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DEWE;
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Massachusetts Institute of Technology
Department of Economics
Working Paper Series
Adjustment
Is
Much Slower Than You Think
Ricardo
J.
Caballero
Eduardo M.R.A. Engel
Working Paper 03-25
July 21, 2003
Room
E52-251
50 Memorial Drive
Cambridge, MA 02142
This
paper can be downloaded without charge from the
Social Science Research Network Paper Collection at
http://ssrn.com/abstract=430040
^SACHUSETTS INSTITUTE
OF TECHNOLOGY
Adjustment
Ricardo
J.
is
Much Slower than You Think
Eduardo M.R.A. Engel*
Caballero
July
21.2003
Absjract
In
most instances, the dynamic response of monetary and other policies
lumpy. The same holds for the microeconomic response of
ables, such as investment, labor
demand, and
prices.
some of
We show
less than half as fast as the estimated response.
shocks
is
infrequent and
we
of estimating
find that the actual response to
For investment, labor demand and prices, the
speed of adjustment inferred from aggregates of a small number of agents
is
likely to be close to instanta-
neous. While aggregating across microeconomic units reduces the bias (the limit of which
some instances convergence
is
extremely slow. For example, even
establishments in U.S. manufacturing, the estimate of
upward by more than SO percent. While
remains significant
at
Codes:
Keywords:
after
prices), in
aggregating investment across
speed of adjustment to shocks
extreme for labor demand and
is
biased
prices,
it
still
high levels of aggregation. Because the bias rises with disaggregation, findings of
microeconomic adjustment
JEL
the bias is not as
its
illustrated
is
by Rotemberg's widely used linear aggregate characterization of Calvo's model of sticky
all
vari-
ARM A procedures substantially over-
estimates this speed. For example, for the target federal funds rate,
is
to
most important economic
that the standard practice
the speed of adjustment of such variables with partial-adjustment
shocks
the
that is substantially faster than aggregate
adjustment are generally suspect.
C22, C43, D2, E2, E5.
Speed of adjustment, discrete adjustment, lumpy adjustment, aggregation, Calvo model.
ARMA process, partial adjustment, expected response time, monetary policy, investment, labor demand,
sticky prices, idiosyncratic shocks, impulse response function.
'Respectively:
MIT
and
and seminar participants
Financial support from
at
NBER;
Yale University and
Humboldt
NSF is
NBER. We
are grateful to
Uruversital. Universidad de Chile
gratefully
acknowledged.
Wold
(CEA) and
representation, time-to-build.
William Brainard, Pablo Garcia, Harald Uhlig
University of Pennsylvania for their comments.
Introduction
1
The measurement of
tance
dynamic response of economic and policy variables
the
macroeconomics. Usually,
in
this
response
to
shocks
is
of central impor-
estimated by recovering the speed at which the variable
is
we
In this paper
of interest adjusts from a linear time-series model.
argue
that, in
many
instances, this
procedure significantly underestimates the sluggishness of actual adjustment.
The
able
severity of this bias
depends on how infrequent and lumpy the adjustment of the underlying
In the case of single policy variables,
is.
such as the federal funds
variables, such as firm level investment, this bias can be extreme.
the discrete adjustment exists,
we show
that regardless of
how
If
rate, or individual
vari-
microeconomic
no source of persistence other than
sluggish adjustment
may
be, the econometri-
cian estimating linear autoregressive processes (partial adjustment models) will erroneously conclude that
adjustment
is
instantaneous.
Aggregation across establishments reduces the
bias, so
we have
the
somewhat unusual
situation
where
estimates of a microeconomic parameter using aggregate data are less biased than those based upon microe-
conomic
data.
Lumpiness combined with
microeconomic adjustments
that
is
linear estimation procedures is likely to give the false impression
significantly faster than aggregate adjustment.
We also show that, when aggregating across units, convergence to the correct estimate of the speed of adjustment
all
is
extremely slow. For example, for U.S. manufacturing investment, even after aggregating across
the continuous establishments in the
LRD
(approximately 10,000 establishments), estimates of speed of
—
adjustment are 80 percent higher than actual speed
percent. Similarly, estimating a
at
the two-digit level the bias easily can
Calvo (1983) model using standard partial-adjustment techniques
to underestimate the sluggishness of price adjustments severely. This
of Bils and
exceed 400
is
is likely
consistent with the recent findings
Klenow (2002), who report much slower speeds of adjustment when looking
at individual price
adjustment frequencies than when estimating the speed of adjustment with linear time-series models.
also
show
The
that estimates of
basic intuition underlying our
adjustment
is
employment adjustment speed experience
is
main
results
is
the following: In linear models, the estimated speed of
associated with lower adjustment speed. Yet this correlation
is infinitely fast.
To see
i.i.d.
that this crucial correlation
in
period.
This means that any non-zero
acts
it
the variable of concern
all
zero
when
always zero
a larger first order correlation
for an individual series that
is
zero, first note that the product of current
is
there
is
no adjustment
serial correlation
is
last
it
adjusted,
it
must be
bias falls as aggregation rises because the correlations at leads
is,
the
common components
realizations in
in the
that the later
in the
which the
unit
it
adjustment only
previous adjustment.
and lags of the adjustments across
adjustments of different agents
different points in time provides the correlation that allows us to recover the
1
preceding
two consecutive periods, and whenever
independent from the shocks included
individual units are non-zero. That
and lagged
in either the current or the
must come from
unit adjusts in
accumulated shocks since
involves the latest shock, which
The
is
two consecutive periods. But when the
catches up with
is
is.
shocks), so that the researcher will conclude, incorrectly, that adjustment
changes
adjusts in
similar biases.
inversely related to the degree of persistence in the data. That
adjusted discretely (and has
We
at
microeconomic speed of ad-
justment.
The
larger this
common component
is
—
as measured, for example,
shocks relative to the variance of idiosyncratic shocks
the
number of agents grows.
we
portance
results
when
the faster the estimate converges to
In practice, the variance of aggregate shocks
of idiosyncratic shocks, and convergence takes place
In Section 2
—
by the variance of aggregate
at a
is
its
true value as
significantly smaller than that
very slow pace.
study the bias for microeconomic and single-policy variables, and illustrate
im-
its
estimating the speed of adjustment of monetary policy. Section 3 presents our aggregation
and highlights slow convergence.
tent with those of investment, labor,
and extensions.
It first
We show
the relevance of this
and price adjustments
phenomenon
in the U.S. Section
for parameters consis-
4 discusses partial solutions
extends our results to dynamic equations with contemporaneous regressors, such as
those used in price-wage equations, or output-gap inflation models.
reduce the extent of the bias. Section 5 concludes and
is
It
then illustrates an
ARMA
method
followed by an appendix with technical
to
details.
Microeconomic and Single-Policy Series
2
When
the task of a researcher
adjustment costs
is
to estimate the
speed of adjustment of a
adjustment cost model (see
in a quadratic
e.g.,
—
state variable
or the implicit
Sargent 1978. Rotemberg 1987)
—
the
standard procedure reduces to estimating variations of the celebrated partial adjustment model (PAM):
Ay,
where y and y* represent the
employment, or
capital),
each period. Taking
first
v,
(1)
actual and optimal levels of the variable under consideration (e.g., prices,
and X
is
a
parameter that captures the extent to which imbalances are remedied
differences and rearranging terms leads to the best
Ay,
with
= X(y? ->,_,),
in
known form of PAM:
= (l-X)Ay,_!+v,,
(2)
=A.Ay*.
In this
model, X
is
thought of as the speed of adjustment, while the expected time
(defined formally in Section 2)
is (1
— A.) /A.
Thus, as
A.
until
adjustment
converges to one, adjustment occurs instantaneously,
while as A decreases, adjustment slows down.
Most people understand
istic
that this
model
is
only meant to capture the first-order dynamics of more
but complicated adjustment models. Perhaps
most prominent among the
latter,
real-
many microeconomic
variables exhibit only infrequent adjustment to their optimal level (possibly due to the presence of fixed
costs of adjustments).
monetary authority
in
And
the
same
is
true of policy variables, such as the federal funds rate set
response to changes
in
aggregate conditions. In what follows,
we
inquire
by the
how good
the estimates of the speed of adjustment from the standard partial adjustment approximation (2) are,
actual adjustment
is
discrete.
when
—
A
2.1
Lumpy Adjustment Model
Simple
Lety, denote the variable of concern
and y* be
optimal counterpart.
its
at
time
We
t
—
funds rate, a price, employment, or capital
e.g., the federal
can characterize the behavior of an individual agent
terms of the
in
equation:
Av,
where
=S,(tf-y,-i),
£, satisfies:
=
=
Pr{L=l)
Pr{^,=0}
From
to
plications of state-dependent models,
suppress
when
While
two basic
the
(ii)
(i)
periods of inaction fol-
increased likelihood of an adjustment with
second feature
not needed for the point
it is
features:
is
central for the
we wish
macroeconomic im-
to raise in this paper. Therefore,
1
it.
follows from (4) that the expected value of
its
(4)
1-X.
accumulated imbalances, and
the size of the imbalance (state dependence).
It
X.
a modelling perspective, discrete adjustment entails
lowed by abrupt adjustments
we
(3)
value
independent of
is X.
£,,
When
h, t
zero, the agent experiences inaction;
is
one, the unit adjusts so as to eliminate the accumulated imbalance.
is
-y,-i)
(y*
(this is the simplification that
Calvo (1983) makes
We
vis-a-vis
assume
more
that
is
h,,
realistic state
dependent models) and therefore have:
E[Ay, |>>;.y,-i]
which
2.2
is
the analog of (1).
The Main
Hence X
= My,*-*-,),
(5)
represents the adjustment speed parameter to be recovered.
Result: (Biased) Instantaneous Adjustment
The question now
arises as to whether,
by analogy
to the derivation
from
(1) to (2), the standard procedure
of estimating
Ay,
when adjustment
recovers the average adjustment speed, X,
answer to
1
show
X denote
that,
the
with
a
OLS
we
estimator
consider
—
i.e.,
ofX
in
equation
without feature
(ii)
(6).
—
is
Let the Ay*
due
to
's
be
i.i.d.
with
convergence
is
One
of our contributions
achieved.
is
to
mean
and variance G
,
Calvo (1983) and was extended by Rotemberg (1987)
continuum of agents, aggregate dynamics are indistinguishable from those of
quadratic adjustment costs.
arise before
lumpy. The next proposition states that the
(Instantaneous Estimate)
'The special model
to
is
(6)
this question is clearly no.
Proposition
Let
= (l-X)Ay,_,+£„
go over the aggregation steps
in
more
a representative agent facing
detail,
and show the problems
that
)
.
and
let
T
denote the time series length. Then, regardless of the value ofX:
=1
plimy-^A.
Proof See Appendix B.l.
|
While the formal proof can be found
in the
appendix,
text If adjustment were smooth instead of lumpy,
But when adjustment
adjustment necessarily
of A. To see
why
is
we would have
intuition in the
main
the classical partial adjustment model, so
— A,
is 1
its
thereby revealing the speed with which
lumpy, the correlation between this period's and the previous period's
is
independent of the true value
consider the covariance of Ay, and Ay,_j, noting that, because adjustment
we may
occurs,
it
instructive to develop
zero, so that the implied speed of adjustment is one,
this is so,
complete whenever
it is
observed adjustments
that the first order autocorrelation of
units adjust.
(7)
= &
Ay,
is
re-write (3) as:
£ Ay?_*
=
Ay,*_,
(8)
{
\
otherwise
where
denotes the number of periods since the
/,
Table
Adjust
in
t
—
Adjust in
)t
Yes
No
5t'oAy;_,I-*
Yes
Yes
sjbiAtfi,I-*
is
,
No
to
Ay,Ay,_,
consider
zero, since at least
Ay,Ay,_,
Ay,*
and/or the
in this
/)
Contribution to Cov(Ay, Ay,- 1
Ay,
Ay,_i
t
Yes
no adjustment
adjustment took place, (as of period
CONSTRUCTING THE MAIN COVARIANCE
No
No
There are four scenarios
adjustment
1:
last
when
Ay,Ay,_i
=0
=0
=0
Cov(Ay,_,,Ay,)=0
Ay;
constructing the key covariance (see Table
1):
If there
was
period (three scenarios), then the product of this and last period's
last
one of the adjustments
is
zero. This leaves the case of adjustments in both
periods as the only possible source of non-zero correlation between consecutive adjustments. Conditional
on having adjusted both
in
/
and
Cov(Ay, Ay,_,
,
|
/
—
1
,
we have
£,=£,_,
= 1) = Cov(Ay,*
,
Ay,*_,
+ Ay,*_ 2 +
•
•
+ Ay?.^ = 0,
)
since adjustments in this and the previous period involve shocks occurring during disjoint time intervals. Every time the unit adjusts,
it
catches up with
shocks anew. Thus, adjustments
-So
that
/,
=
1
if
all
at different
the unit adjusted in period
/
-
1,
previous shocks
moments
2
if
it
in
had not adjusted to and
it
starts
time are uncorrelated.
did not adjust in
r
-
1
and adjusted
in
t
-
2,
and so on.
accumulating
.
Robust Bias: Infrequent and Gradual Adjustment
2.3
Suppose now
place,
it
is
investment
that in addition to the infrequent
only gradual. Such behavior
(e.g.,
Woodford
policy
Our main
(1999)).
ARMA process — a Calvo model
when
observed, for example,
Majd and Pindyck (1987)) or when
(1987), Sack (1998), or
linear
is
adjustment pattern described above, once adjustment takes
there
is
a time-to-build feature in
designed to exhibit inertia
is
result here
is
that the
with additional serial correlation
—
(e.g..
Goodfriend
econometrician estimating a
will only
be able to extract the
gradual adjustment component but not the source of sluggishness from the infrequent adjustment
nent.
That
is,
again, the estimated speed of adjustment will be too fast, for exactly the
same reason
compoas in the
simpler model.
Let us modify our basic model so that equation (3)
now
applies for a
new
variable y, in place of y,,
with Ay, representing the desired adjustment of the variable that concerns us, Ay,. This adjustment takes
place only gradually, for example, because of a time-to-build component.
We
capture this pattern with the
process:
K
Ay,
=
K
X
<bkAy,- k
+ (!-]£ <h)Ay,
k=]
Now
there are
First, the
two sources of sluggishness
(9)
k=\
in the
transmission of shocks. Ay*, to the observed variable. Ay,.
agent only acts intermittently, accumulating shocks
in
periods with no adjustment. Second,
when
he adjusts, he does so only gradually.
By analogy
the
with the simpler model, suppose the econometrician approximates the lumpy component of
more general model
by:
Ay,
Replacing (10) into
(9), yields the
= (l-k)Ay,-i+v,.
following linear equation
in
(10)
terms of the observable. Ay,:
K+\
Ay,
= £a*Ay,_ A + E,,
(11)
.
*=i
with
a\
<j)i
ak
«A'+1
ande,
k
= 2,...,K,
(12)
= Ml-Z*Lj<MArf.
By analogy
tence
=
+ l-A.,
= to-O-XJfc-i,
= -(l-A.)<|>Ki
to the
stemming from
simpler model,
we now show
that the
econometrician will miss the source of persis-
X.
Proposition 2 (Omitted Source of Sluggishness)
Let all the assumptions in Proposition
with all roots of the polynomial
estimates of equation (77).
1
- Xf=
i
1
<|)it^*
hold, with y in the role
of y. Also assume that
outside the unit disk. Let a^.k
=
1
K+\
(9) applies.
denote the
OLS
Then:
plim-r^Ok
phm r ^
<x>
l,...,K
(13)
=
d K+i
=
k
fa,
0.
Proof See Appendix B.l.
Comparing (12) and (13) we see
of X
is
that the proposition simply reflects the fact that the (implicit) estimate
one.
The mapping from
the biased estimates of the
some, but the conclusion
is
similar.
To see
to capture the overall sluggishness in the
this, let
a^s
to the speed of adjustment
is
slightly
more cumber-
us define the following index of expected response time
response of Ay to Ay*:
dAy, +k
(14)
dAy*
*>0
where E,[J denotes expectations conditional on information
/.
This index
the
is
(that
values of
is,
Ay and Ay*) known
at
time
a weighted sum of the components of the impulse response function, with weights equal
number of periods
that elapse until the
response with the bulk of
its
mass
at
corresponding response
low lags has a small value of
to
observed. 3 For example, an impulse
is
x,
since
Ay responds
relatively fast to
shocks.
It
is
easy to
show
Adjustment Model
(1)
(see Propositions
Al and A2
in the
Appendix)
and for the simple lumpy adjustment model (3)
that
both for the standard Partial
we have
l-X
More
generally, for the
response to Ay
model with both gradual and lumpy adjustment described
while the expected response to shocks Ay*
iLiWk
is
equal
1-*
to:
Note
that, for the
models
at
hand, the impulse response
to one, the definition
4
_
lk=iWk
(15)
i-ZLi**
is
always non-negative and adds up to one.
above needs to be modified
t
4
,
*
3
expected
satisfies:
T
does not add up
in (9), the
to:
S*>o*E,[^]
For the derivation of both expressions for t see Propositions Al and
A3
in the
Appendix.
When
the impulse response
Let us label:
It
when
follows that that the expected response
1-A
=
tlum
y
both sources of sluggishness are present
is
the
sum
of the
responses to each one taken separately.
We
can now
Corollary
state the implication of Proposition 2 for the estimated
(Fast Adjustment)
1
expected time of adjustment,
Let the assumptions of Proposition 2 hold
and let x denote
t.
the (classical)
method of moments estimator for x obtained from OLS estimators of
^ a Ay,_
=
Ay,
k
k
+ e,.
k=\
Then:
plim r _„x
with a strict inequality for
X<
Proof See Appendix B.l.
To summarize,
when
this
is lost.
<
Xii„
x
=
xi, n
+ x lum
(16)
.
1.
|
the linear approximation for
approximation
sluggishness
=
is
Ay (wrongly) suggests no sluggishness whatsoever, so
that
plugged into the (correct) linear relation between Ay and Ay, one source of
This leads to an expected response time that completely ignores the sluggishness
caused by the lumpy component of adjustments.
An
2.4
Figure
Application:
Monetary
Policy
depicts the monthly evolution of the intended federal funds rate during the Greenspan era.
1
infrequent nature of adjustment of this policy variable
monetary policy interventions often come
the description of the
fitting
Our
goal
is
to estimate
model we
This procedure
of-fit.
is
the
the
lumpy component
number
shocks
To
in
The
also well
known
that
Sack (1998) and Woodford (1999)).
and
Xi um .
until finding
Regarding the former,
no significant improvement
is
we
in
estimate
AR
thegoodness-
orthogonal to the lagged Ay's.
We
estimated as 2.35 months.
is relevant, the (absolute)
of periods since the last adjustment.
magnitude of adjustments of Ay should increase with
The longer
the inaction period, the larger the
number of
Ay* to which y has not adjusted, and hence the larger the variance of observed adjustments.
test this
the residual
5
x, X| in
number of lags,
T| in
It is
The
just characterized.
warranted since Ay, the omitted regressor,
obtained an AR(3) process, with
If
evident in the figure.
in gradual steps (see, e.g..
both components of
processes for Ay with an increasing
is
5
implication of
lumpy adjustment, we
identified periods with adjustment in
y as those where
from the linear model takes (absolute) values above a certain threshold, M. Next we partitioned
findings reported in this section remain valid
if
we
use different sample periods.
Figure
1:
Monthly Intended (Target) Federal Funds Rate- 9/1987 - 3/2002
The
those observations where adjustment took place into two groups.
where adjustment also took place
of Ay*
in the
group included observations
first
preceding period, so that the estimated Ay only reflects innovations
one period. The second group considered the remaining observations, where adjustments took
in
place after at least one period with no adjustment.
Columns 2 and
3 in Table 2
show
the variances of adjustments in the
above, respectively, for various values of M.
one shock
If there
is
variances, since they
M. We
than
To
mate
at all
would correspond
=
(k
to a
1),
there
random
the fraction of
in
X and
An
where the residual
lumpy adjustment.
to overall sluggishness,
we
y took place. Since
M are in the neighborhood of this value.
t/„ m for different
values of
alternative procedure
used four
criteria:
= >v_i, we
assume
Also note that (he
is
first
t,
that
that
M. For
to extract
First, if y, -£
"reversal" happened at
y,
column
4).
we had
only observe
y,
is
we need
we
larger
6
to esti-
require
some
a criterion to choose the threshold
M,
could be readily done. The fact that the Fed changes rates by multiples of 0.25 suggests that reasonable
choices for
for
than one shock (see
partition of observations
lumpy component
months where an adjustment
reflects only
would be no systematic difference between both
additional information to determine this adjustment rate. If
this
more
therefore interpret our findings as evidence in favor of significant
actually estimate the contribution of the
and second group described
when adjustments
Interestingly, the variance
significantly smaller than the variance of adjustments to
were no lumpy component
first
is,
all
5 and 6 in Table 2 report the values estimated
these cases, the estimated lumpiness
lumpiness from the behavior of y,
yt -\ and y,_]
if y,
Columns
> y,_i
— y,_ 2
and y,_i
no adjustment took place
order autocorrelation of residuals
is
is
directly.
substantial.
For
then an adjustment of y occurred
< y,_2
in
(ory,
period
/.
<
y _i and y,_i
(
Finally, if
this
at
/.
approach,
Similarly
> y,_2). By
we
if
contrast, if
an "acceleration" took place
-0.002, so that our finding cannot be attributed
a
at
to this factor either.
Table
(1)
2:
ESTIMATING THE LUMPY COMPONENT
(3)
(2)
M
Var(Ay,|^
=l
l
^_i = l)
Var(Ay,|^
=
1,^_, =0)
(4)
(5)
(6)
p-value
A
x lum
0.150
0.089
0.213
0.032
0.365
1.74
0.175
0.092
0.221
0.018
0.323
2.10
0.200
0.117
0.230
0.051
0.253
2.95
0.225
0.145
0.267
0.060
0.206
3.85
0.250
0.150
0.325
0.034
0.165
5.06
0.275
0.164
0.378
0.016
0.135
6.41
0.300
0.207
0.378
0.045
0.123
7.13
0.325
0.207
0.378
0.045
0.123
7.13
0.350
0.224
0.433
0.041
0.100
9.00
For various values of
M (see Column
values reported
1),
2 and 3: Estimates of the variance of Ay, conditional
place the preceding period (column 2) and with
on
in the
remaining columns are as follows. Columns
adjusting, for observations
no adjustment
where adjustment
in the previous period
(column
3).
p-value, obtained via bootstrap, for both variances being the same, against the alternative that the latter
Column
t,
so that y
t
Estimates of X.
5:
— y _i >
(
these criteria
we
y,_i
Column
— »_? >
6:
Estimates of
(ory,
Tj um
— >V-i <
adjustment took place
bound
in
in
the (estimated) value of
4:
is larger.
.
y,-\
can sort 156 out of the 174 months
not. This allows us to
also took
Column
A.
—y
t
-i
<
0),
we assume
that
y adjusted
at
r.
With
our sample into (lumpy) adjustment taking place or
between the estimate we obtain by assuming
the remaining 18 periods and that in
which
all
that
no
of them correspond to adjustments
for y.
Table
3:
MODELS FOR THE INTENDED FEDERAL FUNDS RATE
Lumpy component
Time-to-build component
Tl
t2
Kmn
''•max
^lum.min
^lum.max
0.080
$3
0.231
T-lin
0.230
2.35
0.221
0.320
2.13
3.53
(0.074)
(0.076)
(0.074)
(0.95)
(0.032)
(0.035)
(0.34)
(0.64)
Reported: Estimation of both components of the sluggishness index
x.
Data: Intended (Target)
Federal Funds Rate, monthly, 9/1987-3/2002. Standard deviations in parenthesis, obtained via Delta
method.
Table 3 summarizes our estimates of the expected response time obtained with this procedure.
who
searcher
ignores the lumpy nature of adjustments only would consider the
infer a value of x equal to 2.35 months. Yet
x
is
once
we
is
approximately twice as
Note
that these
is,
fast as the true
Consistent with our theoretical results, the bias
7
coincide with estimates obtained for
re-
AR-component and would
consider infrequent adjustments, the correct estimate of
somewhere between 4.48 and 5.88 months. That
to shocks that
A
ignoring lumpiness (wrongly) suggests a response
response.
in the
7
estimated speed of adjustment stems from the
M in the 0.175 to 0.225 range, see Table
2.
.
infrequent adjustment of monetary policy to news.
adjustment accounts for
at least half
As shown
of the sluggishness
above,
this
bias
is
important, since infrequent
modern U.S. monetary
in
policy.
Slow Aggregate Convergence
3
Could aggregation solve the problem
Rotemberg (1987) showed
level? In the limit, yes.
where lumpiness occurs
at the
microeconomic
that the aggregate equation resulting
from individual
for those variables
actions driven by the Calvo-model indeed converges to the partial-adjustment model. That
of microeconomic units goes to
estimate of X, and therefore
But not
all is
even aggregating across
all
we
In this section,
bias vanishes very slowly as the
as the
number
estimation of equation (6) for the aggregate does yield the correct
(Henceforth
x.
good news.
infinity,
is,
return to the simple model without time-to-build).
we show
number of units
that
when
the speed of adjustment
is
already slow, the
in the aggregate increases. In fact, in the case of investment
U.S. manufacturing establishments
not sufficient to eliminate the bias.
is
The Result
3.1
Let us define the N-aggregate change
at
time
t,
Ayf,
as:
^=ii><v,
where Ay,, denotes
the
change
in the variable
of interest by unit
i
in
period
t.
Technical Assumptions (Shocks)
Let Ay*
t
=
v^
+ vf,,
where the absence of
remains after averaging across
with
all f's).
a
subindex
denotes an element
i
common
to
all
i
(i.e, that
We assume:
mean /^ and variance oA2 >
1
the v^'s are
2.
the vj,'s are independent (across units, over time, and with respect to the v^'s), identically distributed
with zero
3.
i.i.d.
mean and
variance oj
>
0,
0,
and
the 2^/s are independent (across units, over time, and with respect to the v^'s
distributed Bernoulli
As
random
in the single unit case,
variables with probability of success
we now
T
(0, 1].
v^'s), identically
|
ask whether estimating
6tf
yields a consistent (as
X G
and
= (l-X)Arf_ +e
1
l
goes to infinity) estimate of X, when the true microeconomic model
following proposition answers
this
(17)
,
is (8).
The
question by providing an explicit expression for the bias as a function of
10
I
the parameters characterizing adjustment probabilities and shocks (X.
fi^,
Ga and G/)
and, most importantly,
N.
Proposition 3 (Aggregate Bias)
Let
XN denote
the
OLS
ofX
estimator
in
T
equation (17). Let
denote the time series length. Then, under
the Technical Assumptions.
plim^J." - X + i=-,
1
(18)
+A
ith
2
K =
It follows
^
V
.
\\
that:
lim plim r _ >00 A
w
N—•<»
Proof See Theorem Bl
To see the source of
correlation, pi
,
Appendix.
in the
the bias
^ Cov(A3 f ,Ay*
,)
and why aggregation reduces
=
1
and 2
in
1)
we
begin by writing the
+
- l)Cov(Ay,. ,Ay 2 .,)]/iV
[A/Var(Ay„)
Ay denote two
/V(N
N
order auto-
1
)]//V
2
2
f
different units.
identical) first-order autocovariance terms,
(also identical) first-order cross-covariance terms,
Likewise, the denominator considers
first
different terms:
+ N{N - l)Cov(Ay u ,Ay 2 ,,_
The numerator of (21) includes A (by symmetry
N(N —
it,
sums and quotients of four
7
and
(20)
[/VCov(Ayi.,.Ayi.,-i)
Var(Ay?)
where the subindex
= X.
I
as an expression that involves
Pl
unit,
(19)
,
identical variance terms
and
one for each
N (N ~
one
each
for
pair of different units.
1) identical
contemporaneous
cross-covariance terms.
From columns
2 and 4 in Table 4
we observe
that the
cross-covariance terms under
adjustment are the same. 8 Since these terms will dominate for sufficiently large
them, compared
8
This
is
to
N
additional terms
somewhat remarkable,
covariance term. In the case of
—
it
follows that the bias vanishes as
adjustments
1
)
=
at all
=
Cov(X £(1 -X) k A/U -k
X
1
N(N —
1)
of
N goes to infinity.
.
-^'^-l-/)
*X0
;>o
X 2 (l-X) w+, Cov(AyI ,_ Jt ,Aj5 _ 1 _ l )
5>
contrast, in the case of the
there are
lags contribute to the cross-covariance term:
*>o
and Ayij-
—
and lumpy
since the underlying processes are quite different. For example, consider the first-order cross-
PAM,
Cov(Ay,,,,Ay 2 ,,_
By
N
PAM
i
:
u-*)
il
2/+
lumpy adjustment model, the non-zero terms obtained when calculating
are due to aggregate shocks included both in the adjustment of unit
II
1
(in ;)
the
and unit 2 (in
I
covanance between
— 1).
Ayj.,
Idiosyncratic shocks
Table 4:
Constructing the First Order Correlation
(1)
(2)
(3)
(4)
Cov(Ayi ,Ayi _i)
Cov(Ay u ,Ay 2 .t-i)
Var(Ay,,,)
Cov(Ay M ,Ay 2i ,)
^(l-XKoJ + of)
&(i-*)<>3
nK+°;)
2-X°A
if
ir
(1)
PAM:
(2)
Lumpy
(fiA
(3)
Lumpy
((iA7^0):
= 0):
&(l-*)°5
°5
&0-*K
-(1-X>2
However, the underlying bias may remain
+ G/
°iW +
2-X°A
^
significant for relatively large values of
2-X°A
tf
A
From Table 4
7
.
it
follows that the bias for the estimated first-order autocorrelation originates from the autocovariance terms
The
included in both the numerator and denominator.
is
zero for the lumpy adjustment model, while
Section
And even though
2).
covanance terms with no
are proportional to
it
PAM
positive under
is
number of terms with
the
first-order autocovariance
ga which
,
considerably smaller in
is
N
significant for relatively large values of
(more on
all
this
in the
(this is the bias
only N, compared with
this bias is
GA + rj
bias, the missing terms are proportional to
2
term
2
we
numerator
discussed in
N(N —
1) cross-
while those that are included
,
applications. This suggests that the bias remains
below) and
that this bias rises with the relative
importance of idiosyncratic shocks.
There
is
a second source of bias once
N>
1,
related to the variance term Var(Ayi
of the first-order correlation in (21). While under
(when A
—
0)
value under
and
GA + a (when A =
2
PAM, GA + o~
2
,
1),
PAM
this
when adjustment
is
variance
lumpy
,)
in the
denominator
increasing in A, varying between
is
variance attains the largest possible
this
independent of the underlying adjustment speed A. This suggests that the bias
more important when adjustment
is
fairly infrequent.
Substituting the terms in the numerator and denominator of (21) by the expressions in the second
concerned.
It
follows that:
Cov[Ayi,, ,Ay 2 ,,-i|E,i,,
=
1, £2,1-1
are irrelevant as far as the covariance
and averaging over
/]
,
is
=
9
is
To
UuA^i]
=
min(l h
,
row of
- l,/ 2 ,,-i)a^,
we obtain:
and /2,1-L both of which follow (independent) Poisson processes,
Cov(Ayi,„Ay2 ,,_
which
is
9
the expression obtained under
further understand
why
1
)=
^(l-X)^,
(22)
PAM.
Var(A>'i,) can be so
much
larger in a
Calvo model than with
partial adjustment,
we compare
the
contribution to this variance of shocks that took place k periods ago, V^,.
For
PAM we have
Var(Ay„) =
Var(
£ X(l - X)*AyJ
f
_k)
k>0
= X2
£
k>0
12
(1
-
X)
a Var(AyJ
,_,),
(23)
'
Table
4,
Pl
This expression confirms our discussion.
creasing in
The reason
we
note that a value of
for this
Whenever the
unit
that
is
to
illustrates clearly that the bias is increasing in
G//ga and de-
biases the estimates of the speed of adjustment even further.
/x^ 7^
its
change
is,
in
is
when estimating X with standard
when
either ct^/g/,
solid line depicts the percentage bias as
slightly
is
it
mean change. When adjustment
absolute value, below the
\ha\.
The
10
clearly negative, inducing negative serial correlation.
N
converges to X. The baseline parameters (solid
By
(24)
3fv
?
introduces a sort of "spurious" negative correlation in the time series of Ayi,.
up, the bias obtained
be significant
10.000,
leads to:
pent-up adjustments are undone and the absolute change, on average, exceeds
product of these two terms
Summing
it
does not adjust,
finally takes place,
expected
— X)
)X/(2
1-X
n-
=
It
1
and N.
A.
Finally,
N=
N(N -
and dividing numerator and denominator by
N
X
or
is
line) are
For
grows.
above 20%. The dash-dot
contrast, the dash-dot line considers jxa
shows the case where X doubles
=
small, or
0.10,
partial
|it.i|
is
adjustment regressions can be
large. Figure 2 illustrates
how XN
= 0, A. = 0.20. Ga = 0.03 and G/ = 0.24. The
N = 1,000, the bias is above 100%; by the time
11,4
line increases
Oa
to 0.04.
speeding up convergence.
which slows down convergence.
Finally, the dotted line
which also speeds up convergence.
to 0.40.
Corollary 2 (Slow Convergence)
The bias
in the
estimator of the adjustment speed
increasing in G/ and
is
I/I4
1
and decreasing
in
Oa
•
N and
Furthermore, the four parameters mentioned above determine the bias of the estimator via a decreasing
X.
expression of K.
Proof
Trivial.
'
I
so that
^PAM
By
contrast, in the case of
v
is
bution
much
iunw
larger
=
Fr{/i,,=Jk}VSff(Ay, il |/, /
=
\(\-\) k
=A-}[AVar(Ay,,|/„
-l
We
"The
=
1,1;,.,
mass
at
also have that
results for a/
this is equivalent to A'
^
/i,*
1
+ (I
-X-)Var(Ay,,,|f,, r
= 0,&
=0)]
t ).
Under PAM. by
at all)
infrequent adjustment the relevant conditional distri-
and
contrast, V;
,
is
a
normal distribution with a variance that grows
generated from
a a
normal distribution with zero
k.
further increases the bias due to the variance term, see entry (3,3) in Table 4.
and k may not hold
>
1)
PAM. With
zero (corresponding to no adjustment
variance that decreases with
=
[lk{<?
A +<$)}.
linearly with k (corresponding to adjustment in
mean and
=t)
under the lumpy adjustment model than under
a mixture of a
is
)
lumpy adjustment we have
Pr{/,.,
Vkt
=X2 (1 _ x) 2* (02 +o2
and therefore
is
if \n A
\
is
large. For the results to
not binding.
I
i
hold
we
need
N > + (2 - X)tiA2 /\aA2 When
1
.
Ma
=
u
.
Bias as a function of
Figure 2:
N for various parameter
configurations
180
160
140
120
to
en
S"
100
a
°-
80
1000
6000
5000
4000
3000
2000
Number
8000
7000
Figure 2 shows that the bias in the estimate of the speed of adjustment
when estimated with very aggregated
for U.S.
data. In this section
we
employment, investment, and price dynamics. These
evidence of their infrequent adjustment
Let us
start
at the
when
N—
LRD.
the bias remains
likely to
1,000, which corresponds to
LRD,
Klenow
above 40 percent. That
is,
significant,
even
series are interesting because there
is
extensive
level.
We use the parameters estimated by Caballero, Engel,
establishments than
Table 5 shows that even
data.
in a typical
estimated speeds of adjustment
two-digit sector of the
at
the sectoral level are
be significantly faster than the true speed of adjustment.
The good news
The
many more
remain
likely to
provide concrete examples based on estimates
microeconorruc
with U.S. manufacturing employment.
is
and Haiti wanger( 1997) with quarterly Longitudional Research Datafile (LRD)
in this case is that for
N=
10,000, which
is
about the size of the continuous sample in
the bias essentially vanishes.
results for prices, reported in
(2002), while a/
Togo from
the a;
is
computed
monopolistic competitive firm
at
10000
Applications
3.2
the
9000
N
of units:
Table
6, are
based on the estimate of X, Ha and
consistent with that found in Caballero et
for
employment
is isoelastic, its
in
Caballero
et al.
production function
al
(1997).
(1997) to that of prices,
is
Cobb-Douglas, and
its
we
,2
The
note that
capital fixed
04.
table
if
the
from Bils and
shows
that the
demand faced by
(which
is
a
nearly correct
high frequency), then (up to a constant):
P*a
where p* and
I*
= w
(
i
-flir)
+
-°-l)1'i
denote the logarithms of frictionless price and employment, w, and
and productivity, and a^
is
the labor share.
demand, which we assume, a/
It is
~ (1 — a/Jo
-
.
a,,
are the logarithm of the nominal
wage
straightforward to see that as long as the main source of idiosyncratic variance
/,.
14
is
.
Table
Slow Convergence: Employment
5:
Average X
Number
10.000
1,000
100
Number
of agents (N)
35
0.901
0.631
0.523
200
0.852
0.548
0.436
0.844
0.532
0.417
of
Time
Periods
(T)
0.400
Reported: average of OLS estimates of X, obtained via simulations.
Number
of
simulations chosen to ensure that numbers reported have a standard deviation less
T = °° calculated from Proposition 3. Simulation parameters:
a A = 0.03, o/ = 0.25. Quarterly data, from Caballero et al.
than 0.002. Case
X: 0.40,
jx
a
=
0.005.
(1997).
bias remains significant even for
N=
10,000. In this case, the main reason for the stubborn bias
is
the high
value of 0/ /oa
Table
6:
SLOW CONVERGENCE: PRICES
Average X
Number
100
Number
of agents (N)
10,000
1,000
60
0.935
(if) 14
0.351
500
0.908
0.542
0.279
0.902
0.533
0.269
of
Time
Periods
(7)
Reported: average
OLS
0.220
estimates of X, obtained via simulations.
Number
of
simulations chosen to ensure that numbers reported have a standard deviation less
than 0.002. Case
T=
«*>
calculated from Proposition
X: 0.22 (monthly data, Bils
and Klenow, 2002), nA
=
3.
Simulation parameters:
0.003,
aA =
0.0054, a,
=
0.048.
Finally, Table 7 reports the estimates for
The estimate of X,
that
found
when
,3
N—
in
fi^ ar) d
Caballero
equipment investment, the most sluggish of the three
Oa< are from Caballero, Engel, and Haltiwanger (1995), and
et al
(1997).
13
Here the bias remains very large and
rj/ is
series.
consistent with
significant throughout.
Even
10,000, the estimated speed of adjustment exceeds the actual speed by more than 80 percent.
To go from
the 07
computed
monopolistic competitive firm
is
for
employment
isoelastic
and
its
in
Caballero
et al
(1997)
production function
15
is
to that
of capital,
Cobb-Douglas, then
we
note that
07,.
~
a/..
if
the
demand
The
faced by a
,
reasons for this
to
is
the combination of a low
A.,
a high
(mostly due to depreciation), and a large
\iA
rj/
(relative
aA ).
Table
SLOW CONVERGENCE: INVESTMENT
7:
Average
A.
Number
of agents (N)
100
1,000
10,000
15
1.060
0.950
0.665
50
1.004
0.790
0.400
200
0.985
0.723
0.305
OO
0.979
0.696
0.274
OO
Number
of
Time
Periods
CO
Reported: average of
OLS
0.150
estimates of X, obtained via simulations.
Number
of simulations chosen to ensure that numbers reported have a standard deviation
less than 0.002. Simulation parameters: X: 0.15 (annual data,
al,
We
1995),
nA =
0.12,
have assumed throughout
aA =
that
0.056, 07
Ay*
=
from Caballero
et
0.50.
Aside from making the results cleaner,
is i.i.d.
it
should be
apparent from the time-to-build extension in Section 2 that adding further serial correlation does not change
the essence of our results.
longer zero, but the bias
subsection, there
1997], Bils and
is
In such a case, the cross correlations
we have described
evidence that the
Klenow
In any event, for
remains.
assumption
is
Can we
fix the
problem while remaining within
that in principle this
4.1
Biased Regressions
series
we have assumed
of interest,
y.
estimating equation
is
some proxy
[1995,
that the
ARMA Correction
the class of linear time-series
possible, but in practice
it
speed of adjustment
is
models? In
this section,
we
unlikely (especially for small N).
estimated using only information on the economic
is
Yet often the econometrician can resort to a proxy for the target y*. Instead of
(2), the
is:
Ay,
with
al
in this
[2002]).
show
far
each of the applications
not farfetched (see, e.g., Caballero et
Fragile Solutions: Biased Regressions and
4
So
i.i.d.
between contiguous adjustments are no
-
( 1
- X) Ay,- + AAy,* + e,
1
available for the regressor Ay*.
16
(25)
.
Equation (25) hints
in principle
procedure for solving the problem.
at a
Since the regressors are orthogonal, A
can be estimated directly from the parameter estimate associated with Ay*, while dropping
the constraint that the
sum
of the coefficients on the right hand side add up to one.
econometrician does impose the
an unbiased and
a
of
latter constraint, then the estimate
A.
biased coefficient, and hence will be biased as well.
will
We
Of
course,
if
the
be some weighted average of
summarize these
results in the
following proposition.
Proposition.4 (Bias with Regressors)
With the same notation and assumptions as in Proposition
Ay?
where Ay* denotes the average shock
defined
+&iAy,*+e„
Ayf_,
fc
period
£Ay*
t,
consider the following equation:
Then, if (26)
/7V.
;
(26)
is
estimated via OLS, and
K
in (19),
without any restrictions on bo
(i)
in
=
3,
and
h\:
plim T _t
J
=
-^-(1-A),
+K
(27)
A,
(28)
1
plim T _J>
(ii)
imposing bo
=
1
—b].
x
u
plinij^^b]
In particular, for
=
N=
and
1
/1.4
Of course,
in the
(at least) the
= A1 H
!
~X
—
Appendix.
in practice the "solution"
Ay* exactly, and
2{K + X)'
= 0:
i
plim T ^„b\
Proof See Corollary Bl
X(l-X)
=A +
above
is
not very useful. First, the econometrician seldom observes
scaling parameters need to be estimated. In this situation, the coefficient es-
timate on the contemporaneous proxy for Ay*
is
no longer useful
for estimating A,
estimated from the serial correlation of the regression, bringing back the bias. Second,
cian does observe Ay*, the adding up constraint typically
,4
The expression
Av* as
of
that
follows
is
a regressor (Proposition 3).
a
is
and the
when
the biased estimator
X and K.
17
must be
the econometri-
linked to homogeneity and long-run conditions
weighted average of the unbiased estimalor X and the biased estimator
The weight on
latter
is
XK/2(K + X), which
in the
regression without
corresponds to the harmonic mean
.
be reluctant to drop (see below).
that a researcher often will
A
Fast Micro - Slow Macro?:
Blanchard (1987) reached the conclusion that the speed of adjustment of prices
In an intriguing article,
to cost changes
much
is
prices adjust faster to
Price- Wage Equation Application
faster at the disaggregate than the aggregate level.
wages (and input
More
specifically,
he found
that
His study
prices) at the two-digit level than at the aggregate level.
considered seven manufacturing sectors and estimated equations analogous to (26), with sectoral prices
the role of y, and both sector-specific
wages and input
condition in this case, which was imposed
prices as regressors (the y*).
Blanchard's study,
in
is
The
classic
in
homogeneity
equivalent in our setting to bo
+ b\ —
1
Blanchard's preferred explanation for his finding was based on the slow transmission of price changes
through the input-output chain. This
ference
speed of adjustment
in
finding can be explained
sition but
is
an appealing interpretation and likely to explain some of the
different levels of aggregation.
at
by biases
Matching Blanchard's framework
0.18 while
at
the aggregate level
Table
8:
However, one wonders how much of the
We
like those described in this paper.
simply highlight the potential size of the bias
it is
to our setup,
0.135.
do not attempt
price-wage equations for
in
we know
dif-
a formal
realistic
that his estimated sectoral
A
decompo-
parameters.
approximately
is
15
Biased Speed of Adjustment: Price-Wage Equations
Average A
Number of firms
(A7 )
100
500
1.000
5.000
10,000
oo
250
0.416
0.235
0.194
0.148
0.142
0.135
oo
0.405
0.239
0.194
0.148
0.142
0.135
No. of Time Periods, T:
Reported: For
T=
250, average estimate of X obtained via simulations.
Number
of simulations chosen to ensure that
T = °°: calculated from (28). Simulation parameters:
= 0.003, aA = 0.0054 and 07 = 0.048 as in Table 6.
estimates reported have a standard deviation less than 0.004. For
X: 0.135
=
and T
250 (from Blanchard, 1987, monthly
Table 8 reports the bias obtained
tions.
It
of firms
assumes
that the true
in the sector.
The
table
shows
We
is
0.135, and considers various values for the
that for reasonable values of
enough
N
there
Blanchard's estimates.
to include
obtained these estimates by matching the cumulative impulse responses reported
Blanchard (1987)
aggregate speed
16
nA
estimating the adjustment speed from sectoral price-wage equa-
speed of adjustment, A,
the estimated value of A, certainly
,5
when
data),
for 5,
we
6 and 7
lags.
For the
sectoral speeds
obtain 0.137. 0.121 and 0.146.
The estimated speed of adjustment
is
The numbers
close to 0.18 for
N=
we
a significant
in the
main
text are the
upward
bias in
16
in the first
two columns of Table 8
obtain, respectively, 0.167, 0.182
1,000.
18
is
number
in
and 0.189, while for the
average X's obtained
this
way.
.
ARMA
4.2
Corrections
Can we
Let us go back to the case of unobserved Ay*.
ARMA
models? In the
first
part of this subsection,
MA
correction amounts to adding a nuisance
However, the second
to
know when
is
warns that
that this is
indeed possible. Essentially, the
nuisance parameter needs to be ignored
this
not encouraging, because in practice the researcher
some
he should or should not drop
we show
term that "absorbs" the bias.
part of this subsection
estimating the speed of adjustment. This
the bias while remaining within the class of linear
fix
is
when
unlikely
MA terms before simulating (or drawing inferences
of the
from) the estimated dynamic model.
On
the constructive side, nonetheless,
the nuisance
we show
when N
that
MA parameter, we obtain better — although
still
is
sufficiently large, even if
biased
—
we do
not ignore
estimates of the speed of adjustment
than with the simple partial adjustment model.
4.2.1
Nuisance Parameters and Bias Correction
Let us start with the positive result.
Let the Technical Assumptions (see page 10) hold}
Proposition 5 (Bias Correction)
an ARMA(l.l) process with autoregressive parameter equal
to
1
—
X.
1
Thus, adding an
Then Ayf follows
MA(1 ) term
to the
standard partial adjustment equation (2):
Ayf
=
/
(l-X)Ayf_ +v,-ey,_ lj
(29)
1
and denoting by XN any consistent estimator of one minus
the AR-coefficient in the equation above,
we have
that:
= X.
plim T -00 /v
The moving average
coefficient, 9. is
tends to
Oa an d
infinity), fa.
°"/-
VM?
a "nuisance" parameter that depends on
N
(it
converges to zero as
N
have that:
Q=~{L-VL -4)>0,
2
with
2
L
K
and
defined
17
in the
Appendix.
Strictly speaking, to avoid the case
1
(2
\-X
in (19).
Proof See Theorem Bl
N- =
+ X(2-X)(K-\)
- \)n^ /Xo^
.
In particular,
where
the
when ma =
I
AR
this
and
MA
amounts
coefficients coincide,
to
19
assuming
N>
1
we need
to rule out the knife-edge case
.
The proposition shows
that
adding an
the bias. This rather surprising result
rection
N, as
not robust for small
is
reduction).
(typically
18
Also, as with
T>
all
is
the
MA(1) term
to the standard partial adjustment equation eliminates
valid for any level of aggregation.
However,
in practice this cor-
MA and AR coefficients are very similar in this case (coincidental
ARMA estimation
procedures, the time series needs to be sufficiently long
100) to avoid a significant small sample bias.
ADJUSTMENT SPEED
Table 9:
X:
WITH AND WITHOUT
MA CORRECTION
Number
Employment
1,000
AR(1):
0.18
0.88
1.40
ARMA(1,1):
ARMA(1,1), ignoring
0.65
1.29
1.47
MA:
AR(1):
ARMA(1,1):
ARMA(1,1), ignoring
Prices
Investment
Reported:
of agents (N)
100
10,000
1.50
1.50
1.50
0.11
0.88
2.72
3.43
1.27
2.83
3.55
3.55
3.55
AR(1):
0.02
0.44
2.65
ARMA(1,1):
ARMA(1,1), ignoring
0.77
3.92
5.29
5.67
5.67
5.67
MA:
MA:
theoretical value of T, ignoring small
sample bias
(7"
=
°°).
"AR(1)" and
"ARMA(1,1)" refer to values of x obtained from AR(1) and ARMA(1,1) representations.
"ARMA(l.l) ignoring MA" refers to estimate obtained using ARMA(l.l) representation, but
ignoring the MA term. The results in Propositions 3 and 5 were used to calculate the expressions for
Next we
x.
Parameter values are those reported in Tables
illustrate the extent to
employment,
applications to
which our
1-X
response that
was defined
is
third
row
N
<-1-X _ x
X~~
'
ARMA
process
(it
could be an
in
For example,
19
See Proposition Al
if jiA
=
MA(«), an AR(°°)
Table 9 illustrates the main result. The estimate of
dropped for x-calculations,
is
(note that in order to isolate the biases that concern us
18
begin by noting that the
or, in
ARMA( 1,1)).
the nuisance term is used in estimation but
of the value of
9_
is:
We
19
Proposition 5, x denotes the correct expected response and x ma the expected
each of the applications
in
MA term
~~~T^e
inferred from a non-parsimonious
our particular case, an
The
in
6.
prices, and investment considered in Section 3.
Tma
>
and
ARMA correction estimates the correct response time in the
expected response time inferred without dropping the
where
4, 5
0,
we have
in the
that the
Appendix
AR
and
MA term are identical for N =
for a derivation.
20
I
unbiased
x,
when
in all the cases, regardless
we have assumed T —
°°).
The
first
and second rows
(in
each application) show the biased estimates. The former repeats our basic
result while the latter illustrates the
bias from not dropping the
tion,
20
yet
it
MA
problems generated by not dropping the nuisance
term
remains significant even
5
Conclusion
The
practice of approximating
is
smaller than that from inferring x from the
at fairly
high levels of aggregation
dynamic models with
linear
ones
is
(e.g., N =
first
The
order autocorrela-
1,000).
widespread and useful. However,
lead to significant overestimates of the speed of adjustment of sluggish variables.
when
MA term (Xma).
The problem
is
it
can
most severe
dealing with data at low levels of aggregation or single-policy variables. For once, macroeconomic
data seem to be better than microeconomic data.
Yet this paper also shows that the disappearance of the bias with aggregation can be extremely slow.
For example,
in the case of investment, the bias
continuous establishments
in the
LRD
While the researcher may think
(N
=
remains above 80 percent even after aggregating across
10.000).
that at the aggregate level
adjustment-cost model generates the data,
all
it
it
does not matter much which microeconomic
does matter greatly for
(linear) estimation of the
speed of
adjustment.
What happened
to
Wold's representation, according
process admits an (eventually infinite)
of Section 4, do
we
stochastic process at
MA
to
representation?
which any
Why,
obtain an upward biased speed of adjustment
hand? The problem
is
that
stationary, purely non-deterministic,
as illustrated
when
by the analysis
at the
end
using this representation for the
Wold's representation expresses the variable of
interest as
a distributed lag (and therefore linear function) of innovations that are the one-step-ahead linear forecast
errors.
When
the relation between the
case when adjustment
is
macroeconomic
variable of interest and shocks
is
non-linear, as
is
the
lumpy. Wold's representation misidentifies the underlying shock, leading to biased
estimates of the speed of adjustment.
Put
somewhat
differently,
when adjustment
is
lumpy, Wold's representation identifies the correct ex-
pected response time to the wrong shock. Also, and for the same reason, the impulse response more generally will
be biased. So will many of the dynamic systems estimated
tests that derive
from such systems.
We
are currently
in
VAR style
working on these
-°This can be proved formally based on the expressions derived
21
in
Theorem BI and
models, and the structural
issues.
Proposition
Al
in the
appendix.
References
[1]
Bils,
Mark and
Peter
J.
Klenow, "Some Evidence on the Importance of Sticky Prices,"
NBER WP #
9069, July 2002.
[2]
Blanchard, Olivier
Activity
[3]
1987,57-122.
Box, George
cisco:
[4]
1,
"Aggregate and Individual Price Adjustment," Brookings Papers on Economic
J.,
and Gwilym M. Jenkins, Time Series Analysis: Forecasting and Control, San Fran-
E.P.,
Holden Day, 1976.
Caballero, Ricardo
Eduardo M.R.A. Engel, and John C. Haltiwanger, "Plant-Level Adjustment and
J.,
Aggregate Investment Dynamics", Brookings Papers on Economic Activity, 1995
[5] Caballero,
Ricardo
J.,
(2),
1-39.
Eduardo M.R.A. Engel, and John C. Haltiwanger, "Aggregate Employment
Dynamics: Building from Microeconomic Evidence", American Economic Review, 87
(1),
March
1997, 115-137.
[6]
Calvo, Guillermo, "Staggered Prices in a Utility-Maximizing Framework," Journal of Monetary Eco-
nomics, 12, 1983, 383-398.
[7]
Engel, Eduardo M.R.A., "A Unified Approach to the Study of Sums, Products, Time-Aggregation and
other Functions of
[8]
ARM A Processes", Journal Time Series Analysis, 5,
1984, 159-171.
Goodfriend, Marvin, "Interest-rate Smoothing and Price Level Trend Stationarity," Journal of Monetary Economics, 19, 19987, pp. 335-348.
[9]
Majd, Saman and Robert
S.
Pindyck, "Time to Build, Option Value, and Investment Decisions," Jour-
nal of Financial Economics, 18,
[10]
Rotemberg, Julio
J.,
"The
New
March 1987, 7-27.
(1987).
Keynesian Microfoundations,"
NBER Macroeconomics Annual,
in
O. Blanchard and
Fischer (eds),
S.
1987, 69-104.
Policy," Federal Reserve
Board Finance
Dynamic Labor Demand Schedules under Rational
Expectations,"
[11] Sack, Brian, "Uncertainty, Learning, and Gradual
Monetary
and Economics Discussion Series Paper 34, August 1998.
[12] Sargent,
Thomas
J.,
"Estimation of
Journal of Political Economy, 86, 1978. 1009-1044.
[13]
Woodford, Michael, "Optimal Monetary Policy
Inertia,"
22
NBER WP # 7261, July
1999.
'
Appendix
A
The Expected Response Time Index:
Lemma Al
(t for
an
Infinite
MA)
x
Consider a second order stationary stochastic process
Ay,
= ^Wk^t-k,
k>0
with
\\Iq
^M
—
=
£*>()¥* <
Yk>o^VkZ has all
1,
and
°°'
^ le
its
roots outside the unit disk.
£ ' 5 uncorrelated,
e,
itncorrelated with Ay,_i,Ay,_2.
...
Assw/Me
?/i«r
Define:
Ik
d&y,+k
=
and
I
lk>okh
—
=
x
-=
de,
[
Then:
Proof That
z=l.
Ik
= ty*
is trivial.
V'(l)
/md
i.
The expressions
from
for x then follow
differentiating *¥(z)
and evaluating
at
I
Proposition Al (x for an
ARMA
Process) Assume Ay, follows an ARMA(p.q):
P
Ay,
9
- Yj
=
fa&yt-k
e,
-
k=l
where
S*>i*V*
__
$>{z)
= — Y,k=\
1
tions regarding the
E, 's
<J*JtS*
a,! ^
®(z)
—
same as
are the
1
in
~~
Sf=i 6*z*
X Q k£,-kk=\
'iflve «// r/iefr
roorr outside the unit disk. The assump-
Lemma A 1.
Define x as in (30). Then:
Proof Given the assumptions we have made about the roots of
Ay
where L denotes the lag operator. Applying
'
Proposition
and x
2
is
A2
(x for a
'More
generally, if the
x
=
(1
=
*(i)
"
*(Z)
we may
write.
E"
with 0(z)/4>(z) in the role of *F(z)
1-zjLi**
we
then have:
i-iLi 6*'
Process) Consider Ay,
in f/ie
simple lumpy adjustment model (8)
- X)/X. 21
number of periods between consecutive adjustments
where
and 0(z),
0(L)
Lemma Al
Lumpy Adjustment
<te/inerf in (/4). 77ien
the particular case
©(i)
<
<J>(z)
interarrival times follow a
Geometric
distribution.
23
are
i.i.d.
with
mean m, thent
= m— 1.
What follows
Proof dAyt+ k/dAy*
and
t
is
equal to one
+ k—1, and is equal
/*
The expression
Proposition
the unit adjusts
dAyl+k
=
Prfe+*
E,
dAy;
for x
A3
when
now follows
(x for a
time
at
t
+k,
not having adjusted between times
t
to zero otherwise. Thus:
=
1
,
£,+*_,
= ^ + ,_ 2 =
...
= &= 0} = X(l - X) k
(31)
.
easily.
Process With Time-to-build and
Lumpy
Adjustments) Consider the process Ay
with both gradual and lumpy adjustments:
K
=
Ay,
K
+ (1 -
Yj tyk&yt-k
A=l
X <WAy,,
k=l
(32)
ith
ll-l
(33)
k=0
where Ay*
is i.i.d.
with zero
mean and variance O2
.
Define x by:
Zjt>o^E,
Si>oE
SAy;
dby, +k
3Ay,-
Then:
Proof Note
that:
9Ay, +k
[
where, from Proposition
H =
k
X{\
-
X) k
.
e
dAy;
Al and
=
(31)
we have
r(i)
7(1)
that the
E*>ofl*Z* and I{z)
infinite series I(z) is equal to 7* in (34),
""
we
=
Gt_j
are such that G(z)
G{z)H(z).
G(l)
+
//'(i)
//(1)
24
sf=1
2,Gk-jHJt
Ay,*
7=0
Gk
Ay, +J
(34)
;=0
=
£*>o<j*z*
=
l/Q>(z), and
Noting that the coefficient of z* in the
have:
g'(i)
"
SB,
[d&yt+j dAyf
'
;
Define H(z)
k
3Ay r+ * 9 Ay, +j
yE
E,
^
l-EJ^fe
i-A
A
t
B
Bias Results
Results in Section 2
B.l
we prove Proposition
In this subsection
which
is
proved
2 and Corollary
1.
Proposition
1
is
a particular case of Proposition 3,
Section B.2. The notation and assumptions are the same as
in
Proof of Proposition 2 and Corollary
The equation we estimate
1
Ay,
=
Y, a k Ayt-k
in
Proposition A3.
is:
+ v,.
(35)
*=i
while the true relation
An argument
that described in (32)
is
analogous to that given
of (32), denoted by w,
what follows,
in
in
and
(33).
Section 2.2 shows that the second term on the right hand side
>
uncorrected with Ay,_£, k
is
1.
It
follows that estimating (35)
is
equivalent to estimating (32) with error term
*»
= = (i-Efo&XAtf-*.
and therefore:
f
plim r _„<5it
=
4> ;.
ifk
[
The expression
pUm^^^t now
for
= K+l.
follows from Proposition A3.
|
Results in Sections 3 and 4
B.2
In this section
we
tion 4
is
Lemma
-X)
proved
in
The notation and assumptions are those in Proposition 3.
of lemmas. Propositions 3 and 5 are proved in Theorem Bl. while Proposi-
prove Propositions
The proof proceeds
X(l
ifk=l,2,-,K,
<
via a series
-\k= 1,2,3,...
In particular, the
Furthermore,
/,.,
and
5.
Corollary Bl.
Bl Assume X] and Xj are
k
3, 4,
and
I, ,
-
lj s
's
i.i.d.
geometric random variables with parameter X, so that Pr{X
= k} =
Then:
m)
=
I
Vkrpf,]
=
^.
(defined in the main text) are all geometric
are independent
Next define, for any integer
s larger
ifi
^
j.
or equal than zero:
Ml =(°. fy
min(Xi —s
-s,X2 )
[
25
ifX <
x
s,
ifX >s.
x
random variables with parameter
X.
Then
EI
Proof The expressions for
To
common
Pr{M,
to X\
= k}
Pr{X,
=
and, since
1
and
X2
this
- s = k,X2 > k + 1} + Pt{X2 = k,Xt -5 > k+
—A)*, with some algebra we
(1
expression to calculate
For any
We
+ Pr{Xi -s = Jfe,X2 = it}
E[MS
]
obtain:
= A-}=A(2-A)(l-X) s+2i - 2
via
strictly positive integer
X;>i &Pr{Mj
= k}
.
completes the proof.
|
s,
= -(l-X) s nl
Cov(&y iih Ayu+s )
Proof
1}
/( J + /:)[l-F(A)]+/(*)[l-F(* + 5 )]+/W/(fc + 5),
— F(fc) =
Lemma B2
trivial.
by F(k) and f(k) the cumulative distribution and probability
Then:
-
Pr{M,
Using
S
known. The properties of the /,/s are also
E[X;] and Var[X,] are well
derive the expression for E[Ms], denote
functions
M-('-«
have:
=
Cov(Ay,+, Ay,)
;
E[Ay, +s Ay t ]
=
Yj
-
XEA
i
>''+^
n\
A>v|/<+5 = q,l,=k£t+s = l£ =
t
l}T?r{lt+s
= q,l,=k& +s =
1,^
= 1} -
ft=lg=l
where
Ay,+ s Ay,
we
second (and only non-trivial) step
in the
add up over a partition of the
set of
outcomes where
+ 0.
The expression above, combined
with:
(
Pr{lt+s
= q,h=k&+s = l& =
l}
=
Lemma B3
For
\ 3 (I-X) k+ 1- 2
,
ifq=l,...,s-l,
if
q
=
s,
|
q^r and any integer s larger or equal than zero we have:
= ^L.(l_A)'ai
Cov(AyV+J ,Av r,)
Proof Denote
.
I
{
and some patient algebra completes the proof.
XA {l-X) k+ i- 2
v,-,
=
Ay*,
=
vf
+v'ir Then:
E[Ay 9 +J Ay r.,|/,y +
.,
.,
_v
,/ r.,]
=
E[^. ;+5 (
26
£
v g;/+5_ 7 )^( J) vr>r _jt)|^ , +J ,Z r,,]
:
&
.
i
= X2
Where
the
first
identity follows
from
possible combinations of values of
to
M
is
based on the expressions
s
in
Lemma Bl, based on
in
the
^
qJ+!
-u
rJ
-y
* 1 I E[4_/_
j=0 i=0
=
+ X2 M
2
l
l
q j +s rJ nA
t]
s (l q j +s ,l rJ
)crA
from conditioning on
the definition of the Ay,/s, the second
i(+5
i.i.d.
and
t, rJ
,
and
M
s (l ql+S ,l rl )
geometric random variables
Lemma Bl
.
the four
denotes the random variable analogous
lq it + s
and straightforward algebra.
and
l rl
The remainder of the proof
.
|
Lemma B4 We have:
=
Var(Ay,,,)
Proof The proof
is
[Ay,,,
|/,-.,]
=
Var^Ay,,,!/,,])
has mean IjjUa and variance
E[Ay
2
similar calculation
shows
X
7=0
v ]=E[^,(
,|J,
]
A
+ <rA + a 2
\r
A
.
based on calculating both terms on the right hand side of the well known
Var(Ay,,,)
Since
-i-^
2
/,,,c>
we
,
2
v,,,_,)
+E „(Var(A.y M |/,,)).
identity:
(36)
/
have:
2
HA(/,,(c^ + c +
)
/,,M.l
that:
E [Ay,-.,
|
/,-,,]
=\l u nA
.
Hence:
Var(Ay,, |/„ )
and taking expectation with respect
to
/,-.,
=
,
(oj
+ a 2 + X( - \%&
1
)
(and using the expressions
E, u Var(Ay„|/„)
An
X/,-
= O2 + O 2 +
X)
The proof concludes by
Lemma B5
For
N>
substituting (37)
A'
=
Bl) leads
1
= l|
corresponds to
Var(Ayf)
i
[
+ _A_
Lemma
-1
(
V
y
that:
= (l-A)/4
in (36).
_
B4. For
1
)
]a
N>
£ £ Cov(Ay,„A
27
2
2
to:
2
(38)
I
1
Var(A } A)
Proof The case
and (38)
Lemma
^' %\
(1
analogous (and considerably simpler) calculation shows
Var,„(E[Ay,,|/,-,])
in
+G 2 + ^l_A)
we
>V ,)
have:
2
/i
.
=
The
first
identity follows
from
Var(Ay, /) does not depend on
i
^
ir
i
we
Recall that in the main text
Proof The proof of
the
first
l)Cov(Ay,,„ Ay 2 ,,)}
.
the bilinearity of the covariance operator, the second
and Cov(Ay,
,
Ay,-^),
The remainder of the proof follows from using
Lemma B6
+N(N -
{/VVar(Ay u )
i
/
does not depend on
j,
the expressions derived in
= 'Z'iL] Ay*, /TV. We
defined Ay*
Var(Ay,*)
=
0*
Cov(Ayf,Ay;)
=
M°A + f]-
To derive
identity is trivial.
i
or
from the
fact that
j.
Lemmas B3
and B4.
|
then have:
+ ^,
we
the second expression
first
note
that:
'u-l
Cov^A^,) = E&(
Y
-
Av* _,)Ay},]
fi
(
k=0
X Efc( k=0
X Ay*,_,)Ay},
k>\
*?
=
with
5y =
1
if i
S ti°A +
k>\
A(a^
|
/,,,
=
+ fi) + (k-l )&]
5,-jo?
=
k^,,
(
1
1]A
-
2
X)
-
(1
k~
X)
-
'
k
^
-
&
A
+ 8,. o 2 ),
= j and zero otherwise.
/
J
The expression
for
Cov(A_vf
Ay*)
,
now
follows easily.
|
Theorem Bl Ay? follows an ARMA( 1,1 ) process: 22
Ay,
where
£,
denotes the innovation process
-
if
Ay, _]
=
e,
-6e,_i,
and
1
_
Also, as
N
result).
also have:
TV
22
+ \{2-X){K-\)
l-X
tends to infinity 9 converges to zero, so that the process for Ayy approaches an AR(1) (and
recover Rotemberg's (1987)
We
2
We have
that
=
9, so that the
{^l^-Dol-^ii-Xr.
process reduces to white noise,
28
if
and only
if
N=
5=1,2,.
1
4-
2
-t-^ ^4-
we
)
--— (i-a.)*;
=
p?
where G^ anrf p^ denote the
s-th
.
5=1,2,3,...
order autocovariance and autocorrelation coefficients of Ayy
,
respec-
23
tively.
Proof
We have:
N
N
£ £ Cov(Ay,,, +i ,Ay
=
Cov(Ay, +5 ,Ay,)
'"=1
;V
)
j=1
A'Var(Ay,,)
+N{N -
l)Cov(Ayi,, +5 ,Ay 2 ,,).
Where the justification for both identities is the same as in the proof of Lemma B5. The expression for
crj' now follows from Lemmas B3 and B4. The expression for the autocorrelations follow trivially using
Lemma B5 and the formula for the autocovariances.
The expressions we derived for the autocovariance function of Ay, and Theorem in Engel (1984) imply
that Ay, follows an ARM A( 1,1) process, with autoregressive coefficient equal to 1 — X. The expression
for 8 follows from standard method of moments calculations (see, for example, equation (3.4.8) in Box
24
Finally, some straightforward calculations prove that 9
and Jenkins (1976)) and some patient algebra.
1
converses to zero as n tends
Corollary Bl Proposition 4
Proof Part
prove
(ii)
(i)
we
to infinity.
is
|
a direct consequence
oj the
preceding theorem.
follows trivially from Proposition 3 and the fact that both regressors are uncorrelated.
first
note that:
v
Cov(Ayf -Ay{ _,
,
h
1
=
1
Ay;-A^_
1
)
Var(Ay;-Ayf_,)
From Lemma B6 and
Ay* and Ay,_i are uncorrelated
the fact that
Cov^-Ay^.Ay'-Ay^)
Var(Ay;
The expressions derived
2,
24
The expressions
A
for
earlier in this
plim 7-_J „X A
'
in
oj
+
^ + Var(Ay?
),
).
appendix and some patient algebra complete the proof.
XN
Proposition 3 follow from noting thai
straightforward calculation shows that
follows that:
#\ + (1 -p?)Var(Ayf
X (<% +
- Ayfl,
it
L >
2,
so that
we do have
29
|0|
<
1
= 1
p^.
|
To
07
NOV
1 1 Z° 03
Date Due
MIT LIBRARIES
3 9080 02613 1497
Ill