iKiiiirMmimim-mimiwi
tp
I
libraries)
Digitized by the Internet Archive
in
2011 with funding from
Boston Library Consortium
Member
Libraries
http://www.archive.org/details/nonlinearaggregaOOcaba
:Mm
H6
St
working paper
department
of
economics
Nonlinear Aggregate Investment Dynamics:
Theory and Evidence
Ricardo
J.
Caballero
Eduardo M.R.A. Engel
January, 1998
massachusetts
institute of
technology
50 memorial drive
Cambridge, mass. 02139
WORKING PAPER
DEPARTMENT
OF ECONOMICS
Nonlinear Aggregate Investment Dynamics:
Theory and Evidence
Ricardo
J.
Caballero
Eduardo M.R.A. Engel
No.
98-1
January, 1998
MASSACHUSEHS
INSTITUTE
OF
TECHNOLOGY
50
MEMORIAL DRIVE
CAMBRIDGE, MASS. 02139
:*^e ''--^yfyn'*
'Ill
MAigAWiUal-TTS iMSTITUTE
JAN
t 8 ^SS9
y^KARtES
Nonlinear Aggregate Investment Dynamics:
Theory and Evidence
By Ricardo
J.
Caballero and Eduardo M.R.A. Engel^
This version: January, 1998
In this paper
we derive a model of aggregate investment that builds from the lumpy microeco-
nomic behavior of firms facing stochastic
fixed
adjustment
costs. Instead of the
standard sharp (5,
s)
bands, firms' adjustment policies take the form of a probability of adjustment (adjustment hazard)
that responds smoothly to changes
in
firms' capacity gap.
The model
properties, and yields nonlinear aggregate time series processes.
occasionally,
more than
offset
The
has appealing aggregation
passivity of normal times
is,
by the brisk response to large accumulated shocks. Using within and
out-of-sample criteria, we find that the model performs substantially better than the standard linear
models of investment
for
postwar sectoral U.S. manufacturing equipment and structures investment
data.
'MIT and NBER;
Universidad de Chile and
NBER. We
are grateful to Olivier Blanchard,
Guy
baroque,
Whitney Newey, James Stock, three anonymous referees and seminar participants at Brown, CEPRChampoussin, Chicago, Columbia, Econometric Society Meetings (Caracas and Tokyo), EFCC, Harvard,
IMPA, LSE, NBER, Princeton, Rochester, SITE, Toronto, U. Catolica (Santiago), U. de Chile and Yale for
their comments. Ricardo Caballero thanks the National Science and Sloan foundations, and Eduardo Engel
FONDECYT (Grants 92-901 and 195-520) and the Mellon Foundation (Grant 9608) for financial support.
This paper is a substantially revised version of "Expl8uning Investment Dynamics in U.S. Manufacturing:
A Generalized (S,s) Approach" (1994).
Introduction
1
repairs aside, investment projects at the plant level are intermittent
Minor upgrades and
and lumpy rather than smooth. This
They
starkly
is
documented
in
Doms and Dunne
(1993).
use the Longitudinal Research Datafile to study the investment behavior of 12,000
continuing (and large) U.S. manufacturing establishments for the seventeen year period
from 1972-1988, and find that:
growth
(i)
more then half of the establishments exhibit
and
close to 50 percent in a single year,
(ii)
capital
over 25 percent, and perhaps as
much
as 40 percent, of an average plant's gross investment over the seventeen year period
concentrated
in a single
year/project.^
Since this basic feature of microeconomic data
vestment equations,
it
is
is
seldom considered
perhaps should come at no surprise that success
testing investment equations
is
so rare.''
At a broad
level,
our goal
in
empirical
in
in-
estimating and
in this
paper
is
to
develop and test a framework to study the dynamic behavior of aggregate investment, subject to the constraint that
it
builds up from microeconomic units generating the
lumpy and
intermittent pattern observed in microeconomic data.
Achieving such a goal requires three methodological ingredients:
model of lumpy adjustment;
testing
method which
is
(ii)
an aggregation procedure; and
not only consistent with
(i)
and
(ii),
(i)
(iii)
A microeconomic
an estimation and
but also able to highlight the
impact of the proposed microeconomic model on aggregate dynamics.
As
in
the standard {S,s) literature, our microeconomic model generates
lumpy behavior
through the presence of a fixed cost of adjusting the firm's capita! stock.
literature, the fixed cost
across firms).
is
random
The optimal
so the '"inaction range"
policy
still
w
(i.e., in
no longer fixed over time (and
takes a simple form under standard assumptions
about the stochastic process of exogenous variables:
a firm's actual and frictionless
is
Unlike this
let z
denote the log-difference between
the absence of adjustment cost) stock of capital, and
be a random variable indexing the adjustment cost faced by the firm at some point in
time, with distribution G{u).
that represents the
not to adjust.
ma.ximum
For any smaller
The
solution to the firm's problem yields a function ^[z)
realization of
u>,
u
for
which a firm with imbalance z chooses
firms adjust fully.
Removing conditioning on
lj,
on the
other hand, yields an adjustment hazard function A{z) which describes the probability
Since plants entry
is
excluded from their sample, these statistics are
likely to represent
the degree of lumpiness in plants' investment patterns.
See Cliirinko (1994) for a survey of the empirical investment literature.
thsit
lower bounds on
.
a firm with imbalance z adjusts. Since
type rule
is
more amenable
more importantly, has the
when
smoothly with
2, this
probabilistic (5, s)-
to aggregation than the standard fixed-bands (5,5)
virtue of nesting a wide variety of models.
G(y)) degenerates into a spike
distribution with plenty of
varies
it
mass
At the extremes,
we recover the (5,5) model, while when
at very
low values of
w and
model and,
it
becomes a
the remaining mass at very
high adjustment costs, we approximately recover a model with linear aggregate dynamics
(the standard partial adjustment model).
Firms' actions are not perfectly synchronized.
ment
costs differ across firms.
On
On one hand,
any point
at
in
time adjust-
the other, differences in initial conditions, idiosyncratic
shocks, and previous actions, yield a non-degenerate cross sectional density of capital imbalances, f{z,i), at
of a large
now
all
number
times. Aggregation proceeds in two steps, both under the assumption
of firms:
First, within
each
-,
the microeconomic adjustment hazard
represents the fraction of units with that imbalance that choose to adjust at any given
moment
across
in time.
~,
Second, to obtain aggregate investment we integrate these adjustments
using as measure the current cross sectional density.
dynamic path
of aggregate investment
assumptions,
In order to describe the
wc characterize the path o{ f{z,t) which, under our
Markovian with a transition operator that depends on the realization of
is
aggregate shocks.
We make
(fairly flexible) distributional
assumptions about aggregate shocks and
esti-
mate the model by Maximum Likelihood using (aggregate) two-digit U.S. manufacturing
investment/capital ratios
for
the period 1918-1992.
generalized {S,s) model, both
tive
in
We
find clear evidence in favor of our
terms of within sample criteria and out-of-sample predic-
power. Our structural interpretation of these nonlinearities indicates that the fraction
of firms' capital subject to fixed costs
important
for both, these features are
When compared
postulate
One
models
is
is
large,
and that these costs are also
more pronounced
for structures
large.
Although
than equipment.
with standard linear models, the forecasting accuracy of the model we
substantially improved.
of the
differs
over the cycle
main mechanisms by which aggregate dynamics generated by (5,5) type
from their linear counterpart,
—
is
that the
number
of active firms changes
a point emphasized by Bar-Ilan and Blinder (1992).
Doms and Dunne's
(1993) confirm the importance of this mechanism by showing that the
number
of plants
going through their primary investment spikes, rather than the average size of these spikes,
tracks closely aggregate manufacturing investment over time. Consistently,
and depending
on the specific sequence of events preceding current events, the nonlinear model we estimate
has the potential to generate brisker expansions than
this feature
which largely explains
Beyond the empirical
two
its
linear counterparts has.
also
It is
enhanced forecasting properties.
findings on investment and
specific methodological contributions to the
costs
its
its
new
integrative nature, this paper has
literature on non-convex
adjustment
and lumpy actions.
On
the microeconomic side, there have been several developments on models of
and intermittent adjustment (the {S,s)
literature).^
lumpy
As we discussed above, here we extend
these models so the adjustment trigger barriers vary randomly across firms and for a firm
over time. This modification
is
a
first
step toward introducing the realistic and empirically
important feature that units do not always wait
for the
and that adjustments are not always of the same
same stock disequilibrium
and
size across firms
for the
to adjust,
same firm
over time, while preserving a fairly parsimonious aggregation setup.
More
recently, there have also
been developments of empirical models of aggregate
dynamics with heterogeneous microeconomic units adjusting intermittently.^ Econometric
implementation of these models, however, has required observing (or estimating separately
in
an often debatable
driving force.
first
stage) a measure of the exogenous
Our nonlinear time
series
component of the aggregate
procedure does not require the
first
stage;
it
only
requires information on the aggregate investment series itself and on the generating process
of the driving force (but not
its
realization).
Somewhat analogously with
procedure of estimating convex adjustment costs parameters from the
order serial correlation of investment, we learn about more complex
the standard
first
(or higher)
lumpy adjustment
cost
functions from the structure of aggregate investment lags and their changes over time.
The next
section presents the basic model.
It is
followed by section
3,
which describes the
econometric method and presents our main empirical results. Conclusions and extensions
are discussed in section
4.
Several technical appendices follow.
See Harrison, Sellke and Taylor (1983) for a technical discussion of impulse control problems. For a good
although with an emphasis on models where investment is infrequent
survey of the economics literature
but not lumpy
— see
—
A model more closely related to a special case of ours is
consumer durable purchases.
^Blinder (1981), Bar-llan and Blinder (1992) and Lam (1992) look at data on inventories (the first
one) and consumer durables (the other two) under the organizing principles of (5,5) models. Bertola and
Caballero (1990) and Caballero (1993) provide a structural empirical framework and estimate {S,s) models
for consumer durable goods. Bertola and Caballero (1994) implement empirically an irreversible investment
model where microeconomic investment is intermittent but not lumpy. Caballero and Engel (1992a, 1993a,
1993b) estimate aggregate models of employment and price adjustments when microeconomic units follow
more general (probabilistic) microeconomic adjustment rules but, contrary to the current paper, they do
not derive these rules from a microeconomic optimization problem.
Dixit and Pindyck (1994).
Grossman and Laroque's (1990) model
of
The Basic Model
2
Overview
2.1
We
model a sector composed of a large but
Each firm faces an
firms.
isoelastic
demand
fixed
number
of monopolistically competitive
for its differentiated
product, which
with a Cobb-Douglas constant returns technology in labor and capital. Both
is
produced
demand and
technology are affected by multiplicative shocks described by a joint geometric random walk.
These shocks have firm
We
work
The
and sectoral (aggregate) components which we specify
specific
in discrete time.
We
sector faces infinitely elastic supplies of labor and capital.
the latter as numeraire and
random walk
let
is
choose the price of
the wage (relative to the price of capital) follow a geometric
process, possibly correlated with
adjust their labor input at will but suffer a loss
our aim
later.
to capture firms' infrequent
demand and technology
when
shocks. Firms can
resizing their stock of capital. Since
and lumpy investment, we assume
this loss takes the
form of a fixed cost; which can be interpreted either as an index of the degree of specificity of
firms' capital, or as a secondary
market imperfection
or as a reorganization cost associated with putting
some of the time
series
if
machines or structures are replaced,
new
capital to work. In order to capture
and cross sectional heterogeneity
in these fixed costs,
we
let
the
extent of the loss due to adjustment vary randomly over time as firms may, for example,
find better or
worse matches or uses
for their old
machines, or
may
face reorganizations of
different degrees of difficulty.
As
in
standard {S,s) models,
tlie
resulting microeconomic policy
interspersed with periods of large in\'cstnicnt or disinvestment.
models, at each point
fixed
in
in
one of inaction
standard search
time the firm decides whether to "accept" the currently offered
adjustment cost or to postpone adjustment and draw a new adjustment cost next
period.
The
than
in
time
for the
interaction between these two
standard {S,s) models, the
same
firm.
disequilibria in
its
size of
mechanisms implies
that,
more
realistically
adjustments varies both across firms and over
During a given time period, firms with identical shortages or
excesses of capital act differently.
Over time, the same firm reacts
differently to similar
stock of capital.
Intuitively, the largest
its
As
is
adjustment cost a firm
is
prepared to tolerate without adjusting
capital stock decreases with the extent of its capital stock imbalance. If the distribution
of adjustment costs
is
non-degenerate, this implies that the probability that a firm adjusts
for a given disequilibrium
—
a concept we describe as the firm's adjustment hazard
—
^
increases smoothly and monotonically with the firm's disequilibrium in
Since we assume the
number
of firms
is
large
its
stock of capital.^
and adjustment costs are independent
across firms, the adjustment /iazarc? described above characterizes actual sectoral investment
at each point in time.
Given firms' capital imbalances at the beginning of a period, the
fraction of units resizing their stock of capital
Sectoral investment
is
the
sum
determined by the adjustment hazard.
is
of the products of the adjustment hazard and the size of
the investment undertaken by those firms that decide to adjust. Equivalently,
it is
the
sum
of the expected investment by firms, conditional on their capital stock imbalances before
adjusting their capital stock.
Sectoral investment depends critically on the
number
of firms at each position in the
space of capital imbalances, thereby motivating our focus on the cross sectional density of
The dynamics
disequilibria.
The path
density.
of sectoral investment are determined by the evolution of this
of this density
is
driven by the interaction of sectoral, firm specific, and
adjustment cost shocks with the history of shocks and actions contained
in
previous cross
sectional densities of disequilibria.
The firm
2.2
Net Profits
2.2.1
When
the firm
is
not investing,
its
flow of net profits
is:
n{K.9) = K''^e-{r + 6)K,
(1)
where
A'
is
the firm's stock of capital;
6^
is
a geometric
random walk shock
to the profit
function which combines demand, productivity and wage shocks; r and S are the discount
and depreciation
rates;
and
/3 is
a parameter which
is
less
tion of decreasing marginal profitability of capital, either
than one, capturing our assump-
due to decreasing returns
in
the
technology or the presence of some degree of monopoly power.
For mathematical convenience, we have written the profit function net of flow payment
on capital,
{r
+ 8)K.
Since there are neither borrowing constraints nor bankruptcy options.
two limiting cases: the standard (S, s) model, where the probabLIity
one at the trigger points, and the standard linear partial adjustment
models, where this probability is independent of the size of the firm's disequilibrium.
For concreteness, let the production function be Cobb-Douglas and homogeneous of degree one with
respect to capital and labor, with capital share o < 1. Let the demand faced by the firm be isoelastic, with
price elasticity minus ?;, 1 < rj < oo. It follows from these assumptions that ^ = ^(t? — 1)/(1 + »(') — 1)) < 1.
^This should be contrasted with
of a firm adjusting
jumps from zero
its
to
.
the solution to the firm's problem
payment
do
unchanged by replacing flow payments
this
for a
lump sum
time of purchase.
at the
useful to replace 6 in the profit function by a variable with
It is
We
is
more economic content.
frictionless stock of capital of the firm,
by defining the
K*
,
as the solution of
the maximization of (1) with respect to capital, so that:
= iK'
where
,f
=
(r
+
\-P)
6)1 (i. Substituting this expression into (1), and defining the disequilibrium
variable
2
=
ln(A7A'*),
allows us to rewrite the profit function as:
Figure
ital,
1
illustrates,
and equation
7r(r)A"
=
-
^ (e^"'
(3e')
IC
(2) implicitly defines, profits
per unit of frictionless cap-
~{z).^
2.2.2
When
in
=
n(z, IC)
(2)
Adjustment Costs
in\'esting, a firm not only
adjustment
costs. Since
commits
we wish
to
pay
for
the capital acquired, but also incurs
and lumpy nature of firms'
to capture the intermittent
investments, we require these costs to exhibit some form of increasing returns.
are
many ways
to do so.
assume the firm
One
sells its old
possibility
is
to follow
Grossman and Laroque (1990), and
stock of capital at a discount
when
kx\ alternative, with similar implications for our purposes,
down operations
is
the one
for a fixed period of
we pursue,
There
is
replacing
to
it
by a new one.
assume firms must shut
time when replacing capital. In the latter case, which
the firm incurs in an adjustment cost proportional to foregone profits
due to reorganization:^
Adjustment Cost =
u;{n(7v',6')
+
(r
+ 6)K] = uK^O,
/? = 0.4, r = 0.06 and i = 0.1.
Cooper and Haltiwanger (1993) for a model where the main cost of reorganization
^Parameters:
See
e.g.
tunity cost.
is its
oppor-
d
ID
d
d
CM
d
CD
£_
q
d
I
CD
d
I
d
I
CD
d
I
CD
d
I
ses'o
S6f0
N
,
where
u
represents the fraction of profits foregone due to the capital stock adjustment.
A
derivation similar to the one that led to (2) allows us to rewrite the adjustment cost in
terms of z and A'*:
Adjustment Cost = u^e^^ K*
where z~ denotes the capital imbalance immediately before adjustment.
Rather than treating
realization
is
— as
standard (5,5) type models
in
—
we
let it
be a
observed at the beginning of each period. With this slight generalization
of the standard fixed cost
framework wc capture
—
in
an admittedly stylized form
—
two
heterogeneity in adjustment costs at any point in time, and time variation
realistic features:
in
as fixed
variable with a distribution function, G{u), independent across firms and over time,
random
whose
u>
these costs for any given firm.^°
More importantly,
it
will
that this extension gives us an important degree of flexibility
be apparent
in section
3
when estimating aggregate
investment equations.
Microeconomic Adjustment
2.2.3
Given the increasing returns nature of the adjustment cost technology, the optimal policy
is
obviously not one of continuous and small investments but rather one of periods of
inaction followed by occasional
lumpy investment. Therefore, the
firm's
problem can be
characterized in terms of two regimes: action and inaction. Finding a solution to the firm's
problem
is
equivalent to characterizing the partition of (w,z)-space into these two regions
and specifying firms' actions when located
we present the
in
the region where they act.
basic steps involved in finding this solution. In
technical aspects and intermediate steps of the solution in
Inheriting the stochastic properties of
6,
K'
appendix
more
In
what follows
A we
discuss the
detail.
follows a geometric
random walk (with
drift):
A7 =
where
is
/X(
is i.i.d.
and, throughout most of the paper, Normal. This implies that
no adjustment,
with the
i.i.d.
A7_,e''-,
Zt
follows a
random walk (with
drift
when there
and Normal innovations). Together
nature of w, this assumption ensures that the firm's decision on whether to
A more realistic formulation would let adjustment costs exhibit some persistence at the individual level.
would also allow for a distribution of adjustment costs that depends on aggregate conditions. We do not
incorporate these features into our model because they complicate substantially both the microeconomic
and aggregation problems.
It
adjust
its
capital stock in period
t,
and
{zt,K' ,ijjt), which we refer to as the
adjustment-disequilibrium
parameters
if
so by
2, frictionless
fully
is
The
'state of the firm'.
determined by the vector
value of a firm with before-
stock of capital A'*, and (current) adjustment cost
— which we denote by V'{z,K',uj) —
does not adjust, V{z,K'), and the value
if it
how much,
if it
is
the
ma^mum
of the value of the firm
does adjust, V{c,K')
- w^e^'A'*; where
the optimally determined return point (see below). In short:
c is
= m^x{V[zt,K;)
V'{zt,K:,u;t)
(3)
The
,
K(c,
A7) - ut^e^"K;}.
evolution of the value of a firm that does not adjust in the current period
described
is
by:
V(z<,A7) = -(r,)A7 +
(4)
+
(l
r)-iEt[l^'(-, + ,,/^r+:,w< + ,)].
Since the profit and adjustment costs functions are homogeneous of degree one with respect
to A'", given z, so are the value functions V{z,h'')
and V'{z, K'
reduce the number of state variables by restating the problem
= V{z,K')/K' and
of frictionless capital. Let v{z)
v'{z,u))
in
=
,u:).
This allows us to
terms of the value per unit
V'{z, K',oj)/ K'
.
Dividing
both sides of equations (3) and (4) by A'' and noting that
1
-(5)e-^-''+\
A7
vields
(5)
v'{zt,u}t)
=
mcix{v{zt), v{c)
(6)
v{zt)
=
-(-<)+v''E,
with c
=
(1
—
(5)/(l
+
r).
now.^' This figure shows
Figure 2 depicts
how
—
v{z), v{c)
-ut^e""},
[t''(r,+,,u;< + ,)e"^'"'"'
an example the basic setup developed up to
in
u>(^e^'
,
and v'{z,uj) determine the trigger points,
given a particular realization of the adjustment cost.
a firm that does not adjust in the current period.
''Parameters:
and
the
0.1;
/?
=
0.4, r
=
0.06, 6
=
0.1;
the distribution of adjustment costs
the
is
—
solid line illustrates the value of
The dashed
a firm that decides to adjust, given a realization of
describes v'{z,u}), and the inaction range
The
uj.
line represents the value of
The maximum between both
for a given
u
—
corresponds to the interval
mean and standard deviation of the logarithm of
with mean 0.17 and coefficient of variation
Gamma
numbers are broadly consistent with our estimates and assumptions
lines
A',*
are
0.16. All
in the empirical part of the paper.
I
<
If'
1
1
CO
1
d
^^,,00^
'^.---'^
^^^^
~
^^^
-
^
CD
d
/
/
/
/
f
d
1
'
f
f
1
,
f
d
f
V
C\J
CD
£_
1
\
f
/
:
\
D
o
d
1
.
1
\
\
\
\
O)
I
1
/
/
1
C\J
-
-
\
\
I
1
-
-
1
\
_
1
/
1
1
1
92"e
I
'
1
^2"e
(X)
\
\
\
\
\
f
1
'
\
\
1
1
I
\
1
-
d
\
1
82-e
d
1
!
1
22"e
suoi^ounj snisA
d
I
d
1
1
02'e
1
8fe
N
between the intersection of the two
follows directly from maximization of the value of a firm that decides to adjust,
It
v{c)
lines.
-
uj^e^^
,
obtained at 2
with respect to the return point
=
c
and that
Al
Proposition
in
this return point
c,
that the
independent of the
is
the functions v{z) we obtained via value iteration
maximum, we have been unable
speaking, the return point
maximum,
its
The
largest
disequilibrium.
when performing estimation always had
show
to
continuous and bounded. Even though
is
this generally.
follows that, strictly
It
should be interpreted as one of the points where v{2) attains
c
solution also can be characterized by the policy function, fl{z), defined as the
adjustment cost factor
hand
z.
for
which the firm finds
it
advantageous to adjust given a
From the value matching condition that equates the two terms on the
side of equation (5),
-'
(7)
initial
is
'^
say the smallest value.
capital imbalance
right
of v{2) and v'{z,u)
Appendix A shows that the Bellman equation obtained by substitut-
ing v{zt) in (5) by (6) has a unique solution, which
a unique
maximum
it
follows that:
n{z)
,.
= r'e-''{v{c)-v{z)),
which implies:
fi(c)
=
0.
Differentiating (7) with respect to _, evaluating the result at 2
=
condition u'(c)
0,
=
close
enough
to 2
and using the
first
order
0.
Figure 3a illustrates the function ^{z) for the example
is
c
yields the additional ''smooth pasting" condition:
n'(c)
of adjustment costs
=
is
=
a
Gamma. As
c,
it
in
Figure
follows from equation (7),
if
2,
where the distribution
the firm's disequilibrium
only will adjust for arbitrarily small adjustment costs.
then on, ^(2) increases with
\z
—
c\.
Figure 3b depicts the inverse of the function Q(z).
of the curve below and above
c,
From
respectively.
We label
L(ii!)
and C/(w) the segments
These functions correspond to the
maximum
shortage and excess of capital tolerated by the firm for any given realization of the adjust-
ment
cost factor w.
In Proposition
'
An {L,c,U)
That
A5 we show
is,
for
any fixed
u;,
they describe a standard {L,c,U) policy.^^
that the set of maxima, and hence of possible return points,
policy corresponds to a two-sided {S,s) model.
The
notation L,
U and
c
is finite.
stands
for
lower
Figure 3a
Figure 3b
0.24
The area enclosed by
the two curves corresponds to the combinations of disequilibria and
adjustment cost factors
A2
Proposition
A3 shows
sition
uj,
is
it
A4 shows
able to
on
that
in
for
which the firm chooses not to adjust.
Appendix
A
derives formally the existence of 0.{z), and Proposition
an analytic function, and therefore has derivatives of
that i^{z) tends to infinity as
show that
rt(z)
is
the optimal policy
tends to
is
of the {L,c,U) type.
when estimating the parameters
will
in section 3
A
G'(cj).
the
all
were unimodal.
depend
inverse, {L{oj),c,U{u!)),
its
on the entire distribution of adjustment cost factors,
given realization of the
not generate the same inaction range for different distribution
In particular, a low value of
functions G{lj).
Yet we have not been
infinity.
should be noted, though, that
It
location of the function Q.{z) and
adjustment cost factor
order. Propo-
unimodal, and therefore have no formal proof that, conditional
policies obtained numerically
The shape and
\z\
all
u)
more
is
likely to lead to action
when
it
comes from a distribution of adjustment cost factors with a high rather than a low average
value.
Adjustment hazard, expected investment and ergodic density
2.2.4
Adjustment hazard
Above we showed
deterministic policy with respect to
that for any given
u
the firm follows a simple
actions are taken only
z:
when
z lies outside the
{L{iL!).U{u)) interval, in which case investment occurs so as to bring z back to
aggregation
mind, here we reduce the amount of information contained
in
Rather than conditioning on w, we only use information on
question: what
is
The answer
X
=
z
—
c
if u;
to this question
< n(x +
if
c)).
is
contained
in
what we
current adjustment cost
Which means
is
its
call
small enough to
+
is
G{u))
is
a
A
Let
firm with deviation
make adjusting
profitable
(i.e.
Gamma
distribution with
bound, upper bound, and "center," respectively. See Harrison
10
its
given by:
c)),
where G{u)) denotes the cumulative distribution function
if
the adjustment hazard.
that the probability of a firm adjusting, conditional on
A(2) = G(n(x
(8)
For example,
the policy.
distribution and ask the
target point.
disequilibrium being equal to x (the adjustment hazard),
uj.
With
the probability that a firm with disequilibrium z adjusts?
denote a firm's imbalance with respect to
X adjusts only
its
in
c.
for the
mean
et al. (1983)
pcj)
adjustment cost factor
and variance
p<^^,
the
adjustment hazard
is:
^W =
^/o'
""'"'""'*''"
<i.'r(p) Jo
Figure 4a shows the adjustment hazard function for three different
adjustment cost factors. These distributions
differ in their
gamma
mean and
distributions of
variances: the solid line
corresponds to an exponential distribution (mean and standard deviation of
0.1);
the long
dashes correspond to a high variance and mean distribution (mean: 80, standard deviation:
282); while the short dashes describe a low variance
and mean distribution (mean:
0.14;
standard deviation: 0.044). These examples illustrate well the range of cases covered by
our setup. Figure 4a shows that when the variance of adjustment costs
is
low, there
is
a
range of adjustment costs where the firm (almost) never adjusts since adjustment costs are
(almost) never small enough to justify
extreme version of
Conversely,
this.
the standard {S,s)
it;
when the variance
is
rather than by the firm's disequilibrium;
in
their
{L,c,U)
of adjustment costs
—
is
case
is
an
high, and so
largely motivated by the adjustment cost
mean, the decision of adjustment
is
— or
draw
the limit, adjustment costs are independent of
the firm's disequilibrium, yielding the standard-linear partial adjustment model.
In Proposition
and lim|^|_^
A(a:)
B2
=
in
Appendix B we show that A(x)
is
differentiable, with A(0)
=
1.
Expected investment
A
stock of capital and,
if it
(e<^
firm with disequilibrium x has a probability A(a:) of adjusting its
does so,
it
- e^)A7 =
Thus the average investment
their capital stock in period
t
invests:
(e-- - Ije-'AY
of firms with disequilibrium x
l)A',(x).
immediately before adjusting
is:
E^[/((x)|i]
(9)
= {e'- -
= A(x)(e-^ -
l)A'((i).
Figure 4b depicts expected investment corresponding to the hazards in figure 4a (with
A't(x)
=1). The nonlinear-convex nature
of expected investment
is
an important feature
of the model, playing a key role in shaping aggregate investment dynamics.
incentives to invest rise
Which
is
more than proportionally with a
a feature not shared by the standard quadratic
firm's disequilibrium.
says that
'"^
adjustment cost model but certainly not
exclusive of models with fixed, or even non-convex, adjustment costs.
11
It
'
1
Figure 4a
CD
—
d
1
'
1
'
1
1
1
\
1
1
<
\
1
1
/
/
00
d
f
r
r-^
d
\
1
\
1
\
CD
1
v
d
1
\
1
.
/
\
^X
^r----^^
in
d
—
< ^
o
*
\
*^^v
\
^
d
^^
X
**>
/
^
\
^
C\I
^^
^
^ ^ -;-/^
""
^
K
/
/
1
'
d
^
d
^V
V
^"--,
o
-0.6
-0.8
il.O
-0.4
^
/
1
/
\.
V
^^->,
-0.0
-0.2
0.2
1
y
0.4
0.6
0.8
X
Figure 4b
1
V
\
\
V
\
\
V.
X
\
x'
S"
UJ
^
oj
Q
^^
"^
~T ^ * ^
C\J
d
^:»s»,^,^
V^'**"^ «•
1
V
<s
CD
d
M.O
-0.8
-0.6
-0.4
-0.0
-0.2
X
0.2
0.4
0.6
0.8
We
Ergodic density
conclude our characterization of microeconomic behavior by stating
that the disequilibria a firm goes through over
have an invariant density. The
its lifetime,
formal proof, which also shows that convergence takes place at an exponential rate,
in
Appendix B. The nonlinear nature
adjustment
of
is
given
microeconomic adjustment, with relatively small
for small imbalances, imprints the opposite pattern
on the ergodic density:
steeper hazards translate into relatively less mass in the tails of the corresponding invariant
density in exchange for more mass in the regions of low values of the adjustment hazard
(i.e.,
relatively platokurtic).^^
Sectoral investment
2.3
2.3.1
Sectoral investment and the cross sectional density
Let Ki",
if^,
^"^'ii^)
a-nd
/((x) denote the aggregate (sectoral) stock of capital
and gross
investment, and the stock of capital and gross investment held by firms with disequilibrium
X at time
t
(before adjustment).
Since adjustment costs shocks are
It{x)
(10)
where
=
i.i.d.
{€-''
across firms,
it
follows directly from (9) that:
-l)\{x)Kt{x),
A'((x) denotes the average stock of capital of firms with imbalance x.
Letting /(i,
take place,
t)
denote the cross sectional density of disequilibria just before adjustments
we can obtain an expression
If
Dividing through by
Kf
for
=/(e""-
aggregate investment:
\)i\(x)h\{x)fix,t)dx.
and rearranging terms, we obtain an expression for the aggre-
gate investment/capital ratio:
(11)
|4
= /(^"' - m{^)f{x,t)dx + |(e-- - l)A(x)
(^^ - ij f{x,t)dx.
^
- l)A(x)
are uncorrelated. Since such an assumption simplifies computations substantially,
we make
The second term on
the right hand side of (11) drops out
'^The U.S. manufacturing plant
level
data studied
distribution of shocks affecting these plants.
12
is
-
A'/*)
and
(e
Engel and Haltiwanger (1995) revealed
considerably more platokurtic than the average
in Caballero,
that the average observed distribution of disequilibria
if (A'((a:)
,
it
and obtain an approximate expression
aggregate investment/capital ratio:
-^- J{e-^-l)A{x)fix,t)dx.
(12)
In
for the
appendix C we describe
in
detail the additional
Monte Carlo simulations showing that the
instead of (12), and present the results of
of the approximation
It is
is
only minor.
computational burden of using (11)
cost
'^
apparent from (12) that since (e~^ - l)A(x)
investment depends not only on the
is
generally nonlinear in x, aggregate
moments
but also on higher
first
of the cross sectional
distribution of disequilibria.
The
2.3.2
An
extreme
linear/partial adjustment
exception to the statement
the previous paragraph occurs
in
hazard does not depend on x and f has most of
its
mass near x =
0,
when the adjustment
so that e"^
—
1
~
—x.
In that case
-|^~-AoA',,
(13)
where Aq denotes the constant hazard, and
A'(
denotes the average disequilibrium before
adjustment. Equation (13) corresponds to the well known partial adjustment model (PAM),
and also coincides with the standard
costs model.
This
is
linear equation arising
from the quadratic adjustment
the only adjustment hazard that does not require cross sectional
information on the right hand side of the aggregate investment equation.
Indeed, a few
steps of algebra allow us to go from equation (13) to the standard expression:''
tA
tA
_
-j^ =
•'(
(14)
where
The
Vi
A
/J
Ao(<5
+
,
..
^
t'()
A
(l-Ao)
+ /I
^
,
''(-1
^^
,
represents an aggregate shock, to be defined in the next paragraph.
and Ha.ltiwanger (1995) are reassuring on this respect. There we used
data for U.S. manufacturing during the 70s and 80s, and documented a
results in CabaJlero, Engel
comprehensive establishment
level
very close empirical
fit between the actual and approximate series during that period.
and k' denote the average of the logarithm of the pre-adjustment stock of capital and the
frictionless stock of capital, respectively. We define u, = A/t*, and note that Afci ~ (/,'1]/A','l]) - 8.
Combining these two expressions with the fact that,
Let
kt
A',
=
k,
-
k:
=
Ak, - Ak:
yields (14).
13
+
.V,-i
Sectoral equilibrium and cross sectional dynamics
2.3.3
Shocks to wages, demand, and productivity drive the dynamics of
decompose these shocks
and firm
into sectoral shocks, vt,
We
frictionless capital.
specific (idiosyncratic) shocks,
cj:
K' = A7_ie Vl+U
which implies that when the firm does not adjust, the disequilibrium measure x evolves
according
to:
Axt = ~{6 +
where capital depreciates
at a rate S
Vt)
-
€t,
from one period to the next.
We
assume these shocks
are exogenous to the firm and the sector.
Between two consecutive periods, the cross section distribution of disequilibria changes
and idiosyncratic shocks. Since we
as a result of firms' adjustments, depreciation, sectoral
are working in discrete time,
We
events within each period.
f{x,
/
—
1).
t
important to describe the timing convention we adopt for
denote the cross section density at the end of period
Depreciation and the aggregate shock corresponding to period
in the density f{x,t).
Period
it is
Next come adjustments,
as
concludes with the idiosyncratic shocks.
The
final
density
is
—
I
hy
follow, resulting
determined by the hazard function
A(2;).
f{x,t), and the cycle
Recalling that a positive shock leads to a decrease in x,
starts again.
t
t
we can summarize
this chain of events as follows:
(15)
f(x,i)
(16)
f{x,t)
where
g^{c)
is
f{x
+
6
+ vt,t-
j A{y)f{y,t)dy
1),
g,{-x) +
/
-
[1
A(a;
+ e)]/> +
the probability density for the idiosyncratic shocks.
€,t)g,{-e) de,
The
integro-difference
equation describing the evolution of the cross sectional distribution from one period to the
next follows directly from equations (15) and (16):
(17)
fix,t)
J Hy)fiy + s +
+
y
[1
- A{x +
vt,t-i)dy g,(-x)
€)]f{x
14
+
€
+
6
+
vt,t- l)g,{-e)
de.
From equations
(12) and (15)
-^
(18)
Combining equations
determined by an
we obtain the following aggregate investment equation:
= J{e-^ - l)Aix)fix +
(18) and (17)
we can determine
+ v„t -
We
Appendix C.
l)dx.
the sequence of aggregate investment
distribution, /(2:,0),
initial cross section
shocks, {vt]. For details see
6
and a sequence of aggregate
turn to estimation issues next.
Empirical Evidence: U.S. Investment
3
Data
3.1
Our data
are constructed from annual gross investment and capital series for 21 two-digit
manufacturing industries from 1947 to 1992.'^ All
of capital correspond to the series used by the
series are in
Bureau of Labor
tivity studies.'^ Since capital stocks are end-of-year,
ratio used in estimation start in 1948.
We
1987 dollars, and the stocks
Statistics for their produc-
our measures of the investment/capital
report separate results for equipment and struc-
tures panels; each has 945 observations.
Econometrics
3.2
The econometric problem
consists of estimating the parameters that characterize (a) firms'
profit functions, (b) the initial distribution of disequilibria, (c) the distribution of adjust-
ment
costs,
(d) the distribution of idiosyncratic shocks,
and
(e) the process
generating
aggregate shocks. In this subsection we outline the main features of the estimation procedure; a detailed description
is
presented
For tractability, we limit the
izing the distribution of
adjustment costs
remaining parameters, we either
shocks),
show that
number
their role
equilibria) or concentrate
is
fix
in
Appendix C.
of parameters being estimated to those characteror, equivalently,
them
(profit function
the hazard function.
As
far as the
and distribution of idiosyncratic
limited within a reasonable range (initial distribution of dis-
them out
of the likelihood function (process generating sectoral
shocks: individual effects and cross sectoral variance-covariance matrix). Since identifying
nonlinearities from a purely time series (as opposed to regressions) dimension requires a
We have 21 rather than 20 sectors because Motor Vehicles is separated from Transportation equipment.
This is one of the three capital stock series reported by the Bureau of Econoniic Activity.
15
large
number of observations, we impose a hazard function or a
costs that
common
is
or structural model
The source
we estimate a semi-structural
across sectors (depending on whether
— see below).
of randomness in our estimation procedure are the sectoral shocks, which
assume multivariate Normal and independent over time,
We
distribution of adjustment
approximate the
initial sectoral cross-sections
for
we
most of our empirical analysis.
by the invariant cross-section of an
individual plant, and proceed to use the Markovian nature of the process generating crosssectional distributions to generate these distributions. For each sector, and at each date, the
cross sectional distribution
is
updated
as a function of the sectoral shock, using an implicit
law of large numbers at the microeconomic (firm)
rate
—
level.
The observed
sectoral investment
a non-linear function of the current shock and the distribution prior to this shock
is
this function
is
one-to-one in the shock.
Conditional on the
initial distribution,
the
sequence of sectoral shocks and the cross section distributions can be recovered from the
time series of sectoral investment rates.
shocks.
We
The
likelihood
calculated using these sectoral
is
also need to calculate the corresponding Jacobian terms,
which correspond to
the elasticities of sectoral investment rates with respect to sectoral shocks. These elasticities
are a byproduct of the calculation of sectoral shocks.
Semi-structural and structural models
3.3
We
estimate two basic models.
In the first
one (semi-structural), we estimate directly
the parameters of an ad-hoc adjustment hazard.
we estimate the adjustment
costs parameters
While
in
the second one (structural),
and obtain the implied liazard via dynamic
Both of these models yield increasing hazards, ^° but include a constant
programming.
hazard as a limit case.
Although our main goal
Semi-structural.
in
the paper
is
to estimate
and assess the
structural model, there are good reasons to start by estimating a less structured version.
It
allows us to search and test for the presence of an increasing hazard
it
facilitates
directly,
and
comparisons with standard linear models. Furthermore, by assuming that the
adjustment hazard
is
an inverted Normal:
A(x) = l-e-^°-^^^'.
(19)
By
more
increasing hazard
increasing for i
>
we mean a hazard that
is
increases with
0.
16
|i|, i.e.,
that
is
decreasing for i
<
and
and A2 >
where Aq >
0,
we
are able to obtain accurate computations and to reduce
estimation time significantly (by a factor of 12) by keeping the cross-section distributions
within a closed family of mixture of Normals (see Appendix C).
Rather than estimating the adjustment hazard
Structural.
we
directly, in this case
estimate the parameters of the adjustment costs function and obtain the hazard from the
solution of the
drawn from
dynamic optimization problem presented
Gamma
a
which has mean
fi^
=
in section 2.
Adjustment
costs are
distribution:
p(f)
and a coefficient of variation cv^
= 1/^. We
estimate
/i^^
and
cv^.
As
for the other structural
markup
of capital, and
parameters, we assume an interest rate, share of each type
of 6, 15 and 20 percent, respectively;^' as well as depreciation rates
equipment and structures of 10 and 5 percent per year,
for
We
respectively.'^^
estimated
the standard deviation of idiosyncratic shocks, c^, obtaining estimates in the range of 5
However, since these were not estimated very precisely, and comparisons
to 15 percent.
across models are easier
wc
liave
imposed
3.4
Main
Table
1
a^
=
if
results
tural results for
The
All estimated
Semistructural
first
two columns present semistructural and struc-
equipment investment, while the
models allow
include a free additive constant that
in
report results where
0.1 (both, in semi-structural as well as structural estimation).
contains our main results.
structures.
we only
idiosyncratic variances are similar,
The semistructural
is
of
/?
two do the same
for individual effects
common
The
investment
for
across sectors.
hazard model,
increasing hazard parameter, A2,
around 0.45
if
in
on the sectoral shocks, and
results allow us to reject the constant
favor of an increasing hazard one.
These parameters imply a value
last
the production function
is
is
significant
constant returns and
all
other factors of production (including the other type of capital) are fully flexible, around 0.3 if all factors
but the other form of capital are flexible, and around 0.15 if all other factors are fixed. Our conclusions
are robust to reasonable variations of
/?,
conjunction with those that we estimate.
but we do not have enough power to identify this parameter
The
results
we report assume
jS
=
in
0.4.
Although there is a slight upward trend in the sample, these depreciation rates are consistent with the
average depreciation rate computed from the ratio of actual depreciation to net capital stocks reported in
Fixed Reproducible Tangible Wealth in The United states, 1925-89.
17
Table
1:
Main Results
Equipment
Parameters
Semi-structural
Ao
A2
constant
0.155
0.000
(0.067)
(0.054)
2.804
2.437
(1.165)
0.878)
-0.006
0.013
0.019
(0.013)
(0.016)
(0.006)
(0.002)
cv^
0.166
0.228
(0.029)
(0.046)
0.327
0.066
(0.109)
(0.009)
2431.4
2430.2
Standard deviations
at the
one percent
level in
both cases.
probability that a firm adjusts
small imbalances to
45% when
small imbalances to about
Structural
0.057
Mcv
LOG-LIK.
Structures
Semi-structural
Structural
32%
its
2612.4
in parenthesis.
The estimated hazard function suggests
capital stock of
its
imbalance
for a
2637.2
is
that the
equipment increases from about 14%
40%, while
40% imbalance,
in
it
for
goes from close to zero for
the case of structures.
The
sharp
nonlinearity can also be captured through the expected invested/capital ratio (conditional
on the imbalance);
for
equipment,
it
goes from close to 0.05 at a 20 percent imbalance to
0.23 at a 40 percent imbalance, while for structures
it
goes from 0.02 to 0.16, for the same
imbalances.
Structural.
The
results of the structural
hazard findings. Moreover, the likelihoods
The estimates
of the
mean
rise,
especially so for structures.
of the distribution of adjustment costs (the
the average adjustment cost drawn
profits for
models confirm the semistructural increasing
is
fi^^s)
indicate that
the equivalent of 16.7 percent of a year's operational
equipment and 22.8 percent
for structures.
18
Since firms can "search" for a low
realization of adjustment costs, these are
upper bounds
for the
average costs effectively paid
by firms when going through a major adjustment episode. Indeed, the average costs paid
are 11.1 percent for equipment and 21.4 percent for structures.
The
difference
between the
unconditional and conditional (on adjustment) means rises with the coefficient of variation
why adjustment
of these costs, which explains
third less than the
structures both
in
means
equipment are one
cost faced by firms (see cv^^) while in the case of
are very similar.
Simple alternatives
3.5
We
mean adjustment
costs actually paid for
view the microeconomic foundation of our approach as one of
this section
we look purely
at the statistical
We
simple linear counterparts.
its
main
virtues,
however
advantage of our structural models over
do not intend to conduct "horse races" against alternative
investment models but want to provide a simple metric to assess the contribution of the
nonlinearities
The
first
we estimated
obvious step
to the time series properties of aggregate investment.
is
to
compare our mode! with the "almost" nested PAM. As
we argued above, the constant hazard model
e~^ -
1
~
PAM, which
-X, yields the standard
each sector's aggregate investment
series.
also run an
we
across sectors:^'^
tA
-'it-i
,
,
•'Mi-l
•'Sf
we
together with the approximation
corresponds to estimating an AR(1) for
same
tA
'if
generally,
0),
For comparability with the structural model,
constrain the correlation coefficient to be the
More
=
(A2
AR(2) with unconstrained autoregressive
coefficients for
each sector:
't
Tm
=
„
°'
I
"'-1
'Sl-1
'^it
In
„
^tt-2
,
+ P^'-Fa- + P^'T'A- +
I
^'•
-'^.(-2
both cases we preserve the assumption of jointly Normal aggregate shocks, and allow
for individual effects.
We
look at within and out-of-sample criteria, and find widespread
evidence supporting the structural nonlinear model over the linear representations for both
equipment and structures.
Within sample
The
criteria.
likelihoods of the linear models are uniformly lower than those of the corresponding
Recall that the structural has the
same
distribution of adjustment costs across sectors.
19
AR(2)s
structural models, even for the
structural models.
— which have 39 parameters more than the nonlinear
But comparing the likelihoods
models (especially the AR(2)s) are not nested
the test for non-nested models developed by
Let
and
and
/[
T
2,
denote the
I2
maximum
in
not strictly correct, since the linear
is
our structural models.
Instead,
Vuong (1989) and Rivers and Vuong
we use
(1991).
value attained by the log-likelihood for models
1
denotes the number of periods considered when calculating the likelihood (42, in
our case),
rti
and
722
denote the number of parameters for models
and S denotes the Newey-West estimate
differences.
The
null hypothesis
is
for the variance of the
1
time
and
2, respectively,
series of likelihood
that both models are vT-asymptotically equivalent;^"*
should this be the case we have that the test statistic
(20)
^^^^
^ h -
I2
- Ijn, - n,)\ogiT)
VST
has a Standard Normal distribution.
model
1;
Positive values of Vi^2 indicate evidence in favor of
negative values evidence in favor of model 2.^^
Table 2 presents Vuong's
statistics
and the
p- values for the test that
both models
(lin-
ear and nonlinear) are equally close to the 'true' model, against the alternative that the
(nonlinear) structural model
is
better.
It is
apparent that the null hypothesis
is
rejected in
favor of the alternative even at very low significance levels.
One
possible reason for the bad relative performance of linear models
assumed that aggregate shocks are Normally distributed.
is
that
we have
Sectoral investment rates, on
the other hand, are clearly not Normal; the skewness and (excess) kurtosis coefficient of
(standardized, for every sector) investment rates are 0.61 and 0.74 for equipment and 0.76
and 0.87
for structures.
Obviously, linear models with Normal errors cannot account for
these departures from Normality.
The
innovations generated by the best partial adjustment
model and best second-order autoregressive models
in
table
3:
also depart
from Normal, as can be seen
Their skewness and kurtosis coefficients are 0.49 and 1.15 for equipment and 0.95
limr-oc VT(/, - h) = 0.
Note that: First, the numerator of (20) contains a penalty term - the Bayesian Information Criterion,
BIC - that corrects for differences in degrees of freedom between both models. Second, denoting the sum
of sectoral likelihoods for model
at time ( by /,,, i = 1,2, and d, = h, — lit, we have that the S in the
denominator of Vi,2 is;
'M.e.,
i
5(r,?)
where
7^
not vary
=
7o
+
2^
?
+
7j.
l
denotes the sample autocorrelation of order j of the d, time series. Since in all cases S{T,
for values of q larger than 7, we choose q = & when calculating S in (20).
much
20
q)
does
Table
2:
Nonnested Models Test: Normal Shocks
Eq uipment
Vuong
statistic
p- value
NOTES: Vuong
St ructures
PAM
AR(2)-UNC
PAM
AR(2)-UNC
2.61
4.25
2.92
4.07
0.0045
< 0.0001
0.0018
<
statistic calculated as in (20). All test statistics
model estimated
in
Table
1
with the model
in
the table.
good', the asymptotic distribution of the statistic
is
0.0001
compare the structural
both models are 'equally
If
Standard Normal. Large positive
values provide evidence in favor of the structural model.
and
1.88 for structures in the partial
and 0.86 and 1.65
different
last
to
adjustment case, and 0.38 and 1.00
for structures in the
from zero (the Normal case)
AR(2)
All these
case.
at the 0.001 level
for
numbers are
equipment,
significantly
(estimated via bootstrap).
The
two rows show that the increasing hazard model generates innovations that are closer
Normal than
its
linear
and constant hazard counterparts. The estimated skewness and
kurtosis coefficients are considerably smaller, and both skewness coefficients do not depart
significantly (at the 0.05 level) from their values
under the Normality assumption.
increasing hazard model does not need to introduce nearly as
in
aggregate shocks to account
Normality
is
for
much skewness and
The
kurtosis
investment behavior.
the natural assumption
when aggregate shocks
are conceived as the
sum
of
a wide variety of small shocks with limited dependence (by the Central Limit Theorem). In
spite of this,
we momentarily
relax this
assumption
in
order to consider shocks which admit
skewness and kurtosis properties similar to those observed
ratios.
in sectoral
investment/capital
For this purpose, we generalized the distribution of the residual to consider convex
combinations of Normal and log-Normal distributions. The log-Normal component did not
add
significantly to the structural model, while the linear
weight to the log-Normal component.
For this reason
models assigned most of the
we compare the
structural models
with NormaJ shocks versus the linear models with log-Normal shocks in table
4.
Although
the likelihood in the linear model improves substantially with this modification, table 4
shows that the
test for
non-nested models
still
21
favor the (nonlinear) structural models by
Table
3:
Skewness and kurtosis
Equi pment
Model
Kurtosis
Skewness
0.49
1.15
0.95
1.88
(0.13)
(0.36)
(0.15)
(0.65)
AR(2)
Structural
0.38
1.00
0.86
1.65
(0.31)
(0.14)
(0.60)
-0.04
0.65
0.17
0.79
(0.20)
(0.13)
(0.34)
in parenthesis:
standard deviation, obtained via bootstrap.
a wide margin. In fact, the reduction in the
the test (the likelihoods
Kurtosis
(0.12)
(0.11)
Shown
Struct ures
Skewness
Part. Adj.
*
for innovations*
become more
denominator due to the increased precision of
correlated across models)
more than outweighs the
increase in the likelihood of the linear model.
Out-of-sample
criteria.
Next, we evaluated the out-of-sample forecasting performance of our model.
For this
purpose we reestimated the nonlinear structural and AR(2) models dropping ten percent of
our observations (the
last five years for
ahead forecast distributions,
for
each of our 21 sectors), and generated the one-step-
each sector and year out of the sample.
the model's performance relative to that of an
We
only evaluated
AR(2) using a standard Mean-Square-Error
criterion, although this reduces the potential forecasting
edge of nonlinear models. ^^
postpone further discussion of forecasts' higher moments
until the conclusion.
Table 5 reports,
year
(first
for
each investment type, the average (across sectors)
two columns), and the percentage increase
model over the nonlinear one (third column).
Except
in
MSE
where the gain
is
for
each
generated by the AR(2)
for 1988, the structural
model systematically outperforms the AR(2) representation. This
structures,
MSE
We
is
nonlinear
particularly true for
over 35 percent during four out of the five years for which
we
generated out-of-sample forecasts.
This favorable evidence for the nonlinear model
See Ramsey (1996)
for
is
reinforced by table
arguments on the bias against nonlinear models inherent
22
in
6.
MSE
It
reports,
comparisons.
Table
4:
Nonnested Models Test: Log-Normal/Normal Shocks
Equipment
test
NOTES: Vuong
PAM
AR(2)-UNC
PAM
AR(2)-UNC
2.42
7.97
3.57
4.53
0.0078
< 0.0001
0.0002
< 0.0001
(Normal)
p-value
Structures
compare the structural
statistic calculated as in (20). All test statistics
model estimated
in
Table
1
with the model
in
the table.
good', the asymptotic distribution of the statistic
is
If
both models are 'equally
Standard Normal. Large positive
values provide evidence in favor of the structural model.
Table
5:
Average
MSE
for
1-Step-Ahead Forecasts: 1988-92
Equipment
AR(2)
MODEL
88
0.591
0.640
89
0.240
90
Structures
AR(2)
MODEL
-0.075
0.306
0.340
-0.092
0.230
0.038
0.371
0.220
0.502
0.308
0.270
0.131
0.207
0.150
0.359
91
0.566
0.540
0.044
0.507
0.350
0.371
92
0.399
0.330
0.194
0.518
0.340
0.425
Year
ln((l)/(2))
The parameters were estimated using data up
23
to 1987.
MSEs
ln((l)/(2))
are multiplied by 10
MSE
each investment type, the average (across years)
for
and nonlinear models
for
each sector for the AR(2)
two columns), and the percentage increase
(first
in
MSE
generated
by the AR(2) model over the nonlinear one (third column). At the bottom of the table,
we
report the percentage increase in average (across sectors and time)
AR(2) model over the nonlinear
the
The
sectoral dimension
in order to gain statistical
same
However, table
sectors) increase.
power
for the nonlinearities
distribution of adjustment costs across sectors, which
to hold too closely in the data.
sectors.
median (across
one, as well as the
generated by
one along which we would have expected the nonlinear model to
is
do relatively poorly, since
forced to impose the
MSE
6
The AR(2), on
this
unfavorable metric the structural
model outperforms the unconstrained AR(2) representation.
MSE
true for structures, where the gain in terms of
and above 30 percent
3.6
Dynamic
for the average
not likely
the other hand, has no constraints across
shows that even under
sector,
is
we were
Again, this
is
particularly
over 15 percent for the median
is
MSE.
implications of increasing hazard models.
Perhaps the main distinctive feature of the model we have estimated, compared with
linear counterparts or a constant hazard model,
that are investing but also the
number
fluctuates over the business cycle. This
level
evidence
number
the
Doms and Dunne
in
is
that not only average investment by those
of firms that choose to invest at
is
its
any point
in
time
a realistic feature according to the establishment-
(1993).
Among many
interesting facts, they
of plants going through their primary investment spikes
(i.e.,
show that
the single year
with the largest investment for the establishment), rather than the average size of these
spikes, tracks closely aggregate
In
terms of our model,
manufacturing investment over time.
this flexibility in
the
number
of firms investing implies that
the extent of the response of aggregate investment to aggregate shocks fluctuates over the
Figure 5 depicts the paths of the median (across sectors) derivatives of
business cycle.
aggregate investment with respect to aggregate shocks for equipment and structures.'^^
is
apparent that
it is
this "index of responsiveness" fluctuates
strongly procyclical:
are, respectively, 0.79
Its
There are traces of the
Evaluated at
widely over the sample. Moreover,
correlations with aggregate shocks
and 0.89
for
equipment, and 0.72 and 0.39
cyclical features of
It
and aggregate investment
for structures.^*
our nonlinear model
in
our out-of-sample
vt-
28^
The
correlations with lagged investment/capital ratios are 0.60 and 0.13, for equipment and structures,
respectively.
24
r
i
i
1
1
o
^;^
'--'
-P
"^
-
05
CD
CD
=5
F
-P
CD
-
k
quipi
true
V
1
LiJ
00
-
1
)
/
"
00
1
GO
^\
'~~~1
~
-
V
V
—
-
LD
^-^^
-
CD
(_
C\J
1
^^\
•• **
CD
00
^^^"^
^^^^^
.^^^
U)
-
-
-
•
—
""
o
V
^
^ ^
CD
CD
''* — — — — —
^
^^"^^
^ ^
-—-«^''
^
"^
~
^
-.
^
CM
CD
V
y
\^
-,—^"l»^-
X
'
4
00
in
- ^ '^^AJ
"" *
^rXf
^.--"^
LD
O
UO
05"o
017-0
oe"o
CD
CD
>
Ll_
-c^irr
^^"""""^TTTi ^
^
r^
^^
^^,„„l«i^
^^^0^^^^
7
/
_
02"o
SS8U8AISU0dS8J ^0 XQpUl
OfO
Table
6:
Sectorial
Average
MSE
1-Step-Ahead Forecasts: 1988-92
for
Equipment
Sector
AR(2)
MODEL
Structures
ln((l)/(2))
AR(2)
MODEL
ln((l)/(2))
20
0.591
0.780
-0.425
0.742
0.150
1.571
21
0.085
0.084
0.003
0.117
0.150
-0.233
22
0.343
0.310
0.091
0.221
0.120
0.612
23
0.243
0.110
0.781
0.278
0.094
1.088
0.333
24
0.160
0.240
-0.405
0.342
0.250
25
0.083
0.092
-0.105
0.560
0.260
0.760
26
0.034
0.068
-0.707
0.633
0.400
0.456
27
1.620
1.710
-0.055
0.386
0.720
28
0.129
0.180
-0.341
0.313
0.400
-0.619
-0.247
29
0.098
0.100
-0.036
0.628
0.570
0.097
30
0.272
0.260
0.035
0.417
0.088
1.560
31
0.082
0.089
-0.097
0.063
0.054
0.159
32
1.433
1.000
0.357
1.483
1.000
0.392
33
0.078
0.076
0.023
0.017
0.024
-0.352
34
1.739
1.410
0.211
0.201
0.270
-0.288
35
0.551
0.730
-0.276
0.353
0.400
-0.127
36
0.506
0.420
0.181
0.328
0.390
-0.170
-0.411
37-1
0.310
0.220
0.338
0.099
0.150
37
0.241
0.270
-0.124
0.494
0.075
1.888
38
0.221
0.200
0.120
0.184
0.210
-0.124
39
0.099
0.072
0.319
0.162
0.100
0.467
-
-
0.047
-
-
0.312
0.241
0.220
0.003
0.328
0.210
0.159
ln(S,(l)/S.(2))
Median
The parameters were estimated using data up
25
to 1987.
MSEs
are multiplied by 10
Table
7:
Standardized Difference
in
Absolute Value of Forecast Errors (AR2-ModeI)
Equipment
Sort by v
Sort by y
Below
-0.286
-0.189
Above
0.255
0.185
Rel. to
Median
forecasts as well.
The
MSE
Structures
^
Sort by
Sort by y
Sort bv dv
-0.043
-0.056
0.100
-0.013
0.155
0.158
-0.033
0.092
gains of our model over the linear
nounced during periods of high
activity.
^
Sort by v
To show
this,
AR(2)
we proceed
in
particularly pro-
is
three steps: First,
we
construct, for each sector, a standardized series of the difference of the absolute values of
the forecast error of the AR(2) and the structural model,
\Jar
^I'^id
e^
Jr.,
—
cr.
where
A./
denotes the forecast errors (ar and nl stand
a^j
respectively), and
A|e-^|:
for
AR(2) and nonlinear model,
denotes the standard deviation of the difference
value of the forecast errors.
Second,
for
in
the absolute
each sector, we sort these standardized series by
one of the following indicators of activity: the sectoral aggregate shock,
u,
the level of the
sectoral investment/capital ratio, y, and the sectoral index of responsiveness, dy/dv.
third,
we average
And
across sectors the sorted standardized series.
Table 7 reports these averages
(below) and above
its
for times
median (above).
when
the sorting variable was below
With only one exception,
all
its
median
the entries suggest
that a substantial fraction of the better performance of the nonlinear model comes from
periods
when the
sectoral indicators of activity (shock, investment
are high. For example,
we
find that periods
when aggregate shocks
achieve a forecasting-performance improvement which
than the average
structures.
MSE
gain for equipment, while
Conversely,
when aggregate shocks
performance improvement
for
equipment, while
it is
is
it
is
is
and
sensitivity index)
are below their median,
0.286 standard deviations lower
0.056 standard deviations lower for
are above their median, the forecasting-
0.255 standard deviations higher than the average
0.158 standard deviations higher for structures.
26
MSE
gain
Final Remarks
4
we derived and estimated a time
In this paper
builds from the realistic observation that
series
model of
sectoral investment that
lumpy adjustments play an important
role in firms'
investment behavior; but that allows for the empirically appealing feature that adjustments
do not need to be of the same
size across adjusting firms
Using a nonlinear aggregate time
fixed
for a firm over time.
procedure, we estimated the distributions of
adjustment costs faced by firms. The adjustment hazards implied by our estimates
they leave a significative range of inaction, and increase sharply there-
are nonconstant:
after.
series
and
Depending upon the history of shocks, the estimated hazards have the potential
dampen
to magnify or
normal times
accumulated
is,
—
the response of investment to aggregate shocks.
occasionally,
more than
offset
The
by the brisk response to large
passivity of
— current or
shocks. These nonlinearities clearly improve the aggregate performance of
dynamic investment equations.
Both the microeconomjc
as well as the aggregate implications of the estimated
largely consistent with the establishment level evidence presented
Haltiwanger (1995)
for U.S.
manufacturing
More importantly, they
density of capital imbalances
in
by Caballero, Engel and
1972-88 period). They found evidence
(for the
of an increasing hazard for the range of disequilibria
their time.
model are
where establishments spent most of
also found an important role for the cross sectional
explaining changes in the marginal response of aggregate
investment to aggregate shocks.
In
the processes of assessing the contribution of the model,
we found an important
forecasting gsi]n over simple linear models. In almost every year and sector, and particularly
so for structures, the nonlinear
model reduced the Mean-Squared-Error by a substantial
amount.
Beyond the current paper, there
tioning at closing:
are three extensions and robustness issues worth
First, the nonlinear
model has non-trivial implications
higher moments.
Our preliminary exploration
viation, skewness
and excess kurtosis of investment/capital
correlated with the business cycle.
to explore
movements
Second,
in the
moments.
working paper version of
A good complement
for
non-structural
ARCH
this
models,
27
for forecasts'
of this issue reveals that the standard de-
These relations hint
in forecasts' higher
men-
at a
ratios' forecasts, are highly
promising structural avenue
^^
paper (Caballero and Engel 1994) we
for
example.
allowed for serial correlation in the rate of growth of aggregate frictionless capital (the
and found very
little
of
it,
which provides support
section. Interestingly, the linear
correlation, especially for
Finally,
(PAM) model we
for
our
estimated
i.i.d.
left
assumption
in
u.t's)
the theory
plenty of unexplained serial
equipment investment.
and also reported
in
Caballero and Engel (1994), we extended the theory and
empirical sections to acknowledge the existence of continuous "maintenance" investment,
which does not require paying sizeable adjustment
lowance was important to obtain a more
and average investment rates
tion of
in
accounting
for
We
found that while such an
realistic distribution of
observed changes
al-
in capital
small changes account for a significant frac-
(in particular,
microeconomic investment changes),
lumpiness
costs.
it
did not diminish the role of microeconomic
the dynamic aspects of aggregate investment.
28
Appendices
A.:
appendix we study the
In this
section
we show
1
Dynamic Optimization
In section 2
dynamic optimization problem.
In
existence and uniqueness of the solution to the firm's Bellman equation.
we study the main properties
of the optimal policy function.
Existence and uniqueness
1
From the main
text
it
follows that the Bellman equation - for the firm's value function
normalized by frictionless capital -
v'{z,u;)
with
its
firm's stochastic
{i
given
is
= m&x(Tr{z + i)-L^^e'^'{i^
7^
0}
+
by:"^*^
i'
f f e'^'v'iz
+ +
i
0} denoting an indicator function that takes the value
Az,ij')dF{Az)dGioj')\
1
when the
,
firm adjusts
and zero otherwise.
capital stock
The operator
associated with the above equation
is
not bounded from below. For this
reason we add a term to v'{z,iv) which does not depend on the choice variable and therefore
does not affect the firm's optimal choice, but which bounds the corresponding operator:
viz,^)
(21)
= v'{z,u)+^ue^'.
Substituting (21) into the expression above, using the expression for n{z) derived in
section 2 of the
main
text,
and performing some straightforward (but tedious) calculations,
leads to the Bellman equation for v{z,i^'):
(22)
v{z,u)
= max{ci(Lj)e^^ -
maXiu
cae^"
-
C2e'
C2e"
+
'A
// e-^'v{z +
'
Az,u')dF{Az)dG{u'),
+ ^ I f e'^'viu + Az,io')dF{Az)dG{J)
},
with
c,{u)
All
=
terms that are not explicitly defined
^ (l
in
+
u-
- ^E
[e^^^] ;.^)
what follows were defined
29
,
in section 2 of
the main text.
where
/^i^
denotes the
mean
C2
=
e/?,
C3
=
^ (l
- VE p^^]
,
of the distribution of the adjustment cost factor G{io).
(22) has a unique solution
To show that
/i.)
we make the following three assumptions and
prove the following two lemmas. The assumptions hold throughout the remainder of this
appendix.
Assumption
1
from below by
0.
The adjustment
=
Assumption
2
VE
Assumption
3
^pfi^^E \e^^^\
[e-'^'l
cost factor,
u), is
jp
J e-'^'dF{Az) <
=
TpfJ-u
bounded from above by
uj
< +00 and
l.^'
f ^^^'dF{Az) <
1,
where
>
0.
11^
denotes the mean of the
distribution of u.
Lemma Al
Let J(z)
P).
=
k-^e^'
Then /(-)
value, which
is
is
- k2e\ with
increasing for z
equal to (1
Proof Elementary
-
calculus.
<
<
<
/3
Zf,j,
1
and
/c]
,
A:2
decreasing for z
P)k\^^^~^\p/k2)'^^^^-'^K at z
Denote z^ = \og{ki0/k2)/{l -
>
z,\},
and attains
its
maximum
= z^.
|
Lemma A2
Consider the operator
This operator
domain
M
x
Then T:
is
T
defined by posing {Tv){z,uj) equal to the right hand side of (22).
defined on the set
B
of
all
real-valued,
bounded, continuous functions with
[0,u;].
(i)
preserves boundedness;
(ii)
preserves continuity; and
(iii)
satisfies
Black-
well's conditions.
Proof
(i)
Consider u ^ B, bounded from below by u and from above by
u.
Then
(Ttz)(z,cj)
is
In the main text we assume that Az follows a Normal distribution, with mean n and variance a
Then this condition is equivalent to fi > ^a^ + log(V') which for r -f 6 << 1 corresponds approximately to
^i >
^cr — T — b. Thus, for the set of parameters we use in the empirical section, a sufficient condition is
that /J > —0.15 for equipment and /j > —0.10 for structures. Both conditions can be expected to hold.
.
30
bounded from above,
since:
<
V'E e-^^
u
+ max
|ci(w)e^^ - C2e'
<
V-E e-^-'
tl
+ max
fc,(u;)e^"'
<
xPE 'e-^'~
u
+ ^{l-P)[l+uJ - VE
where the second inequality follows from the
last inequality
from
That (Tu)(2,cj)
{Tu){z, ij)
Lemma
>
^E[e-^^ u + max
>
i'E [e-^'
u
^-^-'1
V'E e
where we used
(ii)
03
<
ci{u>) in the
and the second
is
constant.
u\(z.u})
in
<
is
maximum
the
T
- C2e^
-
fact that, for
c\{uj)
cae^'^
/?)
(1
second step, and
is
To show that
I
H + ^1 -
follows that {Tu){z,Lo)
(iii)
1/(1-/3)
g^Az'
w >
all
<
0, C3
Ci{lo),
It
,Pz
-
u),
c.e-]}
1/(1-^)
- ^E p^'']
Lemma Al
continuous for u G
(Tu)
the last step.
in
we note that from
first
inherits continuity
The second Blackwell
Proposition
it
from
first
note that
if
Ui,U2 G B, and
then the expected value of any positive random variable,
straightforward calculation shows that, for any u £
+
(22)
continuous.
is
we
/z.)
;B,
of two functions; the
follows that
036oP^
max
,
particular e~^~, preserves the above inequality. Thus:
{T[u
and the
C2e'
satisfies Blackwell's conditions,
U2{z,u)) for aU z and
€26
{Tu,){z,oj)<{Tu2){z,u).
A
1}
follows from:
+ max
To show that the function {Tu)
u(z,u>)
C2e
cae^
w
Al.
bounded from below
is
max
a])(;,u;)
B and any
= {Tu)iz,u) + 0E
condition follows from Assumption
[e-^^]
2.
constant
a.
|
Al
Equation (22) has exactly one solution (and
this solution
Proof
31
belongs to B).
a:
It
follows from
Lemma A2
that
T
defines a contraction
The modulus
(normed with the sup-norm).
of the contraction
now
Existence and uniqueness of a solution to (22)
Theorem
Theorem
(see, e.g.,
mapping on the metric space B
follows from the
Lucas and Stokey (1989)).
3.2 in
mapping
is
i/'E
e~^^
.
Continuous Mapping
|
Properties of the optimal policy
2
We
define the following functions related to the solution of the Bellman equation, v{z,u>),
considered in the preceding section:
(23)
I{z)
=
i)
(24)
Jiz)
=
C3e^'
f f e-^'v{z+ Az,oj)dF{Az)dG{u),
-
C2e'
+
I{z).
Lemma A3
The
function J{z)
is
bounded from above,
i.e.,
sup, J{z)
is finite.
We
denote this
supremum
by Jmax-
Proof
Since v{z,uj) satisfies Bellman's equation, we have:
v{z,u>)
(25)
If
max, J(;) were not
Proposition Al.
It
finite,
= max{^we^* +
J{z)
we would have that
follows that sup, J(c)
,
max J(u)}.
v[z,ijj)
not bounded, contradicting
is
|
is finite.
Lemma A4
The
function J{z) satisfies:
lim
(26)
Z——00
lim
(27)
2—>-|-CX>
J{z)
=
J^ax^E
I
J{z)e-'
=
lim
v{z,uj)
-C2.
Proof
From
(25)
it
[e-^^l
follows that:
32
=
Jmax,
J
,
which, from (23), implies that
lim
I{z)
= JmaxV'E e-^^
Expression (26) now follows from (24).
Expression (27) holds because of (24) and the fact that I{z) inherits boundedness from
v{z,uj).
I
result establishes the existence of a curve in (z,w)-space partitioning this
The next
space into two regions; one where firms adjust their capital stock and another where they
remain inactive.
Proposition
A2
Define:
n{z) = r'iJm..-J{z))e-f'\
(28)
Then
firms adjust
when
remain inactive when
their current
w >
adjustment cost factor, w,
is
smaller than fi(r), and
^{z).
Proof
By equating both terms on
the right hand side of (25)
inequalities that hold for
Proposition
The
obtain:
^n(z)^{J^,,-J{z))e-^\
(29)
The
we
u>
larger
and smaller than
ft{z) follow trivially.
|
A3
function fl{z)
is
analytic on the real line, and therefore has derivatives of
all
order.
Proof
From
the definition of fl{z) and J{z) (see (28) and (24))
that I{z)
is
analytic.
To do
this,
we note that
I{z)
may
we have that
When
uj
=
=
i^
f
K{z + Az)dF{Az),
Q{z) firms are indifferent between adjusting and not adjusting.
33
suffices to
show
be written as the convolution of a
normal density and a continuous, bounded function:
/(z)
it
with
K{z)
=
/
ip
=
-^le
/i
and variance
That the convolution
continuous) function
is
-
/i
a"^
and variance
(where dF{Az)
cr^
is
<7^).
and an integrable
analytic, follows from a well
Theorem
on
9
known property
Lehmann
59 in
p.
(in particular,
(1986)).
a bounded,
of the exponential
|
z tends to -oo:'*'^
^fi(z)~ J„,,,{l-0E[e-^'']} e-'\
(30)
as z tends to oc:
^fi(.-)
(31)
It
=
Jl
A4
Proposition
and
2
of a norma! density
family of distributions (see
As
sl- >^
^
and dF{Az) a normal density with mean
normal with mean
v{z,u))dG(u>),
~
C2e('-'3)^
follows that:
lim
(32)
|j|
— oo
n{z) = 00.
PliOOF
Expression (30) follows from (28) and
lini|.|_,^
A4. Expression (31) follows from (27). Then
n(r) = CO then follows from (30), (31), Assumption
set
2
and the
fact that
/3
<
1.
|
A5
Proposition
The
Lemma
C of
r £ iR such that J{z)
= Jmax
is
a non-empty set with a finite
number
of
points.
Proof
Continuity of J{z)
(it
A4 combined with Assumption
which J{z) attains
maximum
its
2 ensure the existence of a
maximum. Continuity
write a(z)
~
maxima
A3) and
analytic, see the proof of Proposition
of J{z) on the
indeed attained and therefore C
is
we have that
'We
its
is
is
if
lim^^c Inj
34
=
compact
set
set
1-
Q within
Q ensures that the
non-empty. Finally, since J{z)
are isolated, thus showing that
6(2) as : tends to c
bounded, closed
Lemma
C contains a
finite
is
analytic
number
of
elements.
|
Proposition
When
A6
adjusting
capital stock, a firm's optimal choice of z
its
disequilibrium after adjusting does not depend on
its
any element
is
in C.
Thus
its
disequilibrium before adjusting.
Proof
The
result follows
hand
on
from the fact that when the
side expression of (25)
z.
is
maximum between
both terms
in
the right
attained at the second term, this expression does not depend
I
All calculations of ^(z) performed while estimating the distribution of the
cost factor (see section 3.3 in the
element,
c,
main text and Appendix C)
and a function n(~) that
the right of 2
=
c,
is
decreasing to the
a
led to
of 2
left
set
=
C with a unique
and increasing to
c
therefore implying an optimal policy of the {L,c,U) type. Yet
been unable to show formally that C has one element
we have
a unique return point), and also
(i.e.
have not shown formally that, conditional on u, the firm's optimal policy
is
As the following proposition shows, however, we can prove that the
type.
adjustment
of the
{L,c,U)
latter holds in a
neighborhood of a return point.
Proposition
For
(cj,
z) in a
A7
neighborhood of
(0, c),
the optimal policy
of the {L,c,
is
U) type.
Proof
From Proposition A5
is
decreasing to the
follows that there exists a neighborhood
it
of c
left
equation (28) that ^{z)
z
=
c.
Thus,
for all ui
<
is
=
c
and increasing to the right of ;
decreasing to the
max-gv'('^ma.x
policy, conditional on the current
is
of r
=
- J{~))p-~^' and
adjustment cost factor,
B.:
This appendix
left
of the approximation
we
of
=
c.
:;
=
It
c
such that J{z)
then follows from
and increasing to the right of
c
r £
is
V we have
of the
that the optimal
{L,c,U) type.
|
Aggregation
divided into three sections. In Section
sion for aggregate investment
V
and present the
use. In Section 2
1
we
establish the exact expres-
results of simulations to assess the quality
we study the main properties
of the adjustment
hazard. In Section 3 we characterize the average cross-section of firm deviations.
The
following operators, defined on the set of probability measures on the real line, are
35
used
in sections
1
and 3 of
this
appendix and
•
Sectoral (aggregate) shock, A{v):
•
Adjustment shock,
tion characterized
•
v.
HiO): applies adjustments determined by the adjustment func-
by the parameter vector
with zero mean and variance
Full cycle of shocks, T{v
defined above,'^''
6.
i.e.,
+
cr^.
S,9,a): equal to the combination of the three shocks
+
to I{(7)H{d)A{v
Equation (17)
6).
explicit expression for the cross-section that results
•
by
shifts a cross-section
Idiosyncratic shock, I{o^)- convolves the probability measure with a Normal density
•
Appendix C.
in
Aggregate investment functional,
y-.
the main text gives an
in
from applying
T
to f{x,t
—
1).
Assigns to a cross-section the average
vestment rate that results after adjustments take place (see equation (12)
in
in-
the main
text).
Aggregate Investment
1
we derive the exact expression
In this section
tion (12)
is
an approximation.
We
aggregate investment, of which equa-
for
also assess the quality of this approximation.
Lemma Bl
We
introduce the following notation:
•
x,,i:
disequilibrium immediately before period
•
r, (!
number
/,
•
(!
last
A:'(_^:
of periods, as of time
time firm
i
t,
since firm
adjusted (equal to
i
—
desired level of (log) capital of firm
adjustment of firm
t
i
i.
last adjusted.''^
r,
i
immediately before
its last
adjustment
took place.
•
ktj^^:
that
/:,
firm
—
(_
J
t's
(log) capital stock
k'l^
^
+ c and
immediately after the
last
time
that, as changes in capital since time
/,,(
it
adjusted.
have only reflected
depreciation, this quantity completely determines the current capital stock.
Incorporating depreciation.
^^Possible values are: 1,2,3
36
Note
Then, conditional on
on when the firm
we have that
/,,(,
and
i,t
k^j^
^
are independent.
last adjusted, its current disequilibrium
That
is,
conditional
and current capital stock are
independent.
Proof We
have that
=
x,t
k* ^
-
A;*;
and therefore depends on shocks (aggregate and
_
idiosyncratic) that took place during periods
on shocks that took place at
/, (,
t
<
/,,(.
both quantities are independent.
t
>
In-
On
Since shocks are
the other hand,
i.i.d., it
A;,
;^
^
depends only
follows that, conditional on
|
Proposition Bl
Denote:
•
Kt{x\r):
adjusted
•
h'tir)
:
average capital stock of plants with disequilibrium x at time
r
t,
that last
periods ago.
average capital stock of
firms that last adjusted r periods ago. Thus:
all
= ^7r,(r)r,(r).
A-,^
r
•
/((r):
average investment, at time
/,
of those that last adjusted r periods ago. Thus:
r
•
f'{x\r):
Tt
•
cross-section, at time
t,
of plants that adjusted r periods ago.
= n(e)A(vt + S)I{a1) and U = A{vt +
7r((r):
fraction of plants, as of time
t,
<5)I((t2),
we have that
that last adjusted at r periods ago.
Then:
it
= E'^'(^)^(('-)/(e-" -
Proof
It"
=
E''tir)It{r)
37
l)K{x)ft{x\T)dx.
Denoting
- l)A{x)Kt{x\r)f;(x\r)dx
(e-^
E'^'M/c^
E Mr)Kt{T) /(e-^ -
^
Lemma
where we used
It
Bl
the last step.
in
l)Aix)f;ix\r)dx,
|
follows that, to calculate the exact expression for the aggregate investment/capital
ratio,
we need
to keep track of a sufficiently large
and the
ft{x\r)
size distribution of cohorts, 7rt(r).
more burdensome than the approximation we used.
this appro.xjmation
We
note that
if
number
of conditional cross-sections,
Computationally, this
We show
numerically, however, that
mostly affects nuisance (secondary) parameters.
7 denotes the estimated values of the main parameters
transformation of the shocks used when calculating the likelihood
main parameters.
use
is
~^"
substantially
is
It
follows that a
still
7,''^
then a linear
leads to the
same
good measure of the quality of the approximation we
to determine the extent to which the exact expression
imation when we allow for a linear change
in
comes
close to our approx-
the aggregate shocks that determine exact
aggregate investment.
To implement
this idea
we consider 50.000 firms with
and disequilibrium x equal to
zero.
initial capital
All firms belong to the
same
stock equal to one
sector.
We
simulate the
evolution of these firms during 75 time periods, with parameters given by our estimated
structural model.
We
keep track of the aggregate shocks (denoted by
imation to aggregate investment (denoted by yf). Next we
rescaled aggregate shocks
(if,
=
aggregate investment (y^(a,6)).
is
closest to
of
them applied
y^
.
To measure
a
+
We
bvt), this
re-
Vt)
and our approx-
run the whole process with
time keeping track of the exact expression for
find the values of a
and
pro.xjmity between both series
b for
which the
we consider two
series
yj{a,b)
criteria,
both
to the last 45 observations of both series:
^2
^
^_MSiyJ(a^b)-yf)
y^r{yf)
R2
=
1-
V^T{yJ(a,b)-yf)
Var(t/^)
There are three of these parameters in both estimation approaches we use. In both cases we have a free
The two remaining parameters characterize the distribution of adjustment costs in the structural
case and the adjustment hazard in the semistructural case.
This follows from the expression derived for the likelihood (see Appendix C).
constant.
38
Table
8:
Assessing the approximation for aggregate investment
Equipment
Structures
0.0153
0.0030
a
b
R?
d
b
R2
1.054
1.080
0.9803
0.9949
-0.0185
-0.0045
1.225
1.145
0.9962
0.9980
where MS(j/() denotes the average of the squares of the corresponding
its
variance.
These measures capture the
whether they allow or not
our simulations.
It
is
for
fit
of a regression of yj on
an additive constant term."^^
y^
^
and
Var(2/t)
differing in
Table 8 shows the results of
apparent that the excellent quality of the
approximating aggregate investment by
series
fit
we obtained
justifies
(12).
Adjustment hazard
2
In this section
we study
the properties of the adjustment hazard:
A(x) = G(fi(i
where
c is
+
c)),
a fixed element (say the smallest one) in the set
C characterized
in
Proposition
A5.
Assumption
[0,w],
w <
4 The distribution function
G
has a continuous density g(w) with support
CO.
Proposition
B2
Under Assumptions
1,
2
and 3 made
in
Appendix A we have that the adjustment hazard
satisfies:
(a)
|a;|
lim|^|^oo A(2;)
>
M
we have
=
A(a;)
Furthermore, there exists a positive constant
1.
=
M
such that for
1.
It is arguable which criterion is more adequate in our case.
On one hand, we allow for an additive
constant term when estimating our models, on the other hand, it did not vary across sectors.
39
(b)
A(a;)
is
difFerentiable at
all
x and:
A'{x)
= g(a{x +
c)}n'{x
+
c).
Proof
A4 and Assumption
(a) This follows immediately from Proposition
(b)
follows from the fact that A(x)
It
is
1.
the composition of two differentiable functions and
therefore differentiable (see Proposition A3).
|
3
Invariant distribution
Due
to the presence of aggregate shocks, the distribution of disequilibria that determines
aggregate investment (see Section 2.3.3 of the main text) has no invariant distribution. In
Caballero and Engel (1992b) we establish that,
a well defined sense, the average over
in
possible trajectories of aggregate shocks of the cross-section of deviations
invariant distribution faced by an individual firm.''^
distribution exists and that convergence toward
The
on the
•
following operators,
real line, will
We
let 7]
mean
=
all
of
it
2
•
=
We
which are defined on the
We
set
T
+
=
h)I{a'^)7i{e) and Tj
'H{B)A{^i
+
of probability densities
S)I{a'^),
of aggregate shocks, 6 the depreciation rate, a^ the
= G{n{x +
c)).
set of
The operators A,
appendix. For any integer n >
1
H
we denote by
where
sum
fi
denotes the
of the variance of
parameters characterizing the
and I were defined
7^" the n-fold
earlier in
composition of
7i,
1,2.
denote by V(/) the function that associates to / the fraction of firms that adjust
when applying
•
we show that such a
be useful throughout this section.
y4(/j
function A(x)
equal to the
takes place at an exponential rate.
aggregate and idiosyncratic shocks, and 9 the
this
In this section
is
all
72.
denote by Gn-\ the operator that associates to an
of firms that have not adjusted after {n
Thus the aggregate shock is constant and equal
mean and variance a^ = it\2A + a^
-
to
40
1)
/i
initial
density / the cross-section
shocks, normalized to one.
and the idiosyncratic shock
is
Normal with zero
We
The
cLSSume throughout that shocks are Normal.
following
lemma
is
needed to establish
the main result of this section.
Lemma B2
Given a cross-section /o
let
=
/„
7^"(/o).
Denote by nn{fo) the probability that a particular
firm adjusts at time n, conditional on not having adjusted during the
and by
'r„(/o) the fraction of firms that
exists a constant
a £
common
(0, 1),
do not adjust during the
to
all initial
-
^n(/o)
>
1
(34)
r„(/o)
<
a".
by J^n-i the
not adjusted after
(n-
1)
a lower
bound
Ne.xt note that
that A(x)
when
|i|
=
>
for
M
is
a lower
>
M
.
represent those firms that have
7r„(/)
>
inf
=
>nj^i(/)-
bound
n periods. Then there
=
7ri(^„_](/)),
for 7r„,
n >
is
M
such
1.
exists a constant
B2 that there
It
that:
V(/)
Denote by A'(x) the adjustment hazard that
and equal to zero elsewhere.
we have
V(g)
inf
follows from Proposition
it
for |i|
1
also
tT]
=
7r„(/)
inf
Hence
may
T and
C
periods,
Q,
set of all densities that
shocks. Since J^„_i
1
cross-sections, such that:
(33)
Proof Denote
first
n -
first
is
equal to one
ea^y to see that both r„(/o) and
1
—
^i(/o) corresponding to this adjustment hazard are larger or equal than the corresponding
quantities for the original hazard. Tlius
the case of (33), for n
=
it
1.
Applying the operator 72 to a mass point
at x,
which the fraction of firms that does not adjust
distribution before adjustment
is
Q = 2$(M/o-) —
1,
prove (33) and (34) for A'(a:) and, in
suffices to
with
is
(5^,
is
we have that the value
largest,
is
the value such that the
normal with zero mean and variance
< q <
1
and ^ denoting the
c.d.f.
of x for
cr^,'*°
and
this fraction
of a standard Normal.
follows that for any density /(x) the fraction that does not adjusts after applying T2
bounded from above by
"Since
<J>((A/
-
x)/cr)
a.
Hence
- $((-A/ -
>
tti
x)/(t)
is
1
-
q.
maximized
41
Finally, given
at i
=
0.
It
is
any cross-section /o(x), we
have that ^^(/o)
Proposition
=
B3
n^_j(l - ^kifo)), which
Given an arbitrary
bounded from above by q".
is
initial cross-section, /o, let fk
the sequence (/^) as above, but for the particular case where /o
Let
is
= T^{fo).
conditional on not having adjusted during the
first
n —
1
Also, define
a mass point at x
denote the probability that a firm which starts off at i =
7r°
|
=
0.
adjusts at time n,
periods.
Denote:
n;:^(i-^,°)
^'
i
and
define: /'^(x)
=
+ E.>:[nL-d(i-'^?)]'
J2i>oPifi{^)- Then /„ converges to /^
convergence takes place at an exponential
Proof We
consider
At any moment
the
in time,
same number of periods
The weights on
Hence f°
ago.
=
follows from
It
Stokey (1989,
=
z
I
subsets
and
A
=
j
e
a convex combination of /o,/i°,
is
S -
•
•
•
,
fn-\-
Lemma B2
p. 348).
= max(a,
=
-
(1
{0,1,2,3,...}, and transition kernel:
TTs)6s,v--[
and zero otherwise. State
The corresponding
(adjusting).
N
if
1
Sq.
the above densities can be determined by the one-to-one correspondence
Pis,v)
(5,,^
=
f° can be partitioned into groups of firms which last adjusted
with the Markov process with state space
where
the variation distance and
rate.
the case where /o
first
in
probabilities are
+
^s^s,0,
+
s leads to s
1
—
tt,
and
(not adjusting) or to
tt^.
that the above process satisfies Condition
Indeed, with the notation of these authors
1
1
- q) >
0,
with q defined
in
M
in
Lucas and
we have that there
exists
lemma, such that
for all
the preceding
of 5:
rmx{ P^ {s, A), P:{s,A^)) >
e,
since
m..iP,is,A),P,is,A^))
=\'
1
and from
Lemma B2 we
max(7rs,
have max(7rs,
Hence Theorem 11.12
in
'f
1
1
-
-
=
e
^^
^ ^
°^ ^°'^
+
'^
^ ^"^
for all 5.
Lucas and Stokey (1989,
42
+
otherwise,
7r^)
tt^)
^°'^
p.
350) implies that there exists an
invariant distribution, /^, and a constant
/°
(35)
II
Furthermore, /^
is
-n\<
(1
-
e)
G"ii fs
the unique fixed point for the
straightforward calculation show that f^
Extending
G=
=
<
1,
such that:
-n\.
Markov operator.
The
latter
and a
Y^^Pif"-
where /o can be arbitrary
this results to the general case
is
straightforward.
All firms eventually adjust and, once they adjust, the previous case applies (since they
adjust to X
=
0).
Given an arbitrary
/o
we may
write:
T"(/o) =
where /*
is
in
Lemma
.
,
f",^; gn could
the fact that the variation distance
that there exists a G
(0, 1)
iiTVo-rii
Convergence follows by
.
beany
cross-section,
and r„
B2.
From Lemma B2 and
is
- V2)r"/^(/„*) + V2(5n),
a convex combination of Jq, f^,.
was defined
we have
(1
is
bounded from above by
1
such that:
<
(i-r„/2)iiT'^/'(<5o)-rii+v2
<
G"/'
II
/o
-
r
II
+a"/'.
letting n tend to infinity; the rate at
geometric, bounded from above by mcLx(vG, i/q).
which convergence takes place
|
C: Econometrics
This appendix
is
divided into two sections. In Section
and sketch the general approach used
In Section 2
1
we
derive the likelihood function
for calculating this function at given
we describe implementation
parameter values.
details for the semistructural (Section 2.1)
structural (Section 2.2) cases.
43
and
Calculating the likelihood function
1
An
1.1
expression for the likelihood
sources of randomness ("error terms") are the sectoral (aggregate) shocks,
The
where
u,('s,
i
=
and
I,..., I
t
=
We
l,...,r.'"
i.e.,
the
assume that sectoral shocks are Normal
and independent over time, and denote the mean and variance of these shocks
in sector
by ^, and c„, respectively. The (column) vector of sector
is
by
V,;
We
V
[c,j]
aggregate shocks
denotes the column vector with Vi followed by V2 and so on; and fiy
allow for contemporaneous correlation
C=
i's
among shocks from
i
denoted
=
different sectors; the
^[V]-
matrix
denotes the corresponding covariance matrix (which does not vary over time).
Standard change of variable calculations lead to the following expression
minus the
for
log likelihood:
-Log.
where A
lik.
Q B
=
const
+
^
log
§^
dvu
+ ^
log \C\
Log.
lik.
=
const
+ y^log
C
dyxi
dv,t
where
/I'v
hv - /iv)'(C-'
denotes the Kroneckcr product of matrices
Concentrating the likelihood with respect to
[361
+
and
®
hW -
A and B and
y,;
^iv),
=
lal^'^ii-
/iv leads to:
+ 2l°g
corresponds to the vector of sample means.
{V - fIv){V - ^TvY
That the Jacobian
is
well defined
follows from Proposition B2.
Calculating the likelihood
in (36) requires
calculating the sectoral shocks (the
the corresponding partial derivatives (the dyu/dv,i^s). Next
we show
v,t's)
and
that, conditional on
the initial cross-section and the set of parameter values, the relation between sectoral shocks
and sectoral investment rates
is
invertible.
Our proof
is
constructive:
it
describes
how
the
sectoral shocks are actually calculated for given parameter values.
^'By working with a continuum of firms we have that, despite the presence of idiosyncratic and adjustat the micro level, the only source of sectoral randomness are aggregate shocks. That is,
the cross-section that results after adjustments is uniquely determined by the adjustment function and the
cross-section prior to adjustments. Also, the cross-section that results after the idiosyncratic shocks is the
convolution of the density (common across plants and sectors) from which these shocks are drawn with the
cross-section prior to the shock. See equations (15) and (16) in the main text for the corresponding formulas.
ment shocks
44
Calculating the components of the likelihood
1.2
Suppose we know the cross-sections of disequilibria
from the aggregate dynamics
the i-th sector during period
and
(see equations (15), (16)
(37)
=
y,t
determined by the aggregate shocks
is
yltiv^l,v,2,...,v,t,M,0)),
and the cross-section prior to
=
yu
0.
It
follows
in
the
first
t
periods
is
=
i
l,...,/.
a function of only the current aggregate shock
this shock:
y.tiM-J-l),Va)
=
(39)
=
t
(18)):
Furthermore, aggregate investment
(38)
every sector at time
our model that the observed capital-investment ratio in
in
t
in
+
4>{v,t
S
- x)A{x
-vu-S)Mx,t-
l)dx,
I
where
in
our case
=
(p{u)
-
e"
1
but,
more
generally, in the derivation that follows 4>{u)
could be any smooth and strictly increasing function with
above expression with respect to
^(/,(-,^-1),lO = /
d Vu
J
[(h'(v
+
8
v,t
evaluated at v
- x)A(x -
V
-6) +
is
4'(0)
equal
4>{v
+
6
=
0.
The
derivative of the
to:
- x)A'{x
-v-
8)]
f,{x,t-l)dx.
Recalling that f{x,t) denotes the cross section immediately after period Vs sectoral (and
depreciation) shocks,
(40)
It
that:
^iM;t-l),Va)=
[ [<P'{-x)\(x) -
follows from our assumptions on
and increasing
is
we have
(41)
(42)
the fact that A(z)
for positive x (the "increasing
strictly positive
Vii is
d and
when
/,(i,
^
-
1)
<t>{-x)A'{x)]
is
l)dx.
decreasing for negative x
hazard" property) that the above derivative
has support equal to the real line (as in our case).
uniquely determined from (38):
vu
Mx,t -
=
f.((2/.r,/,(-,^- 1))
=
Vaiy^t,v,,t-\,,Vi^l,f,{,0)),
45
Thus
and proceeding inductively we conclude that:
v,t
It
-
v,t{y,t,y,,t-i,--
•
,2/.i,/.(-.0))).
follows that, conditional on the initial cross-sections, the
by the
t;,('s
are uniquely determined
y,('s.
Initial cross-sections
1.3
The
initial cross-section in sector
i
is
measure of the
set equal to the invariant probability
unconditional process describing the evolution of disequilibria for an individual plant
This
that sector.''^
the cross-section obtained
is
paths of aggregate shocks.
Although
when averaging over
in
possible sample
all
'''
this selection
is
arbitrary,
we checked the robustness
of our results by study-
ing the convergence properties of cross sections distribution near our initial distribution.
We compared
the sequence of cross-sections used in our likelihood calculations with those
obtained when we perturbed the mean of the invariant-initial distribution by one standard
The Markov
deviation of the average (across sectors) aggregate shocks.
problem, combined with the contractionary features derived
for
Appendix A, ensures that
any given sequence of aggregate shocks, the distance between both cross-sections tends
to zero over time with probability one; the issue
tlie
in
structure of our
other. Simulations
showed that,
for the
is
how
fast
one distribution converges to
parameter values considered, the distance be-
tween both sequence of cross sections becomes negligible (variation distance
sometime between the second and
we discarded the
1.4
first
third cross-section after the initial
three observations for
all
series
when
less
than 0.01)
For this reason,
one.''''
calculating the likelihood.
Summary
Given a
set of
parameter values, we calculate the likelihood
By unconditional, we mean that we do not condition on actual
variance of shocks relevant for this distribution
sum
in (36) as follows:
sectoral shocks.
For this reason the
and idiosyncratic
we show that if Fo denotes the probability measure describing a particular plant's
deviation at time t = 0, and Ft the corresponding probabihty measure t periods later, then Ft converges in
the variation distance to a distribution F* which does note depend on the initial distribution Fo.
''^See Caballero and Engel (1992b) for a proof.
The parameter on which convergence depends most is the variance of idiosyncratic shocks; convergence
is faster as this parameter becomes larger.
shocks. In
Appendix
is
the
B.l
46
of the variances of sectoral
1.
The
(one for each of the 21 two-digit man-
initial cross-section of firms' disequilibria
ufacturing sectors considered) are set equal to the invariant distribution faced by an
individual plant. These cross-sections are denoted /,(x,0);
2.
For
i
=
1
=
Solve (38) to find
(b)
Calculate dyit/dv^t from (40).
(c)
Determine the next
The next
conducted
in
=
1,...,21.
to T:
(a)
upon
i
and
(15)
u,(,
z
1,
.
,
.
.
21.
set of cross-sections of disequilibria (the /,(-,<)'s)
based
(16).
section provides the details on exactly
how everyone
of the steps above are
both estimation approaches (semistructural and structural).
Implementation
2
Semistructural approach
2.1
This approach estimates the adjustment rate function directly.
ment
rate function
is
common
A(x) =
and A2 >
0.
assume that the adjust-
across sectors and of the form:
(43)
with Aq >
We
1
-e-^°--^^^',
W'e estimate three parameters (besides the
means and
var-cov.
matrix of aggregate shocks); Aq, A2 and an additive constant (common across sectors).
Estimating the
2.1.1
In
what
follows,
we knew
this
initial cross-sections
we do not make any assumptions about the mean
mean, or could estimate
it
of the aggregate shock. If
directly from the observed data, then
we could
de-
termine the invariant density by calculating the invariant probability function of a standardfixed
Markov chain
To compute the
(see Caballero
and Engel (1994)
initial cross-section
we proceed
for details).
as follows.
denote a Normal density with zero mean and variance
47
CTq
=
cr^
For sector
+
c„.
We
i
we
set
let
c,,
5,(1,0)
equal to
0.035 for
all sectors.''^
Given 5,(1, r -
1)
we
calculate g,{x,T) by
y [niXo, A2M(tV +
where
by
y;
v^r-
Then we
1)
and
= y„
i.
The
solution
is
denoted
= T{v,r +
6,Xo,X2,cTo)g,{x,T
-
1).
approaches the unconditional invariant density for an individual
5,(a:,r)
approaches a constant consistent with the mean of sectoral investment/capital
v,t
We
solving for v in
set:
As T grows,
ratio.
-
denotes the average capital-investment ratio of sector
5,(1, r)
plant,
S)]g,{x, r
first
use this density as the
when calculating the
cross-section
initial
Simulations showed that using 30 iterations (for each sector) was sufficient for
likelihood.
all
practical
purposes.
2.1.2
Calculating the likelihood
The family
we work with has the
of adjustment functions
attractive property that the
evolution of the cross-sections can be tracked efficiently using a convex combination of a
number
small
this,
Normal
of
we show next that
densities, then f,{x,t)
densities, thus reducing
if
is
we assume that
f,{x,t
computational time substantially. To see
—
a convex combination of
1) is
A
-|-
a convex combination of A' Normal
1
densities.
We
also derive simple
expressions to update the means, %'ariances and weights assigned to the Normal densities
characterizing /,(x,i).
Consider
variance a^.
for
i',(
first
A
the case A'
=
1
and assume f,{x,t -
1) is
Normal, with mean
fi
and
simple but tedious calculation shows that solving (38) reduces to solving
in:
(44)
y_^
=
e'='''-+*'
-
Ie"''*"'-+*' + ^''''"+*',
-
1
a
where
t'
a^
=
I
+
2A2CT2'
There are two reasons for fixing c„: First, it avoids estimating an additional non-linear parameter.
Second, since we may expect that the variance of idiosyncratic shocks is significantly larger than the variance
of aggregate shocks, the value of the latter is of little relevance when determining the adjustment function
and the invariant density.
48
The
partial derivative in (40)
1^
dv,t
(45)
It
=
e'^(^-+^)
+
1
2
- fi+ -a
c{v)
=
V
d{v)
=
Ao
is
equal
+
,
— A2(v-/i)
to:
4e-''<^"+*'
-
2X2{v,t
+
<5
,cK, + 5)
- /i)4
_
^.
a'-
cr
follows from equation (17) in the
main text that the cross-section density
period's sectoral (aggregate), hazard and idiosyncratic shocks, f,{-,t),
<-th
combination of two Normal densities, one of them with mean
=
tj
and variance r^ + a^^, and the other with zero mean and variance
cr^.
(fi
— vn —
is
^)/(l
after the
a convex
+
2A2cr^)
The former corresponds
to those firms that did not adjust, the latter to those that adjusted their capital stock.
fraction of firms in the group that does not adjust
The
is:
r2
(46)
(7^
In the
of vV
more general
Normal
densities,
combination of terms
y,i
case,
where f,{x,t-l) =
u,( is
obtained by solving an equation analogous to (44) with a linear
like
H
=
partial derivative
is
convex combination
the one on the right hand side of that equation:
Qfc
k
The
'}2k=i (^hf^(,x,i~^) 's a
/ (i'{vii
+
<5
- x)A{x -
u,(
-
S)f^{x,
t
-
l)dx.
•'
equal to a convex combination of terms like those in (45).
also have that f,{x,t) will be a convex combination of
A''
+
1
Normal
densities.
We
Each of
these cross-sections corresponds to a specific cohort, grouping plants that have not adjusted
for the
same number
of periods.
"younger" ones and have
lost
The
"older" cross-sections are
mass monotonically due to the adjustment of their members.
Simulations showed that keeping track of 30 densities
is
extremely conservative: the impact
on aggregate investment of cohorts much older than 30 years
in
every period
more spread out than the
we merge the two
oldest cohorts into one
is
negligible. For this reason,
Normal density with mean and
variance equal to those of the convex combination of the densities being merged.
49
Structural approach
2.2
Instead of estimating the adjustment function directly, as in the semistructural case, here
we
estimate the parameters of the distribution of adjustment costs and obtain the adjustment
function from the solution of the dynamic optimization problem described in section
The
initial distribution is
calculated in a
Adjustment costs are drawn from
a
'""'=
which has mean
^i^
—
p4)
and
Gamma
way analogous
to the semistructural case.
distribution:
54) r
'"''"'**•
coefficient of variation cv^j
=
l/y/p- Again,
parameters (besides the means and var-cov. of aggregate shocks);
constant
(common
2.
/x^,
we estimate three
cv^ and an additive
across sectors).
Adjustment function
2.2.1
The adjustment
function for a given set of parameters
is
obtained by solving numerically
the stochastic dynamic optimization problem described in section
2.
For this purpose -
but not when evaluating the likelihood from the shocks and Jacobian terms- we disregard
sectoral differences in /x^
function
{fi^,
cv^, a^,
and
r, S,
cr^,
and assume the parameters that determine the adjustment
P, in addition to fi^ and cr^
)
are
allows us to calculate only one adjustment function and use
We
common
it
across sectors. This
for all sectors.
use a grid of 800 equally spaced points on the interval [-3.5,3.5] to determine
the value function via value iteration. '^^
The corresponding
simulations showed that 30 iterations were sufficient,
-•
Cn
=
Vr._,{z
+ Az) +
steps, for
which extensive
are:"*^
^e^(''+^^)
/
G{u)du
argmax(u„(2)).
Using 200 points makes no significant difference; we used 800 because the additional time involved was
The reason why we need at least 200 points is that we fix the grid of possible values of x (between
-3.5 and 3.5) in advance, so that often a significant part of this interval becomes irrelevant (the hazard is
small.
almost equal to one on it). Also, the finer the grid, the closer we can get to the case where the adjustment
hazard looks like that of an (5, s) policy.
See Section 2.2.3 for the derivations.
50
The
distribution of
Az
Normal with mean
is
variance faced by an individual firm
(ctq
=
a'^
—
ln(l
+
8)
and variance equal
When
cr^).
calculating Cn
to the total
we
interpolate
with a quadratic polynomial the value function Vn{z) at the 3 points on the grid where
the function
is
largest,
and
polynomial.
By doing
this
set c„ equal to the
argument of the maximum value of
this
maximization over a smoothed function, we avoid having to
work with a discontinuous likelihood function.
We set
the
mean
of the aggregate shocks equal to the
mean estimated with
the semistruc-
tural approach.''^
2.2.2
Family of adjustment functions
The adjustment
function estimated via dynamic optimization
800 points. This makes
it
needed to calculate the
i;,('s
evaluated on a grid of
is
computationally infeasible to solve the 945 non-linear equations
every evaluation of the likelihood. For this reason
in
we work
with a family of adjustment functions characterized by only a few parameters and such that
the derivatives needed for the Jacobian terms do not need to be calculated numerically.
Experimentation with a variety of distributions of adjustment costs showed that the
family of continuous, piecewise inverted Normal adjustment functions approximates well the
adjustment functions obtained via value
purposes, with the middle piece equal to zero.
parameter family
is
A
suffice for
representative
most practical
member
of this four-
of the form:
1
A(x) =
(47)
_ e-\-(x-r-)2
if^<^-^
U X- <
<(
l_e->^(--^)^
We
Three pieces
iteration.
approximated the positive (x >
X
< x+
ifx>x+.
and negative (i <
0)
function obtained via value iteration separately.
,
We
0)
determined
arms of the adjustment
x'^
and
A"*"
by imposing
that the approximation matches the function obtained via value iteration at the (positive) points
where the hazard equals 0.25 and
analogous condition for negative values of
*
We
set this value ex-ante to avoid
non-linear adjustment term 4i[v„
of this drift from the data.
firm deviations,
we allow
+
S
0.75,
We
obtained x~ and A~ imposing an
x.
having to estimate additional non-linear parameters. Because of the
x) in equation (38), there is no simple way to obtain an estimate
—
Also note that, as described
for a firm specific
mean
that
corresponding sector.
51
is
earlier,
when
calculating the invariant density of
approximately equal to the observed mean of the
Calculating the
2.2.3
When
and the corresponding derivatives
keeping track of the cross-section of deviations, we approximate /,(,<
mass points on a
We
t;j('s
why we
grid of equally spaced points (we discuss
The
The operator
by
u,(
+
8.
at each point
A{v^t
+ ^)
is
operator,
m(x) on the pre-shock
at X
H,
is
dard deviation
,
I).
CT;.
The
grid, then after the
/i,, 4- 4cr,(],
where
/i,(
and
a1^
obtained after the idiosyncratic shock. ''^
is
at zero):
if
x
is
a
adjustment shock we have mass
this point at zero.
where the mass was located and stan-
computed
at
33 equally space points on
denote the mean and variance of the cross-section
We
work with
ber of points needed to track the cross-section.
a
dynamic
grid to reduce the
around the mean,
suffices to
accurate estimates.
analytically, thereby avoiding
52
num-
Simulations showed that 33 points on a
grid of width equal to 8 standard deviations, centered
Both the mean and variance are calculated
—
each one of the 34 mass points becomes
to the point
resulting density
fi{-,t
applied next, leading to 34 mass points (one
and mass A(x)m(x) stemming from
Normal density with mean equal
4ct,,
obtained from:
+ 6,X- ,X+ ,x- ,x+ ,a,)f,{x,t -
Finally the idiosyncratic shock takes place:
-
is
yu.
where there was mass before adjustments and a new mass
- .\{x))m{x)
[//,(
chose 33 points shortly).
implemented by shifting the 33 mass points describing
The adjustment
point with mass
a
Next fi{x,t)
partial derivatives are calculated from (40).
f,{xj) = Tiv.t
(1
by 33
1)
solve for v in:
y[ni\-,X+,x-,x+)A{v + S)]Mx,t-l) =
1)
-
any
circularity.
obtain
References
[1]
"Consumer Durables: Evidence on the Opti-
Bar-Ilan, Avner, and Alan S. Blinder.
mality of Doing Nothing." Journal of Money, Credit and Banking, 24, 253-272, 1992.
[2]
Bernanke, Benjamin
"The Determinants of Investment: Another Look," American
S.,
Economic Review 73, 71-75,
[3]
Bertola, Giuseppe, and Ricardo
gregate Dynamics,"
Fischer, eds.
[4]
1983.
in
NBER
Cambridge,
J.
Macroeconomics Annual 1990 O.J. Blanchard and
MA: The MIT
Bertola, Giuseppe, and Ricardo
J.
ment," Review of Economic Studies
[5]
Blinder, Alan
S.,
S.
Press, 1990.
Caballero "Irreversibility and Aggregate Invest61, 1994, p. 223-246.
"Retail Inventory Behavior and Business Fluctuations," Brookings
Papers on Economic Activity
[6]
Caballero, "Kinked Adjustment Costs and Ag-
Caballero, Ricardo
J.
II,
443-505, 1981.
"Durable Goods:
An Explanation
for their
Slow Adjustment."
Journal of Political Economy 101, 351-384, 1993.
[7]
Caballero, Ricardo
J.,
and Eduardo M.R.A. Engel. "Beyond the Partial Adjustment
Model." American Economic Review, Papers and Proceedings 82, 360-364, 1992a.
[8]
Caballero, Ricardo
J.,
and Eduardo M.R.A. Engel, "Price Rigidities, Asymmetries,
and Output Fluctuations,"
[9]
Caballero, Ricardo
J.,
NBER
\VP 4091, 1992b.
and Eduardo M.R.A. Engel, "Microeconomic Adjustment Haz-
ards and Aggregate Dynamics,"
Quarterly Journal of Economics, 108-2, 697-711,
1993a.
[10]
Caballero, Ricardo
J.,
and Eduardo M.R.A. Engel, "Microeconomic Rigidities and
Aggregate Price Dynamics," European Economic Review, 37 (4), 697-711, June 1993b.
[11]
Caballero, Ricardo
in
J.,
and Eduardo M.R.A. Engel, "Explaining Investment Dynamics
US Manufacturing: A
Generalized {S,s) Approach,"
NBER
Working Paper 4887,
October 1994.
[12]
Caballero, Ricardo
J.,
Eduardo M.R.A. Engel and John Haltiwanger, "Aggregate Em-
ployment Dynamics: building from microeconomic evidence," mimeo MIT, 1993.
53
[13]
Caballero, Ricardo
Eduardo M.R.A. Engel and John Haltiwanger, "Plant-Level
J.,
Adjustment and Aggregate Investment Dynamics," Brookings Papers on Economic
Activity
[14]
2, 1-39,
1995.
A
"Business Fixed Investment Spending:
Survey of Mod-
Chirinko, Robert
S.,
elling Strategies,
Empirical Results, and Policy Implications," Journal of Economic
Critical
Literature, 31 (4), 1875-1911, 1993.
[15]
Clark, Peter K., "Tax Incentives and
Economic
[16]
Activity
1,
Equipment Investment,"
Brookings Papers on
317-339, 1993.
Cooper, Russell and John Haltiwanger, "The Aggregate Implications of Machine Replacement: Theory and Evidence," American Economic Review 83-3, 360-382, 1993.
[17]
Dixit,
Avinash, and Robert
S.
Pindyck, Investment Under Uncertainty, Princeton
University Press, 1994.
[18]
Doms, M. and T. Dunne, "An
Investigation into Capital and Labor
Adjustment
at the
Plant Level," mimeo, Center for Economic Studies, Census Bureau, 1993.
[19]
Eberly, J.C., "Adjustment of Consumers' Durables Stocks: Evidence from
Purchases," Journal of Political
[20]
Grossman, Sanford
in
J.,
Economy
102-3,
June 1994,
p.
Automobile
403-436.
and Guy Laroque, "Asset Pricing and Optimal Portfolio Choice
Econometrica, 58 (1), 25-
the Presence of Illiquid Durable Consumption Goods,"
51. 1990.
[21]
Hall,
Robert E. and Dale Jorgenson, "Tax Policy and Investment Behavior," American
Economic Review, 57, 391-414,
[22]
Hansen, Bruce
E., "Inference
1967.
When
a Nuisance Parameter
is
Not
Identified
Under the
Null Hypothesis," mimeo, Rochester, June 1993.
[23]
Harrison,
J.
Michael,
Thomas M.
Sellke
and Allison
J.
Brownian Motion," Mathematics of Operations Research,
[24]
Lam, Sam-Pok, "Permanent Income,
Liquidity,
Taylor, "Impulse Control of
8,
454-466, 1983.
and Adjustment of Automobile Stocks:
Evidence from Panel Data," Quarterly Journal of Economics,106, 203-230, 1991.
[25]
Lehmann, Erich Leo, Testing
Statistical Hypothesis,
Wiley, 1986.
54
Second Edition,
New
York: John
[26]
Lucas, Robert, Stokey, Nancy and Edward Prescott, Recursive Methods in Economic
Dynamics, Cambridge, Mass.: Harvard University Press, 1989.
[27]
Ramsey,
J., "If
Nonlinear Models Cannot Forecast,
What Use Are They?"
Studies in
Nonlinear Dynamics and Econometrics 1(2), July 1996, 65-86.
[28]
Rivers, D.
Toulouse
[29]
and Q.H. Vuong, "Model Selection Tests
Wp
for Nonlinear
Dynamic Models,"
#91-08, 1991.
Rothschild, Michael,
"On the Costs
of Adjustment," Quarterly Journal of Economics,
75, 605-622, 1971.
[30]
Vuong, Q.H., "Likelihood Ratio Tests
ses,"
Econometrica
51, 1989,
for
257-306.
55
7076
^.^\]2
Model Selection and Non-Nested Hypothe-
Date Due
MIT LIBRARIES
3 9080 01972 0967
'f!';ifi;ii!<|S»