Download This PDF is a selection from a published volume from... Economic Research Volume Title: NBER Macroeconomics Annual 2007, Volume 22

This PDF is a selection from a published volume from the National Bureau of
Economic Research
Volume Title: NBER Macroeconomics Annual 2007, Volume 22
Volume Author/Editor: Daron Acemoglu, Kenneth Rogoff and Michael
Woodford, editors
Volume Publisher: University of Chicago Press
Volume ISBN: 978-0-226-00202-6
Volume URL: http://www.nber.org/books/acem07-1
Conference Date: March 30-31, 2007
Publication Date: June 2008
Chapter Title: How Structural Are Structural Parameters?
Chapter Author: Jesús Fernández-Villaverde, Juan F. Rubio-Ramírez
Chapter URL: http://www.nber.org/chapters/c4087
Chapter pages in book: (p. 83 - 137)
2_
How
Are
Structural
Jesus Fernandez-
Villaverde,
Parameters?
Structural
University
of Pennsylvania,
NBER,
andCEPR
Juan F. Rubio-Ramirez,
ofAtlanta
1
Duke University
and Federal Reserve Bank
Introduction
This paper studies the following problem: how stable over time are the
of dynamic
stochastic general equilib
so-called "structural parameters"
rium (DSGE) models?
To answer this question, we estimate amedium
scale DSGE model with real and nominal
rigidities, using U.S. data. In
our model, we allow for parameter
of
drifting and rational expectations
the agents with respect to this drift. We document
that there is strong
our sample. In particular, we
that parameters
evidence
change within
in the parameters
illustrate variations
the monetary
describing
policy
function and in the parameters
characterizing
we show
of firms and households.
Moreover,
reaction
havior
ments
in the
pricing
parameters
are
correlated
with
the pricing be
the move
how
inflation.
Thus,
our
on the empirical
relevance of Calvo models.
are at the core of
Our findings are important because DSGE models
to be a laboratory
modern
macroeconomics.
that re
They promise
to match
searchers can employ
to
with
economic
theory
reality,
design
cast doubts
results
has captured
policy, and to evaluate welfare. The allure of DSGE models
the imagination
a
of many,
inside and outside academia.
In universities,
multitude
and
of economists
fashions.
More
institutions
making
and forecasting.
2006),
2007),
Sweden
implement DSGE models
remarkable
still, a burgeoning
are estimating
DSGE models
The Federal Reserve
the European
Central
Bank
the Bank of Canada
(Murchison
Laseen, Linde,
(Adolfson,
Spain (Andres, Burriel and Estrada
tide, but a dozen other institutions
in their rich varieties
number
of policy
for policy
analysis
Board (Erceg, Guerrieri,
and Gust
and Warne
Coenen,
(Christoffel,
and Rennison
2006), the Bank of
2005), and the Bank of
are
at the leading
2006)
edge of the
are jumping on the
In
bandwagon.
and Villani
84 Fernandez-Villaverde
and
Rubio-Ramirez
are
the good fore
economists
experience with
accumulating
even
of
DSGE
with
record
when
models,
casting
compared
judgmental
and Warne,
from staff economists
Coenen,
(Christoffel,
predictions
addition,
2007).
At
the center
we
of DSGE models
have
the "structural
parameters"
of the economy. We call these
and technology
the preferences
"structural"
in the sense of Hurwicz
(1962): they are invari
parameters
nature.
ant to interventions,
shocks
structural charac
The
including
by
ter of the parameters
is responsible
of the appeal of DSGE
for much
that define
are
the parameters
tive of economic
theory and
models.
Since
models
avoid
the Lucas
fully
invariant
critique
from the perspec
interpretable
to policy
DSGE
interventions,
and can be used
to quantitatively
eval
uate policy.
Our point
of departure
is that, at least at some level, it is hard to be
are really struc
lieve that the "structural parameters"
of DSGE models
are
we
in
interested
for policy anal
the
class
of
interventions
tural, given
us
think, for instance, about technology. Most DSGE models
ysis. Let
a stable production
function, perhaps
a
in
few
papers
Except
(Young 2004),
specify
subject to productivity
the features of the tech
growth.
like the elasticity of output to capital, are constant over time. But
nology,
in a world where
is untenable
this constant
technological
elasticity
can expect that changes in relative input prices
is
We
change
purposeful.
and that those
in the new technologies
will induce changes
developed
ar
to
of
into
elasticities
translate
different
output
may
inputs. Similar
guments
can be made
along nearly
every dimension
of amodern
DSGE
model.
the practice of esti
is not sufficient to dismiss
The previous
argument
as
constant parameter
values. Simplifying
models
with
DSGE
mating
eco
are
to
in
make
like
stable
progress
parameters,
required
sumptions,
as soon as we realize the possible
nomics. However,
changing nature of
we weaken
for inference exer
the justifications
structural parameters,
be
The separation
of DSGE modeling.
the program
cises underlying
form becomes much
and what
is reduced
is "structural"
tween what
more
ambiguous.1
of parameter
but not the necessity
The possibility
drifting motivates
the main question of this paper: how much evidence of parameter drift
is that we find much
is in the data? If the answer
ing in DSGE models
to decide "much" needs to be dis
support for drifting (where the metric
of our estimation
to
reevaluate
the usefulness
we
would
have
cussed),
exercises,
or at least modify
them
to account
for parameter
variation.
How
Structural Are Structural Parameters?
85
a
parameter drifting may also be interpreted as sign of model
as a
our models.
If
and, possibly,
misspecification
guide for improving
the answer is negative?that
is, ifwe find little parameter
drifting?we
as a
in DSGE models
to tackle
increase our confidence
would
procedure
Moreover,
relevant
policy
discussions.
our substantive
this paper also develops
Beyond addressing
question,
new tools for the estimation
of dynamic
with pa
equilibrium models
rameter drifting. We show how the combination
of perturbation
meth
ods and the particle filter allows the efficient estimation
of this class of
can be performed
on
economies.
Indeed, all the required computations
an average
a reasonable
in
amount
of
time.
We
computer
personal
hope
not neces
that those tools may be put to good use in other applications,
that involve time-varying
in
sarily in general equilibrium,
parameters
essential
ways.
Our main
results
changing
preciably
are as follows.
First, we offer compelling
proof of
in
the
Fed's behavior. Monetary
parameters
policy became ap
more aggressive
in its stand against inflation after Volcker's
This agrees with Clarida, Gali, and Gertler
(2000), Lubick
appointment.
and Schorfheide
Boivin
and
Rabanal
(2004),
(2006),
(2007). Our contri
a
bution is to rederive the result within
model where agents understand
and act upon the fact that monetary
policy changes over time.
we
the
the
Second,
expose
instability of the parameters
controlling
level of nominal
and
indexation
of prices and wages.
Those
rigidity
changes are strongly correlated with changes in inflation in an intuitive
way: lower rigidities correlate with higher inflation and higher rigidities
with
lower
ment
of nominal
inflation. Our
finding
suggests
that a more
thorough
treat
rigidities,
possibly
through state-dependent
pricing
models, may yield a high payoff.
We want
to be up front about the
of our exercise. First
shortcomings
and foremost, we face the limitations of the data. With 184
quarterly ob
servations of the U.S. economy,
there is a tight bound on how much we
can learn from the data
(Ploberger and Phillips 2003, frame the problem
of empirical
limits for time series models
in terms of informa
precisely
tion bounds).
The main
of the limitations
of the short
consequence
size
is
estimates.
sample
imprecise
relatively
The second limitation,
forcefully
emphasized
by Sims (2001), is that
we do not allow for
in
the
volatilities
innovations
of the model,
changing
is itself a particular
which
form of parameter
drift. If the innovations
in
the U.S. data are heteroskedastic
we
in
Fernandez-Villaverde
(as
report
and Rubio-Ramirez
to pick up the
2007), the estimation may attempt
86 Fernandez-Villaverde
Rubio-Ramirez
variance
At
by spurious changes in the structural parameters.
and
defend
that
there is still vari
time, Cogley
Sargent (2005)
in the parameters
of a vector autoregression
(VAR), even after con
changing
the same
ation
and
on an exten
We are currently working
trolling for heteroskedasticity.
sion of the model with both parameter
and
drifting
changing volatilities.
We build upon an illustrious
tradition of estimating models with pa
rameter
where
drifting. One classic reference
the authors studied the estimation
are subject to permanent
in this tradition
niques
sion's
framework
tigation.
Our paper
and transitory
are within
the context
linked with
signs of parameter
this class of models
drifting
is a new
relevant
estimates
literature
or stochastic
Mihov
Zha
volatility.
on
of time variation
about whether
of the Cowles
application
Commis
to our inves
a
growing body of research that shows
Since the estimation
of
dynamic models.
undertaking,
VARs with
Examples
and Sargent
(1998), Cogley
(2006). The consensus
evidence
a dispute
are of little direct
and, hence,
is also
is Cooley
and Prescott
(1976),
of regression parameters
that
shocks. Unfortunately,
the tech
the evidence
is scattered.
One
and/
parameters
time-varying
include Uhlig
and
(1997), Bernanke
Primiceri
and
Sims
and
(2005),
(2005),
from these papers
is that there is
emerging
in the parameters
of a VAR, although
there is
comes from changes
the variation
in the au
or from stochastic
This evidence,
toregressive
components
volatility.
is only suggestive,
since a DSGE model with constant param
however,
eters may be compatible with a time-varying
VAR (Cogley and Sbor
done 2006).
A second group of studies has estimated
equilibrium models with pa
rameter variation,
tuations studied.
Justiniano
but it has been
less ambitious
Fernandez-Villaverde
in the extent of the fluc
and Rubio-Ramirez
and Primiceri
the importance
(2005) demonstrate
for U.S. data using aDSGE model. King
to account
volatility
with a simple real business
with
(2007) and
of stochastic
(2006) works
parameter drift in
cycle (RBC) economy
his approach
four parameters. However,
relies on particular properties
it
is
too
to
of his model
and
cumbersome
be of general applicability.
a small-scale New
Canova
model with pa
(2005) estimates
Keynesian
rameter drifting but without
the parameters.
He uncovers
that enter
estimates
roborates
with
the agents being aware
important movements
of these changes in
in the parameters
into the Phillips curve and the Euler equations.
Boivin (2006)
a
rule
with
real-time
data.
He cor
parameter-drifting
Taylor
in
of
rule
the
obtained
coefficients
previous
findings
changes
final data. Benati
(2006), elaborating
on an argument
by Woodford
Structural Are Structural Parameters?
How
87
inNew Keyne
introduced
the indexation mechanisms
(2006), questions
and shows that they are not structural to changes inmone
sian models
and Sichel (1996) find unstable pa
tary policy rules. Oliner, Rudebusch,
of
rameters even investment models with more intricate representations
and
than those found in current DSGE models. Owyang
spending
and
models
of
estimate
(2004)
monetary
policy
Ramey
regime-switching
of the monetary
the evolving
authority
through
preferences
identify
their interaction with the structural parameters.
capital
are also numerous
papers that tell us about parameter
drifting,
in an indirect way. A common practice when
estimating models
has been to divide the sample into two periods, usually before and after
in the inference re
1979, and argue that there are significant differences
There
albeit
is Clarida,
of this method
representative
a
we
will
discuss
later.
(2000),
paper
with our analysis
Finally, a literature that has connections
sults. One
celebrated
Gali,
and
Gertler
is the one
in dif
that deals with DSGE models with aMarkov-switching
process
or fiscal policy
like monetary
ferent aspects of the environment,
(Davig
and Leeper 2006, and
and Leeper 2006a and 2006b, Chung, Davig,
and Zha 2006). The stated motivation
of these papers
Farmer, Waggoner,
is that Markov
omy
switches
So far, none
better.
us understand
of the econ
the dynamics
help
an estimated
of these papers has produced
model.
The
discuss
different
equilibrium
parameter
medium-scale
duce
as follows. First, in section 2, we
is organized
in dynamic
to
think
about
ways
parameter
drifting
In section 3, we develop
two simple examples
models.
of
our investigation.
a
drift that motivate
Section 4 defines
rest of the article
this model
model
of the U.S. economy
and discusses
how to intro
to the data. Section 5 introduces parameter
drifting and
to adapt the approach
in section 4 to handle this situation.
explains how
We report our results in section 6. Section 7 concludes.
vides the interested reader with technical details.
2
Parameter
Drifting
and Dynamic
Equilibrium
An appendix
pro
Models
are at least three ways
to think about parameter
in an es
drifting
we
timated DSGE model.
The simplest
which
call
the pure
approach,
as a conven
econometric interpretation, is to consider parameter
drifting
There
ient phenomenon
to fit the data better or as the consequence
of a capri
nor forecast.
cious nature that agents in the model neither understand
its simplicity,
this interpretation
violates
the spirit of rational
Despite
88 Fernandez-Villaverde
not having
free parameters
expectations:
we will not investigate
with. Consequently,
and Rubio-Ramirez
that the researcher
can
play
this case further.
is as a characteris
The second way to think about parameter
drifting
tic of the environment
that the agents understand
and act upon. Let us
come back to our example of the production
function.
Imagine that the
function Yt = AK?
is given by a Cobb-Douglas
aggregate
technology
a
output Yt is produced with capital Kt and labor Lt given
L,1_a',where
with
and
the
The
difference
level
A
share
ar
parameter
only
technology
is that at is indexed by time (neither the realism
standard environment
our example
nor the empirical
is crucial for the argument,
justification of
we could argue in favor of both features). Let us also assume
although
at
that at evolves over time as a random walk with reflecting boundaries
function satisfies the usual prop
0 and 1, to ensure that the production
the new
erties. We could imagine that such drift comes about because
so
a random requirement
of
have
capital. The
technologies
developed
one
as
are
of
their
that
have
rules
decision
lution of the agents' problems
current
the current ar Why?
First, because
at determines
arguments
and
hence
to
values
future
forecast
because
Second,
at helps
prices.
at+j
is our favorite one, and itwill
to predict future prices. This interpretation
frame our reading of the results in section 6.
about parameter drifting
The final perspective
is as a telltale of model
(1971) and Rosenberg
by Cooley
point,
misspecification.
These
models.
DSGE
when
is
(1968),
estimating
cogent
particularly
for
are
them
useful
To
make
models
policy pur
complex constructions.
of the
that affect the dynamics
poses, researchers add many mechanisms
so
on.
In
addi
and
and
costs,
wages,
economy:
adjustment
sticky prices
for
the
models
DSGE
tion,
utility
assumptions
require right parametric
of shocks,
costs, distribution
function, adjustment
function, production
one di
the model
and so forth. Ifwe seriously misspecified
along at least
This
raised
as the only possibility
left to
parameter
drifting may appear
this point
in section 3 illustrates
to fit the data. Our example
results and as
in our empirical
in detail. We will exploit this possibility
the degree of nominal
sess how the drift in the parameters
determining
mension,
the model
implies
rigidity in the economy
decisions may be flawed.
3
that time-dependent
models
of pricing
Two Examples
that generate parameter
two simple examples
In this section, we present
to il
the examples
We
chosen
have
models.
DSGE
in estimated
drifting
Structural Are Structural Parameters?
How
89
our points as clearly as possible,
and not based on their rele
are not far-fetched:
or plausibility.
the
However,
examples
they
recurrent
in
deal with
the literature and are linked (albeit we
themes
lustrate
vance
do not explore
this connection
to its fullest)
to relevant
features
of the
economy.
3.1
Parameter
as a Consequence
Drift
first example
policy to different
deals with
The
variables.
Policies
in the elasticity
the changes
of monetary
It is common to postulate
that the monetary
to set the short-run nominal
in
operations
authority uses open market
to a Taylor
terest rate Rt according
Rt
of Changing
rule:
/
"<*? >
(Rt-i\lR\(nt\Hy^1-^
7f-hr)l(n)(?)J
II represents
the target levels of inflation of the monetary
R
the steady-state
gross return of capital, yt is output, and yt a
authority,
measure
ac
of target output. The term emt is a random shock distributed
The variable
cording to Jf(0,1).
In an influential
tracted the attention
7n before
Clarida, Gali, and Gertler
(2000) at
to changes in the elasticity parameter
as Fed chairman in 1979.
appointment
They
contribution,
of economists
and after Volcker's
of the Taylor rule, that
document, with a slightly different
specification
more
than
doubles
after
1979.
This
has
in
been corroborated
7n
finding
and
found resilient tomodifications
in the empirical
many studies
spec
ification
of the sample be
(Lubick and Schorfheide
2004). The division
the time before and after 1979 has also been exploited by Boivin
and Giannoni
(2006), who find that the point estimates of the structural
also
the two periods.
parameters
vary between
substantially
tween
are one particular
in the policy coefficients
Changes
example of pa
rameter drift. They can be the consequence
of the shifting priorities
of
the policymakers
of
in
the
or, as emphasized
by Sargent (1999),
changes
of the effectiveness
of monetary
perception
policy. Once we recognize
that there is evidence of the parameter
7n drifting over time, it is natural
to assume that agents are aware of the
changes and act upon them. Such
an environment
some
of
the insights of Sims (1980) about
may capture
a
the difference
between
change in policy regime (in our Taylor rule, a
in
the
rate is determined)
the
interest
and the evolution
of
way
change
the policy within one regime, which
could be represented
in our context
as the drift of the parameters
of the rule.
90
3.2
Parameter
Drift
and
Fernandez-Villaverde
as a Telltale
ofModel
Rubio-Ramirez
Misspecification
example revisits several of the themes in Browning, Hansen,
and Heckman
for inference of an
(1999). We explore the consequences
a model with
econometrician
lived agents when
estimating
infinitely
Our
second
are
an
model.
actually generated
by
overlapping-generations
show how our estimate of the discount
factor will be a function of the
the data
We
true discount
factor, the elasticity of output to capital, and the changing
of the population.
This example
is relevant because
distribution
age
in the age structure of the U.S. population
have been contin
variations
uous due to shifts in fertility and mortality.
3.2.1
World
We begin by creating a simple artificial
are two generations
In each period
of households
there
t,
alive,
maximizes
and old. Each household
the life utility
An Artificial
world.
young
logc;
+
pE,logc;+1
was born in
that the household
the superindex
denotes
where
t,
period
it consumes,
and Et is the conditional
the period in which
the subindex
for
factor, p, captures the preference
operator. The discount
expectations
current consumption.
We pick a log utility function to simplify the alge
bra that follows.
young and get a wage wt for a unit of time
live off their savings when
that they supply inelastically. Households
are c\ + st = wt and c\+1 =
are old. The period budget constraints
they
Households
Rt+1st,
where
ital. From
work
when
st is the household
the first order
savings
condition
=
[1/(1+ P)H and c\ [0/(1 + p)]wr
and
Rt+1
the
of households,
gross
we
return
have
on
cap
that c\ =
is born. For the mo
In each period, a number nt of new households
some random pro
assume
we
the
of
that
is
realization
will
ment,
lt
only
that the
cess. Nothing
is lost by assuming
of substance for our argument
is exogenous.
size of the new generation
is defined by a Cobb-Douglas
side of the economy
The production
=
in the econ
function, yt
kt is the total amount of capital
k^l)-*, where
in
we assume total depreciation
omy and lt the total amount of labor. If
the
condition
and
to
the
the economy,
lt
impose
algebra,
simplify
again
=
=
=
(1 a)/^/"-" and Rt
afcj^n J-".
nt, we get by competitive
pricing wt
in the econ
Total consumption
All that remains is some accounting.
omy
plus
in period
t, Ct, is equal to the consumption
of the young generation.
the consumption
of the old generation
all of
The old consume
Structural Are Structural Parameters?
How
91
is equal to the capital income of the economy, Rtkt =
their income, which
a fraction
consume
[1/(1 + P)] of their income,
akfn]-". The young
which
is equal to the labor income of the economy wtlt = (1 - a)fc"nf1_a.
is:
Then total consumption
1 + a(3
* l
1 + (3
'
resource
By the aggregate
ital in period t + 1) is
(1
a)P
=
=
t
L<
k
constraint,
investment
(or, equivalently,
cap
-_?r
t+1
1 + a(3 Lt
Finally, we
find per capita consumption
cvtcas:
Q
=
CPC-i
*
nt + *t-i
3.2.2
An
Econometrician
Let
us now
amodel
suppose
with
that we
have
an
a
who aims to estimate
in
representative
our
T
lived
and
observations
from
To
agent
economy.
finitely
generated
do so, the econometrician
that the agent has a utility function:
postulates
?
r t
econometrician
11(1+7*)
maxE,?p*
where
t -1
1 +
yt is the (random)
and t:
7
=
nt
nt-l
+
logcf
growth
rate of the population
between
periods
nt_x
+
nt-2
and 70 = 0. This utility function is the same as in the canonical presenta
tion of the RBC model
in Cooley
and Prescott
(1995) except that the
is stochastic
instead of constant. The pro
growth rate of the population
duction
side of the economy
is the same as before, yt = k^l]~a. Thus, the
the artificial world we have created and the
between
only difference
the econometrician
model
estimates
is that, instead of
two gen
having
erations alive in each moment,
a model
the econometrician
estimates
with a representative
agent.
are the consequences
on the estimated
What
parameters?
Imagine
that the econometrician
knows a and that the depreciation
factor
to estimate
the only remaining unknown
Then, a simple procedure
is 1.
pa
92
in the world,
rameter
Fernandez-Villaverde
and
factor p, is to build
the discount
Rubio-Ramirez
the population
moment:
+
^W
TU^r
W W+i
and substitute
_Vr-i_
. _
z"-?
r-i
Pt~ i
by the sample mean:
the expectation
&tc
r^7
;
evolves over time. First, note
study how this expression
the expressions
found before, we get:
substituting
We
a+
7l+1
1+ p 1
a
)R>^- (n'+i+n')2
nt + nt_x
c?+l
that, by
1-a
3
C(
Then:
.
=
3r
!_a
i
a
1+ 3
3-~
We want
to work
consumption
gregate
Ia(?. +
^ +
yT_1("t+l
on the previous
First, we substitute
expression.
for its value in terms of capital and labor:
T=
1-a ^"o^^^-^fc^p
1
ex
1+ P
(nt+1+ ntf
B-??
^t==0
nt +nf_1
The only remaining endogenous
nate it,we recursively
substitute
T(l-a)p
ag
1
Vt-i
BT
",)2 1
1
fcfnj"a
element
in this equation
is kt. To elimi
kt_{ to find:
VI
'"VCL-cOP
Then:
Vt
HT
nt+nt_l
IT
_ /(l-a)P
1 1-a ^"^Fin?_T
o
=o-i-y=-\-f_j.?i
Kl+p
a
_
^=0
(nt+,+ n)2 IT ?
'_1
(n,+ n,_1)nJ-alL
1-a
Wl
A 1+ *p<T_7 PI
/(l-a)p
+
\ 1 <*P
UI
/ J
V?
J
Structural Are Structural Parameters?
How
which
93
a PT,which
is biased and drifts over time according
to the
This expression
of the population.
is composed
of three parts.
the true parameter,
bias,
p, second the deterministic
delivers
evolution
First,
1
1+ P
1 -a
a
over time.
and finally the term involving
the nts and kQ,which fluctuates
further structure on population
Without
growth over time, it is diffi
cult to say much about PT. In the simple case where yt = 7 is constant, as
T ?> 00, the only factor dominating
is:
(1)
1+ P a
Pr-P-r^-1?^d+7)-2
To explore the behavior
of PT in the general case where
7, varies, we
simulate
and estimate
the model
the parameter
recursively with data
from an economy with a = 0.3 and p = 0.96. The growth rates of popu
lation are 2,4,3,1,2,
and 5 percent each for 50 periods
(i.e., for period 1
to 50, growth rate is 2 percent, for
period
so
our
and
We
results
forth).
percent
plot
evolution
cilitate
51 to 100, the growth rate is 4
in figure 2.1, where we see the
over
time of PT and how it inherits
(1),we superimpose
comparison with
the properties
of yt. To fa
the value of (1) that would
1.14i-.-1-.-
i
1.12-
-Estimate
of p
-Long-run
Limit
|
r-11-'
1.1=_,
j
1.06-
L-,
L_|
|
1.04!__:
1.02'-'-'-'-'-'
0
50
Figure 2.1
Estimate
of 0 versus
100
long-run
limit
150
200
250
300
94 Fernandez-Villaverde
and Rubio-Ramirez
be implied if the growth rate in a period stayed constant over time. The
to (1)within
each block of 50 periods.
graph shows how PT converges
4
The Baseline
Model
our
around a baseline New Keynesian
investigation
business
it is the paradigmatic
cycle model. We pick this model because
of
the
DSGE
economies
estimated
Since
representative
by practitioners.
we
on
on
occa
have
the
record
other
(Fernandez-Villaverde
2005)
gone
structure
We will
of this framework, we do not feel obliged
the problems
sions, criticizing
to repeat those shortcomings
here. Suffice it to say as amotivation
that
in
it
interest
institutions
the
level
of
this
model,
given
by policymaking
to see amore appropriate
is difficult
vessel for our exploration.
de
model
is well known
(see the book-length
we
our
in
will
be
brief
2003).
pres
scription
Consequently,
entation and will omit some of the technical aspects. On the other hand,
we need to discuss
at a certain level of de
the model
for concreteness,
of the model
tail. The interested reader can access the entire description
The New
Keynesian
inWoodford
at a complementary
at www.econ.upenri
posted
In
to clarify our
this
section,
.edu/~jesusfv/benchmark_DSGE.pdf.
we
in
the
model
without
will introduce
the
ideas,
parameters.
changes
In section 5, we will introduce the parameter
change over time.
4.1
technical
appendix
Households
house
is as follows. A representative
of the economy
hold consumes,
saves, holds real money
labor, and
balances,
supplies
curve and Calvo's
a demand
to
sets its own wages
pricing. The
subject
The basic
structure
a
is manufactured
final output
producer,
final-good
competitive
by
a
as
uses
continuum
of intermediate
which
goods manufactured
inputs
rent
The intermediate-good
producers
competitors.
by monopolistic
capital
and
labor
to manufacture
their good. Also,
that they can only
the intermediate
change prices fol
a
that fixes the
rule. Finally, there is monetary
authority
interest rate through open market operations with
nominal
producers
a Calvo's
lowing
good
face the constraint
one-period
of two unit
is induced by the presence
growth
public debt. Long-run
one
in
the
investment
and
one
in
neutral
the
of
level
roots,
technology
trends will allow us to estimate the
stochastic
These
specific technology.
model
with
We have
data.
the raw, undetrended
a continuum
in the economy
of households
indexed
by;. The
How
95
Structural Are Structural Parameters?
maximize
households
the following
lifetime utility
real money
balances,
cjt,
in consumption,
separable
worked,
ljt:
hZ
P'd, tog(c;,
-
+ v logM
hcjt_,)
which
function,
mjt/pt,
is
and hours
"
j ^7^
% is the in
factor, h controls habit persistence,
P is the discount
of Frisch labor supply elasticity, dt is a shock to intertemporal
pref
erence with the law of motion:
where
verse
log dt
and
tog ft
=
pd log dt_t +
where
(jdedrt
<ptis a labor supply
=
P9tog
~
zdJt Jt(0,1),
shock with
cp,_i+ <W where
the law of motion:
e9#,
~ >f
(0,1).
con
set of Arrow-Debreu
trade on the whole
securities,
on
events.
and
Our
indi
notation
a.t+1
tingent
aggregate
idiosyncratic
cates the amount of those securities that pay one unit of consumption
in
event
at time t at (real) price
To
by household/
wjt+lt purchased
qjt+u.
save on notation, we
on
the
event.
the
House
drop
explicit dependence
holds also hold an amount,
of government
bonds that pay a nominal
bjt,
- th
household's
gross interest rate of Rt and invest xt. Then, the/
budget
is:
constraint
Households
=
Wjtht
+
mit
K+i
rt
rt
c
lrt?jt K^jtWjt-i
~+
+
Rt-i-^
+
fy
+ Tt +
Ff
is the real wage,
rt the real rental price of capital, ujt > 0 the in
z^
use
of capital,
is the physical
cost of
in resource
tensity of
ujt
\L~l<l>(ujt)
shock
be
described
terms, |n, is an investment-specific
(to
technological
a
is
and
is
the
of
the firms
transfer,
Tt
momentarily),
lump-sum
F,
profits
in the economy. We assume that <J>(1)= 0, <&'and <?>"
> 0.
where
Investment
*,
=
where
(l-8)Vi
induces
+*
a law of motion
1~V(-^A
for capital:
xj*
8 is the depreciation
rate and V(-) is a quadratic
cost
adjustment
such that V(AX) = 0, where Ax is the growth rate of investment
the balance growth path. Note that we index
capital by the time its
function
along
xjt
96
level is decided.
autoregressive
jl,
=
The investment-specific
+
The first order
-
technological
shock
follows
an
process:
fVi exP(\
zM)
zM
with
conditions
and ?M - X(0,1)
=
where
hc^Y1
dt(cjt
and Rubio-Ramirez
Fernandez-Villaverde
a^
respect
b$ltdt+1(cjt+1
to
cjt, bjt, ujt, kjt,
and
xjt
are:
=
\;?
hep)'1
=
rt
%
ix-'c&'Cu,,),
-
=
PM-j^-K1
L
8Hm
+
~
r?+iVi
Cw.%)]
L
and
\ Xjt-l / Xjt-1 J
\ *;t-l /
con
the budget
associated with
A.., is the Lagrangian multiplier
asso
is the marginal
Tobin's Q, the Lagrangian multiplier
straint and
qjt
constraint normalized
ciated with the investment
by \jt.
adjustment
in
ismore
The first order condition with respect to labor and wages
is
The labor employed
volved.
sup
producers
by intermediate-good
where
a representative,
plied by
The
by each household/.
bor of households
l*=
\\^-^dj\
with
firm that hires the labor supplied
competitive
la
the differentiated
labor supplier aggregates
function:
the production
, (2)
y\ controls the elasticity of substitution
among different
types of
labor demand.
labor and ldtis the aggregate
func
The labor "packer" maximizes
profits subject to the production
and
the wage
labor wages wjt
tion (2), taking as given all differentiated
we
maximization
this
From
wt.
get:
problem
where
How
Structural Are Structural Parameters?
Then, to find the aggregated wage,
to deliver:
tion wtldt = IJ
wjtljtdj
we
97
again use
the zero profit
condi
\i/(i-m)
wt = (A
^ wjr^djj
a Calvo's
set their wages
following
setting. In each pe
their wages. All other
riod, a fraction 1 6Wof households
reoptimize
can
their
In
households
index
wages
by past inflation.
only partially
Households
dexation
is controlled
the household
cannot
ric
where
\w e (0,1). This implies
by the parameter
t
its
its normalized
for
wage
periods,
change
that if
wage
is nj=1(n**^y
after t periods
n,+s)w;;.f.
and separable utility in labor (see
Since we assume complete markets
we
on a symmet
and
will concentrate
Levin, 2000),
Erceg, Henderson,
equilibrium
?
=
=
c-t
ct, ujt
ut,
kjt_x
=
=
xt,
kt, xjt
Xjt
=
Xt, qjt
qt,
and w*t = w*. In anticipation
and after a fair amount
of that equilibrium,
we arrive at the recursive equations:
of manipulation,
n - 1 / 11*- \!-JW*
\ri-l
and:
that determine
the evolution
of wages.
a
in
fraction 1 - dw of households
set wf as their
Then,
every period,
the remaining
fraction dwpartially
index their price by past
wage, while
inflation. Consequently,
the real wage
index evolves:
4.2
The Final-Good
Producer
is one final good produced
function:
lowing production
There
(A\e/(e-l)
(ytmdi\
where
e controls
Final-good
using
intermediate
goods with
the fol
, (4)
the elasticity
are
producers
of substitution.
perfectly
competitive
and maximize
profits
98
Fernandez-Villaverde
and
Rubio-Ramirez
to the production
function
(4), taking as given all intermediate
and
the
final
the same steps as
goods' prices pit
good price pr Repeating
we
obtain the demand
for wages,
functions
for each intermediate
good:
subject
where
ydt is the aggregate
\\pityitdi to deliver:
Pt
=
1
/V
and the zero profit
demand
condition
ptydt
=
:,.V/(1~e)
\\oPl'edi\
4.3
Intermediate-Good
is a continuum
There
Producers
of intermediate-good
Each
producers.
access to a technology
represented
i has
producer
ate-good
duction
function:
intermedi
by a pro
of the
kit_x is the capital rented by the firm, lditis the amount
labor input rented by the firm, the parameter
<J>
corresponds
"packed"
to the fixed cost of production,
and where At follows:
where
=
At
At_x
exp(A^
+
=
zAt) wherezAtt
vAzAt
and
~
eAt
Jt(0,1).
=
We can
<f>is scaled by the variable zt
A]/{1~<l)iL<?/il~a).
a
as
two
the
levels
index
of
think of zt
At and jx,,
weighted
technology
function. The
is the share of capital in the production
where
the weight
zero
are
that
economic
4>z, guarantees
profits
roughly equal to
product
in the steady state. Also, we rule out the entry and exit of intermediate
that zt evolves over time as zt = zM exp(A2 + zzt)
good producers. Note
=
=
+
+
where zzt
a). We will see
a) and A2
(AA aA J/(l
(zAt azM)/(l
in the following
that Az is the mean growth rate of the economy.
solve a two-stage problem.
First, given
Intermediate-good
producers
in
factor markets
wt and rt, they rent ldtand kit_x in perfectly
competitive
amarginal
cost of:
real costs, which
order tominimize
implies
The
fixed cost
on
cost does not depend
The marginal
shocks and rent inputs at the same price.
i: all firms
receive
the same
How
Are
Structural
Parameters?
Structural
99
the price that maxi
choose
producers
under the same pricing scheme as house
of firms reoptimize
their prices. All
holds. In each period, a fraction 1
dp
is
other firms can only index their prices by past inflation. Indexation
= 0 is no indexation
and
controlled by the parameter
x g (0,1), where x
= 1 is total indexation.
X
The problem of the firms is then:
Second,
intermediate-good
mizes discounted
real profits
max
Pit
~
^s-A~
mct+\ytt+r
T=0
+
T
Etjr(pe,)^[(ri
\
Pt )
|_W
to
subject
\S
where
= 1
Pt + T/
value
the marginal
ogenous
of a dollar
to the household
by the firm. Since there are complete
value is constant across households
markets
is treated
in securities,
as ex
this
and, consequently,
Xt+r/\
on future profits.
in terms of two recursive equa
We write the solution of the problem
tions in g) and g2:
marginal
is the correct valuation
=
g) \mctyd + ped
j^- J
g]+1
=
=
(e l)g2 and n*
eg]
pf/pr
Given Calvo's pricing, the price index evolves:
where
pxr
=
%(^uy~zp)-i
or, dividing
by p]~e,
1=9'(ir)
4.4
+ (i %)pr~e
+o-vn
The Government
The government
rule:
sets the nominal
interest
rates according
to the Taylor
100
Fernandez-Villaverde
and
Rubio-Ramirez
yd
r -(-rj
J
L(nj lv/
exp<''<5)
that are financed with
open market
operations
lump-sum
ensure
to
the
that
is balanced period by
Tt
government
budget
the target levels of inflation (equal to
period. The variable II represents
inflation in the steady state), R is the steady-state
gross return of capital,
through
transfers
the steady-state
gross growth rate of ydt.With a bit of abuse of
Kyd
we will refer to the term
as the growth gap. The
language,
(yd/yd_^/Ayd
=
term mt is a random shock to monetary
cimemt
policy that follows mt
and
where
period
over time observed
4.5
in the United
States.
Aggregation
First, we begin with
yf
to N(0,1). We introduce the previous
the smooth profile of the interest rate
emt is distributed
according
interest rate, Rt, to match
=
the aggregate
demand:
ct + xt + p.,"1*(uf)fc,_r
function for intermediate-good
Then, using the production
producers,
the fact that all the firms pick the same capital-labor
ratio, and market
we find the aggregate demand
clearing in the output and input markets,
must
be equal
to aggregate
supply:
y'
=-^
where
--&'
induced by price
is the aggregate
loss of efficiency
of the index under Calvo's pricing:
properties
=
1)f
+
dispersion.
By the
(i-e;)nr.
8i^"V1
Finally,
we
integrate
!>-w:(^)>?.
where
lt is the aggregate
labor demand
labor supply
over all households;
of households.
Hence
to obtain:
ifwe
define
we
101
Structural Are Structural Parameters?
How
get:
and:
, njs
(W,
if
4.6
\-n
if")
vr-i+ a ej(nr*)-"
Equilibrium
in this economy
is standard and the equa
of equilibrium
it are determined
tions that characterize
the
first order conditions
of
by
A definition
the household,
the government,
the first order
of the firms,
conditions
the Taylor
rule of
and market
clearing.
our quantitative
the
analysis, we must
approximate
a
is
of
the
Ours
model
(even
economy.
equilibrium
dynamics
large
the version without
parameter
drifting has 19 state variables). More
over, we will need to solve the model
repeatedly
during our estimation
To undertake
elsewhere
Rubio
(Fernandez-Villaverde,
process. We have
argued
a non
to
is
and
Santos
that
there
much
be
from
Ramirez,
2006)
gained
linear estimation
of the model,
both in terms of accuracy and in terms of
true ifwe want to allow the agents in
identification.
This is particularly
to ensure themselves
the economy
against future changes in the param
eters of the model. Hence, we require a nonlinear
solution method
that
In previous work
is fast and accurate.
(Aruoba, Fernandez-Villaverde,
and Rubio-Ramirez
2006), we have found that a second order perturba
tion around
vious
the deterministic
steady
state of the model
fulfills
the pre
desiderata.
solving the model, we clear up some technical issues. First,
of technological
are growing
in
change, most of the variables
To achieve the right accuracy in the computation,
we make
the
But before
because
average.
variables
Hence,
and solve
stationary
we
define
=
=
ct
=
ct/zt,
Xt
in the transformed
the model
=
=
Xtzt,
ft
=
=
rt\x,t, qt
variables.
=
qt\xt,
xt
=
xjzt,
wt
=
=
wjzt, wf
w*/zt, kt
kt/zt\it, and yd
ydt/zr Also note that Ac
Ax
=
=
=
forms for $( ) and
Aw
Aw*
A2. Second, we choose functional
kyd
=
we
+
For
V(-).
O(m)
1)
l)2. We normalize
pick $(m)
^(u
(02/2)(w
102
and
Fernandez-Villaverde
Rubio-Ramirez
u = 1 in the
f = <S>'(1)= ^
and O(l) = 0. The in
steady state. Hence,
- A
=
vestment
cost function
is V(xt/xt_1)
adjustment
(K/2)[(xt/xt_1)
J2.
=
= 0.
Then, along the balanced
growth path, V(AX)
V'(AX)
We will perform our perturbation
in logs. For each variable vart, we
as
define vart = log vart
the
var,
log
log deviation with respect to the
state.
the
states of the model
Then,
steady
St are given by:
jbx_v g)_v g2t_v kt_v At_v pu,
f= (iit_v
a a ft
U\\t-VJl-V i Xt-V A-t-1' ^t-V 2M-1; a^t-V
and
the
shocks
exogenous
are
et_v v*_v vtv
$t-V
ZA,t-l
\'
J
=
et
(e^,
zdt,
e^t,
zAt,
?mt).'
As a first step, we parameterize
the matrix of variances-covariances
of
the exogenous
shocks as fl(x) = x^X where ft(l) = Cl is a diagonal ma
on that assumption,
trix. However,
and we
nothing
really depends
an
handle
of variances-covariances.
Then, we
arbitrary matrix
state
take a perturbation
solution around the deterministic
of the
steady
that is, x
0.
model,
we build the law of motion
From the output of the perturbation,
for
could
the states:
=
st+1 *js;,
e;y +
\(sft,
z't)%2(s\,e;y + *s3,(6)
is a 24 X 24 matrix. Theterm %(S't,
is a 1 X 24 vector and %
where %
the
linear
the model,
constitutes
of
solution
(S't, zft)%2(Sft, eft)r is the
e[)f
X 24 vector of constants
a
is
1
added by
and
^s3
component,
quadratic
that corrects for precautionary
behav
the second order approximation
ior. Some of the entries
From
W
=
% will be zero.
find the law of motion
for the observables
of the matrices
the same output, we
A log yt, A log lt, log n?
(A log ^\
log R,)'.
=
Now, define St
(S't, S't_x,e^). We keep track of the past states, S't_u be
in the following measurement
cause some of the observables
equation
in
first
differences.
will appear
Then, we write to the observation
equa
tion:
^T =
W,e;y
+1
e;y + *o3(7)
(s;, e't)%2(S't,
X 48 matrix.
X 48 vectors and
V0l and ^o3l
%2 is a 48
to
for states is unique
the law of motion
(or at least equivalent
same
the
a class of different
the
for
all
of
which
have
states,
implications
where
While
How
Structural Are Structural Parameters?
103
on what we
the observation
of the model),
equation depends
dynamics
assume the researcher actually observes.
In our case, we have chosen the
first differences
of the relative price of investment,
output, hours, infla
we do not know much
tion, and the federal funds rate. Unfortunately,
and how they may
the right choice of observables
results (for one of the few articles on this topic,
Giannoni
2006).
about
mation
4.7
The Likelihood
affect our esti
see Boivin
and
Function
the state space representation
of our
(6) and (7) constitute
Equations
can
ex
model. One convenient
is that we
property of this representation
a DSGE model,
an
to
it
of
the
likelihood
chal
evaluate
otherwise
ploit
lenging
assigns
eter
^
task. The likelihood,
that the model
??(YT; ^P), is the probability
to a sequence of realizations
of the observable YT given param
values:
=
(P, h, v, #, 8, x\, e, a, <(>,0W,Xw, %, XP, $2/ 7r/ V
7n> n/K*
Aa/ ft*/ ftp/
that 0: is not included in W because
it is a function of the other pa
rameters in the economy
to ensure that f = 4>rWith ??(YT; ty), we can
estimate W by maximizing
the likelihood or by combining
itwith a prior
Note
to form a posterior distribution.
for the model parameter
do we evaluate
the likelihood
the Markov
!?(YT; \P)? Given
our
structure of
we begin by
state space representation,
the
factorizing
likelihood
function as:
density
How
?{YT; V)
=
T
J] ^(Vr IY'-1;?).
t=i
Then, conditioning
on the states:
?(YT;?) = /X(YX|S0;V)dS0U /2(Vt ISf;V)p(StV1;
| V)dSt
(8)
li we know St, computing ?(Yt ISt; Mf) is
relatively easy. Conditional
on St, the measurement
a
is
from et to
(7)
equation
change of variables
W. Hence, we can apply the
to
formula
the
evaluate
change-of-variable
ifwe know S0,we can
and
required probabilities.
(6)
Similarly,
employ
the measurement
(7) to compute
equation
S?(Yt |S0; ty). Consequently,
of the sequence
|
knowledge
(p(St Y'"1;
<*'))/l1and of p(S0; Mf) allows us to
104
Fernandez-Villaverde
and
Rubio-Ramirez
(or at least drawing
from) p(S0> ^) is usually
often
2005).
straightforward,
although
costly (Santos and Peralta-Alva
to
is
The difficulty
characterize
the sequence of conditional
distributions
the integrals in (8).
|
V)]J=1 and to compute
[p(St Y'-1;
find i?(YT; ^).
Evaluating
for doing so (but not the only one; see the technical ap
algorithm
to
and Rubio-Ramirez
Fernandez-Villaverde
2007, for alterna
pendix
is to use a simulation
tives and references)
technique known as the par
An
and Rubio-Ramirez
ticle filter. Fernandez-Villaverde
(2005 and 2007)
shown that the particle filter can be successfully
applied to the es
timation of nonlinear
DSGE models.
The particle fil
and/or nonnormal
ter is a sequential Monte
the [p(SjY'_1;
Carlo method
that replaces
an
distribution
of
draws
W)]J=1 by
empirical
generated
by simulation.
have
in the particle filter is that the simulation
is generated
The bit of magic
a
as
known
(SIR).
importance
through
procedure
sequential
resampling
that
the
Monte
Carlo
guarantees
importance
resampling
Sequential
in a reasonable
amount of time,
sufficient
accuracy
cannot
that
be achieved without
(Arulampalam,
something
resampling
in further
describes
and Clapp 2002). The appendix
Gordon,
Maskell,
detail the working
of the particle filter.
method
achieves
A Bayesian
4.8
Approach
our model
The
the data using Bayesian methods.
for the esti
is a powerful
and flexible perspective
Bayesian paradigm
An
and
Schorfheide
the
mation
of DSGE models
2006).
(see
survey by
on a
a
to
based
inference
is
coherent
First, Bayesian
approach
analysis
a
in
natural
handles
the
set
of axioms. Second,
clear
Bayesian
approach
and
lack
both serious concerns in
of
identification,
way misspecification
it has
of DSGE models
the estimation
(Canova and Sala 2006). Moreover,
even when
evalu
and
desirable
properties,
asymptotic
small-sample
We will
confront
with
and Rubio-Ramirez
criteria
(Fernandez-Villaverde
by classical
in
to introduce presample
2004). Third, priors are a flexible procedure
the
with
associated
formation and to reduce the dimensionality
problem
in our
attractive
This property will be especially
number of parameters.
the
number
since parameter
practical
drifting will increase
application,
of our model.
of dimensions
ated
the likelihood of the model
The Bayesian approach combines
a posterior
a prior density
for the parameters
p(^) to form
with
p(^IYT)oc?g(YT;^)p(^).
??(YT;W)
How
Structural Are Structural Parameters?
105
the uncertainty
summarizes
the parameters,
The posterior
regarding
For example, under a quadratic
and it can be used for point estimation.
loss function, our point estimates will be the mean of the posterior.
is also difficult to characterize, we generate draws
Since the posterior
it using a Metropolis-Hastings
algorithm.
to obtain point estimates,
empirical distribution
in the appendix.
We describe
this algorithm
from
5
Parameter
We
use
variances,
the resulting
and so on.
Drifting
we are
drifting. Since the extension
ready to deal with parameter
we pres
to other cases of parameter variation
is rather straightforward,
our model.
ent only one example of drift within
Now
in section 3, we will
by the first example
the Taylor rule is specified as:
Motivated
uation where
t'[-t)
Note
Un) UJ
the difference
elasticities
with
of the response
J
investigate
the sit
(9)
exp(w,)-
in (5): in the new equation
the specification
the
of the interest rate (yRt, ym, 7 t)are indexed by
time.
that the parameters
follow an autoregressive
postulate
ensure
in
to
that
the
is positive:
(AR[1])
parameter
logs
=
+ eRt, 0] (10)
log yRt min[(l
pR)log 7^ + pK log 7^
We will
tog 7ro
tog yyt
=
=
(1~
(1
~
Pn) log 7n + Pn log 7m-i + em
model
(U)
+
+
Pytog yyt_,
Eyt (12)
py) log yy
[zRt, Ent, eyJ are i.i.d. normal shocks and Q is a 3 X 3 matrix of co
variances.2 We allow for arbitrary correlation
in the innovations,
since it
is plausible
that the reasons why the monetary
more
becomes
authority
are
same
to
reasons
inflation
it
the
will
less
become
(less) responsive
where
to the growth
the
responsive
gap. Also, we could generalize
or
in
in
II
the
of
variance
changes in parameters
by allowing
mt
changes
(R and Kyd are not chosen by the monetary
authority but they are im
of the model and by II). Finally, we impose
plied by the other parameters
the stability condition
that the smoothing
coefficient
yRtmust be less
than 1 in levels (or less than 0 in logs).
Our specification
of parameter
drift emphasizes
the continuity
of the
(more)
106
and
Fernandez-Villaverde
Rubio-Ramirez
in opposition
to the discrete changes
in the parameters
process,
a
and
(see
process
captured by Markov-switching
Davig
Leeper 2006a
and 2006b). We do not have a strong prior preference
for one version or
it is parsimonious
the other. We prefer our specification
because
and
as
to
it
and
such
the Fed's gradual
handle,
easy
captures phenomena
change
learning
about
the behavior
of the economy.
to our favorite
interpretation of parameter drifting, we will
understand
that policy evolves over time following
(10)-(12). Consequently,
equations
they react to it and make their decisions
based on the current values of yt and on the fact that ytwill evolve over time.
According
assume that agents
The
drift
that the economy
will
of the parameters
travel
implies
the Taylor principle
is not satisfied. However,
this
through
mean
con
not
not
In
that
is
the
the
may
unique.
necessarily
equilibrium
zones where
text of Markov-regime
changes
and Leeper
(2006b) have
Davig
in the coefficients
what
of the Taylor rule,
they call the general
developed
ized Taylor principle. Davig and Leeper argue that a unique equilibrium
is in
if the Taylor rule is sufficiently
the economy
survives
active when
time
the
of
the active policy regime or if the expected
economy
length
small. To keep this
policy regime is sufficiently
we
on
to
not
will
dwell
results
focused,
paper
generating
equivalent
one
our
to
in
it
and
Suffice
that
further
environment.
say
Davig
Leeper's
is that we can handle restrictions on
of the Bayesian approach
advantage
will
be in the nonactive
the use of the priors. For example, we can
the parameter
drifting with
a
on (10) by putting a zero prior on the
reflecting boundary
implement
in our empirical
that boundary.
of violating
Also,
analysis,
possibility
we estimate 7n as being bigger than 1. This suggests
that the Taylor prin
on average.
ciple will be satisfied, at least
Our
formulation
we do not model
of parameter
one important drawback:
drifting has
over time. In sec
the parameters
change
in the policy parameters
could be a re
explicitly why
tion 3, we discuss
that changes
or evolving
about
flection of changing political priorities
perceptions
include ex
of policy. A more complete model would
the effectiveness
we discipline
of the pa
the movement
through which
plicit mechanisms
can be incorporated
into
rameters over time. Many of those mechanisms
our framework,
since we are rather flexible with
drift that we can handle.
forms for the parameter
in section 4 carries on except with
The model
and the fact that all the conditional
expectations
process
(10). Thus,
the states of the model
with
the type of functional
the modification
now
parameter
of (9)
the
incorporate
are:
drifting
Structural Are Structural Parameters?
How
?
I *h-i'
=
'
107
V
t-i'8t-i'8t-i'*t-i'Rt-i'yt-i'ct-i'vt-i'vt-i'
I?
?
?
x-v
\\t-vh-v
?3 j '
~
*-t-v
zt-v
Zv.,t-i'
"-t-v
9f-i/
zA,t-v
^Rt-v
7m-i/
7yJ
as three additional
states. We
included yRt, ym, and
yyt
from the
of separating
the convention
drifting parameters
an
are
a
interest
semicolon
other states with
(;) since they
object of
by
we
to
the
likeli
filter
the
evaluate
themselves.
Similarly,
apply
particle
where
will
we
have
follow
hood
of the model
and
the Metropolis-Hasting
algorithm
to simulate
from the posterior.
6
Empirical
This
Analysis
we report the
our empirical
presents
point
analysis. First,
fixed over the
of the model when we keep all parameters
sets a natural benchmark
for the rest of the
This estimation
section
estimates
sample.
study. Second,
the parameters
we
discuss
the results
of an exercise
where
we
allow
to change
of the Taylor rule of the monetary
authority
over time. Third, we analyze the evolution of the parameters
that control
In the interest of space we select
the level of price and wage
rigidities.
these two exercises as particularly
illustrative of the procedure we pro
we
have
could
the
However,
pose.
performed many other exercises within
of our methodology.
estimate
the model
using
framework
five time series
States:
con
to
the
of
respect
price
price
real
hours
worked
(2)
(3)
per
output per capita growth,
sumption,
capita, (4) the CPI, and (5) the federal funds rate. Our sample goes from
1955:Q1 to 2000:Q4. We stop our sample at the end of 2000 because of the
on the relative price of investment
absence of good information
after
We
(1) the relative
of investment
for the United
with
series compatible with
the observed
the model, we
and
real
investment
in
real
gross
compute
output
consumption
units. For that purpose, we use the relative price of investment
defined
as the ratio of an investment
a
deflator and
deflator
for consumption.
that time. To make
both
is constructed
deflator
from the deflators
of non
consumption
durable goods and services reported in the national
income and product
accounts
are
mea
investment
deflators
(NIPA). Since the NIPA
poorly
The
deflator constructed
sured, we rely on the investment
a series that ends at 2000.-Q4. The
appendix
provides
tion on the construction
of the data.
by Fisher (2006),
further informa
108 Fernandez-Villaverde
6.1
Point
and Rubio-Ramirez
Estimation
parameters.
reporting results, we specify priors for the model's
We adopt flat priors for all parameters. We impose boundary
constraints
to
out
to
make
the
and
rule
values
that are
proper
priors
parameter
only
a
a
either incompatible with the model
for
value
variance,
(i.e., negative
Calvo parameters
outside the unit interval) or implausible
(the response
to inflation in the Taylor rule being bigger than 100). The looseness
of
Before
in
is shown by the fact that the simulations
performed
the following never travel even close to the bounds. Also, we fix four pa
demand v is ir
rameters,
(v, $, 3>2,8). The parameter
controlling money
will supply
the government
relevant for equilibrium
because
dynamics
such constraints
as much
as required to implement
the nominal
interest rate de
money
to
fix
We
the
the
rule.
<\> zero, since we do
parameter
Taylor
by
on
not have information
pure profits by firms (in the absence of entry/
termined
exit of firms, there are no serious implications
for equilibrium
dynam
is set to 0.001,
The
of
the
investment
cost, <$>2,
ics).
parameter
adjustment
are difficult
to identify. Our
and depreciation,
8, to 0.0149 because
they
that
ratio in the data (remember
the capital-output
choice of 8 matches
in our model
economic
we
have
depreciation,
controlled
both physical depreciation,
by 8, and
induced by the change in the relative price of
capital).
Our choice of flat priors ismotivated
that, with this
by the observation
to
likelihood
function.3
Conse
the
the
is
posterior
proportional
prior,
a classical exercise
as
can
our
results
be
interpreted
Bayesian
quently,
function (the point estimate under an
the mode of the likelihood
where
is the maximum
likelihood
absolute value loss function for estimation)
more
a researcher who prefers
informative
estimate. Moreover,
priors
to accommodate
his
can always
the posterior
draws
from
the
reweight
favorite priors (Geweke 1989).4 We repeated our estimation with an in
in the results.
formative prior without
finding important differences
our results by reporting the mean and the stan
Table 2.1 summarizes
coincide
of the posterior.5 Most of our point estimates
the
standard
exercises
and
other
estimation
with the typical findings of
on a few of them. We
are small. Hence, we comment
deviations
only
we have a Frisch
is 0.88?and
have a high degree of habit persistence?h
dard
deviation
the bounds of find
elasticity of labor supply of 0.74 (1 /1.36), well within
and
in
recent
literature
microeconomic
the
(Browning, Hansen,
ings
n
are
e
and
of substitution
of elasticities
Heckman
1999). The estimates
around
8, implying
average markups
of around
14 percent.
How
Structural Are Structural Parameters?
Table
2.1
Point
estimates
Point
Estimate
Parameter
S.D.
Parameter
Point
Estimate
109
Point
S.D.
Parameter
0.001Wr
0.999
0
0790
0.012
h
0.877
0.190
0.056
1.260
0.075
1.008
3.6e-4
i|i8.942
fi1.359
0.009
yy
0.045
\yu
0.004 n
Estimate
S.D.
0951
0.006
p?
0.942
0.015
a^
0.101
0.007
0.006
0.002
|^
\<rA
k
7.679
0.600
a
0.255
0.011
am
0.003
8.4e-5
0.451
ew
0.0923
e
7.957
0.1593
ad
0.060
0.003
0.849
Xw
0.1231
t)
7.965
0.2984
0.070
0.011
0.907
0.012
0.010
2.86e-4
0.151
Xp
0.100
0.0005
4.57e-4
6p
A^
\AA
<r9
is a relatively high 0.91,
for price adjustment,
parameter
0p,
is
It
the indexation
0.15.
is
level, xp/
tempting to compare our esti
on the average duration
of
with
the microeconomic
evidence
or
Nakamura
and
and
Klenow
Steinsson
How
2004,
2006).
(Bils
The Calvo
while
mates
prices
ever, the comparison
is difficult
because
we
prices change every quarter for all producers,
and a fraction 1-0 because
ducers reoptimize
have partial
indexation:
a fraction
because pro
0p
of indexation. The Calvo
for wage adjustment,
parameter
Qw, is 0.45, while
0.85. Our point estimates
imply stronger nominal
inwages,
in line with Rabanal and Rubio-Ramirez
banal
much
from Smets
(2004) but diverging
more informative priors.
the indexation,
xw, is
in
rigidities
price than
(2005) or Gali and Ra
and Wouters
(2003), who have
The policy parameters
(yR, yn, 7 , II) are quite
the interest rate over time (7R is estimated
standard.
The
Fed
to be 0.79), and re
is
to
the output growth
(7R 1.25) and weakly
that the Fed has a target for quarterly
infla
smooths
sponds actively to inflation
gap (yy is 0.19). We estimate
tion of 0.78 percent
(or around 3 percent yearly).
The growth rates of the investment-specific
technological
and of the neutral
U.S.
change, A ,
of the growth
in the
technology, AA, imply that most
is induced by improvements
in the capital
(83 percent)
This
result
corroborates
the
of mod
technology.
importance
economy
producing
elling biased
technological
change for understanding
growth and fluc
tuations that Greenwood,
and Krusell
Herkowitz,
(1997 and 2000) have
so
The
defended.
estimated
forcefully
long-run growth rate of the econ
+
aA
is
0.4
a)
omy, (AA
percent per quarter, or 1.6 percent annu
J/(l
110
Fernandez-Villaverde
and
Rubio-Ramirez
in the sample. Also, the standard devi
ally, roughly the observed mean
ation c^ ismuch higher than vA.
serves different
our model
as a
Our estimation
roles. First, it validates
our
for
exercises
with
promising
parameter
laboratory
drifting. Since in
case we
the benchmark
with
the basic growth
results with parameter
obtain
results
compatible with the literature and
of
the
U.S. economy, we know that the
properties
come from that feature of the
will
indeed
drifting
to initialize the parame
Second, we use our point estimates
ters in the exercises with parameter
drifting.
In the next two subsections, we will report our findings when we al
to vary at a time. We do this for convenience.
low one parameter
First,
estimation.
to move
the com
makes
parameters
simultaneously
and estimation
of the model much more costly. Second, the in
putation
in the sample is limited and it is difficult
to obtain stable esti
formation
allowing
mates
several
in our second
Third, especially
to have the richest possible model
are allowed
that as soon as parameters
otherwise.
is not so much
to show
exercise, our objective
to fit the data well but
to change
of
will
We
continue
appear.
strong signs
misspecification
in the near future.
of parameters
ration of joint moves
6.2
Our
Evolution
of Policy
first exercise
studies
over
time,
the explo
Parameters
the evolution
in the
of the policy parameters
how much evidence
there is in
evaluates
Taylor rule. This investigation
over time. As we discussed
in
the data of a changing monetary
policy
debated
the topic (e.g., Clarida,
section 3, the literature has extensively
and Sargent 2001, Lubick and Schorf
Gali, and Gertler 2000, Cogney
the empirical
heide 2004, Sims and Zha 2006, Boivin 2006). However,
on
so far are unsatisfactory
because
they rely either
applied
forecast
of the sample that do not let the agents in the model
divisions
of reduced forms.
the changes in policy or on the estimation
is 7m_a, since this parameter
the most
interesting parameter
Arguably
methods
controls
how
In addition,
to inflation.
the monetary
authority responds
aggressively
of
with
the
issue
of
is
linked
7m_a
multiplicity
intimately
a source of insta
of monetary
and the possibility
policy being
of ynt_1 over
bility. Figure 2.2 plots our point estimate of the evolution
interval to gauge the uncertainty
time plus the two standard deviations
in the estimation. We report the smoothed values of ynt_x using
present
and West
the whole
2004). We find it con
(Godsill, Doucet,
sample
some
to eliminate
of the quarter-to
for expositional
venient,
purposes,
equilibria
How
Structural Are Structural Parameters?
12
Ill
I-1-1-1-1-r-j
10
_21-1-1-1-1-1?'
1960
Figure 2.2
of Response
Evolution
1970
. %
L Greenspan
1980
Period
1990
-j
2000
to Inflation
this goal, in figure 2.3
quarter variation of the parameter. To accomplish
we compute
we
graph the trend of the change of the parameter where
that this trend is
the trend using aHodrick-Prescott
filter. We emphasize
a device to read the graph more clearly and lacks a formal statisti
only
cal interpretation.
figures 2.2 and 2.3,7m-1 starts low, slightly above 1 during the
itwas even below
1.
1960s, and early 1970s, with periods when
in the mid-1970s,
and especially
after Volcker's
However,
appointment
In both
1950s,
as chairman
to in
of the Board of Governors,
7m_1 soared. The response
was
as
as
in
its
it
flation reached
the early 1980s, where
6 in
peak
high
one quarter. After that maximization,
7m_1 slowly decreases
during the
the Fed's more permissive
1990s, perhaps
reflecting
modate
the strong productivity
associated
growth
attitude
with
to accom
the Internet
boom.
our model
to
has parameter
it is not straightforward
drifting,
with
in
estimates
obtained
compare
fixed-parameter
we
models. However,
clearly confirm the findings of Clarida, Gali, and
Gertler
(2000), Lubick and Schorfheide
(2006)?that
(2004), and Boivin
Since
these numbers
monetary
policy
has become
much
more
active
in the last 25 years. Our
112
Fernandez-Villaverde
and
-1-r
4.5
I-1-1-1-/^
35
/
3"
/
2
5.
Volcker
Rubio-Ramirez
\
Y
Greenspan
/
1'-'-'-'-'-L
1960
Figure 2.3
HP-Trend
Evolution
of Response
1970
1980
Period
1990
2000
to Inflation
finding is also consistent with the results of figure 12 in Cogley and Sar
as
of the activism coefficient
gent (2001), where
they trace the evolution
a
VAR.
measured
by
parameter-drifting
of importance
Another
is the inflation target of the mone
parameter
II. Histories
like those in Taylor (1998), Sargent (1999), or
tary authority,
(2006) explain that the inflation target may have changed over
time as a reflection of the Fed's varying beliefs about the trade-off be
tween unemployment
of the
and inflation. Figure 2.4 plots the evolution
Primiceri
interval. From the start
target over time plus the two standard deviation
of the sample until the early 1970s and, later, for the 1990s, II hovers
is
around 1.004, or, in annual terms, around 1.6 percent. This number
zone
or
to
comfort
close to the informal
that, according
many
target
the Fed's behavior. During
the intermediate
describes
commentators,
the views the Fed
years, the inflation target increases, reflecting perhaps
the Phillips curve or illustrating
had about the possibility
of exploiting
the changing
features of the economy
the information
lags regarding
the similarity of
(2002). We find intriguing
by Orphanides
emphasized
figure 2.4 to Romer and Romer's
accounts
and internal Greenbook
based on narrative
(2002) hypothesis,
forecasts of the Fed, that monetary
How
Structural Are Structural Parameters?
1.025
i-.-.-.-.-n
1-02"
0.99'-'-'-'-'-^
"
, *
H
i\i\ i A "
i j
/ VV\j,l <^|
s
1.015-J
IjLj %4
i
1960
2.4
Figure
of inflation
Evolution
policy
113
1970
1980
Period
1990
I
2000
target
in the United
States has gone through
and renewed
temperance.
a
long cycle of moderation,
aggressiveness,
a reality
of the inflation target provide
Our estimates of the evolution
In figure 2.5, we plot the inflation target versus
check on our procedure.
realized inflation. If the estimation
isworking
properly, part of the vari
ation in the inflation target needs to be accounted
for, in a purely me
chanical fashion, by changes
in inflation. That is precisely what we ob
serve: as inflation
increases and then falls during
the late 1960s and
1970s, the target inflation estimated goes up and down.
that the inflation target fluctuates
Note, however,
roughly between 40
and 50 percent
less than inflation. Particularly
during the 1970s, the in
flation
is accounted
target iswell below actual inflation. This difference
in two ways. First, by the form of our Taylor rule. We assume
that one
into
the
rule
is
the
between
the
input
growth gap
growth of output ydJ
rate of the economy
The 1970s were
yf_1 and the long-run growth
Ayd.
in comparison with
our model
in
Thus,
years of very low growth
Ayd.6
as
the
behavior
as
a
of
Fed
re
the
rates
the
interest
terprets
lowering
in exchange
for higher inflation. Second,
sponse to low output growth
our model
backs up large negative
shocks in the 1970s that
technology
114
Fernandez-Villaverde
and
Rubio-Ramirez
1.0451-.-.-
n
?
104" ,
1.035-
Inflation Target!
j-Inflation
\
!;
103"
1.025-
i
I
ri
(\
f !
1.02J
f !(i|
1015
}'|
^ y^jj
q 9951-1-1-1-1-1-l
2.5
Figure
Inflation Target
1960
versus
1970
i
1980
1990
2000
Inflation
an alternative way to think
that the big rise in inflation
during the 1970s had less to do with changes in the inflation target than
shocks.
with a series of unfavorable
aggregate
our results. First, the Fed's response
toward inflation
We summarize
push inflation above the target level. Hence,
about this result is that our model
suggests
became more
in the late 1970s and early 1980s and has stayed
aggressive
a small decline. Second,
the inflation tar
with
since
then,
perhaps
high
to account for the high in
get was relaxed in the 1970s, but not enough
flation
trust our results not only because
they come
are
of a coherent DSGE model, but also because
they
uses
that
alterna
literature
the findings of the existing
of that decade. We
from the estimation
consistent
with
with narrative accounts
procedures,
the reality check explained
previously.
tive estimation
and with
6.3
Evolution
of Price
and Wage
of monetary
policy,
Rigidities
are those determining
the extent of
in our model
A key set of parameters
These
four
and
gener
parameters
wage
rigidities,
price
(8p/ xp, 0W, xJthe impulse
ate the nominal
required to match
rigidity in the economy
How
Structural Are Structural Parameters?
response functions
Evans 2005).
documented
115
by VARs
(Christiano,
Eichenbaum,
in the model,
it is unfortunate
their importance
link with microeconomic
have only a tenuous
Given
rameters
and
that these pa
foundations.
form of a con
are the reduced
adjustment probabilities
vex adjustment
cost model,
that produces
this reduced
the environment
form has changed over the years in our sample. We have gone from pe
to ris
riods of high inflation and low response of the monetary
authority
if the Calvo
Even
to periods of much
at
lower inflation and amore aggressive
the U.S. economy
has
the Fed toward inflation. Moreover,
a notable
in
of
level
deregulation,
increasing
competition
experienced
rates.
and
lower
internal markets
unionization
from international
trade,
or their relation to the
The justification
of the indexation
parameters
ing prices
titude by
is even less clear. Why do agents index
probabilities
adjustment
their prices and wages? And if they do, to which
quantity? Past infla
tion? Current
inflation? Steady-state
inflation? Wage
inflation? Conse
it
to
is
the
that
natural
examine
the
parameters
quently,
possibility
(0p,
drift over time, both as a measure
of how strong nominal
XPf 0a,/ Xw)
and as a tool to assess the
rigidities have been in each different moment
Calvo
extent
of the model
of possible misspecification
along this dimension.
in the case of policy parameters, we specify an AR(1) as the law of
motion
for the parameters:
As
log %t
log xpt
tog
6^
-
=
min[(l
=
min[(l
=
tog Xwt
min[(l
=
p6p)log 9p
+
-
mint(l
+
p6p log 0^
Px log xpt-i
+
eQpt,0]
+
?*?*' ?1
pjlog
xp
PeJlog
Qw+ pQwlog 0^
PeJtog
Xw + Pxtog Xwt-i zxwt>0]
~
+ eQwt,0]
are i.i.d. normal shocks and where we take the
(eM, eXpt,eewt/ exwt)
of the value of the parameter
induced by the autoregressive
and 0 to be sure that the two standard deviation
interval and
component
where
minimum
2.6 its HP-trend
the HP-trend
of the CPI superim
(again, with
an
in
to
evolves
it starts
posed).
opposite way
price duration:
low in the 1950s and 1960s but raises very strongly
during the late 1960s.
figure
Indexation
in the mid-1970s
and
Then, it drops dramatically
20 years (except for a temporary
increase in the
part of the sample, during the 1990s, \pt steadily
in the second half of the 1970s may be
dexation
stays low over the next
early 1980s). In the last
drops. The drop in in
accounted
for by firms
116
and
Fernandez-Villaverde
0.4
0.35 -
/
0.3/
/
*
0.25
\
0.2 - s'<SJ
0.15'-'-'-'-'-^
I-.-.-.-.-rn
0.025
"0.02
^
-0.015
\\jr-'
O
5\
/
/
/
\
\
\^7\
Rubio-Ramirez
/
\
- 0.01
/~\
"
?
"0.005
/
0
1960
1970
1980
1990
2000
Period
Figure 2.6
HP-Trend
Price
switching
Indexation
to more
vs. HP-Trend
often
rules. Firms were
and less automatic
price adjustments
induced
the
volatile
inflation of
perhaps
by
a
indexation
partial
costly option. Mechani
optimal
pricing
those years, which made
finds
cally, our estimation
sistent
Inflation
less indexation
because
inflation
is less per
in the 1970s.
to combine the evolution of the Calvo parameter
We find it illuminating
0 ,and of indexation
We do so in figure 2.7 (for their levels) and in fig
xpt
ure 2.8 (for their HP-trends).
shows
The comparison
of both parameters
that periods of high price rigidities are also periods of low indexation. The
This result points out
is true as well, except for the mid-1970s.
to increase the level of in
that adding indexation as an ad hoc procedure
in price adjustments.
flation inertia may hide important dynamics
for wages.
We repeat our two experiments
Figure 2.9 (in levels) and
converse
with
inflation superimposed)
plot the evolu
figure 2.10 (in HP-trends,
workers
before
the
duration
of
the
of
tion
average
reoptimize
spell
ismore dif
1(/1
wages,
Qwt), in quarter terms. In this case the evidence
ficult to interpret, with a big spike in the second half of the 1980s, which
it is still the case that
is probably due to sampling uncertainty. However,
as
as
went
inflation
the
1970s,
up, wage
rigidity went down, and
during
more
inflation was tamed in the early 1980s, wages
rigid.
again became
Figures
2.11 and 2.12 draw
the evolution
of wage
indexation.
Here,
in
1,-,-,-,-.-rn 0.6
if
0.6- \M
v iI
v
ni \
'!
^|
I
-'-
0.41-
-?-J
?-2
0
1980
1970
1960
1
1990
2000
Period
Figure 2.7
Price Rigidity
vs.
Indexation
0.9 i- -1-?-1-n
A
0.85 \
0.8-
""
\
\\\ \
0.75-
0.65'-'-'-'-'-^
\
/
\
0.7-
/
/"\
\
\
/
r\
.-'''I
/
^^
0.35
/
- 0.3
*
/
/\
//
- 0.2
\j\
0.4
/ "
?'25
/
0.15
1960
1970
1980
Period
Figure 2.8
HP-Trend
Price Rigidity
vs. HP-Trend
Indexation
1990
2000
51-1-1-1-1-1~
4.5
w
35-
I
u
|
1970
1960
1980
1990
2000
Period
2.9
Figure
Wage
Average
Duration
3
0.03
l-.-.-.--yr.-rn
"
2.5-
y
\
/
1.5'-'-'-'-'-J
* /
\
- 0.02 _
o
\
0
1970
1960
1980
Period
Figure 2.10
HP-Trend
Wage
I
Rigidity
versus
HP-Trend
Inflation
1990
2000
IfI
O.51-1-1-1-1-J
KilVI
/JV%
1970
1960
!/
1990
1980
2000
Period
Figure
Wage
2.11
Indexation
0.025
0.91-,-,-.-,-=-rn
0.85-
..
\
\
0.750.7-
/
\
** / /
\
\/
x/ 5=
/
\/
y^\
^~'
/
^x
/
\^
- 0.02
.?.
- 0.01 C
- 0.005
^/
0.65'-'-'-'-'-^
1960
1970
1980
Period
Figure 2.12
HP-Trend Wage
Indexation
versus
HP-Trend
Inflation
0
1990
2000
120
Fernandez-Villaverde
and
Rubio-Ramirez
1
1|-.-1-.-.-m
I
1970
1960
v_,_J
1980
1990 2000
?'6
2000
Period
2.13
Figure
Wage
Rigidity
vs.
Indexation
1|-.-.-,-,-rn 1
cr/
""^^-'l
^^^^^
?-51 -?--^^-^v^_
?-8
>f
^~~~~^^__--^
I
0.6
0'-'-'-'-'-J
1960
1970
1980
1990
2000
Period
Figure 2.14
HP-Trend Wage
20
Rigidity
vs. HP-Trend
Indexation
r?-,-,-,-,-rn
0I-.-.---.-.-J
10
o
1960
1970
1980
1990
2000
Period
Figure 2.15
Price vs. Wage
Rigidity
indexation
is
the clarity of the result is embarrassing:
wage
comparison,
we did for prices, we interpret
As
mirror
of
inflation.
the
nearly
perfect
tomore of
of workers
this finding as the natural consequence
switching
indexation
less of an interesting
that make
ten wage
reoptimizations
indexation
iswhat
the model
rule in times of high inflation.8 Less wage
the higher volatility
of inflation in the data.
we finish our graphical display with figures 2.13 to
For completeness,
of the different parameters
2.18, where we plot the evolution
controlling
needs
to capture
we refrain
rigidities against other. Because of space constraints,
the reader can appreciate
of the plots. However,
from further discussion
nominal
Structural Are Structural Parameters?
How
10
121
3
rr-1-'-1-7^-n
o'-?-'-'-'-^
??
**?._-?* ^
1970
1980
\
1
1960
1990
2000
Period
Figure 2.16
Price
HP-Trend
vs. HP-Trend
Wage
Rigidity
1
1|-,-,-,-,-m
1960
o.6
0l-1-1-1-1-J
1970
1990
1980
2000
Period
Figure 2.17
Price Indexation
vs.
Wage
Indexation
1
-n
o.41-.-x^<^_^zr~n-
^
0.2p^^><C^^
q\-.-.-.-1-J
0-8
^
\ ^~~^
0.7
0.1 h
^~"~^_--"^
06
1960
1990
1980
1970
2000
Period
Figure 2.18
HP-Trend
Price vs. HP-Trend
Wage
Indexation
over time solidifies
in the evolution
that the similarity
of the parameters
our confidence
a systematic pattern of relation
that we are uncovering
ships between nominal
rigidities and inflation.
to be strong proof of the changing nature
We consider our findings
of the nominal
in
the
and of a strong indication of
economy
rigidities
model misspecification
dimension
of price and wage
the
along
adjust
ment.
cause we
We
cannot capture the evolution
price adjustment
are less than 1 in levels (they will
always be more
are taking logs).
Calvo's
rameters
first report the experiment
where
we
let
8^,
the Calvo
of the pa
than 0 be
parameter
of
122
Fernandez-Villaverde
and Rubio-Ramirez
20j-.-.-.-.-r-i
15
0I-,-,-,-1-J
1970
1960
1980
1990
2000
Period
2.19
Figure
Price Duration
Average
over
informative
(and more
to
to the micro evidence)
report the average dura
directly comparable
in
tion of the spell before the producers
1(/1
reoptimize,
Qpt), quarter
the
Hodrick
terms. Figure 2.19 plots that duration while
2.20
figure
plots
the
HP-trend
for
of
and Prescott
(HP)-trend and,
purposes,
comparison
as
consumer
in
In
rest
of
the
the
all
this figure,
the
fig
price index (CPI).
price
ures
changes,
evolve
time. We
find itmore
of the paper where we plot two different
line represents
that the continuous
convention
variables, we
the parameter
follow
the
on the left
one the parameter on the right y-axis.7
a clear pattern: average duration was high
Figures 2.19 and 2.20 reveal
in the late 1950s, dropped
in the 1960s, and only started to pick
quickly
y-axis
and the discontinuous
trend until today. Inter
up in the late 1970s, continuing with an upward
in
the
of the spell before
duration
the
average
changes
estingly enough,
in infla
are
the producers
strongly correlated with changes
reoptimize
trend inflation (late
tion. In figure 2.20 we see how times of increasing
are
and
duration
vice versa: how
of
times
1960s, 1970s)
falling average
trend inflation (the 1980s and the 1990s) are times of
times of decreasing
increasing average duration.
Our
rameter
second
experiment
that controls price
the pa
is with
regarding price rigidities
\pt,
2.21
of
the
evolution
indexation.
Figure
plots
10
v.
.
\
o
x^
I->-1-i-n
0.04
c
r^y
-
\
>~^
-'-^
o'-'-'-
0
1970
1960
1980
1990
2000
Period
Figure 2.20
Price Rigidity
HP-Trend
vs. HP-Trend
t\I Am 11A.
l\l lMI n
05
*
Inflation
Ailh
A
l i
03 r Jl aa/ n n\l! Xl\ H H W
/.
*H aK
A^NT
?'3jiltxllN
. u
V
0.1- W " ' f V
0I-,-1-,-,-L.
1960
1970
1980
Period
Figure 2.21
Price Indexation
1990
2000
124 Fernandez-Villaverde
and Rubio-Ramirez
the parameter over the sample plus the fundamentals
that determine
results
underscore
decisions
firms
and
households.
Our
of
pricing
the
that
this problem
is relevant empirically. Also, they suggest that the evidence
in Klenow
and Kryvtsov
the intensive margin
of price
(2005)?that
accounts
inflator?
for
95
the
variance
of
of
percent
monthly
changes
may be a product of the sample
of inflation limits identification
the low level
(1988-2003), where
period
it eliminates
because
the source of vari
Indeed, in our figures 2.7 and 2.8, ifwe look at the pe
in the pricing parameters.
less variation
riod 1988-2000, we observe
sources for this misspecification
of the
There are at least two possible
our
that
rationalize
could
mechanism
of
the
model
findings.
pricing
First, time-varying
price and wage rigidity parameters may be revealing
a
a
change in the probability
problem of omitted variables. For example,
ation of the data.
translates
into a different
slope of the (implicit)
adjustment
a variation of inflation. How
curve in our model
into
and
thus,
Phillips
ever, in the data, there are other shocks that affect inflation, like the price
or
rate fluctuations.
Since
of energy, the price of commodities,
exchange
of price
we
do not
include
these shocks, we may be capturing
of inflation through variations
of these sources
fluence
the changing
in the Calvo
in
pa
rameters.9
source of misspecification
may be the time-dependent
of pricing (either a la Calvo as in the model we have presented
or a la Taylor). Thus, we can read our results as favoring models
of
and
and
(Caballero
Leahy
Engel 1999, Caplin
state-dependent
pricing
duration
1991 and 1997), since those have an endogenously
changing
The
second
structure
The extra analytical
and wages.
difficulty
implied by state
and
Wolman
models
1999)
may be a price we
(Dotsey, King,
dependent
are forced to pay. Another
strand of the literature that may consider our
of prices
is the one that deals with sticky information
(Mankiw
interesting
inflation increases the incentives
and Reiss 2002, and Sims 2002). Higher
it is likely to imply more
to gather
information
and, hence,
frequent
and
wage adjustments.
price
results
for optimal policy de
Finally, our findings have relevant implications
like 0 , as exoge
we interpret the evolution
of
if
First,
parameters
sign.
con
the
that
be
it
monetary
authority may
may
something
nously given,
a
it
how
would
of
on
not
enter
into
discussion
its
do
behavior
dition
(we
them in real time, we only raise this as a theoretical possibility).
amount of
that the measured
ifwe real our results as showing
Second,
are endogenous
to monetary
policy, optimal design be
price rigidities
comes tougher than in the basic Ramsey exercises.
estimate
How
7
Structural Are Structural Parameters?
125
Conclusion
Less so
of DSGE models?
parameters
are
indicates
there
that
large varia
analysis
tions in the estimated values of several of the key parameters
of a bench
our
macroeconomic
model during
mark medium-scale
sample period.
How
are the structural
structural
than we
often
claim. Our
to in
authority
changes in the response of the monetary
and in the inflation target that confirm previous
other
findings by
In particular, we report a move by the Fed toward a much
researchers.
more aggressive
stand against raising prices in the late 1970s. Also, we
We document
flation
find that changes in the inflation target account, at most,
cent of the increase in inflation in the 1970s. Our results
for 40 to 50 per
are remarkable
in a context where agents understand
that pol
they are derived
over
to
time
and
that
evolution.
evolves
icy
respond
We uncover
that the parameters
controlling nominal
rigidities drift in
a substantial way, and more
are
correlated
with in
important,
strongly
because
These findings cast serious doubts on the usefulness
on Calvo
and invite deeper
pricing
investigations
models.
dependent
pricing
flation.
of models
based
of
state
to be interpreted as a sweeping
criticism of
it is not. The literature has
the estimation
of DSGE models,
because
made
impressive progress over the last years and has contributed much
our understanding
to improving
of aggregate
and the ef
fluctuations
We
do not want
our work
in this re
fects of economic
have been engaged
policy. We ourselves
search agenda and plan to continue doing so. We hope, instead, that our
to further estimation
of DSGE mod
paper will be read as an invitation
both as amecha
parameter
drifting. This avenue is promising,
as
a
for incorporating
richer dynamics
and
tool for de
diagnostic
gross
tecting
misspecifications.
In fact, as our discussants
have rightly pointed out, much
remains to
be done. We have only scratched
the surface of the problem of estimat
els with
nism
with parameter
the
ing DSGE models
drifting. We have not explored
model when we have different sources of variations
in the parameters
at
the same time or when
there is stochastic volatility
in the shocks. Also,
we have not studied the consequences
of drifting parameters
for the dy
functions of the
cycle or for the impulse-response
we
not
model.
have
evaluated
different
of param
Finally,
specifications
eter drift or analyzed
reasons for parameter
the possible
in de
drifting
namics
of the business
tail.
Our
skepticism
about
the structural
nature
of most
structural
param
126
Fernandez-Villaverde
and
Rubio-Ramirez
is not a call to perform
exercises. Along with Tom
reduced-form
and
Mark
Watson
and Rubio-Ramirez
(Fernandez-Villaverde
Sargent
we
some
out
have
of
the
of
reduced
2007),
singled
estimating
problems
eters
form models.
But there are many
other papers emphasizing
the weak
is
that
The fundamental
every
point
has strengths and limitations. As Hurwicz
(1962)
empirical procedure
us many years ago, just because we name
warned
"struc
something
tural/7 we should not believe we have taken the theoretical high ground.
nesses
of reduced-form
inference.
Acknowledgments
author: Juan F. Rubio-Ramirez,
Corresponding
Duke
NC
27708, USA.
Durham,
University,
We thank the editors, Daron
[email protected].
213 Social
Sciences,
E-mail:
Juan.Rubio
Kenneth
Acemoglu,
two discussants,
Tim Cogley
and
our
and Mike Woodford;
Rogoff,
Frank Schorfheide;
Pau Rabanal,
Schmitt
Stephanie
Garey Ramey,
at
and
the
NBER
Macroeconom
and
Martin
Uribe,
Grohe,
participants
ics Annual
for comments.
the usual disclaimer, we
conference
Beyond
must note that any views expressed herein are those of the authors and
not necessarily
those of the Federal Reserve Bank of Atlanta or the Fed
eral Reserve System. Finally, we also thank the NSF for financial sup
port.
Endnotes
of the definition
of struc
1. Indeed, Hurwicz
the contingency
(1962) himself
emphasized
antici
is relative to the domain of modifications
"the concept of structure
tural parameters:
are willing
to consider,
differ with
they
regard tomodifications
pated"; "If two individuals
as structural,"
and "this rel
differ with
accepted
regard to the relations
they will probably
not a property of the ma
is due to the fact that it represents
ativity of the concept of structure
of those
of the anticipations
terial system under observation, but rather a property
the state of the system"
(p. 238; italics in the original).
concerning
asking
for
predictions
in this formulation
coefficients
(pR, pm, p ) and the matrix Q become
autoregressive
are also
true structural
their
about
We
structural
nature, but
skeptical
parameters.
our doubts
we will
for the moment.
to avoid the infinite regression
ignore
problem,
2. The
the new
is a small
3. There
ference
classical
by
the bounded
qualifier:
about those bounds
thinking
of the priors. We can fix this small
support
as frontiers of admissible
values
parameter
in a
perspective.
After
do not argue that our flat priors are uninformative.
a flat prior may
curved. Moreover,
become
the model,
highly
or to compare
itwith,
like forecasting
model
for other purposes
need to elicit our priors more
would
carefully.
4. We
dif
a
reparameterization
to use
if we wanted
for example,
of
the
a VAR, we
How Structural Are Structural Parameters?
127
with
filter
of the standard
deviation
the particle
the estimates
and Rubio-Ramirez
2007, and Dejong,
(Fernandez-Villaverde
us from
and Richard
constraints
Liesenfeld,
2007). Computational
preclude
Dharmarajan,
a simulation
sufficiently
long to fully avoid this problem.
running
5. A word
are
of caution
6. This
observation
models
here:
unstable
relatively
as
are,
a model
have motivated
may
argued
by Bansal
and Yaron
where
(2004),
Ayd changes
quite difficult
over time, but such
to estimate
in small
samples.
7. We
do not plot
deviations
interval for the average price duration
the standard
(nor later
because
the transformation
1(/1
duration)
wage
generates
implausi
Qpt)
as soon as the simulation
of
travels close to 1. The standard
de
bly large upper bounds
Qpt
that the parameter
too
viations
interval for
itself is estimated
without
show,
however,
Qpt
much
uncertainty.
for the average
of cost-of-living
allowance
the early 1970s, there was a raise in the prevalence
in collective
and
Kahn
This
(Hendricks
(COLA) escalators
1985).
agreements
bargaining
our result. However,
even at their
observation
could be used to undermine
peak, COLAs
8. During
a small
6 million
it is diffi
of the labor force. Moreover,
workers,
percentage
only covered
since they had many
cult to map COLAs
rules
from the 1970s into our model,
contingent
indexation
that make
them quite different
from the naive
rules that we use. In fact, it could
even
be possible
to think
a
about
state-contingent
COLA
as an
implicit
form
of reopti
mization.
9. Similarly,
in the Calvo
be accounted
for by
may
part of the variation
parameters
an
role in models
like Smets and Wouters'
shocks, which
(2003).
important
markup
play
it is difficult
to see which
shocks will have the level of persist
However,
type of markup
ence that we observe
in the movements
of the Calvo parameters
that we estimate.
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Appendix
This appendix offers further details about the technical aspects of the pa
per. First, we discuss some general computational
aspects and elaborate
on the solution of the model.
the particle filter that
Second, we describe
evaluates
the likelihood
function
the estimation
procedure.
of the data.
struction
8.1
Computation
of the model.
Fourth, we
close with
on
Third, we comment
the details of the con
of the Model
to
to be described
below
feature of the algorithm
important
on a good
is that it can be implemented
and estimate
the model
for the perturbation
of the
computer. We coded all programs
desktop
The most
solve
model
Visual
the particle
filter in Fortran 95 and compiled
run
on Windows-based
machines
Fortran 9.1 to
and
Mathematica
Processor
programs
5160 EMT64
to generate analytic derivatives).
at 3.00 GHz with 16 GB of RAM.
them
in Intel
(except some
We use a Xeon
is challenging
because we have 19 state
that we allow in each empirical
variables plus the drifting parameters
we need to recompute
for
the solution of the model
exercise. Moreover,
The only non-linear
values in the estimation.
each new set of parameter
in a reasonable
amount
this computation
that accomplishes
procedure
and Rubio
of time is perturbation
(Aruoba, Fernandez-Villaverde,
The
solution
of the model
we
have
already
eliminated
our solution
the equilib
by perturbing
one where
of the model
the
(i.e.,
two
the
determinis
the
unit roots) around
2006). We implement
rium conditions
of the rescaled
Ramirez
version
How
Structural Are Structural Parameters?
133
tic steady state. This means
that the solution
is locally accurate regard
in the economy. Also, note that the steady
less of the level of technology
au
state will depend on the level of inflation targeted by the monetary
thority.
We use Mathematica
to compute the analytical derivatives
and to gen
the corresponding
analytical expression. Then,
the solution
into a Fortran 95 code that evaluates
erate Fortran 95 code with
we
load that output
value as implied by the Metropolis
for each parameter
to be described
below. The solution will have the
Hastings
algorithm
of the model
form:
(s;+1,/;)'
=
r5l(s;,e;)' +
|(s;/e;)rs2(s;,e;)'
+ rs3 (13)
recalling our notation,
St are the states, et are the shocks, Jt is a
of variables of interest in the model
that are not states, and the TJs
are matrices
of the right size. With
the appropriate
(13), and by selecting
where,
vector
rows, we
build
the state space
=
sf+1 %&,
e;y +
=
e;)' +
w
where
vjs;,
S,
=
|(s;,
|(s;,
(S't, S[_v e^)
representation:
e;)*s2(s;, e;y + %3
e't)%2(s't,e/y + %3
and W
=
(A log \l~\ A log yt, A log /? log Ylt, log
R,)'.
8.2
Description
of the Particle
Filter
of the particle filter. We will delib
provide now a short description
on
the
intuition
focus
of
the
and we will gloss over
erately
procedure
are
a
technical
issues
relevant for successful
that
of the
many
application
filter. We direct the interested
reader to Fernandez-Villaverde
and Ru
We
(2007), where we discuss most of those issues in detail, and
in Doucet,
de Freitas, and Gordon
im
(2001), which present
Carlo algorithms,
like Pitt and Shephard's
sequential Monte
bio-Ramirez
the articles
proved
(1999) auxiliary particle filter.
As we described
in the main
state space representation,
T
i
t=i
we
structure of our
text, given the Markov
can factorize the likelihood
function as:
134
and obtain
Fernandez-Villaverde
and
Rubio-Ramirez
the factorization:
?(YT;?) = j?(Ya |S0;V)dS0U \2{\
t=2
Consequently,
could evaluate
ifwe had
IS,;V)p(St Y";
I V)dSt
(14)
the sequence
we
I
(p(St Y'_1;
yir))J=1and p(S0; 'VP),
of the model.
and
Santos
Peralta-Alva
under which we can draw the numerical
solu
the likelihood
(2005) show conditions
tion of the model
to approximate
of evalu
p(S0; yir).The two difficulties
are
to
then
the
characterize
of
conditional
distribu
(14)
sequence
to
and
the
different
in the
tions[p(SJ Y'-1; ^)]J=1
compute
integrals
ation
expression.
The particle filter begins
from the observation
that, if somehow we
can get N draws of the form
from the sequence
\
(p(St Yf_1;
[(s^.JJlJJlj
and substitute
the inte
^))J=i, we can appeal to a law of large numbers
in the empir
likelihoods
evaluated
grals with amean of the conditional
ical draws:
XOT;?) ? 1 f
where
2>(%Is<|o;
?)?
?
-J-
Isj,,.,; <J>)
<g(Y,
our notation
in the subindex
for the draws indicates
the condi
a
means
set
on infor
t\t
-1
moment
t
draw
at
conditional
(i.e.,
tioning
mation until t-1) and the superindex
the index of the draw. The
denotes
is that we substitute
intuition of the procedure
the exact but unknown
sequence
counterpart.
by its empirical
[p(St\ Yf_1; <lIr)]fs=1
How do we draw from [p(St Y'-1;
\
^)],T=1? The second
particle filter is that we can extend importance
sampling
to a sequential
environment.
The following proposition,
inal form to Rubin (1988), formalizes
the idea:
key
idea of the
(Geweke 1989)
due in its orig
a
1. Let
V). Let the sequence
I
Proposition
fsj|f_1JJl1 be draw from p(St Y'~2;
a
[s\]y=1 be draw with replacement from ls\\t_^=1 where the resampling proba
bility isgiven by
q'i=
Jg(Yjs;u_i;^)
iJli^Xls;,^;^)'
Then (s,f)Jl1is a drawfrom p(St Y';
| ?).
1 shows how to recursively use a draw
The proposition
from
(sjj^)^
a
to
I
draw
from
is
This
result
crucial.
Y';
Y'"1;
|
V)
Mf).
p(St
get
(s\ u)^
p(St
How
Structural Are Structural Parameters?
135
us to incorporate
in Yt to change our current es
the information
timate of St. This iswhy this step is known
in filtering theory as update
reader
has
is
(the discerning
probably
already realized that this update
It allows
an
of Bayes' theorem).
nothing more than
application
success of the filter. A naive exten
is
The resampling
for
the
step
key
sion of Monte
Carlo
of
sequence
just draw a whole
techniques will
to
without
stopping period by period
resample according
[(sJU-i)/ljr=i
to proposition
1. Unfortunately,
is that all the sequences
quence of states, which
this naive
scheme
diverges.
far away from
become
arbitrarily
is a zero measure
set and
The reason
the true se
the sequence
that is
ones in weight.
closer to the true states dominates
all the remaining
A
even
simulation
shows
that
the
after
appears
simple
very
degeneracy
few
steps.
we draw N exogenous
(s\ \t)^=1,
=
the shocks in our model
e|+1
Given
shocks,
quite simple,
something
are
(e'M+1, e^+1, e^+1, e^+1/ e^+1)'
we
distributed.
the
law
of
states
motion
for
that
Then,
normally
apply
relates the s\\t and the shocks e|+1 to generate
known
This
(sj+1 \t)^Lv
step,
since
as forecast,
difference
of proposition
put us back at the beginning
one period
that we have moved
forward
tioning.
The following
rithm:
pseudocode
summarizes
1, but with the
in our condi
the description
of the algo
Set t ~> 1. Sample N values
(s[)|0)Jl1fromp(S0;^r).
N
Prediction:
values
Step 1,
Sample
(s\ |f_1)Jl1using (sj_21t^)^, the law of
motion
for states and the distribution
of shocks er
Step 0, Initialization:
to each draw
Step 2, Filtering:
tion 1.
Assign
Step 3, Sampling:
the probabilities
Sample N times with
(q\)^=v Call each draw
to step 1.Otherwise
With
the output
(sj,^)
the weight
q\ in proposi
from (s\ \^_1)J11
replacement
using
< T set t ~> t + 1 and
t
If
(sj, t).
go
stop.
of the algorithm,
we
just substitute
into our formula
^(Y^;?) - 1 f 2(X ^|0;
I ?) n
M>)(15)
^ I 2(Y,I S}|t_i;
and get an estimate
(2002) and Kiinsch
hand
of the likelihood
(2005) show weak
side of the previous
equation
limit theorem applies.
and a central
of the model.
Del Moral and Jacod
conditions
under which
the right
is a consistent
estimator of !?(YT;V)
136
8.3
Estimation
Rubio-Ramirez
Procedure
in the main
We mention
and
Fernandez-Villaverde
part of the text that the posterior
of the model
p(M'|YT)oc??(Yr;M')p(M')
we can draw
to characterize.
is difficult,
if not impossible,
However,
from it and build its empirical counterpart using aMetropolis-Hastings
is as follows:
The algorithm
algorithm.
Set i ~> 0 and an initial Mr. Solve the model
for M*.
Step 0, Initialization:
and build the state space representation.
Evaluate prior p(M^) and ap
+
1.
i->
i
with
Set
MO
(15).
proximate ^(YT;
Step 1, Proposal
2, Solving
Step
state space
Step
draw: Get a draw M'f from a proposal
the Model:
Solve
the model
density
for M'f and build
q{yt_lf yf).
the new
representation.
3, Evaluating
the proposal:
Evaluate
p(Vf)
and ^(YT;
M'f) with
(15).
~
If x, ^ [^(VT; ^*)p(^*)^w/
U(0,1).
=
Mr - %v
M'f )]/[^(YT; M'f_1)p(%1)(/(M'f, %_,)] set Mr M'f, otherwise
Step 5, Iteration: If i< M, set i~*i + 1 and go to step 1. Otherwise
Draw
Step 4, Accept/Reject:
Xi
stop.
This algorithm
requires us to specify a proposal density q{-, ).We fol
low the standard practice and choose a random walk proposal, M'f = M^
+ k .,Kf~ >f(0, XK),where ZK is a
is selected to
scaling matrix. This matrix
get the appropriate
acceptance
ratio of proposals
(Roberts, Gelman,
and
Gilks 1997).
To reduce the "chatter" of the problem, we will keep the innovations
in the particle filter (i.e., the draws from the exogenous
shock distribu
across different passes
constant
tions and the resampling
probabilities)
out by McFadden
As pointed
of the Metropolis-Hastings
algorithm.
(1989), this is required to achieve stochas
(1989) and Pakes and Pollard
in a
is not strictly necessary
and even if the condition
tic equicontinuity,
it
variance
of
the
reduces
the
numerical
framework,
proce
Bayesian
dure.
8.4
Construction
of Data
in the text, we compute both real output
in consumption
units to make
the observed
As we mention
investment
ible with
the model.
We
define
the relative
and real gross
series
price of investment
compat
as the ra
How
Structural Are Structural Parameters?
tio of the investment
deflator
The con
for consumption.
the deflators
of nondurable
and the deflator
is constructed
deflator
137
from
sumption
goods and services reported in the NIPA. Since the NIPA investment de
we use the investment deflator constructed
flators are poorly measured,
by Fisher (2006). For the real output per capita series, we first define
nominal
gross invest
output as nominal
consumption
plus nominal
as the sum of personal
con
consumption
on
and
national
de
nondurable
services,
sumption
goods
expenditures
fense consumption
federal nondefense
expenditures,
consumption
state
and
local
and
government
consumption
expenditures,
expendi
ment.
We
tures. We
define
define
nominal
nominal
gross investment
on durable
goods,
sumption
expenditures
federal government
nondefense
vestment,
local government
investment,
gross
private
ment,
and private residential
fixed
as
is defined
the ratio between
put
civilian
as the sum of personal
con
national
defense
gross in
state and
gross investment,
fixed invest
nonresidential
investment.
our nominal
Per capita nominal out
output series and the
between
16 and 65. Since we need
population
real output per capita in consumption
units, we deflate the
series by the consumption
deflator. For the real gross investment
per
nominal gross investment
capita series, we divide our above mentioned
series by the civilian noninstitutional
between
16 and 65 and
population
noninstitutional
to measure
the consumption
constructed with
deflator.
per capita series is
Finally, the hours worked
the index of total number of hours worked
in the busi
ness
sector and the civilian noninstitutional
between
16 and
population
65. Since our model
are
that
hours
worked
0
between
per capita
implies
and 1,we normalize
series of hours worked per capita such
the observed
that it is, on average,
0.33.