THE EFFECTS OF NEWS SHOCKS AND BOUNDED RATIONALITY ON
MACROECONOMIC VOLATILITY
by
BRIAN JOSEPH DOMBECK
A DISSERTATION
Presented to the Department of Economics
and the Graduate School of the University of Oregon
in partial fulfillment of the requirements
for the degree of
Doctor of Philosophy
June 2017
DISSERTATION APPROVAL PAGE
Student: Brian Joseph Dombeck
Title: The Effects of News Shocks and Bounded Rationality on Macroeconomic Volatility
This dissertation has been accepted and approved in partial fulfillment of the requirements
for the Doctor of Philosophy degree in the Department of Economics by:
Bruce McGough Co-Chair
George Evans Co-chair
Jeremy Piger Core Member
Arafaat Valiani Institutional Representative
and
Scott L. Pratt Dean of the Graduate School
Original approval signatures are on file with the University of Oregon Graduate School.
Degree awarded June 2017
ii
c© 2017 Brian Joseph Dombeck
iii
DISSERTATION ABSTRACT
Brian Joseph Dombeck
Doctor of Philosophy
Department of Economics
June 2017
Title: The Effects of News Shocks and Bounded Rationality on Macroeconomic Volatility
This dissertation studies the impact embedding boundedly rational agents in real
business cycle-type news-shock models may have on a variety of model predictions,
from simulated moments to structural parameter estimates. In particular, I analyze the
qualitative and quantitative effects of assuming agents are boundedly rational in a class of
DSGE models which attempt to explain the observed volatility and comovements in key
aggregate measures of U.S. economic performance as the result of endogenous responses
to information in the form of “news shocks”. The first chapter explores the theoretical
feasibility of relaxing the rational expectations hypothesis in a three-sector real business
cycle (RBC) model which generates boom-bust cycles as a result of periods of optimism
and pessimism on the part of households. The second chapter determines whether agents
forming linear forecasts of shadow prices in a nonlinear framework can lead to behavior
approximately consistent with fully informed individuals in a one-sector real business cycle
model. The third chapter analyzes whether empirical estimates of the relative importance
of anticipated shocks may be biased by assuming rational expectations.
By merging the two hitherto separate but complementary strands of literature
related to bounded rationality and news shocks I am able to conduct in-depth analysis
of the importance of both the information agents have and what they choose to do with it.
At its core, the study of news in macroeconomics is a study of the specific role alternative
iv
information sets play in generating macroeconomic volatility. Adaptive learning on the
other hand is concerned with the behavior of agents given an information set. Taken
together, these fields jointly describe the input and the “black box” which produce model
predictions from DSGE models. While previous research has been conducted on the effects
of bounded rationality or news shocks in isolation, this dissertation marks the first set of
research explicitly focused on the interaction of these two model features.
v
CURRICULUM VITAE
NAME OF AUTHOR: Brian Joseph Dombeck
GRADUATE AND UNDERGRADUATE SCHOOLS ATTENDED:
University of Oregon, Eugene, OR
Louisiana State University, Baton Rouge, LA
DEGREES AWARDED:
Doctor of Philosophy, Economics, 2017, University of Oregon
Master of Science, Economics, 2013, University of Oregon
Bachelor of Science, Economics, 20012, Louisiana State University
AREAS OF SPECIAL INTEREST:
Applied Structural Macroeconomics
Bounded Rationality
Information Choice
GRANTS, AWARDS AND HONORS:
Kleinsorge Summer Research Fellowship, University of Oregon, 2016
Kleinsorge Summer Research Fellowship, University of Oregon, 2014
Kleinsorge First Year Fellowship, University of Oregon, 2012
Graduate Teaching Fellowship, University of Oregon, 2012-2017
vi
ACKNOWLEDGEMENTS
The completion of this dissertation would not have been possible without the
support of many groups of people. I thank Professors Bruce McGough, George Evans,
Jeremy Piger, Van Kolpin, Anne van den Nouweland , Kaj Gittings, Bulent Unel, Naci
Mocan, Carter Hill, and Charles Roussel for all of their guidance and mentorship, and for
teaching me to think like an economist. I thank Savannah Dombeck, Kellie Geldreich, and
the rest of the University of Oregon Department of Economics staff for their assistance
in navigating the myriad of requirements necessary for completing the Doctoral program.
Finally, I thank my family for their love, patience, and understanding over the years.
vii
For Savannah
viii
TABLE OF CONTENTS
Chapter Page
I. INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
II. EXPECTATIONAL STABILITY AND THE “COMOVEMENT PROBLEM” . . 4
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
Incorporating News Shocks into DSGE Models . . . . . . . . . . . . . . . . 9
Adaptive Learning . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
Application to Beaudry and Portier (2004) . . . . . . . . . . . . . . . . . . 17
Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
III. SHADOW PRICE LEARNING IN A NEWS-SHOCK MODEL . . . . . . . . . . 28
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
The Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
Equilibrium . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
News Shocks and Expectations Formation . . . . . . . . . . . . . . . . . . . 41
Simulation and Model Performance . . . . . . . . . . . . . . . . . . . . . . . 50
Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
IV. LEARNING VS NEWS: WHAT DRIVES BUSINESS CYCLES? . . . . . . . . . 57
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
The Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
Adaptive Learning . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72
Estimation Methodology . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75
Main Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82
Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88
ix
Chapter Page
APPENDICES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90
A. APPENDIX FOR CHAPTER II . . . . . . . . . . . . . . . . . . . . . 90
B. APPENDIX FOR CHAPTER III . . . . . . . . . . . . . . . . . . . . . 100
C. APPENDIX FOR CHAPTER IV . . . . . . . . . . . . . . . . . . . . . 105
REFERENCES CITED . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123
x
LIST OF FIGURES
Figure Page
1. IRF for Accurate News . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
2. IRF for Inaccurate News . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
3. Model Simulation Under SP-learning . . . . . . . . . . . . . . . . . . . . . . . . . 51
4. Relative Importance of News: Key Macroeconomic Variables, RE vs NRE . . . . . 86
5. Relative Importance of News: Government Spending, RE vs NRE . . . . . . . . . 87
xi
LIST OF TABLES
Table Page
1. Calibrated Parameters for Three-sector Model . . . . . . . . . . . . . . . . . . . . 23
2. Alternate Parameterizations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
3. Calibrated Parameters for One-sector Model . . . . . . . . . . . . . . . . . . . . . 47
4. t-tests for Data Generating Process, News, 230 Periods . . . . . . . . . . . . . . . 52
5. t-tests for Data Generating Process, No News, 230 Periods . . . . . . . . . . . . . 52
6. t-tests for Data Generating Process, News, 1000 Periods . . . . . . . . . . . . . . 54
7. t-tests for Data Generating Process, No News, 1000 Periods . . . . . . . . . . . . 54
8. Predicted Business Cycle Statistics . . . . . . . . . . . . . . . . . . . . . . . . . . 55
9. Calibrated Parameters for Estimated One-sector Model . . . . . . . . . . . . . . 81
10. Estimated Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83
11. Model Predictions: Volatility and Persistence, Data vs NRE vs RE . . . . . . . . 88
xii
CHAPTER I
INTRODUCTION
A primary objective of applied structural macroeconomics is to identify the
determinants of business cycles and create empirically relevant dynamic stochastic
general equilibrium (DSGE) models. While most modern DSGE models assume agents
have rational expectations and restrict attention to the effect of unanticipated shocks as
drivers of the business cycle, my dissertation studies the theoretic and empirical effects
of embedding boundedly rational agents in real business cycle-type news-shock models to
determine whether and when these assumptions influence the predictions generated by our
models.
News-shock models seek to generate positive comovement between consumption,
investment, total labor supply, and output - which is consistent with postwar U.S.
macroeconomic data - in response to news about the future state of the economy. In
Chapter II I examine whether such models are compatible with a relaxation of the RE
assumption in the context of the three-sector RBC model introduced in Beaudry and
Portier (2004) by endowing agents with a perceived law of motion (PLM) for the evolution
of the economy. Given their beliefs about the economy’s transitional path, agents take
actions which result in an actual law of motion (ALM) for the economy. Forecasting errors
as captured by differences between agents’ PLM and the ALM are then used to update
their forecasting model via recursive least squares. This process of adaptive learning
about the laws of motion for the economy plays out each period. If agents acting in this
way eventually learn to behave in the same way as fully rational agents, the rational
expectations equilibrium (REE) is said to be expectationally stable (E-stable). I find that
1
the RE solution for this specific news shock model is E-Stable, and that this finding is
robust to a variety of alternative parameterizations and timing assumptions. Furthermore
I show that under certain assumptions regarding agents’ information sets, if the REE is
E-stable in any model without news shocks it will remain E-stable when news shocks
are included. This implies the effects of news shocks on model predictions can be safely
considered in a wide variety of existing models whose REE have been shown to be E-
stable.
In Chapter III I explore the qualitative and quantitative effects of allowing agents
to be boundedly optimal in the sense of Evans and McGough (2015) in the well-known
news-shock model of Jaimovich and Rebelo (2009). Boundedly optimal agents are similar
to their boundedly rational counterparts in that they are not endowed with knowledge
of the conditional distribution of all variables in the model. However, instead of merely
learning about the laws of motion for endogenous variables they are also endowed with
forecasting models for the shadow prices of endogenous state variables. Forecasts of
shadow prices are updated recursively which gives rise to a rich set of behavior as agents
incorporate news shocks into their forecasts. By embedding this behavioral process in
a nonlinear DSGE model, I am able to show through simulations that agents’ behavior
will converge close to that of fully rational agents, a remarkable feat considering how
simplistic the process is. Furthermore I compare actual US business cycle statistics with
those obtained by calibrating and simulating the model with shadow-price learning agents
and find that bounded optimality has a significant effect on the properties of simulated
data as compared to that generated under RE.
Finally, in Chapter IV I utilize Bayesian estimation techniques to determine whether
the estimated relative importance of news in the news-shock model of Schmitt-Grohe
and Uribe (2012) is robust to the inclusion of adaptive learning. News shocks and
2
adaptive learning both act as expectations-based sources of business cycle activity in
macroeconomic dynamic stochastic general equilibrium (DSGE) models. This chapter
explores whether these sources act as complements or substitutes to each other by
comparing estimates of the relative importance of news shocks obtained across expectation
formation mechanisms. I find the relative importance of news shocks is amplified by up to
35% for key macroeconomic variables under learning as compared to rational expectations,
implying that news shocks may be more important for driving business cycles than
existing estimates would otherwise suggest. The chapters primary contributions are to
clarify which assumptions the importance of news is robust to and to produce a new
estimate of the constant-gain learning parameter from a novel joint news-and-learning
estimation routine.
3
CHAPTER II
EXPECTATIONAL STABILITY AND THE “COMOVEMENT PROBLEM”
Introduction
Starting with Kydland and Prescott (1982) the economic literature seeking to
explain stylized facts and comovements observed in postwar macroeconomic U.S. data
through supply-side innovations has flourished. So-called real business cycle (RBC) theory
maintains innovations to productivity are responsible for the booms and busts which
comprise the business cycle. This implies a peculiar interpretation of cyclical activity:
while expansions are driven by technological progress, recessions must be caused by
technological regress.1 In recent years a subset of RBC literature has begun exploring
alternate mechanisms through which the economy might experience business cycles,
focusing in particular on the potential role of information and expectations. For example
in Beaudry and Portier (2004), the model of which is the focus for this chapter, news
about the future generates expectationally driven business cycles of the sort described
in Pigou (1927) in which the economy experiences boom-bust periods resulting from
the actions of forward-looking agents responding to noisy signals of future technological
innovation.
As noted in Krusell and McKay (2010), business cycles are characterized by positive
comovement in aggregate consumption, investment, employment, and output. While
standard RBC models generate positive pro-cyclical comovements in key variables in
1 Stiglitz (2014) points out that “... by implication, the Great Depression was marked by an episode of
acute amnesia, where in large parts of the world, people got less productive!”
4
response to contemporaneous productivity shocks, they are unable to generate these
positive comovements in response to anticipated productivity shocks. Such models
typically imply news about future total factor productivity (TFP) will cause consumption
to move opposite investment, employment, and output. For example, the baseline
calibration of Kydland and Prescott’s original model suggests households will substitute
investment for consumption in anticipation of higher future marginal factor productivity
stemming from anticipated future technological growth. Because leisure is a normal good,
households also reducing employment, causing a drop in the level of output. Thus, good
news about the future state of the economy causes a recession today.
Several dynamic stochastic general equilibrium (DSGE) models have arisen in
an effort to fix this “comovement problem”.2Beaudry and Portier (2004) show that
standard one and two sector RBC models are incapable of generating qualitatively
realistic expectationally driven business cycles, that is, business cycles featuring positive
comovement amongst key macroeconomic variables in response to news about the future.
They propose a three-sector RBC model which utilizes a CES production function for
consumption and short-term substitutability constraints on consumption and investment.
Jaimovich and Rebelo (2009) augment the canonical RBC model to include variable
capital utilization, investment adjustment costs, and a novel preference specification
which allows the modeler to manipulate the strength of the wealth effect on labor
supply. Lorenzoni (2009) studies a version of the Phelps-Lucas “island model” capable
of generating empirically realistic comovements in response to anticipated demand
shocks where firms receive a noisy public signal of future aggregate productivity and a
private signal of their own productivity. Krusell and McKay (2010) modify a version of
2DSGE models are fully microfounded models of agent-level behavior in an economy. In addition to
describing the decision making and expectation formation processes for households, firms, governments,
central banks, and other economic agents.
5
the Diamond-Mortensen-Pissarides search-and-matching model to allow the number of
firms, which is endogenous and interpreted as investment, to react to news about future
productivity. Guo et al. (2015) show variable capital utilization and increasing returns to
scale production are sufficient modifications to allow a standard RBC model to produce
qualitatively and quantitatively realistic expectationally driven business cycles in response
to news about future aggregate demand shocks.
Each of these approaches to fixing the comovement problem do so by changing the
structure of the economy considered. But these departures from canon have not been
accompanied by an analysis of whether the resulting rational expectations equilibrium
(REE) are expectationally stable (E-stable); that is, whether the REE is “learnable”.
E-stability of an REE can be seen as a measure of the model’s reliance on the rational
expectations hypothesis. This paper proceeds by analyzing the expectational stability
properties of the (unique) RE solution to the three-sector RBC model presented in
Beaudry and Portier (2004). The goal is to discover whether news shock models can
deliever REE which are simultaneously E-stable and capable of generating qualitatively
realistic expectationally driven business cycles. My main results suggest that E-stability
properties are robust to the inclusion of news shocks.
To this point the news-shock literature has exclusively relied on the assumption
championed by Muth (1961) that agents have model consistent or rational expectations
(RE). In other words, agents are perfectly informed as to the precise laws of motion
governing the evolution of the economy. Critically, this implies their forecasts will be
correct on average. This assumption elegantly addresses the critique of Lucas (1976) that
changes in policy may naturally result in changes to the behavior of agents. While Lucas
and Muth both carefully pointed out RE is not meant to be taken as a realistic description
of the way in which individual agents behave, it is natural to ask whether weaker
6
assumptions about the capabilities of agents could result in similar model predictions.3
In particular, it is worth asking if an alternative expectation formation mechanism can be
both a behavioral description and a useful tool for analyzing economic models.
One attractive alternative proposed in Marcet and Sargent (1989a,b,c) is to assume
agents are engaged in “adaptive learning”. The literature on boundedly rational agents4
supposes agents are endowed with a perceived law of motion(PLM) about the true
structure of the economy, but are unsure of the precise coefficients governing the laws
of motion. Regarding their PLM as true the agents make optimal choices given their
information set. Since the actions of agents today influence the state of the economy
tomorrow, this gives rise to the actual law of motion (ALM) for the economy. Agents
behave as econometricians and update their estimates of the laws of motion using a
learning algorithm based on the observed forecast errors each period. A REE is said to
be E-Stable if the PLM converges (asymptotically) to the REE, that is, if agents can
eventually learn the true dynamics of the economy. Learning thus provides one way of
checking the sensitivity of models to the standard rational expectations assumption: if a
3Specifically, Muth (1961) states
“[Rational Expectations] does not assert that the scratch work of entrepreneurs resembles
the system of equations in any way; nor does it state that predictions of entrepreneurs are
perfect or that their expectations are all the same”
while Lucas (1978) states
“...[Rational Expectations] does not describe the way agents think about their environment,
how they learn, process information, and so forth. It is rather a property likely to be
(approximately) possessed by the outcome of this unspecified process of learning and
adapting.”
4See Evans and Honkapohja (2001) for a comprehensive guide to the theory.
7
particular REE is not E-stable, then the predictions of the model may be perceived as less
robust than a similar model featuring a learnable REE.
Given the rich set of behavior observed in models which contemplate changes to
information sets and/or expectations formation mechanisms, it is natural to check whether
the innovations necessary to generate expectationally driven business cycles are consistent
with relaxing the assumption of rational expectations. This is particularly important in
the context of new-shock models, as stability results have been shown to be quite sensitive
to specific assumptions regarding the information sets of agents. For example, Bullard and
Eusepi (2014) show that excluding the value of contemporaneous endogenous variables
from agents’ information sets breaks the link between determinacy and E-stability found
in McCallum (2007). Since by construction news-shock models consider novel assumptions
regarding agents’ information sets while simultaneously altering modifying well-known
DSGE models, an exploration of the E-stability properties in this class of models appears
called for.
I begin by describing news shocks and how they are implemented into
macroeconomic DSGE models. Next I introduce adaptive learning and demonstrate the
procedure for determining whether a given REE is E-stable in the presence of news shocks.
Finally, I apply the learning analysis to the 3-sector RBC model of Beaudry and Portier
(2004). The chapter concludes with a brief discussion of the main results.
8
Incorporating News Shocks into DSGE Models
Consider an economy with temporary equilibria described by the stationary system
of non-linear expectational difference equations
Eˆt [f (Xt+1, Xt, Xt−1, νt)] = 0
where X a vector of endogenous and exogenous variables and ν a vector of anticipated and
unanticipated exogenous white-noise shocks. While non-linear solutions to these types of
problems are increasingly being computed numerically, typically the system is linearized
via a first-order Taylor-series expansion about the steady state X¯. The model now takes
the form
xt = A+BEˆtxt+1 + Cxt−1 +Dνt
where x a vector of endogenous variables in terms of deviation from steady-state.
Partitioning xt into endogenous variables yt and auxiliary state variables wt, the model
can be rewritten
yt = α + βyt−1 + χwt + δEˆtyt+1 (2.1)
wt = ϕwt−1 +Mνt (2.2)
News shocks are exogenous stochastic shocks which are realized today but do not impact
economic fundamentals until some time in the future. Unanticipated or “surprise” shocks
thus can be thought of as a particular form of news shocks in which the realization and
impact occur contemporaneously. In either case, the shocks are incorporated easily into
9
the model by adding new auxiliary variables to the vector wt, and appropriately defining
the matrices ϕ and M .5
Adaptive Learning
E-Stability: Contemporaneous Expectations
I focus on the minimum state variable (MSV) solutions to the system given by (2.1)
and (2.2), which corresponds to a solution with the same set of variables obtained by
solving the model under rational expectations. Restricting attention to the MSV solutions
helps facilitate a direct comparison of solutions obtained under different expectation
formation assumptions.
Before continuing it is worth noting that under rational expectations it is irrelevant
whether agents time t information sets include the values of the contemporaneous
endogenous variables yt or if they observe only lagged endogenous variables yt−1 and must
instead forecast them: because the expectations are model consistent, the forecast will not
be systematically incorrect. When considering bounded rationality, however, this is no
longer the case; differences between the PLM and the ALM will drive a wedge between
expected and realized values and these residuals may well be serially correlated. This
5For example, suppose agents receives news zero and one periods in advance. Then we can write
equation (2.2) as (
ε0t
ε1t
)
=
(
0 1
0 0
)(
ε0t−1
ε1t−1
)
+
(
1 0
0 1
)(
ν0t
ν1t
)
where wt =
(
ε0t , ε
1
t
)
and νt =
(
ν0t , ν
1
t
)
. In this case the total time t innovation to the exogenous variables
in equation (2.1) is given by ε0t = ν
0
t + ε
1
t−1 = ν
0
t + ν
1
t−1. Agents receive the information about the partial
innovation ν1t−1 in period t−1, but it does not impact the economy until period t; the partial innovation ν0t
is a surprise shock learned about in period t which affects fundamentals contemporaneously.
10
wedge can significantly impact the E-stability property of REE, and thus I consider both
cases in what follows.
Regardless of the specific assumptions on the information set of agents, the MSV
solutions of the system above have the form
yt = a+ byt−1 + cwt (2.3)
wt = ϕwt−1 +Mνt (2.4)
Suppose agents are endowed with the form of the solution given in equation (2.3) as their
PLM. Then given the information assumptions above, expectations are given by
Eˆtyt = yt
Eˆtyt+1 = a+ bEˆtyt + cϕwt
= a+ byt + cϕwt
which can be substituted into equation (2.1) to yield the ALM
(I − δb) yt = (α + δa) + βyt−1 + (χ+ δcϕ)wt (2.5)
This provides a mapping from beliefs captured by the PLM to the ALM. This “T-map” is
given by
T (a, b, c) =
{
(I − δb)−1 (α + δa) , (I − δb)−1 β, (I − δb)−1 (χ+ δcϕ)}
11
E-stability of a solution is determined by the matrix differential equation
d
dτ
(a, b, c) = T (a, b, c)− (a, b, c) (2.6)
It is worth noting that a REE
(
a¯, b¯, c¯
)
can be interpreted as the solution to the fixed point
problem
0 = T
(
a¯, b¯, c¯
)− (a¯, b¯, c¯)
i.e. a set of beliefs which are self-fulfilling and coincide on average with realizations such
that
(
I − δ (I + b¯)) a¯ = α (2.7a)
δb¯2 − b¯+ β = 0 (2.7b)(
I − δb¯) c¯− δc¯ϕ = χ (2.7c)
Inspection of equation (2.7b) shows the coefficient matrix b is independent of the assumed
structure for anticipated and unanticipated shocks; that is, the coefficient matrix b is
independent of ϕ and M from (2.2). Furthermore, for a given b equations (2.7a) and
(2.7c) uniquely determine the coefficient matrices a and c. However, b is the solution of
a matrix-quadratic which is not in general unique, and hence solution techniques such as
the well-known methods of Blanchard and Kahn, Uhlig, Klein, or Sims, must be employed
to obtain the REE.
12
The Jacobian of the vectorized matrix differential equation (2.6) evaluated at the
REE a¯, b¯, c¯ can be shown to be comprised of three blocks6
DTa(a¯, b¯) = I ⊗
(
I − δb¯)−1 δ (2.8a)
DTb(b) =
[(
I − δb¯)−1 β]′ ⊗ [(I − δb¯)−1 δ] (2.8b)
DTc(b¯, c¯) = ϕ
′ ⊗ (I − δb¯)−1 δ (2.8c)
Proposition 10.3 of Evans and Honkapohja (2001) suggests the REE is E-Stable if all
eigenvalues of the matrices DTa(a¯, b¯), DTb(b¯), and DTC(b¯, c¯) have real parts less than 1.
This can be easily established given a particular model, but the objective of this paper
is to determine what effect - if any - the inclusion of anticipated shocks has on the E-
stability of a given REE. This is established by Proposition 2.3.1
Proposition 2.3.1. If agents’ information sets include the value of contemporaneous
endogenous variables and the REE corresponding to the core of a model featuring news
shocks, defined as the system of equations which exists when news shocks are shut down,
is E-stable, then the REE corresponding to the model with news shocks included is also
E-stable.
6For example, since the equation governing the evolution of b is independent of a and c we may start
with the equation
Tb(b) = (I − δb)−1 β
the matrix differential of this equation with respect to b is obtained by applying the rule dF−1 =
−F−1(dF )F−1, and hence
dTb(b) = (I − δb)−1 δ(db) (I − δb)−1 β
Since dvecx = vecdx and vec(ABC) = (C ′ ⊗ A)vecB the vectorized Jacobian DTb = ∂vecTb/∂(vecb)′-
which determines local stability of this equation evaluated at a particular b- is given by
DTb(b) =
[
(I − δb)−1 β
]′
⊗
[
(I − δb)−1 δ
]
Similar operations can be employed to obtain DTa and DTc.
13
Proof. Suppose the REE is E-stable without anticipated shocks included in the model.
Then since the value of b¯ is independent of ϕ and M (which define the structure of
exogenous shocks), and since a is uniquely determined for given b, it follows the structural
modifications necessary to incorporate news shocks will have no effect on the eigenvalues
of the matrices DTa and DTb. Furthermore, these modifications can always be done such
that ϕ a block-upper triangular matrix with upper-left block corresponding to the no-
news model and lower-right block a nilpotent lower-shift matrix. Since the eigenvalues of
DTc are the set of pairwise-products of eigenvalues of ϕ
′ and
(
I − δb¯)−1 δ, the inclusion of
the nilpotent sub-matrix implies DTc will acquire a set of zero eigenvalues in addition to
exactly the same eigenvalues as in the no-news model, thereby preserving E-stability of the
REE.
Proposition 2.3.1 suggests that to check whether an REE for a particular model is
E-stable it will be sufficient to consider the model without news shocks. If it is E-stable
without news then it will continue to be E-stable in the presence of news. One immediate
implication of this is that any models with well known unique and E-stable REE can be
modified to include news. However, the conditions which imply a REE is E-stable depend
critically on the information assumed to be known to agents at the time of making their
forecast. In the next section I consider the effect of introducing an informational friction
to the decision making process of agents.
E-Stability: Delayed Information Assumption
In contrast to the preceding subsection, I now assume the values of contemporaneous
endogenous variables are unknown to agents at the time they must make their
decisions. Instead, agents must form forecasts of these values using their PLM. This
14
approach emphasizes the (possible) distinction between individual and aggregate-level
endogenous variables in macroeconomic models; for example, individual households
consumption/savings decisions are typically functions of future real interest rates, which
are themselves determined by the aggregate consumption/savings decision of the economy
as a whole. Thus each household must form a forecast of the future aggregate action.
I again consider the model given by equations (2.1) and (2.2). The PLM is again
assumed to take the form of the MSV solution, which is still
yt = a+ byt−1 + cwt (2.9)
wt = ϕwt−1 +Mνt (2.10)
However, the information assumptions now imply that expectations are given by
Eˆtyt = a+ byt−1 + cwt
and Eˆtyt+1 = a+ bEˆtyt + cϕwt
= (I + b)a+ b2yt−1 + (bc+ cϕ)wt
As before these can be inserted into equation (2.1) to obtain the ALM
yt = [α + δ (I + b) a] +
[
β + δb2
]
yt−1 + [χ+ δ (bc+ cϕ)]wt (2.11)
Thus the mapping of beliefs from the PLM to the ALM is given by
T (a, b, c) =
{
α + δ (I + b) a, β + δb2, χ+ δ (bc+ cϕ)
}
15
and expectational stability of a solution is determined by the matrix differential equation
d
dτ
(a, b, c) = T (a, b, c)− (a, b, c) (2.12)
where again an REE can be seen as the solution to the fixed point problem
0 = T
(
a¯, b¯, c¯
)− (a¯, b¯, c¯)
It is simple to verify the solution to this problem is identical to that implied by equations
(2.7a), (2.7b), and (2.7c), establishing that the specific informational assumptions are
equivalent under RE: when agents are perfectly informed to the true ALM for the
economy it matters not whether they know the values of contemporaneous endogenous
variables or forecast them, as their forecasts will by assumption be correct on average.
However, the additional complication for boundedly rational households can have
non-trivial effects on the conditions necessary to ensure asymptotic convergence to
the REE. In particular, Proposition 10.1 in Evans and Honkapohja (2001) states the
RE solution (a¯, b¯, c¯) is E-Stable if all eigenvalues of the Jacobian of the vectorized T-
map, which again is comprised of three blocks, have real parts less than 1. Under this
16
alternative timing regime, the blocks are given by7
DTa(a¯, b¯) = I ⊗ δ + I ⊗ δb¯
DTb(b¯) = b¯
′ ⊗ δ + I ⊗ δb¯
DTc(b¯, c¯) = ϕ
′ ⊗ δ + I ⊗ δb¯
As with the contemporaneous timing assumption, the matrices DTa and DTb are
unaffected by the modifications to ϕ and M necessary to include news shocks, and hence
their eigenvalues are similarly unaffected. However, DTc is clearly affected; furthermore,
this matrix is the sum of two matrices. Very little can be said of the eigenvalues of the
sum of matrices in general, and thus analysis of E-stability under this delayed information
set must rely on numerical exercises. In what follows I apply the analytic E-stability
results from this section to the model of Beaudry and Portier (2004), and find that the
REE are E-stable under both informational assumptions for a wide range of parameter
constellations and informational structures
Application to Beaudry and Portier (2004)
Any solution to the comovement problem must address two issues. First, good news
about the future must lead to increased demand for investment when the information is
obtained as opposed to when the shock actually materialize. And second, the increased
investment must be financed by increasing employment as opposed to decreasing
7Since the Kronecker product is a bilinear map, we have
DTa
(
a¯, b¯
)
= I ⊗ δ (I + b¯)
= I ⊗ δ + I ⊗ δb¯
17
consumption. There are a variety of methods available for achieving these desired
results. In what follows I will apply the E-stability analysis of the previous section to the
news-shock model of Beaudry and Portier (2004), in which good news about the future
generates a boom in the current period: all major macroeconomic variables - consumption,
investment, labor supply, and output - rise upon receipt of the positive information
regarding future economic fundamentals.
The economy of Beaudry and Portier (2004) is comprised of three sectors:
investment or “durable” goods, intermediate “nondurable” goods, and final composite
consumption goods. Households make a consumption/savings decision using income
earned by supplying labor to the durable and nondurable good markets and renting
capital to the consumption good market; this decision is partially informed the arrival
of information about a future technological innovation i.e. a news shock. All markets are
competitive. The durable and nondurable goods are produced using CRTS technology
from household labor and a fixed factor, and production is augmented by sector-specific
technology. The consumption good is produced from the capital stock and the nondurable
good using CES technology featuring complementarity between the inputs.
The keys to generating positive comovement in this model are twofold. Because
the nondurable good and capital are complements in production of the consumption
good, news about the technology used in producing the nondurable good causes a
contemporaneous change to the demand for investment. And because the consumption
and investment decisions are essentially decoupled from each other - the value of
investment is directly related only to the loss of leisure by increasing labor in the durable
good sector, as opposed to the loss of utility from reducing consumption - the increased
investment is purchased by households working more rather than less. Thus, the model
18
produces qualitatively realistic expectationally driven business cycles in which good news
about the future generates a boom in all key macroeconomic variables today.
The Model
Formally, the composite consumption good Ct is produced from the nondurable good
Xt and the (predetermined) capital stock Kt−1 according to
Ct =
(
aXνt + (1− a)Kνt−1
) 1
ν (2.13)
where ν ≤ 0 to ensure the inputs are complements in production. The nondurable good is
produced from household labor lx,t, a fixed factor Fx, and technology θx,t according to the
constant returns to scale (CRTS) production function
Xt = θx,tl
αx
x,tF
1−αx
x (2.14)
where 0 ≤ αx ≤ 1 captures labor’s share of nondurable-good production. The law of
motion for the aggregate capital stock is
Kt = (1− δ)Kt−1 + It (2.15)
where 0 ≤ δ ≤ 1 the depreciation rate and It gross private investment produced by
the durable goods sector from household labor lk,t, a fixed factor Fk, and technology θk,t
according to the CRTS production function
It = θk,tl
αk
k,tF
1−αk
k (2.16)
19
where 0 ≤ αk ≤ 1 captures labor’s share of durable-good production. The fixed factors are
inelastically supplied by households and have the effect of introducing diminishing returns
in labor supply while maintaining CRTS in overall production. These fixed factors can be
thought of as any scarce resource that constrains production such as privately held land or
managerial capital.
Households are infinitely lived and receive utility from consumption and disutility
from supplying labor. The lifetime utility function is assumed separable in consumption
and labor, and is given by
U = Eˆ0
[ ∞∑
t=0
βt
{
log(Ct)− v0
(
l¯ − lx,t − lk,t
)}]
(2.17)
where 0 < β < 1 the discount factor, v0 > 0 scales the disutility of supplying labor, and l¯
total disposable time. Eˆt denotes the subjective expectations of the household given their
time t information set. The flow budget constraint is
Ct + PtIt ≤ Wx,tlx,t +Wk,tlk,t +RtKt−1 + Πx,t + Πk,t (2.18)
where Pt the price of the investment good in terms of the consumption good and Πx,t and
Πk,t the returns to renting the fixed factor in the nondurable and durable goods sectors,
respectively. The aggregate resource constraint is defined to be
Yt = Ct + PtIt (2.19)
20
where Yt the total output of the economy. Technology of the durable good exhibits a
deterministic trend such that
θk,t = g0,ke
g1t (2.20)
while the technology of the nondurable grows stochastically according to
θx,t = g0,xe
g1tθˆx,t (2.21)
θˆx,t = θˆ
λ
x,t−1e
ε0t (2.22)
where 0 < λ < 1. One may interpret innovations to nondurable good-specific technology
as capturing the process of product differentiation e.g. the creation of higher quality or
entirely new products. New goods will require a higher stock of infrastructure, and this
complementarity between nondurable TFP and the capital stock is key to the model’s
ability to generate qualitatively realistic expectationally driven business cycles.
News itself is assumed to take the form of an anticipated shock to the growth
of nondurable technology. In particular, ε0t = ν
0
t + ν
n
t−n where n is the horizon for
which the shock is anticipated. This is in contrast to the “noisy news” view, which
models news shocks as noisy signals of future innovations. Both formulations in some
sense support the notion of revisions and agents being surprised by realizations which
differ from their expectations. But the anticipated shocks version has the advantage of
being straightforward to incorporate into the systems of equations comprising modern
macroeconomic models; furthermore, it is simple to consider alternative assumptions for
the information flows e.g. the length and/or density of the anticipated shock structure.
21
E-Stability
Solving this model for a particular parameterization involves detrending the non-
stationary variables, calculating the non-stochastic steady state, and then log-linearizing
the system around this non-stochastic steady state; see Appendix A for details. Abusing
notation, the (linear) system of expectational difference equations can be put in standard
form
yt = α + βyt−1 + χwt + δEˆtyt+1 (2.23)
wt = ϕwt−1 +Mνt (2.24)
where yt =
(
C˜t, l˜x,t, l˜k,t, I˜t, K˜t, X˜t, Y˜t, P˜t, R˜t, W˜x,t, W˜k,t, Λ˜t, Q˜t, θ˜k,t, θ˜x,t,
˜ˆ
θx,t
)′
the endogenous
variables, wt = (ε
0
t , ε
1
t , · · · , εnt )′ a vector of auxiliary state variables designed to pass news
shocks through time, and νt = (ν
0
t , ν
1
t , · · · , νnt )′ a vector of anticipated and unanticipated
exogenous stochastic shocks.8 The elements of the matrices α, β, χ, and δ correspond to
the log-linearized equations which describe a temporary equilibrium and are functions of
the structural parameters; the non-zero elements of ϕ, and M are selected to imply the
desired structure for the arrival of anticipated and unanticipated shocks.
Beaudry and Portier (2004) calibrate the model parameters using results from
previous studies or to achieve specific steady-state values. My analysis of the E-stability
of their news-shock model begins with their baseline calibrations, which are summarized in
Table 1.
8Technology for the durable good is assumed to be deterministic. Incorporating a stochastic element
would simply require modifying wt and νt to include the appropriate auxiliary and exogenous variables
and making the corresponding revisions to the filtering matrices ϕ and M .
22
Baseline Parameterization
Parameter Values
β 0.98
δ 0.05
αx 0.60
αk 0.97
l¯ 2
v0 1
Fx 1
Fk 1
λ 0.999
ν -3.78
TABLE 1. Calibrated Parameters for Three-sector Model
Inclusion of News Shocks
Putting the model in standard form allows an immediate application of the E-
stability results from the section which introduced adaptive learning. In particular, it
is simple for any given parameterization to calculate the eigenvalues of the T-map’s
vectorized Jacobian which indicate whether or not the system is E-stable. I begin by
setting n = 0, that is, the case where there is no news and all shocks are completely
unanticipated. The largest real roots of DTa(a¯, b¯), DTb(b¯), and DTc(b¯, c¯) under the
contemporaneous information assumption are 0.5199, 0.51947, and 0 respectively, while
the corresponding numbers are 0.1547, 0.1538, and 0 under the delayed information
assumption. Thus the RE solution for this model is E-Stable under both informational
assumptions without news.
Proposition 2.3.1 implies that the eigenvalues should be unaffected by the inclusion
of news shocks under the contemporaneous timing assumption. Indeed, setting n = 1 so
that households receive information about the technological innovation 1-period ahead
and keeping all other parameters fixed at their baseline levels results in no change to the
23
eigenvalues for DTa(a¯, b¯), DTb(b¯), or DTc(b¯, c¯), as expected. Furthermore, while there is
no corollary for the delayed timing assumption, numerical results suggest the E-stability
of the REE is similarly unaffected by the inclusion of news under the delayed timing
assumption.
The main contribution of this chapter is the finding that news shocks do not alter
the E-stability properties for REE. This is important because E-stability results are,
in general, sensitive to the assumptions regarding the timing and flow of information.
While news shocks are essentially a modification of the exogenous driving forces for
the economy which would not in general be expected to alter E-stability results, their
impact is felt through the revision of expectations and subsequent behavioral changes
on the part of households from the arrival of new information. This informational
feedback effect is responsible for the fact that some REE which are E-stable under the
contemporaneous timing assumption may fail to be E-stable under the delayed timing
assumption. The main results of this chapter affirm the robustness of REE to news shocks
and suggest modelers wishing to incorporate anticipated shocks into their models may do
so without sacrificing on the robustness of their REE to alternate expectations formations
assumptions.
Robustness
In what follows I seek to characterize what role, if any, the specific calibration
choices for structural parameters and the new shock structure play in determining
whether the REE is E-stable. I proceed by first considering alternate calibrations for
the structural parameters and then exploring the effect of lengthier forecasting horizons
and denser informational structures. I find that the E-stability of the REE is robust to
24
all constellations consistent with the model being able to generate qualitatively realistic
expectationally driven business cycles, with and without news shocks.9
To analyze whether alternative parameterizations may impact the E-stability results,
I adopt the following strategy: I change the value of structural parameters one at a time
while holding all others at their baseline calibration, searching only within the parameter
space consistent with the model generating qualitatively realistic expectationally driven
business cycles. The REE is E-stable for all parameterizations considered and under both
informational timing assumptions. Table 2 displays the smallest and largest considered for
each parameter.
Robustness Checks
Parameter Smallest Value Largest Value
β 0.001 1
δ 0 1
αx 0.001 0.999
αk 0.001 0.999
ν -100 -0.001
a 0.001 0.999
TABLE 2. Alternate Parameterizations
In general, the largest real parts of eigenvalues are larger when households discount
the future less (smaller β), when the depreciation rate of capital is smaller (smaller δ),
when the decreasing returns to producing the nondurable good are smaller (larger αx),
when the decreasing returns to producing the durable good are larger (smaller αk), when
the complementarity between capital and nondurable goods in producing the consumption
good is weaker (larger ν), and as the relative importance of the nondurable good to capital
in producing consumption decreases (smaller a). Even so, setting these parameters jointly
9The ability of the model to solve the comovement problem is dependent only the value of ν,
which governs the substitutability between capital and the nondurable good in the production of the
consumption good. As long as these factors are complements in production of the consumption good i.e.
ν ≤ 0, the model will be capable of generating qualitatively realistic expectationally driven business cycles.
25
to values which on their own would tend to imply relatively large eigenvalues does not
cause the REE to become expectationally unstable.
One might also be concerned about assumed structure for news shocks themselves,
that is, the length and density of the forecasting horizon. To analyze what role this
structure plays in the E-stability of the REE I proceed as follow. First, I extend the
forecasting horizon while keeping the structural parameters at their baseline levels by
considering n = 1, 2, ..., 10. Second, I allow for multiple anticipated shocks, that is,
households receive multiple pieces of information for the eventual innovation. For example,
households may receive news about the innovation four and eight periods ahead. Neither
changes to the length of the forecasting horizon nor its density have any impact on the
largest real parts of eigenvalues of DTa(a¯, b¯), DTb(b¯), or DTc(b¯, c¯) under either timing
assumption.
Conclusion
That the three-sector news-shock model of Beaudry and Portier (2004) can generate
qualitatively realistic expectationally driven business cycles while the RE solution is E-
stable suggest there is nothing fundamental about news shocks per se to imply models
incorporating them will be expectationally unstable. In fact, Proposition 2.3.1 shows
that the REE of any model be E-stable when news shocks are included in the time t
information set as long as it is E-stable when news shocks are not included. While no
analytic result exists for alternative timing specifications, numerical experimentation
suggests this may be true more generally, as alternate informational structures did not
impact the largest eigenvalues of the associated T-map’s vectorized Jacobian.
26
This is a particularly important finding given the well-known sensitivity of E-
stability of REE to specific assumptions regarding agents information sets. Since news-
shock models rely critically on novel changes to agent’s information sets, and since no
other studies have examined E-stability in the context of a news-shock model, this study
suggests a potential justification for RE on the grounds that boundedly rational reduced-
form learning agents can learn the RE coefficients over time. More work will need to
be done to examine whether the REE of such news-shock models demonstrate similar
learnability properties under alternative forms of bounded rationality, such as Euler-
equation learning, infinite-horizon learning, or shadow-price learning.
27
CHAPTER III
SHADOW PRICE LEARNING IN A NEWS-SHOCK MODEL
Introduction
The relatively recent interest in using news to explain macroeconomic fluctuations
began largely from the asset-price bubbles experienced in the late 1990s with the tech-
boom. The “irrational exuberance” exhibited by the market was difficult to generate in
most standard macroeconomic DSGE models, and it is this gap which the original news-
shock model of Beaudry and Portier (2004) attempted to fill. As shown in Chapter II,
by modifying the structure of a RBC model in a particular way the authors were able to
generate boom-bust cycles fueled by households experiencing periods of optimism and
pessimism which created liquidation cycles in the capital stock. Yet this is just one way of
solving the “comovement problem” found in RBC models, and it comes at the expense of
making significant changes to the standard RBC model. In particular, it requires three
sectors and very strong complementarity of inputs to final goods production, both of
which are somewhat unusual in the broader context of typical workhorse macroeconomic
models.
Jaimovich and Rebelo (2009) on the other hand provide a news-shock model capable
of solving the comovement problem while remaining approachable to researchers familiar
with standard RBC models. The core of the model is essentially a discrete-time Ramsey
model with elastic labor supply. Savings are converted to investment according to some
technological process, and investment is used to create a depreciating capital stock. This
textbook structure is augmented by assuming variable capacity utilization of capital, a
28
cost to adjusting investment from its previous level, and a novel preference specification
nesting the well known specifications of Greenwood et al. (1988) and King et al. (1988).
While these features taken together represent a significant departure from standard RBC
models, the individual elements are fairly standard.
The relative simplicity of the model makes it an excellent candidate for exploring
the qualitative and quantitative effects of relaxing the RE assumption in favor of a weaker
assumption. Furthermore, since it is quite similar to well-known RBC models, the results
of such an experiment exist in an environment in which some context has already been
provided. While many alternatives to RE exist, I ultimately choose to consider bounded
optimality of the sort described in Evans and McGough (2015). To place this decision in a
broader context, I begin by describing the two benchmark theories of bounded rationality:
Euler-equation learning and infinite-horizon learning.
Under Euler-equation learning, found e.g. in Evans and Honkapohja (2006) and
Honkapohja et al. (2012), agents behave in accordance with their Euler equations
which relate control decisions today with the forecasted values of decisions tomorrow.
This approach eliminates the reliance on RE in deriving the equilibrium dynamics of
the economy found in the reduced-form learning approach of Chapter II. Rather than
assuming agents are rational, solving their dynamic programming problem, and then
relaxing the RE assumption, Euler-equation learning studies the evolution of agent’s
beliefs and the path of the economy when their expectations are boundedly rational even
when solving their dynamic programming problem. One of the benefits of Euler-equation
learning is its simplicity: agents need only make one-period ahead forecasts to determine
their optimal control decisions.
29
However, a possible shortcoming is this approach requires agents to forecast the
future value of control variables, a somewhat strange exercise in a representative agent
model where these are completely under the control of the agents. The infinite-horizon
approach to learning of Preston (2005) avoids this by formally expressing decisions today
as being determined by agent’s expectations of their future lifetime budget constraint
and transversality conditions. Behavior is thus based on the relationship between future
wealth and the implied control decisions. One of the main benefits of the infinite-horizon
approach is the strict adherence to microfoundations which implies behavior is truly
optimal given beliefs.
In Branch et al. (2012a) a “hybrid” of the infinite-horizon and Euler-equation
approaches called N-step optimal learning is developed. N-step optimal learning assumes
agents explicitly take current and future expected values of wealth over a finite range
into account when making their control decisions, which are themselves anchored to
the Euler equation as a behavioral primitive. This learning mechanism has the infinite-
horizon approach as a limiting case, and may be viewed as its finite-horizon version. N-
step optimal learning captures both the simplicity of Euler-equation learning and the
rigorous adherence to fully optimal behavior imparted by infinite-horizon learning, but
it still requires considerable sophistication on the part of agents in the model.
A recent alternative similar in spirit but requiring less agent-level sophistication is
the shadow-price learning (SP-learning) approach of Evans and McGough (2015). Rather
than taking the behavioral primitive to be the Euler equations, which often embody
complicated nonlinear relations between contemporaneous control decisions and the
evolution of the determinants of future income, SP-learning assumes agents base their
decisions on the standard first-order necessary conditions (FONCs) derived from the
intertemporal Lagrangian. These FONCs describe a set of conditions which must be
30
satisfied by the agents control decisions in order to be considered optimal, and they are
functions of the expected present value of shadow prices for endogenous state variables.
SP-learners form forecasts of these shadow prices using a linear PLM and update their
beliefs over time using recursive least squares; thus the behavior is by definition optimal
given beliefs.
SP-learning is an attractive alternative to RE for many reasons. First, the
informational assumptions are quite natural: agents need understand little more than
their preferences and budget constraint and take as given many of the same variables that
real households do e.g. wages and interest rates. Second, the behavior of agents is quite
intuitive: they simply contemplate how changing their behavior today would impact key
variables tomorrow based on their beliefs and take action accordingly. Third, the updating
of beliefs occurs via recursive least squares, an exercise which any student of introductory
econometrics could conduct. Finally, it lends itself to considering heterogeneity of agents
along some dimension such as information or initial wealth, as these differences will cause
different transition paths under learning for different households.
While much attention in the bounded rationality literature has focused on
determining whether agents endowed with boundedly rational expectations may learn
to behave rationally, there is a somewhat smaller literature which examines the effect
of relaxing RE on the quantitative predictions generated by a given DSGE model. For
example, Williams (2003a) compares data generated by simulations of an RBC and a
NK model under RE and reduced-form learning. He finds that reduced-form learning
has an extremely small effect on generated moments. He then considers an alternative
learning approach where agents’ decisions incorporate uncertainty about the structure
of the economy into their decision rules. This “structural learning” approach is shown
to substantially increase the volatility and persistence of key macroeconomic variables
31
generated by an otherwise standard RBC model. Given the discussion above this is not
too surprising: the “structural learning” approach embodies to varying degrees the same
core idea behind Euler-equation and infinite-horizon learning, namely that agents should
try to act optimally given their beliefs, and that these beliefs may be quite different from
those prescribed by RE.
This chapter proceeds in the spirit of Williams (2003a) by trying to determine the
quantitative effects on business cycle statistics generated by the Jaimovich and Rebelo
(2009) news-shock model when agents are assumed to be SP-learners. The central goal
of this chapter is to address what effect relaxing the assumption of RE in favor of SP-
learning has on the empirical relevance of this particular news-shock model, and whether
the behavior of SP-learning households will come to approximate that of their fully
rational counterparts. I first describe the economic environment, including the calibration
for the model, the equations governing a temporary equilibrium, and the way in which
news shocks are modeled. I then present the results of simulation exercises to generate and
compare data on business cycles within the model to those found in US data. The chapter
concludes by reviewing the main results and discussing paths for future research.
The Model
The representative household chooses consumption Ct and hours worked ht to
maximize the lifetime utility function
Eˆ0
∞∑
t=0
βtU(Vt) (3.1)
32
where 0 < β < 1 the household’s discount factor and U the period utility function which
takes the CRRA form
U(Vt) =
V 1−σt − 1
1− σ (3.2)
where σ > 0 the inverse intertemporal elasticity of substitution and the argument Vt is
given by
Vt = Ct − ψh
1+ 1
θ
t St
1 + 1
θ
(3.3)
where ψ > 0 scales the disutility of labor supply and θ > 0 governs the Frisch elasticity
of labor supply. St is a geometric average of current and past habit-adjusted consumption
and takes the form
St = C
γ
t S
1−γ
t−1 (3.4)
where 0 ≤ γ ≤ 1 governs the magnitude of the wealth elasticity of labor supply. This
preference specification, often referred to as “JR preferences”, allows the modeler to
calibrate the wealth effect of labor supply to be “small” while permitting a balanced
growth path. The comovement problem in typical RBC models is partially caused by
the wealth effect of labor supply dominating the substitution effect, thereby causing
consumption and hours worked to move in opposite directions upon receipt of good
news, and hence this utility function makes it simple to develop qualitatively realistic
expectationally driven business cycles.
Note this specification nests two well-known and important preference specifications.
γ = 0 corresponds to the preferences of Greenwood et al. (1988) in which labor supply
33
depends only on current real wages and is independent of the marginal utility of income,
while γ = 1 corresponds to the preferences of King et al. (1988) which are compatible
with a balanced growth path at the optimal steady state of the economy. Small values for
γ allow the economy to be consistent with a balanced growth path while also implying
a very weak wealth effect of labor supply, both of which are important features for any
dynamic stochastic general equilibrium (DSGE) model hoping to explain movement in key
macroeconomic variables through news shocks.1
Households are assumed to own physical capital Kt and rent it to firms in a
competitive market. Each household’s stock of capital evolves according to the law of
motion
Kt = (1− δ (ut))Kt−1 + It
[
1− Φ
(
It
It−1
)]
(3.5)
where It gross private investment. Note the timing convention: subscripts correspond to
the time period in which the variable is decided, hence in time t the household chooses Kt
and takes as given the predetermined capital stock Kt−1, which was chosen in period t− 1.
Φ
(
It
It−1
)
imposes a cost to adjusting investment from its previous level while δ (ut)
implies capital depreciation is a function of its utilization rate ut. Both functions are
convex in their arguments, and I follow SGU in assuming the quadratic functional forms
δ(ut) = δ0 + δ1(ut − 1) + δ2
2
(ut − 1)2 (3.6)
1Note that the case of γ = 0 is not consistent with a balanced growth path: along a steady state the
real interest rate must be growing at a constant rate, and thus the marginal utility of income will also be
growing. A balanced growth path requires the wealth and substitution effects from the constantly growing
real wage to cancel, but γ = 0 ensures that only the substitution effect exists, and hence labor supply will
be changing at a rate inconsistent with the rest of the economy.
34
where δ0 > 0 the steady-state depreciation rate, δ1 > 0 determines the steady-state value
of ut, and δ2 > 0 captures the rental rate elasticity of capacity utilization and
Φ
(
µIt
)
=
κ
2
(
µIt − µI
)2
(3.7)
where µIt ≡ ItIt−1 the growth rate of investment, κ > 0 a scaling parameter, and µI the
steady-state growth rate of gross private investment. This specification implies Φ
(
µI
)
=
Φ′
(
µI
)
= 0 and Φ′′
(
µI
)
> 0, i.e. there are no investment-adjustment costs on a balanced
growth path.
Each period the household receives labor income from working ht hours at rate Wt,
rental income from from renting utKt−1 units of effective capital at gross rental rate Rt,
and lump sum firm-profits of Πt. The household uses this income to purchase consumption
and investment goods. The flow budget constraint is given by
Ct + AtIt ≤ Wtht +Rt (utKt−1) + Πt (3.8)
where At is an exogenous process representing the current state of technology for
producing investment goods from consumption goods with stationary growth rate µAt ≡
At
At−1
and steady-state value µA. In a decentralized equilibrium At may be interpreted as
the relative price of investment goods in terms of consumption goods, that is a unit of the
investment good may be exchanged for At units of the consumption good.
Households are assumed to make their choices to maximize their expected lifetime
utility, and hence each period they solve a constrained optimization problem to maximize
expected discounted utility. Formally, households choose a set of stochastic processes
35
{Ct, ht, ut, It, Kt, St}∞t=0 to maximize 3.1 subject to the constraints given by equations
3.2-3.8 and initial conditions for the endogenous state variables I−1, K−1, and S−1. This
can be written as a standard dynamic constrained maximization problem:
max
Ct,ut,ht,It,Kt,St
Eˆt
∞∑
t=0
βtU (Vt)
subject to U (Vt) =
V 1−σt − 1
1− σ
Vt = Ct − ψh
1+ 1
θ
t St
1 + 1
θ
St = C
γ
t S
1−γ
t−1
Ct + AtIt ≤ Wtht +Rt (utKt−1) + Πt
Kt = (1− δ(ut))Kt−1 + It
[
1− Φ
(
It
It−1
)]
The household’s time t control decisions uˆt = (Ct, ut, ht, It, Kt, St)
′ will depend
on their expectations of the future shadow price of the endogenous state variables
x1t = (It−1, Kt−1, St−1)
′, which are variables the household can directly affect in the
future through their actions today.2 Under RE all agents within the economy know the
conditional distributions of all variables, and hence the laws of motion for the endogenous
and exogenous state variables are known and the expected future value of these shadow
prices is straightforward to compute. Standard solution techniques under RE involve
deriving Euler equations relating today’s control decisions to those of tomorrow to describe
the behavior of households and combining these and other optimality conditions, resource
constraints, and laws of motion to arrive a system of expectational difference equations.
2While agents will also use their expectations of future realizations of the exogenous state variables
in their decision making process, their actions do not affect the marginal values of these variables; hence,
they need not concern themselves with the expected future shadow prices of these exogenous states.
36
Typically this system is nonlinear, and hence the solution to the first-order approximation
is considered.
In what follows I will replace RE with a version of bounded rationality based
on shadow prices called SP-learning. Under SP-learning, households do not know the
conditional distribution of all variables, and in particular they do not know the precise
way in which shadow prices depend on their behavior. Instead they estimate the value of
these shadow prices using a linear forecasting rule which is a function of observables.
The solution technique under SP-learning stands in stark contrast to that of RE.
A major difference is that while agents are endowed with linear forecasting rules, their
behavior is embedded within the nonlinear specification of the model. This implies
except for special cases, the relevant equilibrium notion is that of a restricted-perceptions
equilibrium (RPE) as opposed to a REE. A RPE can be thought of as an equilibrium
arising from agents’ optimally misspecified beliefs which are consistent with the stochastic
processes realized in the economy. Given their forecasting model agents are unable to
detect a misspecification. The first task of this chapter is determining how similar this
RPE is to the implied REE.
Denote the shadow prices of investment, capital, and habit-adjustment by λIt , λ
K
t and
λSt , respectively. These have the simple interpretation as the time t value of an additional
unit of their corresponding endogenous state in time t. For example, λKt is the marginal
value of an additional unit of preinstalled capital Kt−1 in time t. With this interpretation
the FONCs describing optimal household choices of Ct, ht, and ut given beliefs about the
future values of these shadow prices can be obtained via simple variational arguments.3
3Note that given optimal decisions for consumption, labor supply, and capacity utilization the choices
for the geometric average of habit-adjusted consumption St, gross investment It, and hence next period’s
capital stock Kt+1 are pinned down by equations (3.4), (3.8), and (3.5) respectively.
37
The details of this derivation are presented in Appendix B. The end result is the following
three FONCs in the controls
UCt(Vt) +
∂St
∂Ct
βEˆtλ
S
t+1 =
∂It
∂Ct
(
βEˆtλ
I
t+1 +
∂Kt
∂It
βEˆtλ
K
t+1
)
−Uht(Vt) =
∂It
∂ht
(
βEˆtλ
I
t+1 +
∂Kt
∂It
βEˆtλ
K
t+1
)
∂Kt
∂ut
βEˆtλ
K
t+1 =
∂It
∂ut
(
βEˆtλ
I
t+1 +
∂Kt
∂It
βEˆtλ
K
t+1
)
and the following three FONCs in the endogenous state variables
λIt =
∂Kt
∂It−1
βEˆtλ
K
t+1
λKt =
∂It
∂Kt−1
(
βEˆtλ
I
t+1 +
∂Kt
∂It
βEˆtλ
K
t+1
)
+
∂Kt
∂Kt−1
βEˆtλ
K
t+1
λSt = USt−1(Vt) +
∂St
∂St−1
βEˆtλ
S
t+1
It can be shown that the endogenous shadow prices can be expressed in terms of the
contemporaneous controls and states, and hence their actual value is determined as a
function of the contemporaneous decisions of the household. However, the household
makes these decisions based on their expectations of the value of future shadow prices
without knowing the nonlinear way in which the actual values are determined. Given a
set of beliefs and a linear PLM for the evolution of these shadow prices, SP-learners take
optimal actions given their forecasts of future shadow prices. This behavior results in the
ALM for the shadow prices. Beliefs are updated as weighted averages of current beliefs
and forecast errors via the recursive least squares algorithm as is standard in the adaptive
learning literature. A crucial question addressed by studying household behavior under
38
this less stringent specification for expectation formation is whether the household’s beliefs
will (approximately) converge to those of a rational agent.
The model is closed by describing the production side of the economy. It is
important to keep in mind that the only agents operating as SP-learners are the
households; firms behave in a manner consistent with rational expectations. One possible
justification for this discrepancy in assumptions about expectations formation is that these
types of agents may be more likely to have the means to act in a highly sophisticated
manner, whereas the average household likely does not.
The representative firm pays for ht worker-hours and rents utKt−1 units of effective
capital to produce output Yt using CRT technology according to the production function
Yt = zt (utKt−1)
1−α hαt (3.9)
where α ∈ (0, 1) governs the labor share of output in steady state. As is typical in RBC-
type models, the supply side of the economy is subjected to exogenous stochastic shocks,
given here by a transitory shock zt. Factor markets are competitive and hence the gross
rental rate equals the value of the marginal product of effective capital
Rt = (1− α) Yt
utKt−1
(3.10)
and the wage paid is equal to the value of the marginal product of labor
Wt = α
Yt
ht
(3.11)
39
Output is fungible and may be used for private consumption or gross investment. Total
demand is thus given by
Yt = Ct + AtIt (3.12)
Equilibrium
A temporary equilibrium for all periods t ≥ 0 is described collectively by the optimal
behavior of households given beliefs:
UCt(Vt) +
∂St
∂Ct
βEˆtλ
S
t+1 =
∂It
∂Ct
(
βEˆtλ
I
t+1 +
∂Kt
∂It
βEˆtλ
K
t+1
)
(3.13a)
−Uht(Vt) =
∂It
∂ht
(
βEˆtλ
I
t+1 +
∂Kt
∂It
βEˆtλ
K
t+1
)
(3.13b)
∂Kt
∂ut
βEˆtλ
K
t+1 =
∂It
∂ut
(
βEˆtλ
I
t+1 +
∂Kt
∂It
βEˆtλ
K
t+1
)
(3.13c)
Kt = (1− δ (ut))Kt−1 + It
[
1− Φ
(
It
It−1
)]
(3.13d)
Ct + AtIt ≤ Wtht +Rt (utKt−1) (3.13e)
St = C
γ
t S
1−γ
t−1 (3.13f)
I−1,K−1, S−1 given (3.13g)
where the utility function, its argument, and the functional forms for δ(ut) and Φ
(
It
It−1
)
are given by equations (3.2), (3.3), (3.6), and (3.7), respectively. Appendix B expresses
these equations in terms of the underlying variables as opposed to derivatives, which I
present here for the sake of expositional clarity. The market clearing conditions, aggregate
40
resource constraint, production function, and factor-market prices are given by
Yt = Ct + AtIt (3.14a)
Yt = zt (utKt−1)
1−α (ht)
α (3.14b)
Wt = α
Yt
ht
(3.14c)
Rt = (1− α) Yt
utKt−1
(3.14d)
The laws of motion for the exogenous stochastic processes At and zt along with the precise
way in which news shocks are included in the model are described below.
News Shocks and Expectations Formation
It remains to describe the particular way in which shadow price learning agents
form their expectations. The variables relevant to household decision making but
outside of their control in time t - that is, the predetermined control variables, factor
prices, and the exogenous state variables (plus a constant) - are collected into xt =
(1, It−1, Kt−1, St−1,Wt, Rt, At, zt)
′, and households are assumed to have a PLM for the
endogenous-state shadow prices given by λt = Htxt, where λt =
(
λIt , λ
K
t , λ
S
t
)′
and
Ht denotes the household’s time t estimates of the coefficient-matrix relating the state
variables to the value of the shadow price.
There is one issue that must be addressed: Wt and Rt are exact functions of ht,
utKt−1, and the exogenous processes zt and At, and hence including all of these variables
in the regression will lead to perfect multicollinearity. To avoid this I drop Wt and Rt from
41
the set of regressors utilized by the household and denote this subset of state variables as
x˜t = (1, It−1, Kt−1, St−1, At, zt)
′.4
The household’s PLM for the shadow prices is thus given by the linear model
λt = H
′
tx˜t (3.15)
where Ht is updated via recursive least squares according to the dynamic system
RH,t = R
H
t−1 + gt
(
x˜t−1x˜′t−1 −RHt−1
)
(3.16)
Ht = Ht−1 + gtR−1H,tx˜t−1
(
λt−1 −H ′t−1x˜t−1
)′
(3.17)
RH,t is the household’s time t estimate of the second-moment matrix for the regressors
while the “gain” parameter gt controls how much weight households put on new
information. Much of the theoretical work exploring whether agents can learn the
coefficient-matrix of an economy’s REE, such as Evans and Honkapohja (2001) assumes
a decreasing gain such as gt = t
−1 , which implies the household response to forecast
errors vanishes asymptotically. Alternatively, studies focused on simulation or estimation
of DSGE models under learning, such as Williams (2003b) and Milani (2007), employ
a constant gain where gt = g¯, which implies the household is a lifelong learner and
continually revises its coefficient estimates placing the most weight on the most recent
observations. I will simulate the model under both assumptions. Under constant-gain
4One could also consider assuming the household does not observe some of the exogenous processes
and instead uses wages and the gross rental rate in their forecasts. This would require specifying a PLM
and dynamic system for updating beliefs for these price variables. It can also lead to serious issue with
asymptotic multicollinearity as household’s beliefs converge to those of rational agents, because as the
residuals in the dynamic system for updating beliefs go to zero the actions of the household become
perfectly collinear with market prices.
42
learning I calibrate the gain to be 0.0152, which is in line with recent estimates from the
empirical adaptive learning literature.
The value of shadow prices in time t is not known ex-ante to the household, and is
determined as a result of their control choices. Hence, time t beliefs are updated using
information available through time t − 1. This clarifies the central idea of shadow price
learning: households make optimal decisions given their (misspecified) beliefs and the
information available to them in the moment. That beliefs may be misspecified highlights
a particularly salient feature of SP-learning in particular and bounded rationality in
general by embracing the notion that agents in the economy may certainly fail to fully
understand the dynamics governing its evolution and yet an equilibrium may still exist.
Taking expectations of equation (3.15) we have Eˆtλt+1 = HtEˆtx˜t+1, and hence
households must forecast the future values of regressors.5. The values of the time t + 1
endogenous state variables It, Kt, and St are pinned down by the time t flow budget
constraint, capital accumulation equation, and geometric-average identity respectively,
and it is natural to assume the household knows these values. In addition the transition
equations for the exogenous processes zt and At are assumed to be known to the
household.6
5I have assumed certainty equivalence on the parts of households; that is, they believe their time
t estimate of the coefficient matrix is the correct estimate now and in the future, and hence behave
accordingly. Thus household’s do not need to attempt to forecast the way in which their estimates will
change in the future.
6Since there is no feedback between the household’s decisions and the evolution of these processes this
is a very simple estimation exercise from an econometric standpoint if the households actually observe
all of this data. An alternative approach would carefully consider the variables a household seems likely
to observe. Exogenous variables that directly influence the household, such as the technology available
for converting consumption goods into investment goods At in the flow budget constraint seem quite
natural; however it is less obvious that households would directly observe productivity zt. As mentioned
previously, one could drop these as regressors in favor of e.g. wages and/or the rental rate of capital, but
this may cause the coefficient estimation routine to suffer from asymptotic multicollinearity and/or bias
from sampling error present in using estimated data as an explanatory variable.
43
These expectations are augmented by the inclusion of news shocks to the information
set. A news shock can be thought of as information which arrives exogenously to the
household about the value of some future innovation to an exogenous process, but
(importantly) does not impact any economic fundamentals contemporaneously. The action
generated by news shocks is therefore entirely in the response of agents to this information
about the future. News is modeled as an anticipated shock to the exogenous processes and
hence has the interpretation of imparting incomplete (but accurate) information to the
household about future economic fundamentals.
In particular let the law of motion for the exogenous processes, indexed by w =
(A, z), take the form
ln(wt) = ρw ln(wt−1) + ε0w,t
ε0w,t = ε
1
w,t−1 + ν
0
w,t
[
ν0w,t ∈ It
]
ε1w,t = ε
2
w,t−1 + ν
1
w,t
[
ν1w,t−1 ∈ It
]
... =
...
εnw,t = ε
n−1
w,t−1 + ν
n
w,t
[
νnw,t−n ∈ It
]
where It the time t information set of the representative household, νkw,t−k the k-period
ahead anticipated shock, and
[
νkw,t−k ∈ It
]
= 1 if νkw,t−k ∈ It and 0 otherwise. This
specification allows the modeler to easily and parsimoniously consider a variety of
assumptions regarding the precise details of informational acquisition by households.7
7In fact, this requires the creation of only as many auxiliary state variables as the lengthiest forecasting
horizon. For example, if a household receives information four and eight periods in advance the longest
forecasting horizon is eight periods, and thus eight auxiliary state variables must be generated. An
alternative but equivalent specification would generate a set of k state variables for each signal vkw,t−k so
that e.g. if a household received information four and eight periods in advance one would need to generate
12 additional state variables.
44
The system above makes clear the sense in which news shocks here are being
modeled as partial information about the total value of the innovation. This permits a
compact autoregressive representation
ln(wt) = ρw ln(wt−1) + ε0w,t (3.18)
εw,t = ϕwεw,t−1 +Mwνw,t (3.19)
where νw,t =
(
ν0w,t, ν
1
w,t, ..., ν
n
w,t
)′
distributes i.i.d normal with mean zero and variance-
covariance matrix equal to the identity matrix. The matrix ϕw and vectors εw,t, νw,t, and
Mw are given in Appendix B.
To see how introducing news as anticipated shocks can have real effects, suppose
households were to receive information four and eight periods in advance about the time
t + 1 shock, and that the exogenous process is subjected to an unanticipated shock which
arrives in time t+1. Then the time t expectations about the evolution of process w in time
t+ 1 would forecast its value to be
Eˆt ln(wt+1) = ρw ln(wt) + ν
4
w,t+1−4 + ν
8
w,t+1−8
while the actual value realized in time t+ 1 would be
ln(wt+1) = ρw ln(wt) + ν
0
w,t+1 + ν
4
w,t+1−4 + ν
8
w,t+1−8
This enables the consideration of a variety of interesting behavior as households revise
their expectations in response to new information. For example, the expected value
for the innovation prior to time t + 1 could fail to materialize, that is, ν0w,t+1 =
− (ν4w,t+1−4 + ν8w,t+1−8), or could materialize exactly as expected i.e. ν0w,t+1 = 0. More
45
generally one may think of the household continually adjusting expectations throughout
time in response to the receipt of new information, and this information being descriptive
of the innovation’s final value to a varying degree in any given period.
Recall the household’s PLM for shadow prices is a linear function of the time t + 1
endogenous state variables and exogenous processes. Since households are assumed to
know the law of motion for all exogenous processes they will incorporate the anticipated
shocks directly into their decision rules for time t controls via their forecast of future
shadow prices. That is, the time t expectation of future shadow prices based on the PLM
λt = H
′
tx˜t is
Eˆtλt+1 = H
′
tEˆtx˜t+1
where Eˆtx˜t+1 =

1
EˆtIt
EˆtKt
EˆtSt
EˆtAt+1
Eˆtzt+1

=

1
It
Kt
St
ρAAt + Eˆtε
0
A,t+1
ρzzt + Eˆtε
0
z,t+1

where the household’s expectation of the future shock conditional on all information
received up to time t, Eˆtε
0
w,t+1, is described above.
Calibration
The model is calibrated using a combination of commonly used values in the
literature, estimates obtained from Schmitt-Grohe and Uribe (2012), and steady-state
targets for some endogenous variables. σ = 1 which corresponds to logarithmic utility, θ =
46
Parameter Value Description
σ 1 Intertemporal Elasticity of Substitution
θ 1.4 Frisch-labor Supply Elasticity
γ 0.001 Wealth Elasticity of Labor Supply
β 0.985 Subjective Discount Factor
α 0.64 Steady-state Labor Share
δ0 0.025 Steady-state Depreciation Rate
u 1 Steady-state Capacity Utilization Rate
h 0.2 Steady-state Labor Supply
κ 1.3 Adjustment Cost Acceleration
ρz 0.5 Persistence of Investment-specific TFP Growth
ρA 0.9 Persistence of TFP Growth
TABLE 3. Calibrated Parameters for One-sector Model
1.4 so that the wage elasticity of labor supply is 2.5 when the wealth effect of labor supply
is shut off, and γ is set to 0.001 which is simultaneously consistent with an extremely
small wealth effect of labor supply and a balanced growth path. β is assumed to be 0.985
so that the steady-state gross real interest rate is 1.5 percent, and α is set to 0.64 so that
labor’s share of output in steady-state is 64 percent. Steady state quarterly depreciation δ0
is set to 2.5 percent and δ2 is chosen so that the elasticity of δ(ut) is 0.15. δ1 is calibrated
to ensure steady-state capacity utilization equals 1, while the disutility-scale parameter
ψ is chosen so that household’s spent 20% of their time working. The second derivative
of the investment adjustment cost function κ is set to 1.3, though this is subjected
to robustness checks since the literature has little to say about this parameter. The
autoregressive parameters for growth rates of zt and At are set to 0.5 and 0.9, respectively,
which are consistent with the estimated values obtained in Schmitt-Grohe and Uribe
(2012). This is summarized in Table ?? The relative importance of anticipated vs surprise
shocks for each exogenous process is set consistent with estimates from Schmitt-Grohe
and Uribe (2012). In particular, the standard deviations for the surprise components of zt
and At are set to 0.21 and 0.65, respectively, while the corresponding standard deviations
47
for the (cumulative) anticipated components are 0.32 and 0.2. This implies the majority
(60%) of variation in the growth rate of investment-specific technology is anticipated,
while just under 25% of variation in TFP is anticipated.
To develop a benchmark against which to compare the results from SP-learning, I
linearize the temporary equilibrium around the non-stochastic steady state. The resulting
system of first order expectational difference equations can be easily solved under rational
expectations, and the resulting equilibrium is the REE.
The Response to News
I turn now to an exploration of the response to news shocks by key macroeconomic
variables in the model. Figure 1 shows the responses by consumption, investment, hours
worked, and output when a fully rational household learns at time t = 0 that there will
be a 1 unit increase in the value of investment-specific or total factor productivity in time
t = 3, and the expected innovation arrives as expected. In both cases the model generates
positive comovement amongst all variables at the time the news is received: good news
about the future causes all aggregate variables to rise.
The total response to news about a shock to investment-specific technology zt is
more subdued than that regarding a shock to TFP At, because the price of investment
affects output only through its effect on capital accumulation while an increase in TFP
directly increases output and factor prices everything else held constant. Interestingly this
gives rise to a discrepancy in the relative impact of news on the overall response: most
of the movement in key variables stemming from news about zt occurs in the period the
news is received, while most of the movement from news about At occurs in the period
the shock occurs. Clearly including additional exogenous processes will lead to a richer set
48
FIGURE 1. IRF for Accurate News
of possible reactions to news of each; indeed Schmitt-Grohe and Uribe (2012) include five
additional exogenous disturbances and allow agents to receive information of their future
values, in which case one should no longer speak generically of “news”.
Now suppose the news received by households turns out to be completely false. That
is, at t = 0 households come to expect a 1 unit increase in the exogenous processes will
occur in t = 3 which does not materialize. Figure 2 plots the IRF resulting from this
thought experiment on rational households. Again, the receipt of news generates positive
comovement in the key macroeconomic variables. However, once the news is shown to
have been erroneously optimistic all variables tend back towards their initial steady
state values. Contrary to the prediction of Beaudry and Portier (2004), in this model
all variables remain above their steady-state levels for an extended period of time. Thus
even in the face of incorrect news the positive comovement amongst variables remains is
preserved.
49
FIGURE 2. IRF for Inaccurate News
Simulation and Model Performance
Having demonstrated the ability of the model to generate qualitatively realistic
expectationally driven business cycles in response to news about the future I now turn to
an evaluation of the model’s ability to generate quantitatively realistic empirical moments
under SP-learning. I begin by determining whether the behavior of SP-learners will cause
the economy to converge (approximately) to the REE. Figure 3 shows a simulation in
which the initial beliefs of agents are perturbations of the linearized RE solution, the
economy begins in steady state, and agents update their beliefs with a constant gain. For
each variable the solid red line is the RE steady-state value, while the dashed line is the
average of the actual variable realizations.
50
FIGURE 3. Model Simulation Under SP-learning
Since households employ a linear forecasting model in a nonlinear environment,
convergence to the rational expectations equilibrium is not to be expected. Indeed,
Figure 3 provides strong evidence that the economy converges to some RPE - the key
macroeconomic variables all fluctuate around some stationary value - but the implied
RPE seems quite far from the REE as judged by the respective central tendencies of the
variables.
Tables 4 and 5 perform tests for equality of means for key model variables from
simulating the model 1000 times each for 230 periods under varying assumptions regarding
households expectations formation mechanism and their access to news. Table 4 displays
the cross-sectional means of consumption, investment, hours worked, output, and the
endogenous shadow prices under RE and SP-learning when agents receive news. Generally
the cross-sectional means are higher under SP-learning than under RE, and the difference
51
RE Mean SPL Mean pval 95% Low 95% High
Consumption 0.533 0.568 0.000 -0.036 -0.032
Labor Supply 0.200 0.212 0.000 -0.013 -0.012
Investment 0.154 0.172 0.000 -0.019 -0.018
Output 0.687 0.736 0.000 -0.052 -0.046
Investment SP 0.000 0.060 0.000 -0.061 -0.059
Capital SP 4.250 4.043 0.000 0.193 0.219
Habit-adjustment SP -154.107 -155.120 0.000 0.704 1.321
TABLE 4. t-tests for Data Generating Process, News, 230 Periods
RE Mean SPL Mean pval 95% Low 95% High
Consumption 0.534 0.563 0.000 -0.031 -0.027
Labor Supply 0.200 0.210 0.000 -0.011 -0.010
Investment 0.154 0.169 0.000 -0.016 -0.014
Output 0.688 0.728 0.000 -0.043 -0.038
Investment SP -0.000 0.034 0.000 -0.035 -0.034
Capital SP 4.248 4.068 0.000 0.167 0.192
Habit-adjustment SP -154.085 -154.949 0.000 0.558 1.170
TABLE 5. t-tests for Data Generating Process, No News, 230 Periods
is highly statistically significant: the p-value for the t-test of mean equality across
expectations formation mechanisms imply rejection of the null hypothesis that the means
are equal at any reasonable level of significance. Put another way, I am able to reject
the null hypothesis that the data comes from the same data generating process, which
of course is true. Similar findings are found for the case where households do not receive
news in Table 5. Again the null hypothesis that the simulated data came from the same
data generating process can be rejected at any meaningful level of significance.
While the discussion above suggests the rejection of mean equality should not be
surprising, it is is worth noting each simulation is run for a relatively short amount of
time. In Tables 6 and 7 I increase the simulation length to 1000 periods in order to allow
the learning algorithm more time to converge. No significant changes arise: the RPE is
statistically quite distinct from the REE.
52
A deeper inspection of the exact mechanisms driving this wedge between the RPE
and REE may proceed from several points of observation. First, the data-generating
process under SP-learning is inherently nonlinear. This contrasts with the REE which
is obtained via a first-order approximation to the equilibrium dynamics of the model.
Exploring higher-order approximations may shed light on precisely how important the
nonlinearities are to the data-generating processes. Second, comparing tables 4 and 5
suggest that including anticipated shocks in agents’ information sets increases the size of
the wedge between the RPE and REE. While the cross-sectional mean values under REE
are the same whether news is included or not, those obtained under SP-learning are closer
to that of REE when there is no news. This suggest something about the news pushes
SP-learning household behavior away from that of their ration counterparts, which is
especially interesting given the results of the previous chapter showing news shocks should
not matter to the learnability of an REE.
Finally, it appears that SP-learning agents systematically behave in such a way as to
cause the shadow price of investment to be positive. This will occur if household behavior
is such that the gross level of investment is increasing (It > It−1) and capital is being
utilized at a rate above the rational expectations steady state (ut > 1), or if the opposite
cases are true. It is telling that this is exactly the behavior prescribed by the model in
the event that the household receives news about future productivity, and likely explains
why news drives a wedge between RE and NRE: under NRE households may go through
periods in which they respond too strongly (relative to RE) to news of the future.
In contrast to Williams (2003a), where reduced-form learning converged to the REE
and simulated data did not differ meaningfully whether agents were rational or boundedly
rational, these results suggest that the convergence to the RPE should cause simulated
business cycle statistics to differ from those obtained under RE. To explore this further,
53
RE Mean SPL Mean pval 95% Low 95% High
Consumption 0.533 0.635 0.000 -0.103 -0.101
Labor Supply 0.200 0.238 0.000 -0.038 -0.037
Investment 0.154 0.192 0.000 -0.039 -0.039
Output 0.687 0.824 0.000 -0.138 -0.136
Investment SP -0.000 0.117 0.000 -0.118 -0.116
Capital SP 4.251 3.620 0.000 0.625 0.638
Habit-adjustment SP -154.125 -162.404 0.000 8.017 8.542
TABLE 6. t-tests for Data Generating Process, News, 1000 Periods
RE Mean SPL Mean pval 95% Low 95% High
Consumption 0.533 0.600 0.000 -0.068 -0.066
Labor Supply 0.200 0.223 0.000 -0.024 -0.023
Investment 0.153 0.177 0.000 -0.023 -0.023
Output 0.686 0.774 0.000 -0.088 -0.087
Investment SP -0.000 0.038 0.000 -0.038 -0.038
Capital SP 4.251 3.797 0.000 0.449 0.460
Habit-adjustment SP -154.070 -158.385 0.000 4.056 4.573
TABLE 7. t-tests for Data Generating Process, No News, 1000 Periods
Figure 8 compares simulated data from the model with and without news under the
assumption of RE and SP-learning against that of actual quarterly U.S. data. The data is
from Jaimovich and Rebelo (2009) and covers the range 1944:Q1-2004:Q4. The logarithm
of the simulated data is detrended by the Hodrick-Prescott (HP) filter using a smoothing
parameter of 1600, consistent with quarterly data. I conducted 1000 simulations of 230
periods each, which is the length of the US data sample. I report the statistics for the
cross-sectional average standard deviations of hours worked, investment, and consumption
relative to that of output across all simulations along with the correlations between output
and hours worked, investment, and consumption.
US data suggests investment is the most volatile, while hours worked tends to be
roughly as volatile as output and consumption is much smoother. Significant differences
exist in the volatility of simulated data between RE and SP-learning, whether news
54
US Data RE (News) SPL (News) RE (No News) SPL (No News)
σh/σY 0.968 0.714 0.716 0.714 0.715
σI/σY 3.103 2.386 2.167 2.282 2.023
σC/σY 0.712 0.737 0.706 0.746 0.725
corr(Y, h) 0.860 1.000 1.000 1.000 1.000
corr(Y, I) 0.890 0.850 0.879 0.922 0.912
corr(Y,C) 0.770 0.969 0.969 0.977 0.986
TABLE 8. Predicted Business Cycle Statistics
shocks are included or not. All model simulations except SP-learning with news shocks
suggest hours worked are much too smooth, and relative volatility of consumption under
SP-learning is extremely close to the data. Only the specification with SP-learning and
news shocks matches the stylized facts pertaining to the ordering of relative volatilities
of investment, hours worked, and consumption. The simulated correlation between hours
worked and output is unit, which does not match the data, but is expected: output and
hours worked are perfectly correlated by assumption in the RBC model. The correlation
between output and investment under SP-learning is closer to that of the data than under
RE regardless of whether news is included or not, while both expectations formations
mechanisms do a poor job capturing the correlation between output and consumption.
Conclusion
The analysis of this paper has shown the behavior implied by SP-learning households
is quite distinct from that of their rational counterparts. While agents in this news-shock
model can not converge to the exact REE, the process of making decisions based on linear
forecasting rules of shadow prices and updating these forecasts via adaptive least squares
leads agents to converge to an RPE. The difference between the implied RPE and the
REE seem to be exacerbated by the inclusion of anticipated shocks in agents’ information
sets. Furthermore, this behavior causes the model to generate simulated moments for key
55
business cycle statistics which are quite distinct from those obtained under RE. In some
cases the model under SP-learning appears to better approximate US data, while in others
the RE approach is more accurate.
One interesting extension would be to consider informational heterogeneity in this
environment. Angeletos and La’O (2010) considers the qualitative and quantitative effects
of informational heterogeneity in a Lucas-Phelps “island” model where households on a
continuum of islands receive public and private signals about the productivity of aggregate
and island-specific productivity. They are somewhat surprised to find that information
heterogeneity does not have any serious effect on welfare or quantitative features of the
model. But Hellwig (2010) observes this goes back to an insight going back to Hayek
(1945) suggesting that markets parsimoniously convey all relevant information through
prices, and thus heterogeneity by itself is not likely to represent a significant source
of amplification or persistence in simulated data. However, SP-learning introduces a
behavioral friction in the way agents utilize information which is likely to be exacerbated
by informational frictions stemming from heterogeneity. While it is purely speculative, I
suspect SP-learning (and potentially other forms of bounded rationality) may provide a
simple means of making informational heterogeneity matter in DSGE models.
56
CHAPTER IV
LEARNING VS NEWS: WHAT DRIVES BUSINESS CYCLES?
Introduction
Having established in Chapter II that news shocks do not necessarily impinge
upon the E-stability properties of a REE, and having conducted a calibrate and simulate
exercise in Chapter III to understand the interaction between information flows, bounded
rationality, and restricted perceptions equilibrium, I turn now to an empirical exploration
of the relationship between news shocks and expectation formation. The central goal
of this chapter is to determine whether the estimated importance of news - measured
as the relative contribution of anticipated shocks versus surprise shocks in generating
macroeconomic volatility - is affected by the way in which agents are assumed to form
their expectations.
In what is widely regarded to be the seminal work in the modern empirical news
shock literature, Beaudry and Portier (2006) explores the relationship between stock
prices, the growth rate of technology, and business cycles using several specifications
of VARs. They find evidence that innovations to total factor productivity (TFP) are
often anticipated, and further that responses to this anticipation are responsible for a
large fraction of business cycle fluctuations. This contrasts with the typical RBC view
of technological innovation by surprise which had begun to be questioned as early as
Basu et al. (2004) in which improvements to technology are shown to typically have no
contemporaneous impact on output.
57
Beaudry and Lucke (2010) allows news shocks to compete with more traditional
sources of volatility in a VAR identified using short-run and long-run timing assumptions
and find that surprise shocks play little role in economic volatility compared to news
shocks. Barsky and Sims (2012) allow traditional news shocks to compete with “animal
spirit shocks” a la Lorenzoni (2009). The authors conclude that much of the innovation
in measured consumer confidence is due to traditional news shocks and that most of the
relation between changes in consumer confidence and economic activity are due to news.
In Beaudry et al. (2011) the focus is on what the authors refer to as “optimism shocks”
which are essentially sunspots that exogenously shift expectations. These shocks are
identified using sign restrictions and are shown to permanently affect current and future
economic activity in a way strongly resembling news shocks. The narrative approaches of
Alexopoulos (2011) and Ramey (2011) have similarly found support for the notion that
news is an important determinant of volatility in aggregate economic variables.
Schmitt-Grohe and Uribe (2012) modifies the well-known RBC-type news-shock
model of Jaimovich and Rebelo (2009) to include seven exogenous variables which are
subjected to both anticipated and unanticipated shocks. Using likelihood-based classical
and Bayesian estimation techniques they find anticipated shocks account for about half of
observed volatility in output and consumption. Conversely, Khan and Tsoukalas (2012)
allow news shocks to compete against traditional sources of macroeconomic volatility in a
small-scale monetary RBC model similar to that considered in Smets and Wouters (2007).
Their results imply anticipated shocks are responsible for only 15% of volatility in output.
These strongly contrasting results highlight the preeminent role of model specification in
DSGE estimation.
While much of the empirical news shock literature has approached the topic from
a reduced form or structural VAR perspective, Blanchard et al. (2013) and Beaudry and
58
Portier (2014) point out estimation via VARs in news-rich environments may suffer from a
nonfundamentalness problem. Alternatively, one may estimate the relative importance of
anticipated shocks in a news-rich dynamic stochastic general equilibrium (DSGE) model.
However, the construction of such a model requires the modeler to make a number of
assumptions. One of the most important is the way in which agents’ expectations are
formed, and all previous authors on the subject have utilized the rational expectations
hypothesis.
While much early work on adaptive learning focuses on whether particular rational
expectations equilibrium (REE) are expectationally stable (E-stable), Williams (2003a) and
Adam (2005) each explores the impact of adaptive learning on macroeconomic volatility
and persistence in calibrated and simulated DSGE models.1 Their results suggest that
relaxing rational expectations (RE) in favor of an adaptive learning approach not only
increases persistence and volatility in key macroeconomic variables, but may also provide a
better fit to the data than models based on RE.
More recently, adaptive learning has been used in estimated DSGE models. Milani
(2007) estimates a monetary-NK DSGE model in which agents are assumed to have
what he refers to as “near-rational expectations” (NRE), that is, they form expectations
using a correctly specified forecasting model but are unsure of the precise coefficients
governing the laws of motion of the economy.2 In stark contrast to the results obtained
under RE, the model under NRE simultaneously fits the data better and ascribes little
importance to mechanical sources of volatility and persistence such as habit formation
1A REE is said to be E-stable if the beliefs of boundedly rational agents converge (asymptotically)
to those of fully rational agents. Besides reflecting the sensitivity of models to the specific assumptions
placed on agents’ expectation formation mechanisms and serving as an asymptotic justification for the use
of RE, E-stability has been proposed as a selection criterion for indeterminate models.
2This is also referred to as “Euler equation learning” because agents are learning about endogenous
variables which appear in the Euler equations of otherwise rational agents.
59
and inflation indexation. Similar results obtain in e.g. Milani and Rajbhandari (2012)
and Slobodyan and Wouters (2012) in various DSGE models and under various forms of
bounded rationality.
Relatedly, Eusepi and Preston (2011) show that periods of optimism and pessimism
arising from the erroneous forecasts (and subsequent updating of beliefs) of boundedly
rational agents in an otherwise standard stochastic growth model generate the same
type of expectationally driven business cycles as those found in news-shock models. This
observation presents a natural question: to what extent are estimates of the importance
of anticipated shocks impacted by the specific assumptions governing agents’ expectation
formation mechanism? Do news shocks serve as proxies for an underlying learning process
in the data? Or could it be that the combination of news shocks and adaptive learning
could actually enhance the relative importance of anticipated shocks by exacerbating the
waves of optimism and pessimism?
My main results from this chapter suggest the importance of news is robust to the
inclusion of adaptive learning. In particular, the relative importance of news is estimated
to be up to thirty-five percent greater for key macroeconomic variables such as output,
consumption, investment, and hours worked under NRE than under RE. However, these
differences arise from movement in endogenous variables caused by the arrival of the
anticipated event as opposed to movement caused by the anticipation itself, what Sims
(2016) refers to as the impact of “realized news” and “pure news”, respectively.
These findings speak to current differences in structural estimates of the importance
of news, which appear to be sensitive to the structure of the model considered. Since
parameter estimates obtained from DSGE models under learning often attenuate the
importance of specific mechanical sources of volatility and persistence relative to RE, this
60
study will clarify whether the importance of news is also sensitive to specific assumptions
governing expectation formations.
Furthermore, it provides additional evidence regarding the merits of using adaptive
learning in empirical applications. Despite a preponderance of evidence suggesting its
implausibility as a description of individual or aggregate behavior, RE is still the paradigm
for describing expectations in dynamic macroeconomic models.3 It is noteworthy that
while the present study suggests the model under learning and RE produce similar
model predictions for volatility and persistence, the models produce dissimilar estimates
regarding the relative importance of news, implying forecasts generated from the
underlying models may be quite different.
The chapter proceeds as follows. I first describe the news-shock RBC-type model
which is to be estimated. Next, I describe the specific assumptions made for agent-
level expectation formation. This is followed by a detailed description of the estimation
methodology and the presentation of the main results for the chapter. The chapter
concludes with a discussion of the main results which attempts to place them in the
context of recent empirical macroeconomic literature.
The Model
The economy considered is that of Schmitt-Grohe and Uribe (2012) which itself
augments the news-shock RBC model of Jaimovich and Rebelo (2009) considered in
the previous chapter. Aggregate demand consists of a government which levies lump-
sum taxes to finance expenditure and a representative household making the usual
3See e.g. Assenza et al. (2014) for a summary of the state of learning-to-forecast and learning-to-
optimize laboratory experiments, which have largely failed to find support for RE as a description of
aggregate behavior.
61
consumption/savings and labor supply decisions while also choosing the rate at which
to utilize their existing capital stock. Adjustments to gross investment are assumed to be
costly. In addition, household preferences incorporate direct and indirect internal habit-
formation.
Aggregate supply consists of competitive firms which produce a fungible final good
from labor supplied through monopolistically-competitive labor unions and physical
capital rented from the household. The model features seven exogenous variables which
have been shown in the literature to be empirically important mechanical sources of
volatility and persistence. Finally, each of the exogenous variables are subjected to
disturbances in the form of anticipated and unanticipated exogenous stochastic white-noise
shocks, that is, news and surprise shocks.
Households
The representative household maximizes lifetime utility which is given by
E0
∞∑
t=0
βtζtU(Vt) (4.1)
where β ∈ (0, 1) the household’s discount factor, ζt a stationary exogenous stochastic time-
preference shock, and U the period utility function which takes the CRRA form
U(Vt) =
V 1−σt − 1
1− σ (4.2)
62
where σ > 0 the inverse intertemporal elasticity of substitution. The argument of the
utility function is given by
Vt = Ct − bCt−1 − ψhθtSt (4.3)
where Ct and ht are private consumption and hours worked, respectively. b ∈ [0, 1)
controls the degree of internal habit formation, ψ > 0 scales the disutility of labor supply,
and θ > 1 governs the Frisch elasticity of labor supply.4 St is a geometric average of
current and past habit-adjusted consumption and evolves according to
St = (Ct − bCt−1)γ S1−γt−1 (4.4)
where γ ∈ (0, 1] governs the magnitude of the wealth elasticity of labor supply.
This preference specification was first used in Jaimovich and Rebelo (2009). The
comovement problem in RBC models is partially caused by the wealth effect of labor
supply dominating the substitution effect, thereby causing consumption and hours worked
to move in opposite directions upon receipt of good news about the future. The JR
specification for utility allows the modeler to change the strength of the wealth effect of
labor supply by calibrating γ.
Households are assumed to own and rent physical capital to firms in a competitive
market. The capital stock accumulates according to
Kt = (1− δ (ut))Kt−1 + zIt It
[
1− Φ
(
It
It−1
)]
(4.5)
4In the special case where there is no internal or external habit formation i.e. b = γ = 0, the Frisch
labor supply is ηλ = 1θ−1 , hence the restriction that θ > 1
63
where Kt−1 the predetermined capital stock determined in period t − 1, It gross
private investment, and zIt a stationary exogenous stochastic shock to the technology for
converting investment into installed capital.
The function Φ (·) imposes an investment-adjustment cost on households while δ (·)
implies capital depreciation is a function of the current-period capital utilization rate ut.
Both functions are increasing, convex and take quadratic functional forms. In particular
we have
δ(ut) = δ0 + δ1(ut − 1) + δ2
2
(ut − 1)2 (4.6)
where δ0 > 0 the steady-state depreciation rate, δ1 > 0 determines the steady-state value
of ut, and δ2 > 0 captures the rental rate elasticity of capacity utilization. The time-
varying investment-adjustment cost function is given by
Φ
(
µIt
)
=
κ
2
(
µIt − µI
)2
(4.7)
consistent with Christiano et al. (2005), where µIt ≡ ItIt−1 the growth rate of investment
with steady state µI and κ > 0 a scaling parameter which implies the growth rates of
investment and output are autocorrelated. This specification implies Φ
(
µI
)
= Φ′
(
µI
)
= 0,
i.e. there are no investment-adjustment costs on a balanced growth path, and κ = Φ′′
(
µI
)
.
The household receives labor income from supplying ht hours at rate W
∗
t , rental
income from from renting utKt−1 units of effective capital at gross rental rate Rt, and
lump-sum dividends from labor-union membership and firm-profits of Πt. The household
uses this income to pay lump-sum taxes Tt and purchase private consumption and gross
64
private investment. The flow budget constraint is thus given by
Ct + AtIt + Tt ≤ W ∗t ht +Rt (utKt−1) + Πt (4.8)
where At a difference-stationary exogenous stochastic process representing the current
state of technology for producing investment goods from consumption goods with growth
rate µAt ≡ AtAt−1 . In a decentralized equilibrium At may be interpreted as the relative price
of investment goods in terms of consumption goods; that is, a unit of the investment good
may be exchanged for At units of the consumption good.
Labor Market
Households are assumed to have market power, albeit in an indirect way. There is
a continuum of monopolistically competitive labor unions which households supply labor
to, and these unions negotiate wages with final goods producers. The market power of
these unions is captured by a stationary exogenous stochastic process ωt, which can be
interpreted simply as a time-varying markup in wages. The solutions to the final-good
producers’ cost-minimization and unions’ profit-maximization problems together imply
all unions charge the same price for labor services and pay the same wage to member-
households, and hence the wage received by households is given by W ∗t = Wt/ωt where
Wt the price charged by unions to firms for labor services and µ
W
t ≥ 1 the exogenously
time-varying wage markup which controls the level of market power the labor unions
have with steady-state value µW . This friction drives a wedge between the wage paid by
firms and that received by workers. Profits of the labor unions are remitted to all member
households as a lump-sum dividend. Details of the solution to the optimization problems
of firms and unions are presented in Appendix C.
65
Firms, Government, and Market Clearing
The representative firm contracts ht worker-hours and rents utKt−1 units of effective
capital to produce a final good Y outt using CRTS technology according to
Y outt = zt (utKt−1)
αk (Xtht)
αh (XtF )
1−αk−αh (4.9)
where the parameters αk, αh ∈ (0, 1) govern the steady-state output shares of effective
capital and effective labor, respectively, and satisfy αk + αh ≤ 1. F is a fixed factor
(e.g land or managerial capital) which introduces production exhibits diminishing returns
to scale in effective capital and labor while ensuring production exhibits overall CRTS.
Production is augmented by a stationary exogenous stochastic Hicks-neutral productivity
shock zt and a difference-stationary exogenous stochastic Harrod-neutral (i.e. labor-
augmenting) productivity shock Xt with growth rate µ
X
t =
Xt
Xt−1
and steady-state growth
µX .
The market for effective capital is competitive and hence the gross rental rate equals
the value of the marginal product of effective capital
Rt = αk
Y outt
utKt−1
(4.10)
and the wage paid by firms to labor unions is similarly equal to the value of the marginal
product of labor
Wt = αh
Y outt
ht
(4.11)
66
The final good is fungible and may be used for private consumption, private investment, or
government spending Gt, and hence the aggregate resource constraint is
Y outt = Ct + AtIt +Gt (4.12)
where Y outt is total output. Because At and Xt are non-stationary, output displays a
stochastic trend which can be shown to be XYt = XtA
αk/(αk−1)
t . The government maintains
a balanced budget such that Gt = Tt. Furthermore, government spending exhibits a
stochastic trend XGt =
(
XGt−1
)ρxg (
XYt−1
)1−ρxg
which co-integrated with the trend in output
so that the share of government spending in output is stationary; however ρxg ∈ [0, 1)
allows the trend to be smoother than that of output. Detrended government spending
gt ≡ Gt/XGt is a stationary exogenous stochastic process with steady state g. Since XGt is
predetermined, and since detrended government spending is stationary, the current level of
government spending is independent of contemporaneous changes to exogenous variables;
however, these may affect government spending with a lag via corresponding changes to
the trend path of output.
Temporary Equilibrium
The temporary equilibrium for this economy obtains where all households maximize
utility given expectations, all firms maximize profits, and all markets clear. This can be
67
described the nonlinear system of expectational difference equations
Kt = (1− δ (ut))Kt−1 + zIt It
[
1− Φ
(
It
It−1
)]
(4.13a)
Y outt = Ct + AtIt +Gt (4.13b)
Y outt = zt (utKt−1)
αk (Xtht)
αh (XtF )
1−αk−αh (4.13c)
St = (Ct − bCt−1)γ S1−γt−1 (4.13d)
Vt = Ct − bCt−1 − ψhθtSt (4.13e)
Λt =
(
ζtV
−σ
t −
γSt
Ct − bCt−1 Πt
)
− bβEˆt
[(
ζt+1V
−σ
t+1 −
γSt+1
Ct+1 − bCtΠt+1
)]
(4.13f)
Λt =
θζtV
−σ
t ψh
θ−1
t St
Wt/ωt
(4.13g)
Πt = ζtV
−σ
t ψh
θ
t + βEˆt
[
(1− γ)St+1
St
Πt+1
]
(4.13h)
QtΛt = βEˆtΛt+1 [(Rt+1ut+1 +Qt+1(1− δ(ut+1)))] (4.13i)
Rt = δ
′(ut)Qt (4.13j)
AtΛt = QtΛtz
I
t
(
1− Φ
(
It
It−1
)
− Φ′
(
It
It−1
)(
It
It−1
))
(4.13k)
+ βEˆt
[
Qt+1Λt+1z
I
t+1Φ
′
(
It+1
It
)(
It+1
It
)2]
(4.13l)
Wt = αh
Y outt
ht
(4.13m)
Rt = αh
Y outt
utKt−1
(4.13n)
where Λt the marginal value of income, Πt the marginal value of household’s stock of
current and past habit-adjusted consumption (i.e. the Lagrange multiplier from relaxing
the constraint on St), and ΛtQt the marginal value of pre-installed capital. Qt can thus be
interpreted as marginal Tobin’s Q. Eˆt denotes the (possibly non-rational) expectations of
the household given their time t beliefs and information set. The temporary equilibrium
68
can be concisely described by
Eˆt [f (Yt+1, Yt, Yt−1, νt)] = 0 (4.14)
where Yt and νt are collections of endogenous and exogenous variables and exogenous
stochastic white-noise shocks, respectively.
Stationarity, Information Flows, and the Log-linearized System
Because At and Xt are non-stationary the system (4.14) must be transformed to
be in terms of stationary variables. The details of this transformation and the resulting
stationary nonlinear system of expectational difference are presented in Appendix C.
Letting Yˆt be a collection of (detrended) endogenous and exogenous variables, the resulting
stationary system can be concisely describe as
Eˆt
[
f
(
Yˆt+1, Yˆt, Yˆt−1, νt
)]
= 0 (4.15)
Yˆt includes the seven stationary exogenous variables x =
{
z, ζ, zI , g, ω, µA, µX
}
which are
assumed to follow AR(1) processes such that
lnxt = ρx lnxt−1 + 0x,t
where ρx ∈ (0, 1). In addition to an unanticipated or “surprise” shock, households receive
information about each of the exogenous variables 4 and 8 periods in advance. These
news shocks can be parsimoniously modeled by introducing auxiliary variables εnx,t for
69
n = 0, 1, ..., q where q the longest forecasting horizon and
ε0x,t = ε
1
x,t−1 + σ
0
xν
0
x,t
ε1x,t = ε
2
x,t−1
ε2x,t = ε
3
x,t−1
ε3x,t = ε
4
x,t−1
ε4x,t = ε
5
x,t−1 + σ
4
xν
4
x,t
ε5x,t = ε
6
x,t−1
ε6x,t = ε
7
x,t−1
ε7x,t = ε
8
x,t−1
ε8x,t = σ
8
xν
8
x,t
where νkx,t for k = 0, 4, 8 are assumed to be Gaussian white-noise shocks with unit variance
and effective standard deviation σkx. Repeated substitution of the auxiliary variables
implies ε0x,t = σ
0
xν
0
x,t + σ
4
xν
4
x,t−4 + σ
8
xν
8
x,t−8, and hence household expectations are given
by
Eˆtε
0
x,t+n =

0 if n > 8
σ8xν
8
x,t+n−8 if 8 ≥ n > 4
σ4xν
4
x,t+n−4 + σ
8
xν
8
x,t+n−8 if 4 ≥ n > 1
i.e. household’s expectations of future exogenous variables are constantly updated as news
shocks flow into their information sets.
70
While the anticipated components do not affect contemporaneous economic
fundamentals, the information reveals something about the future state of the economy.
Households incorporate this information into their decision rules, and it is precisely
the fact that news shocks affect various endogenous variables in different ways at
different times which allows the econometrician to identify the effect of a particular news
shock.5 Note that the inclusion of such news shocks implies no departure from rational
expectations: The anticipated shocks are fundamental to the model, and hence rational
households must incorporate them into their forecasts of the future.
The system describing the informational flow structure can be represented by
collecting auxiliary variables into vectors εxt =
(
ε0x,t, ε
1
x,t, ..., ε
8
x,t
)′
, the Gaussian white-noise
shocks into vectors νxt =
(
ν0x,t, ν
1
x,t, ..., ν
8
x,t
)′
, and writing
wt = ϕwt−1 +Mνt (4.16)
where wt =
(
εzt , ε
ζ
t , ε
zI
t , ε
g
t , ε
ω
t , ε
µA
t , ε
µX
t
)′
and νt =
(
νzt , ν
ζ
t , ν
zI
t , ν
g
t , ν
ω
t , ν
µA
t , ν
µX
t
)′
. Given the
assumed structure of information flows, ϕ is an upper-shift matrix with 1’s on the super-
diagonal and zeros elsewhere while M is simply a sparse matrix containing the effective
standard deviations of the anticipated and unanticipated shocks.6
Following the majority of applied structural macroeconomic literature, I log-linearize
the stationary system (4.15) around its non-stochastic steady state. Denote by y˜t the
collection of endogenous and exogenous variables in terms of their percent-deviation from
5See Schmitt-Grohe and Uribe (2012) or Beaudry and Portier (2014) for more on the issue of
identification in news-shock DSGE models.
6In what follows I assume the exogenous stochastic shocks are pure white-noise and hence are
uncorrelated across forecasting horizons and across exogenous variables. It would be simply to modify
these assumptions, however, and would amount to appropriately changing ϕ and M to capture the new
relationships.
71
non-stochastic steady state. Then the log-linearized system of equations, including those
defining the information flow structure, can be written compactly as
y˜t = α + βy˜t−1 + χwt + δEˆty˜t+1 (4.17a)
wt = ϕwt−1 +Mνt (4.17b)
A variety of well-known solution techniques exist for obtaining the solution to a linear
system of first-order expectational difference equations like that defined by equations
(4.17a) and (4.17b) under the assumption of rational expectations e.g. Blanchard and
Kahn (1980), Uhlig et al. (1995), Klein (2000), or Sims (2002). However, the objective of
this chapter is to compare and contrast the estimated importance of anticipated shocks
when agents are assumed to be rational versus when the rational expectations hypothesis
is slightly relaxed. Thus, the next section describes the assumed expectation formation
mechanism and adaptive learning process for the boundedly rational agents.
Adaptive Learning
As is clear from the temporary equilibrium described by (4.17a), household actions
are conditional on their expectations of the future state of the economy. I depart from RE
by assuming households behave as econometricians equipped with a forecasting model.
Their beliefs, referred to as the perceived law of motion (PLM), take the form of the
minimum-state variable (MSV) solution obtained under rational expectations; that is,
agents form forecasts according to
y˜t = at−1 + bt−1y˜t−1 + ct−1wt
wt = ϕwt−1 +Mνt
(4.18)
72
This model is “correct” in the sense that its structure is consistent with the REE; there
are no omitted or included irrelevant explanatory variables.7 However, the exact values
may be different from those implied by the REE. The model is closed by assuming
households update their beliefs over time according to the constant-gain least squares
(CGLS) algorithm. Denoting household beliefs as ξ′t = (at, bt, ct) we have
ξt = ξt−1 + g¯Rˆ−1t Zt−1
(
y˜t − ξ′t−1Zt−1
)′
(4.19)
Rˆt = Rˆt−1 + g¯
(
Zt−1Z ′t−1 − Rˆt−1
)
(4.20)
where the data used to forecast is captured by Z ′t−1 =
(
1, y˜′t−1, w
′
t
)
and g¯ the “constant-
gain” parameter which describes the relative importance of recent forecast errors. The
system (4.19)-(4.20) defines a household’s time t estimate of the coefficients ξt and the
matrix of second-moments Rˆt as a weighted average of their previous estimates and the
new information contained in the forecast error t = y˜t − ξ′t−1Zt−1. Note that setting g¯ =
t−1 would result in agents using the well-known recursive least squares (RLS) algorithm
for updating beliefs; furthermore, if the REE is expectationally stable (E-stable) then it
obtains as a special limiting case where t→∞ and g¯→ 0.
7I am implicitly assuming household’s understand the informational flow structure; that is, they
understand which news is relevant for which exogenous variables and that they receive news 0, 4, and 8
periods in advance. This is an intuitively appealing assumption, and also a weak one, given that the lack
of a feedback-loop implies agents would very quickly learn the true laws of motion governing the evolution
of this exogenous system.
73
Given the PLM described by equation (4.18) and assuming agents do not observe the
contemporaneous value of endogenous variables i.e. Eˆty˜t 6= y˜t8, expectations are given by
Eˆty˜t = at−1 + bt−1y˜t−1 + ct−1wt (4.21)
Eˆty˜t+1 = at−1 + bt−1Eˆty˜t + ct−1ϕwt
= [(I + bt−1) at−1] + b2t−1y˜t−1 + [bt−1ct−1 + ct−1ϕ]wt (4.22)
which can be substituted into the system (4.17a) to yield the actual law of motion (ALM)
for the economy
y˜t = [α + δ (I + bt−1) at−1] +
[
β + δb2t−1
]
y˜t−1 + [χ+ δ (bt−1ct−1 + ct−1ϕ)]wt
wt = ϕwt−1 +Mνt
(4.23)
Equation (4.23) emphasizes the feedback mechanism implied by adaptive learning:
household perceptions of the economy - captured by at−1, bt−1, and ct−1 - directly impact
the actual state of the economy through their influence over household decision making.
Collecting the endogenous, exogenous, and auxiliary variables into a vector St = (y˜t, wt)′,
the ALM for the economy can be written
St = At + FtSt−1 +Gtνt (4.24)
where νt
i.i.d∼ N (0, I) and At, Ft, and Gt are time-varying coefficient matrices formed from
deep structural parameters of the economy and household beliefs. The central goal of this
chapter is to compare estimates of the deep parameters obtained under RE against those
8This distinction does not matter under RE as all agents are endowed with model-consistent
expectations, and at first blush may appear strange in the context of a representative agent model.
However, it is a natural assumption to make in general equilibrium settings where individual agents must
forecast the aggregate behavior of all other agents to determine their individual behavior.
74
obtained under near-rational expectations, specifically those corresponding to the effective
standard deviation of the anticipated and unanticipated shocks.
Estimation Methodology
Following the recent literature in estimating DSGE models with learning e.g. Milani
(2007), Slobodyan and Wouters (2012), and Milani and Rajbhandari (2012), I choose to
estimate the parameters of (4.27) using Bayesian Markov Chain Monte Carlo (MCMC)
methods. In the present context there are three main advantages to Bayesian versus
classical maximum likelihood estimation.
First, as described in An and Schorfheide (2007), maximum likelihood procedures
do not allow an econometrician to avoid estimates which are known to be unlikely
given previously acquired institutional knowledge. Bayesian estimation allows for
the augmentation of the data through a joint prior distribution which re-weights the
likelihood. In this way the econometrician can easily and transparently allow information
which is not contained in the data to have an impact on the estimated results.
Second, Bayesian methods result in an entire posterior distribution, as opposed to
point estimates obtained under maximum likelihood. This emphasizes the fact that our
models merely approximate the economy by allowing the econometrician to report results
in terms of probabilities rather than a single “true” parameter value. This facilitates an
honest communication of results in line with the norms in a variety of disciplines ranging
from political science to meteorology.
Finally, the likelihood functions associated with DSGE models can present major
problems for optimization routines. Because these models are approximations of a data
75
generating process, the corresponding likelihood functions often feature many peaks
and cliffs. MCMC approaches such as the Random-Walk Metropolis Hastings (RWMH)
algorithm are robust to these topographic challenges as their success derives precisely
because they explore the entire parameter space, thereby eliminating the well-known
problem of Newtonian optimizers getting stuck in inappropriate spaces due to initial
starting values.
The object of interest is the posterior distribution p(Θ|Y T ) which describes the
probability of a particular parameter constellation Θ given data Y T . The posterior
distribution is obtained through Bayes’ Law as
p(Θ|Y T ) ∝ p(Y T |Θ)p(Θ) (4.25)
where the likelihood function p(Y T |Θ) captures the probability of observing the data in
Y T given Θ and the prior distribution p(Θ) summarizes the econometrician’s a priori
knowledge of the parameters Θ. In general the posterior distribution does not take
any known form, and hence I utilize the RWMH algorithm to sample from the target
distribution.
Because the resulting sequence of random samples is a Markov chain, the proposal
distribution should be chosen to ensure proposals are accepted neither too often nor
too infrequently. If samples are accepted too frequently then the sequence will be highly
serially correlated; if samples are accepted too infrequently than the algorithm will fail to
explore the entirety of parameter space. But choosing a proposal distribution to yield an
acceptance rate which satisfies this criteria can be extremely difficult, especially in high-
dimensional settings, because the various elements may behave in strikingly different ways.
76
To overcome this challenge I utilize the Log Adaptive Metropolis (LAP) procedure
of Shaby and Wells (2010). In addition to recursively updating the variance-covariance
matrix of the proposal distribution vis-a-vis the adaptive Metropolis algorithm of Haario
et al. (2001), LAP multiplicatively updates the tuning parameter proportionally to
the deviation of the actual acceptance rate from its target. This results in a proposal
distribution which quickly establishes an acceptance rate in close proximity to the desired
one, while also respecting the relationships between estimated parameters revealed by the
history of draws.
The Observables
To estimate the DSGE model given by (4.24) I must specify a link between the
unobserved state-variables of the model and their observed empirical counterparts. I
utilize quarterly U.S. data provided by Schmitt-Grohe and Uribe (2012) on the demeaned
growth rates of real per-capita output, consumption, investment, government expenditure,
and hours worked along with the growth rates of total factor productivity (TFP) and
the relative price of investment denoted gYt , g
C
t , g
I
t , g
G
t , g
h
t , g
TFP
t , and g
PI
t , respectively.
I use this data to facilitate comparisons between my results and previous estimates of the
relative importance of anticipated shocks.
The data runs from 1955:Q2 to 2006:Q4 which corresponds to 207 observations.9
The growth rate of real per-capita output is assumed to be measured with error, which is
required by the fact that the RBC model implies a linear restriction between these seven
9Because the model does not feature structural change I am implicitly assuming the nature of “news”
has remained the same over the entire period. This may not be an entirely satisfying assumption,
and future work will examine the impact of splitting the sample a la Beaudry et al. (2011) in which
optimism shocks are found to be much more important in their post-1983 subsample than in their pre-
1983 subsample.
77
observables which is not satisfied by the data.10 Formally, the observable variables are
linked to the non-stationary variables from (4.13a)-(4.13n) by

gYt
gCt
gIt
ggt
ght
gTFPt
gPIt

=

∆ log (Yt)
∆ log (Ct)
∆ log (AtIt)
∆ log (Gt)
∆ log (ht)
∆ log
(
ztX
1−αk
t
)
∆ log (At)

+

σYme 0 0 0 0 0 0
0 0 0 0 0 0 0
0 0 0 0 0 0 0
0 0 0 0 0 0 0
0 0 0 0 0 0 0
0 0 0 0 0 0 0
0 0 0 0 0 0 0


eYt
eCt
eIt
egt
eht
eTFPt
ePIt

or concisely as
Yt = HSt +Det (4.26)
where et
i.i.d∼ N (0, I) a vector of white-noise measurement error shocks and σYme the
effective standard deviation of measurement error in output growth. H is a selection
matrix of which defines the observed variables in terms of the unobserved states. The
system of transition equations (4.24) together with the system of measurement equations
(4.26) imply estimation can be conducted by analyzing the (potentially time-varying)
state-space model
St = At + FtSt−1 +Gtνt
Yt = HSt +Det
(4.27)
10This is the well-known stochastic singularity problem in estimating linearized DSGE models is called.
See e.g. Ruge-Murcia (2007) for more on this and its impact on estimation.
78
which is a linear-Gaussian system in the econometric-state variables St and variables with
directly observed counterparts Yt.
The Likelihood Function
Given the state-space model in (4.27), the value of the likelihood function evaluated
at a given parameter constellation Θ can be easily calculated using the Kalman Filter.
The introduction of adaptive learning requires the coefficient-matrices of the transition
equations to be updated using estimates of the state variables based on the CGLS
algorithm. I follow Slobodyan and Wouters (2008) and use the simple filtered estimates
for the states rather than the smoothed estimates.
The additional updating step presents two potential computational difficulties. First,
agents updated estimates may imply the system is non-stationary; thus I incorporate a
projection facility as in Marcet and Sargent (1989a) which causes agents to skip their
updating step when they realize their new estimates are non-nonsensical. Second, MSV
learning in linearized-DSGE models can produce estimated second-moment matrices with
very small minimum eigenvalues due to perfect-multicollinearity from the models’ implied
linear restrictions; hence, I incorporate a ridge correction mechanism as in Slobodyan and
Wouters (2012) which conditionally adds an arbitrarily small constant to the diagonal of
the estimated second-moment matrix.11
Provided initial values for the states S0|0, the estimated mean square error (MSE)
matrix P0|0, beliefs for the second moment matrix Rˆ0, and beliefs for the coefficient
matrices x0 = (a0, b0, c0), the Kalman Filtering routine proceeds by repeating 5 steps
11I set the value of this small ridge to be 1e− 6 times a conformable identity matrix.
79
1. Calculate the a priori estimates of the time t states St|t−1 and MSE matrix Pt|t−1
2. Use these estimates to produce a priori estimates of the time t observables Yt|t−1
and corresponding variance-covariance matrix Ωt|t−1
3. Calculate the a posteriori time t value of the (log) likelihood function
4. Update the estimated coefficient matrices At, Ft, and Gt implied by agents updating
their beliefs Rˆt and ξt according to the CGLS algorithm
5. Calculate the a posteriori estimates of the time t states St|t and MSE matrix Pt|t
for t = 1, 2, ...T . The projection facility and ridge-correction mechanism are conditionally
applied during step 4, which does not exist under RE. Initial values for the states and
estimated MSE matrix are set to the unconditional mean and variance-covariance matrix
of the states, respectively, while initial beliefs are assumed to be equal to the REE. Since
the REE is E-stable, this implies agents’ estimates will fluctuate around the REE, thereby
facilitating a simple comparison between the estimated posterior densities.
The Prior
The parameter vector Θ contains all of the parameters of the model, some of which
are calibrated. In particular, the discount factor β is set to 0.99 implying a steady-state
quarterly real-interest rate of 1% if δ1 = δ2 = 0. σ is set to 1 to yield logarithmic utility
while αk and αh are set to 0.225 and 0.675, respectively, implying mild decreasing returns
to scale. δ0 is the steady-state depreciation rate and is set to 2.5 percent. The fixed factor
F and steady-state values for the Hicks-neutral technology shock z, preference shock ζ,
and investment-specific technology shock zI are normalized to 1. The steady-state value
80
Calibrated Parameters
Parameter Values Description
β 0.99 Discount factor
σ 1 Logarithmic Utility
αk 0.225 Effective Capital Share of Output
αh 0.675 Effective Labor Share of Output
δ0 0.025 Steady-state Depreciation Rate
F 1 Fixed Factor
z 1 Hicks-neutral technology shock
ζ 1 Preference shock
zI 1 Investment-specific technology shock
ω 1.15 Wage-markup shock
µA 0.9972 Growth rate, Relative Price of Investment shock
µX 1.0032 Growth rate, Harrod-neutral technology shock
h 0.2 Steady-state Hours Worked
u 1 Steady-state Capacity Utilization Rate
TABLE 9. Calibrated Parameters for Estimated One-sector Model
for the wage-markup shock ω is set to 1.15 and g is set such that government’s share of
output is 0.20 in steady state. The steady-state values of the growth of Harrod-neutral
technology µX and the relative price of investment µA are set to 1.0032 and 0.9957,
respectively, to match the long-run averages implied by the data. Finally, δ1 and ψ are
calibrated to ensure capacity utilization and hours worked in steady state are 1 and 0.2,
respectively. This information is summarized in Table 9.
Each of the remaining parameters which are to be estimated require an prior
distribution. These priors may reflect knowledge about the parameters obtained from
previous studies, and often help ensure the estimation routine focuses on “sensible”
values e.g. ensuring standard deviations are non-negative. I choose my priors to be
consistent with those in Schmitt-Grohe and Uribe (2012). In particular, the parameter
θ governing the Frisch elasticity of labor supply is distributed Gamma with mean 5 and
standard deviation 2, the curvature of the depreciation function δ2 is distributed inverse-
Gamma with mean 0.01 and standard deviation 0.05, and the curvature of the investment
81
adjustment cost function κ is distributed Gamma with mean 10 and standard deviation
2. These correspond to rather loose priors, and the inverse-Gamma on κ ensures it will
be strictly positive. The standard deviations for the structural shocks are assumed to
follow Gamma distributions to allow for the possibility that some shocks are completely
unimportant i.e. have standard deviations equal to zero. The surprise shocks and output’s
measurement error have mean and standard deviation equal to 0.1, and the anticipated
shocks have mean and standard deviation such that surprise shocks account for 50% of
total variance in each exogenous variables at the prior.
The internal habit adjustment parameter b as well as the autoregressive parameters
for each shock ρx are assumed to follow Beta distributions with mean 0.5 and standard
deviation 0.2 which correspond to hump-shaped prior distributions over the unit interval.
The habit-adjustment term γ from the JR preferences is uniformly distributed over the
unit interval. Finally, the constant-gain parameter g¯ is distributed Gamma with mean and
standard deviation 0.035, which is consistent with priors used in recent learning-estimation
literature while placing non-trivial probability at near-zero values.
Main Results
Table 10 reports the median and 95% highest posterior distribution interval (HPDI)
implied by the priors and 500,000 draws from the estimated posterior distribution under
RE and NRE. The most notable differences between models occur with respect to the
estimated standard deviations for the wage-markup shock (σ0ω,σ
4
ω,σ
8
ω) and the growth
rate of the relative price of investment (σ0µA ,σ
4
µA ,σ
8
µA). Under RE, surprise shocks are
responsible for about half of the variance in the wage-markup shock at the posterior
median, while under NRE they are responsible for almost none. A similar shift occurs
82
Prior and Posterior Distribution for Estimated Parameters
Prior Posterior - RE Posterior - Learning
Param. Distr. 95% Low Med. 95% High 95% Low Med. 95% High 95% Low Med. 95% High
σ0z Gamma 0.000 0.069 0.303 0.649 0.730 0.814 0.639 0.733 0.824
σ4z Gamma 0.000 0.049 0.214 0.000 0.057 0.252 0.000 0.062 0.283
σ8z Gamma 0.000 0.049 0.212 0.000 0.051 0.194 0.000 0.051 0.249
σ0ζ Gamma 0.000 0.069 0.300 2.795 3.398 4.081 2.741 3.352 4.043
σ4ζ Gamma 0.000 0.049 0.214 0.000 0.051 0.215 0.000 0.049 0.219
σ8ζ Gamma 0.000 0.049 0.215 0.000 0.050 0.219 0.000 0.051 0.205
σ0zI Gamma 0.000 0.069 0.302 4.701 5.599 6.563 4.634 5.602 6.560
σ4zI Gamma 0.000 0.049 0.214 0.000 0.051 0.230 0.000 0.047 0.215
σ8zI Gamma 0.000 0.049 0.214 0.000 0.051 0.211 0.000 0.047 0.207
σ0ω Gamma 0.000 0.069 0.302 0.000 0.206 1.115 0.000 0.079 0.410
σ4ω Gamma 0.000 0.049 0.212 0.000 0.153 1.103 0.000 0.054 0.245
σ8ω Gamma 0.000 0.049 0.215 0.000 0.133 1.091 0.917 1.038 1.169
σ0g Gamma 0.000 0.069 0.302 0.000 0.073 0.309 0.000 0.067 0.279
σ4g Gamma 0.000 0.049 0.214 3.686 4.236 4.816 3.718 4.289 4.900
σ8g Gamma 0.000 0.049 0.212 0.000 0.051 0.223 0.000 0.053 0.246
σ0µA Gamma 0.000 0.069 0.303 0.015 0.277 0.372 0.001 0.214 0.352
σ4µA Gamma 0.000 0.049 0.212 0.000 0.115 0.341 0.000 0.180 0.352
σ8µA Gamma 0.000 0.049 0.213 0.000 0.083 0.306 0.000 0.096 0.337
σ0µX Gamma 0.000 0.069 0.304 0.032 0.078 0.149 0.024 0.076 0.148
σ4µX Gamma 0.000 0.049 0.214 0.000 0.021 0.065 0.000 0.024 0.074
σ8µX Gamma 0.000 0.049 0.215 0.000 0.044 0.124 0.000 0.044 0.125
σYme Gamma 0.000 0.069 0.305 0.541 0.594 0.651 0.542 0.594 0.652
ρx,g Beta 0.125 0.500 0.871 0.117 0.420 0.733 0.131 0.439 0.738
ρz Beta 0.131 0.501 0.873 0.797 0.858 0.914 0.798 0.861 0.920
ρζ Beta 0.129 0.502 0.869 0.015 0.109 0.231 0.013 0.103 0.225
ρzI Beta 0.127 0.499 0.868 0.729 0.798 0.868 0.724 0.796 0.864
ρω Beta 0.115 0.500 0.858 0.917 0.950 0.980 0.923 0.955 0.985
ρg Beta 0.132 0.499 0.873 0.903 0.942 0.979 0.906 0.944 0.979
ρµA Beta 0.128 0.500 0.869 0.351 0.465 0.575 0.359 0.469 0.580
ρµX Beta 0.132 0.500 0.873 0.889 0.951 0.990 0.877 0.950 0.988
θ Gamma 1.503 4.760 8.961 2.936 3.696 4.490 3.057 3.803 4.645
κ Gamma 3.171 9.479 18.048 4.709 5.889 7.203 4.625 5.713 7.017
δ2 InvGamma 0.000 0.000 0.002 0.030 0.035 0.042 0.029 0.035 0.041
γ Unif 0.001 0.499 0.951 0.000 0.001 0.008 0.000 0.001 0.006
b Beta 0.503 0.707 0.885 0.813 0.841 0.867 0.812 0.840 0.865
g¯ Gamma 0.000 0.024 0.106 0 0 0 0.000 0.007 0.025
TABLE 10. Estimated Parameters
83
in the implied variance for the growth rate of the relative price of investment: under RE
the surprise shock is responsible for about 80% of total variance, while it is responsible
for only about 50% under NRE. Thus, the learning model ascribes greater importance to
anticipated shocks for driving exogenous variables than the model with RE.
That the values for mechanical sources of volatility and persistence such as the habit
formation parameter b and the rental rate elasticity of capacity utilization δ2 are so similar
across models is somewhat surprising. Previous estimation-with-learning exercises as in
Milani and Rajbhandari (2012) and Slobodyan and Wouters (2012) found that mechanical
sources of volatility and persistence failed to maintain their importance when agents were
assumed to be boundedly rational. However, my results are consistent with the earlier
finding from Slobodyan and Wouters (2007) that differences in estimates between RE and
models with learning are to a large extent driven by differences in the information sets of
agents. Because agents’ initial beliefs correspond to the REE the information set of agents
under NRE is quite similar to that of their rational counterparts.
To assess the quantitative importance of anticipated shocks versus surprise shocks,
I conduct a forecast-error variance decomposition (FEVD) exercise on the implied growth
rates of output, consumption, investment, hours worked, and government spending. For
surprise shocks typical FEVDs have a simple interpretation: because the shock is realized
in the same period it is announced, the induced volatility can be directly attributed to the
change in fundamentals.
Anticipated shocks, however, have two separate impacts on the system. Sims
(2016) refers to these impacts as “pure news’ and “realized news”, respectively. Pure
news captures volatility resulting from the change in agents’ expectations of future
fundamentals before those fundamentals have actually changed. Realized news captures
84
volatility resulting from the actual arrival of the anticipated event. For instance, if news
arrives four periods in advance, horizons up to four periods will represent pure news, while
horizons after four periods will represent both pure and realized news.
Since an unconditional FEVD is an asymptotic result, it will mix these two effects.
If we are interested in the relative importance of anticipated versus surprise shocks then
we should look at FEVDs conducted on shorter horizons. Figure 4 shows the 95% HPDI of
the estimated variance for the growth rates of output, consumption, investment, and hours
worked explained by anticipated shocks across models and FEVD horizons from 7500
draws from the posterior distributions, while Figure 5 displays the same information for
the growth rate of government expenditure.12 The vertical axis gives the median percent
of volatility explained by anticipated shocks, while the horizontal axis gives the FEVD
horizon. The light-shaded (dark-shaded) region corresponds to the 95% HPDI from the
model under NRE (RE).
In all cases the importance of news is greater under the learning model. Furthermore
the HPDIs are larger under NRE than under RE. This suggests that news is not merely
a proxy for the endogenous learning process, but that in fact the combination of news
and learning leads to additional volatility than would otherwise be implied. However it is
worth noting that the entirety of this difference occurs from the realized news component:
consistent with the results from Sims (2016), the relative importance of pure news is
similarly low under both models.
Given the observed differences in the relative importance of anticipated vs
unanticipated shocks across expectation formation mechanisms, it is natural to wonder
12I calculate FEVDs for horizons h = {0, 1, 2, 3, 4, 5, 6, 7, 8, 16}. Technically an unconditional FEVD is
taken in the limit as the forecasting horizon goes to infinity. In practice this convergence occurs relatively
quickly. Horizons longer than 16 result in variance shares which are nearly identical to those generated at
16.
85
Share of Volatility Due to Anticipated Shocks, RE vs NRE
0 1 2 3 4 5 6 7 8 16
0
20
40
60
FEVD Horizon
P
e
rc
e
n
t
E
x
p
la
in
e
d
Variance of Output Growth
0 1 2 3 4 5 6 7 8 16
0
10
20
30
40
FEVD Horizon
P
e
rc
e
n
t
E
x
p
la
in
e
d
Variance of Consumption Growth
0 1 2 3 4 5 6 7 8 16
0
20
40
60
FEVD Horizon
P
e
rc
e
n
t
E
x
p
la
in
e
d
Variance of Investment Growth
0 1 2 3 4 5 6 7 8 16
0
20
40
60
80
FEVD Horizon
P
e
rc
e
n
t
E
x
p
la
in
e
d
Variance of Growth in Hours Worked
FIGURE 4. Relative Importance of News: Key Macroeconomic Variables, RE vs NRE
whether there are differences in each model’s ability to fit the data. Table 11 compares
the standard deviation (relative to output), correlation with output, and first-order
autocorrelation coefficients for the growth rates of output, consumption, investment, hours
worked, government expenditure, TFP, and the relative price of investment obtained from
the actual data to that obtained from 7500 simulations of each model. The deep structural
parameters for each simulation were generated via a random draw of the respective
model’s posterior distribution. The summary statistics are the median value across these
simulations. The variables Y,C, I, h, g, TFP and A refer to the growth rates of output,
consumption, investment, hours worked, government expenditure, TFP, and the relative
86
Share of Volatility Due to Anticipated Shocks, RE vs NRE
0 1 2 3 4 5 6 7 8 16
0
20
40
60
80
100
FEVD Horizon
P
er
ce
n
t
E
x
p
la
in
ed
Variance of Government Expenditure Growth
FIGURE 5. Relative Importance of News: Government Spending, RE vs NRE
price of investment, respectively, and relative standard deviations are the ratio of the
standard deviation of the variable to that of the growth rate of output.
The summary statistics produced by both models are extremely similar to each
other and are broadly consistent with the data. The growth rates of consumption and
investment are estimated to be overly volatile, while the growth rate of the relative price
of investment is somewhat less volatile; however, the underlying standard deviation of
output growth is almost identical to that of the data. Furthermore the relative standard
deviations of growth in hours worked, government spending, and TFP are all quite
accurate compared to the data. The estimated correlations with the data are weaker than
those observed in the data for the growth rates of investment, hours worked, and TFP,
but match quite well for consumption and government spending. Finally, the estimated
87
Model Predictions: Volatility and Persistence, Data vs NRE vs RE
Y C I h g TFP A
Relative Standard Deviation
Data 1.00 0.56 2.52 0.92 1.25 0.83 0.45
Model - NRE 1.00 0.70 3.06 0.97 1.15 0.85 0.35
Model -RE 1.00 0.70 3.05 0.98 1.18 0.88 0.37
Correlation with Output
Data 1.00 0.50 0.69 0.72 0.25 0.40 -0.12
Model - NRE 1.00 0.51 0.57 0.40 0.26 0.27 0.01
Model -RE 1.00 0.49 0.55 0.37 0.27 0.29 0.01
Autocorrelation
Data 0.28 0.20 0.53 0.60 0.05 -0.01 0.49
Model - NRE 0.32 0.46 0.63 0.18 0.03 -0.01 0.46
Model -RE 0.34 0.48 0.65 0.19 0.03 -0.00 0.46
TABLE 11. Model Predictions: Volatility and Persistence, Data vs NRE vs RE
persistence of consumption is too high, while the estimated persistence of hours worked is
too low.13
Conclusion
This chapter has demonstrated that the estimated relative importance of anticipated
news versus unanticipated news, as interpreted by the previous literature, is robust to the
inclusion of learning. In particular it appears that news and learning serve to augment
each other: in general, the estimated macroeconomic volatility in key variables is more a
function of news under NRE than under RE. This is especially interesting given that both
models produce quantitatively similar predictions for overall volatility and persistence.
13My estimates are broadly similar but generally closer to the data than those produced in Schmitt-
Grohe and Uribe (2012). For example, the sum of squared deviations of data vs estimated relative
standard deviation is around 0.33 under both NRE and RE under my estimates compared to 1.6 under
theirs. This is driven to a large extent to the relative standard deviation of investment to output growth
which is much greater under their estimates due to a much lower estimated standard deviation of output
(0.91 in the data and my estimates, 0.73 under their estimates).
88
However, pure news is responsible for almost none of the estimated volatility in
either model; it is only once the anticipated shock materializes that news begins to be
important. It is possible that this finding is sensitive to the particular form of learning
assumed. The NRE approach undertaken in this chapter represents a relatively small
departure from RE.
Future research will look at the relative impact on model predictions and estimates
across a variety of bounded rationality assumptions. Departures from RE such as the
infinite-horizon learning of Preston (2005), finite-horizon learning of Branch et al. (2012a),
and shadow-price learning of Evans and McGough (2015) all reflect larger departures from
RE, and provide a natural avenue for future projects studying the empirical importance of
both news shocks and adaptive learning in DSGE model estimation. It will be particularly
interesting to observe whether differences emerge between between bounded rationality of
the “learning-to-forecast” and “learning-to-optimize” types emerge.
89
APPENDIX A
APPENDIX FOR CHAPTER II
Competitive Equilibrium
Firms
The representative firm in the competitive durable goods sector solves
max
lk,t
PtIt −Wk,tlk,t −Rk,tl˜x
s.t. It = exp (θk,t)l
αk
k,t l˜
1−αk
k
where Pt the price of the investment good in terms of the consumption good, Wk,t and
Rk,t the rents paid to labor lk,t and the fixed factor l˜k, respectively, and θk,t is a measure of
technology.1 The FONCs are given by
Wk,t = αkPt
It
lk,t
(A.1)
Rk,t = (1− αk)pt It
l˜k
(A.2)
Thus labor and the fixed factor in this sector are paid their marginal products in terms of
the numeraire good.
1 Technology in the durable goods sector evolves deterministically according to
log θk,t = g0,k + g1 ∗ t
90
Consumption Good
The representative firm solves
max
Xt,Kt−1
Ct − Px,tXt −RtKt−1
s.t Ct = (aX
v
t + (1− a)Kvt−1)
1
v
where Px,t the price of the nondurable good Xt in terms of the consumption good and Rt
the gross real interest rate. The FONCs are given by
Rt = (1− a)C1−vt Kv−1t−1 (A.3)
Px,t = aC
1−v
t X
v−1
t (A.4)
This will be used below to determine the value of labor and the fixed factor in the non-
durable goods sector Xt.
The nondurable good is an intermediate good in production, and hence is only
consumed by final goods producers. The representative firm in the competitive nondurable
goods sector solves
max
lx,t,l˜x
Px,tXt −Wx,tlx,t −Rx,tl˜x
s.t Xt = exp (θx,t)l
αx
x,t l˜
1−αx
x
91
where Wx,t and Rx,t the rents paid to labor lx,t and the fixed factor l˜x, respectively, and
θx,t is a measure of technology.
2
The FONCs are given by
Wx,t = αxXt/lx,t (A.5)
Rx,t =
rx,t
(1− αx)Xt/l˜x
(A.6)
(A.7)
We can use the demand from the representative final good producer to determine the
wage and rental rates. Equations (A.5) and (A.4) together imply
Wx,t = aC
1−v
t αx
Xvt
lx,t
(A.8)
and equations (A.6) and (A.4) together imply
rx,t = aC
1−v
t (1− αx)
Xvt
l˜x
(A.9)
2 Technology in the nondurable goods sector evolves according to
log θx,t = g0,x + g1 ∗ t+ log θˆx,t
log θˆx,t = λ log θˆx,t−1 + t
where t an i.i.d mean zero shock which is known to the firms when making their production decision and
vt−n is the news shock revealed to households.
92
Households
The household problem is given by
max
Ct,lx,t,lk,t,It,Kt
logCt + v0(l¯ − lx,t − lk,t) (A.10)
subject to the law of motion for capital and the flow budget constraint
Kt = (1− δ)Kt−1 + It (A.11)
Ct + PtIt = Wx,tlx,t +Wk,tlk,t +RtKt−1 + Πt (A.12)
The LaGrangian is thus
L = Eˆ0
∞∑
t=0
βt
{
logCt + v0(l¯ − lx,t − lk,t)
}
+ Λt {Wx,tlx,t +Wk,tlk,t +RtKt−1 + Πt − Ct − PtIt}
+ ΛtQt {(1− δ)Kt−1 + It −Kt}
where Λt the marginal value of income, Qt Tobin’s marginal Q, and hence ΛtQt the
marginal value of capital in terms of consumption. The first-order necessary conditions
93
for an interior solution to the household’s problem are given by
∂L
∂Ct
= 0 ⇐⇒ Λt = 1
Ct
(A.13a)
∂L
∂lx,t
= 0 ⇐⇒ v0 = ΛtWx,t (A.13b)
∂L
∂lk,t
= 0 ⇐⇒ v0 = ΛtWk,t (A.13c)
∂L
∂It
= 0 ⇐⇒ Pt = Qt (A.13d)
∂L
∂Kt
= 0 ⇐⇒ ΛtQt = βEˆt [Λt+1 (Rt+1 +Qt+1(1− δ))] (A.13e)
94
Equilibrium
A temporary equilibrium is a set
{
Ct, lx,t, lk,t, It, Kt, Xt, Yt, Pt, Rt,Wx,t,Wk,t,Λt, Qt, θk,t, θx,t, θ˜x,t
}
such that consumers and firms are utility maximizing and markets clear:
Yt = Ct + PtIt (A.14a)
Ct =
(
aXνt + (1− a)Kνt−1
) 1
ν (A.14b)
Xt = θx,tl
αx
x,tF
1−αx
x (A.14c)
It = θk,tl
αk
k,tF
1−αk
k (A.14d)
Kt = (1− δ)Kt−1 + It (A.14e)
Λt =
1
Ct
(A.14f)
v0 = ΛtWx,t (A.14g)
v0 = ΛtWk,t (A.14h)
Pt = Qt (A.14i)
ΛtQt = βEˆt [Λt+1Rt+1 + Λt+1Qt+1(1− δ)] (A.14j)
Rt = (1− a)
(
Ct
Kt−1
)1−v
(A.14k)
Wx,t = aαxC
1−v
t
Xvt
lx,t
(A.14l)
Wk,t = Ptαk
It
lk,t
(A.14m)
θk,t = g0,ke
g1t (A.14n)
θx,t = g0,xe
g1tθ˜x,t (A.14o)
θ˜x,t = θ˜
λ
x,t−1e
t (A.14p)
(A.14q)
95
where Λt and ΛtQt the marginal value (or shadow price) of income and installed capital,
respectively. Qt is the ratio of these marginal values and can be interpreted as the relative
price of installed capital in terms of the consumption good, i.e. marginal Tobin’s Q.
Equation (A.14a) is the economy’s aggregate resource constraint and equations
(A.14b)-(A.14d) are the production functions for the consumption, nondurable, and
durable goods respectively. Equation (A.14e) is the capital accumulation equation.
Optimal household decision making for consumption, nondurable good labor supply,
durable good labor supply, investment, and capital yields equations (A.14f)-(A.14j),
respectively. Markets are competitive and hence factors of production are paid the value of
their marginal products, as specified by equations (A.14k)-(A.14m). Finally, technology for
producing the durable and nondurable goods is given by equations (A.14n)-(A.14p).
96
Detrending
Define the for any non-stationary variable x the stationary counterpart xt = xˆte
g1t.
The temporary equilibrium in terms of detrended variables is
Yˆt = Cˆt + PtIˆt (A.15a)
Cˆt =
(
aXˆνt + (1− a)
(
Kˆt−1
eg1
)ν) 1ν
(A.15b)
Xˆt = θˆx,tl
αx
x,tF
1−αx
x (A.15c)
Iˆt = θˆk,tl
αk
k,tF
1−αk
k (A.15d)
Kˆt = (1− δ)Kˆt−1
eg1
+ Iˆt (A.15e)
Λˆt =
1
Cˆt
(A.15f)
v0 = ΛˆtWˆx,t (A.15g)
v0 = ΛˆtWˆk,t (A.15h)
Pt = Qt (A.15i)
eg1ΛˆtQt = βEˆt
[
Λˆt+1Rt+1 + Λˆt+1Qt+1(1− δ)
]
(A.15j)
Rt = (1− a)
(
Cˆt
Kˆt−1
eg1
)1−v
(A.15k)
Wˆx,t = aαxCˆ
1−v
t
Xˆvt
lx,t
(A.15l)
Wˆk,t = Ptαk
Iˆt
lk,t
(A.15m)
θˆk,t = g0,k (A.15n)
θˆx,t = g0,xθ˜x,t (A.15o)
θˇx,t = θˇ
λ
x,t−1e
t (A.15p)
97
Log-Linearization
The entire detrended, log-linearized system of equilibrium conditions is given by
Y˜t −
(
C
Y
)
C˜t −
(
PI
Y
)
P˜t −
(
PI
Y
)
I˜t = 0 (A.16)
C˜t − a
(
X
C
)v
X˜t = (1− a)
(
K
C
)v
K˜t−1 (A.17)
X˜t − θ˜x,t − αxl˜x,t = 0 (A.18)
I˜t − θ˜k,t − αk l˜k,t = 0 (A.19)
K˜t −
(
I
K
)
I˜t =
(
1− δ
eg1
)
K˜t−1 (A.20)
Λ˜t + C˜t = 0 (A.21)
Λ˜t + W˜x,t = 0 (A.22)
Λ˜t + W˜k,t = 0 (A.23)
Q˜t − P˜t = 0 (A.24)
Λ˜t + Q˜t = EˆtΛ˜t+1 +
(
1
R +Q(1− δ)
)
EˆtR˜t+1 +
(
Q(1− δ)
R +Q(1− δ)
)
EˆtQ˜t+1 (A.25)
R˜t + (R(v − 1)) C˜t = (R(v − 1)) K˜t−1 (A.26)
l˜x,t + W˜x,t − (1− v) C˜t − vX˜t = 0 (A.27)
l˜k,t + W˜k,t − P˜t − I˜t = 0 (A.28)
θ˜k,t = 0 (A.29)
θ˜x,t − ˜ˇθx,t = 0 (A.30)
˜ˇθx,t = λ
˜ˇθx,t−1 + t (A.31)
98
where a ˜tilde over a variable indicates the percent deviation from steady state and
variables without a time subscript indicate their steady state value.3
3Note that the gross interest rate is expressed in “deviation from steady state” rather than “percent
deviation from steady state”, since the variable is already expressed in percentage terms.
99
APPENDIX B
APPENDIX FOR CHAPTER III
Solving the Model
Suppose the household was acting optimally and contemplated reallocating a small
amount of consumption for investment. The marginal cost of the reduction in consumption
is the direct loss of utility today as well as the expected discounted value of changing the
geometric average of habit-adjusted consumption. The marginal benefit is the expected
discounted value of the additional investment, which affects future investment adjustment
costs and increases tomorrow’s capital stock. Hence the household’s first-order necessary
condition for Ct is
UCt(Vt) +
∂St
∂Ct
βEˆtλ
S
t+1 =
∂It
∂Ct
(
βEˆtλ
I
t+1 +
∂Kt
∂It
βEˆtλ
K
t+1
)
(B.1)
Now suppose the household considered a small increase in their labor supply. The
additional labor income could be used to increase investment, and hence the marginal
benefit is the expected discounted value of the additional investment they can afford,
which again affects future investment adjustment costs and increases tomorrow’s capital
stock. The marginal cost is the disutility associated with higher labor supply, and hence
the household’s first-order necessary condition for ht is
−Uht(Vt) =
∂It
∂ht
(
βEˆtλ
I
t+1 +
∂Kt
∂It
βEˆtλ
K
t+1
)
(B.2)
100
If the household were to utilize more of their predetermined capital stock they would
receive rental income which could be used to finance additional investment, again affecting
tomorrow’s capital stock and investment adjustment costs. However working capital more
intensely also increase the rate of depreciation, and thus the first-order necessary condition
for ut is
∂Kt
∂ut
βEˆtλ
K
t+1 =
∂It
∂ut
(
βEˆtλ
I
t+1 +
∂Kt
∂It
βEˆtλ
K
t+1
)
(B.3)
The optimal choices of Ct, ht, and ut pin down the value for It through the flow
budget constraint, which pins down the value for Kt through the capital accumulation
equation. Likewise, the optimal choice of Ct generically determines the value of St. Thus
equations (B.1), (B.2), and (B.3) fully describe the optimal behavior of the household as a
function of their beliefs about the future shadow prices.
The expected value of these shadow prices may also be derived using variational
arguments. The time t value of additional It−1 in time t holding everything else constant is
the resulting change to investment adjustment costs which changes the size of tomorrow’s
capital stock, that is
λIt =
∂Kt
∂It−1
βEˆtλ
K
t+1 (B.4)
The time t value of additional preinstalled capital in time t is the expected discounted
value of the additional rental income (which could be invested) plus the value of directly
increasing tomorrow’s capital stock, that is
λKt =
∂It
∂Kt−1
(
βEˆtλ
I
t+1 +
∂Kt
∂It
βEˆtλ
K
t+1
)
+
∂Kt
∂Kt−1
βEˆtλ
K
t+1 (B.5)
101
Finally, the time t value of a small change to the predetermined level of habit-adjusted
consumption in time t is the direct change to utility today, as well as the expected
discounted value of the resulting change in the level of habit-adjusted consumption, that is
λSt = USt−1(Vt) +
∂St
∂St−1
βEˆtλ
S
t+1 (B.6)
The solution to the household’s optimization problem satisfies the FONCs for the controls
given by
(
Ct − ψhθtSt
)−σ (
1− γψh
θ
tSt
Ct
)
+
γSt
Ct
βEˆtλ
S
t+1 =
1
At
βEˆt
(
λIt+1 +
(
1− Φ
(
It
It−1
)
− Φ′
(
It
It−1
)(
It
It−1
))
λKt+1
)
(B.7a)
(
Ct − ψhθtSt
)−σ ( θψhθ−1t St
Wt
)
=
1
At
βEˆt
(
λIt+1 +
(
1− Φ
(
It
It−1
)
− Φ′
(
It
It−1
)(
It
It−1
))
λKt+1
)
(B.7b)
δ′(ut)
Rt
βEˆtλ
K
t+1 =
1
At
βEˆt
(
λIt+1 +
(
1− Φ
(
It
It−1
)
− Φ′
(
It
It−1
)(
It
It−1
))
λKt+1
)
(B.7c)
λIt = Φ
′
(
It
It−1
)(
It
It−1
)2
βEˆtλ
K
t+1 (B.7d)
λKt = (Rtut)
1
At
βEˆt
(
λIt+1 +
(
1− Φ
(
It
It−1
)
− Φ′
(
It
It−1
)(
It
It−1
))
λKt+1
)
+ (1− δ(ut))βEˆtλKt+1 (B.7e)
λSt =
(
Ct − ψhθtSt
)−σ (− (1− γ)ψhθtSt
St−1
)
+ β
(1− γ)St
St−1
Eˆtλ
S
t+1 (B.7f)
It can be shown that the shadow prices can be calculated as
λIt = Φ
′
(
It
It−1
)(
It
It−1
)2(
Rt
δ′(ut)
)(
Ct − ψhθtSt
)−σ (θψhθ−1t St
Wt
)
(B.8a)
λKt = (1 + utδ
′(ut)− δ(ut))
(
Rt
δ′(ut)
)(
Ct − ψhθtSt
)−σ (θψhθ−1t St
Wt
)
(B.8b)
λSt =
[(
Ct − ψhθtSt
)−σ ((1− γ)St
St−1
)][(−Ct
γSt
)(
1− γψh
θ
tSt
Ct
− θψh
θ−1
t St
Wt
)
− ψhθt
]
(B.8c)
The full dynamic system under SP-learning is described by the following equations,
comprised of the optimality conditions for households and firms, resource constraints, and
102
identities:
(
Ct − ψhθtSt
)−σ (
1− γψh
θ
tSt
Ct
)
+
γSt
Ct
βEˆtλ
S
t+1 =
1
At
βEˆt
(
λIt+1 +
(
1− Φ
(
It
It−1
)
− Φ′
(
It
It−1
)(
It
It−1
))
λKt+1
)
(
Ct − ψhθtSt
)−σ (− θψhθ−1t St
Wt
)
=
1
At
βEˆt
(
λIt+1 +
(
1− Φ
(
It
It−1
)
− Φ′
(
It
It−1
)(
It
It−1
))
λKt+1
)
δ′(ut)
Rt
βEˆtλ
K
t+1 =
1
At
βEˆt
(
λIt+1 +
(
1− Φ
(
It
It−1
)
− Φ′
(
It
It−1
)(
It
It−1
))
λKt+1
)
λIt = Φ
′
(
It
It−1
)(
It
It−1
)2 ( Rt
δ′(ut)
)(
Ct − ψhθtSt
)−σ ( θψhθ−1t St
Wt
)
λKt =
(
1 + utδ
′(ut)− δ(ut)
)( Rt
δ′(ut)
)(
Ct − ψhθtSt
)−σ ( θψhθ−1t St
Wt
)
λSt =
[(
Ct − ψhθtSt
)−σ ( (1− γ)St
St−1
)]
×[(−Ct
γSt
)(
1− γψh
θ
tSt
Ct
− θψh
θ−1
t St
Wt
)
− ψhθt
]
Kt = (1− δ (ut))Kt−1 + It
[
1− Φ
(
It
It−1
)]
Ct +AtIt ≤Wtht +Rt (utKt−1)
St = C
γ
t S
1−γ
t−1
Yt = Ct +AtIt
Yt = zt (utKt−1)1−α (ht)α
Wt = α
Yt
ht
Rt = (1− α) Yt
utKt−1
Eˆtλ
I
t = H
I
t x˜t
Eˆtλ
K
t = H
K
t x˜t
Eˆtλ
S
t = H
S
t x˜t
RH,t = RH,t−1 + gt
(
x˜t−1x˜′t−1 −RH,t−1
)
Ht = Ht−1 + gtR−1H,tx˜t−1
(
λt−1 −H′t−1x˜t−1
)′
103
Recursive Representation of News-Shocks
We can write the evolution of any exogenous variable indexed by w as
ln(wt) = ρw ln(wt−1) + ε0w,t (B.9)
εw,t = ϕwεw,t−1 +Mwνw,t (B.10)
where εw,t =
(
ε0w,t, ε
1
w,t, .., ε
n
w,t
)
a vector of auxiliary state variables which carry the
news shocks through time, ϕ a lower-shift matrix with 1’s on the super-diagonal and
zeros elsewhere, νw,t =
(
ν0w,t, ν
1
w,t, ..., ν
n
w,t
)′
a vector of (anticipated and unanticipated)
shocks, and Mw =
([
ν0w,t ∈ It
]
,
[
ν1w,t−1 ∈ It
]
, ...,
[
νnw,t−n ∈ It
])
a row-vector of 1’s and 0’s
respecting the specific assumptions regarding the information obtained by households.
104
APPENDIX C
APPENDIX FOR CHAPTER IV
The Labor Market
Final-goods producers demand a composite labor good hct =
[∫ 1
0
ht(j)
1
1+µt dj
]1+µt
where ht(j) denotes labor of type j ∈ [0, 1] and µt ≥ 0 is the exogenous stochastic wage
markup. The cost minimization problem for a final-goods producer is
min
ht(j)
∫ 1
0
Wt(j)ht(j)dj
subject to hct ≤
[∫ 1
0
ht(j)
1
1+µt dj
]1+µt
the solution to which yields the conditional factor demand functions h∗t (j) =
hct
(
Wt(j)
W ct
)− (1+µt)
µt where W ct =
[∫ 1
0
Wt(j)
− 1
µt dj
]−µt
the cost of a single unit of the composite
labor input.
Labor ht(j) is provided by monopolistically competitive unions. Since labor is freely
mobile all unions pay their members the same wage rate Wt and charge firms Wt(j). The
profit maximization problem for these unions is
max
Wt(j)
(Wt(j)−Wt)ht(j)
subject to ht(j) = h
∗
t (j)
the solution to which is W ∗t (j) = (1 + µt)Wt. It is clear from this expression all unions will
charge firms the same price denoted W ∗t , and that this wage is marked up in accordance
105
with the value of market power captured by µt. This implies firms will demand identical
quantities of each type of labor, that is ht(j) = h
c
t for all j.
The profits of each union j are thus Πt(j) = µtWth
c
t , which is the same for all
unions. These profits are rebated to member-households as a lump sum. Since the unions
determine the hours of labor allocated, firms choose their labor demand, and households
choose how much labor to provide, an equilibrium requires
∫ 1
0
ht(j)dj = h
c
t =
∫
ht(ω)dω.
106
Inducing Stationarity
Many of the endogenous variables inherit the non-stationarity of At and Xt. In
particular, it can be shown that
– Ct, St, Y
out
t , and Wt grow at rate µ
Y
t =
XYt
XYt−1
– Kt and It grow at rate µ
K
t =
XKt
XKt−1
where XKt =
At
XYt
– Gt grows at rate µ
G
t =
XGt
XGt−1
– Rt and Qt grow at rate µ
A
t
– Πt and Λt grow at rate
(
µYt
)−σ
Thus we define the stationary variables Yˆ outt =
Y outt
XYt
, Cˆt =
Ct
XYt
, Wˆt =
Wt
XYt
, Iˆt =
AtIt
XYt
,
Kˆt =
AtKt
XYt
, gt =
Gt
XGt
, Rˆt =
Rt
At
, Qˆt =
Qt
At
, Πˆt =
(
XYt
)σ
Πt, and Λˆt =
(
XYt
)σ
Λt. ut and ht
are already stationary variables. By manipulating the equations defining a non-stationary
temporary equilibrium one can express the system in terms of the stationary variables. In
particular, one can show the stationarized dynamic system of equilibrium conditions to be
107
given by
Kˆt = (1− δ (ut)) Kˆt−1
µKt
+ zIt Iˆt
[
1− Φ
(
Iˆt
Iˆt−1
µKt
)]
Yˆt = Cˆt + Iˆt + gtX
G,Y
t
Yˆt = zt
(
ut
Kˆt−1
µKt
)αk
(ht)
αh (F )1−αk−αh
Vˆt = Cˆt − bCˆt−1
µYt
− ψhθt Sˆt
Sˆt =
(
Cˆt − bCˆt−1
µYt
)γ (
Sˆt−1
µYt
)1−γ
Λˆt =
ζtVˆ −σt − γSˆt
Cˆt − b Cˆt−1µYt
Πˆt
− bβEˆt
(µYt+1)−σ
ζt+1Vˆ −σt+1 − γSˆt+1
Cˆt+1 − b CˆtµYt+1
Πˆt+1

Λˆt =
θζtVˆ
−σ
t ψh
θ−1
t Sˆt
Wˆt/ωt
Πˆt = ζtVˆ
−σ
t ψh
θ
t + β(1− γ)Eˆt
[
Sˆt+1
Sˆt
Πˆt+1
(
µYt+1
)1−σ]
QˆtΛˆt = βEˆt
[
µAt+1
(
µYt+1
)−σ
Λˆt+1
(
Rˆt+1ut+1 + Qˆt+1(1− δ(ut+1))
)]
Rˆt = δ
′(ut)Qˆt
Λˆt = QˆtΛˆtz
I
t
(
1− Φ
(
Iˆt
Iˆt−1
µKt
)
− Φ′
(
Iˆt
Iˆt−1
µKt
)(
Iˆt
Iˆt−1
µKt
))
+ βEˆt
Qˆt+1Λˆt+1zIt+1µAt+1 (µYt+1)−σ Φ′
(
Iˆt+1
Iˆt
µKt+1
)(
Iˆt+1
Iˆt
µKt+1
)2
Wˆt = αh
Yˆt
ht
Rˆt = αk
Yˆt
ut
Kˆt−1
µKt
108
where the functional forms for the depreciation rate and investment-adjustment costs are
given by
δ (ut) = δ0 + δ1 (ut − 1) + δ2
2
(ut − 1)2
Φ
(
Iˆt
Iˆt−1
µKt
)
=
κ
2
(
Iˆt
Iˆt−1
µKt − µI
)2
The laws of motion for the exogenous stochastic processes can be written
ln (zt/z) = ρz ln (zt−1/z) + 0z,t
ln (ζt/ζ) = ρζ ln (ζt−1/ζ) + 0ζ,t
ln
(
zIt /z
I
)
= ρzI ln
(
zIt−1/z
I
)
+ 0zI ,t
ln (gt/g) = +ρg ln (gt−1/g) + 0g,t
ln (ωt/ω) = ρω ln (ωt−1/ω) + 0ω,t
ln
(
µAt /µ
A
)
= ρµA ln
(
µAt−1/µ
A
)
+ 0µA,t
ln
(
µXt /µ
X
)
= ρµX ln
(
µXt−1/µ
X
)
+ 0µX ,t
µYt = µ
X
t
(
µAt
)αk/(αk−1)
µKt = µ
X
t
(
µAt
)1/(αk−1)
XG,Yt =
(
XG,Yt−1
)ρx,g
µYt
=
XGt
XYt
109
where the shock structure for any stochastic process x =
{
z, ζ, zI , g, ω, µA, µX
}
is given by
0x,t = 
1
x,t−1 + ν
0
x,t
1x,t = 
2
x,t−1
2x,t = 
3
x,t−1
3x,t = 
4
x,t−1
4x,t = 
5
x,t−1 + ν
4
x,t
5x,t = 
6
x,t−1
6x,t = 
7
x,t−1
7x,t = 
8
x,t−1
8x,t = ν
8
x,t
Finally, the growth rates of the seven key macroeconomic variables considered in terms of
the stationary variables is
gYt =
Yt
Yt−1
=
Yˆt
Yˆt−1
XYt
XYt−1
=
Yˆt
Yˆt−1
µYt
gCt =
Ct
Ct−1
=
Cˆt
Cˆt−1
XYt
XYt−1
=
Cˆt
Cˆt−1
µYt
gIt =
AtIt
At−1It−1
=
Iˆt
Iˆt−1
XYt
XYt−1
=
Iˆt
Iˆt−1
µYt
ght =
ht
ht−1
gGt =
Gt
Gt−1
=
gt
gt−1
(
XG,Yt−1
)ρxg−1
gTFPt =
zt
zt−1
(
µXt
)1−αk
gAt = µ
A
t
110
Steady State Computation and Calibration
Let x be the steady-state value of the stationary random variable xˆt. The steady
states for z, ζ, and zI are normalized to 1, g is set such that the output share of
government spending is 0.20, ω is set to 1.15, and µY and µA are set to the mean
gross growth rate of per capita output and relative price of investment observed in the
data which, given the stochastic trends in output and government spending, implies
µX = µY µA
αk
1−αk and XG,Y =
(
µY
)1/(ρx,g−1)
. The parameter µI is set to µK so that
investment-adjustment costs (and their derivative) are zero in steady state, while δ1 is set
to 1
βµA µY −σ
− (1− δ0) to ensure the steady-state capacity utilization rate u is 1. The scale
parameter ψ governing the disutility of labor in the utility function is calibrated to ensure
the steady-state value of h is 0.20. With these calibrations for the steady-state values of
exogenous processes and free parameters, the non-stochastic steady state can be easily
computed from the equilibrium conditions and constraints given above. In particular, the
111
steady state can be shown to be
R = δ1
h = 0.2
u = 1
Q = 1
K =
αkz
(
µK
u
)1−αk
hαh
R

1
1−αk
I =
(
1− 1− δ0
µK
)
K
Y =
RuK
αkµK
W = αh
Y
h
g = 0.2
C = Y − I − gXGY
S = C(1− b/µY ) (µY ) γ−1γ
ψ =
[(
θhθ−1S
W/ω
)(
1− bβ (µY )−σ)−1 + ( γS
(C − bC/µY )
)(
hθ
1− β(1− γ) (µY )1−σ
)]−1
V =
(
C − bC
µY
)
− ψhθS
Π =
(
ζV −σψhθ
) (
1− β(1− γ) (µY )1−σ)−1
Λ =
θζV −σψhθ−1S
W/ω
112
Log-Linearization
The stationarized system can be log-linearized around its non-stochastic steady state.
Denote by a t˜ilde the deviation from steady state, that is, for any variable x we have x˜t =
log
(
xt
x
)
.
1. The capital accumulation equation is
Kˆt = (1− δ (ut)) Kˆt−1
µKt
+ zIt Iˆt
[
1− Φ
(
Iˆt
Iˆt−1
µKt
)]
Taking logs of both sides we have
log Kˆt = log
(
(1− δ (ut)) Kˆt−1
µKt
+ zIt Iˆt
[
1− Φ
(
Iˆt
Iˆt−1
µKt
)])
Noting that logK = log
(
(1− δ(u)) K
µK
+ zII
(
1− Φ ( I
I
µK
)))
, the first-order Taylor
series expansion around the steady state can be shown to be
K˜t =
(
1− δ(u)
µK
)
K˜t−1 −
(
δ′(u)u
µK
)
u˜t −
(
1− δ(u)
µK
)
µ˜Kt +
(
zII
K
)
z˜It +
(
zII
K
)
I˜t
2. The aggregate resource constraint is
Yˆt = Cˆt + Iˆt + gtX
G,Y
t
Taking logs of both sides we have
log Yˆt = log
(
Cˆt + Iˆt + gtX
G,Y
t
)
113
Noting that log Yˆ = log
(
Cˆ + Iˆ + gXG,Y
)
, the first-order Taylor series expansion
around the steady state can be shown to be
Y˜t =
(
C
Y
)
C˜t +
(
I
Y
)
I˜t +
(
gXG,Y
Y
)
X˜G,Yt +
(
gXG,Y
Y
)
g˜t
3. The production function is
Yˆt = zt
(
ut
Kˆt−1
µKt
)αk
(ht)
αh (F )1−αk−αh
Taking logs on both sides yields
log Yˆt = log zt + αk log ut + αk log Kˆt−1 − αk log µKt + αh log ht + (1− αk − αh) logF
Since the equation is log-linear, subtracting each side by log Y = log z + αk log u +
αk log Kˆ − αk log µK + αh log h + (1− αk − αh) logF , the log-linearized production
function can be shown to be
Y˜t = z˜t + αku˜t + αkK˜t−1 − αkµ˜Kt + αhh˜t
4. The utility function’s argument is
Vˆt = Cˆt − bCˆt−1
µYt
− ψhθt Sˆt
Taking logs of both sides yields
log Vˆt = log
(
Cˆt − bCˆt−1
µYt
− ψhθt Sˆt
)
114
Noting that log Vˆ = log
(
Cˆ − b Cˆ
µY
− ψhθSˆ
)
, the first-order Taylor series expansion
around the steady state can be shown to be
V˜t =
(
C
V
)
C˜t −
(
bC/µY
V
)
C˜t−1 +
(
bC/µY
V
)
µ˜Yt −
(
ψhθSθ
V
)
h˜t −
(
ψhθS
V
)
S˜t
5. The identity for the geometric average of current and past habit-adjusted
consumption is
Sˆt =
(
Cˆt − bCˆt−1
µYt
)γ (
Sˆt−1
µYt
)1−γ
Taking logs of both sides yields
log Sˆt = γ log
(
Cˆt − bCˆt−1
µYt
)
+ (1− γ) log
(
Sˆt−1
µYt
)
Noting that log Sˆ = γ log
(
Cˆ − b Cˆ
µY
)
+ (1− γ) log
(
Sˆ
µY
)
, the first-order Taylor series
expansion around the steady state can be shown to be
S˜t =
(
γ
1− b/µY
)
C˜t −
(
γb/µY
1− b/µY
)
C˜t−1 + (1− γ)S˜t−1 +
(
γb/µY
1− b/µY − (1− γ)
)
µ˜Yt
6. The FONC for consumption is
Λˆt =
ζtVˆ −σt − γSˆt
Cˆt − b Cˆt−1µYt
Πˆt
− bβEˆt
ζt+1Vˆ −σt+1 − γSˆt+1
Cˆt+1 − b CˆtµYt+1
Πˆt+1
(µYt+1)−σ

115
Taking logs of both sides yields
log Λˆt = log
ζtVˆ −σt − γSˆt
Cˆt − b Cˆt−1µYt
Πˆt
− bβEˆt
ζt+1Vˆ −σt+1 − γSˆt+1
Cˆt+1 − b CˆtµYt+1
Πˆt+1
(µYt+1)−σ

Noting that log Λˆ = log
((
ζVˆ −σ − γSˆ
Cˆ−b Cˆ
µY
Πˆ
)
− bβEˆ
[(
ζVˆ −σ − γSˆ
Cˆ−b Cˆ
µY
Πˆ
)(
µY
)−σ])
,
the first-order Taylor series expansion around the steady state can be shown to be
Λ˜t = φ1
(
ζ˜t − σV˜t − φ3
(
Eˆtζ˜t+1 − σEˆtV˜t+1
))
− φ2
{
S˜t + Π˜t +
(
b/µY
1− b/µY
)
C˜t−1 −
(
b/µY
1− b/µY
)
µ˜Yt −
(
1 + b2β
(
µY
)−σ−1
1− b/µY
)
C˜t
−φ3
(
EˆtS˜t+1 + EˆtΠ˜t+1 −
(
1
1− b/µY
)
EˆtC˜t+1
)}
+
[
φ3
(
σ (φ1 − φ2)− φ2b/µ
Y
1− b/µY
)]
Eˆt+1µ
Y
t+1
where φ1 =
ζV −σ
Λ
, φ2 =
γSΠ
ΛC(1−b/µY ) , and φ3 = bβ
(
µY
)−σ
.
7. The FONC for labor supply is
Λˆt =
θζtVˆ
−σ
t ψh
θ−1
t Sˆt
Wˆt/ωt
Taking logs of both sides yields
log Λˆt = log
(
θζtVˆ
−σ
t ψh
θ−1
t Sˆt
Wˆt/ωt
)
This equation is linear in logs; subtracting log Λˆ = log
(
θζVˆ −σψhθ−1Sˆ
Wˆ /ω
)
directly yields
Λ˜t = ζ˜t − σV˜t + (θ − 1) h˜t + S˜t − W˜t + ω˜t
116
8. The FONC for the geometric average of current and past habit-adjusted
consumption is
Πˆt = ζtVˆ
−σ
t ψh
θ
t + βEˆt
[
(1− γ) Sˆt+1
Sˆt
Πˆt+1
(
µYt+1
)1−σ]
Taking logs of both sides yields
log Πˆt = log
(
ζtVˆ
−σ
t ψh
θ
t + βEˆt
[
(1− γ) Sˆt+1
Sˆt
Πˆt+1
(
µYt+1
)1−σ])
Noting that log Πˆ = log
(
ζVˆ −σψhθ + β
[
(1− γ) Sˆ
Sˆ
Πˆ
(
µY
)1−σ])
, the first-order Taylor
series expansion around the steady state can be shown to be
ΠΠ˜t =
(
ζV −σψhθ
) (
ζ˜t − σV˜t + θh˜t
)
+
(
β(1− γ)Π (µY )1−σ)(EˆtS˜t+1 − S˜t + EˆtΠ˜t+1 + (1− σ)Eˆtµ˜Yt+1)
9. The FONC for capital is
QˆtΛˆt = βEˆt
[
Λˆt+1
(
Rˆt+1ut+1 + Qˆt+1(1− δ(ut+1))
)(
µAt+1
(
µYt+1
)−σ)]
Taking logs of both sides yields
log Qˆt + log Λˆt = log
(
βEˆt
[
Λˆt+1
(
Rˆt+1ut+1 + Qˆt+1(1− δ(ut+1))
)(
µAt+1
(
µYt+1
)−σ)])
117
Noting that log Qˆ + log Λˆ = log
(
β
[
Λˆ
(
Rˆu+ Qˆ(1− δ(u))
)(
µA
(
µY
)−σ)])
, the
first-order Taylor series expansion around the steady state can be shown to be
Q˜t + Λ˜t =
(
βµA
(
µY
)−σ
Q
){
(Ru+Q(1− δ(u)))
(
EˆtΛ˜t+1 + Eˆtµ˜
A
t+1 − σEˆtµ˜Yt+1
)
+ (Ru)EˆtR˜t+1
+ ((R−Qδ′(u))u) Eˆtu˜t+1 + (Q(1− δ(u))) EˆtQ˜t+1
}
10. The FONC for the capacity utilization rate of capital is
Rˆt = δ
′(ut)Qˆt
Taking logs of both sides yields
log Rˆt = log (δ
′(ut)) + log Qˆt
Thus the first-order Taylor series expansion around the steady state can be shown to
be
R˜t =
δ′′(u)u
δ′(u)
u˜t + Q˜t
11. The FONC for investment is
Λˆt = z
I
t
(
1− Φ
(
Iˆt
Iˆt−1
µKt
)
− Φ′
(
Iˆt
Iˆt−1
µKt
)(
Iˆt
Iˆt−1
µKt
))
QˆtΛˆt
+ βEˆt
[(
zIt+1Φ
′
(
It+1
It
µKt+1
)(
It+1
It
µKt+1
)2
Qˆt+1Λˆt+1
)(
µAt+1
(
µYt+1
)−σ)]
118
Taking logs of both sides yields
log Λˆt = log
(
zIt
(
1− Φ
(
Iˆt
Iˆt−1
µKt
)
− Φ′
(
Iˆt
Iˆt−1
µKt
)(
Iˆt
Iˆt−1
µKt
))
QˆtΛˆt
+βEˆt
[(
zIt+1Φ
′
(
It+1
It
µKt+1
)(
It+1
It
µKt+1
)2
Qˆt+1Λˆt+1
)(
µAt+1
(
µYt+1
)−σ)])
Noting that at steady state
log Λˆ = log
(
zI
(
1− Φ
(
Iˆ
Iˆ
µK
)
− Φ′
(
Iˆ
Iˆ
µK
)(
Iˆ
Iˆ
µK
))
QˆΛˆ
+βEˆ
[(
zIΦ′
(
I
I
µK
)(
I
I
µK
)2
QˆΛˆ
)(
µA
(
µY
)−σ)])
the first-order Taylor series expansion around the steady state can be shown to be
Λ˜t = z
IQ
{
z˜It + Q˜t + Λ˜t
+ Φ′′
(
µK
) (
µK
)2 [
I˜t−1 − µ˜Kt − I˜t
+βµA
(
µY
)−σ
µK
(
EˆtI˜t+1 + Eˆtµ˜
K
t+1 − I˜t
)]}
12. The wage valuation equation is
Wˆt = αh
Yˆt
ht
Taking logs of both sides yields
log Wˆt = logαh + log Yˆt − log ht
119
This is linear in logs. Subtracting both sides by log Wˆ = logαh + log Yˆ + log h the
log-linearized equation can be shown to be
W˜t = Y˜t − h˜t
13. The effective-capital valuation equation is
Rˆt = αk
Yˆt
ut
Kˆt−1
µKt
Taking logs of both sides yields
log Rˆt = logαk + log Yˆt − log ut − log Kˆt−1 + log µKt
This is linear in logs. Subtracting both sides by log Rˆ = logαk + log Yˆ − log u −
log Kˆt−1 + log µK the log-linearized equation can be shown to be
R˜t = Y˜t − u˜t − K˜t−1 + µ˜Kt
14. Output growth is given by
µYt = µ
X
t
(
µAt
)αk/(αk−1)
Taking logs of both sides yields
log µYt = log µ
X
t −
(
αk
1− αk
)
log µAt
120
This is linear in logs. Subtracting both sides by log µY = log µX −
(
αk
1−αk
)
log µA the
log-linearized equation can be shown to be
µ˜Yt = µ˜
X
t −
(
αk
1− αk
)
µ˜At
15. Capital (and investment) growth is given by
µKt =
µYt
µAt
Taking logs of both sides yields
log µKt = log µ
Y
t − log µAt
This is linear in logs. Subtracting both sides by log µK = log µY − log µA the log-
linearized equation can be shown to be
µ˜Kt = µ˜
Y
t − µ˜At
16. The stochastic trend for government spending is
XG,Yt =
(
XG,Yt−1
)ρx,g
µYt
Taking logs of both sides yields
logXG,Yt = ρx,g logX
G,Y
t−1 − log µYt
121
This is linear in logs. Subtracting both sides by logXG,Y = ρx,g logX
G,Y − log µY the
log-linearized equation can be shown to be
X˜G,Yt = ρx,gX˜
G,Y
t−1 − µ˜Yt
17. The stationary exogenous stochastic processes x =
{
z, ζ, zI , g, ω, µA, µX
}
all evolve
according to
xt = x
1−ρxxρxt−1
0
x,t
Taking logs of both sides yields
lnxt = (1− ρx)x+ ρx lnxt−1 + ε0x,t
where ε0x,t = ln 
0
x,t. This is linear in logs. Subtracting both sides by lnx =
(1− ρx)x+ ρx lnx the log-linearized equation can be shown to be
x˜t = ρxx˜t−1 + ε0x,t
122
REFERENCES CITED
Adam, K. (2005). Learning to Forecast and Cyclical Behavior of Output and Inflation.
Macroeconomic Dynamics, 9(01):1–27.
Aiyagari, S. R. (1994). ”Uninsured Idiosyncratic Risk and Aggregate Saving”. The
Quarterly Journal of Economics, pages 659–684.
Alexopoulos, M. (2011). “Read All about It!! What Happens Following a Technology
Shock?”. American Economic Review, 101(4):1144–79.
Allen, S. G. (1985). “Why Construction Industry Productivity is Declining”. The Review
of Economics and Statistics, 67(4):661–69.
An, S. and Schorfheide, F. (2007). Bayesian Analysis of DSGE Models. Econometric
reviews, 26(2-4):113–172.
Angeletos, G.-M., Hellwig, C., and Pavan, A. (2007). “Dynamic Global Games of Regime
Change: Learning, Multiplicity, and the Timing of Attacks”. Econometrica,
75(3):711–756.
Angeletos, G.-M. and La’O, J. (2010). ”Noisy Business Cycles”. In NBER
Macroeconomics Annual 2009, Volume 24, pages 319–378. University of Chicago
Press.
Angeletos, G.-M. and Pavan, A. (2004). “Transparency of Information and Coordination
in Economies with Investment Complementarities”. Technical report, National
Bureau of Economic Research.
Assenza, T., Bao, T., Hommes, C., Massaro, D., et al. (2014). Experiments on
expectations in macroeconomics and finance. Experiments in macroeconomics,
17:11A`70.
Barsky, R. B. and Sims, E. R. (2012). “Information, Animal Spirits, and the Meaning of
Innovations in Consumer Confidence”. American Economic Review, 102(4):1343–77.
Basu, S., Fernald, J., and Kimball, M. (2004). Are Technology Improvements
Contractionary? Technical report, National Bureau of Economic Research.
Basu, S., Fernald, J. G., and Kimball, M. S. (2006). Are Technology Improvements
Contractionary? The American Economic Review, 96(5):1418–1448.
123
Beaudry, P. and Lucke, B. (2010). Letting Different Views about Business Cycles
Compete. In NBER Macroeconomics Annual 2009, Volume 24, pages 413–455.
University of Chicago Press.
Beaudry, P., Nam, D., and Wang, J. (2011). Do Mood Swings Drive Business Cycles and
Is It Rational? Technical report, National Bureau of Economic Research.
Beaudry, P. and Portier, F. (2004). “An Exploration into Pigou’s Theory of Cycles”.
Journal of Monetary Economics, 51(6):1183–1216.
Beaudry, P. and Portier, F. (2006). “Stock Prices, News, and Economic Fluctuations”.
American Economic Review, 96(4):1293–1307.
Beaudry, P. and Portier, F. (2014). “News-Driven Business Cycles: Insights and
Challenges”. Journal of Economic Literature, 52(4):993–1074.
Becker, R. A. (1980). ”On the Long-run Steady State in a Simple Dynamic Model of
Equilibrium with Heterogeneous Households”. The Quarterly Journal of Economics,
95(2):375–382.
Benhabib, J. and Farmer, R. E. (1994). “Indeterminacy and Increasing Returns”. Journal
of Economic Theory, 63(1):19–41.
Bernanke, B. S. and Mishkin, F. S. (1997). “Inflation Targeting: A New Framework for
Monetary Policy?”. Technical report, National Bureau of Economic Research.
Blanchard, O. J. and Kahn, C. M. (1980). The Solution of Linear Difference Models under
Rational Expectations. Econometrica: Journal of the Econometric Society, pages
1305–1311.
Blanchard, O. J., L’Huillier, J.-P., and Lorenzoni, G. (2013). “News, Noise, and
Fluctuations: An Empirical Exploration”. American Economic Review,
103(7):3045–70.
Blinder, A. S. (1999). “Central Banking in Theory and Practice”. Mit press.
Blinder, A. S. (2013). “After the Music Stopped: The Financial Crisis, the Response, and
the Work Ahead”. Penguin.
Branch, W., Evans, G. W., and McGough, B. (2012a). Finite-horizon learning. In CDMA
Working Paper.
Branch, W., Evans, G. W., and McGough, B. (2012b). “Finite Horizon Learning”. CDMA
Working Paper Series 201204, Centre for Dynamic Macroeconomic Analysis.
124
Bullard, J. and Eusepi, S. (2014). ”When Does Determinacy Imply Expectational
Stability?”. International Economic Review, 55(1):1–22.
Burnside, C., Eichenbaum, M., and Rebelo, S. (1995). “Capital Utilization and Returns to
Scale”. In NBER Macroeconomics Annual 1995, Volume 10, pages 67–124. MIT
Press.
Cho, S. (2015). “Determinacy and E-stability Under Reduced-form Learning”. Economics
Letters.
Christiano, L. J., Eichenbaum, M., and Evans, C. L. (2005). Nominal Rigidities and the
Dynamic Effects of a Shock to Monetary Policy. Journal of political Economy,
113(1):1–45.
Dombeck, B. (2016). “Implications of Solving the Co-movement Problem vis-a-vis
Expectational Stability”.
Ellison, M. and Pearlman, J. (2011). “Saddlepath Learning”. Journal of Economic
Theory, 146(4):1500–1519.
Eusepi, S. and Preston, B. (2010). “Central Bank Communication and Expectations
Stabilization”. American Economic Journal: Macroeconomics, 2(3):235–71.
Eusepi, S. and Preston, B. (2011). “Expectations, Learning, and Business Cycle
Fluctuations”. American Economic Review, 101(6):2844–72.
Evans, G. W. and Honkapohja, S. (2001). Learning and Expectations in Macroeconomics.
Princeton University Press.
Evans, G. W. and Honkapohja, S. (2006). “Monetary Policy, Expectations and
Commitment”. The Scandinavian Journal of Economics, 108(1):15–38.
Evans, G. W. and McGough, B. (2015). ”Learning to Optimize”. Working paper.
Farmer, R. E. and Guo, J.-T. (1994). “Real Business Cycles and the Animal Spirits
Hypothesis”. Journal of Economic Theory, 63(1):42–72.
Greenwood, J., Hercowitz, Z., and Huffman, G. W. (1988). “Investment, Capacity
Utilization, and the Real Business Cycle”. The American Economic Review, pages
402–417.
Guerro´n-Quintana, P. and Nason, J. M. (2012). ”Bayesian Estimation of DSGE Models.
Guo, J.-T., Sirbu, A.-I., and Weder, M. (2015). “News About Aggregate Demand and the
Business Cycle”. Journal of Monetary Economics, (0).
125
Haario, H., Saksman, E., and Tamminen, J. (2001). An Adaptive Metropolis Algorithm.
Bernoulli, pages 223–242.
Hellwig, C. (2005). “Heterogeneous Information and the Benefits of Transparency”.
Technical report, UCLA mimeo.
Hellwig, C. (2010). ”Comment on ”Noisy Business Cycles”. Technical report, National
Bureau of Economic Research.
Hommes, C., Sonnemans, J., Tuinstra, J., and Van De Velden, H. (2007). “Learning in
Cobweb Experiments”. Macroeconomic Dynamics, 11(S1):8–33.
Honkapohja, S., Mitra, K., and Evans, G. W. (2012). ”Notes on Agents Behavioral Rules
Under Adaptive Learning and Studies of Monetary Policy”.
Jaimovich, N. and Rebelo, S. (2009). “Can News about the Future Drive the Business
Cycle?”. American Economic Review, 99(4):1097–1118.
Karnizova, L. (2010). ”News versus Sunspot Shocks in a New Keynesian Model”.
Economics: The Open-Access, Open-Assessment E-Journal, 4.
Karnizova, L. V. (2007). ”News versus Sunspot Shocks in Linear Rational Expectations
Models”. Department of Economics, University of Ottawa= De´p. de Science
e´conomique, Universite´ d’Ottawa.
Khan, H. and Tsoukalas, J. (2012). “The Quantitative Importance of News Shocks in
Estimated DSGE Models”. Journal of Money, Credit and Banking, 44(8):1535–1561.
King, R. G., Plosser, C. I., and Rebelo, S. T. (1988). “Production, Growth and Business
Cycles: I. The Basic Neoclassical Model”. Journal of monetary Economics,
21(2):195–232.
Klein, P. (2000). Using the Generalized Schur Form to Solve a Multivariate Linear
Rational Expectations Model. Journal of Economic Dynamics and Control,
24(10):1405–1423.
Krusell, P. and McKay, A. (2010). “News Shocks and Business Cycles”. FRB Richmond
Economic Quarterly, 96(4):373–397.
Kydland, F. E. and Prescott, E. C. (1982). “Time to Build and Aggregate Fluctuations”.
Econometrica: Journal of the Econometric Society, pages 1345–1370.
Leeper, E. M., Richter, A. W., and Walker, T. B. (2012). “Quantitative Effects of Fiscal
Foresight”. American Economic Journal: Economic Policy, 4(2):115–44.
126
Lorenzoni, G. (2009). “A Theory of Demand Shocks”. American Economic Review,
99(5):2050–84.
Lucas, R. E. (1976). “Econometric Policy Evaluation: A Critique”. In Carnegie-Rochester
conference series on public policy, volume 1, pages 19–46. Elsevier.
Lucas, R. E. (1978). “Asset Prices in an Exchange Economy”. Econometrica: Journal of
the Econometric Society, pages 1429–1445.
Marcet, A. and Sargent, T. J. (1989a). “Convergence of Least-Squares Learning in
Environments with Hidden State Variables and Private Information”. The Journal of
Political Economy, pages 1306–1322.
Marcet, A. and Sargent, T. J. (1989b). “Convergence of Least Squares Learning
Mechanisms in Self-Referential Linear Stochastic Models”. Journal of Economic
theory, 48(2):337–368.
Marcet, A. and Sargent, T. J. (1989c). “Least-Squares Learning and the Dynamics of
Hyperinflation”. In International Symposia in Economic Theory and Econometrics,
edited by William Barnett, John Geweke, and Karl Shell, pages 119–137.
McCallum, B. T. (2007). “E-stability vis-a-vis Determinacy Results for a Broad Class of
Linear Rational Expectations Models”. Journal of Economic Dynamics and Control,
31(4):1376–1391.
McFadden, D. (1989). “A Method of Simulated Moments for Estimation of Discrete
Response Models Without Numerical Integration”. Econometrica: Journal of the
Econometric Society, pages 995–1026.
McGough, B. (2006). “Shocking Escapes”. Economic Journal, 116(511):507–528.
Milani, F. (2007). “Expectations, Learning and Macroeconomic Persistence”. Journal of
Monetary Economics, 54(7):2065–2082.
Milani, F. and Rajbhandari, A. (2012). Expectation Formation and Monetary DSGE
Models: Beyond the Rational Expectations Paradigm. Advances in Econometrics,
28:253.
Mitra, K., Evans, G. W., and Honkapohja, S. (2013). “Policy Change and Learning in the
RBC Model”. Journal of Economic Dynamics and Control, 37(10):1947 – 1971.
Morris, S. and Shin, H. S. (2002). “Social Value of Public Information”. The American
Economic Review, 92(5):1521–1534.
127
Morris, S., Shin, H. S., and Tong, H. (2006). “Social Value of Public Information: Morris
and Shin (2002) Is Actually Pro-Transparency, Not Con: Reply”. The American
Economic Review, pages 453–455.
Muth, J. F. (1961). “Rational Expectations and the Theory of Price Movements”.
Econometrica: Journal of the Econometric Society, pages 315–335.
Pigou, A. C. (1927). “Industrial Fluctuations”. Macmillan.
Preston, B. J. (2005). ”Learning About Monetary Policy Rules When Long-horizon
Expectations Matter”. 1.
Ramey, V. A. (2009). “Identifying Government Spending Shocks: It’s All in the Timing”.
Working Paper 15464, National Bureau of Economic Research.
Ramey, V. A. (2011). “Identifying Government Spending Shocks: It’s all in the Timing”.
The Quarterly Journal of Economics, 126(1):1–50.
Ruge-Murcia, F. J. (2007). Methods to Estimate Dynamic Stochastic General Equilibrium
Models. Journal of Economic Dynamics and Control, 31(8):2599–2636.
Schmitt-Grohe, S. and Uribe, M. (2012). “What’s News in Business Cycles”.
Econometrica, 80(6):2733–2764.
Shaby, B. and Wells, M. T. (2010). Exploring an adaptive metropolis algorithm. Technical
report.
Sims, C. A. (2002). Solving Linear Rational Expectations Models. Computational
economics, 20(1):1–20.
Sims, E. (2016). What’s news in News? A cautionary note on using a variance
decomposition to assess the quantitative importance of news shocks. Journal of
Economic Dynamics and Control, 73(1):41–60.
Slobodyan, S. and Wouters, R. (2007). Adaptive Learning in an Estimated Medium–Size
DSGE Model. Technical report, mimeo.
Slobodyan, S. and Wouters, R. (2008). Estimating a Medium–scale DSGE Model with
Expectations Based on Small Forecasting Models. Technical report, mimeo.
Slobodyan, S. and Wouters, R. (2012). Learning in an Estimated Medium-scale DSGE
Model. Journal of Economic Dynamics and control, 36(1):26–46.
Smets, F. and Wouters, R. (2007). “Shocks and Frictions in US Business Cycles: A
Bayesian DSGE Approach”. American Economic Review, 97(3):586–606.
128
Stiglitz, J. E. (2014). “Reconstructing Macroeconomic Theory to Manage Economic
Policy”. Working Paper 20517, National Bureau of Economic Research.
Svensson, L. E. O. (2006). “Social Value of Public Information: Comment: Morris and
Shin (2002) Is Actually Pro-Transparency, Not Con”. American Economic Review,
96(1):448–452.
Tong, H. (2004). “Do Transparency Standards Improve Macroeconomic Forecasting?”.
Unpublished Paper, 455.
Uhlig, H. et al. (1995). A Toolkit for Analyzing Nonlinear Dynamic Stochastic Models
Easily.
Veldkamp, L. L. (2011). “Information Choice in Macroeconomics and Finance”. Princeton
University Press.
Williams, N. (2003a). ”Adaptive Learning and Business Cycles”.
Williams, N. (2003b). ”Adaptive Learning and Business Cycles. Manuscript, Princeton
University.
Woodford, M. (2007). “The Case for Forecast Targeting as a Monetary Policy Strategy”.
Journal of Economic Perspectives, 21(4):3–24.
Woodford, M. (2013). “Fedspeak: Does it Matter How Central Bankers Explain
Themselves?”.
Yang, S.-C. S. and Traum, N. (2011). “When Does Government Debt Crowd Out
Investment?”. In 2011 Meeting Papers, number 479. Society for Economic Dynamics.
129