Cyclical Fluctuations in Oligopolistic Industries under
Heterogeneous Information:
¤
An Empirical Analysis
Michael Binder
University of Maryland
November 1995
Abstract
In this paper I investigate whether industry-speci¯c cyclical °uctuations have a common structure across four-digit-SIC level U.S. manufacturing industries. Taking the
industry structure as oligopolistic, I construct and estimate a structural model of
industry-speci¯c cycles allowing for
a priori
unconstrained `°exible functional form'
production and demand functions and for dynamic strategic interaction among heterogeneously informed ¯rms within an industry.
Using data for various producer
goods four-digit-SIC level industries within the U.S. manufacturing sector, I ¯nd
that my model captures cyclical movements in the industries investigated by and large
fairly well. The industries investigated therefore have a common cyclical structure in
the sense that industry cycles can be explained by a common model framework. The
results reveal, however, that theories that try to explain industry-speci¯c cyclical °uctuations solely by changes in ¯rms' real cost of production or ¯rms' markup-behavior
cannot capture the nature of these °uctuations across all industries analyzed.
Keywords:
Industry Cycles, Oligopolistic Pricing, Heterogeneous Information.
JEL-Classi¯cation:
¤
Mailing Address:
20742.
E-Mail:
D82, E32, L16.
Department of Economics, University of Maryland, Tydings Hall, College Park, MD
binder@Glue.umd.edu.
I have substantially bene¯ted from comments, suggestions and
encouragement by Frank Diebold, George Mailath, Lee Ohanian, and Jose-Victor Rios-Rull.
Valuable
comments were also obtained from Dirk Bergemann, Lutz Hendricks, Matthias Kahl, Sumantra Sen, and
participants in University of Pennsylvania Econometrics and Shadow Workshops.
The usual disclaimer
applies. The German National Scholarship Foundation provided ¯nancial support for this project.
1. Introduction
The cyclical behavior of production and pricing decisions of ¯rms operating in imperfectly competitive industries has been the subject of a large body of both theoretical
and empirical work in macroeconomics and industrial organization.
Many of the more
recent contributions have been spurred by the work of Hall (1986, 1988), who found
that a substantial number of (two-digit-SIC level) U.S. industries exhibit important noncompetitive characteristics in that marginal cost in these industries is well below output
price, and who speculated that
aggregate U.S. cyclical °uctuations may be closely linked
to industries' deviations from perfectly competitive market structures.
It is for many aspects of economic policy analysis clearly equally important to understand the working mechanism of
industry-speci¯c cyclical °uctuations, taking into account
the ¯nding that a large number of U.S. industries exhibit important non-competitive
characteristics.
tuations.
In this paper my concern is with such industry-speci¯c cyclical °uc-
I investigate whether industry-speci¯c cyclical °uctuations have a common
structure across four-digit-SIC level U.S. manufacturing industries. Taking the industry
structure as oligopolistic, I construct and estimate a structural model of industry-speci¯c
cycles allowing for an
a priori unconstrained `°exible functional form' production struc-
ture of ¯rms and for dynamic strategic interaction among ¯rms within an industry. Industry demand is assumed to be given by a `°exible functional form' demand function,
incorporating industry-speci¯c as well as aggregate demand °uctuations.
I capture the
strategic interaction among ¯rms by modeling ¯rms' strategies to be Markov, so that
¯rms' past behavior in°uences current decisions only through the evolution of a state
1
vector that summarizes `directly' pro¯t relevant variables.
This in turn makes it feasible
to allow for heterogeneous information across ¯rms: Firms are modeled as having complete information about their own cost structure, but having limited information about
2
other ¯rms' costs of production in any given period.
Heterogeneous information is an
important element in modeling the production and pricing decisions of oligopolistic ¯rms
since the uncertainty of ¯rms about other ¯rms' behavior it introduces is an important
1
I thus do not consider strategies for which past behavior of ¯rms may a®ect their current behavior
only because the ¯rms
believe
that such past behavior a®ects current decisions. See section 2 for further
details.
2
Throughout this paper, I will refer to ¯rms' information as being `heterogeneous', as is the standard
terminology for the information structure studied in this paper in the macroeconomic literature. In the
game theoretic literature, the terms `asymmetric', `private' or `incomplete' information are more commonly
used.
2
3
factor in the model-internal propagation of supply- and demand shocks.
That ¯rms en-
gage in a `forecasting the forecasts of others' in the spirit of Townsend (1983) is also what
is observed in a ma jority of U.S. industries: Firms typically have only a limited amount
of information about their competitors' production structure, strategies and management
techniques, but continuously try to gain further information about the latter, using the
services of market research or consulting companies.
The model's structure is in summary rich enough to account for many characteristics of ¯rms' technology, ¯rms' strategic behavior, and determinants of industry demand,
which together build the basis for ¯rms' production and pricing decisions.
The model
structure therefore makes it sensible to empirically implement the model for a variety
of U.S. industries. I obtain the model's quantitative predictions about industry-speci¯c
cycles in oligopolistic industries by estimating the model parameters using four-digitSIC level (producer goods) U.S. manufacturing data, and subsequently simulating the
equilibrium laws of motion for a linear-quadratic approximation of the model. For these
simulated data as well as the actual U.S. data I conduct a Kydland-Prescott style comovements analysis, and investigate impulse responses with respect to industry-speci¯c cost
and industry-speci¯c as well as aggregate demand shocks.
Conditional on these model
evaluation statistics, a recession-recovery analysis for the actual and simulated industry
cycles is then conducted. The latter particularly allows me to assess how ¯rms' pricing
decisions, their real cost of production, and their markup-behavior vary across the industry cycles. I ¯nd that my model captures industry-speci¯c cyclical movements in the
industries investigated by and large fairly well.
These industries therefore have a com-
mon cyclical structure in the sense that industry cycles can be explained by a common
model framework. The recession-recovery patterns reveal, however, that industry-speci¯c
cyclical °uctuations cannot be attributed to changes in ¯rms' real cost of production or
¯rms' markup-behavior alone.
The derivation of the model equilibrium laws of motion requires the solution of a
heterogeneous information rational expectations model with strategic interaction, and
the solution algorithm presented in section 3 of the paper, which builds on techniques
recently suggested by Binder and Pesaran (1995a, 1995b), can be viewed as a contribution
3
I do not provide in this paper an explicit comparison of the quantitative properties of an industry
structure with heterogeneous information to a corresponding one under homogeneous information, but
con¯ne myself to an analysis of the case of heterogeneous information. See Binder (1995) for a detailed
analysis of the quantitative importance of ¯rms' uncertainty about other' ¯rms behavior for the propagation of supply- and demand shocks within the context of a real business cycle model.
3
of interest independent from the economic question addressed in this paper.
The paper complements other recent studies investigating industry-speci¯c and aggregate cyclical °uctuations in imperfectly competitive industries, in part adding to the
methodological framework of some of the previous literature.
The focus of the major-
ity of this previous work has been on the cyclical behavior of prices and/or markups
in imperfectly competitive industries.
Among the empirical investigations, Domowitz,
Hubbard and Petersen (1988) ¯nd - using Hall's (1988) Solow residual approach to measure marginal cost - that ¯rms' markups in non-durable goods four-digit-SIC level U.S.
manufacturing industries are procyclical, while durable goods four-digit-SIC level U.S.
manufacturing industries are found by them to exhibit acyclical or slightly countercyclical markups.
A limitation of their study, though, is that Hall's methodology is only
applicable if production technologies are characterized by constant returns to scales. Bils
(1987), employing only variations of labor input to measure marginal cost, ¯nds that
markups in two-digit-SIC level U.S. manufacturing industries on average are markedly
countercyclical. Noting that none of these studies uses an
a priori
unrestricted technol-
ogy structure as o®ered by `°exible functional forms', Morrison (1992, 1994) employs a
generalized Leontief cost function to describe ¯rms' production environment, and ¯nds
weakly procyclical markups in (two-digit level) Canadian manufacturing industries. Her
study uses a static oligopolistic industry structure, however, neglecting any strategic interaction among ¯rms. The available empirical evidence regarding the behavior of prices
in four-digit-SIC level U.S. industries seems to indicate that these are countercyclical,
and °uctuating in a rather smooth fashion over time (Domowitz, Hubbard and Petersen
(1987)).
4
Theoretical models focusing on ¯rms' strategic interactions within the setting
of oligopoly supergames and having implications about the cyclical behavior of prices and
markups have been developed by Green and Porter (1984) and Rotemberg and Saloner
5
(1986).
Both models focus on trigger price strategy equilibria in which collusive output
levels can be enforced by the credible threat of retaliation in the event of defection of one
of the oligopolists.
While Green and Porter's supergame implies procyclical, discretely
jumping prices (and, in the absence of strongly countercyclical marginal costs, also procyclical markups), Rotemberg and Saloner's model (which assumes constant marginal
4
See also Cooley and Ohanian (1991) who establish a similar pattern for the cyclical behavior of
aggregate U.S. prices.
5
See also Rotemberg and Woodford (1992) who embed Rotemberg and Saloner's (1986) oligopoly
supergame within a real business cycle model.
4
6
costs) predicts countercyclical markups.
That Green and Porter's as well as Rotem-
berg and Saloner's ¯ndings are sensitive to the production technology assumed has been
pointed out by Staiger and Wolak (1992), who develop an oligopoly supergame in which
capacity constraints are a crucial determinant for the cyclical behavior of markups. They
argue that procyclical markups are likely to be found in oligopolistic industries facing
signi¯cant capacity constraints.
The remainder of this paper is organized as follows.
In section 2, I develop a het-
erogeneous information rational expectations model for an oligopolistic industry along
the lines outlined above.
The techniques which make it feasible to solve and estimate
this model are described in section 3. Section 4 presents the empirical results, evaluating
the model's empirical performance and conducting industry recession-recovery analyses.
Finally, section 5 o®ers some concluding remarks and suggestions for future research.
2. A Theoretical Model of an Oligopolistic Industry under Heterogeneous
Information
Firms in this paper's model are viewed as operating within an oligopolistic industry
consisting of
N
¯rms.
7
Each ¯rm maximizes its stream of expected current and future
discounted real pro¯ts by choosing optimal levels of capital services, hours, intermediate materials, and energy usage (all as contingency plans in output), as well as output.
Firms' revenues depend on an inverse demand function, which is taken to have a generalized Leontief functional form in the demand factors.
Since the model's empirical
implementation will be at the level of four-digit-SIC industries, treating industry demand
as exogenous appears not unreasonably restrictive. As demand factors considered will be
(aggregate) military spending (
G
t ), and a price index of related goods (
prel
t , at the level
of two-digit-SIC industries). The cost structure is made up by a variant of the generalized
Leontief cost function of Diewert and Wales (1987), augmented by ¯rm-speci¯c, serially
8
correlated cost shocks and external, quadratic adjustment costs in the level of output.
Both the generalized Leontief cost function and the adjustment costs are taken as being
identical for all ¯rms in the industry. The current realization of the cost shock is private
6 These
opposing predictions can be traced to the models' assumptions about whether or not demand
shocks are observed by ¯rms before production decisions have to be made.
7
To simplify matters, the number of ¯rms is taken as ¯xed over time. This assumption is less restrictive
than it may appear at ¯rst sight, as ¯rms will face adjustment costs to changes in the level of output.
8
Note that given the choice of demand factors above, it is reasonable to postulate the innovations in
these demand factors as being orthogonal to the ¯rm-speci¯c cost shocks.
5
information to each ¯rm, giving the model a heterogeneous information structure under
which each ¯rm is `forecasting the forecasts of others': In making its current output decision, each ¯rm not only makes a projection about its competitors' current and future
output decisions, but also takes into account its pro jection about its competitors' mean
beliefs about its own output decisions, and so on ad in¯nitum.
Firm
i i
's (
; ;::: ;N
= 1 2
) problem of maximizing the expected present discounted
value of real pro¯ts is consequently:
ÃX
E ¯t+s ¼i;t+s
!
1
q
max
+
i;t
where current period pro¯ts
N
X
¼it p
=
qi
(
j =1
¼it
s=0
s (¢)
1
)
¡
(
0 given, with the inverse demand function
(
N
X
j =1
N qjt; zt
1
AC qit; qi;t
adjustment costs in output
(
(
(
) =
ÃX X
r u
=1
,
=
=
µi
0 and
µi;
¡1
1
¡1
+
³
½µ µi;t
2
¡2
9
industry output price,
f =
z
PN
(2.2)
1
(2.3)
j =1 N jt
=
´q q
it i;t
2
(
¡
2
¡1 )
;
(2.4)
) is speci¯ed as
!
5
+
+
$it;
X
r
br wrt
j =1 N qj
1
µit
+
X
r
°r wrt qit;
(
)
»
´ qi
denotes ¯rm
0
, where
(2.5)
are speci¯ed as
(1
i
2
)
's level of output,
(2.6)
p
the
z is a 2 £ 1-dimensional vector
9 All prices are in real terms, being divided by an aggregate output price index.
6
2
$it iid N ;¾$ ;
given. In equations (2.1) to (2.6),
0
)
N qjt ; zt ) having the functional form
, and ¯rm-speci¯c cost shocks
µit ½µ µi;t
(
ÃP
:5 ! 2
·
f
h
h
ht
;
PN 1 q
bru wrt: wut: qit
5
µit C qit; wt ;
¡
are
¡1 )
(
(
¡1 )
bru bur r; u k; l; m; e
=
(2.1)
¡
C qit; wt
the current period cost function
p PNj
) =
¡1 )
AC qit; qi;t
C qit; wt
it
are speci¯ed as
N qjt; zt qit AC qit; qi;t
p
j ­
of demand factors, namely
costs of production,
k
l
w
G
and
prel µi
,
£ 1-dimensional
the 4
( ), hours ( ), intermediate materials (
i
represents a shock to ¯rm
's time-invariant discount factor.
m
i
's instantaneous
vector of factor prices (for capital services
e t
·h ´ bru
) and energy usage ( )),
is time, and
,
All remaining letters such as
,
etc.
¯
¯rm
denote
parameters in respectively the demand factors, adjustment costs, and instantaneous costs
of production.
As speci¯ed in (2.6), all ¯rms' cost shocks have the same stochastic speci¯cation,
the innovations
$it
being (cross-sectionally) independent realizations from an
¾$
2
distribution with variance
iid
-Normal
, however, rendering cost shocks ¯rm-speci¯c.
The information available to ¯rm
i
it = ªt [ Áit , where
is assumed to be given by ­
t consists of
the `public information history' ª
t = (q1 ; q2 ;:::; qt¡1 ;
ª
with
qt
history'
= (
Áit
q1t q2t ::: qNt
µ1;µ2;: ::;µt¡1
µt
0
)
and
z1; z2; :::; zt; w1 ; w2; :::; wt ) ;
;
µ1t µ2t ::: µNt
= (
0
) , and ¯rm
i
(2.7)
's `private information
is characterized as follows:
Áit
= (
µit ;
i
)
; ; :::;N:
= 1 2
(2.8)
The structure of the game is taken to be common knowledge to all ¯rms.
In choosing its optimal output level,
qit
, ¯rm
i
has to take into account that its cur-
rent choice of output may alter its rival oligopolists' choices of future output levels.
In
modeling this strategic interaction, I restrict each ¯rm's strategy to be Markov, and each
¯rm's strategy is a continuously di®erentiable function of a state vector which includes
that ¯rm's payo®-relevant private information contained in
t
lic information history ª
only.
Áit
and elements of the pub-
To characterize a Markov strategy more precisely, the
following additional notation will be helpful: Let
t = (q1 ; q2; :::; qt¡1 ;
¡
denote the `history of the game',
and
i
¹i µtj t
; ;::: ;N
(
ª ) represents ¯rm
i
µ1;µ2;: ::;µt
s
s1 s2
= (
;
z1 ; z2;: ::; zt ; w1 ; w2;: ::; wt )
: :: sN
(2.9)
0
)
be the vector of ¯rms' strategies,
's subjective belief about all ¯rms' current `types'
µit
,
t
¡t and
si is Markov with respect to ¹i (¢) if 8 t and two histories
³
´
µtjª(¡t)
¡t , si (¡t ) = si (¡t ) whenever (i) Áit (¡t ) = Áit (¡t ), and (ii) ¹i (µt jª(¡t )) = ¹i
= 1 2
, conditional on the information history ª . Then, following Maskin and
Tirole (1994), a strategy
0
0
0
7
0
for all
µt . In verbal description, if a ¯rm's strategy is Markov, the ¯rm will always behave
the same way in reaction to two histories for which that ¯rm's own current type is the
same, and for which its (subjective) assessment of the distribution of types conditional
on the public history ªt coincides.
There are a variety of reasons why a restriction to Markov strategies may be desirable
for my purposes here.
Apart from the technical attribute that the Markov restriction
reduces the set of equilibria signi¯cantly and may therefore be argued to enhance the
model's predictive power, other reasons for studying Markov perfect equilibria include
their relative ease of technical tractability, and that a standard econometric analysis
typically presumes the equilibrium process to exhibit covariance stationarity, which may
be violated for certain trigger strategy equilibria.
10
Restricting ¯rm
Markov, it is a (continuously di®erentiable) function of ¯rm
i's strategy to be
i's state. The determination
of this state is in dynamic games under heterogeneous information not without problems,
i in principle may condition its behavior on some variable
however. This is because ¯rm
µjt of all other ¯rms
which contains information about the unobservable current type
6
j = i, even if that variable is not immediately pro¯t relevant. Inclusion of such a variable
in the state vector may warrant inclusion of yet other variables which are not immediately
pro¯t relevant but give information about the former variable, and so on
ad in¯nitum.
As
discussed in Maskin and Tirole (1994), in games (such as the one studied here) in which
­it is su±cient for ¯rm
i to infer the distribution of all other ¯rms' current types, however,
¯rms have no incentive to base their strategies on any non-public information except their
directly payo®-relevant type
µit . I describe in section 3 below how the elements in the state
vector can be determined using techniques from control theory.
the su±cient state statistic for ¯rm
which I will later denote by
s
¸i .
strategies is given by pairs ( i ,
i is characterized by a (¯nite-dimensional) vector,
Then, a Bayesian perfect equilibrium under Markov
¢
¹i ( )),
8 i, such that the strategy si
is (i) an optimal response to the strategies
subjective beliefs
For now, suppose that
sj
of all other ¯rms
adopted by each ¯rm
6
j = i, and (ii) ¯rms'
¢
¢
¹i ( ) about other ¯rms' types are consistent with objective beliefs ¹i ( )
determined on the basis of Bayes' rule and some updating restrictions introduced below,
s
and the other ¯rms' equilibrium strategies: The pairs ( i ,
¢
¹i ( )),
8 i, constitute a Bayesian
perfect equilibrium under Markov strategies if
10
Arguments along these lines are made for example by Du±e, Geanakoplos, Mas-Colell and McLennan
(1994) and Maskin and Tirole (1994). These references explore these issues in some further detail.
8
X
(i)
¡t
e
e b
si ; s¡i j ¡t ) ¡ ¼i (si ; s¡i j ¡t ))
¹ (¡t j­it ) (¼i (
for all t, ­it , and Markov strategies
¸ 0
(2.10)
bs
i,
and
(ii) ¹i (¢ j ­it ) = ¹i (¢ j ­it ) for all t and ­it .
e
si ; s¡i j ¡t ) denotes ¯rm i's expected present discounted value of (real) pro¯ts
In (2.10), ¼i (
si
at time t if ¯rm i's and its rival oligopolists' strategies are given respectively by
s¡i ,
and
and the current history is ¡t .
The Bayesian perfect equilibrium under Markov strategies satis¯es the following ¯rstorder conditions for all i and t:
E(p( Qt ;
+
P1=1
s
where Qt =
zt ) j
­it ) + qit ¢ E(@ p( Qt ;
³P
¡ µit ¢ @ C (qit ;
¯
s
1
N
¢E
P =1
N
j
j
6=
i
wt ) = @ qit
+
qi;t
s
zt)
= @ qit j ­it ) ¡ @ AC (qit ; qi;t
+1 ; q
¡ ¯ ¢ E (@ AC (qi;t
¢ @ p( Qt
+
s;
zt+s )
+
= @ qj;t
s
it )
= @ qit j ­it )
+
¢ @ qj;t
¡1 ) = @ q
s
= @ qit j ­it
it
´
= 0;
(2.11)
qj t .
The ¯rst-order conditions (2.11) are the standard Euler equa-
tions of a heterogeneous information rational expectations model, augmented by `dynamic
+
reaction' terms in the last line of (2.11): The terms @ qj;t
s
= @ qit , s = 1; 2; : : :, identify
the change in ¯rm j's period t + s optimal level of output induced by a change in ¯rm
i's period t output choice.
Such a change in ¯rm j's period t + s optimal output level
+
alters ¯rm i's period t + s pro¯ts by qi;t
s
+
¢ @ p( Qt
s;
zt+s )
+
= @ qj;t
s,
and thus needs
to be taken into account by ¯rm i when maximizing its stream of expected current and
+
future discounted real pro¯ts. The next section describes how the terms @ qj;t
s
= @ qit ,
s = 1; 2; : : :, are endogenously determined as part of the model equilibrium computation.
3. Solution and Estimation Method
To solve the model described in the previous section, suppose for now that all parameters
in equations (2.1) to (2.6) were known. I will turn to a description of how I estimate these
parameters after discussing the solution algorithm. The steps in my solution algorithm are
as follows.
11
To keep the necessary computations at a tractable level, I work with a second-
order Taylor series approximation of ¯rms' revenue and instantaneous cost functions in
11
For techniques of computing Markov equilibria in the context of dynamic games under homogeneous
information, see the excellent review by Pakes (1994).
9
relative deviations from the model's non-stochastic steady state. The model's objective
function is then linear-quadratic, which, given the unbounded support of the cost shocks
12
µit , implies that ¯rms' strategies are linear.
The ¯rst step of the algorithm is to postulate
a state vector ¸i for all ¯rms i = 1; 2; : : : ; N , and specify an initial strategy coe±cient
vector for all ¯rms conditional on this state vector. I then compute the model equilibrium
by iterating on the initially speci¯ed strategy coe±cient vector until the latter is consistent
with the actual equilibrium laws of motion.
At the core of the necessary loops is the
solution of a linear rational expectations model under heterogeneous information.
To
¯nd a determinate solution to that model, I use the techniques of Binder and Pesaran
(1995a, 1995b). These steps are described in detail in (i) to (v) below. When the ¯xed
point of the mapping from ¯rms' strategies to the actual equilibrium laws of motion is
computed, it remains to be checked whether ¯rms in this equilibrium are making optimal
use of all payo® relevant private information and the public information available, or
whether there is some unexploited information in ­it which is not part of the initially
speci¯ed state vector ¸i .
The methodology used to conduct this check, which adapts a
technique suggested by Sargent (1991), is described in (vi) below. The algorithm's steps
in more detail are as follows:
(i) Specify the variables entering ¯rm i's state vector ¸i (which is taken to be of dimension
g
£ 1) and, conditional on ¸i , specify its strategy as
qit = a ¸it ;
where a is a 1
8i;
(3.1)
£ g-dimensional coe±cient vector.
(ii) Compute the non-stochastic steady state of the (untransformed) system (2.1) to (2.6)
by setting all exogenous stochastic variables equal to their unconditional means.
13
(iii) Obtain a linear-quadratic approximation of ¯rms' objective function (2.1) in relative
deviations from non-stochastic steady state by substituting ¯rms' revenue and instantaneous cost functions by second-order Taylor-series expansions in relative deviations from
non-stochastic steady state.
12
See Mailath (1989) for a discussion why strategies satisfying sequentiality in general are nonlinear in
signaling models with bounded support of players' types.
13
This computation of course presumes knowledge of the non-stochastic steady state values for all
exogenous variables in
z
t and
w
t , which can only be obtained once a speci¯cation of the laws of motion
generating these processes has been made.
10
iv) Obtain the Euler equations corresponding to the transformed linear-quadratic problem obtained in step (iii), which can for all i and t be written in the following form:
(
qe
it
=
P
P
'0 + '1 qe ¡1 + 1=1 '2 E (qe + j­ ) + 1=0 '3 E (Qe + j­
P1=0 P '4 E (ze + j­ ) + P '5 we + '6 µe ;
+
i;t
s
s
s
hs
h
h;t
i;t
s
s
it
it
s
s
r
r
rt
t
s
it )
(3.2)
it
'0; '1 ; : : : ; '6 are complicated nonlinear functions of the `deep' parameters in (2.1) to (2.6), as well as the coe±cient vector a in (3.1), and a (e
¢) denotes
e = (q ¡ q)=q, q
a variable in relative deviation from non-stochastic steady state (i.e. q
denoting the non-stochastic steady state value of q , etc.). It should be noted that the
are already incorporated in (3.2), their values be`dynamic reaction' terms @q + = @q
where all coe±cients
it
it
it
j;t
s
it
ing computed using ¯rms' strategies as speci¯ed in (3.1). The system of Euler equations
(3.2) constitutes a heterogeneous information linear rational expectations model which
can be solved in a third step using the methods suggested in Binder and Pesaran (1995b).
To apply their framework, I make the following assumptions:
Assumption 1:
Firms' mean forecast of future industry-wide output levels can be de-
composed as follows
E (Qe +
t
for
s j­it )
=
E (Qe +
t
s jªt )
+
1
N
E (qe + j­
(
i;t
it )
s
¡
E (qe +
i;t
s jªt ))
;
(3.3)
i = 1; 2; : : : ; N , and s = 1; 2; : : :.
Assumption 1 says that the beliefs of ¯rm
i about other ¯rms' future output decisions
are based on the public information history ªt only.
Assumption 2:
In forming expectations about future industry-wide output levels, ¯rms'
second-order mean beliefs satisfy
E (E (Qe +
t
for
+
s j­i;t
s
¡
u )j­j t )
=
E (Qe +
t
; i 6= j;
s jªt )
(3.4)
i; j = 1; 2; : : : ; N , s = 1; 2; : : :, and 0 · u < s.
While Assumption 1 is a restriction on how the oligopolists' mean beliefs about their
rival oligopolists' types and implied output decisions are formed, Assumption 2 restricts a
given ¯rm's mean beliefs about other ¯rms' mean beliefs. It postulates that in forecasting
other ¯rms' expectations about industry-wide output, a given ¯rm only uses the public
11
information contained in ªt . As is discussed in Binder and Pesaran (1995b), Assumption
1 and Assumption 2 make it possible to derive determinate solutions for a broad class of
(possibly multivariate) linear rational expectations models under heterogeneous information. While it is in general di±cult to assess how `restrictive' the two assumptions are,
Binder and Pesaran compare for a simple model without strategic interaction the solution
derived using the equivalents of Assumption 1 and Assumption 2 with the solution computed using a numerically exact procedure of Sargent (1991) (the latter being feasible for
certain special model cases) and ¯nd only small di®erences in the two solutions' dynamic
properties.
Making Assumption 1 and Assumption 2, the solution to (3.2) can be computed by
Qe t , and then using this solution to
e g, i = 1; 2; : : : ; N .
obtain the disaggregate solutions (for ¯rm-speci¯c output levels) fq
Aggregating (3.2) across all ¯rms i, one obtains
¯rst deriving the solution for industry-wide output
it
Qe
P
P
'1 Qe ¡1 + 1=1 '2 1 =1 E (qe + j­ )
P1=0 '3 1 P =1 E(Qe + j­ ) + 1 P =1 ºe¤ ;
+
t
=
t
where
ºe¤
it
=
'0 +
N
i
s N
s
1X
X
'
s
=0
N
s N
s
t
e
s
it
4 E (z + j­
hs
h;t
i;t
i
it )
s
+
s
it
N
i
N
(3.5)
it
X ' we ' µe :
5
6
fqe g i
; ;:::;N
r
+
rt
(3.6)
it
r
h
The following propositions derive the disaggregate solutions
it
,
= 1 2
. In par-
ticular, Proposition 3.1 shows that (3.5) may be transformed to a rational expectations
model in which the expectations about the endogenous variable
Qe
t
depend only on the
public information history ªt . The transformed model incorporates a process measuring
the disparity of expectations about exogenous variables based on the private information
history ­it versus the same expectations based on the public information history ªt .
ne o
Proposition 3.2 and Proposition 3.3 below will then show how to compute the aggregate
solution
Q
t
and the disaggregate solutions
e
fq g, 8i, using this transformed model.
it
A
reader less interested in technical details may skip the precise statement of these propo-
v
sitions given in what follows, and turn immediately to the description of step ( ) of the
solution algorithm below.
Proposition 3.1. (Equivalent Representation)
Suppose
(i) Assumption 1 and Assumption 2 hold,
12
and
(ii) all eigenvalues of the matrix
where
F
I
¡
M2C
N
+
= (
He 2 =
J1
He 2)¡1(
+
M3 ) fall inside the unit circle,
M4
+ M3 C;
N
(3.7)
M2
N
M1 , M2 , M3 and M4 are J1 £ J1 -dimensional coe±cient matrices given by
0
'1
B
B
B 0
M1 = B
B
B
@
0
¢¢¢
0
¢¢¢
.
.
.
.
.
.
.
0
0
¢¢¢
1
C
C
0 C
C
;
C
C
A
0
0
0
BB (N'21 + '31) (N'22 + '32)
0
0
B
M2 = B
BB
@
0
BB (N
B
M3 = B
BB
@
¡
and
C
is a
equation
¡
1)
'21
(3.8)
.
.
.
..
N'2
(
¢¢¢
+
J1
¢¢¢
0
.
.
.
.
.
.
..
.
.
.
.
0
0
:::
0
N ¡ 1)'22
¡(
N ¡ 1)'2 1 ¡1
¡(
¢¢¢
;J
'3
J1
)
1
C
C
C
C
;
C
C
A
N ¡ 1)'2
¡(
J1
1
0
¢¢¢
0
0
.
.
.
.
.
.
.
..
.
.
.
.
.
.
0
0
:::
1
0
0
'30
B
B
B 0
M4 = B
B
B
@
0
¢¢¢
0
¢¢¢
.
.
.
.
.
.
..
0
0
:::
1
0
C
C
0 C
C
;
C
C
A
(3.9)
1
CC
CC ;
CC
A
(3.10)
(3.11)
.
.
.
.
0
J1 £ J1 -dimensional matrix de¯ned as the solution to the quadratic
matrix
M2 + M3 )C 2 + (M4 ¡ I 1 )C + M1 = 0 1 £ 1 ;
(
J
J
J
(3.12)
J1 being a scalar denoting a truncation point for ¯rms' forecasting horizon.
Then an equivalent representation of the heterogeneous information model (3.5) is
e
H1 U
t
=
e +1jª ) + ´ ;
H2 E (U
t
13
t
t
(3.13)
where
Ue
t
=
³
Ze +1 ª )
Ze t = Xe t f1
Xe t = xe t
Ze
E(
0
t
j
t
0
t
E(
¢¢¢
¡C
e t+1 ªt)
xet = e t
0
BB J1
B 0J1 J1
1=B
BB
@
I
E (Q
Q
f2( J1 f1)
¡M
I
.
0J1 £J1
0J1 £J1
0
BB 0J1 J1
B J1
2=B
BB
@
I
H
1
and H
f1 = (
M
2
¡M 2
¢¢¢
2
M C
1 jªt )
I
0
(3.17)
;
¡
£
J
I 1
0J1 £J1
0J1 £J1
0J1 £J1
¢¢¢
0J1 £J1
0J1 £J1
.
.
.
.
.
.
J
0J1 £J1
0J1 £J1
(3.16)
¡M
¢¢¢
0J1 £J1
´
.
.
.
..
.
(3.14)
;
fJ +1( J1 f1) 1 1
C
C
0 J1 J 1
C
C
C
C
A
0J1 £J1
.
.
.
0
¡
¢¢¢
.
.
.
´
(3.15)
e t+J1
¢¢¢
.
.
.
0
;
¡
E (Q
J
I 1
£
(both H
1
¡
¡M
t)
2 jª
;
j
¢¢¢
.
.
.
£
H
j
t
Xe t ªt)
xet 1
¡ M E(
³
Ze +J
..
¢¢¢
I 1
;
1
C
C
C
C
C
C
A
(3.18)
(3.19)
1 (J2 + 1) £ J1 (J2 + 1)),
are of dimension J
e 1)
+ M4 )(IJ1
f2 =
¡H
1 ( M2C
¡
N
+
4
M
N
3
M C
+
N ¡
1
)(N
¡
1) + M3 C;
2(IJ1 ¡ He 2 ) 1 ( MN2 C + MN4 )(N ¡ 1)
e 1) 1 MN2 (IJ1 ¡ He 2) 1( MN2 C + MN4 )(N ¡ 1)
+(M2 C + M4 )(IJ1 ¡ H
e 1) 1 ³ MN2 + NM31 ´ (N ¡ 1) + M3;
+(M2 C + M4 )(IJ1 ¡ H
M
(3.20)
¡
M
¡
¡
(3.21)
¡
¡
fr =
e 2) 1( MN2C + MN4 )(N ¡ 1)
2 r 2 (IJ1 ¡ H
e 2) 1 MN2 (N ¡ 1)
+M2 F r 3(IJ1 ¡ H
e 1) 1 MN2 F r 2(IJ1 ¡ He 2) 1( MN2 C + MN4 )(N ¡ 1)
+(M2 C + M4 )(IJ1 ¡ H
e 1) 1 MN2 F r 3(IJ1 ¡ He 2) 1 MN2 (N ¡ 1);
+(M2C + M4)(IJ1 ¡ H
M
M F
¡
¡
¡
¡
¡
¡
¡
¡
r
= 3; 4; : : : ; J2 + 1;
J
(3.22)
¡
¡
2 is a truncation point for ¯rms' forecasting horizon,
e1 =
H
and
t
´
=
³
t
´
¤
0
2
M C
N
0
0J1 £1
14
+
4;
M
¢¢¢
(3.23)
N
0
0 J1 £ 1
´
0
;
(3.24)
with
P =1 P F 1(I ¡ He 2) 1E(ºe + jª )
³
´ P P =1 e 1
1
1
+
=1
³+
´ P =1(I ¡ eH1) 1 F (I ¡ He12)P E(ºe + jª )
+
+
=1 (I ¡ H1) E (ºe jª ) + » +
=1 ºe ;
³
´
e = ºe 0 ¢ ¢ ¢ 0 ;
º
P P
e
» =
³
´ P P=1 =1 F e 1(I 1 ¡ He 2)1 1" + e 1 e
+
+
³ =1 =1(I´ P¡ H1) eF 1(Ie ¡ H2) " +
+
+
=1 (I ¡ H1 ) " ;
M2
´t¤ =
M2 C
N
M4
N
N
1
i
s
N
1
M4
N
N
N
¡
M2 C
N2
M4
N2
M2
N
2
s
s
M2 C
N
and
1
i
2
t
¡
J1
t
s
t
t
t
t
N
it
i
0
s¡
N
e
s¡
N
N
i
2
i;t
M2
¡
¡
J1
(3.26)
ºi
¡
J1
J1
M4
(3.25)
N
¤
i
1
s
it
N
N
t
s¡
N
it
t
M2
¡
J1
i
¡
J1
J1
s
i
N
M2 C
s¡
s
J1
¡
ºi
i;t
s
(3.27)
ºi
it
e
e
º
"i;ti +s = E (ºi;t+s j­it) ¡ E (ºi;t+s jªt ):
(3.28)
Proof of Proposition 3.1:
See Appendix 1.
e
To check the approximation incurred by truncating the future expectations of Qt
e
and Zt at respectively dates t + J1 and t + J2 , in the computational implementation I
evaluate the expectations at t + J1 + 1 and t + J2 + 1, and re¯ne the initial choice of
J1 and J2 in an iterative procedure. It should be noted that the term ´t in (3.13), which
absorbs all exogenous variables, contains only realizations of these exogenous variables
and expectations about them based on the public information history ªt , augmented
by the vector »t which measures the disparity of expectations about exogenous variables
based on the private information history ­it versus the public information history ªt .
Since E (»t+s jªt ) = 0;
8s ¸
0, this term does not pose any di±culty in solving (3.13),
however, as is shown in the following proposition.
Proposition 3.2. (Aggregate Solution)
1
Suppose all eigenvalues of the matrix H3 = H1 H2 fall inside the unit circle.
¡
Then the solution to the equation system (3.13)
e
e
H1 Ut = H2 E (Ut+1 jªt ) + ´t
15
is given by
Ue
t
Xc
1
=
r=0
1r H1
1 E (´t+r jªt ) +
Xc
2r
Á2 1
¡
¡
M
where
c1r =
M
c2r =
M
t
¿
»
t
=
T
¿
t
=
T
=
T
0
@
T
0
@ 0Á1
1H 1
1
¡
³ PN
¡
1H 1
1
¡
r=0
¡
'6
i=1 N
³
³
³e
M
1
H
¡
1 E (´t+r jªt ) + T »
¤r1
0Á1 £Á2
0Á2 £Á1
0Á2 £Á2
Á1
0Á1 £Á2
£
¤r
2
0Á2 £Á1
T
1;
(3.30)
¡
1;
(3.31)
0
´
0
´
0 J1 £ 1
0
¢¢¢
0 J1 £ 1
t
0
0J1 £1
t
1
A
´
it ¡ E (µeit jªt)
µ
0
¢¢¢
0
;
(3.32)
;
(3.33)
0
0
´
0
(3.34)
;
1 (J2 + 1) £ J1 (J2 + 1)-dimensional matrix such that ¤ = T 1 H3 T
normal form of H3,
0
1
¤1
0Á1 Á2
A;
¤=@
0Á2 Á1
¤2
T
being a
(3.29)
¡
¢¢¢
0J1 £1
¿
T
0
0
»
1
A
t + T ¿ t;
¡
J
is the Jordan
£
(3.35)
£
1 £ Á1 ) denoting the block associated with the Á1 nonzero
eigenvalues of H3 which lie inside the unit circle, and ¤2 (which is of dimension Á2 £ Á2 )
representing the block associated with the Á2 (Á1 + Á2 = J1 (J2 + 1)) eigenvalues of F
with ¤1 (which is of dimension
Á
which are zero.
Proof of Proposition 3.2:
See Appendix 1.
To compute the ¯rst component of the left hand side of (3.29),
P
c
r=0 M1r H1
1
ne o
1 E (´t+r jªt ),
¡
in a tractable manner, one may apply the forward recursive method described in Binder
Ut , one can extract the solution process
ne o
e t given in (3.14), using that Xe t = Ze t + f1( J1 ¡
t observing the de¯nition of U
f1) 1 (Ze tjªt), and using the de¯nitions of Xe t and xe t given in (3.16) and (3.17). Proposition 3 shows how to compute the disaggregate solutions f eitg, 8 , conditional on having
n o
derived the aggregate solutions e t .
and Pesaran (1995a). Using the solution for
Q
M
M
¡
E
q
Q
16
i
I
Proposition 3.3. (Disaggregate Solution)
A2 Cb )¡1A2 fall inside the
where the J3 £ J3 -dimensional coe±cient matrices A1 and A2 are given by
Suppose all eigenvalues of the matrix
B
= (IJ3
0
'1
B
B
B 0
A1 = B
B
B
@
0
'21
B
B
B 1
A2 = B
B
B
@
and
and
Cb
is a
J3 £ J3 -dimensional
equation
¡
0
¢¢¢
0
¢¢¢
.
.
.
.
.
.
.
0
0
¢¢¢
'22
¢¢¢
1
C
C
0 C
C
;
C
C
A
unit circle,
0
(3.36)
.
.
.
..
0
'2 3 ¡1 '2
J3
;J
0
¢¢¢
0
0
.
.
.
.
.
.
..
.
.
.
.
.
.
.
0
0
¢¢¢
1
0
1
C
C
C
C
;
C
C
A
(3.37)
matrix de¯ned as the solution to the quadratic matrix
A2 Cb 2 ¡ Cb + A1 = 0
J3 £ J 3
;
(3.38)
J3 being a scalar denoting a truncation point for ¯rms' forecasting horizon.
Then the solution to (3.2) is given by
e
vit =
Cb ve
X
1
i;t¡
1+
s
where
e
vit =
#
³
=0
B E (# +
s
E (qe +1 j­
qe
it
i;t
it )
e
¢¢¢
= £1 (ºit ; E (´t+u¡r jªt¡r ));
¤
it
i;t
sj
­it ) + Ât ;
E (qe +
i;t
J3 ¡
1
j­it )
(3.39)
´
r; u = 0; 1; 2; : : :) ;
0
(3.40)
(3.41)
and
Â
t
= £2 (» t¡r ; ¿ t¡r ;
r = 0; 1; 2; : : :);
(3.42)
where £1 and £2 are linear functions.14
Proof of Proposition 3.3:
14
The (lengthy) exact expressions of these functions are contained in a note available upon request.
17
See Appendix 1.
The solution
fqe g is given by
it
the ¯rst element in
ve
it .
(v) The ¯fth step in the solution algorithm is to use the disaggregate equilibrium laws
e
of motion for qit and compare them to the ¯rms' initially postulated strategies (in step
(i)). One will have to repeat steps (i) to (iv) (for a given state vector ¸i ) until the laws
of motion implied by ¯rms' strategies (3.1) coincide with the actual equilibrium laws of
motion generated by those strategies and computed in step (iv).
(vi) Even when a ¯xed point in step (v) has been obtained, this ¯xed point is still
conditional on the (arbitrary) initial speci¯cation of the elements entering the ¯rms' state
vector ¸i ,
8i.
The remaining step is therefore to verify that ¯rms use all information
potentially available to them for the purpose of inferring all other ¯rms' current and
expected future types if their state is speci¯ed as ¸i .
To conduct this check, I adapt a
method suggested by Sargent (1991) in the context of heterogeneous information rational
expectations models without strategic interaction. The idea is as follows. If ¯rm i's initial
strategy is given by (3.1), its `perceived law of motion' can be written as
e
¡1 + ³
¸it = H ¸i;t
e
where H is a g
15
¸it .
£ g-matrix,
and ³it is a g
(3.43)
it;
£ 1-dimensional
vector of the innovations in
e
Since ¸it contains only µit and current and past elements of ªt , the matrix H can
be constructed from a and the parameters entering the stochastic processes generating
the exogenous terms. Steps (i) to (v) of the solution algorithm then compute the ¯xed
point of the mapping from the perceived laws of motion (3.43) to the actual laws of
motion for the elements in ¸it for all i. Mapping these actual laws of motion into state
space representations for all ¯rms i with the elements in ¸it making up the vector on the
left-hand side of the measurement equation in the state space representation for ¯rm i, it
is a simple step to deduce the innovations representations for these state space systems
using the Kalman ¯lter, which in turn yield the mean-squared error optimal forecasts of
the next period realizations of the elements in ¸it , conditional on the information history
¨it = (¸i1 ; ¸i2 ; : : : ; ¸i;t
¡1 ).16
One can then compare these forecasts of the ¸it 's to the ones
15
Writing the perceived laws of motion in the form of (3.43) presumes that qi;t¡1 is an element of ¸it .
In (the unlikely) case that it was not, qit would have to be added to the vector of elements in the perceived
laws of motion (3.43), 8i.
16
See for example Anderson and Moore (1979) for a discussion of these state space techniques.
18
based on the perceived laws of motion (3.43) using the transfer function representations
of the state space systems.
If the two sets of forecasts are equivalent, that is, if their
impulse response functions match for the innovations in the endogenous variables and
if the innovation variance-covariance matrices match, then the ¯rms' state vectors are
optimal in the sense that adding further past values of elements in ¸it would not enable
¯rms to gain further information about other ¯rms' current types and output decisions.
The above discussion of the solution algorithm presumed knowledge of all the parameters in (2.1) to (2.6).
I turn next to a discussion of how the model's parameters
may be estimated. In this discussion, three issues have to be kept in mind: sample size,
identi¯ability and feasibility. The relatively small sample size of the data set described in
section 4 below (96 observations) requires a relatively parsimonious parameterization of
the model's demand and instantaneous cost function, which is why both are speci¯ed as
nonlinear `°exible functional forms' rather than as linear-quadratic functions. The sample
size also limits the number of lagged endogenous and/or exogenous variables potentially
available as instruments and requires a fairly low truncation horizon for the expressions
containing rival ¯rms' expected future responses to a given ¯rm's current output decisions,
making an entirely (linear) Euler-equation based estimation of the model parameters undesirable. The latter (as well as a maximum-likelihood estimation based on the solution
process for Qt ) may in fact not be feasible in the ¯rst instance due to a potential lack
of identi¯cation in the absence of further restrictions on the model's `deep' parameters.
Feasibility is an issue here since due to the complexity of the computations involved in
steps (i) to (vi), augmenting the model solution algorithm by an estimation procedure
involving iteration to convergence of
all
model parameters would computationally be an
enormously demanding procedure. Analogous to Pakes (1994), I therefore break the estimation of the model's parameters into two steps: that of estimating the parameters of the
demand function, the instantaneous cost function and the cost shock (which is essentially
a static estimation problem), and that of estimating the adjustment cost parameter ´ in
(2.4).
The parameters of the demand function, the instantaneous cost function and the cost
shock are simultaneously estimated by applying the Generalized Method of Moments
(GMM) to a (just identi¯ed) system consisting of the demand function p(Qt ;
input demand equations derived from µit C (qit ;
19
w
z
t ), the
t ) using Shephard's Lemma, and the
stochastic speci¯cation of
µit ,
where
µit
with
¤
Cit
=
wkt kit
+
wlt lit
+
is constructed as
µit
=
wmt mit
¤
Cit = C (qit ;
+
wet eit ,
w)
t
(3.44)
;
kit , lit , mit ,
and
e it
denoting the input
quantities of capital services, hours, materials and energy usage of ¯rm i. As instruments
used are lagged values of the endogenous and exogenous variables. Three problems arise
in estimating the parameters of the demand function, the instantaneous cost function and
the cost shock in this way: First, the input demand equations need to be augmented by an
error term which is not part of the model and therefore has no clear interpretation. Second,
given that
fµitg
is obtained through (3.44), adjustment costs are restricted
not
to be in
capital services, labor, materials or energy usage as measured by the data counterparts of
these model variables. This essentially leaves only °uctuations in non-production workers
(as
lit
and
wlt
are based on the hours and earnings of production workers) to account
for adjustment costs. Third, since the estimation of the parameters of the instantaneous
cost function and the cost shock are based on ¯rm-speci¯c rather than industry-wide
relationships, but the data to be used are at the industry level, the estimation in e®ect
presumes that the cost shocks
µit
have the same realization across all ¯rms for all
t.
This appears an unavoidable assumption for estimation purposes, since it is not possible
to aggregate the input demand equations and
¤
Cit = C (qit ;
w ) across ¯rms in a manner
t
which would allow estimation of the relevant parameters using industry-wide data only, as
these relations are nonlinear. Clearly, a satisfactory resolution of this problem would be
possible if ¯rm-speci¯c data for all model variables were available, which unfortunately is
not the case. To get some understanding of the possible severeness of this third problem,
I also estimated the model parameters using simulated model data for both
an arbitrary
fqit g,
fQt g
and
generated on the basis of the parameter values obtained using the
estimation procedures described in this section. Most of my ¯ndings reported in section
4 below are surprisingly robust to which set of simulated model data is used for the
re-estimation of the model parameters in such an experiment.
The estimation of the adjustment cost parameter
´
in
AC
industry-wide analog of the linear Euler equation (3.2), as
given the quadratic structure of
AC
(qit ;
qi;t
¡1 )
(qit ;
´
¡1) is based on the
qi;t
is identi¯ed in the latter
in (2.4). Since the obtained parameter
estimate is of course conditional on the vector of strategy coe±cients
(GMM-) estimation needs to be repeated as
20
a
a
in (3.1), this
is updated as part of the model solution
algorithm. However, this updating requires iteration to convergence with respect to just
one parameter, which is a computationally feasible procedure.
4. Evaluating the Model's Quantitative Predictions
To study the empirical implications of the model developed in section 2 in the absence of
appropriate ¯rm-level data, I use industry data at the four-digit-SIC level. The industries
analyzed are from the set of industries considered by Domowitz, Hubbard and Petersen
(1987). I separately estimate and evaluate the model for four industries: SIC 3312 (blast
furnaces and steel mills), SIC 3351 (copper rolling and drawing), SIC 3612 (transformers, except electronic), and SIC 3621 (motors and generators).
17
These four industries
were classi¯ed by Domowitz, Hubbard and Petersen (1987, 1988) to be producer-goods
industries, the ¯rst two producing non-durable goods, and the latter two durable goods.
18
All of these four industries were found by Domowitz, Hubbard and Petersen (1987) to
exhibit substantial non-competitive behavior in that output price in these industries is on
average well above marginal cost. Domowitz, Hubbard and Petersen judged two of these
industries (SIC 3312 and SIC 3351) to exhibit relatively low markups, whereas the other
19
two were classi¯ed by them as industries with a relatively high markup.
For these four
industries, I collected a data set for all model variables, on a quarterly basis from 1967:Q1
to 1990:Q4. It is described in Appendix 2 how the individual series are constructed. As
all quantities and prices in the model exhibit a trend behavior over the sample, but the
model speci¯ed in section 2 does only consider cyclical °uctuations around trend, it is
necessary to detrend the data. All results reported below (parameter estimates as well as
model evaluation statistics) are based on the cyclical component of the data as obtained
from an HP-¯lter with smoothing parameter `
¸'= 1600.
Clearly, this is only one of many
possible detrending procedures, and the sensitivity of the results obtained to this choice
of detrending procedure is a critical issue.
I therefore also extracted a cyclical compo-
nent from the data using the unobserved components speci¯cation of Harvey and Jaeger
(1993). The model evaluation statistics reported below appear fairly insensitive to which
The choice of industries is restricted by the availability of data over a su±ciently long horizon from
the18sources described in Appendix 2.
Domowitz, Hubbard and Petersen de¯ne producer-goods industries as industries whose shipped output
is to 50% or more used for investment or materials purposes. Durable goods are de¯ned by them as capital
goods
for use either by households or ¯rms.
19
The numerical values for the markups as reported by Domowitz, Hubbard and Petersen are given in
Table 3. See also the further discussion below.
17
21
of these two detrending procedures is applied, and therefore only results based on the
HP-¯lter are reported.
The model parameter estimates (based on the cyclical component of the data series)
are given in Table 1. While there are too many parameters estimated to make their individual discussion useful, a few points should be noted. For all industries, the hypothesis
of constant returns to scale for Cit , i.e. br = °r = 0, r = k, l, m, e, is strongly rejected
at conventional levels of signi¯cance. All exogenous laws of motion except that for military spending, Gt , are found to be well captured by AR(2)-processes, whereas military
spending is parameterized as an AR(1)-process. Adjustment costs are in absolute value
fairly low across industries, which, given the remarks made in the previous section about
what they may capture in light of the estimation procedure used, is not too surprising.
Using the
fµit g-processes
computed on the basis of (3.44), I ¯nd the correlations of these
cost shocks across industries to be close to zero in all cases, except for the correlation of
the cost shocks between industries SIC 3612 and SIC 3621, which is :4863.
The
fµitg-
processes therefore indeed appear not to measure cost shocks which are common across
industries. Finally, regarding the number of ¯rms, N , a sensitivity analysis is conducted.
Since my model is set up such that the quantity of industry output in the absence of
strategic interaction would be independent of the number of ¯rms, the results reported
below are non-surprisingly qualitatively insensitive to values of N ranging from 20 to 75.
All results given below are based on a value of N = 50.
Employing the estimated parameter values, the equilibrium laws of motion are computed as described in section 3, yielding for all industries a unique solution to the linear
Euler equations (3.2).
As model evaluation statistics I ¯rst compute sets of unconditional second moments
for both actual U.S. and model simulated industry data. These moments provide a check
of the model's consistency with some crucial features of industry cycles.
20
Table 2, which
for both sets of data gives the standard deviation of output as well as the standard
deviations of output price, capital services, hours, materials and energy usage relative to
output, indicates that across all four industries considered here, the actual data's standard
deviation of output, and the relative standard deviations of output price and materials
are quite well matched by the corresponding model statistics.
The relative standard
deviations of the other inputs (capital services, hours and energy usage) are typically
20
The model statistics are based on averages across 100 simulations based on 96 observations each (the
length of the sample 1967:Q1 to 1990:Q4).
22
underpredicted by the model, though still not being grossly inconsistent with the actual
U.S. data. Turning to the autocorrelation function for output, Table 2 shows that the
model predicts for all industries for all leads and lags signi¯cantly stronger correlations
than actually observed in the data. The qualitative pattern of a monotonic decay of
the autocorrelation function found in the data is, however, preserved in the model data.
Overall, the comparison of data and model unconditional second moments reveals no gross
inconsistencies of the model with observed U.S. data for the four industries considered
here.
Table 3 gives the average markups (de¯ned as the di®erence between output price
and marginal cost, divided by the output price) in the four industries. All markups are
signi¯cantly greater than zero. In contrast to the ¯ndings of Domowitz, Hubbard and
Petersen (1987) (who used average variable cost rather than marginal cost), however,
industry SIC 3312 is here found to have a relatively large markup, in fact the largest
across all industries analyzed. This comparison with the ¯ndings of Domowitz, Hubbard
and Petersen should be viewed with some caution, though, as their sample is an annual
one from 1958 to 1981.
To evaluate the impact of supply- and demand shocks on the production decisions of
¯rms in the four industries, I next consider impulse response functions with respect to the
various model shock components (cost shocks, military spending shocks, and shocks to the
price index of related goods). Figures 1 to 4 plot the impulse response functions for actual
U.S. data and simulated model data. The impulse responses are computed as follows: For
both the actual data and model impulse responses, the time path of the exogenous variable
shocked (that is the cost shocks µ , 8i, military spending G , and the price of related
goods prel ) in response to a one standard deviation shock in the respective innovation
at time t = 1 is obtained. In a second step, this time path of the shocked exogenous
variables is used in an autoregressive speci¯cation containing Q and the current as well
as lagged values of all exogenous variables shocked to determine the resulting `actual
data' impulse response to a one standard deviation shock to the exogenous variable under
consideration.21 Correspondingly, for the model impulse response the model equilibrium
laws of motion are used in the second step. Also plotted in Figures 1 to 4 are 95%con¯dence intervals for the actual data impulse responses. The latter are computed using
standard Monte Carlo methods (based on 500 simulations) as described for example in
it
t
t
t
21
The number of lags in the autoregressive process in
the basis of Akaike's model selection criterion.
23
Q
t and the exogenous variables is determined on
Hamilton (1994). All impulse responses are in percentage deviations from steady states.
Figures 1a to 4a show that in response to a cost shock, ¯rms decrease output temporarily,
restoring the quantity produced to its original level in monotonic fashion, this being true
for both model and U.S. data impulse responses (the exception being the actual data
impulse response for industry SIC 3612, for which one observes an `overshooting'-type
readjustment of the industry's output to its original level).
However, in all cases the
model underpredicts the amplitude of the response to a one-standard deviation shock,
typically signi¯cantly so. The response to shocks in military spending in contrast di®ers
widely across industries (Figures 1b to 4b). The actual data impulse responses suggest
for all industries at least initially a
spending.
probably
decrease
in ¯rms' output to an increase in military
While this qualitative pattern is at ¯rst sight rather surprising (one would
a priori
expect in the `worst case' there to be little to no
increase
in industry
output in response to a raise in military spending), it may just indicate that government
military spending is countercyclical for the industries under consideration, an increase in
its quantity having so little positive e®ect on industry demand that it is far outweighed
by a simultaneous drop in the level of other demand components, potentially not part of
the model studied here. The model impulse response functions predict a positive e®ect of
military spending shocks on industry output for two industries (SIC 3351 and SIC 3612).
The e®ect for both industries is rather small, however, re°ecting the magnitude and (for
SIC 3351) lack of signi¯cance of
·G
in those industries' demand function. While in SIC
3312 actual data and model impulse responses match relatively well, industry SIC 3621's
model impulse response suggests an initial decrease in output followed by a monotonic
increase back to the pre-shock level. This is at odds with the corresponding actual data
impulse response, which exhibits an overshooting-type readjustment of ¯rms' output level
after the initial decline in output in response to a one standard deviation shock to
Gt
.
Taken together, the impulse responses in reaction to a military spending shock exhibit
su±ciently di®erent patterns for the model as compared to the U.S. data that the model's
prediction in this direction probably has to be disregarded. Finally, the responses to a one
standard deviation shock in the price of related goods (Figures 1c to 4c) are for most parts
qualitatively similar to those for a cost shock. Across all four industries output declines
(suggesting that the price index of related goods mostly captures complements rather
than substitutes), and reaches its pre-shock level after about 16 quarters. For industries
SIC 3312 and SIC 3612, the amplitude of the model's impulse response is signi¯cantly
lower than that found in the data, thus that the model's quantitative predictions are to
24
be viewed with caution in these industries.
Taking unconditional second moments and impulse responses together, the model performs quite well (by the standard of quantitative-theoretic models) in explaining industryspeci¯c cycles.
The four industries investigated here therefore have a common cyclical
structure in the sense that industry cycles can be explained by one common model framework. It remains to be seen, though, whether the model's performance could be further
improved if some alternative demand factors (which can be treated with con¯dence as
being orthogonal to ¯rms' cost shocks) were considered.
While the impulse responses are informative about how ¯rms' output decisions vary
in response to a (one-time) shock in
one
of the model's exogenous variables, it is also
of signi¯cant interest to study the cyclical behavior of production and pricing decisions
if
all
exogenous processes are simultaneously in°uencing ¯rms' behavior.
By analyzing
the average behavior of output price, marginal cost and markup across industry cycles,
it is possible to assess whether these cycles may have a simple common explanation,
such as changes in ¯rms' real cost of production or ¯rms' markup-behavior alone, as has
been suggested in some of the previous literature.
I therefore simulate time-paths for
the model's output, output price, marginal cost and markup, and use recession-recovery
patterns to extract the average cyclical behavior of the various series. Industry cycles are
dated by applying the algorithm of Bry and Boschan (1971) to the actual and simulated
22
output series for the various industries.
Table 4 reports the number of industry cycles
and average duration from peak to trough (in months) for the cycles so obtained.
For
all four industries, the number of cycles in the actual versus simulated series deviates
only by one, for all industries (except SIC 3621) there being one more cycle in the actual
data than in the simulated model data. This in turn leads to the average duration from
peak to trough being longer in the simulated output series than in the corresponding
U.S. output series. Figures 5 to 8 plot recession-recovery patterns for the four industries
for actual and simulated model output and price of output, as well as (model) marginal
cost and the (model) markup for all four industries. The recession-recovery patterns are
constructed as follows.
For each industry recession-recovery sequence, the changes in
the variable under consideration's smoothed two-quarter growth rates (that is changes of
the current quarter's realization of a variable relative to the average of the four preceding
22
The Bry-Boschan program used here was generously provided to me by Mark Watson. Monthly series
are created from the actual and simulated quarterly ones by setting the value in each month equal to the
corresponding quarterly value.
25
quarters) from their level at the cyclical peaks are computed, and then averaged across the
23
industry recessions.
Also graphed in the right columns of the ¯gures are one-standard-
error bands (computed using the levels of the smoothed two-quarter growth rates) as
measures of intercycle dispersion.
The vertical line in each graph depicts the middle
month of the quarter in which the peak occurred.
By default, output in the recession-recovery patterns (Figures 5a,b to 8a,b) exhibits a
procyclical pattern, its behavior in the quarters before the peak and during the recession
in the U.S. data being closely matched by the simulated model data. For some industries,
output rises from its level at the bottom of the recession slightly faster in the actual data
than in the simulated model data, this simply being an artifact of the models' average
peak to trough duration being larger than that observed in the actual data.
It should
also be noted that the spread as measured by one-standard-error bands is quite high for
certain quarters, as there is partially substantial dispersion in the strength and length
of the recessions in each industry with respect to which the recession-recovery patterns
average.
For all four industries considered, actual output prices (Figures 5c,d to 8c,d) peak
two quarters after the peak in the output series occurs and then fall, not recovering until
seven to eight quarters after the industry cycle peak. This pattern is fairly well matched
by the output price series for the simulated data from SIC 3312, even though prices there
decline not as sharply in the recession as the corresponding U.S. output price series. For
the other industries, model output prices replicate the patterns found in the data, but
the peak occurs two to three quarters before the peak in the corresponding actual output
price series. The actual and model recession-recovery patterns thus indicate that ¯rms in
the oligopolistic industries studied here achieve the highest level of output prices during
industry booms.
The behavior of marginal cost (Figures 5e to 8e), on the other side, di®ers quite
substantially across the four industries. The patterns range from leading procyclical (SIC
3612) to lagged procyclical (SIC 3312) to almost acyclical (SIC 3351 and SIC 3621). In
the latter two industries, marginal cost peaks seven quarters after the corresponding peak
in output, starting to rise sharply three to four quarters into the recession.This di®ering
pattern of marginal cost's cyclical variation across the industries analyzed reveals that a
theory which aimed to explain industry-speci¯c cycles mainly by changes in ¯rms' real
23 This methodology is similar to standard NBER-procedures used in the context of aggregate cyclical
°uctuations. See for example Moore (1983) for further details.
26
cost of production would not fare well.
Given the substantial variations in the cyclical behavior of marginal cost across industries, it is not surprising that the cyclical time-path of markups (Figures 5f to 8f ) is
also anything but uniform across the four industries, ranging from procyclical with early
recovery (SIC 3312 and SIC 3612), to (one-quarter) leading procyclical with a substantial drop only late in the recession (SIC 3351), to almost acyclical with an average level
during the recession slightly higher than in the months preceding the peak in output
(SIC 3612).
Since all four industries considered are within the class of producer goods
manufacturing ones, this suggests that previous ¯ndings of a uniform cyclical variation
of markups in producer goods manufacturing industries may have to be reconsidered.
Theories of industry-speci¯c cycles based on uniform cyclical variations of markups are
clearly rejected in light of the results obtained here.
5. Conclusion
In this paper I have constructed a model to study the cyclical behavior of production and
pricing decisions in oligopolistic industries, allowing for
a priori
unconstrained `°exible
functional form' production and demand functions, and strategic dynamic interaction
among heterogeneously informed ¯rms. The empirical implementation of the model for
a few four-digit-SIC level producer goods U.S. manufacturing industries has found the
model to capture cyclical movements in these industries by and large fairly well.
The
results indicate, however, that industry-speci¯c cycles cannot be explained by changes in
¯rms' real cost of production or ¯rms' markup behavior alone.
Future work should try to assess whether and to what extent industry-speci¯c elements may add to the explanatory power of the model in the directions in which it
does not do very well.
In particular, it may be possible to identify alternative and/or
additional demand factors which would enhance the model's performance regarding the
response of ¯rms' production decisions to demand shocks.
The reduced-form empirical
literature has also established an important role for the industries' degree of unionization
and ¯rms' ¯nancial structure in explaining oligopolistic ¯rms' production and pricing decisions. Their incorporation into the model could thus lead to an even closer match of the
model's variable comovements with those found in the data. It will clearly also be helpful
in this regard to broaden the database constructed so far, and conduct a rather exhaustive industry analysis.
Finally, it will furthermore be interesting to assess the cyclical
27
behavior of ¯rms' pro¯t outlook across industries.
Calculation of the expected present
discounted value of ¯rms' pro¯t requires solving a nonlinear heterogeneous information
rational expectations model conditional on the solution path for output, however, and
will require an innovative technical approach.
28
Appendix 1: Proof of Propositions 3.1 to 3.3
Proof of Proposition 3.1:
Using Assumption 2, (3.5) can be rewritten as
P
P
1 N
e
Qe t = '1 Qe t¡1 + 1
s=0 (N'2s + '3s ) N i=1 E (Qt+s j­it )
P
P
1s=0(N ¡ 1)'2sE(Qe t+sjªt) + N1 Ni=1 ºeit:
¡
(5.1)
As is customary for homogeneous information rational expectations models, the next step
is to reduce (5.1) to a ¯rst-order system. To accomplish such a reduction, de¯ne the
e
dimensional vector xt by (3.17).
Then, truncating future expectations of endogenous
variables in (5.1) after a forecasting horizon of
et
x =
where
J1 £1-
J1 periods, (5.1) can be rewritten as
P
M1xe t¡1 + MN2 Ni=1 E (xe t+1 j­it ) + M3E (xe t+1 jªt)
M PN E (x
1 PN
+ N4
i=1 e t j­it ) + N i=1 ºeit ;
(5.2)
M1 , M2, M3, and M4 are de¯ned in (3.8) to (3.11), and ºeit is given by (3.26).
To
e ¡1 in (5.2), one may use the transformation
e t = et C e t¡1 speci¯ed in (3.16) above and employed extensively in Binder and Pesaran
eliminate the lagged dependent variable xt
X
x ¡
x
(1995a) for the solution of homogeneous information multivariate rational expectations
models.
In (3.16),
C
is a
J1 £ J1-dimensional
matrix de¯ned as the solution to the
quadratic matrix equation (3.12). Applying the transformation (3.16) to (5.2), the latter
equation system becomes
et
X =
PNi=1 E ( e t+1 ­it) + ³ M C + M ´ PNi=1 E( e t ­it)
N
N
1 PN ºeit :
e
e
+M3 CE ( t ªt) + M3E ( t+1 ªt) + N
i=1
M2
N
X
X j
Advancing (5.3)
s
2
j
X
4
X j
(5.3)
j
periods and taking conditional expectations with respect to ­it ,
29
j
e t ­it ) to obtain
one can solve recursively forward for E (X
j
¡ He ¡ F ¡´ PE e + j
³
e 1 ¡1
I ¡H
6= E ´e j
¡1
³
P Ee j
P ¡1 I ¡ He ¡1 F ¡1 I ¡ He ¡1
1
2
+
6=
=1
³
´
P
¡1
e1
E e +1j
I ¡H
P ¡1 I ¡ He ¡1 F ¡1 I ¡¡1He ¡16= P E e
1
2
+ +1j
6=
=1
³
´
e
1
e 1 ¡1 E ºe j
"
I ¡H
´
P ¡1 I ¡ He ¡1 F ¡1 I ¡ He ¡1 ³E ºe j
e
1
"
1
2
+
=1
+ :
E e +1 j
i
F
T !1
e
E
+1j
e t ­it) = (IJ
E (X
1
+(
+
J1
(
s
)
J1
+(
J1
+
s
(
J1
+
s
(
J1
+
N
M2
s
(
N
s
M2
s
M2 C
N
j
j
N
(Xt
j
N
)
J1
­it )
M4
M2
ªt ) +
t
J1
(Xt ªt )
i
+
i
)
J1
(
(
N
j
N
N
)
J1
+
M3
+
(
N
)
N
(Xt
M3
)
N
M2
)
M4
1
J1
N
J1
M2
)
J1
+(
J1
M2 C
)
J1
1 M2
1)
(Xt
i
e
(Xt
r
f
=0
(5.4)
ªt )
s
N
(
it
t
s
ªt ) +
ºi
i;t
N
(Xt
s
­it ). Observing
and substituting
­it ) back into (5.3) yields
X2 f
J
Xt =
ªt )
ºi
fall inside the unit circle, letting
(5.4) and the forward solution for
s
ªt )
In analogous fashion one may also solve recursively forward for
that by ( ) all eigenvalues of
(Xt
i
j
e
Mr+1E (Xt+r ªt ) + ´t¤ ;
(5.5)
where the matrices Mr , r = 1; 2; : : : J2 are de¯ned in (3.20) to (3.22), and ´t¤ is given by
e
e
(3.25). Using the de¯nitions of Zt in (3.15), Ut in (3.14), ´t in (3.24), and the coe±cient
matrices H1 in (3.18) and H2 in (3.19), it is a simple step to rewrite (5.5) as (3.13).
Proof of Proposition 3.2:
Using the Jordan normal form ¤ of H3 , (3.13) may be rewritten in the form of two
uncoupled blocks of equations
Ut
t
;
and
Ut
e
j
1 = ¤1E (U +1 1 ª ) + ´ 1;
j
t
(5.6)
t
2 = ¤2E (U +1 2 ª ) + ´ 2 ;
¡1
where Ut = T ¡1Ut, ´t = T ¡1 H1 ´t , Ut
0
1
U1
A, and ´
= @
t
;
t
t
t
Ut
30
2
0 1
´1
A.
= @
t
t
´ t2
(5.7)
Also de¯ning
»t
0
=@
» t1
»t
2
1
A, the solutions to (5.6) and (5.7) are respectively given by
U1=
t
1
X
r
=0
¤1 E (´t+r;1jªt ) + » t1 ;
r
and, since ¤2 is a nil-potent matrix of order
U2=
X¡1
Á
(5.8)
2,
Á2
t
r
=0
¤2 E (´t+r;2 jªt ) + » t2:
r
(5.9)
From (5.8) and (5.9) the solution (3.29) readily follows.
Proof of Proposition 3.3:
Upon writing the solution for
e
Qt
in autoregressive form, the remaining steps necessary to
derive (3.39) are conceptually similar to the techniques used in the proofs of Proposition
1 and Proposition 2, and are therefore skipped here.
31
Appendix 2: Description of the Data Set
To conduct the empirical analysis, data series for all model variables in the four fourdigit-SIC level industries chosen are constructed. The sample for all series except
1967:Q1 to 1990:Q4 (
G
t
G
t
is
is only available from 1972:Q1 onwards).
All price indices are in real terms, being divided by an aggregate U.S. GDP-de°ator (as
reported by the Bureau of Economic Analysis, National Income and Product Accounts),
and are normalized to a value of one in the ¯rst quarter of 1967. Unless otherwise noted,
quantity indices are obtained by dividing the nominal value of a series by its real price
index.
Some series are available only on an annual basis, and in these cases linear interpolation
is used to generate quarterly observations. Some other series are reported on a monthly
basis, and for these three-monthly averages are employed as quarterly values.
Where
necessary, series are seasonally adjusted using a variant of the X-11 ¯lter.
A brief description of how each series is constructed is given in what follows.24
Q:
t
Value added plus cost of materials, obtained from the Bureau of the Census,
Census of Manufactures and Annual Survey of Manufactures.
p : Producer price index, reported by the Bureau of Labor Statistics, Producer Price
t
Indices.
k : Using four-digit-SIC level time series on new and used capital expenditures (from
t
the Bureau of the Census, Census of Manufactures and Annual Survey of Manufactures),
as well as two-digit-SIC level ¯xed nonresidential private capital stock and corresponding
investment series (from the Bureau of Economic Analysis, Fixed Reproducible Tangible
Wealth in the United States), capital stock series at the four-digit-SIC level are computed.
Capital services series are extracted from these stock data in a second step using a proportionality factor between capital services and capital stock data for U.S. manufacturing
(the latter two obtained from E.R. Berndt and D.O. Wood (1975): Technology, Prices and
the Derived Demand for Energy, Review of Economics and Statistics, 57, 259 - 268, and
Bureau of Economic Analysis, Fixed Reproducible Tangible Wealth in the United States).
wkt:
Assuming strict proportionality between capital purchase price and the rental
price of capital, this series is obtained by dividing two-digit-SIC level current-cost valuation ¯xed nonresidential private capital by the corresponding series in constant-cost
24
A more detailed description is contained in a note available upon request.
32
valuation (both series from the Bureau of Economic Analysis, Fixed Reproducible Tangi-
ble Wealth in the United States).
l
t:
Computed on the basis of the number of production workers and their average
weekly hours, as reported by the Bureau of Labor Statistics, Employment, Hours, and
Earnings.
wlt
:
Average hourly earnings for production workers, obtained from the Bureau of
Labor Statistics, Employment, Hours, and Earnings.
mt
: Materials minus electric energy purchased, from the Bureau of the Census, Census
of Manufactures and Annual Survey of Manufactures.
wmt
: Price index for intermediate materials, supplies and components of durable goods
manufacturing, collected by the Bureau of Labor Statistics, Average Price Data.
et
Electric energy purchased (in kwh), from the Bureau of the Census, Census of
:
Manufactures and Annual Survey of Manufactures.
wet
:
Cost of electric energy purchased, as reported by the Bureau of the Census,
Census of Manufactures and Annual Survey of Manufactures.
Gt
: National defense purchases of durable goods (real value), taken from Bureau of
Economic Analysis, National Income and Product Accounts.
prelt
: Price index of `related goods', constructed as the (value added plus materials)
weighted producer price index of other four-digit-SIC level industries within the same twodigit-SIC level industry (data taken from Bureau of the Census, Census of Manufactures
and Annual Survey of Manufactures, and Bureau of Labor Statistics, Producer Price
Indices).
33
Table 1
Parameter Estimates
(t-statistics in brackets, where applicable)
bkk
bkl
bkm
bke
bll
blm
ble
bmm
bme
bee
bk
bl
bm
be
°k
°l
°m
°e
½µ1
SIC 3312
SIC 3351
SIC 3612
SIC 3621
.0606 (2.38)
.0148 (1.91)
.0190 (1.40)
.0116 (.70)
-.0253 (-2.00)
.0135 (2.62)
.0233 (3.64)
.0255 (4.52)
-.0153 (-1.65)
-.0143 (-4.49)
-.0058 (-1.98)
-.0021 (-.54)
.0145 (2.54)
-.0050 (-1.76)
-.0081 (-4.56)
-.0041 (-1.33)
.1778 (1.62)
.2025 (4.64)
.1038 (.53)
.0678 (.27)
.0696 (1.55)
-.1070 (-3.81)
.1350 (3.94)
.1182 (3.99)
.0134 (2.57)
-.0090 (-2.51)
.0083 (3.76)
.0099 (4.56)
.6667 (4.84)
.7499 (5.79)
-.0964 (-.49)
-.0299 (-.17)
-.0010 (-.24)
.0176 (7.56)
.0034 (2.39)
.0016 (1.16)
.0013 (.17)
.0029 (.93)
.0027 (.50)
5.21E-05 (.01)
231.80 (1.94)
17.27 (6.92)
.8103 (.13)
12.46 (.97)
-166.88 (-.20)
34.28 (1.17)
-32.55 (-.29)
61.40 (.24)
-309.01 (-.27)
-30.43 (-.26)
247.76 (2.10)
400.50 (2.27)
48.87 (.89)
8.25 (4.43)
2.15 (.69)
5.61 (1.17)
-9.99E-07 (-2.48)
-2.16E-06 (-2.86)
-1.02E-05 (-2.34)
-6.95E-06 (-2.25)
-4.73E-06 (-1.67)
-1.25E-05 (-1.43)
-4.22E-05 (-.54)
-3.85E-06 (-.06)
-6.06E-06 (-1.53)
3.59E-05 (1.03)
.0002 (1.96)
7.25E-05 (1.71)
-4.68E-07 (-2.55)
1.54E-07 (.27)
-1.09E-06 (-.51)
-1.05E-06 (-.92)
½µ2
.8489 (8.61)
1.0915 (11.58)
.7873 (7.67)
.8335 (8.44)
-.3246 (-3.29)
-.3821 (-4.07)
-.1620 (-1.58)
-.3161 (-3.21)
¾$
.0155
.0184
.0138
.0143
30727.15 (5.31)
30727.15 (5.31)
30727.15 (5.31)
30727.15 (5.31)
.4471 (4.31)
.4471 (4.31)
.4471 (4.31)
.4471 (4.31)
3190.27
3190.27
3190.27
3190.27
.2259 (4.00)
.2798 (4.79)
.2090 (4.01)
.1783 (4.08)
1.2825 (12.64)
1.3482 (14.40)
1.3629 (14.05)
1.3748 (14.32)
-.5046 (-4.97)
-.6034 (-6.43)
-.5592 (-5.75)
-.5652 (-5.91)
.0238
.0173
.0105
.0116
½G
0
½G
1
¾G
prel
½0
prel
½1
prel
½
2
¾prel
34
SIC 3312
SIC 3351
SIC 3612
SIC 3621
wk
.2305 (3.45)
.2305 (3.45)
.3135 (4.09)
.3135 (4.09)
wk
.9421 (9.18)
.9421 (9.18)
.8670 (8.47)
.8670 (8.47)
wk
-.1612 (-1.57)
-.1612 (-1.57)
-.1705 (-1.68)
-.1705 (-1.68)
.0074
.0074
.0063
.0063
wl
.3609 (4.47)
.3609 (4.65)
.4392 (5.07)
.3054 (3.76)
wl
1.0184 (10.24)
.8752 (8.77)
.7299 (7.12)
.7866 (7.55)
wl
-.3181 (-3.20)
-.2326 (-2.34)
-.1982 (-1.91)
-.0891 (-.8447)
½0
½1
½2
¾wk
½0
½1
½2
.0145
.0112
.0096
.0125
wm
.2421 (4.5026)
.2421 (4.50)
.2421 (4.50)
.2421 (4.50)
wm
1.2774 (14.08)
1.2774 (14.08)
1.2774 (14.08)
1.2774 (14.08)
wm
-.5033 (-5.54)
-.5033 (-5.54)
-.5033 (-5.54)
-.5033 (-5.54)
¾wm
.0177
.0177
.0177
.0177
we
.3120 (3.34)
.3409 (3.83)
.3972 (4.35)
.2702 (3.69)
we
1.0050 (9.78)
.9844 (9.60)
.9899 (9.86)
1.0073 (9.88)
we
-.2059 (-2.00)
-.2483 (-2.40)
-.2868 (-2.86)
-.2278 (-2.25)
¾we
.0393
.0361
.0431
.0198
·G
-14.67 (-7.13)
1.04 (2.57)
.1239 (1.01)
-.7693 (-3.46)
-11728.30 (-19.41)
-1549.89 (-16.53)
-1211.25 (-35.41)
-1632.23 (-26.01)
240.49 (77.13)
74.58 (94.73)
71.12 (106.69)
82.33 (96.20)
.0280 (1.58)
.1445 (2.04)
.3876 (1.66)
.3022 (2.31)
¾wl
½0
½1
½2
½0
½1
½2
·prel
·Q
´
Not estimated were the discount factor,
the number of ¯rms,
N,
¯,
which was set to a value of
¯
for which a sensitivity analysis was conducted.
35
= 1:03(¡:25) , and
Table 2
Second Moments (Standard Errors in Brackets)
Standard Deviation of Output and Standard Deviations Relative to Output
SIC 3312
U.S. Data
Model Data
¾Q
¾p / ¾Q
¾K / ¾Q
¾L / ¾Q
¾M / ¾Q
¾E / ¾Q
10.94
.29
.39
.76
.87
.65
13.19 (2.23)
.34
.20
.56
.90
.62
Standard Deviation of Output and Standard Deviations Relative to Output
SIC 3351
U.S. Data
Model Data
¾Q
¾p / ¾Q
¾K / ¾Q
¾L / ¾Q
¾M / ¾Q
¾E / ¾Q
12.08
.38
.34
.80
1.15
.72
11.16 (1.67)
.29
.21
.70
1.13
.65
Standard Deviation of Output and Standard Deviations Relative to Output
SIC 3612
U.S. Data
Model Data
¾Q
¾p / ¾Q
¾K / ¾Q
¾L / ¾Q
¾M / ¾Q
¾E / ¾Q
6.57
.44
.57
1.07
1.05
.82
5.86 (1.28)
.43
.43
.81
1.02
.59
Standard Deviation of Output and Standard Deviations Relative to Output
SIC 3621
U.S. Data
Model Data
¾Q
¾p / ¾Q
¾K / ¾Q
¾L / ¾Q
¾M / ¾Q
¾E / ¾Q
6.74
.33
.44
1.08
.98
.68
8.28 (1.48)
.26
.18
.80
.93
.44
36
Autocorrelations of Output - SIC 3312
U.S. Data
Model Data
Q
Q(+/-1)
Q(+/-2)
Q(+/-3)
Q(+/-4)
1.00
.76
.46
.14
-.16
1.00 (.00)
.91 (.03)
.73 (.09)
.52 (.15)
.34 (.19)
Autocorrelations of Output - SIC 3351
U.S. Data
Model Data
Q
Q(+/-1)
Q(+/-2)
Q(+/-3)
Q(+/-4)
1.00
.78
.53
.23
-.09
1.00 (.00)
.87 (.04)
.62 (.10)
.38 (.16)
.20 (.19)
Autocorrelations of Output - SIC 3612
U.S. Data
Model Data
Q
Q(+/-1)
Q(+/-2)
Q(+/-3)
Q(+/-4)
1.00
.84
.62
.38
.13
1.00 (.00)
.95 (.02)
.83 (.08)
.67 (.14)
.52 (.19)
Autocorrelations of Output - SIC 3621
U.S. Data
Model Data
Q
Q(+/-1)
Q(+/-2)
Q(+/-3)
Q(+/-4)
1.00
.83
.58
.28
-.02
1.00 (.00)
.93 (.03)
.77 (.08)
.58 (.14)
.40 (.19)
37
Table 3
Average Markups25
25
My Model
DHP
SIC 3312
.330
.202
SIC 335
.103
.143
SIC 3612
.230
.271
SIC 3621
.284
.270
Note: The average markups computed by Domowitz, Hubbard and Petersen (1987), reported in the
second column under the heading `DHP', are based on annual data for the sample 1958 to 1981.
38
Table 4
Actual versus Simulated Cycles - Number of Cycles and Average Duration from Peak to
Trough (in Months)
SIC 3312
Number of Cycles
Average Duration Peak to Trough
U.S. Data
6
20
Model Data
5
31
SIC 3351
Number of Cycles
Average Duration Peak to Trough
U.S. Data
7
15
Model Data
6
26
SIC 3612
Number of Cycles
Average Duration Peak to Trough
U.S. Data
7
19
Model Data
6
22
SIC 3621
Number of Cycles
Average Duration Peak to Trough
U.S. Data
5
26
Model Data
6
26
39
The ¯gures are not contained in this downloadable version of the paper.
They will be included with the next version posted.
40
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