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http://www.archive.org/details/asymmetricbusineOOacem
DEWEY
!
HB31
.M415
working paper
department
of economics
ASYMMETRIC BUSINESS CYCLES:
THEORY AND TIME-SERIES EVIDENCE
Daron Acemoglu
Andrew
95-24
Scott
Aug. 1995
massachusetts
institute of
technology
50 memorial drive
Cambridge, mass. 02139
)i
ASYMMETRIC BUSINESS CYCLES:
THEORY AND TIME-SERIES EVIDENCE
Daron Acemoglu
Andrew Scott
95-24
Aug. 1995
MASSACHUSETTS INSTITUTE
,
--.•ogy
NOV 1 4
1995
95-24
ASYMMETRIC BUSINESS CYCLES:
THEORY AND TIME-SERIES EVIDENCE
Daron Acemoglu
Department of Economics
Massachusetts Institute of Technology
and
Andrew
Institute of
Scott
Economics and Statistics and
Oxford University
AH
Souls College,
August 1995
JEL
Classification:
E32
Keywords: Asymmetries, Cyclical Indicator, Intertemporal Increasing Returns
Regime Shifts, Temporal Agglomeration, Unobserved Components.
Abstract
We
offer a theory of
who have been
economic fluctuations based on intertemporal increasing
returns: agents
active in the past face lower costs of action today. This specification explains the
observed persistence
in individual
and aggregate output fluctuations even
shocks, essentially because individuals respond to the
same shock
in the presence of i.i.d
differently
depending on
their
A feature of our model is that output growth follows an unobserved
components process with special emphasis on an underlying cyclical indicator. The exact process for
output, the sharpness of turning points and the degree of asymmetry are determined by the form that
heterogeneity takes. We suggest a general formulation for models with latent cyclical variables which,
under certain assumptions, reduces to a number of state space models that have been successfully
used to model U.S. GNP. Using our general formulation we find that allowing for richer heterogeneity
recent past experience.
fit to the data and also that U.S. business cycles are asymmetric, with
asymmetry manifesting itself as steep declines into recession. We estimate a strongly persistent
cyclical component which is not well approximated by discrete regime shifts nor by linear models.
Our estimates of the relative size of intertemporal returns needed to explain U.S. GNP vary but on
the whole are not implausibly large.
enables us to obtain a better
this
We owe a special debt to James
Hamilton whose insightful comments improved an
earlier version
of
this paper.
We also thank
Philippe Aghion, Charlie Bean, Ricardo Caballero, Paul David, Steve Durlauf, Vittorio Grilli, Christopher Harris, Rebecca
Henderson, Per Krusell, Jim Malcomson, John Moore, Chris Pissarides, Danny Quah, Kevin Roberts, Donald Robertson,
various seminar participants and two
anonymous
referees for useful
comments. Remaining errors are the authors'
responsibility.
1.
Introduction
Aggregate economic fluctuations are characterized by successive periods of high growth
followed by consecutive periods of low activity. The transitions between these periods of high and
low growth are often marked by sharp turning points and considerable evidence suggests
moments
the stochastic properties of the
Neftci (1984), Diebold and
economy change and
paid to a variety of coincident and leading indicators
(e.g.
reflected in the considerable attention
is
Stock and Watson (1989), and the papers
and Moore (1991)).
in Lahiri
A
to
display asymmetries, see inter alia,
Rudebusch (1989), Hamilton (1989) and Acemoglu and Scott (1994). The
importance of tracking these movements in the business cycle
is
that at these
natural
way
1
to
model temporal agglomeration and asymmetries
assume non-convexities, such
in
economic fluctuations
as discrete choice or fixed costs at the individual level, because
such non-convexities imply that individuals concentrate their activity in a particular period.
It is this
implication which has been analyzed with considerable success in the (S,s) literature. However, while
the presence of fixed costs can account for the discreetness of
naturally lead to persistence
in the
-
once an individual undertakes an action they are
near future. The key reason
is that
exploited within a period
.
As
it
less likely
does not
to
do so
economies of scale arising from fixed costs can be
a consequence persistence in aggregate fluctuations relies
aggregation across heterogeneous agents: either
profitability
points,
although the presence of fixed costs leads to increasing
returns, these are infratemporal; the full extent of
2
economic turning
of investment for others
(e.g.
more agents investing
on
in the past increases the
Durlauf (1991) and (1993)) or aggregate shocks affect
agents differently, leading to a smoothed response over time (e.g. Caballero and Engel (1991)).
This paper emphasizes an alternative explanation of persistent individual and aggregate output
fluctuations
returns
by suggesting
from an
an agent
that fixed costs introduce intertemporal increasing returns, that
activity this period are higher if the activity occurred last period
who was
active in the recent past
is
more
likely to
is,
the
and as a consequence
be active now. In the presence of such
intertemporal linkages temporal agglomeration arises from two sources; active agents maintaining a
high/low activity level in successive periods because of intertemporal increasing returns and agents
acting together, either in response to a
show how a model of
common
shock or because of strategic complementarities.
the former explains a
fluctuations and also offer a
number of empirical
framework which enables an economic
component models of U.S. output which emphasize the underlying
The relevance of intertemporal increasing
is
1
2
persistent
We
and
(ii)
returns depends
We
features of business cycle
interpretation of
unobserved
cyclical indicator.
on
(i)
whether individual behavior
whether firm's technology and costs contain important intertemporal aspects. The
use this term, following Hall (1991), for the bunching of high economic activity over time.
To overcome
this
problem empirical implementations of
(S,s)
models sometimes include time-to-build
considerations or decreasing returns at high levels of investment, e.g. Caballero and Engel (1994).
answer
to these questions will vary
depending upon the type of
instance, in the case of radical changes in the capital stock, e.g.
under consideration. For
activity
Cooper and Haltiwanger's (1993)
automobile retooling, intertemporal effects are unlikely to be important, and
lack of persistence in these cases.
investment
is
persistent
new
investment in
supported by the
However, Section 2 surveys micro evidence
that firm level
and findings from technology studies, the management science
many
theory which supports the notion that
organizational
this is
literature
important discrete decisions
technology, product development, innovation, maintenance) exhibit
and
(e.g.
some degree
of intertemporal increasing returns. These empirical findings motivate the theoretical model of Section
3 in which a firm has to decide each period whether to undertake both maintenance and investment.
Maintenance
modelled as having two
is
technologies and
(ii)
facilitating the
effects:
new
adoption of
innovations.
leads to intertemporal increasing returns: firms find
it
The
interaction of these
when
the firm has invested last period, and a natural
it
roles
a result, investment
asymmetry
individual behavior; in response to a range of shocks, the agents will find
if
As
two
newly adopted
profitable to maintain the
technologies and this in turn reduces the costs of adopting future innovations.
costs are lower
of existing
increasing the productivity
(i)
is
introduced in
profitable to invest only
they have invested in the recent past.
Section 4 examines
aggregate economic fluctuations
the
how our model
intertemporal increasing returns, and illustrates
implied by
individual
level
accounts for the main features of
temporal agglomeration and asymmetric fluctuations: the significance and persistence of a cyclical
component
in
asymmetries.
output fluctuations,
An
important feature of our model
the persistence of business cycles
make
and the importance of turning points and business cycle
direct contact with a
model of output growth,
is its tractability.
and the sharpness of turning points are determined but we also
number of econometric
e.g.
Not only can we characterize how
studies
which
rely
upon an unobserved component
Harvey (1985), Watson (1986), Clark (1987). This generality enables
us to nest a variety of different types of business cycle asymmetries.
Section 5 where, in the context of our model,
points between
booms and
we
find that the
this
theme
the circumstances under
is
pursued in
which turning
recessions are most marked, so that output growth approximates a shifting
regime process as in the discrete
exercises,
we examine
And
state
Markov model popularized by Hamilton
form of heterogeneity
(1989). In
(the distribution of idiosyncratic shocks)
is
all
these
the
main
determinant of the time-series properties, and also of the asymmetric nature of fluctuations.
Section 6 takes our model to data. First,
model and a number of estimated regime
shift
ability to nest different
form of heterogeneity.
exercise yields a
look
models
returns necessary to explain U.S. business cycles.
its
we
at the
connection between our theoretical
to infer the size of intertemporal increasing
However, the
real
advantage of our formulation
is
unobserved component models with different specifications regarding the
We
thus estimate
number of interesting
two versions of our general model on U.S.
results. First,
we
find that a version of our general
data. This
model with
normally distributed idiosyncratic shocks does considerably better than the version with uniformly
distributed shocks, but this latter
model
a
is
precisely
what previous empirical work has used. Therefore, our
offers an alternative time-series representation
grounded
in
economic theory
that outperforms
the difference between the versions can be heuristically
number of popular models. Moreover,
thought to be the importance of asymmetry terms in the version with normal shocks. Thus, our
and
investigation also establishes that U.S. business cycles are asymmetric
illustrates that these
asymmetries manifest themselves by sharp downswings in economic activity which cannot be captured
by linear models.
Since our formulation
make
estimation results to
derived from an underlying economic model,
is
inferences about the nature of the aggregate
find that the cyclical dynamics of the U.S.
GNP
instance,
shifts
we
between
is
also confirmed
by the
fact that
estimate the variance of idiosyncratic shocks to be larger than that of the aggregate shock.
We
modest and
that
by the estimation
also again find that the size of non-convexity implied
overall the
2.
also use the
that the idiosyncratic uncertainty is playing
an important role in the propagation of economic shocks and this
we
economy. For
cannot be characterized as discrete
model implies
different regimes, which, in our theoretical
we
model provides a good
fit
result is
to the data.
Individual Persistence and Intertemporal Increasing Returns
Our
interest in temporal
agglomeration focuses attention on problems in which fixed costs are
important and so agents have to
activity.
A
make
a discrete choice about whether or not to perform a certain
crucial feature of our analysis will be the investigation of
qualitative choice
(i.e.
whether or not
to
perform a certain
how
an agent's current
activity) influences the
same decision
tomorrow. In the standard case, an activity that requires a fixed cost will be bunched within a period
of time, essentially
scale.
This observation
period of activity
is
because fixed costs imply the existence of infratemporal increasing returns to
is
lies at the heart
of (S,s) models
followed by periods of inactivity
(e.g.
Scarf (1959)) and implies that a brief
at the individual level.
considerable evidence supporting (S,s) models (e.g. Bertola and Caballero (1990), Caballero and
Engel (1994),
Doms
and Dunne (1994), Cooper, Haltiwanger and Power (1994)), there
substantial evidence that firm level investment decisions are characterized
For instance, using
UK
firm level data,
effects in investment behavior.
= a + 0.856
is
However, while there
I,.
1
/K .,(l-0.122
t
Bond and Meghir (1994)
They estimate equations
I t.,/Kt. 1 )+pZ,
+
e where
t
Z
t
is
French,
German and
Bond
et al (1994) estimate
UK firm panel
find significant autoregressive
and find I/K,
a vector of firm relevant variables and
its
AR(1) and AR(2) models using Belgian,
of the autoregressive coefficients around 0.3. Even the evidence in
concentrated investment bursts are spread
e,
sample mean yields an AR(1)
data finding strongly significant investment lags, with the
that there is significant investment occurring in
also
significant persistence.
for the investment-capital ratio
a white noise disturbance. Evaluating the quadratic term at
coefficient of around 0.75.
by
is
Doms and Dunne
sum
(1994) reveals
most years of the sample for each firm and
that
over several years. These findings suggest that while
infratemporal economies of scale (lumpiness) are important there
which can profitably be exploited. This
raises the question,
may
also be intertemporal linkages
what form of investment produces these
intertemporal linkages?
The most obvious form of intertemporal economies of
past experiences increase productivity or reduce costs.
new knowledge
incorporating
More
scale
is
learning-by-doing, whereby
a slow and costly process, limiting the degree to
is
which productive
investments can be undertaken within a period. However, the more familiar an individual
most recent technology vintages, the cheaper
adopt the
latest version. In contrast
with the
is
an individual
most recent vintage faces compatibility problems when dealing with the
not using the
technology.
to
it is
The
cost incentives
where
explicitly, consider the case
frontier
be that only limited innovative moves are taken within each period, with
result will
making forward
steps
more
likely
to
come from
active agents.
The idea
intertemporal linkages might lead agents to take a sequence of small investment steps
empirical support in studies of technological innovation. In this literature a distinction
given
is
is
that
drawn
between incremental improvements of an existing technology and radical improvements which destroy
the existing technology and start an alternative production or organizational method, e.g.
"incremental change" versus
and Anderson's (1986)
widespread agreement that the
less
"technological discontinuities".
Tushman
There
is
spectacular incremental changes account for most of the
productivity gains (see Abernathy (1980),
Myers and Marquis (1969) and Tushman and Anderson
(1986)).
More
more
In
importantly for our paper, there
come from
likely to
Freeman's (1980,
discoveries
words
and opening up new
frontier by one
who have been
firms
p. 168)
is
"the
also a consensus that incremental innovations are
active at the earlier stages of product development.
advance of scientific research
technical possibilities, a firm which
means or another may be one of the first
to realize
is
is
constantly throwing up
new
able to monitor this advancing
a new possibility". Arrow (1974)
and Nelson and Winter (1982) also emphasize the advantages possessed by incumbent innovators
in
being able to further cope with incremental changes. For instance, Nelson and Winter stress the
importance of engineers understanding
new
technologies, suggesting that firms
who
innovate have
an advantage in doing so again. Abernathy (1980, p.70), using evidence from industries diverse as
automobiles, airlines, semi-conductor and light bulb manufacturing, notes that "Each of the major
companies seems
to
have made more frequent contributions
previous innovations in a field facilitate future innovations.
are fixed effects:
industry wide
some
firms
may simply be good
at
in
a particular area" suggesting that
One possible explanation of these findings
innovating in certain areas. However, the
work of Hirsch (1952), Lieberman (1984) and Bahk and Gort (1993) suggests
than just individual fixed effects
form of investment and
its
is
operating. In the remainder of this paper
implications for aggregate fluctuations.
4
we
that
shall focus
more
on
this
Individual Behavior in the Presence of Intertemporal Increasing Returns
3.
(i)
The Environment
We
assume
that agents (or firms) are risk-neutral
their return (either profit or utility).
from technological progress. More
The key decision
explicitly,
technology has a stochastic productivity that
is
and forward looking, and seek
whether or not
period
3
To
.
each period a new technology becomes available. This
is
revealed at the beginning of the period.
obtain the highest return from this innovation,
productivity
y,=>Vr a o
if
a,,,
+
end of the
investment
(s t )
compatibility with existing technologies
installation period there
(
no maintenance
effort at the
adopted
4
(e.g.
s,
is
fine-tuning).
(s t =l),
the productivity of the firm
end of the period (m =0),
t
and equal zero otherwise.
this period,
is
higher permanently.
avoided when m,=l).
assumed
to
is
on decisions
is
that
(i.e.
C m =y^). A
why
the productivity shock in (1)
is
there
denoted by Oq (our results are
F(.).
Maintenance costs
have a discrete nature,
The
e.g.
whether to adopt the
discrete nature of the choice
multiplicative with the investment decision; productivity
shocks are associated with one vintage of technology and therefore only affect output when the
technology
is
performance of the
latest innovation.
More
specifically,
we assume
and y2 are positive so
next period are lower. Here y2
is
that
when equipment
maximum
investment costs are given by:
cr(YrY2™,-i>,
y,
new
adopted.
In this model, maintenance also has an additional role other than ensuring the
where both
is
distinctive feature of (1) is that
t
recent technology vintage or whether to maintain existing machines.
explains
If the
We assume that u, is a serially uncorrected random
be equal to a positive constant, y
as in (S,s) models, our focus
When
1
this increase in productivity is not as large as
shock to the productivity of investment with distribution function denoted by
are
Under these
and m, are binary decision variables that equal
t
depreciation
the option to gain
0)
could be (a,-0C2+u instead of o^+u,). Deterministic depreciation
if
is
a i +M />V a:2y /( 1 - m /)
and maintenance (n\) are undertaken
is
unchanged
software. If the
is
a, and o^ are positive parameters and
new technology
it
at the
its
from the innovation by maintenance
assumptions the firm's output
where
new
The agent
adopted, the productivity of the agent increases permanently, starting from the current
is
needs to be monitored. In particular,
additional
maximize
to invest in order to benefit
decides whether to adopt this technology or not, for instance whether to install a
technology
to
(2)
is
maintained the firm's investment costs
the extra investment cost incurred by a plant
which did not maintain
Time-to-build considerations can easily be incorporated in the return function and only serve to change the timing
of returns.
4
See Pennings and Buitendam (1987) for the importance of maintenance type
activities.
last period. In
terms of our computing example,
assistance
new program
is
removed
is
beginning of the period for installation
at the
not maintained, technical
(to
remove bugs) and
at the
(for fine tuning to increase productivity if so decided).
Each period
the firm decides whether to invest (s
t
=0
or 1) and also whether to maintain (m,=0
Denoting the discount factor of the representative firm by P and the per period return by
1).
we
computer
less costly. In contrast, if the
is
necessary both
end of the period
or
existing bugs in the system need to be
implementation of the new software, implying that next period the instalment
to ensure the optimal
of another
all
r(.),
have:
r(s
a a +M
>'/y-r o ( i
+
jn lv;m tij _ vyltj _ v u lv)=
lv
/^-a^v<1 "m^" Y m<v-(YrY2m
o
and the maximization problem of the firm
max
E,{ Jl^r(s
subject to (1) and taking
y,.,,
the choice variables are
s I+j
Whether
u
t
,
at
time
y
;-i> / +;
/+
t is:
^ntv;ml+j_^H.vu ^ }
lv
t
m,., as given. In
and m,+j
(4)
period
t+j, the state
variables are m,^.,, u t+j , y t+j ., and
.
the firm invests in any period depends
on u In
.
t
contrast, the return to
only depends on whether or not investment occurs. If no current investment
benefit of maintenance
is
the cost reduction that
occurs. In contrast, if there
is
it
when
there
is
undertaken, the only
brings in the following period
if
investment then
current investment, future productivity also increases by
there are three possibilities regarding maintenance:
maintain only
is
investment. Because
we
rather than a fixed characteristic,
always maintain
(i)
we wish
concentrate on
(iii).
to
(ii)
make maintenance
To
this
maintenance
ctj.
As
a result
never maintain,
(iii)
a decision of the firm
end we assume
5
:
Assumption A:
0Y 2 < Y ° <A_
H
(5)
1-/3
The
first
in the
inequality can be understood by noting that
absence of current investment:
otherwise there
Assumption
j3y 2
I
A
is
is
no
if
there
is
Py2
is
the
maximum
benefit
investment next period costs are lower by y2
benefit. Therefore, the first part of the inequality implies
stronger
dF(u)<y <
we
than
+/3y
7
|
require
dFCu)
but
,
from maintenance
simpler
where
a\,
to
and
understand
go,
than
the
maintenance
necessary
are constants defined below.
is
,
not
condition,
worthwhile just to obtain future cost savings. In the presence of current investment, the
from investment
is
minimum
gain
the present value of the productivity increase due to maintenance, (^/(l-P).
Consequently the second part of the inequality
profitable to maintain if there
is
current investment.
It
follows that
where the firm only maintains when
attention to the case
even without future cost savings,
states that
it
when
(5) holds,
invests, thus
m =s„
t
we can
is
it
limit our
and the per period
return simplifies to:
Kv^iVJ^^ai^"Mv^
where 5 =Y +Yi and 5 =y2 Assumption
-
l
A therefore enables
(6)
us to write our problem in a
way which
focuses on the intertemporal increasing returns arising from the interactive term 5,s t+j st+j.,.
generally, (6) can be interpreted as the period return for a firm
investment costs,
firms
(ii)
who
if it
invested last period.
Under
More
which benefits from lower current
this interpretation
our model captures the idea that
are used to adapting to a changing environment face lower costs of further changes.
Optimal Decision Rules
Using
(6) the
maximization problem
at
time
t
can be written in a dynamic programming form
with the value function V(.) satisfying:
V(yt_^ _ v u)=sup
t
{/-(^;y / . 1 ^. 1 ,u / )+/3£; / K(y/ . 1 -a0+ (a 1+ v /
Solving the agent's optimization problem
Proposition
s
1:
we
>s^^ /+1 )}
(7)
obtain (proof in the appendix):
The value function
r \ and
^(y
M ^ M ,u><V<^M +<to-i +te
*/"^<«V w rVi
s=0 and
K^Qo-to^
if
is
the unique function that satisfies (7)
To understand the dichotomous
does not invest
it
(8)
Vfy,.^,.^,)^^,-!
(s t =0).
(<|)'s
denote constant coefficients).
nature of the value function, consider the case where the agent
The disturbance u
t
is
irrelevant to future profits
does not matter whether the firm invested/maintained
only upon
y,.,.
However,
The optimal choice of
value.
Shocks below
technology.
function.
The
if the
st
this
critical
last
and with no investment
costs,
period and so the value function depends
firm invests the value function depends linearly upon y .„
t
depends on whether the investment shock, u„
is
s t.j
above a certain
and u
.
t
critical
value lead to insufficiently high productivity to be embodied in the firm's
value depends on
To understand what
s t.,
due to the intertemporal non-separability
determines these
critical
in the cost
values note that the firm invests this period
iff
between the
the difference
Appendix and
(8), the
left
hand side of
agent invests
^a'xl^o,^
this inequality
It is
which
Appendix). The intuition behind
model. The firm
all
is
expression
comparing the strategy
periods compared to s,=0,
Using the
positive.
/ dF(uhl) + J- / WM ^(u,J] +S lVl+rU,>0
and
implicitly defines the coefficients co
this
is
<">0
oo
/
and
iff
(Jg
[1-/8
(7) evaluated at s,=l
is
a
good way of
6
s,=l with s t =0
.
co,
in (8) (see
illustrating the
If s t=l
(9)
(A3)
in the
main features of our
production increases by oc,+u for
t
which has a net present value (including costs) of
(i-/?)> 1+u ,)-(S -SrVi)
Any
from choosing
further benefits
(
s
t
=l depend upon future values of u
.
t
10 >
There are three
possible cases:
u t+1 e
(i) If
beyond
the agent will not invest regardless of
and there are no consequences
st
(10).
(ii)
is
(-oo,(0 -C0i)
If
u t+1 e
[cOq-co^cOo) the
household's t+1 investment decision depends upon
only favorable enough for investment
investment
i.e. s t+1
=l
if s t =l
and
s t+I
=0
if
the household benefits
st .
The shock
from cost savings arising from past
otherwise. In this case, investing today
means
a difference in
expected discounted value next period of
u°
<->o
P { dF(u m )x{(l-P)\a
f u^dFiu^y^-SJ}
+
(11)
u -u)[
where the
value of
first integral
u, +1
represents the probability that u t+1 e
conditional on u t+1 e
[o) -a),,{0 ). If
and the second
[co -cO|,co )
both u t+1 and u t+2
fall in
in turn will
benefit at t+2
when
only be the case
is (1 1)
s t =l.
As
{u
t
}
multiplied by pProb(co -a>,<u t+1 <a>
assumed an
is
).
the expected
the region [(%-<»,, co
additional benefit accrues in t+2, suitably discounted. In other words, s t+2 only equals
which
is
i.i.d
sequence
1
)
this
when
same
s t+1 =l,
this additional
A similar logic holds for all future periods
and summing over time yields
{1-0
Our
results are
/ dF(u hl )} x[{p / iF(uM )}x{^ +5 rS
l
unchanged
if
m, and
(9) holds as a strict inequality agents
s,
lie in
the
continuum
}+
J-
/
[0,1] rather than take discrete values. In this case, if
choose one corner, s,=l and
m =l;
t
if (9) is strictly
holds as an equality agents are indifferent between any choice that has m,=s t If in this case
.
chooses s,=m,=l our results hold exactly.
8
(12)
u^dFiu,^)]
negative, s t =m =0. If (9)
t
we impose that
the agent
Equation (12)
from only
[co -co,,to
if
it
is
the net expected value of
invests today. If the firm does not invest at time
However,
the firm will not invest in the future.
),
future investment projects the firm benefits
all
t,
and future shocks
if
fall in
the region
same
the firm invests today the
sequence of shocks leads the firm to invest and reap the benefits given by (12). Thus there
important asymmetry in the
way firms respond
to investment
and as a consequence the marginal propensity
dependence
relies entirely
on
0),>0,
shocks in high and low activity
to invest varies
which from (A3)
in the
between these
appendix
is
states.
is
an
states,
This state
equivalent to 8,>0, or in other
words, the presence of intertemporal increasing returns.
Finally, if u t+1 e [co
(iii)
benefit
s,.„
from lower
costs.
=
costs are not. If s t
value of (38,(l-F(co
expected value of
Summing over
2
8
1
the shock
so favorable agents invest regardless of whether they
is
However, while investment decisions are the same
1
the cost of choosing s t+1 =l
The same
)).
f3
.°°).
is
lower by 8„ which has expected present
)],
e [©
and u t+2 e
[co ,°°),
with
with similar expressions holding for t+3,
etc.
benefit accrues at t+2, if both
[F(a) )-F(co -co,)][l-F(aj
irrespective of
u, +1
-co,,co
)
time gives
Wq
CO
/ dF(u JY^jdF{u J
{1-/3
(13)
t
t
"o
"o-"i
This expression represents the reduction in future costs arising from current investment and again
reflects the persistence in s„ captured
The sum of
and (13)
(10), (12),
The most important
households.
lead to investment
for an agent
4. Cyclical
(i)
who
if
integral
between
Due
t-1,
q
x
and
u)
.
summarized by
(9), is the
dependence
to intertemporal increasing returns, shocks in the range
received by an agent
has not invested at
(£> -(js
equal to (9) and characterizes the optimal decision rule of
is
feature of the decision rule,
of current actions on past decisions.
[co ,co -a>,)
by the
who
has been active in the past
making individual behavior
Fluctuations in the Aggregate
(s,.,=l)
but not
persistent.
Economy
Characterizing Output Fluctuations
We now
turn to the implications of individual level intertemporal increasing returns for
aggregate economic fluctuations.
normalized to
1,
each of
whom
We
assume the economy consists of a continuum of agents,
faces the technology described above. Because the intertemporal
increasing returns are internal to the firm,
decisions (denoted
for agent
s,'
productivity shock v
.
t
i)
allow for correlation between the different investment
of heterogenous agents by assuming the existence of an aggregate
Heterogeneity
agent receives a shock u^Vj+e,'.
we
We
is first
introduced via an idiosyncratic shock,
assume
that
£,'
G(.) (with associated density function g(.)) and that
density function,
h(.). Finally,
we assume
£,'
is
is
drawn from a common
v, is i.i.d
£,'
such that each
distribution function
with distribution function, H(.), and
uncorrected between individuals and over time. Both
shocks are normalized to have zero mean and are assumed to be observed before agents
make
their
investment decisions.
The decision
cOjS,.,'.
rule of the individual
due
given in section 3 and takes the form of invest
Conditioning on the aggregate shock, this can be written
>w
€t
where
is
co
and
to,
-(i>
s;_
1
rv
-
t
(i4)
t
t
to the idiosyncratic shock, requiring s t to
t
u '>(0
as, invest iff:
are defined by the distribution of u \ F(.). Investment decisions vary
that invest in period
iff
be indexed by
i.
among
firms
Defining S as the proportion of agents
t
(equivalently, the aggregate propensity to invest),
we
have:
Proposition 2:
Aggregate output follows the process
Ay =|Ay/^=a°+a 5 /+ f
1
u/di
/
ie[s'=i\
=a° + (a 1+ v /
)V
(15)
"nV,
oo
/ «8(*>k+ S„
eg(e)de
f
where
M1^(<V w i^,)>*m
-v,)} + {GK-v>G(Q -<D v,)}S
r
M
S,={1-G(<vvJ>(1-Sm
={l-G(<D
To
understand (16), note that of the
firms
S,.,
who
(l6)
invested last period only those with an
idiosyncratic shock greater than cOo-co^v, will invest now. Therefore, the
in
two successive periods
is
(l-G(co -co,-v ))S
t
with an idiosyncratic shock greater than
Therefore, as
shown
idiosyncratic shocks,
weights depend on S
(ii)
in Fig.l,
S
t
is
t. 1
.
Of
co -v t will
the (1-S,.,) firms
do so now and
their
measure of firms who invest
who
did not invest only those
measure
is
(l-G(co -v,))(l-S t.
1
).
a weighted average of points on the distribution function of
where the location of these points depends upon the aggregate shock and the
t.,.
The Nature of Business Cycle Fluctuations
Proposition 2 outlines an unobserved components model for
GNP,
as the
output consists of both a measurement equation, (15), and a state equation, (16).
the state equation
is
that
due
its
non-linear nature,
it
law of motion for
The importance of
enables us to explain the asymmetries and the
temporal agglomeration discussed in the introduction. S represents the average number of firms
t
are investing and
is
most naturally interpreted as the
in generating output fluctuations (e.g.
Watson
cyclical
component of output, which
is
who
crucial
(1986), Hamilton (1989)). Variations in S not only alter
t
the growth rate of output via (15), but also provide persistence because S, follows a time varying
10
AR(1) process with autoregressive
shocks
in the
optimal for
in the case
whereas with
s t'=l, s t+I '=l
where
s
t
'=0 a value of u t+1 in this region
i.i.d
S, relies
increasing returns, if 8,=0 S,
is
upon
is
8,
last
did not. In the aggregate this
being non-zero and so depends entirely on intertemporal
needed because each type of agent responds to shocks
As
a result,
which impact on the economy, but also who
AR
who
is
growth requires tracking the number of agents who are/are not
displaying persistence in their behavior.
The
it
white noise. The need for an unobserved components approach arises
fact that monitoring output
investing. This
means
shocks are converted into persistent cyclical fluctuations. The autoregressive nature
of the cyclical component
from the
caused by
would be optimal. Therefore, agents who invested
period have a higher propensity to invest this period than those
implies that
is
1
[co -u)„a) ):
region
s l+1 '=0,
coefficient equal to G(o) -v,)-G(to -co -v t). Persistence
parameter in (16)
is
will
it is
differently,
each group
not sufficient to monitor only the shocks
be affected by them.
and
in general time varying
this feature
enables us to account
for the empirical findings of business cycle asymmetries; essentially, the impact effect of aggregate
shocks on output varies over the business cycle. Referring back to Fig.l, changes in v
position of the
two points along the horizontal
axis: a
high v shifts the chord
t
be higher. However, the exact impact that v has depends upon both
t
distribution of shocks,
i.e.
upon both the slope of the chord
and the weights on the two points (determined by
of (16)
is
a source of path dependence in our
S _, both affects S through the
t
t
AR
S,.,).
model
As
AB
(which
is
and S
t
will
and the cross sectional
determined by G(.) and
as well as persistence; a
future shocks due to the interaction between v and S
shift the
Vj)
a result, the non-linear autoregressive form
coefficient but also alters the
t
S,.,
AB down
t
way
that the
shock which changes
economy responds
to
t_,.
However, under the assumption of uniformly distributed idiosyncratic shocks these asymmetric
interactions
between S
t
_,
and v are absent. In
this
t
case the
and (16) approximate the standard "return to normality"
been used
More
to successfully
model U.S.
GNP
generally, (15) and (16) allow for a
AR coefficient in (16)
state
is
constant and (15)
space model, variants of which have
by Harvey (1985), Watson (1986) and Clark (1987)
number of
alternative
7
.
component based models which
account for a wide range of asymmetries and cyclical fluctuations. Each of these different models of
the cycle arise
from
alternative assumptions
on the
distribution of idiosyncratic shocks.
although our model allows an important role for heterogeneity,
we do
However,
not need to monitor
complicated changes in the cross-sectional distributions to keep track of the state of the economy:
what matters for business cycles
is
not the exact relative position of each agent over time but the
distribution function for idiosyncratic shocks
statistic for all distributional issues,
around particular ranges. The
latter serves as
considerably simplifying the analysis of the next subsection.
The uniform distribution case only approximates the return to normality model due to
measurement equation disturbance and because, if v, is such that either o^-v, or a^-Wi-v,
£j,
the autoregressive coefficient
is
a sufficient
no longer constant.
11
the non-normality of the
is
outside the support of
Motivated by these observations,
we now
turn to investigate
how
the degree of intertemporal
increasing returns and heterogeneity interact to determine the time series properties of the aggregate
economy, and of
(Hi)
specifically the cyclical
component S
.
t
Determinants of the Time Series Properties of the Business Cycle
The non-linear nature of our model means
candidate
is
Given G(.)
Corollary
there
is
no unique definition of persistence, but one
the degree of serial correlation in S, conditional
is
1:
a distribution function
we
Business cycle persistence
upon v
:
t
have:
is
non-decreasing in
8,,
the degree of intertemporal increasing
returns.
Persistence
is
due
to the fact that
some agents have shocks
invest in this period only because investment costs are lower
determines the measure of these marginal agents, and
correlation
To
is
strengthened
illustrate
how
when
to recent high activity.
itself a positive
As
the nature of the business cycle varies with different degrees of increasing
uniformly distributed, and
let
the gains
Assume aggregate and
from learning-by-doing,
idiosyncratic uncertainty
assumptions for the case 5,/5 =l/3 and
a,,
be equal to 1.52 (see Section 6.1
Given the strong path dependence
1/2.
in our
and 3 are drawn for the same (suitably scaled) sequence of random shocks. Both figures
intertemporal increasing returns convert
far
i.i.d
more persistent in Fig.3 than
to continue to invest
even
latter
and 3 show the cyclical component arising from these
for a justification of this choice). Figs. 2
is
co,
function of 8„ serial
be equally important with both having a variance of 0.25, the former being normally and the
indicator
and
-g>i> co )
intertemporal increasing returns are higher.
returns consider the following illustrative simulation.
to
is
due
in the interval [(D
illustrate
how
shocks into cyclical fluctuations, however, the cyclical
in Fig.2.
in the presence of
system Figs.2
The increased learning-by-doing persuades firms
mediocre productivity shocks, significantly reducing the
noise in the cyclical component.
To understand the impact of distributional
affects
general measure of persistence than (17), which
Integrating across
all
on the
cyclical
component we
turn to a
more
was defined conditional on a given value of v
possible values of the aggregate shock,
we
arrive at a global
measure of
persistence;
H-zrkww
M
(18)
as
12
.
t
Taking a
By
Taylor expansion of P around
first-order
zero,
and
P can be approximated by
g(oo
definition the second term
around
co
,
Corollary
2:
The
is
and
(0
behind
intuition
from shocks
shocks
For given
is
co,,
important because
,
it
of
around
g(.)
g(co
)
economy
Thus we can
co -co,).
are linked to the
results in section 6,
The
)
zero,
we
obtain
will increase P.
again related to the fact that individual persistence arises
economy
determines the density of agents
)co, is
is serial
the distribution of idiosyncratic
who
are around this critical region.
number of such marginal
a measure of the
correlation. Corollary
agents,
2 therefore implies that spreads
in heterogeneity, represented
by
lower the number of agents
intuition of Corollary 2, that the persistence in the
form of the
where we
assumed
state
will reduce persistence to the extent they
e')
is
take a further first-order Taylor expansion of G(.)
In the aggregate
(Oq-co,).
the greater
increases in the variance of
,
mean of v, which
(which will often be produced by increases
co
in the region [co
the
an increase in g((0
For our global measure of persistence, g(co
is
we
)co,.
this corollary is
in the region [co
so that the higher
if
m,,
dynamics of
this
distribution of the idiosyncratic shocks, is important for our
will investigate the empirical
performance of different versions of
this
general models distinguished by the form of the heterogeneity and thus by the degree of asymmetry
in the
law of motion for output growth.
Now,
to further analyze the
impact of aggregate uncertainty on the business cycle
we
take a
second-order Taylor expansion of (17) around u^ followed by an additional first-order Taylor
expansion around
co
.
This gives:
P^Gi^yGi^-^yigX^yg'^o-^Wariy,)
s %o)^( o- u i)V'W u
M)
ffl
so
we
have;
Corollary 3: Increases in the variance of aggregate shocks reduce (increase) the persistence of the
cyclical
component
if g(.) is
concave (convex) around
oo
.
Corollary 3 states the surprising result that for a large class of idiosyncratic distributions, the
more
volatile are aggregate investment shocks, the less important is the business cycle
that the cyclical
component becomes
less persistent
t
a>
,
in the sense
and aggregate output fluctuations are increasingly
driven directly by v and not by the state equation.
concave around
-
The
intuition
behind
this is that
when
g(.) is
increased volatility of the aggregate shock leads to the critical investment
threshold being located at points with low density. In contrast, in the case of a convex g(.) function
13
at co
,
more aggregate
variability will take us to values of the density function that are
on average
higher and thus increase serial correlation due to the increased weight of marginal agents.
(iv)
Structural Heterogeneity
The purpose of
subsection
this
is
show
to
when we extend our model
that
component and
different types of heterogeneity across agents, the importance of the cyclical
degree of non-linearities associated with the business cycle
that increasing the dispersion of agents can,
between agents,
this is a surprising result
In particular
the
we show
under certain conditions, increase the amount of
Given
persistence provided by our cyclical indicator.
may be enhanced.
to allow for
which
that {S
t
}
represents the extent of
re-iterates the limitations
co-movement
of representative agent
models.
We
have so far only considered what
terminology of Caballero and Engel (1991), that
Whereas
may be termed
is
stochastic heterogeneity using the
heterogeneity in the form of idiosyncratic shocks.
in practice, agents also differ significantly in their technological possibilities/preferences as
well as the opportunities they face, often captured in microeconometric
effects.
To
include these influences
functions to be firm specific.
Using the solution outlined
We
we
work by
the inclusion of fixed
consider structural heterogeneity by allowing investment cost
assume agent
in Section 3
i
has investment costs given by
each agent invests
8
iff
eJ^Oo-w^i-v,
where the distribution function of
distribution of the cost parameter,
8
to mean-preserving spreads of IX).
co
'.
'
is
(
T(.) with support set
and
is
determined from the
Increases in the degree of structural heterogeneity are equivalent
The law of motion
^KlA ){/(l- G ("i.-v )yr(a)i )} +5
1
U
/
)
/
u
.1
for S, is
now
{/(l-G(a)i -o> 1 -v )yr(a>i )}
)
/
)
u
={/(l-G(o);-v ))dr(a>;)} + 5
u
/
22 )
M
(23)
{/{G(a);-v )-G(a)^-(o 1 -v )Mr(a);)}
u
/
Applying the same two successive Taylor expansions as
/
in the last subsection,
our global measure
of persistence becomes
P~f„ [^-Gtoj-aOWui-tt!/,, S(*>uW<d
Corollary 4:
The
If g(.) is
(24)
)
convex (concave), mean preserving spreads of
intertemporal increasing returns parameter, 5,, can similarly be
14
made
T(.) increase (decrease) P.
individual specific.
Therefore, an increased dispersion of agents in the form of greater structural heterogeneity can
interact with stochastic heterogeneity to increase the persistence generated through the cyclical
component. The intuition here
higher than g(.)
itself.
where investment
that if g(.) is convex, the averages of neighboring points will be
is
Since the higher
when
only profitable
is
is g(.)
more
structural heterogeneity leads to a
the higher
is
the measure of agents in the critical region
costs are low, in the case of
serially correlated process for
a mean-preserving spread (an increase in structural heterogeneity),
distribution of co
g(.)
is
Regime
The message from
Shifts in
One
t
}.
We can therefore see that
by extending the cross sectional
economic
state.
globally concave, an increase in structural heterogeneity will reduce
business cycle, heterogeneities
5.
{S
the presence of
g(.),
to regions of g(.) that are higher, increases persistence in the
'
Conversely when
persistence.
convex
this section is clear, far
from removing the non-linear nature of the
have a key determining influence on economic fluctuations.
Economic Fluctuations
of the attractions of fixed cost models
is
that the discrete individual behavior they
imply
can explain the sharpness of business cycle turning points. While our general model implies that the
component
cyclical
turning points.
A
is
always
number of
serially correlated,
it
does not impose any conditions on the nature of
Hamilton (1989), Acemoglu and Scott (1994), Suzanne
studies (e.g.
Cooper (1994), Diebold and Rudebusch (1994)) have achieved empirical success
fluctuations
by assuming the business cycle
moves from recession
to
(i)
shifts serve as
in our
modelling output
shifts, that is
we
by abrupt
investigate the factors
model, establishing the circumstances under which
a good approximation.
The Nature of Turning Points
To
analyze this issue
on the probability of a given
p(5|5>
where p(S
of
be characterized by regime
expansion (or vice versa). In the next subsection
which determine the nature of turning points
models with regime
to
in
S,
by
|S')
we
S,
use our basic model with only stochastic heterogeneity, and focus
occurring conditional upon
p(S
|S')
shifts.
is
:
(25)
(i-s^K-v ) +s'sK-<v v )
denotes the density of S conditional on S,.,=S' and
t
v, is
written as an implicit function
(16).
mass concentrated
its
which
*G2
The extreme case of regime
has
S,.,,
at
two
shifts
corresponds to the case where S =0 or
t
particular points. Similarly in
more general
1
so that p(S |S')
characterizations, if
has marked peaks, then transitions between different states take on the character of regime
A
necessary (but not sufficient) condition for p(S|S') to have
different intervals is that
at p'(S |S')=0,
we
will
it
should be non-unimodal. Thus,
have located a
local
minimum which
15
if
we can
its
mass concentrated
establish
p"(S
in
two
|S') is positive
implies p(S |S') cannot be single peaked.
From
(25)
we
have:
While no general statement can be made regarding the
transition
between
we have
states
the
following:
Proposition 3: The conditional density of
S, is
non-unimodal
if g(e) is
more concave than
h(v) in the
neighborhood of p'(S |S')=0.
Proposition 3 suggests turning points tend to be abrupt
h(.).
Though
this is
particular point,
is
it
not a necessary condition, naturally
g(.)
more
likely to
To
the case
economic
states
become more
approximate the discreetness of regime
investigate further this point,
where
8,/8 =l/2
we
If the idiosyncratic
is
at
a
shock
distributed
As
a consequence, turning points
shifts.
utilize a simulations
approach. Recall that Fig.3 showed
and the aggregate and idiosyncratic shocks were equally uncertain with a
variance of the idiosyncratic shock to 2.5.
The very
readily apparent: in Fig.3, there are never less than
numbers of agents
mass concentrated
distinct in the sense that the conditional
variance of 0.25. Fig. 4 maintains the degree of increasing return at the
large
its
the distribution function is concentrated in
if
distribution of {S,} is concentrated in particular intervals of (0,1).
are
has
more concave than
a straight line and output growth, though persistent,
is
uniformly along the continuum (a^OofCC,). However,
the middle (as in Fig.l)
when
g(.) is locally
be more concave. This can be seen in Fig.l.
will tend to
uniformly distributed, G(.)
when
are
still
active,
same
level but increases the
different cyclical patterns in the
60%
two
figures are
of agents investing so that even in recession
and turning points are extremely sharp, particularly the
observations at around 37 and 73. In contrast, the greater importance of idiosyncratic shocks adds a
considerable amount of noise to the cyclical indicator in Fig.4.
observations 37 and 73 can
the cyclical
The above
results
while the turning points
at
be detected, they represent only two of several observations where
still
component changes
And
direction.
and discussion show again
that the
form of heterogeneity
is crucial, this
time in determining the sharpness of turning points and they also suggest that the limiting case where
aggregate uncertainty
is
much more
important than idiosyncratic uncertainty
is
useful to analyze as
a benchmark.
(ii)
A Model
of Regime Shifts
Proposition 3 and related simulations suggest that the
distribution the
more appropriate
increases in heterogeneity
may
is
more concentrated
the idiosyncratic
a regime shift characterization. Further, Corollary 2 implies that
reduce the persistence of cyclical fluctuations by reducing g(o)
16
).
This
suggests that the business cycle will display extreme regime shifts and possess a highly persistent
component
cyclical
in a version of our
such a model by setting u '=v
t
.
t
The
model with no heterogeneity. In
particular interest in this special case
this
is
subsection
that
it
we
focus on
offers a theoretical
widely used discrete Markov state space models.
justification for the
Because the model only contains an aggregate shock, the laws of motion for the aggregate
economy
are the
same
as those for the individual firm.
From our
results of Section 3(ii)
and
introducing the notation
Prob[S =l\S ^\]=p
t
t
Prob[S =Q\S _^]=q
t
t
we have
Proposition 4: The stochastic process for the change in aggregate output
is
AY,=-a0+ (a 1+v,)S,
where S
t
is
a
Markov chain with
T
(28)
transition matrix
M
p
(29)
and
p=
f
dH(v)
,
(30)
q=fdH(y)
<>„-&>,
As
hitting the
in the
state
random shocks
economy behaves
the durations of
in a
manner
space formulation for output growth. The distinguishing feature of (28)-(30)
that fluctuations take the
the
cyclical fluctuations are the result of
economy. These shocks are propagated by intertemporal increasing returns
which requires a
is
model with heterogeneity,
form of
shifts
differently, not only
booms and
between
distinct
economics
does the growth rate differ
recessions, features
common
states. In
(oc,?K))
but
each of these
if
states
p*l-q then so do
with the model of Hamilton (1989) which
closely corresponds to (28)-(30).
This regime shift model also shares a number of similarities with the model of Durlauf (1993)
who
also explicitly
models the
transition
technology adoption. In both models
between
this
states
of the business cycle and focuses on
choice involves a non-convexity and intertemporal
increasing returns to scale with the end result being that white noise productivity shocks are converted
into
serially
correlated output fluctuations.
Both models also suggest a strong role for path
17
dependence, in (28)-(30) shocks which
persistent as they affect the
way
shift the
economy responds
the
between ours and Durlauf s work comes
in the
Durlauf s model the intertemporal linkage
words firms have a higher propensity
impact of
economy from one
form
to future shocks.
However, a key difference
that the intertemporal non-separability takes. In
arises through localized technological spill-overs; in other
Due
to the externality
Durlauf s model
generates multiple long run equilibria in the sense that the stochastic process for output
ergodic.
However,
in
our model, because increasing returns to scale are internal, and
implies that the equilibrium path
which
certainty in
6.
state the
is
uniquely determined,
economy
The
to invest if their neighbors invested in the recent past.
focus of Durlauf s paper.
this externality is the
state to another are highly
i.e.
given
{AY,}
will be, but also that
and
v,
is
{S,.,,S t . 2 ,..},
non-
is
this not
only
we know
with
always ergodic.
Econometric Evidence
we
In this section
likelihood estimates of our
use the econometric results of other researchers as well as
own
maximum
general model of Section 4 to assess what scale of intertemporal
increasing returns and what form of heterogeneity are necessary to account for the business cycles
We
in the U.S.
also use the analysis of Sections 3 and 5 to infer
of U.S. business cycle asymmetries and
from our estimates the exact form
the relative importance of idiosyncratic and aggregate
sources of uncertainty.
(i)
Regime
Shift
Models
Hamilton (1989) estimates a two
identical to (28)-(30)
state discrete
Markov model
for
US GNP
which
is
nearly
and finds p=0.9, q=0.76 and oc,=1.52. To calculate the implied degree of
we
intertemporal increasing returns
solve the equations in (A3) in the appendix using these estimated
parameter values and making assumptions regarding the distribution and variance of aggregate shocks
plus the magnitude of the learning-by-doing effects (6,/8
5,/5
and
o
obtain the
2
v
same persistence
(Do-CD,.
Fig.5 shows different combinations of
in
S
t
a higher variance requires
,
variance of shocks, the
more
more increasing
returns.
distributed.
To
The reason
for
likely agents are to receive a future
shock
less
This lessens the expected future benefits from increasing returns and makes agents both
less likely to invest
returns of
.
which generate p=0.9 and q=0.76 when aggregate shocks are normally
this is that the greater the
than
9
)
22%
and
less likely to
(8,/8 =0.22) to
when G^O.Ol we need only
remain investing once they have done
produce the appropriate values of p and q
8,/8
=ll%. As
cVa
Oq does not influence the values of
matches mean
US
i
=
cOq
0.03, the variance of
and
go,
we
Thus with increasing
require
a2 v =0.05,
but
well as providing persistence, intertemporal increasing
returns generate considerable amplification of the productivity shock.
and ^=0.05, even though
so.
AY
t
is
For the case where 8,/8 =0.22
equal to 0.634. Relying only on a
and so (without loss of generality)
output growth.
18
it is
set to ensure the
model
we need
single productivity shock to drive output fluctuations,
effects of
53%
to explain
Hamilton's
assumed by Hamilton and
results
all
results.
implausibly large learning-by-doing
But with an additional additive disturbance
in (1), as
econometric implementation of unobserved component models, his
can be explained with very small amounts of intertemporal increasing returns and aggregate
uncertainty.
Results from alternative studies confirm this finding that only small scale intertemporal
increasing returns are necessary to generate empirically observed regime shift behavior. Suzanne
Cooper (1994) uses monthly
to (28).
To match
industrial production
from 1931
to estimate a transition matrix similar
her estimates of the transition probabilities (p=0.55 and q=0.46) while also
US GNP
matching the variance of
growth (again without resorting
we need
disturbances other than a unique aggregate shock),
to
any additional productivity
the saving in fixed costs arising
from
3% (i.e. 5[/5 =0.03). Diebold and Rudebusch (1994) estimate
US industrial production (allowing for a time dependent T).
learning-by-doing to be only around
equations analogous to (28) using
Assuming only one disturbance and using
their estimated standard error for
underestimate as this implicitly sets S,=l for
all t),
we
find 8,/8
=0.8%
AY,
to calibrate
is sufficient to
ov 2
(an
explain their
results.
(ii)
The General Unobserved Components Model
In this subsection
we
use (15) and (16) to obtain estimates of our structural parameters. These
equations represent a general state space model and require assumptions regarding the distribution of
idiosyncratic shocks before they can be estimated.
idiosyncratic shocks are uniformly distributed
We
and then
examine these equations assuming
that they are
first that
normally distributed. As well as
enabling us to assess the robustness of our estimates of structural parameters these assumptions allow
As
us to examine the importance of asymmetries in U.S. output fluctuations.
assumption of uniform idiosyncratic shocks removes any asymmetry in the
(15)
and (16) under different
discussed above, the
state equation.
Estimating
distributional assumptions therefore enables a simple heuristic test of
the importance of asymmetries in U.S. business cycles.
Assuming
that idiosyncratic
shocks are uniformly distributed over the range
[-a,a],
(15) and
(16) can be written as
a-oi n
„
l
'
2a
v,
o>,
1
2a
M
(31)
2a
=c+ rs,_ 1+ u t
where the coefficients
ty
are functions of the structural parameters u)
the squared disturbance in the
(e.g.
Harvey (1989),
ch.3).
measurement equation
Various versions of
this
19
this is the
,
co„ a,
o^,
and a, and aside from
standard return to normality model
model have been used
to estimate the cyclical
component of U.S. GNP, with Watson (1986) and Clark (1987) both estimating models similar
10
(31)
If instead
.
we assume
that idiosyncratic
order Taylor approximations for
shocks are distributed normally, then" (taking second-
higher order terms,
all
to
full details
available from the authors)
12
;
Ay,=VV, +V, +V/2+VA/-i
5r(l-G(»o)) + [ G ( u o)- G! ( w o- u i)] 5i-i^ u o) v /
(32)
fe(«o- w i)-S( <0 o))v A-i
Equations (31) and (32) show that a simple
measurement equation
We
is
to test the significance of the
test for
asymmetry
in either the
v Stl term.
t
estimate (31) and (32) using the growth rate of quarterly U.S. real
period 1954:1 and 1987:4. Because the equations contain only one shock,
measurement equation
in
state or
GNP
for the
we augment
the
each case with a normally distributed additive measurement error.
Estimation was performed using
maximum
Kalman
likelihood via the
filter.
We
also ignored the
squared disturbance term which considerably simplified the estimation. In neither case did our
estimates reveal any indication of heteroscedasticity, suggesting that dropping the squared term
was not an important omission. Because
in
(31)
and (32) the measurement and
have a correlated disturbance a simple alteration
Kalman
Harvey (1989), p.112). Table
(see
filter
6 shows the (smoothed) estimates of
idiosyncratic disturbances.
component
S,
is
1
required to the standard recursions of the
contains our estimation results, and Figure
emerge from the
that
Both versions of the model suggest
R
2
is
persistent, with
that
on
a persistent cyclical
the structural parameters that
av 2 was
,
(31)),
we
find that the cyclical
an autoregressive coefficient of 0.52 and the model has a
=0.595, thus explains just under
shock,
different assumptions
successfully accounts for serial correlation in U.S. output growth.
Examining the uniform distribution case (equation
component
state equation
60%
GNP growth. The
of the variation in U.S.
we uncovered were
estimates of
as follows: the variance of the aggregate
0.06, the variance of the idiosyncratic shock,
ae 2 was
,
1.45. This implies that the
10
Watson and Clark estimate their model using the level of the U.S. real GNP allowing for a trend, a cycle and an
component. Thus our comparison is purely with their specification of the cyclical component, our equation
(16). They both actually estimate the cyclical component as an AR(2) process. Acemoglu and Scott (1993) show this
can easily be allowed for by letting intertemporal increasing returns operate with longer lags.
irregular
11
Equation (32)
is
an approximation which
is
valid for any distributional assumption regarding idiosyncratic shocks.
Different assumptions regarding G(.) lead to different estimates of the model's structural parameters.
12
The exact expression
for b 4 is as follows:
btfG(<»($-G(u Q-**j\[l
—
[/«T° "'
W
"°
"^( M ) rf"] x{ 5( a) o- a) i)-S( w o)}
—
[G(fc)
This term disappears
when
)-G(a>
-o 1 )] 2
idiosyncratic shocks are uniform.
20
MWo-WiteK-^ih^oSK)-
variance of idiosyncratic shocks
being relatively
is
around 25 times that of aggregate shocks, thus heterogeneity
much more important
than aggregate uncertainty. This result
is
in line with those
of Schankerman (1991) and Davis and Haltiwanger (1992) from micro data that idiosyncratic
compared
variations are large
to
comovement
why
section, this finding also explains
across
agents.
all
Given our
results in the last
of Hamilton's (1989) model
statistical tests
Hansen
(e.g.
(1992) and Garcia (1992)) reject the notion of regime shifts between distinct states as an
appropriate characterization of U.S. business cycles.
implies that
compared
when
the business cycle indicator S,
per
(SJSq) of around
The
GNP
peak,
growth
use equations (A3) to calculate estimates for 5 and S x
We find that S =
27.6%
2%
is
higher
1.96 and S x
=
,
the learning-by-
0.54 implying intertemporal increasing returns
(as a proportion of fixed costs).
estimates of equation (32) which allows for asymmetries reveal considerably greater
persistence in the cyclical
component — the autoregressive
0.67 as opposed to 0.52 in the uniform case.
0.712, thus the
model with normally
significant as equation (31)
statistical
R2
goes up to
in this case
70%
version of the model
this
of the
is
also
that allows for asymmetries outperforms the
model.
Further, the interaction term between vt and
measurement equations, strongly so
S,. x
is
significant in
in the former, confirming the
both the state and the
asymmetric nature of U.S.
business cycles and thus indicates where the improved performance of (32)
6 shows the estimated sequence for the cyclical indicator arising
comes from. Figure
from assuming normal and
uniform idiosyncratic distributions. The strong correspondence between the two sequences
encouraging since
in
it
is
what Clark, Harvey and Watson have estimated, thus
more general model suggested by our theory
conventional
importantly,
distributed idiosyncratic shocks explains over
essentially
is
More
coefficient in the state equation
The superior performance of
variation in U.S. business cycles.
the
at its
be 0.0201, which
to
x
our estimated structural parameters and assuming a real interest rate of
annum we can
doing parameters.
now
is
a
also recovered
to a trough.
Finally, using
4%
We
implies that
both exercises. However, there
we
is
are basically uncovering the
is
same underlying component
a major difference between the two series: the asymmetric
model enables much sharper downward swings
and
into recession
this feature
it is
of the data
which accounts for the success of the more general model. These sharper downswings are
captured by the positive coefficient on the
(1984)
who used
a different
statistical
Again uncovering our
v,
that this
structural estimates,
we
find that
boom and
a x = 0.017 which
recession that
we
term vt S,_x The presence of
this
we
of Neftci
also have the
added
flexibility
less
than the
at the
peak of the
of having the asymmetry
term implies that as long as we remain
21
is
obtained from (31).
does not imply growth to be higher by only 1.7%
cycle because in contrast to (31),
.
in line with the findings
methodology than the one here.
estimate of the growth difference between
However, note
S ul term and are
in a
boom
(i.e.
positive
values of v,)
we
obtain an added growth effect and this
of the cycle growth
(v,<0), this can
cr
e
2
is
happen rather
around
3.5,
of the order of 0.2%, thus
higher by 1.9%. But also this term implies that
is
The
still
fact that
av 2 was 0.035 and
our asymmetric model, (32), leads to a
may be due
interactions across individual investment decisions.
specific effects translated into
As a
to the absence in
result,
clearly requires a richer data set than the aggregate time series
we
our
what may be firm
co-movements by economic interactions across agents
estimated as aggregate shocks in our specification. Examining whether this
that
a downturn
giving a substantial role to idiosyncratic
large estimate of the relative variance of aggregate shocks
show
is
peak
considerably less than the approximate estimate of 10 from Schankerman
disturbances in generating uncertainty.
estimates
there
at the
ratio of the variance of idiosyncratic shocks to aggregate
(1991) and Davis and Haltiwanger (1992), though
model of any
when
sharply. Returning to the rest of the estimates,
was 0.1207, so our estimate of the
shocks
is
is
will
be
actually the case
use here. However, our
even without spillover effects we can have a model where idiosyncratic
shocks are the major source of uncertainty yet output growth shows a clear cyclical pattern with
sharp swings into recession. Resorting again to (A3)
to estimates of S
we
find that our asymmetric
model leads
and S l of 1.81 and 0.63 respectively, so that intertemporal effects of around
1/3 are required to explain U.S. fluctuations.
It
also interesting to note that the increased complexity of (32) implies that our
is
structural parameters are overidentified (see for instance the expression in footnote 11)
whereas
they were exactly identified in (31). This gives us two overidentifying restrictions with which to
test the plausibility
statistic
of 4.1 which
of our model. Performing a
is
Wald
test
on these
asymptotically distributed chi-squared with two degrees of freedom, thus
comfortably accepting the overidentifying restrictions. This result
piece of evidence that our
7.
restrictions gives a test
model provides a good match
is
promising as
it is
another
to the cyclical dynamics of U.S.
GNP.
Conclusion
We
have outlined a theory of economic fluctuations based on internal intertemporal
increasing returns in a
model of
discrete investment choice. This
microeconometric findings of persistence
in the
management
in firm level
new
is
motivated by
investment as well as the emphasis placed
science and economics of technology literature
learning-by-doing in the adoption of
model
on the importance of
technologies. Incorporating these effects into a
model
of a firm's investment choice leads to a tractable model of business cycle fluctuations.
This tractability enables us to
fluctuations.
Our
fully analyze the
theoretical findings are:
(i)
determinants of aggregate economic
Intertemporal increasing returns naturally lead to
temporally agglomerated and asymmetric economic fluctuations, in the sense of persistent
periods of low and high growth separated by business cycle turning points
(ii)
Cross-sectional
considerations play a key role in determining the non-linear and asymmetric nature of
22
fluctuations
and distributional issues are found to be
crucial in determining the sharpness of
turning points and deciding whether aggregate output experiences regime
heterogeneity plays a key role,
variable, the average
all
number of
shifts, (iii)
the business cycle relevant information
active agents, thus
is
Although
captured in one
our model enables a synthesis between
representative agent models and those stressing the importance of heterogeneity.
A
major attraction of our model
is its ability,
under certain simplifying assumptions, to
justify the
popular unobserved component models of Watson (1986), Clark (1987) and Hamilton
(1989)
of which place a special emphasis on an underlying cyclical indicator.
all
attraction
is
An
additional
the model's ability to offer a general formulation for cyclical components, and
provide a simple test for the presence of business cycle asymmetries.
general model that incorporates asymmetric effects provides a good
that these are characterized by asymmetries, with particularly fast
captured by a linear model; that the cyclical component
is
The data
fit
suggest that our
to U.S. business cycles;
downturns that cannot be
highly persistent; and, that despite
the asymmetries, business cycles are not characterized by sharp swings such as required by
regime
shift
models. In our context,
this latter fact
corresponds to the aggregate sources of
uncertainty being dominated by idiosyncratic shocks, which
is
what investigations on micro data
also find. Estimates of the degree of internal intertemporal increasing returns necessary to
account for U.S business cycles vary between 3 and 53%, suggesting that under some
circumstances only very modest amounts of intertemporal increasing returns are needed. Adding
other sources of uncertainty to the model or allowing for spillovers between agents would serve
to reduce
even further the extent of internal increasing returns required. Assessing the
importance of internal and external intertemporal increasing returns
further research.
returns
At
this stage,
is
relative
an obvious topic of
our results lead us to conclude that intertemporal increasing
may be an important channel of persistence,
fluctuations.
23
amplification and asymmetries in economic
\AJ
CD
CO
i
>
1
^—
£
«p +
o
o £
4-»
CO
^_
O
4-»
CO
+
+
£
S
Figure 2
1
12 16 20 24 2S 32 36 40
5,/5 =0.33
:
a 2 v=0.25 a 1
=0.25
-4-
4 4S & 2 5 6 ©O 6 4 ©S 7 2 76 SO
a 2 s =0.064 a%=0.841 a\=0.025
.00
0.95
O.QO
O.SS
O.SO
0.7S
0.70
0.65
o.eo
0.5S
Figure 3
:
-
-
-
12 16 20 2* 28 32 36
5,/5 =l/2
4
Figure 4
:
&
a2 v=0.25 a2
12 16 20
6,/5 =l/2
2-4-
=0.25
.4-0
4A
.*S
a2 s =0.014 a 2
52 S6 60
=0.541
6-4-
68 72
SO
2 7©
SO
a 2 ,=0.032
2S 32 36 40 44 48 5 2 56 60 ©4 GS
^=0.25 a2 =2.5 a2 s =0.012 ^=4.855 a\=0.001
25
7©
"7
Figure 5
:
Learning-by-Doing and Volatility of Aggregate Shocks
26
Figure 6
:
The
Cyclical
Component of U.S.
GNP
Growth
27
Figure 6
:
The
Cyclical
Component of U.S.
GNP
Growth
27
Table
1
:
Estimates of General and Uniform Model
Normal Case
Uniform Case
Parameter
Estimate
Tstatistic
Estimate
Tstatistic
bo
0.0004
1.691
0.0020
1.957
bt
0.0201
3.107
0.0173
3.284
0.1027
1.985
b4
c
0.1826
2.232
0.1769
2.173
T
0.5213
5.669
0.6653
6.339
0.3229
3.496
0.0065
2.842
K
°u
R
2
2
0.0070
2.703
0.595
0.712
Sample period 1957:1,1987:4. Data = Quarterly growth
Maximum
likelihood via
Kalman
filter.
28
in
U.S GNP. Estimation by
Appendix
Proof of Proposition
We
1
need to establish
values of
a>
To
and
and
(Oj
establish
that
<f>'s
we
(i)
the value function defined in (8) satisfies (7) for particular
(i)
the value function satisfies a transversality condition.
(ii)
substitute (8) into (7) to derive:
I/u^Wq-w^^,
then
00
+
/5[
/
+
(4> Q <l>Jiyl -i**o
+" +
l
uWs+ il>u u*i) dfl u>i)
(Ai)
>0-G>l
+
where a
/
(0o
4
^//n + «o + ai + ^) d ^
similar equation applies for
/
I^-V^.
P
^o-^o=
~
1-/8
/
'M)]
vt < (Oq-co^!. Equating coefficients gives
+
/ ^("m)
0)0-0);
l
A
HU
/ "m^("m)
(a2)
-"l
<*F(u, +1 )
OJQ-O)!
To
find expressions for
of (5)
is
(o
and
(Oj
we
use (8) to see under what conditions the
left
hand side
greater evaluated at s,=l than evaluated at s,=0. This gives
0)^(1-0)
0)
G) =[l-)3
/
dF{u ^Y.\{\-P)h,-*,t
(A3)
ao-S^l-P)
Defining the right hand side of the definition of
point. Defining z(u) as t(u)-u
lim o,- + = z(«) = -°°
o>
as t(g)
)
we need
to prove t(g)
)
has a fixed
we have
and
lim^.^
z(o>)=+oo
(A4)
Continuity of z(u) follows from continuity of t(.) and by the intermediate value theorem z(u)
must have a zero. Thus a fixed point,
a)
,
of
t(.) exists.
29
(ii)
The value
function defined in (8) satisfies a transversality condition
iff
oo
(A5)
Km^jS'/ K« M .'M.«0< ff W=°
since V(.)
is
linear condition
is
Establishing uniqueness of
everywhere differentiable and
B<1.
satisfied for
its
requires proving z(o)
<o
)
has a unique fixed point. z(co
) is
derivative equals
00
zK)=iW(<>- w /K-(l-WMli3
"0
oo
+[(1-/3)50-^^(1-^
^(u M )-^
/
u
dF(u t+l )Y l
/
ul+1dF(u„)]
/
(A6)
o>o-(l#i
-(l-P)Sj
<->o
x 8[/-(Q )^(o)
j
dF(«
/
-(l-)8)5 )][l-i8
1
2
M )]- -l
"o-(l-^)«i
Substituting z((o )=0,
true at a unique value of
We now
<d
we have
z'(a)
)=-l which establishes that z(o) )=0 can only be
.
establish by contradiction that there can
recursion (7). Let
W^^s^Um)
be a solution to
variables let s w be the optimal choice of s
t
t
.
(7)
be no non-linear solutions to the
and for given values of the
state
Thus
(A7)
+
J%l
+
V(a +M i><V.«wWM M)
i
-oo
Observation
under both
We
u,
s,
1:
w =0 and
^"(y^s^.u,) cannot depend on
as returns
from a higher y
t. x
accrue
s tw =l.
know by assumption
where V(.) gives
y,.!
s tv =0.
The
that both V(.,.,.) in (8)
and W(.)
satisfy (7).
Take a value of
difference between (5) evaluated at W(.) and V(.), for given
u,, is
30
^{-B/ RO'M -»« +(a 1 +i0*r.*,
,r
,«Vi)
,
<
AO ,
A8
>
-/(*0^J C^M^^^M)^O-/(^jC^-l +a b))^ l,M)>
From Observation
Thus
W(.,.,.)
linear in
is
repeat this exercise for
s,.
holds for
1 this
x
=0
y,.!
or
1
with a constant coefficient.
see this note that
we can
.
s tv
= 1 and
get the
same
relationship.
2: If
E,T^ ^{y .^
t
is
To
and get the same coefficient kt Similarly we can repeat
the exercise for values of u, at which
Observation
for fixed u„ implying that
all y,.!
true for u,=u,
t
._^
it is
.
\s
t
r l)>E^ i*r(y
also true for
t
^H
_
u,=u+A, such
t
that s w =l (=0).
t
a,.!
A>0. The
that
increasing in u whereas the right-hand side does not
a value of u„ u^s,^) (independent of
(A10)
v u N |*,=0)
left-hand side of (A10)
depend on u This implies there
.
t
by Observation 1) such that u
Suppose u^Sn^Oo-^iSt-i (the argument
w
t
>(<)u
(s t . 1 )
for the reverse inequality
is
exists
implies
is
analogous).
First,
Varying
consider
u, in
the interval (-oo^o-cojSi.j).
F° r aH u
m this interval,
t
w =s v =0.
t
u, in this interval gives:
^(yM^",)-/^(y,-i +a
where Kj does not depend upon
that W(.)
is
independent of
Consider a value of
Considering variations of
u, in
u, in
u,.
As
oM i)=*2
<
+
there
is
However
the interval (-00,0^0)^).
the interval
u, in this interval
W(.,.,.) is linear in
Thus over
We
this
range too
also
not depend on
a v which
t.
(u^Sn^+oo). This
implies
s tv
=s w =l.
t
gives
is
linear in
W(.,.,.) is linear in u,,
need to consider the
u,
A11 )
no investment we see from Observation 2
%i^,)-^%i+v«iV.vJ=Wu-^>,
W(.,.,.)
s,
u,.!
(
with coefficient
with coefficient
over (coo-ajjSujU^s,.!)). Thus the coefficient of
31
thus
<£ u =l/(l-j3).
interval ((Oo-OjSn.v^s,.!)) in
has the same form as the value function in (7)-(8).
<f> u ,
A12 )
which
u,
s,.
1
w =0
and W, does
of the value function
Finally
we need
takes the values
and
any value function that
to
1.
check how W(.)
is
influenced by
Thus any general function
satisfies (7)
h(s t
.
x)
s,.
x
.
However
can be written as k4 s
must have the same form as
32
this variable
V(.).
QED
t .!,
only
implying
Abernathy,
W.
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35
Date Due
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