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EXPLAINING INVESTMENT DYNAMICS
A GENERALIZED
(S,s)
Ricardo
94-32
J.
IN U.S.
MANUFACTURING:
APPROACH
Caballero
Sept. 1994
massachusetts
institute of
technology
50 memorial drive
Cambridge, mass. 02139
EXPLAINING INVESTMENT DYNAMICS
A GENERALIZED
(S,s)
Ricardo
94-32
J.
MANUFACTURING:
IN U.S.
APPROACH
Caballero
Sept. 1994
^vttlSETTSlNSTm*
•^^OFTECHNOUOeV
OCT
21«W
6
Explaining Investment Dynamics in U.S. Manufacturing:
A
Generalized (S,s) Approach
By Ricardo
Caballero and Eduardo M.R.A. Engel
J-.
1
First version: September, 1993
This version: September, 1994
In this paper
we
derive a
model of aggregate investment that builds from the lumpy
microeconomic behavior of firms facing stochastic fixed adjustment
standard (5,
s)
costs.
Instead of the
bands, firms' optimal adjustment policies are probabilistic, with a probability
of adjusting (adjustment hazard) that grows smoothly with firms' disequilibria.
upon the
Depending
specification of the distribution of fixed adjustment costs, the adjustment hazards
approach encompasses models ranging from the very non-linear (S,s) model to the linear
partial adjustment model.
Except
for the latter
ment obtained from adding up the
shocks,
is
highly non-linear.
extreme, the processes for aggregate invest-
actions of firms subject to aggregate
Estimating the aggregate model by
find clear evidence supporting non-linear
models over linear ones
manufacturing equipment and structures investment.
maximum
for
itive
its linear
likelihood,
we
postwar sectoral U.S.
For a given sequence of aggregate
shocks, the nonlinear model estimated generates brisker expansions and
— sharper contractions than
and idiosyncratic
counterpart. These features
fit
— to a
lesser extent
well the observed pos-
skewness and large kurtosis of U.S. manifacturing sectoral investment/capital
ratios.
'MIT and NBER; Universidad de Chile and NBER. We are grateful to Olivier Blanchard, Whitney
Newey, James Stock, and seminar participants at Brown, CEPR-Champoussin, Econometric Society Meeting
Harvard, IMPA (Rio), NBER EFCC, Princeton, Rochester, U. Catolica (Ing. Sist.), U. de Chile
(Centro de Economia Aplicada) and Yale for their comments. Ricardo Caballero thanks the National Science
and Sloan foundations, and Eduardo Engel FONDECYT (Chile), Grant 92/901, for financial support.
in Caracas,
Digitized by the Internet Archive
in
2011 with funding from
Boston Library Consortium
Member
Libraries
http://www.archive.org/details/explaininginvestOOcaba
Introduction
1
Characterizing the dynamic behavior of aggregate investment has not been easy. Variables that should be significant, such as cost of capital and q, seldom are; variables that
should not appear in investment equations, such as cash flows and income, often do. Moreover, estimates are typically unstable, differing across samples, estimation procedures,
researchers.
2
Not everything
is
so dismal, however, for recent work has improved the state
of affairs, mainly by refining the
measurement and instrumenting of
by incorporating credit constraints into the analysis.
Common
somewhat
to the
new and
and
old literatures
is
cost of capital, 3
and
4
a limited treatment of dynamics. This
is
surprising, for there seems to be consensus on adjustment costs playing an im-
portant role in determining aggregate investment dynamics. Indeed, investment functions
— rather than stock demands — are interesting and well defined objects only
if
frictions in
stock adjustments are significant. In spite of this consensus, adjustment costs and invest-
ment dynamics have been mostly treated
as a nuisance to deal with
when
trying to estimate
the effect of cost of capital (or other relevant variables) on capital accumulation.
example of this tradition can be found
in Hall
same
lags,
criticism applies to
tures: First, barring
future
demand and
clear
and Jorgenson's (1967) seminal paper, where
an elegant frictionless demand for capital theory
by means of a few ad hoc
A
is
transformed into an investment theory
which then turn out to play a crucial role in estimation. The
most q models. The elegance of
measurement
issues, q
is
a
q- theory
derives from two fea-
sufficient statistic for all
productivity conditions affecting the firm.
And
information about
second, the mapping
from q to investment depends exclusively on the adjustment cost function. Although the
first
of these features
is
regularly stressed, the latter
limit their study to linear or log-linear relations
now we know how
is
often disregarded and researchers
between q and investment. Of course, by
to justify these specifications, both the lags and the linear g-models; the
empirical workhorse
is
the quadratic adjustment costs model which, besides "validating"
these specifications, delivers simple smooth and linear dynamics.
5
Quadratic adjustment costs may be a useful approximation when investment
2
See Chirinko (1994) for a survey of the empirical investment literature.
(1983), and Blanchard, Rhee,
3
4
See
See
e.g.
Summers
is
just a
See Clark (1979), Bernanke
(1993) for "horse races."
Auerbach and Hassett (1992), Clark (1993).
Hubbard and Petersen (1988), Hubbard and Kashyap (1992), Bernanke, Gertler and
e.g. Fazzari,
Gilchrist (1993).
s
See Rothschild (1971) for a compelling discussion of the relevance of more general adjustment costs,
particular non-convexities, for investment dynamics.
in
part of a
more general model, but they
even more
in isolation. It is
are difficult to justify
difficult to give
when studying investment
any structural interpretation to the resulting
estimates, for their implications are at odds with the basic microeconomic investment facts.
At the plant
level, leaving aside
minor upgrades and
and lumpy rather than smooth. This
They use the Longitudinal Research
starkly
is
repairs, investment
documented
in
is
intermittent
Doms and Dunne
(1993).
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
more then half of the establishments exhibit
(i)
growth between 40 and 60 percent
a single year, and
in
plant's gross investment over the seventeen year period
we focus our attention
In this paper
(ii)
is
capital
between 25 and 40 percent of a
concentrated in a single year. 6
dynamic aspects of aggregate
entirely on the
in-
vestment. In doing so we impose the constraint that the theory must be consistent with
the basic
lumpy and intermittent nature
several methodological advantages.
of microeconomic data. These constraints yield
Among
these, they facilitate a meaningful structural
interpretation of our findings, therefore inheriting the standard advantages of structural
parameters.
7
The
results in this paper, however,
and show that modeling inaction
gregate level.
We
at the
go beyond methodological considerations
microeconomic
makes a
level
difference at the ag-
characterize the aggregate nonlinearities implied by the model and study
their role in shaping aggregate investment dynamics.
At the microeconomic
level, there
and intermittent adjustment (the
has been extensive development of models of lumpy
(5, s) literature).
8
Here we extend these models so the
adjustment trigger barriers vary randomly across firms and
modification
is
a
first
for
a firm over time.
This
step toward introducing the realistic and empirically important fea-
ture that units do not always wait for the
adjustments are not always of the same
same stock disequilibrium
size across firms
and
for the
to adjust,
and that
same firm over
time.
Recently, there have also been developments of empirical models of aggregate dynamics
6
Since plants entry
is
excluded from their sample, these statistics are
likely to represent lower
bounds on
the degree of lumpiness in plants' investment patterns.
7
See Bertola and Caballero (1990) for a similar motivation. Another important advantage of paying
when modeling aggregate adjustment, is that this facilitates integrating microeconomic and macroeconomic data for aggregate purposes, see Caballero and Engel
attention to microeconomic aspects of adjustment
(1993b), Caballero, Engel and Haltiwanger (1993), and Eberly (1994).
8
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
Pindyck (1994). A model more closely related to a special case of ours is
Grossman and Laroque's (1990) model of consumer durable purchases.
survey of the economics literature
but not lumpy
— see Dixit and
9
with heterogeneous microeconomic units adjusting intermittently. Econometric implemen-
tation of these models, however, has required observing (or estimating separately in a
In the current context, this
stage) a measure of the aggregate driving force.
first
amounts to
But undoubtedly many of the problems of the
constructing a cost of capital measure.
empirical investment literature are due to the difficulties of constructing a proper measure
of the cost of capital, a variable that despite the current efforts
simultaneity and omitted variables problems.
10
In this paper
time series procedure that- does not require the
on the investment
series itself
it
be plagued by
we implement a nonlinear
only requires information
and on the generating process of the driving force (but not
Somewhat analogously with
its realization).
stage;
first
likely to
is
adjustment costs parameters from the
(or higher) order serial correlation of invest-
first
ment, we learn about more complex and
the standard procedure of estimating convex
realistic
lumpy adjustment
cost functions
from the
structure of aggregate investment lags and their changes over time.
We
estimate nonlinear dynamic panel data models for two-digit U.S. manufacturing
investment/capital ratios for the period 1948-1992.
in favor of
tors)
all
find clear
and widespread evidence
our generalized (5, s) model over simple linear models. Perhaps the most reveal-
ing result occurs
across
We
when a
generalized (S,s) model with parameters constrained to be equal
sectors (3 parameters), outperforms a linear
AR(2) processes
(42 parameters).
Our
model with unrestricted (across
structural interpretation of these non-linearities
indicates that the fraction of firms' capital subject to fixed costs
costs are also large.
Although important
sec-
is
for both, these features are
large,
and that these
more pronounced
for
structures than equipment.
One
models
of the main mechanisms by which aggregate dynamics generated by (S,s) type
differs
from their linear counterpart,
is
that the
number
of active firms changes
9
Blinder (1981), Bar-Ilan 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 (S, s) models. Bertola and
Caballero (1990) and Caballero (1993) provide a structural empirical framework and estimate (S,s) models
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, 1993b,
1993c) 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.
10
For a discussion of some of these biases see Shapiro (1986) and Blanchard's (1986) comment on that
paper. Also see Cummins, Bassett and Bubbard (1994) for an argument relying on measurement error
and noise to focus estimation on periods where shocks are known to be large. They find that estimated
for
adjustment costs are much smaller in these periods than in non-tax credit reform periods.
may be due to the nonlinearities we describe in this paper.
finding
Part of their
— a point emphasized by
over the cycle
Doms and Dunne's
Bar-Ilan and Blinder (1992).
number
(1993) confirm the importance of this mechanism; they show that the
through their primary investment spikes, rather than the average
of plants going
size of these spikes, tracks
manufacturing investment over time. For a given sequence of aggregate
closely aggregate
shocks, the nonlinear model estimated in this paper generates brisker expansions and
a lesser extent
— sharper contractions than
its
These features
linear counterpart.
— to
fit
well
the observed positive skewness and large kurtosis of U.S. sectoral investment/capital ratios.
The next
Our main
basic
section presents the basic model. Section 3 describes the econometric method.
empirical results are presented and discussed in section
model
Section
4.
Section 5 extends the
Conclusions are presented in
to allow for flexible as well as fixed capital.
6.
The Basic Model
2
Overview
2.1
We
model a sector composed of a large but
firms.
Each firm faces an
isoelastic
demand
fixed
number
of monopollstically competitive
for its differentiated
product, which
with a Cobb-Douglas constant returns technology on labor and capital. Both
is
produced
demand and
technology are affected by multiplicative shocks described by a joint geometric random
walk process. These shocks have firm specific and sectoral (aggregate) components which
we
specify later.
The
We
work
in discrete time.
sector faces infinitely elastic supplies of labor and capital.
the latter as numeraire and
let
the
wage
(in
We
choose the price of
terms of capital) follow a geometric random
walk process, possibly correlated with demand and technology shocks. Firms can adjust
their labor input at will but suffer a loss
that this loss
is
when
resizing their stock of capital.
an increasing function of the firm's scale of operation;
11
it
We
assume
can be interpreted
either as an index of the degree of specificity of firms' capital, or as a secondary market
machines or structures are replaced, or as a reorganization cost associated
imperfection
if
with putting
new
capital to work.
For a given scale of operation, the extent of the loss due to adjustment varies over time
as firms may, for example, find better or worse matches or uses for their old machines, or
11
The
cost.
precise
meaning of
"scale of operation"
depends on the
specific interpretation of the
adjustment
may
face reorganizations of different degree of difficulty.
we model the proportional
of realism,
firms and time, and
or not to resize
As
in
its
we assume
its
loss factor as
realization
For simplicity, but at the cost
a random variable independent across
known by the
is
firm
when
it
decides whether
is
one of inaction
current stock of capital.
standard (S, s) models, the resulting microeconomic policy
As
interspersed with periods of large investment or disinvestment.
in
standard search
models, at each point in time the firm decides whether to "accept" the currently offered
adjustment cost (proportional
loss factor) or to
postpone adjustment and draw a new
adjustment cost next period. The interaction between these two mechanisms implies that,
more
realistically
firms
and over time
than
in
standard (S,
for the
same
s)
firm.
models, the size of adjustments varies both across
During a given time period, firms with identical
shortages or excesses of capital act differently; over time, the same firm reacts differently
to similar disequilibria in
its
Intuitively, the largest
stock of capital.
adjustment cost for which a firm does not adjust
capital decreases with the extent of its capital stock imbalance.
adjustment costs
is
If
its
Since the adjustment cost factors are independent across firms, and
is
large, the adjustment hazard described
investment at each point in time.
hazard. Sectoral investment
the
sum
is
the
stock of capital. 12
we assume that the
above characterizes the sectoral
Given firms' capital imbalances at the beginning of a
period, the fraction of units resizing their stock of capital
size of the
the distribution of
— a concept we describe as the firm's adjustment hazard —
increases smoothly and monotonically with the firm's disequilibrium in
of firms
stock of
non-degenerate, this implies that the probability that a firm adjusts
given the firm's disequilibrium
number
its
sum
is
determined by the adjustment
of the products of the adjustment hazard and the
investment undertaken by those firms that decide to adjust. Equivalently,
it is
of the expected investment by firms, conditional on their capital stock imbalances.
Except
for the degenerate case
where the adjustment hazard
justment model), sectoral investment depends
critically
is
constant (partial ad-
on the number of firms at each
position in the space of capital imbalances, thereby motivating our focus on the cross sectional density of disequilibria.
12
The dynamics
This should be contrasted with standard
from zero to one at the trigger points, and
(S, s)
of sectoral investment are then determined
models, where the probability of a firm adjusting jumps
adjustment models, where
also with standard linear partial
this probability is independent of the size of the firm's disequilibrium. Later we argue that these polar
extremes correspond to two degenerate cases: the standard (5, s) model to the case where the distribution
of adjustment costs collapses to a mass point, the partial adjustment model to the case where the distribution
of adjustment costs degenerates in the opposite direction (infinite variance keeping the location parameter
fixed).
by the evolution of the cross sectional density of
is
disequilibria.
The path
of this density
driven by the interaction of sectoral, firm specific, and adjustment cost shocks with the
history of shocks
and actions contained
2.2
The firm
When
the firm
is
not investing,
where
K
is
flow of profits
the firm's stock of capital; 9
/?
and
T)
=
(a(77— 1))/(1 +a(r)— 1)) with
the elasticity of
demand
monopolist solution to be
do
and
r
a the
elasticity of
which implies that
(3
this
by defining the
rates;
production with respect to capital
faced by the monopolist.
interior;
and depreciation
6 are the discount
<
As
1,
is
standard, n
since
useful to replace 9 in the profit function by a variable with
It is
We
a shock to the profit function which combines
is
demand, productivity and wage shocks; 13
and
is:
K O-(r + 6)K,
U(K,O) =
-
(1)
its
in previous cross sectional densities of disequilibria.
>
1
for the
we assume that a <
1.
more economic content.
frictionless stock of capital of the firm,
K*, as the solution of
the maximization of (1) with respect to capital, so that:
9
where £
=
(r
+
= tir^-v,
6)/P- Substituting this expression into (1), and defining the disequilibrium
variable
z
=
In
K/K*,
allows us to rewrite the profit function as:
U(z, A")
(2)
Figure
1
illustrates,
and equation
=
tt(z)K'
=
z
f
{f
-
z
(Se )
K\
(2) implicitly defines, profits
per unit of frictionless cap-
ital, tt(z).
At times of adjustment, the firm incurs
profits
13
due to reorganization. 14 Assuming
in an
adjustment cost proportional to foregone
this cost corresponds to the
opportunity cost
For convenience, we have written the profit function net of flow payment on capital, (r+6)K. Since there
are neither borrowing constraints nor bankruptcy options, the solution to the firm's problem
by replacing flow payments for a lump sum payment at the time of purchase.
14
See e.g. Cooper and Haltiwanger (1993).
is
unchanged
CD
D
3
9
of installed capital during reorganization, a derivation similar to the one that led to (2)
implies that:
where
u is the realization
of a positive
immediately before adjustment.
15
K
=
Adjustment Cost
random
6
=
variable
u£e 0z ~K',
and z~ denotes the capital imbalance
Realizations of u; are generated by a
common
distribution
function, G(ui), and are independent across firms and over time. 16
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
be characterized
firm's
in
lumpy investment. Therefore, the
terms of two regimes: action and inaction. Finding a solution to the
problem means finding the function separating these two regions
characterizing the actions taken by the firm
region.
The
We
problem can
firm's
when
in (w,z)-space,
and
crossing from the inactive to the active
turn to this next.
value of a firm with disequilibrium
adjustment cost parameter
u>,
2, frictionless
V'(z, K*,u),
does not adjust, V(z,K*), and the value
the
is
if it
stock of capital
maximum
K
m
,
and (current)
of the value of the firm
does adjust, V(c, K*)
— u^e^'K*; where
if it
c is
the optimally determined return point (see below). In short:
(3)
The
V'{z ,IC ,u
t
t
t
)
= max{V(z K' ) V{c,K' - u ie^K' }.
t ,
,
t
t
)
t
t
evolution of the value of a firm that does not adjust in the current period
is
described
by:
(4)
V(zt ,K;) = z(zt )K; +
(1
+
r)-
1
Et [V(zt+1 ,AT+1 ,u;t + i)].
Since the profit and adjustment costs functions are homogeneous of degree one with respect
to K*, given 2, so are the value functions
V(z,K*) and V*(z, K*,u). This
allows us to
reduce the number of state variables by restating the problem in terms of the value per unit
of frictionless capital. Let v(z)
= V(z,K*)/K m
and v*(z,u)
= V'(z,K*,oj)/K\
Dividing
15
Of course many other specifications of lumpy adjustment costs are possible. For example, these could
be proportional to the frictionless or new capital rather than the old one. The former specification would
simplify the problem since it would make the upgrading and downgrading decisions symmetric; the latter
specification, on the other hand, would complicate the problem somewhat since the return points from
upgrading and downgrading would be different. None of these modifications would change anything fundamental in what follows, however.
16
We take this as a first step toward a more realistic formulation where at the individual level adjustment
costs exhibit some persistence and, at any point in time, the distribution of adjustment costs depends on
aggregate conditions.
both sides of equations (3) and (4) by
K* and
noting that
k:t+i
ti = (l-*)e- A *-H,
K'
l
t
yields
(5)
v'(z ,ut)
=
majt{«(2 t ),r(c)-^ete },
(6)
v(z t )
=
w(zt )
t
with
i/}
=
(1
—
S)/(\
+ r).
+
z
[v'(z t+u u t+1 )e-* '+>],
Tl>E t
Before going further, we use figure 2 to illustrate the basic setup
This figure shows how v(z), v(c)
developed up to now.
— w£e^ z and v*(z,u) determine
,
The
the trigger points, given a particular realization of the adjustment cost.
illustrates the value of
dashed
a firm that does not adjust
line represents the value of
The maximum between both
ijj
v(c)
— u>£e@ z
a firm that decides to adjust, given a realization of u.
lines describes u*(z,w),
obtained at z
We
=
let il(z)
and that
c
this return point
c,
is
that the
maximum
independent of the
z.
From the
equates the two terms on the right hand side of equation
Q(z)
(7)
solution to the firm's problem
=
r
l
e- p*(v(c)
is
easily obtained.
can be obtained by substituting equation (5) into
of v(z) and v*(z,u)
-
v(zt )
=
ir(z t )
+ VE
t
[e-
Ai '+'
u>t+i-
Using the definition of
it
ad-
value matching condition that
(5), it follows that:
v(z)).
The Bellman equation
is
known, the
for this function
(6):
max{t;(zt+1 )
Q.(z)
is
initial disequilibrium.
,
v(c)
- u t+l
^ ^}}
z
where E t denotes the expectation, based on information available at time
Azt+i and
a given
lines.
should be apparent from the formulae and figure 2 that once v(z)
(8)
for
denote the largest adjustment cost factor for which the firm finds
vantageous to adjust given a capital imbalance
It
—
and the inaction range
from maximization of the value of a firm that decides to adjust,
with respect to the return point
,
The concave
in the current period.
— corresponds to the interval between the intersection of the two
It follows directly
solid line
t,
,
with respect to
and calculating the expectation with respect
CM
CD
N
£-
O
CD
i
WO
29"0
09'0
8S*0
99*0
suo^oun^ 8n|eA
*S*0
2S*0
09'0
u
to
t
+i leads to:
=
v(z)
(9)
*{z)
+
V'E
e"
A*
+ Az) +
v(z
{
^
r Q(z+Az)
(2+Az)
"I
G(w) dw
/
!
where G(w) denotes the cumulative distribution function of the adjustment cost
the expectation
taken over Az.
is
The
first
factor,
and
term inside the expectation can be interpreted
as tomorrow's value of the firm for those realizations of the adjustment cost factor
and
shocks to z that lead to no investment, weighted by the probability of no adjustment for
the corresponding Az; the second term corresponds to the probability weighted value of
those situations where the firm adjusts
The
value function
when not
its
stock of capital.
adjusting, v(z), can be found by solving the functional
equation that results from replacing Q(z) from (7) in
v'(c)
=
0.
we can replace
Alternatively,
(9),
and using the
first
order condition
v(z) and v(c) from (9) in (7), in order to directly
find a functional equation for the policy function fl(z).
After a few algebraic steps, the
latter strategy yields:
=
ft(z)
if>E
The optimal
e
ril(c+Az)
-(l-0)Az UQ(z+Az)
policy ft(z)
is
condition with respect to
]
the solution to this functional equation subject to the
is
v'(c)
=
=
0,
obtained from differentiating (7) with respect to
pression at z
=
c,
order
c:
fi'(c)
which
first
+
using the fact that fi(c)
=
0,
z,
evaluating the resulting ex-
and imposing the
first
order condition
0.
Figure 3a illustrates the function fi(z) for an example where the distribution of adjust-
ment
costs
is
exponential: dG(u>)
firm's disequilibrium
costs.
From then
is
close
= (l/\)e~ u / x dw. As
enough to z
=
\z
shortages) than to the right (excess of capital). This
since depreciation
is
is strictly
thus there
is less
— c\, more
sharply to the
asymmetry has
the
less likely to reverse
left (capital
several sources: First,
positive (and possibly net productivity
on average positive), shortages are
if
adjust for arbitrarily small adjustment
c, it will
on, fi(z) increases with respect to
follows from equation (7),
and demand growth
by themselves than excesses,
value in delaying upgrading than downgrading. Second, by making the
Figure 3a
Figure 3b
S
m
^r
^^
CM
3
33
^^^^
3
o
k
^^^^
^^
7
CM
6.00
0.04
0.08
0.12
0.16
u
0.20
0.24
0.28
0.32
a given disequilibrium) we have
adjustment cost an increasing function of old capital
(for
made
K* and
the total adjustment cost asymmetric: for given
from the right than from the
latter case.
z
of
=
17
left
of c because old capital
Third, the profit function
maximum
c than to the left of its
asymmetry outweigh the
is
asymmetric;
c,
L(u>)
it is
more
costly to adjust
is
larger in the former than in the
it
decreases faster to the right of
(see figure 1). In our example, the first
two sources
last one.
Figure 3b depicts the inverse of the function
low and above
u,
and U(u),
respectively.
il(z).
We label the segments of the curve be-
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
That
u>.
The area enclosed by
for
any fixed w, they describe a standard (L,c,U) policy. 18
the two curves corresponds to the combinations of disequilibria and
adjustment cost factors
The shape and
is,
for
which the firm chooses not to adjust.
location of the function fi(z) and
on the entire distribution of adjustment cost
adjustment cost factor
functions G(uj).
will
(L(u),c, U(uj)), depend
its inverse,
A
factors, G(u>).
given realization of the
not generate the same inaction range for different distribution
In particular, a low value of
u> is
more
likely to lead to action
when
it
comes from a distribution of adjustment cost factors with a high rather than a low average
value.
We
in the
next section, when we characterize aggregate investment and
discuss the relation between
G(u) and microeconomic adjustment
in
more
detail
connection with the
its
underlying distribution of adjustment cost factors.
Sectoral investment
2.3
2.3.1
Let x
The adjustment hazard and
=
z
—
c
cross sectional distribution
denote a firm's imbalance with respect to
frictionless stock of capital. Also let
Kf,
//*,
its
target point rather than its
Kt(x) and It(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).
Those firms with deviation x whose current adjustment cost
adjusting profitable
(i.e.
for
which
u <
fi(x
+
17
c)) adjust. Since
is
small enough to
adjustment costs are
This "irreversible investment" feature of the adjustment cost function seems
support to our choice of adjustment cost specification.
18
An (L, c, U) policy corresponds to a two-sided (5, s) model. The notation L,
bound, upper bound, and "center," respectively. See Harrison et al. (1983).
10
realistic,
U
and
c
make
i.i.d.
lending further
stands for lower
and the number of firms
approximately equal
is
large, the fraction of firms with deviation x that adjusts
to:
A(x)
(10)
=
G(Sl(x
+ c)),
where G(u>) denotes the cumulative distribution function
For example,
u.
if
adjustment hazard
G(uj)
is
Gamma
a
distribution with
for the
mean
=
—L—
1
is
2
the
stock of capital and,
if it
p</>
uf-^e-^+du.
aggregate investment
so, its contribution to
investment
and variance
,
p<f>
fO(x+c)
/
firm with capital stock x has a probability A(i) of adjusting
does
adjustment cost factor
is:
A(x)
A
is
its
equal to (e~ x
is
— \)K
t
{x).
Thus aggregate
given by:
l*
=
e
~*
_
J(
l)A( x )K t (x)r(x,t)dx,
where Kt{x) denotes the average stock of capital of firms with imbalance x and f(x,i) the
cross sectional density of disequilibria just before adjustments take place.
Assuming that the average
independent of
we have
x,
capital stock of firms with disequilibrium x
that:
Equation (11)
is
approximately
19
±t
(11)
is
= J(e-*-l)\(x)f(x,t)dx.
the fundamental aggregate investment/capital ratio expression;
how, conditional on the cross-section of
disequilibria, current investment
is
it
shows
determined
by the adjustment hazard function. Figure 4a shows the adjustment hazard function
the three different
gamma
distributions of adjustment cost factors depicted in figure 4b.
These distributions have the same median but
to
p =
1
for
(exponential distribution) and
<f>
different variances: the solid line corresponds
=
0.1, the
long dashes correspond to a high
variance distribution, while the short dashes describes a low variance distribution. Figure
4a shows that as the variance increases the hazard
shifts
from an (5, s) policy, to a smoothly
increasing hazard, and eventually approaches a constant probability of adjustment/partial
adjustment model. 20 The connection between these results and the height of the density of
K
19
This independence assumption can be motivated by the fact that I is a stationary variable while
not (because K' is not). Provided firms have been in the industry for a long time, knowing a firm's x
conveys no information about its level of capital.
is
20
Strictly speaking,
we obtain the
partial
adjustment model only
11
if
c~*
—
1
ss
—x
for
most relevant
i's.
Figure 4a: Adjustment hazards
'''
**
'»
*
r
\
CO
o
'
(D
a
1
>W
d
'
"*
1^
1
N»
(D
X
d
X.
d
/
/
v
in
/
/
\l
"'
1
I
d
o
a
{
/
i%
/•
/
(M
\
M
O
/
/
d
a
\
w
\
—- «xpon«n
— Nghv«r
vr
\
y
-"
|
i
low
.
.
«UJ
-0.8
-0.8
-0.4
-0.2
-0.0
0.2
0.4
0.6
0.8
1.0
Z
Figure 4b: Adjustment costs
a
d
a
a
jo
O
I
a
d
- - Km vw
a
a
too
—
0.04
0.08
0.12
0.16
0.20
0.24
hiQh vtr
0.28
0.32
the adjustment cost around zero
is
apparent.
If
the density around zero
is
small, there
is
a range where the firm almost never adjusts since adjustment costs are almost never small
— case an extreme version of
Conversely, when the density around zero
large — as
frequently the case when
the variance grows keeping the median constant — the decision of adjustment
largely
enough to
justify
the standard (S,s)
it;
— or
this.
(L,c,U)
is
is
is
is
motivated by the adjustment cost draw rather than by the firm's disequilibrium. The limit
when adjustment becomes independent
of this,
partial adjustment
of the disequilibrium, corresponds to the
model.—
Sectoral equilibrium and cross sectional dynamics
2.3.2
Shocks to wages, demand, and productivity drive the dynamics of
decompose these shocks into
sectoral shocks, v t
,
and firm
frictionless capital.
We
specific (idiosyncratic) shocks, ( t
:
which implies that when the firm does not adjust, the disequilibrium measure x evolves
according
to:
Ai| = -v +
t
We
In (1
-
6)
assume that these shocks are exogenous
-
e t as
-(6
to the firm
+
vt )
-
and the
et .
sector.
21
Between two consecutive periods, the cross section distribution of disequilibria changes
as a result of firms' adjustments, depreciation, sectoral shocks,
Since
we are working
each period.
We
in discrete time,
it is
and idiosyncratic shocks.
important to describe the timing of events within
denote the cross section density at the end of period
Depreciation and the aggregate shock corresponding to period
t
t
—
t
concludes with the idiosyncratic shocks. The
starts again.
by f{x,t
—
1).
follow, resulting in the
density f(x,t). Adjustments, as determined by the hazard function A(x)
period
1
come
final density is f(x,t),
next, and
and the cycle
Recalling that a positive shock leads to a decrease in x, we can summarize
21
The former assumption needs no explanation while the latter requires additional restrictions in the
underlying monopolistic competitive model in order to eliminate the sectoral price from the demand curve
faced by the firm. This requires precise offsetting of the income and substitution effects induced by sectoral
price changes.
of such a model can be found in Caballero and Engel (1993a). In any event, we
more general) equilibrium setup as a limitation of the current approach.
An example
view the lack of a
full (or
12
this chain of events as follows:
(12)
f(x,t)
=
(13)
f(x,t)
=
where g t (e)
is
f{x
+ 6 + vu t-l),
A(y)f(y,t)dy\
[J
+ J(l-A(x +
gi (-x)
())f(x
the probability density for the idiosyncratic shocks.
+
e,t)g c (-()dt,
The
integro-difference
equation describing the evolution of the cross sectional distribution from one period to the
next follows directly from equations (12) and (13):
=
/(*,*)
(14)
[jA(y)f(y
+
Combining equations
J(\
(11), (12)
+S+
- A(x +
e))f(x
and (14) we can
any given sequence of aggregate shocks {v
We
v t ,t-l)dy gc (-x)
t
}
+e+6+
v t ,t- l)5c(-0 de.
fully characterize
and
aggregate investment, for
initial cross section distribution
/(i,0).
turn to estimation issues next.
3
Econometrics
Overview
3.1
The econometric problem
consists of estimating the parameters of: (a) firms' profit func-
tions, (b) the initial distribution of disequilibria, (c) the distribution of
(d) the distribution of idiosyncratic shocks,
The data
We
(e) the
process generating aggregate shocks.
available are sectoral (aggregate) investment data.
assume that the stochastic nature of the problem (the "error term")
aggregate shocks.
would be
case
and
adjustment costs,
perfect.
we have
vector of
all
If
22
to posit
these shocks were
known
exactly, the
The aggregate shocks may
some underlying time
for the "true"
due to
parameters
either be totally unobserved, in which
series process
parameters in the model; or they
fit
is
may be
captured by a subvector of 0, the
related to observables via an auxiliary
economic model or long run relation (both of which include residual error terms). Even
in
a straightforward manner, and
so later in the paper, to simplify the exposition here
we consider the case where the
though the approach we describe next can be generalized
we do
"Conditional on also knowing the initial cross sectional distribution. We assume there is no measurement
was the only source of error allowed for in previous analyses (see footnote 9). Unfortunately,
integrating both approaches is computationally too burdensome.
error in yt, which
13
aggregate shock
is
unobserved and
i.i.d.
The observed investment data may now be viewed
if/Kf by
of the unobserved aggregate shocks. Denoting
(15)
=
yt
T
where
as a highly nonlinear transformation
yt(v 1 ,...,v t ;Q);
yt
we have
,
t
=
that:
l,...,T,
denotes the number of time periods considered.
Whatever estimation
we compute the
criterion
we
use,
23
it
will require that, for
a given set of parameters,
shocks that correspond to the observed sectoral investment
set of aggregate
data (and some functions of these shocks). This amounts to finding the inverse functions
of those described in (15): 24
vt
(16)
The
in period
t
-
1
summarized
vt is
vt
=
vt (f(;t
recursive nature of this problem
calculated without knowing
summarized
all
-
=
l,...,T.
in the cross-section density after adjust-
t=l,...,T.
l),yt );
clear.
is
The aggregate shock
at time
t
cannot be
the preceding shocks. Yet the effect of preceding shocks
is
in the cross sectional density of firms' deviations before the current shock, as
can be seen from replacing f(x,t) by f(x,t
(18)
It
t
takes place, so that:
(17)
The
vt(v 1 ,...,v t -i,y t ;0);
previous shocks on
effect of
ment
=
= J(e~ x+S+V '
yt
follows that the computational
—
1) in (11)
l)A(i - 6
-
and using
v )f(x,t
t
-
(12):
l)dx.
problem of calculating the v defined
t
in (16)
corresponds
to the following recursive problem: 25
•
23
Find
Without
v\
by solving (18) with f(x,t -
loss of generality as far as the
1)
replaced by the initial density (determined
methodology we descibe next
is
concerned,
we consider maximum
likelihood estimation.
24
And making
2s
To ensure uniqueness when
sure that this inverse
is
uniquely determined, which we do shortly.
show that the right hand side of this equation
solving (18), we
function of v for any density f(x,t
—
1).
This
is
equivalent to showing that 4>(u)A(—u)
is
is
an increasing
increasing in u,
in our case <j>(it) — e — 1, but more generally, 4>(u) could be any increasing function with 0(0) = 0.
straightforward calculation shows that a sufficient condition for <j>(u)A(—u) to be increasing in u is that
A(u) be decreasing (A'(u) < 0) for negative values of u and increasing (A'(ti) > 0) for positive values of a.
where
u
A
All the adjustment hazards used in the estimation section satisfy this condition.
14
by the subvector of
that characterizes this distribution).
•
Determine f(x,
•
Find V2 by solving (18) with f(x,t
•
Determine /(x,2) by using equation
•
Find U3 by solving (18) with f(x,2)
1)
by using equation
(14).
-
1)
replaced by /(i,l).
(14).
in
the place of f(x,t
—
1).
This problem entails a highly nonlinear but structured relation between the data and
aggregate shocks. At each point in time, given the distribution of disequilibria, there
nonlinear relation between the current aggregate shock, v t
,
and the observation, y
t
is
a
The
.
current shock also enters nonlinearly in the equation determining the cross sectional distri-
bution of disequilibria relevant for next period's investment decisions; thus, establishing a
nonlinear relation between observations and lagged aggregate shocks.
For given parameters, the sequence of aggregate shocks and the derivatives of aggregate
investment with respect to these shocks (the inverse Jacobian), which are the inputs to
the likelihood function, can in principle be obtained from the combination of a routine
to solve the nonlinear static equation relating current shocks to current observations, and
a stochastic Markov chain designed to track down the evolution of the cross sectional
In practice, however, this approach
distribution.
Markov chain
grid fine
enough
to prevent
is
computationally burdensome for any
a host of numerical problems from arising.
Obviously, computational considerations are particularly important
considered are long or
when multiple time
in the paper. In order to
cope with
a family of hazard functions which
model and,
at the
this
is
when the time
series are analyzed simultaneously, as
problem, in the estimation section we
series
we do
first
later
consider
general enough to test the basic implications of our
same time, simple enough so that an accurate approximation method
that reduces the estimation time by several orders of magnitude can be designed. Next
consider a
for
more general family of hazard
functions, which provides
adjustment hazards obtained via value iteration
This family
is
for
used to estimate structural parameters.
15
we
good approximations
a wide range of parameter values.
A
3.2
simple family of hazards
Estimating a structural model
computationaly burdensome. Thus,
is
by estimating faster approximations which may shed
start
light
it
seems sensible to
on the potential success of
the proper structural model.
The algorithm
if
we work with a
number
for
computing the aggregate shocks described above
(sufficiently rich) family of
is
considerably faster
adjustment hazards, characterized by a small
of parameters, for which the cross-section densities remain within the family.
makes finding such a family of
undergo three very
densities difficult
is
that the cross section distributions
different transformations
from one period to the next, two of which
new
period, the aggregate shock shifts the cross
are non-trivial. At the beginning of the
section; this
is
What
followed by firms' adjustments (the hazard shock); and the period concludes
with the convolution of the distribution resulting from the previous transformations with
the distribution of idiosyncratic shocks.
The family
It is
easy to see that this family
as long as
we assume that the
remains within
is
of cross sectional densities
is
we use
is
the family of mixture of normal densities.
closed under aggregate shocks and idiosyncratic shocks,
latter are also normal.
this family after the
To ensure that the
cross section
hazard shock, we assume that the adjustment hazard
an inverted normal:
A(x)
(19)
where A >
We
will
and A 2 >
0.
= \-e- x °- X2X \
26
show that tracking the evolution of the
cross sectional distribution reduces to
keeping track of the mean, variance, and weights of a
finite
number
parameters can be obtained from a simple recursive structure.
of Normals.
These
Furthermore, the static
nonlinear equation relating the contemporaneous aggregate shock to the current observation
and cross sectional density
is
also
of (18) for different values of v t
In
is
easier to solve, since evaluating the right
We
focus on a
maximum
issues
is
likelihood procedure
important
and assume that the aggregate shock
fi v
and standard deviation
for the substantive issues discussed below.
modifications, the procedure can be adapted to
'It is
side
and procedure with a simple univariate ex-
unobservable and distributed Normal with mean
of these assumptions
hand
considerably faster.
what follows we describe the basic
ample.
is
much
straightforward to add a linear term,
— X\x,
more robust (but
to the exponent.
16
er„.
None
With minor
less efficient)
estimation
procedures
(e.g.
GMM),
to other distributional assumptions about aggregate shocks, to
partial observability of the aggregate shock,
We
later).
and to a multivariate context (which we do
assume that idiosyncratic shocks are normal with zero mean and variance a\
and, for simplicity, proceed conditionally on the
this section
The
(v\
.
,
.
.
,
how we
discuss
basic algorithm
vt) to
(j/i
,
...
,
is
deal with initial conditions.
conceptually simple.
is
A
change of variable calculation relating
yx) shows that minus the log-likelihood
-/(0| yi ,..., yT ) = const +
(20)
which
we
£lngi
dvt
t
t
section density before the aggregate shock, current investment
The exact shape
and
(16).
The
first
term
Conditional on the cross
is
a non-linear, increasing
of this non-linearity depends on
Thus dyt/dvt not only
the cross sectional density at the time of the shock.
,
to:
*]*fc<*Z]£)
+
to be minimized over 0, with y t and v t defined in (15)
function of the current aggregate shock.
equal
is
captures the rich dynamics of the model under consideration.
vt
the end of
initial cross section density; at
varies with
but also over time. This should be contrasted with a partial adjustment model or an
ARIMA
model, where dyt/dv does not depend on the current shock and remains constant
t
over time.
Next we show that assuming that the cross section density
is
a convex combination
of normal densities, and the hazard an inverted normal, simplifies the calculation of the
likelihood.
Let us begin by considering the simplest possible case, where the cross section density at
time
t is
normal with mean
\i
and variance a 2
based upon (18), shows that finding the current shock
consideration)
(21)
is
equivalent to solving for v t
yt
=
e
c
^-
1
.
A
(for the
in:
- l e d{x" )+ £ c(v '\
where
2
(22)
1
(23)
c(v)
=
(24)
d(v)
=
simple but tedious calculation,
+
2X 2 a 2
'
v-» + ±a\
r2
Xo+.-Xiiv-nf
a*
17
parameter vector
under
The corresponding
derivative needed in the
pi =
Once v
t
is
e*»)
+
4
e_d(U,)
term of the log-likelihood
first
" 2A
^
~
eC(V,)
M)4
[
" *~
equal
is
to:
l
found by solving (21), equation (14) can be used to show that the cross section
density after the f-th period's aggregate, hazard and idiosyncratic shocks will be a convex
combination of two normal densities, one of them with mean
variance r
2
-f
of
,
77
and the other with zero mean and variance
=
a\.
(/z
—
v t )/(l
+
2A2<7
2
)
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
equal
is
and
The
to:
5In the
normal
more general
densities, v t
is
combination of terms
where f(x,t)
=
a kfk( x ,t)
J2k
is
a convex combination of
obtained by solving an equation analogous to (21) with a linear
like the
Vt
The
case,
one on the right hand side of that equation:
= ^2 a k
l(S
+v t
x)A(x -
6
-
v t )fk (x, t)dx.
evolution of the cross- section density can be tracked by considering a linear combination
of expressions like those considered above. For example, in the case where f(x,t)
to the convex combination of Nt normal densities and A(x)
a convex combination of Nt
+
1
normal
densities.
is
given by (19), f(x,t
Each of these corresponds to a
cohort, grouped according to the last time they adjusted.
more spread out than the "younger" ones and have
lost
is
The
equal
+
1) is
specific
"older" distributions are
mass monotonically due to the
adjustment of their members. In order to keep the number of normal densities considered
manageable, we reduce the number of densities tracked down at each point
in
time by one,
by merging the two older cohorts.
Finally,
is,
in
we need
to determine the initial cross sectional density.
The ergodic
density
a precise sense, the best guess for this density (see Caballero and Engel, 1992b).
disregard the
first
To observations when calculating the likelihood, thus allowing for the
possible distortions introduced by this approximation to
We
We
wash away.
approximate the ergodic density by a convex combination of normal densities as
18
follows. Let /o denote the density of idiosyncratic shocks,
27
from those firms that do not adjust
one
idiosyncratic shocks.
28
after /g is subject to
and ff the density that
results
and
set of aggregate, hazard,
Let /| denote the density that results after applying an aggregate,
hazard, and idiosyncratic shock to /f, and so on. Standard Markov chain arguments show
that the ergodic density has the form:
where the
number
<^t's
correspond to the stationary distribution of the Markov chain describing the
of periods since a firm last adjusted. Since the
</>jt's
decrease at a geometric rate,
a relatively small number of densities usually provides a good approximation.
depicts the corresponding weighted densities and the resulting ergodic density.
Figure 5
The
largest
density corresponds to the density of idiosyncratic shocks, the density that follows to the
left is
the density of those that do not adjust after a sequence of aggregate, idiosyncratic
and hazard shocks, and so on.
A
3.3
general family of hazards
Experimentation with a variety of distributions of adjustment costs shows that the family
of continuous, piecewise inverted normal hazards approximates well the hazard functions
Three pieces
obtained via value iteration.
suffice for
middle piece corresponding to a hazard that
member
is
most practical purposes, with the
identically equal to zero.
of this four-parameter family of adjustment hazards
l-e- A "( r - x r ifx<
A(x)
(25)
=
i
!
we
In this case
_ e -A+ (*-*+)»
is
A
representative
of the form:
x -,
if
x~ < x < «,+
x
if
x>1 +
track the evolution of the cross section density and solve the non-linear
We
equations that provide the v^s by working with a flexible discretization in i-space.
describe this discretization procedure next.
Assume
(ii,Wi),
i
the cross section at the beginning of period
=
l,...,n, where
i,-
t
—
1 is
a collection of mass points:
denotes where the i-th mass point
27
is
located and
Here idiosyncratic shocks are as faced by an individual firm. Thus the corresponding variance
+ <t 2v See Caballero and Engel (1992b) for details.
28
The size of the aggregate shock is equal to the mean of this process.
to a\
.
19
u>,
is
the
equal
Pl&utLE
0.035
0.025-
5~
corresponding weight (£j Wi
=
The aggregate shock
1).
discrete counterpart of equation (18).
Having found
proceeds as follows: After the aggregate shock
is
of
characterized by (i,
—v
t
,Wi).
hits,
and variance.
The
Even though
preceding subsection,
at zero arises.
functions, thereby
ment
is
this
idiosyncratic shock follows,
discretized
by evaluating the cross
provides a practical
making
again.
considerably slower than the one described in the
way of approximating a
structural estimation possible.
costs, the corresponding "exact" hazard
approximate
is
The
n points so that the cycle described above begins
methodology
it
the evolution of the cross section
vt,
whose width and location are determined by the current mean
grid has
this
obtained by solving the
the resulting collection of mass points
leading to a continuous distribution. This distribution
29
is
Next follows the hazard shock, which increases the number
mass points by one, since a mass point
section density on a grid
vt
is
rich family of
hazard
Given a distribution of adjust-
We
obtained via value iteration.
hazard by the "closest" hazard within the family described by
then
(25).
We
use this hazard in the iterative procedure described above, which calculates the sequence
of aggregate shocks corresponding to particular parameter values.
4
4.1
Empirical Evidence: U.S. Investment
Overview and Data
Our data
are constructed from annual gross investment and capital series for 21 two-digit
manufacturing industries from 1947 to 1992. 30 All
capital correspond to the series used by the
studies.
31
series are in
Bureau of Labor
1987 dollars, and the stock of
Statistics for their productivity
Since capital stocks are end-of-year, our measures of the investment/capital ratio
used in estimation start in 1948.
We
report separate results for equipment and structures
panels; each has 945 observations.
We
begin this section by estimating simple adjustment hazard functions for sectoral
(aggregate) U.S. manufacturing equipment and structures investment, without imposing
the tight theoretical restrictions of the microeconomic model developed in section
this semi-structural approach,
we learn that there
is
2.
From
clear trace in aggregate data of microe-
conomic adjustment hazard functions that are increasing rather than constant with respect
In the estimation section we use 33 equally spaced points between fi — 4<r and ft + 4<r, where fi and a
denote the mean and standard deviation of the density being approximated.
30
We have 21 rather than 20 sector because Motor Vehicles is separated from Transportation equipment.
31
This is one of the three capital stock series reported by the Bureau of Economic Activity.
20
to firms' disequilibria.
We
average adjustment cost
and 59 percent
then proceed to estimate structural models, and find that the
(in
terms of annual revenues)
for structures,
percent, respectively.
is
about 16 percent
for
and that the standard deviation of these costs
Our estimate
equipment
is
6 and 8
of the standard deviation of idiosyncratic (within each
sector) shocks ranges from 5 to 12 percent
and
is
quite stable across investment categories
and estimation methods (structural and semi-structural).
Results: Semi-structural hazard
4.2
Using the panels described above, we estimate the following "inverted-normal" hazard:
A(z)
(26)
= l-e- A °- A2X\
which has the potential to capture the increasing nature of the adjustment hazards depicted
in section 3,
and converges to the partial adjustment model as A2 approaches zero (together
with the approximation (e~ x
—
1)
a
—x).
We
restrict the
hazard to be the same across
sectors within each panel, but allow for different means, variances
of sectoral shocks.
We
and
also allow for a general variance-covariance
first
order correlations
matrix across sectors'
innovations.
For each category (equipment and structures), we solve at each point in time 21 static
nonlinear equations (one for each sector) to find the (sectoral) aggregate shocks; with these
on hand, we use 21 dynamic equations to generate new sectoral cross sectional distribuAfter repeating these steps for
tions.
coefficients, the
observations,
we compute the
serial correlation
corresponding sectoral innovations, the variance-covariance matrix of
toral innovations,
sec-
and the derivatives of aggregate investment with respect to the current
shock. Using these results,
We
all
we maximize the multivariate
likelihood over Ao, A2
and
<r £
32
.
constrain the initial distributions of disequilibria to be the sectoral ergodic distri-
butions corresponding to the parameters being estimated. In order to reduce the effect of
the approximation introduced by this criterion,
we exclude the
from the joint likelihood. Finally, at each point
for each sector
in
first
3 shocks of each sector
time we keep track of 20 distributions
and merge the distributions of those that have spent more than 20 years
without adjusting. 33
32
I.e.,
we concentrate out the variance-covariance matrix and
"Remember
2, 3,...,
serial correlation coefficients.
that each distribution represents a cohort of the firms that have not adjusted for the last
20 (or more) periods.
21
1,
Table
1
The two major
displays the results.
and structures. Within each of
these, the first
subdivisions correspond to equipment
column shows the estimates
for
a partial
adjustment model, the second column reports the results for the inverted-normal hazard
in equation (26),
and the third column shows the estimates obtained with unconstrained
(across sectors) second-order autoregressions for every series.
For equipment, the
first
column shows a
from zero and implies a partial adjustment
coefficient Ao
34
which
coefficient of 0.44.
is
35 36
'
significantly different
The mean
order
first
0.47.
The
second column contains the main results of this section. These clearly indicate the
rele-
serial correlation of
the seetoral shocks
is
0.19, with a range
between —0.03 and
vance of considering non-constant hazards: The likelihood function exhibits a substantial
improvement when A2
is
allowed to be positive, 37 while the linear term, Ao, becomes indis-
tinguishable from zero (as implied by our model, see equations (7) and (10)). 38 At the
time, the average serial correlation of sectoral shocks
—0.13 to
0.23).
Comparing these
is
(on average) reduced
results with those in the third
(it
same
ranges from
column, reveals that the
likelihood obtained with unconstrained second-order autoregressions leads to a likelihood
that
is
If
ally,
substantially lower than that of the constrained (across sectors) non-linear model.
we take
the setup
we used
in the section describing the
then we should not allow for
Interestingly, if
first
we re-estimate the increasing hazard model
partial adjustment
model with unconstrained
is still
setting the 21
er t ),
common
first
order se-
substantially better than in the
serial correlation (2,438 versus 2,405)
the non-structural second order-autoregressions model (2,438 versus 2,434).
parameter (A2 and
liter-
order correlations in the increasing hazard model.
correlation coefficients to zero, the likelihood
rial
microeconomic model
Two
and
in
nonlinear
across sectors, do better than 21 and 42 free sectoral serial
correlation coefficients.
34
And contemporaneous correlation between innovations, as in the preceding columns.
The partial adjustment coefficient is equal to 1 — e -A °.
36
The likelihood has a second maximum which is slightly better than the one reported
3S
here (likelihood
but which corresponds to an implausible partial adjustment coefficient close to one and
very large unexplained serial correlation of aggregate shocks. Since the third column has a substantially
improvement of
2.7)
better likelihood than either one of these maxima, and the underlying model
of a partial adjustment with adjustment parameter equal to one and
the structurally meaningful estimates in the case of the
first
may be
interpreted as that
AR(2) innovations, we concentrate on
column.
2
not identified in column 1, we cannot formally use the standard \ test to assess the
improvement in the fit as we allow for A2 > 0. This is the "nuisance" parameter problem discussed in, e.g.,
Since
<r t
is
Hansen (1993). This
is
a second order problem in our case, however, where the likelihood differences are
large.
38
Strictly speaking, calling Ao the "linear"
(e~
in
x
—
what
1)
a —x
in equation (11). Since
term is appropriate only in conjunction with the approximation
our main conclusions do not depend on this caveat, we disregard it
follows.
22
Table
Equipment
Parameters
P.A.M.
Ao
A2
Semi-structural results
1:
Incr.
Haz.
Structures
AR(2)
P.A.M.
Incr. Haz.
0.582*
0.000
0.501*
0.000
(0.125)
(0.031)
(0.098)
(0.203)
—
1.035
—
0.644
—
(0.318)
—
0-£
0.116
—
—
—
0.051
(0.019)
&!
0.19
0.04
0.75
0.07
-0.10
0.63
a2
—
—
-0.14
—
—
-0.07
2404.8
2444.2
2434.4
2551.7
2643.0
2588.6
Likelihood
*
—
(0.198)
(0.036)
Note:
AR(2)
- stands for estimates obtained with the approximation € -( x v s )
— \siv-\-S — xin
equation (11). The autoregressive model for the sectoral residuals
vu
=
const.
+ auvu-i +
23
a2i"ti-2
+ tit-
is:
The
The
results for structures are qualitatively similar to those for equipment.
column shows a
adjustment
coefficient Ao
coefficient of 0.39.
which
is
significantly different
The mean
from zero and implies a partial
order serial correlation coefficient
first
first
is
0.07, with
a range between —0.12 and 0.27. The second column indicates the relevance of considering
non-constant hazards:
A2
likelihood function exhibits a substantial
improvement when
allowed to be positive, while the linear term, Ao, again becomes indistinguishable
is
from
The
zero.
The
order correlations of sectoral shocks remain on average close to zero
first
(they range from —0.30 to &07). Finally, the third column shows that unconstrained (across
sectors) second-order autoregressions lead to a worse
fit
than the simplest increasing hazard
model. 39
Results: Structural estimation
4.3
Rather than estimating the adjustment hazard
directly, in this section
we estimate the
parameters of the adjustment costs function and obtain the hazard from the solution to the
dynamic optimization problem presented
We
parameters, however.
assume an
and 20 percent, respectively; 40
in section 2.
We
do not estimate
interest rate, share of capital,
as well as depreciation rates for
all
the structural
and markup of
6,
30
equipment and structures
of 10 and 5 percent per year, respectively.
Adjustment
costs are
drawn from a
G <"> =
which has mean
and
ct £
.
As
before,
and sectoral
is
\i
first
w
=
we
pep
and a
Gamma
distribution:
*4)i> v
*T(p)
"/ *'""
=
coefficient of variation cv w
order serial correlation coefficients.
series in the likelihood in order to reduce
39
As
We
estimate
fi
w cv^,
,
allow for a general variance-covariance matrix for sectoral innovations,
The
initial cross section distribution
taken to be the ergodic one but, as before, we exclude the
The
l/y/p.
results are reported in Table 2.
any systematic
first
three observations of each
effect of this
approximation. 41
For equipment, adjustment costs are on average
before, even if the 21 first order serial correlation coefficients axe set to zero in the increasing hazard
model, the likelihood
is
substantially larger than in the partial adjustment
model with unconstrained
serial
correlation (2,634 versus 2,552), and in the non-structural second-order autoregressions model (2,634 versus
2,589). Once again, two nonlinear parameters, common across sectors, do better than 21 and 42 free sectoral
serial correlation coefficients.
*°These parameters imply /? = 0.6.
41
We approximate the ergodic density by the density that results after 30 iterations, beginning with a
point mass at zero.
24
Table
Parameters
2:
Structural results
Equipment
Structures
0.165
0.587
(0.016)
(0.080)
0.342
0.143
(0.082)
(0.027)
flu,
cv w
cr c
0.108
0.074
(0.060)
(0.013)
P\
0.131
-0.005
<T
(0.118)
(0.095)
2446.6
2649.2
A"
3.68
4.525
A+
12.64
20.96
x~
-0.472
-0.850
x+
0.398
0.693
P1
Likelihood
16.5 percent of annual revenues (net of labor
0.34,
payments) and their
which implies a standard deviation of 5.6 percent. Average
and the likelihood
is
slightly larger
coefficient of variation is
serial correlation is small
than that obtained with the semi-structural inverted-
normal hazard of the previous section. 42 Indeed, our semi-structural representation of the
structural hazard takes the form given in equation (25).
The rows
of the lower panel in
Table 2 report the implicit value of the parameters of this representation.
For structures, adjustment costs are on average 58.7 percent of annual revenues (net
of labor payments) and their coefficient of variation
deviation of 8.4 percent. Average serial correlation
42
For consistency with the microeconomic model,
reasonably close to zero.
it is
25
is
is
0.143, which implies a standard
negligible,
important that the
and the likelihood
is
larger
serial correlation coefficients
be
than that obtained with the semi-structural inverted normal distribution of the previous
section.
43
As with equipment, estimating the semi-structural hazard
constraints imposed by the microeconomic
in (25)
without the
dynamic optimization problem does not
raise
the likelihood significantly.
Figure 6 shows the implied adjustment hazards (panel a) for equipment (solid line) and
structures (dashed line), and the corresponding estimated distribution of adjustment costs
(panel b).
The hazards, which
are depicted in the space of percentage deviations from
the target levels, imply a substantial range where no adjustment occurs; they are sharply
increasing thereafter.
This difference
Clearly, there
is
more inaction
for structures
than for equipment.
larger than can be inferred directly from the figure, since the depreciation
is
and idiosyncratic uncertainty are smaller
for structures
than for equipment, thereby making
large disequilibria less likely for structures than for equipment (for a given hazard).
model's explanation for this difference
in inaction is
distributions of adjustment costs are illustrated:
larger for structures than for
What
4.4
is
found
in the
Our
bottom panel, where the
adjustment costs are on average much
equipment investment.
special of increasing hazard models?
Perhaps the main distinctive feature of the model we have estimated, compared with
linear counterparts,
also the
number
business cycle.
(1993),
who
is
its
that not only average investment by those that are investing but
of firms that choose to invest at any point in time fluctuates over the
That
this
is
realistic is
confirmed by the evidence in
Doms and Dunne
use the Longitudinal Research Datafile to study the investment behavior of
12,000 continuing U.S. manufacturing establishments for the seventeen year period from
1972-1988.
Among many
interesting facts, they
through their primary investment spikes
(i.e.,
show that the number of plants going
the single year with the largest investment
for the establishment), rather than the average size of these spikes, tracks closely aggregate
manufacturing investment over time. In terms of our model,
this flexibility in the
number
of firms investing implies that, contrary to the case of the linear model, the extent of the
response of aggregate investment to aggregate shocks fluctuates over the business cycle.
Figure 7 depicts the paths of the derivatives of aggregate investment with respect to aggregate shocks for equipment and structures, evaluated at
"Therefore substantially larger than
for the linear partial
models.
26
Vt. It is
apparent that this "index
adjustment and second-oder autoregression
Figure 6a: Adjustment Hazards
09
d
^\
s*
l\
oo
d
I
d
1
d
X
\
I
m
d
/
\
1
o_
V
^1.0
-0.6
'
'
/
\
I
/
/
/
\
d
/
/
\
1
/
/
\
1
/
/
\
I
10
<°
//
\
1
/
V
02
0.2
0.6
/
1.0
e*-1 1% disequilibria]
Figure 6b: Distributions of adjustment costs
u
—
—
1.4
•quip.
itruet
1.8
Figure ^a: Equipment-index of responsiveness
Figure fb: Structures-index of responsiveness
Table
Skewness and kurtosis
3:
Model
Equipment
Kurtosis
Skewness
Kurtosis
0.36
1.18
0.74
1.75
(< 0.01)
(< 0.01)
(<0.01)
(< 0.01)
0.35
0.85
0.81
1.72
(<0.01)
(< 0.01)
AR(2)
(<
(< 0.01)
0.01)
-0.10
0.65
0.07
0.33
(> 0.20)
(< 0.01)
(> 0.20)
(> 0.07)
Structural
Shown
Structures
Skewness
Part. Adj.
*
for innovations*
in parenthesis: p-values
obtained via bootstrap.
of responsiveness" fluctuates widely over the sample. Moreover,
Its correlation
Our model
with aggregate shocks
is
it is
strongly procyclical:
0.79 for equipment and 0.72 for structures.
implies a complex non-linear process for investment.
To a
first
approxi-
mation, however, the contemporaneous response of investment to aggregate shocks can be
characterized in simpler terms.
above implies that
for
The
procyclicality of the index of responsiveness described
any given sequence of shocks, the increasing hazard model has sharper
(non-linear) cyclical features. In particular,
expansions (the asymmetry
is
it
will
due to the strong
generate fatter
drift
tails,
especially so during
induced by depreciation). Figure 8
shows the mean difference (normalized by average I/K) between actual investment and
the predictions of the linear model; the latter supplied with the sequence of shocks inferred
from the non-linear model. The largest absolute values of
of large departure between investment and its mean.
An
.
is
alternative
way
to reach a similar conclusion with respect to the role of nonlinearities,
to look directly at the departures of
normality.
that I/K
for
44
ib
is
this series occurs during periods
44
I/K and
shocks generated by each model from
Figure 9 shows the histogram of standardized sectoral I/K.
not normal;
its
45
It is
apparent
skewness and (excess) kurtosis coefficients are 0.61 and 0.74
equipment and 0.76 and 0.87
for structures.
Obviously, linear models with normal
we normalize by actual rather than average I/K, the difference during recessions
Standardized within each sector.
If
27
is
accentuated.
Figure ga: Equipment-mean-difference
Figure fib: Structures-mean-difference
Figured^: Histogram of Equipment
l/K
[standardized]
Figure Ob: Histogram of Structures l/K [standardized]
errors cannot account for these departures.
The innovations generated by
the best partial
adjustment model and best second-order autoregression models also depart from normal,
be seen
as can
Their skewness and kurtosis coefficients are 0.36 and 1.18 for
in table 3:
equipment and 0.74 and 1.75
for structures in the partial
0.85 for equipment, and 0.81 and 1.72 for structures in the
adjustment case, and 0.35 and
AR(2)
are significantly different from zero (the normal case) at the 0.01 level.
show that the increasing hazard model generates innovations that
The estimated skewness and
linear counterparts.
its
and three out of four of the
smaller,
level)
coefficients
numbers
case. All these
The
last
two rows
are closer to normal than
kurtosis coefficients are considerably
do not depart significantly (at the 0.05
from their values under the normality assumption. The non-linear model does not
need to introduce nearly as much skewness and kurtosis
in aggregate shocks to
account for
investment behavior.
Small Changes
5
The models estimated
in the preceeding section (see Figure 6)
imply that firms always
adjust their capital by large amounts (at least 30%). Strictly interpreted, this implication
microeconomic
unrealistic at the
is
level, for in
addition to large projects, plants often
experience small adjustments. 46
In this section
we extend the model developed
We
of small adjustments.
and incorporate the
consider two kinds of capital, one which
another which can be adjusted costlessly.
that, even though
earlier
now most adjustments
We
is
possibility
costly to adjust
and
estimate the extended model and conclude
at the firm level are small, the occasional
lumpy
nature of adjustments continues playing a central role for understanding aggregate dynam-
We
ics.
for
conclude that
lumpy investment, that
The model
of Section 2
costs ("fixed")
latter
by
k.
Sec
and
extended to allow for two kinds of capital, one with adjustment
Firms' technology
Doms and Dunne
Fixity
is
and the other without
inputs instead of two.
4
drives the results in the preceeding sections.
Model Extension
5.1
46
not excluding the possibility of small adjustments, but allowing
it is
flexibility
The
is
("flexible").
47
The former
is
denoted by
K
and the
Cobb-Douglas with constant returns, now with three
elasticity of
production with respect to both kinds of capital
(1993).
only refer to the presence or absence of adjustment costs.
28
is
ar and q(1 —
r), respectively,
the particular case where r
It is
from
=
<
with
r
<
1.
Thus, the model of section 2 corresponds to
1.
easy to show that a firm with a stock of fixed capital equal to
its frictionless
optimal stock of fixed capital of z maximizes
with
/?=
(28)
1
its profits
Ta ^-
1]
+
—
tq(i/
flexible capital, k,
its flexible
even
is
if it
too high to
make adjustment
does not adjust
magnitude of
Expressing
A;
this
change
as a function of
is
(i.e.,
when
0)
and the
In every period the firm adjusts
by the current change
larger
Az <
profitable, the firm will adjust its
its fixed capital.
capital in the direction determined
capital - the
by choosing:
1)'
Thus, when the firm's desired stock of fixed capital increases
current adjustment cost
and a deviation
= e-^-W'K,
k
(27)
K
fixed capital
K, and using the expression
for
/3
in desired frictionless
is
also adjusted.
in (28), equation (1)
and
the arguments that follow in section 2.2 apply directly to the more general case considered
here.
Thus the shape
of the firm's optimal policy
is
the
same
as before: conditional
current draw of adjustment cost factor, the firm follows an (S, s) rule for adjusting
capital. Its flexible capital adjusts according to
5.2
on the
its fixed
(27).
Sectoral Investment
Even though calculations are somewhat more cumbersome, aggregate equations are obtained following analogous steps to those of section 2.3.
of firms' disequilibria
is
identical to
what we saw
We
evolution of the cross-section
in that section.
only requires modifying the expressions relating
translates into aggregate investment.
The
how a
The extension
of the model
given cross-section of disequilibria
sketch the steps required to do this next.
Let:
v
=
1
-r'"
,-(i-/9)<=
T
with c denoting the disequilibrium immediately after firms adjust their stock of fixed capital.
Let
if and
respectively,
J* denote aggregate investment during period
and
let
ij and
Kj denote total investment and
Then:
29
t
in fixed
and
flexible capital,
total capital stock, respectively.
h
W
-
(ttxt)/^-
=
(
)A(x)f(x,t)dx,
1
^/(e- I -l)A(x)/(x,Odx],
TT^)[(^-^-i) +
where:
J =
(29)
uJe-^-^
t
with f(x,t) and f(x,t) defined in section
this expression at
when deriving the non-linear equations
,
hand,
that must be
are straightforward.
Results
5.3
4 3
= iJ/Kj. With
yt
solved to find the aggregate (sectoral) shocks, v t
We
f(x,t)dx,
2.3.2.
Adding both expressions above we obtain
the modifications needed in section 3
x
estimate structural models for equipment and structures separately, as in section
48,49
Table 4 presents tne
The parameter measuring
from
results.
50
the share of both kinds of capital, r,
significantly different
(the value implicit in section 4), but large enough so the fixed capital continues
1
playing a key role, especially so for structures.
and
is
coefficient of variation of the
Gamma
The remaining
(mean
structural parameters
distribution characterizing adjustment costs,
and the standard deviation of idiosyncratic shocks) take similar values to those obtained
The
earlier.
first
equipments. This
if
order correlations are larger than in the preceeding section, specially for
may be due to assuming that
flexible capital involves
no adjustment
costs;
the corresponding adjustment costs are smaller than for fixed capital, but non-negligible,
then a
first
order correlation term
More than the
may be
expected.
what
structural interpretation,
though over 90 percent of yearly investment
at the
microeconomic and aggregate
48
Thus
The
interest rate, (total) share of capital
50
We
include an additive constant
important to stress here
is
that even
smaller than 20 percent, the nonlinearities
level are fully
49
in
is
is
accounted for by the large adjustments.
each case there are two kinds of capital: with and without adjustment costs.
(or),
(common
and markup are fixed
at the
same
values as before.
across sectors) which partly captures investment that
totally unrelated to our model; the estimated values are 0.045 for
equipment and 0.020
earlier, given the non-linear nature of the model, it is computationally infeasible to
unobserved, stochastic aggregate shocks with measurement error.
mentioned
30
is
As
combine
for structure.
Table
4:
Parameters
tau
Hw
cv w
Extended model:
Equipment
results
Structures
0.563
0.829
(0.103)
(0.098)
0.184
0.522
(0.011)
(0.099)
0.235
0.172
(0.029)
(0.101)
0.038
0.055
(0.006)
(0.010)
0.283
0.069
a P\
(0.111)
(0.087)
Likelihood
2451.3
2660.4
Ol
P\
31
And
these are the ones that explain the improved aggregate
fit
of the models presented in
paper over standard linear ones.
this
6
In this paper
we
Final Remarks
derived and estimated a model of sectoral investment that builds from
the realistic observation 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 non-linear time
and
series procedure,
for a firm over time.
we estimated
the distributions of fixed ad-
justment costs faced by firms that maximizes the likelihood of aggregate (sectoral) data.
Restricting these distributions to the gamma-family,
we found that
their
means and
stan-
dard deviations are 16.5 and 5.6 percent of a years' revenue (net of labor payments) for
equipment, and 58.7 and 8.4 percent for structures.
More importantly,
the adjustment
hazards implied by these findings are clearly non-linear: they leave a significative range of
complete inaction, and increase sharply thereafter.
At the aggregate
linearities noticeably
The
level,
the estimated hazards imply brisk cyclical features. These non-
improve the aggregate performance of investment equations.
empirical methodology developed in this paper should serve as a useful comple-
ment to microeconomic
studies of investment, as well as in
intermittent microeconomic adjustment
is
suspected.
32
many
other applications where
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