
Luca Benzoni, Pierre Collin-Dufresne, Robert S. Goldstein, and Jean Helwege
WP-12-04
Signaling Effects of Monetary Policy
Leonardo Melosi
WP-12-05
Empirical Research on Sovereign Debt and Default
Michael Tomz and Mark L. J. Wright
WP-12-06
Credit Risk and Disaster Risk
François Gourio
WP-12-07
From the Horse’s Mouth: How do Investor Expectations of Risk and Return
Vary with Economic Conditions?
Gene Amromin and Steven A. Sharpe
WP-12-08
Using Vehicle Taxes To Reduce Carbon Dioxide Emissions Rates of
New Passenger Vehicles: Evidence from France, Germany, and Sweden
Thomas Klier and Joshua Linn
WP-12-09
Spending Responses to State Sales Tax Holidays
Sumit Agarwal and Leslie McGranahan
WP-12-10
3
Working Paper Series (continued)
Micro Data and Macro Technology
Ezra Oberfield and Devesh Raval
WP-12-11
The Effect of Disability Insurance Receipt on Labor Supply: A Dynamic Analysis
Eric French and Jae Song
WP-12-12
Medicaid Insurance in Old Age
Mariacristina De Nardi, Eric French, and John Bailey Jones
WP-12-13
Fetal Origins and Parental Responses
Douglas Almond and Bhashkar Mazumder
WP-12-14
Repos, Fire Sales, and Bankruptcy Policy
Gaetano Antinolfi, Francesca Carapella, Charles Kahn, Antoine Martin,
David Mills, and Ed Nosal
WP-12-15
Speculative Runs on Interest Rate Pegs
The Frictionless Case
Marco Bassetto and Christopher Phelan
WP-12-16
Institutions, the Cost of Capital, and Long-Run Economic Growth:
Evidence from the 19th Century Capital Market
Ron Alquist and Ben Chabot
WP-12-17
Emerging Economies, Trade Policy, and Macroeconomic Shocks
Chad P. Bown and Meredith A. Crowley
WP-12-18
The Urban Density Premium across Establishments
R. Jason Faberman and Matthew Freedman
WP-13-01
Why Do Borrowers Make Mortgage Refinancing Mistakes?
Sumit Agarwal, Richard J. Rosen, and Vincent Yao
WP-13-02
Bank Panics, Government Guarantees, and the Long-Run Size of the Financial Sector:
Evidence from Free-Banking America
Benjamin Chabot and Charles C. Moul
WP-13-03
Fiscal Consequences of Paying Interest on Reserves
Marco Bassetto and Todd Messer
WP-13-04
Properties of the Vacancy Statistic in the Discrete Circle Covering Problem
Gadi Barlevy and H. N. Nagaraja
WP-13-05
Credit Crunches and Credit Allocation in a Model of Entrepreneurship
Marco Bassetto, Marco Cagetti, and Mariacristina De Nardi
WP-13-06
4
Working Paper Series (continued)
Financial Incentives and Educational Investment:
The Impact of Performance-Based Scholarships on Student Time Use
Lisa Barrow and Cecilia Elena Rouse
WP-13-07
The Global Welfare Impact of China: Trade Integration and Technological Change
Julian di Giovanni, Andrei A. Levchenko, and Jing Zhang
WP-13-08
Structural Change in an Open Economy
Timothy Uy, Kei-Mu Yi, and Jing Zhang
WP-13-09
The Global Labor Market Impact of Emerging Giants: a Quantitative Assessment
Andrei A. Levchenko and Jing Zhang
WP-13-10
Size-Dependent Regulations, Firm Size Distribution, and Reallocation
François Gourio and Nicolas Roys
WP-13-11
Modeling the Evolution of Expectations and Uncertainty in General Equilibrium
Francesco Bianchi and Leonardo Melosi
WP-13-12
5
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