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Massachusetts Institute of Technology
Department of Economics
Working Paper Series
Three Strikes and You're Out:
Reply to Cooper & Willis
Ricardo
J.
Caballero
Eduardo M.R.A. Engel
Working Paper 04-10
March 2004
Room
E52-251
50 Memorial Drive
Cambridge,
This
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1
iMASSACHUSETTS INSTITUTE
C'i^ TECHNOLOGY
1
MAR
1
1
200^
Three Strikes and You're Out:
Reply
By Ricardo
J.
to
Cooper and Willis
Caballero and Eduardo M.R.A. EngeP
March, 2004
Abstract
Cooper and Willis (2003)
is
the latest in a sequence of criticisms of our methodology for
estimating aggregate nonlinearities
when microeconomic adjustment
based on "reproducing" our main findings using
artificial
microeconomic agents face quadratic adjustment
costs.
sults
lumpy. Their case
That
Their mistakes range from misinterpreting their
is
data generated by a model where
is,
they supposedly find our re-
where they should not be found. The three claims on which they base
incorrect.
to
is
own
their case are
simulation results to failing
understand the context in which our procedures should be applied. They also claim that
our approach assumes that employment decisions depend on the gap between the target and
current level of
unemployment. This
is
incorrect as well, since the 'gap approach' has been
derived formally from at least as sophisticated microeconomic models as the one they present.
On
a
more
positive note, the correct interpretation of
Cooper and
Willis's results
shows
our procedures are surprisingly robust to significant departures from the assumptions
that
made
in
our original derivations.
JEL Codes:
Keywords:
C22, C43, D2, E2, E5.
Adjustment hazard, aggregate nonlinearities, lumpy adjustment, observed and
unobserved gaps, quadratic adjustment.
*Caballero: Department of Economics, Massachusetts Institute of Teclmology, 50
MA
Memorial Drive, Cambridge,
02142, and National Bureau of Economic Research (e-mail: caball@MIT.EDU); Engel: Department of Eco-
nomics, Yale University, P.O.Box 208268,
mail: eduardo.engel@yale.edu).
New
Haven,
CT
06520, and National Bureau of Economic Research
(e-
Summary
1
of the case
Cooper and Willis (2003), henceforth CW,
is
Labor Adjustment: Mind the Gap." In
comment
this
"The Economics of
the third version of the authors'
CW argue that our finding (in Caballero and
Engel (1993), henceforth CE, and Caballero, Engel and Haltiwanger (1997), henceforth
lumpy microeconomic adjustment matters
They base
their case
for aggregate
on "reproducing" our main findings using
where microeconomic agents face quadratic adjustment
we show
In this reply
that the three claims
mistakes range from misinterpreting their
own
on which they base
is,
model
they supposedly find our
employment decisions depend on the gap between
is
the target
incorrect as well, since the 'gap approach' has
phisticated
CW's
results
partures from the assumptions
CE
Throughout,
microeconomic
distribution of
and
CEH
shows
made
and current
On
more
a
microeconomic gaps are
CE and CEH
=
(first
other studies that at the
is:
'kMi^^+yMi^\
(1)
Ad^'^ is
and desired employment
the equation above simplifies to a linear
on aggregates
at least as so-
explaining aggregate employment fluctuations.
the cross section distribution of gaps between actual
variable depends only
of unemployment.
positive note, the correct
many
represents the rate of growth of aggregate employment, and
0,
level
and examine whether the implied features of the
usefiil in
A£',
and y =
approach assumes that
in our original derivations.
level adjustments are lumpy,*
When A, >
Their
that our procedures are surprisingly robust to significant de-
take as an assumption validated in
Specifically, the basic regression in
AE
that our
been derived formally from
microeconomic models as the one they present.
interpretation of
their case are incorrect.
simulation results to failing to understand the con-
which our procedures should be applied. They also claim
text in
where
That
not warranted.
data generated by a
artificial
costs.
is
where they should not be found.
results
This
employment dynamics,
CEH) that
moments of the
the i-th
moment of
at the firm level.
model where the
left
hand
side
cross section distribution of gaps). This
case can be obtained either from a microeconomic model where agents adjust infrequently but with
a probability that
is
independent of their gap (the constant hazard model of Calvo, 1983) or from a
model where agents face quadratic adjustment
When Y >
'CW
0,
in favor
and adjust
all
the time (Sargent, 1978).^
on the other hand, higher moments of the cross-section
(2003) seem to agree with
evidence"
costs
this
assumption,
in particular, in their
distribution of gaps matter
conclusion they refer to "overwhelming
of it.
"See Rotemberg (1987) for a formal proof of the aggregate equivalence of Calvo 's lumpy adjustment model and
the quadratic adjustment cost model.
1
from a scenario where microeconomic adjust-
for aggregate dynamics.^ This case can be obtained
ment is lumpy and the probabiUty of such adjustment is increasing
model of CE). There
it
is
ample microeconomic evidence
matters for aggregate adjustment.
very significant y
>
and a
We
find that
it
in the
gap (the increasing hazard
for this behavior, the question is
whether
does, since our aggregate regressions
large contribution of yvW/
to aggregate
CW's critique has changed over time, but as of today,
it
can be
employment
show
a
fluctuations.
claims, all of them
split into three
based on applying our procedures to data generated with a model with smooth microeconomic
adjustment:
When our measure
• Claim 1:
of microeconomic gaps are computed from
there exist parameter configurations for
there
the
•
is
Claim
which estimates of y are similar to
no microeconomic lumpiness or
common denominator in
nonlinearities. This has
microeconomic
been
ours, even
their
though
main claim, and
CW (2001, 2002, 2003).
When the microeconomic gaps
2:
their artificial data,
are not directly observed but can be estimated with
used in
data, the procedures
CEH
give nonsensical results
when
applied to
their data.
When
• Claim 3:
only aggregate data are used, coupled with the Kolmogorov equations
quired to keep track of the simulated cross section distribution of gaps, as in CE, our
mates can be found even when their
Not only
are these claims incorrect, as
we will
argue below, but they also reflect a fundamental
We developed a methodology to study whether lumpy
microeconomic adjustment has aggregate implications, not
In section
incorrect.
microeconomic adjustments
II
we show
In section III
that
we
should not be used
if
to infer
from aggregate data whether
are lumpy.
due to a basic interpretation error of
their
argue that since the identification strategy
gaps with microeconomic data
is
built
on the observation
that
own
we
results,
Claim
1
is
adopt for estimating
microeconomic data are lumpy,
microeconomic data are not lumpy. Therefore Claim 2
Furthermore, the fact that
esti-
(linear) data are used.
misunderstanding of the point of our papers.
the underlying
re-
is
it
not surprising.
CW find nonsensical results while we find meaningful and statistically
significant results, indicates that our findings
do not
arise
when microeconomic adjustments
are
smooth.
^The higher moment
it
is
that matters in specification (1)
simple and shows up often both
in
our
work and
in
is
the third
CW's
moment.
critique.
We
focus on this specification because
Yet there are other specifications
our work that involve higher moments different from the third moment, which explains
'higher moments'.
why we
in their
and
generically refer to
IV we show
In section
that
Claim 3 has nothing
to
nomic adjustment. Their finding comes from relaxing
do with lumpy
to
what we found with
to
Somewhat
to departures
paradoxically, the
non-lumpy microeoco-
an extreme the maintained assumption of
our analysis that the driving forces are random walks."* This result
comparable
vs.
is
neither
new nor quantitatively
actual data.
CW
work of
can be used to show that our approach
from the random walk assumption. In
is
robust
nothing can be found with the serial
fact,
CW (2002), and (almost) nothing with the low serial correlation of 0.47
CW (2003) dropped further to 0.28, and even then the gain in R^ from
correlation of 0.81 used in
assumed
in
CW (2001).
adding higher moments
it
is
substantially less than half of what
we
found. Section
V concludes.
Their main critique
2
In the
main part of their critique,
CW compute from their artificial data the cross-sectional moments
of static gaps and esfimate an equation analogous to
(1):
A£P^ = ;iywJ')'^^ + YyW;^)-^^,
where A£'^^ and
M^')'*^^ stand for the rate
(2)
of growth of aggregate employment and the
/th
moment
of the cross section distribution of static gaps respectively, when the underlying data are generated
with
CW's
quadratic adjustment cost model.
Their main finding
ent from the one
is
that they estimate a positive
and
statistically significant y,
we find using actual data. Cooper and Willis then argue that this is evidence that a
researcher testing for aggregate nonlinearities on their data
are important for aggregate dynamics.
our results
may
not very differ-
It
would conclude, erroneously,
follows, they argue, that our
methodology
is
that these
flawed and
well be due to misspecification error.
However, finding similar values of y does not mean that a researcher will conclude that the
nonlinear term
to also look at
cases.
It
is
equally important for aggregate dynamics in the two cases. For
whether the regressor that
turns out that
much smaller
it
is
one needs
multiplied by y has similar variability in the two
does not: The variability of M^^^ in
than that of the corresponding
this,
moment when
CW's
quadratic adjustment
model
is
micro-adjustments are lumpy. Thus,
''in our derivations, and as is standard in much of the (5,5)-literature, we assumed that the driving forces follow a
random walk, an assumption that cannot be easily rejected in the data. In this case, one can show that the static gap (the
difference between cuiTent employment and the optimal level of employment if there are no adjustment costs) is equal
gap (the difference between current employment and the optimal level of employment if adjustment
removed only today) plus a constant that depends on the drift. This is a very usefiil result since the static gap
to the frictionless
costs are
is
straightforward to calculate while
its
frictionless counterpart involves
more complex dynamic
calculations.
the contribution of yM^^^
simulated data, while
is
it
minuscule in explaining aggregate employment
is
large
and economically
reported values of neither R^ nor
A,
The
first
column
in Table
1
when
table that the R'^ reported
is
and R^
falls
based on Table la in
estimating (2)
is
the
CW
(2003).
the second
By
column of Table
here, the
moment of
y,
even though
statistically
less than
0.013 over the
CEH,
reported in
R^ increases by 0.15 when adding a non-linear parameter
and the variation of the speed of adjustment over the relevant range
as that in
CW's comment,
the third
same with or without
contrast, in the corresponding exercise in Table 3 of
1
put, the
apparent from their
It is
economically irrelevant, as the adjustment speed varies by
relevant range of gaps.^
in
CW's
rises).
gaps: 0.90. Similarly, the estimated value of the non-hnear parameter
significant,^ is
Simply
significant in our findings.
change when adding the nonlinear term
while they change substantially in our setting (k
volatility in
is
more than
ten times as large
CW's model.
The economic
striking in the
irrelevance of the non-linearities estimated
by Cooper and
2002 version of their comment, where they used a more
Willis
is
even more
realistic value for the first
order correlation of productivity shocks (0.81 at an annual level). ^ There they report an I^ of 0.97,
both for the model with and without the non-linear parameter,^ and the adjustment speed implied
by
their non-linear
^The
model
statistical significance
their simulations,
^Where
varies only
by 0.005 over the relevant range of values taken by the gap.
they find possibly reflects the fact that they use time series with 1000 observations in
while CEH's estimates are based on 35 observations.
the 'relevant range'
is
defined as
the cross-section of static gaps, respectively.
^q ± 2<3a, with //q and Oq denoting the mean and standard deviation of
A tedious but straightforward calculation from first principles shows that
_ StgOfi
2
2
with:
Gk
"-^o-^'
where
G=
( 1
A+
p-
p^
— 5)/(l — 5pj, X denotes the speed of adjustment in the partial adjustment representation of the quadratic
adjustment cost model, 5 denotes the discount rate that results
target as a present value of future static targets,
in this
model when calculating
the (con-ect)
dynamic
Og denotes the standard deviation of firm-specific productivity shocks
CW
a is defined on p. 23 in
(2002).
^As we pointed out the en-ors in the first and second versions of CW's
and
critique, they reacted
by looking for a new
parameter configurations and new model specifications that might help their case. Their lack of success, despite two
major revisions of their original comment, possibly
^This
is
for the
reported for R^ are
is
benchmark with high adjustment
1
.00.
the best evidence of the robustness of our findings.
costs.
For the benchmark with low adjustment
costs, both values
Estimating unobserved gaps with microeconomic data
3
The second and
third points of
CW's
critique
stem from the
fact that in practice the
gaps are not
observed and hence neither are the cross-sectional distributions of these gaps. They argue that
our procedures to estimate these gaps and moments introduce
positives
and nonsensical
tion), since
Cooper and
claims 2 and
In
(in this section)
distinction
and
between the procedure
that in Caballero
Willis' specific critique differs
Engel
in Caballero,
and Engel (1993)
(in the
next sec-
between these cases (corresponding
to their
observe the microeconomic data but have no direct observation of the gaps. In
order to construct the microeconomic gaps,
when hours exceed
are
certain
normal
below normal.
we
relies heavily
context, the relationship
The
use information on hours.
level, there is a shortage
idea being that
of labor while the opposite
true
is
one needs to estimate the mapping from the hours-gap
Still,
the employment-gap, and the equation that does this suffers
Our way out
to false
3, respectively).
CEH we
when hours
which can lead
errors
Again, Cooper and Willis are mistaken.
results.
To explain why, we begin by making a
and Haltiwanger (1997)
new
on our observation
that
from
to
classic simultaneity problems.
microeconomic adjustment
is
between hours and employment gaps can be estimated
lumpy. In
if
this
one only uses
observations where large adjustments took place; the basic logic behind this procedure being that
during these episodes the variability of the regressor
regression. Yet if one
and
Willis' data,
knows
that
swamps the
variability
microeconomic data are not lumpy, as
of the error term in that
is
the case with
no sensible researcher would use our procedure. Cooper and Willis make the
mistake of not understanding that the microeconomic estimation procedure in
on the observation
reality,
4
Cooper
that
microeconomic behavior
a fact explicitly acknowledged
is
CEH is
lumpy. Fortunately for us, the
conditional
latter
holds in
by CW.^
Estimating unobserved gaps using only aggregate data
In Caballero and Engel (1993)
cross-sectional
we do
moments from an
not observe microeconomic data and hence generate the
internally consistent model. This
tablished fact that microeconomic adjustment
Kolmogorov/Markov functional equation
sponding to a given
^"[There
Conclusion.
is]
set
is
model
starts
from the well
es-
lumpy, and uses this information to construct the
for the evolution of the cross-section distribution corre-
of parameters. Cooper and Willis apply our procedure to data generated by
ovei-whelming evidence that plant level adjustment
is
nonlinear",
CW (2003),
first
paragraph
in the
adjustment cost model, and find evidence that y in equation (2)
their quadratic
is
when
positive
it
should be zero.
Here
CW
fail
assumed
correlation
we assumed
that the driving forces follow a
the Introduction, in this case the static gap
longer
and
within the {S,s) literature that
is
now depends on
gap
They then drop the
is
if the
equal to the frictionless gap plus a constant.
random walk assumption
The
the state.
serial correlation
step in
first
serial
with a quadratic adjustment cost model.
It is
But
results qualitatively similar to ours.
CW
in
well
gap no
between
static
to rediscover this result."^
is
around 0.28 (we report
to
to generate
microeconomic data
only then that they find, under some circumstances,
new (we
this is neither
random walk assumption
go on
It is
relaxed, the static
is
of the driving forces from one
correlation coefficients at annualized rates) and
departures from the
low
random walk. As mentioned
a sufficient statistic for the probability of adjustment, so that the difference
frictionless
all serial
the very
is
in their driving processes.
In our derivations
known
which
to identify the real reason behind their finding,
and
static
knew
already
frictionless
that for very large
gaps could not be exchanged)
nor quantitatively comparable to our findings.
Paradoxically, the findings in
the serial correlation
is
CW are encouraging for the gap approach, since
dropped to very low
the values of serial correlation used in
levels that things start breaking
CW (2001),
which
we found
CE
it is
only
down. In
when
fact, for
would be no
are already low," there
significant false positive finding.
Table 2 reports the gains in R^ that
the (absolute) gap, versus those that
in
from adding a hazard term increasing
would be obtained from doing the
degrees of serial correlation in the driving processes.'^ Clearly, there
of finding an increasing hazard when there
(i.e.,
assumed random walk.
then the gain in
"'Although they
the difference
fit
was
fail to
none)
CW exercise with different
is
no
risk
if serial correlation is
of false positives
not too far from the
CW had to stretch things a lot to find parameters similar to ours, and even
less than half
of the gain
we
found.
highlight the connection between their sharp departure from the
between both gap measures. Also, beginning
claiming that our approach assumes
CW, 2003)
is
in
tliat
in their abstract,
the optimal policy depends
random walk assumption and
by repeatedly
they mislead their readers
on the gap.
In the final sentence of Section
they finally acknowledge that the "gap approach" can be derived from optimizing behavior
follow a random walk, yet credit a previous version of their
comment
for this well
known
II
(in
when shocks
result (see, for example,
Nickell, 1985).
"The
aimualized serial correlation they use in
driving force
we
RBC
(see, e.g.,
models
'"We
used
replicate
in
Cooley andPrescott, 1995)
CW's
CW
Caballero and Engel (1993)
is
is
(2001)
is
close to 0.50.
above 0.80. Also note
CE
for this
model
procedure and use 1000 observations as they
(0.75).
serial correlation in the actual
used
to calibrate
0.81.
aggregate shock in order to calibrate the R^ of the constant-hazard
in
The
that the standard value
/
did.
We
also add
i.i.d.
quadratic adjustment cost
nonnal noise
model
to
match
to the
the R'-
Final
5
The
first
Remarks
paragraph in the conclusion of
CW
illustrates the
flawed logic of their approach.
concludes that "despite the overwhelming evidence that plant-level adjustment
is
It
nonlinear, the
question of whether this matters for aggregate employment dynamics remains an open issue."
But
if the
goal
is
to
show whether
clearly established microeconomic lumpiness matters at the
aggregate level, then the natural approach
to start fi"om a
is
and determine whether aggregation removes
what our methodology
is
all
when
that satisfy the
little
at all,
we
have made an
wrong, or
irrelevant, or driven
test
is
is
precisely
with simulated data that
whether a procedure designed to
test
false positives
twisted logic at best.
effort to take their claims seriously.
than can be rescued from the sequel of
either
and
start
microeconomic lumpiness condition provides
applied to their counterfactual data. This
In our reply, however,
of micro nonlinearities, which
designed to do. Instead, Cooper and Willis
does not resemble actual microeconomic data
competing hypotheses
traces
model with microeconomic lumpiness
CW's
attempts.
by an extraneous
The
But there
is
very
results they claim to find are
ingredient. Let us recap
what they did and
the conclusions they should have drawn:
1
CW relax both of our maintained assumptions — that microeconomic adjustments are lumpy
and
that driving forces follow a
random walk
lumpy microeconomic adjustment
it
in
first
an extreme fashion. The evidence on
overwhelming, even Cooper and Willis acknowledge
and our assumption of a random walk
at times,
assumption of an annual
2.
is
—
is
definitely closer to reality than their
order correlation as low as 0.28.
Correctly interpreted, their main result implies the exact opposite of their Claim
microeconomic gaps are observed, our methodology does not detect
ities
when
When
the
nomic
data,
microeconomic gaps are not observed but need
one should not use our
identification strategy
is
know
to
be estimated from microeco-
(which
sharp contrast with those
when only
we found with actual
relies
to be smooth. In
rameter estimates they find with their counterfactual data are not
Finally,
significant nonlinear-
CW.
lumpiness) with their data, where adjustment
4.
When the
applied to data generated even with the major departures from our assumptions
considered by
3.
1.
on microeconomic
any event, the pa-
statistically significant, in
data.
aggregate data are available and the path of the cross-sectional
distri-
bution needs to be simulated, the assumptions about the serial correlation of the driving
processes
become more
important. This
is
not new. The surprising feature of
CW's
results
is
that
even
after
sion of their
is less
dropping the
serial correlation as
as they
do in the most recent ver-
comment, the explanatory power of the nonlinear terms
than half of what
we found
in the data. If one adopts
the persistence of the driving processes,
essentially
much
no
realistic
and uses the values assumed
no gain from adding higher moments
their claim, there is
more
false positive finding
in their
to their regressions.
in
experiments
assumptions on
CW (2002), there
is
Again, and contrary to
even when applying our methodology
to highly
unrealistic data.
The other two paragraphs
proach"
First,
is
voodoo-economics and
what they
us from
in their conclusion carry the implicit
call "the
that they are
messages
ready to deliver a superior gap-free alternative.
microeconomic models
as the
one they present
extensive literature on the optimality of {S,s) models).'^ Second, and perhaps
in published
from dynamic optimization
work. In
gap ap-
gap approach" has been derived formally by us and many others before
at least as sophisticated
the methods derived
that "the
fact, the difficulties in
that
(for this, see the
more
importantly,
do not "rely on gap measures" already
exist
measuring gaps was our own motivation for Caballero
and Engel (1994, 1999).'^
To end on a more positive
note,
CW's approach contrasts with more constructive and interesting
recent developments in the literature on the
adjustments. For example,
RBC model,
macroeconomic implications of lumpy microeconomic
Kahn and Thomas (2003) conclude
matter for actual investment. But this
gate data generated
is
fixed costs if the interest rate
is
be modified for
first
invest-
as a demonstration that fixed costs
do not
And
it
among
they also
show
that the aggre-
features of actual aggregate data, such as the
such spikes can be generated by microeconomic
and
fruitful area
of research:
to capture the nonlinearites that are
this
proof of optimality of (S,s) policies
discussed in this reply see,
that
fact,
not endogenized (confirming the results in Caballero and Engel,
Let us hope that energy will be spent on
'^The
on aggregate
not what they did. In
999). This finding points to an interesting
to
many
by such a model misses important
skewness caused by investment spikes.
need
an otherwise standard
fixed costs of adjusting caphal do not have a significant impact
ment.'^ This finding has been misinterpreted by
1
that within
is
How does the RBC model
observed in aggregate investment?
type of question.
in
Scarf (1960). For important extensions, relevant to the models
others, Harrison, Sellke
and Taylor (1983), Grossman and Laroque (1990), and the
pedagogical survey in Dixit (1993).
'"'in
these papers
we extended the
(S,s) literature to incorporate stochastic adjustment costs
model via maximum likelihood.
'^See Veracierto (2002) for a similar conclusion in a model of iiTeversible investment.
and estimate a
structural
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Table
1:
ESTIMATION WITH STATIC Gap
CW (quadr. adj
.)
CEH
(lumpy
R- without non-linear parameter
0.90
0.65
R^ with non-linear parameter:
0.90
0.79
Increase in R^ after adding non-linear parameter:
0.00
0.14
Minimum adjustment speed (non-linear model):
Maximum adjustment speed (non-linear model):
0.19
0.31
0.20
0.46
Range of adjustment speeds (non-linear model);
0.01
0.15
adj .)
CW column based on Table la in CW (2004). CEH based on Table 3 in CEH (1997). Maximum,
minimum and range of adjustment speeds are calculated considering adjustment hazards in the range
jjG ± 2aa, where /jq and Oa denote the mean and standard deviation of the cross-section of static
gaps for the model under consideration.
11
Table
2:
Estimation with Inferred Static Gap: Macroeconomic Data
Data
Driving force p (annual)
BLS, as
in
CE
Increase in R-
OTS
(1993)
Sim. Quadr. Adj.
1.00
0.00
Sim. Quadr. Adj.,
0.75"
0.01
0.47
0.03
028
0.05
CW (2001)
Sim. Quadr. Adj., as in CW (2004)
Sim. Quadr. Adj. as in
'Increase in Br' denotes the difference between the R- obtained
constant hazard and the E}
when imposing
a constant hazard, in
when
estimating a
model with a non-
both cases using the methodology in
CE
(1993). 'Sim. Quadr. Adj.' stands for 'Simulated Quadratic Adjustment'.
This
value of p
calibrating
RBC
somewhat below both the value in the driving force used
models (see Cooley and Prescott, 1995).
is
3837 023
12
in
CE
and the values used when
MIT LIBRARIES
3 9080 02617 6898
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