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Chapter 1
Introduction
Science is facts; just as houses are made of stones,
so is science made of facts; but a pile of stones is not
a house and a collection of facts is not necessarily
science.
—Henri Poincaré
1.1 Background
The seminal contribution of Kydland and Prescott (1982) marked the crest
of a sea change in the way macroeconomists conduct empirical research.
Under the empirical paradigm that remained predominant at the time, the
focus was either on purely statistical (or reducedform) characterizations of
macroeconomic behavior, or on systemsofequations models that ignored
both generalequilibrium considerations and forwardlooking behavior on
the part of purposeful decision makers. But the powerful criticism of this
approach set forth by Lucas (1976), and the methodological contributions
of, for example, Sims (1972) and Hansen and Sargent (1980), sparked a
transition to a new empirical paradigm. In this transitional stage, the for
mal imposition of theoretical discipline on reducedform characteriza
tions became established. The source of this discipline was a class of mod
els that have come to be known as dynamic stochastic general equilibrium
(DSGE) models. The imposition of discipline most typically took the form
of “crossequation restrictions,” under which the stochastic behavior of a
set of exogenous variables, coupled with forwardlooking behavior on the
part of economic decision makers, yield implications for the endogenous
stochastic behavior of variables determined by the decision makers. Never
theless, the imposition of such restrictions was indirect, and reducedform
specifications continued to serve as the focal point of empirical research.
Kydland and Prescott turned this emphasis on its head. As a legacy of
their work, DSGE models no longer serve as indirect sources of the
oretical discipline to be imposed upon statistical specifications. Instead,
they serve directly as the foundation upon which empirical work may be
conducted. The methodologies used to implement DSGE models as foun
dational empirical models have evolved over time and vary considerably.
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4
1 Introduction
The same is true of the statistical formality with which this work is con
ducted. But despite the characteristic heterogeneity of methods used in
pursuing contemporary empirical macroeconomic research, the influence
of Kydland and Prescott remains evident today.
This book details the use of DSGE models as foundations upon which
empirical work may be conducted. It is intended primarily as an instruc
tional guide for graduate students and practitioners, and so contains a dis
tinct howto perspective throughout. The methodologies it presents are
organized roughly following the chronological evolution of the empirical
literature in macroeconomics that has emerged following the work of Kyd
land and Prescott; thus it also serves as a reference guide. Throughout,
the methodologies are demonstrated using applications to three bench
mark models: a realbusinesscycle model (fashioned after King, Plosser,
and Rebelo, 1988); a monetary model featuring monopolistically compe
titive firms (fashioned after Ireland, 2004a); and an assetpricing model
(fashioned after Lucas, 1978).
The empirical tools outlined in the text share a common foundation: a
system of nonlinear expectational difference equations derived as the solu
tion of a DSGE model. The strategies outlined for implementing these
models empirically typically involve the derivation of approximations of
the systems, and then the establishment of various empirical implications
of the systems. The primary focus of this book is on the latter component
of these strategies: This text covers a wide range of alternative methodolo
gies that have been used in pursuit of a wide range of empirical applications.
Demonstrated applications include: parameter estimation, assessments of
fit and model comparison, forecasting, policy analysis, and measurement
of unobservable facets of aggregate economic activity (e.g., measurement
of productivity shocks).
1.2 Overview
This book is divided into three parts. Part I presents foundational material
included to help keep the book selfcontained. Following this introduc
tion, chapter 2 outlines two preliminary steps often used in converting a
given DSGE model into an empirically implementable system of equations.
The first step involves the linear approximation of the model; the second
step involves the solution of the resulting linearized system. The solution
takes the form of a statespace representation for the observable variables
featured in the model.
Chapter 3 presents two important preliminary steps often needed for
priming data for empirical analysis: removing trends and isolating cycles.
The purpose of these steps is to align what is being measured in the data
with what is being modelled by the theory. For example, the separation of
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1.2 Overview
5
trend from cycle is necessary in confronting trending data with models of
business cycle activity.
Chapter 4 presents tools used to summarize properties of the data.
First, two important reducedform models are introduced: autoregressive
moving average models for individual time series, and vector autoregressive
models for sets of time series. These models provide flexible characteriza
tions of the data that can be used as a means of calculating a wide range of
important summary statistics. Next, a collection of popular summary statis
tics (along with algorithms available for calculating them) are introduced.
These statistics often serve as targets for estimating the parameters of struc
tural models, and as benchmarks for judging their empirical performance.
Empirical analyses involving collections of summary statistics are broadly
categorized as limitedinformation analyses. Finally, the Kalman filter is
presented as a means for pursuing likelihoodbased, or fullinformation,
analyses of statespace representations. Part I concludes in chapter 5
with an introduction of the benchmark models that serve as examples in
part II.
Part II, composed of chapters 6 through 9, presents the following empir
ical methodologies: calibration, limitedinformation estimation, maximum
likelihood estimation, and Bayesian estimation. Each chapter contains a
general presentation of the methodology, and then presents applications
of the methodology to the benchmark models in pursuit of alternative
empirical objectives.
Chapter 6 presents the most basic empirical methodology covered in the
text: the calibration exercise, as pioneered by Kydland and Prescott (1982).
Original applications of this exercise sought to determine whether models
designed and parameterized to provide an empirically relevant account of
longterm growth were also capable of accounting for the nature of short
term fluctuations that characterize businesscycle fluctuations, summarized
using collections of sample statistics measured in the data. More generally,
implementation begins with the identification of a set of empirical mea
surements that serve as constraints on the parameterization of the model
under investigation: parameters are chosen to insure that the model can
successfully account for these measurements. (It is often the case that cer
tain parameters must also satisfy additional a priori considerations.) Next,
implications of the duly parameterized model for an additional set of statis
tical measurements are compared with their empirical counterparts to judge
whether the model is capable of providing a successful account of these
additional features of the data. A challenge associated with this method
ology arises in judging success, because this secondstage comparison is
made in the absence of a formal statistical foundation.
The limitedinformation estimation methodologies presented in chap
ter 7 serve as one way to address problems arising from the statistical
informality associated with calibration exercises. Motivation for their im
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6
1 Introduction
plementation stems from the fact that there is statistical uncertainty asso
ciated with the set of empirical measurements that serve as constraints in
the parameterization stage of a calibration exercise. For example, a sample
mean has an associated sample standard error. Thus there is also statis
tical uncertainty associated with model parameterizations derived from
mappings onto empirical measurements (referred to generally as statisti
cal moments). Limitedinformation estimation methodologies account for
this uncertainty formally: the parameterizations they yield are interpretable
as estimates, featuring classical statistical characteristics. Moreover, if the
number of empirical targets used in obtaining parameter estimates exceeds
the number of parameters being estimated (i.e., if the model in question is
overidentified), the estimation stage also yields objective goodnessoffit
measures that can be used to judge the model’s empirical performance.
Prominent examples of limitedinformation methodologies include the
generalized and simulated methods of moments (GMM and SMM), and
indirectinference methods.
Limitedinformation estimation procedures share a common trait: they
are based on a subset of information available in the data (the targeted
measurements selected in the estimation stage). An attractive feature of
these methodologies is that they may be implemented in the absence of
explicit assumptions regarding the underlying distributions that govern the
stochastic behavior of the variables featured in the model. A drawback is
that decisions regarding the moments chosen in the estimation stage are
often arbitrary, and results (e.g., regarding fit) can be sensitive to parti
cular choices. Chapters 8 and 9 present fullinformation counterparts to
these methodologies: likelihoodbased analyses. Given a distributional
assumption regarding sources of stochastic behavior in a given model,
chapter 8 details how the full range of empirical implications of the model
may be assessed via maximumlikelihood analysis, facilitated by use of the
Kalman filter. Parameter estimates and model evaluation are facilitated in
a straightforward way using maximumlikelihood techniques. Moreover,
given model estimates, the implied behavior of unobservable variables
present in the model (e.g., productivity shocks) may be inferred as a by
product of the estimation stage.
A distinct advantage in working directly with structural models is that,
unlike their reducedform counterparts, one often has clear a priori guid
ance concerning their parameterization. For example, specifications of sub
jective annual discount rates that exceed 10% may be dismissed outof
hand as implausible. This motivates the subject of chapter 9, which details
the adoption of a Bayesian perspective in bringing fullinformation pro
cedures to bear in working with structural models. From the Bayesian
perspective, a priori views on model parameterization may be incorporated
formally in the empirical analysis, in the form of a prior distribution. Cou
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1.2 Overview
7
pled with the associated likelihood function via Bayes’ Rule, the corre
sponding posterior distribution may be derived; this conveys information
regarding the relative likelihood of alternative parameterizations of the
model, conditional on the specified prior and observed data. In turn, con
ditional statements regarding the empirical performance of the model
relative to competing alternatives, the implied behavior of unobservable
variables present in the model, and likely future trajectories of model vari
ables may also be derived. A drawback associated with the adoption of a
Bayesian perspective in this class of models is that posterior analysis must
be accomplished via the use of sophisticated numerical techniques; special
attention is devoted to this problem in the chapter.
Part III outlines how nonlinear model approximations can be used in
place of linear approximations in pursuing the empirical objectives de
scribed throughout the book. Chapter 10 presents three leading alterna
tives to the linearization approach to model solution presented in chapter
2: projection methods, valuefunction iterations, and policyfunction
iterations. Chapter 11 then describes how the empirical methodologies pre
sented in chapters 6–9 may be applied to nonlinear approximations of the
underlying model produced by these alternative solution methodologies.
The key step in shifting from linear to nonlinear approximations involves
the reliance upon simulations from the underlying model for characterizing
its statistical implications. In conducting calibration and limitedinforma
tion estimation analyses, simulations are used to construct numerical
estimates of the statistical targets chosen for analysis, because analytical
expressions for these targets are no longer available. And in conducting full
information analyses, simulations are used to construct numerical approx
imations of the likelihood function corresponding with the underlying
model, using a numerical tool known as the particle filter.
The organization we have chosen for the book stems from our view that
the coverage of empirical applications involving nonlinear model approx
imations is better understood once a solid understanding of the use of
linear approximations has been gained. Moreover, linear approximations
usefully serve as complementary inputs into the implementation of nonlin
ear approximations. However, if one wished to cover linear and nonlinear
applications in concert, then we suggest the following approach. Begin
exploring modelsolution techniques by covering chapters 2 and 10 simul
taneously. Then having worked through chapter 3 and sections 4.1 and 4.2
of chapter 4, cover section 4.3 of chapter 4 (the Kalman filter) along with
section 11.2 of chapter 11 (the particle filter). Then proceed through chap
ters 5–9 as organized, coupling section 7.3.4 of chapter 7 with section 11.1
of chapter 11.
In the spirit of reducing barriers to entry into the field, we have devel
oped a textbook Web site that contains the data sets that serve as examples
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means without prior written permission of the publisher.
8
1 Introduction
throughout the text, as well as computer code used to execute the method
ologies we present. The code is in the form of procedures written in the
GAUSS programming language. Instructions for executing the proce
dures are provided within the individual files. The Web site address is
http://www.pitt.edu/˜dejong/text.htm. References to procedures avail
able at this site are provided throughout this book. In addition, a host
of freeware is available throughout the Internet. In searching for code,
good starting points include the collection housed by Christian Zimmer
man in his Quantitative Macroeconomics Web page, and the collection of
programs that comprise DYNARE:
http://dge.repec.org/
http://www.cepremap.cnrs.fr/∼michel/dynare/
Much of the code provided at our Web site reflects the modification of
code developed by others, and we have attempted to indicate this explic
itly whenever possible. Beyond this attempt, we express our gratitude to
the many generous programmers who have made their code available for
public use.
1.3 Notation
A common set of notation is used throughout the text in presenting models
and empirical methodologies. A summary is as follows. Steady state values
of levels of variables are denoted with an upper bar. For example, the steady
state value of the level of output yt is denoted as y. Logged deviations of
variables from steady state values are denoted using tildes; e.g.,
�
�
yt
�
.
y t = log
y
The vector xt denotes the collection of model variables, written (unless
indicated otherwise) in terms of logged deviations from steady state values;
e.g.,
yt
xt = [�
�
ct
�
n t ]′ .
The vector υt denotes the collection of structural shocks incorporated in
the model, and ηt denotes the collection of expectational errors associ
ated with intertemporal optimality conditions. Finally, the k × 1 vector µ
denotes the collection of “deep” parameters associated with the structural
model.
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9
1.3 Notation
Loglinear approximations of structural models are represented as
Axt +1 = Bxt + C υt +1 + Dηt +1 ,
(1.1)
where the elements of the matrices A, B, C , and D are functions of the
structural parameters µ. Solutions of (1.1) are expressed as
xt +1 = F (µ)xt + G(µ)υt +1 .
(1.2)
In (1.2), certain variables in the vector xt are unobservable, whereas
others (or linear combinations of variables) are observable. Thus filter
ing methods such as the Kalman filter must be used to evaluate the system
empirically. The Kalman filter requires an observer equation linking observ
ables to unobservables. Observable variables are denoted by Xt , where
Xt = H (µ)′ xt + ut ,
(1.3)
with
E(ut ut′ ) = u .
The presence of ut in (1.3) reflects the possibility that the observations of
Xt are associated with measurement error. Finally, defining
et +1 = G(µ)υt +1 ,
the covariance matrix of et +1 is given by
Q (µ) = E(et et′ ).
(1.4)
Given assumptions regarding the stochastic nature of measurement er
rors and the structural shocks, (1.2)–(1.4) yield a loglikelihood function
log L(X |), where collects the parameters in F (µ), H (µ), u , and
Q (µ). Often, it will be convenient to take as granted mappings from µ to
F , H , u , and Q . In such cases the likelihood function will be written as
L(X |µ).
Nonlinear approximations of structural models are represented using
three equations, written with variables expressed in terms of levels. The
first characterizes the evolution of the state variables st included in the
model:
st = f (st −1 , υt ),
(1.5)
where once again υt denotes the collection of structural shocks incorpo
rated in the model. The second equation is known as a policy function,
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10
1 Introduction
which represents the optimal specification of the control variables ct in
cluded in the model as a function of the state variables:
ct = c(st ).
(1.6)
The third equation maps the full collection of model variables into the
observables:
Xt = �
g (st , ct , υt , ut )
≡ g(st , ut ),
(1.7)
(1.8)
where once again ut denotes measurement error. Parameters associated
with f (st −1 , υt ), c(st ), and g(st , ut ) are again obtained as mappings from
µ, thus their associated likelihood function is also written as L(X |µ).
The next chapter has two objectives. First, it outlines procedures for
mapping nonlinear systems into (1.1). Next, it presents various solution
methods for deriving (1.2), given (1.1).
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