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Seasonality and Equilibrium
Business Cycle Theories
R. Anton Braun and Charles L. Evans
Working Papers Series
Issues in Macroeconomics
Research Department
Federal Reserve Bank of Chicago
December 1991 (WP-91 -23)
FEDERAL RESERVE BANK
OF CHICAGO
ll
Seasonality and Equilibrium Business Cycle Theories
R. Anton Braun
Charles L. Evans
•k-k
March 1990
Revised November 1991
Abstract
Barsky-Miron [1989] find that the postwar U.S. economy exhibits a
regular seasonal cycle, as well as the business cycle phenomenon.
Are
these findings consistent with current equilibrium business cycle theories
as surveyed by Prescott [1986]?
We consider a dynamic, stochastic
equilibrium business cycle model which includes deterministic seasonals and
nontime-separable preferences. We show how to compute a perfect foresight
seasonal equilibrium path for this economy.
An approximation to the
stochastic equilibrium is calculated.
Using postwar U.S. data, GMM
estimates of the structural parameters are employed in the perfect
foresight and simulation analyses.
The nontime-separable model predicts most of the seasonal patterns
found in aggregate quantity time series; a notable exception is the
seasonal pattern in labor hours. An evaluation of the model's predictions
for deseasonalized second moments finds support for the parameterization.
This model broadly displays a seasonal cycle as well as the business cycle
phenomenon.
JEL Classification(s):
E32, E20
Department of Economics
University of Virginia
Charlottesville, VA 22901
(804) 924-7845
Research Department
Federal Reserve Bank of Chicago
P.0. Box 834
Chicago, IL 60690-0834
(312) 322-5812
For helpful comments, we thank Fabio Canova, Marty Eichenbaum, Eric
Ghysels, Jim Hamilton, Valerie Ramey and seminar participants at the 1990
NBER Summer Institute, Rutgers, Queens', the Federal Reserve Bank of
Chicago, and the Universities of Montreal, South Carolina, and Virginia. An
earlier version of this paper was presented at the 1990 Winter Econometric
Society meetings.
Any opinions, findings, conclusions, or recommendations
expressed herein are those of the authors and not necessarily those of the
Federal Reserve Bank of Chicago, or the Federal Reserve System.
1.
Introduction
The postwar U.S. economy exhibits a regular seasonal cycle, as well as
the business cycle phenomenon:
Miron [1989].
this is the principal finding by Barsky and
These researchers analyze aggregate data which has not been
adjusted for seasonality and find that deterministic seasonals account for
between 50 - 95% of the variation in the growth rates of aggregate quantity
variables such as GNP,
consumption,
and investment.
Are these findings
consistent with current equilibrium business cycle theories as surveyed by
Prescott
[1986]?
technological
Prescott
change
are
an
concludes
that
variations
important
source
of
accounting for about 70% of cyclical fluctuations
[1989]).
economic
the
rate
of
fluctuations,
(Kydland and Prescott
Theory predicts cyclical fluctuations, but does it also predict
seasonal cycles?
in
in
assessing
Answering this question is a potentially important step
the
validity
of
equilibrium
theories.
The
similarities
between the seasonal cycle and the business cycle suggest that the economic
mechanism generating business
cycles.
Consequently,
cycle fluctuations also generates
seasonal
a proper theory should predict seasonal cycles as
well as business cycles.
We
which
consider a dynamic,
includes
parsimonius:
stochastic
deterministic
seasonals.
Our
seasonal
specification
Hansen,
and Singleton
As in Kydland and Prescott
[1989],
and Braun
[1990],
exhibit nontime-separability in consumption goods and leisure.
is tractable.
seasonal
is
[1982],
preferences
This model
In particular, we show how to compute a perfect foresight
equilibrium path
for
this
economy.
An
approximation
stochastic equilibrium is also calculated around this
cycle model
we include only a technology seasonal, a preference seasonal,
and a government spending seasonal.
Eichenbaum,
equilibrium business
2
to
the
equilibrium path.
Using
a
Generalized
Method
of
Moments
(GMM)
estimator,
the
model's
structural parameters and the seasonals are estimated using postwar U.S.
data.
The over identifying restrictions of the model are not rejected at
conventional
levels;
and the
technology seasonal
estimates
indicate
a
strong seasonal pattern, sufficient to drive an equilibrium seasonal cycle.
Given
cycle
the
parameter
properties
of
estimates,
the
the
seasonal
equilibrium model
patterns
accord
well
and business
with
the
data.
First, the model replicates many of the seasonal patterns in the aggregate
data,
particularly
for
output,
consumption,
productivity, and the real interest rate.
capital,
two,
three,
and
four.
The
contribution of deterministic seasonals
labor
The principal shortcomings are
with respect to second and fourth quarter investment,
quarters
average
model
also
and labor hours in
captures
the
large
for the total variation in most
aggregate variables. Second, we find that seasonal variation in technology
is essential
for explaining the seasonal patterns
variation in preferences
seasonals
no
larger
in output.
and government purchases
than
0.3%
per
alone
quarter.
Seasonal
generate
Third,
output
without
nontime-separabilities in preferences, seasonal variation in output, hours,
and investment is much too large.
Habit-persistent preferences for leisure
are an important element in the model's ability to match the magnitude of
seasonal variation in aggregate hours and local durability of consumption
services also proves to be important for matching the seasonal properties
of consumption.
Fourth, the model's predictions for deseasonalized second
moments match the data's second moments with about the same accuracy as
existing nonseasonal real business cycle models.
Does
theory
fluctuations?
The
predict
seasonal
fluctuations
as
well
equilibrium model with nontime-separable
3
as
cyclical
preferences
displays the seasonal patterns emphasized by Barsky and Miron
well
as
the
business
cycle
phenomenon.
successfully offers predictions
frequencies
is perhaps
The
fact
across both business
its greatest strength.
The
that
[1989]
the
as
model
and seasonal
cycle
fact that the model
requires
large seasonal variations
in technology to achieve this match,
however,
suggests that this model is simply a benchmark.
Other theories
which deliver seasonal variations in technology endogenously may encompass
the findings here.
The paper
is organized as
follows.
Section 2 presents
economy which includes seasonality and exogenous growth,
the model
and the perfect
foresight seasonal equilibrium path is defined.
Section 3 presents the GMM
estimation
discusses
strategy,
parameter estimates.
equilibrium paths
relative
seasonals.
describes
the
Section 4 analyzes
implied by
contributions
of
the
and
the
structural
the perfect foresight seasonal
parameter
technology,
estimates
and
preferences,
assesses
and
the
government
Section 5 presents and analyzes the simulation results for the
stochastic economy with seasonality.
2.
data,
Section 6 offers conclusions.
An Equilibrium Business Cycle Economy with Seasonality
This section presents a one-sector, real business cycle economy which
is subjected to
seasonal variation
government purchases.
in the
technology,
preferences,
and
The model is similar to the models considered by
Christiano-Eichenbaum [1990] and Braun [1990].
2.1 The economy with growth and seasonality
Consider
an
economy
composed
of
a
large
number
infinitely-lived households each of which seeks to maximize
4
of
identical,
cu
'.I
log cfc +
«■{
t—
0
*
where c and
1
*
72
1
log
7 2>0
[2.1]
represent consumption and leisure services,
Consumption services are related to private consumption
respectively.
(cp)
and public
consumption (g) as follows:
C* -
cpt +
7^
where
gt + a (cpt l + 7X
governs
the
consumption goods.
nonseparability:
complements
substitutability
preferences.
of
< 1, |a|<1
public
goods
[2.2]
for
private
The parameter a governs the character and degree of
if
a
is
(substitutes)
complementarity
71
0 <
case
can
negative
across
also
The variable r
(positive),
adjacent
be
consumption
time
interpreted
as
goods
are
periods.
The
habit-persistence
in
is a deterministic preference seasonal which
follows:
rt =
T1 Qlt + T2 Q2t + r3 Q3t + r4 Q4t ’
and the variable
rj>0 for
J
[2'3]
is a dummy variable taking on the value of
period t corresponds to season j, and zero otherwise;
the preference seasonal in season j .
labor
a11
1
consequently,
when
is
Leisure (1) is time not devoted to
(n) , leading to the time allocation constraint that n^_ + 1^ — T,
where T is the maximum number of hours available per period.
are defined over leisure services
1^
Preferences
:
Ib|<1.
[2.4]
The parameter b governs the character and degree of nonseparability:
if b
is negative (positive), then leisure choices are complements (substitutes)
across adjacent time periods.
Finally, the operator
is the mathematical
expectations operator conditional on all information known at time t.
Each household has access to a production function of the form:
, ,d J
yt - ( kt )
.
d Nl-<
( zt nt )
where y is output and
[2.5]
and n^ are the quantities of capital and labor
5
services demanded by the entrepreneur-household.
The household's output
can be consumed (privately or publicly) or stored in the form of additional
capital next period.
Each period, the existing capital stock depreciates
at the geometric rate 8.
The variable z^_ is a labor-augmenting technology
shock which includes deterministic seasonal components:
[2.6]
Zt =
Zt-1 eXp ( At }
At =
A1 Qlt + A2 Q2t + A3 Q3t + A4 Q4t + €t
where c
is a purely indeterministic, white noise random variable.
that log zt is a random walk with seasonal drift:
Notice
when the seasonal growth
rates A^ do not sum to zero, this economy experiences growth.
The
economy
services:
possesses
suppliers
competitive
markets
in
labor
of labor services receive a wage w^_,
capital services receive the rental rate r .
each household in a lump-sum fashion, TL^.
and
capital
suppliers
of
Finally, the government taxes
This leads to the household's
period budget constraint:
cpt + kt+i = yt + (1'*)kt • V
v
V
- rt(kt*kt) • TLt
[2-7]
where k and n represent the supply of capital and labor services by the
household.
The
government
chooses
a
stochastic
process
uncontrollable from the household's perspective.
for
g^_
which
is
Government purchases are
assumed to contain a permanent and a transitory component.
The permanent
component is related to the technology shock z^_;
the transitory component
is
a
an
autoregressive
process
of
order
1 with
seasonal
mean.
The
stochastic process for g^_ is:
log ^
where
g^
- log gjt -
p [log
^
- log
[2.8]
is the seasonal mean of transitory government purchases when
period t corresponds to season j;
+ ut , 0<P<1
6
and u^ is an indeterministic,
white
noise
random variable.
A
specification such as
this
one,
but without
seasonality, was adopted in Christiano-Eichenbaum [1990] and Braun [1990].
In this Ricardian environment, we assume without loss of generality
that the government's budget constraint is g^ — TL^.
This leads to the
economy-wide resource constraint (in per capita terms):
cpt + kt+i + st = yt + (1'5)kt • V
When
the
supply of
v
labor and capital
V
• rt(kt'kt)
equals
the
demand
[2-9]
for
labor
and
capital (respectively), equation [2.9] is the familiar per capita national
income accounting identity for this closed economy.
are
identical
in
this
economy,
each
individual's
Since all individuals
net
supply
of
either
factor will be zero, in equilibrium.
As
in
King-Plosser-Rebelo
[1988],
an
empirical
analysis
of
this
economy is facilitated by rescaling the economy in a way which induces a
stationary
environment.
To
this
end,
define
the
following
scaled
variables:
where
i
is gross
investment.
Under
the
assumption that
the unsealed
economy exhibits balanced growth, the scaled variables are stationary;
the
remaining unsealed variables are leisure services (1 ), labor (n) and the
rental
rate
(r),
which
are
stationary
without
household's problem in the scaled economy becomes:
7
any
rescaling.
The
max
!o I
*'
{ rt log K + 7 2
[2.1']
+ Tt l°g z
l°g
t=0
subject to the constraints
K
=
[2.2']
Hpt + 7i gt + a (Hpt-i + 7i it-i) e
It =
T ’nt
+
[2.4']
b (T-nt-l>
. rd ,0 , d ,1-0 -0A
yt = ( kt > ( nt >
e
p
cpt + kt+i “ yt +
' V
where the uncontrollable g
poses
no
[2.5']
analytical
v
V
• rt(kt * V e"At - gt
[2-7 ']
has replaced TL^, and the presence of log z^_
difficulties
since
it
is
uncontrollable
and
stochastically dominated.
Given
initial
values
for
the
capital
stock,
leisure,
government
purchases and private consumption, as well as a law of motion for g , the
equilibrium in this economy is essentially a sequence of contingency plans
( cp^,
necessary
n^; t>0 } which satisfies:
conditions
transversality
resource
markets.
for
condition
constraint,
and
a
on
(4)
(1) the household's first-order
constrained
capital,
(3)
maximum
the
market-clearing
of
[2.1'],
economy-wide
in
the
labor
per
and
(2)
capita
capital
The system of equations which characterize the equilibrium of
this scaled economy are:
~ i + b p Et—
1t
a
=
Mt (1 -9) Y.[ nt" e "At
1t+l
8
[2.10]
,, .. 'At+1
Mt+1 ( . r5-l 1-5 '*At+l
i
+ (1-5) e
I * kt+l nt+l
IL
V
P E+
'At+1
Mt -
-Z3T
ct
=
1
[2.11]
rt+l
[2.12]
a P Et '
+
ct+l '
rtf
k
Hpt + Et+i + it
1-tf
nt
e
+
/i
e\
t
(1-5) k
[2.13]
e
-
g -i-fl "*At
kt nt
e
[2.14]
Ct
=
_
_
cpt + 71 gt+ a (cpt-l + 71 8t-l) e
[2.15]
lt
-
T - nt
[2.16]
wt
-
(1-5) kfc nt
rt
-
s-5-1 1-5 ~°Xt At
8 k
nt
e
e
yt
_*
+
b ( T - V l
)
■e
Pt Ht ^
lim
t-*»
The variable
constraint.
2.2
1
-
(1-5) yt/nt
e
[2.17]
[2.18]
<* ?tA t) e
[2.19]
0
is the Lagrange multiplier associated with the resource
These conditions determine the equilibrium of the economy.
Characterizing the Seasonal Equilibrium
In
general,
equations
[2.10]
attention
to
a
perfect
-
[2.19],
a particular
foresight
with
path
would
the uncertainty
perfect
solve
removed.
foresight path.
This
the
system
We
of
restrict
path has
the
characteristic that the value of a variable x in quarter j is always equal
9
to its realization four quarters ago.
For our economy with
(quarterly)
seasonals, equations [2.10] - [2.16] can be reduced to twelve restrictions
which
{ cp , kj, n^
; j
=
1,2,3,4
} must
satisfy.
These
seasonal
restrictions (without uncertainty) can be written as:
f2
f2
-4
rkr—
'k + b ^p — 1.
1.^J
J+l
~
/)\ rO -6 -6X .
a. (1-0)
k. n. e
j
J
J J
J
. r e-i
p
j+i
i-e
kj+i V i
$
' * Aj + i
e
i-e
+
n.
J
d-5) e ' V i
[2.21]
(1-5) kj
[2.22]
where the seasonal index j runs from 1 to 4.
convention that when j=5,
r_
[2.20]
We adopt a wrap-around dating
this represents the first quarter
iff
r
(j=l).
The
_
variables a., c., and 1. are functions of the essential variables cp and n,
J J
J
as well as the exogenous variables g, r, and A:
~k
Cj
“
~
~
—
Cpj + 71 Sj+ 3 (Cpj-l + 71 gj-l^ 6
T - n.
J
r.
i
+
b(T-n^_1 )
f
+
a
-j
a
P
C.
J
[2.24]
\
\
- e
[2.23]
j+l
J
^
rj+l
[2.25]
►
Cj+1 ^
This leads to the following definition:
Definition
A sequence
{ cp^,
k^,
n^
; i = 1,2,3,4
) which satisfies
equations [2.20] - [2.22] is a perfect foresight seasonal equilibrium.
10
For all parameterizations of the model which we consider below,
able to numerically calculate a unique seasonal equilibrium:
of multiple equilibria was found.
Furthermore,
we were
no evidence
in the course of solving
for the stochastic equilibrium (in Section 5), we calculate roots for the
log-linear system which exhibit the proper characteristics to ensure local
stability of the perfect foresight seasonal equilibrium.
state space representation,
That is, in the
the fundamental matrix had equal numbers
roots inside and outside the unit circle.
of
Also, Chatterjee and Ravikumar
[1989] characterize existence and uniqueness in a model similar to ours.
Given the perfect foresight path for { cp^, k^, n^ ; i — 1,2,3,4 } and
w . , rj } can be computed using seasonal counterparts to equations [2.14],
[2.13], [2.17], and [2.18]:
[2.26]
i. =
J
[2.27]
w. = (1-0 y./ n.
[2.28]
rj - '
/ (V
[2.29]
’Aj)
To compare these seasonal means with the Barsky-Miron seasonal results, the
data set must be precisely defined,
the seasonals in { Aj ’ rj ’ &j
> j **
1,2,3,4 } must be estimated, and the model's other parameters set.3
3.
Estimation of the Structural Parameters
Given seasonally unadjusted time series data for the U.S economy, the
Euler equation methods of Hansen-Singleton [1982] can be used to estimate
the model's structural parameters and test the overidentifying restrictions
implied by the model and choice of instruments.
11
The parameter vector to be
estimated is:
7^,
= ( $, a, b, Ax> X2 , A3> A^,
where
the
seasonals
are
related
- p log g^
relationship d^ = log
d3 , d4> p, o^, a g ),
t 4>
d^
to
the
log
seasonals
by
the
Notice further that there are only
two t *s. In estimating the model we restrict attention to preferences that
vary only in the fourth quarter in response to Christmas. Thus in quarters
one through three r takes on the value
7^, 7^,
the value r^. The parameters /?,
in
depreciation
rate
Kydland-Prescott
parameter
7^
Christiano-Eichenbaum
S
was
[1982]
chosen
and
and T are set a priori,
to
[1990]
be
and
2.5%
King-Plosser-Rebelo
Braun
per
[1990].
quarter,
[1988].
The
The
as
private consumption was chosen to be 0.4, as found by Aschauer [1985].
7^
on
leisure
was
normalized
to be
1.
The
in
utility
governing the substitutability of government purchases
utility weight
in
The discount factor ft was chosen to be
accordance with previous studies.
- 25
1.03 * , as
and in quarter four r takes on
total
for
The
time
endowment for the household is 1369 hours per quarter.
The
moment
stochastic
Euler
government
Specifically,
equations
equations,
spending
chosen
the
for
the
production
autoregression,
and
estimation
function,
two
consist
the
variance
of
transitory
estimates.
the household's time t decision for n^_ and kt+^ yield the
conditions:
two
[3.1]
[3.2]
12
where c
and 1
values of
7^,
can be constructed from equations
a, and b.
[2.2]
and
[2.4]
given
The technology specification [2.5] - [2.6] yields
the stochastic equation:
---- | A log yt • 6 A log kt - (1-0) A log nt
j
[3.3]
_A1 Qlt ‘ A2 Q2t ' A3 Q3t ' A4 Q4t
where
c
is an unconditionally mean zero
observed
by
the
econometrician.
The
random variable which
law
of
motion
for
government spending [2 .8] yields the stochastic equation:
g t -1
gt
log —
- P log —
dl ^lt - d2 Q2t ' d3 Q3t - d4 <*4t = U t
Zt
"-t-1
where u^
is an unconditionally mean zero
observed by the econometrician.
is not
transitory
I3 ’4 !
random variable which
is not
The d^ seasonals are related to the log g^
seasonals by the relationship d^ - log g ^ ’ P log
Finally,
the
residual errors e and u are used to estimate the standard deviations a
t
t
c
and a :
u
[3.5]
E(
\
■ "l
)
[3.6]
- 0
Equations [3.1] - [3.6] are estimated simultaneously.
The instruments for equations [3.1] - [3.6] were selected as follows:
in equation [3.1], four seasonal dummies, and the time t growth rates of
private consumption, leisure, output-capital ratio and output-labor ratio;
in equation
rates
of
[3.2],
private
output-labor ratio;
[3.4],
four
four seasonal dummies and the time t and t-1 growth
consumption,
leisure,
in equation [3.3], four seasonal dummies;
seasonal
dummies
and
the
equations [3.5] and [3.6], only unity.
output-capital
13
logarithm
of
g^ ^/zt
ratio
and
in equation
an(*
A total of 31 instruments are used
to
estimate
yielding
the
13
18
parameters
overidentifying
of
the
nontime-separable
restrictions.
For
specification,
a
time-separable
specification in which the parameters a and b are set to zero a priori,
only 16 parameters are estimated, yielding 15 overidentifying restrictions.
The
[1989]
original
data:
data
U.S.
seasonality.
set
employed
quarterly
in this
data
which
For the empirical analysis
constructs of our model,
however,
Investment
Government
capita.
(g)
The
is
not
the
been
Barsky-Miron
adjusted
for
theoretical
we redefine some of the variables
as
Output (y) is Gross National
Private consumption (cp) is nondurables plus services
consumption expenditures per capita.
Fixed
has
is
to conform to the
follows (and convert to per capita values).
Product per capita.
study
plus
Durable
Federal,
capital
consumption
State,
stock
Investment (i) is the sum of Business
is
and
expenditures,
Local
computed
government
using
the
per
capita.
purchases,
flow
per
investment
expenditures, a quarterly depreciation rate of 2.5%, and an initial capital
stock value for 1950.
Labor hours are computed as the product of total
nonagricultural employment times average hours per week of nonagricultural
production workers times 13 weeks per quarter (per capita).
Average labor
productivity and the capital rental rate are constructed from the output,
labor, and capital data.
The data is converted to per capita values by
using the civilian population, 16 years and older.
Table
3.1
presents
two
sets
of
parameter
estimates
of
tf.
The
time-separable (TS) estimates set the parameters a and b equal to zero a
priori;
the nontime-separable estimates (NTS) allow a and b to be nonzero.
The NTS estimates display local durability (or adjacent substitutability)
in preferences for consumption goods and habit-persistence in preferences
for
leisure
hours;
that
is,
a
is
14
estimated
to be
positive
and
b
is
estimated to be negative.
Habit-persistence
in leisure
is consistent with previous
analyses using these preference specifications.
quarterly data,
leisure.
Braun
[1990] has
In
empirical
seasonally adjusted
found evidence of habit-persistence
In seasonally adjusted monthly data, Eichenbaum-Hansen-Singleton
[1989] have also found evidence of habit-persistence in leisure hours.
number
in
of
researchers
have
consistent with a>0.
returns,
estimated
consumption preferences
which
are
For example, using monthly data on consumption and
Eichenbaum-Hansen-Singleton
[1989],
Gallant-Tauchen
[1990]
Heaton[1990]
find evidence of local durability
in consumption.
other hand,
using
Braun
habit-persistence
A
quarterly
consumption
data,
finds
and
On the
evidence
of
and Constantinides [1990] shows that negative values of
a can help explain the equity premium puzzle.
In Table 3.1,
the remaining parameter estimates are similar across
both sets of estimates.
to
be
around
.28
The capital and labor shares are tightly estimated
and
.72,
respectively.
transitory government spending are similar:
low, while first quarter spending is high.
government
spending
approximately .88.
autoregressive
The
seasonal
patterns
in
fourth quarter spending is
For each parameterization the
coefficient
p
is
estimated
to
be
Finally, the estimated standard deviations of u^_ and €
are roughly similar across both parameterizations.
Both estimated parameterizations display a large degree of seasonal
variation
in
technology.
The
fourth
quarter
growth
in
total
factor
productivity is estimated to be 6% (or 24% on an annualized basis).
The
first quarter experiences technical regress (on average), growing at a rate
of
-7%
in
one
quarter.
If
the
technological
specification
[2.4]
is
correct, and the factor inputs and output are properly measured, then these
15
seasonals represent true seasonal variation in aggregate technology.
[1990]
Barro
suggests that, weather conditions and seasonality in construction
probably
account
for
technological growth.
measured
seasonal
technology,
some
of
the
seasonal
patterns
in
aggregate
In this paper we adopt the interpretation that the
variation
in technology
represents
true variation
in
and consider the ability of the equilibrium model to capture
the seasonal patterns uncovered by Barsky-Miron.^
Finally,
Hansen's
[1982]
J-statistic indicates
that neither model's
overidentifying restrictions can be rejected at conventional significance
levels.
Since
the
nontime-separable
specification
nests
time
preferences we can directly test the additional restrictions
separable
imposed by
time-separability by estimating the TS specification using the converged
NTS
weighting
matrix.
Following Eichenbaum,
This
estimation
produced
Hansen and Singleton (1988)
a
J-statistic
of
888.
the difference between
this statistic and the J-statistic from the nontime-separable estimates is
distributed
asymptotically
x
2
with
two
degrees
of
freedom.
Thus,
the
additional restrictions imposed by time-separable preferences are sharply
rejected at conventional significance levels.
4.
The Perfect Foresight Seasonal Equilibrium Analysis
This section examines the roles of seasonal technology, preferences,
and government purchases
for
generating
equilibrium
seasonality
in
our
Our research strategy is to answer the following question:
in the context
of a standard neoclassical model, what factors account for the seasonal
fluctuations in aggregate data? If seasonal variation in preferences r and
government g are important, while technology A is not, then explaining the
estimated seasonality in A is unlikely to be a high priority. On the other
hand, if all of the seasonality in aggregate data is explained by seasonal
variations in A, then an explanation for seasonal A is of first-order
importance (although beyond the scope of the current paper).
16
parameterized
aggregate
model.
variables
Figure
along
4.1
the
PFSE
plots
the
seasonal
growth path
for
growth
three
rates
cases:
of
(1)
technology seasonals only (TECH), (2) preference seasonals only (PREF), and
(3) transitory government seasonals only (GOVT).
indicates,
this
As the discussion below
decomposition analysis highlights
the
important role
of
technology seasonals in generating seasonals in output and hours.
Case 1:
Technology Seasonals
In each panel of Figure 4.1, the TECH case presents the results from
solving the perfect foresight NTS model for the scenario in which the only
source of seasonality is in technology.
set at their estimated values.
their weighted average value
transitory
government
The technology seasonals have been
The preference seasonals r have been set to
of
purchases
.1879.
Although
process
(g^_)
the
have
seasonals
been
set
to
in the
their
nonseasonal mean value, the government purchases panel still shows seasonal
variation in the growth rate of g^_.
This variation is inherited from the
2
seasonal variation in the permanent component of government purchases.
Along the seasonal equilibrium path, seasonal variation in technology
has important effects on hours, output, investment and labor productivity.
All
of
these variables
are
proseasonal,
exhibiting positive
(negative)
growth rates in seasons where the growth rate of technology (A) is positive
(negative).
The seasonals
in output are approximately the same as
seasonals
technology;
this
in
is
similar
to
the
observation made
the
in
Christiano [1988] that the movements in output and the Solow residual (in
Our specification for government purchases embodies the assumption that g^
and
z^_ cointegrate.
This
assumption
is
necessary
for
the
perfect-foresight equilibrium path to exhibit balanced growth.
17
economy's
adjusted data) are very close.
volatile as output.
output;
as
in
Seasonal investment is about four times as
Seasonal hours are only about one-half as volatile as
the
RBC
literature,
since
seasonal
output
variability
exceeds that of labor hours, labor productivity is strongly seasonal.
Private consumption purchases are smooth but slightly counterseasonal.
There are two reasons for this.
First, since the technology seasonal is an
anticipated event, there is no wealth effect.
variability,
Second,
fourth
quarter government purchases grow due to the growth in technology.
Since
7 ^=.4
and
unanticipated
government
consumption,
wealth
shocks,
So one source of consumption
purchases
consumption
demand
substitute
declines
though the real interest rate falls,
7^=0
the
is
in
absent.
imperfectly
the
fourth
for
private
quarter.
Even
equilibrium consumption falls.
If
and government purchases do not substitute for private consumption,
model
Therefore,
predicts
that
technology
equilibrium
seasonals
consumption
do not
contribute
would
rise
slightly.
significantly
to
the
seasonality in consumption expenditures.
Habit-persistence in leisure preferences have the effect of smoothing
seasonal
labor
hours
movements.
technology seasonals
times
as volatile
volatile.
economies:
The
only,
as
for
pattern
output,
For
seasonal
the NTS
of
time-separable
labor hours
economy;
seasonality,
investment,
the
are approximately
output
however,
labor hours,
economy
is
is
the
about
same
with
three
twice
as
for both
and labor productivity are
proseasonal, while consumption and the interest rate are counterseasonal.
Given our
estimates
of the
structural parameters,
habit persistence
in
leisure plays an important role in matching the magnitude of a seasonal
cycle, but not the pattern.
In summary,
seasonal variation
18
in
technology
leads
to
substantial
seasonal
fluctuations
fluctuations
are
in
output.
essentially
For
the
same
the
size
NTS
as
economy,
the
the
estimated
output
technology
seasonals.
Case 2:
Preference Seasonals
In each panel of Figure 4.1, the PREF case presents the results from
solving the perfect foresight NTS model for the scenario in which the only
source
of
seasonality
is
in
preferences.
Technology
purchases do not contain any seasonal effects:
and
government
the quarterly growth rates
of technology and government purchases are set at the baseline average of
0.17%.
In our economy with only preference seasonals, the equilibrium effects
essentially involve only consumption and investment.
The first and fourth
quarter movements in consumption growth closely parallel the two seasonal
movements
in
preferences
(r^
and
r^).
Since
consumption
preferences
exhibit adjacent substitutability (a>0), second quarter consumption growth
is positive due to the first quarter reduction in consumption expenditures.
A similar explanation holds for the third quarter drop in expenditures.
The
transitory
nature
of
these
shifters
produces
seasonal
changes
in
investment that almost exactly offset the changes in consumption,
leaving
the level of output unchanged.
and the
Labor hours,
labor productivity,
interest rate are also largely unchanged over this seasonal equilibrium
path.
and
These findings also hold for the TS economy:
productivity
preferences.
consumption
display
Attempts
preference
to
no
perceptible
match
shifters
the
would
seasonal
seasonal
output, labor hours,
variation
variation
dramatically
magnify
responses of consumption and investment documented here.
19
in
due
GNP
the
Thus,
to
with
large
seasonal
fluctuations in technology are crucial for generating seasonal variation in
output.
Case 3.'
Transitory government spending seasonals
In each panel of Figure 4.1, the GOVT case presents the results from
solving the perfect foresight NTS model for the scenario in which the only
source of seasonality is in transitory government purchases.
Technology
and
Along
preferences
seasonal
have
no
growth path,
seasonal
variation
consumption and
in
this
case.
investment move
together;
these
movements run counter to the seasonal pattern in government purchases.
TS
results
are
about
equilibrium responses
analysis:
changes
interest rate
the
same
as
for
the NTS
economy.
The
the
The
seasonal
to government seasonals are similar to the Case 2
in output,
are virtually
labor hours,
imperceptible.
labor productivity,
Again,
this
and the
reinforces
the
important role of seasonal variations in technology.
Case Analysis Summary
Collecting the results from Cases 1 - 3
labor
hours,
labor
productivity,
and
suggests that seasonal output,
interest
determined largely by the technology seasonal.
driven primarily by the preference shifter,
rate
movements
will
be
Seasonal consumption is
and investment patterns will
depend on the joint configuration of technology, preference, and government
spending seasonals.
and
anticipated
This decomposition is due primarily to the transient
nature
of
the
seasonal
technology.5
5.
Evaluation of the Stochastic Model
20
shifters
in
preferences
and
This section presents results from solving a stochastic version of the
seasonal business cycle model.
In the course of describing the empirical
characteristics of this model two issues will be addressed.
models'
predicted
seasonal
patterns
will
be
compared
First,
with
the
the
data.
Second, (deseasonalized) cyclical properties of the seasonal business cycle
models will be examined and compared with the data.
5.1
Solving the Stochastic Model
Before discussing the results we will briefly outline the methodology
used in solving for the stochastic equilibrium.
In the last section the
perfect foresight seasonal equilibrium was calculated and analyzed.
this section we approximate the true stochastic equilibrium.
In
In solving
the model we modify the standard method (see Kydland and Prescott [1982],
King,
Plosser and Rebelo
[1988] and Christiano and Eichenbaum
[1990]) of
approximating the true stochastic equilibrium with a Taylor expansion about
the steady-state.
The modifications arise because the perfect foresight
equilibrium we consider exhibits seasonal cycles.
The nonlinear model has
the characteristic that technology and preferences change with the season.
Varying the location of the Taylor approximation with the season allows the
linearized system to inherit this characteristic of the nonlinear model.
We
consider
this
approach
to be
the appropriate
strategy adopted by Hansen and Sargent
[1990],
generalization
Ghysels
[1989],
of
the
and Todd
[1990] who analyze seasonality in explicitly linear-quadratic frameworks.
The first step
[2.10]-[2.16]
in solving the model
about
the
calculated in section 2.
stochastic
difference
perfect
is to linearize
foresight
seasonal
the equations
equilibrium
path
The linearized system can be reduced to twelve
equations
governing
21
the
evolution
of
the
capital
stock,
hours,
equations
and
are
[2.20]-[2.22].
private
consumption
linearized,
in
stochastic
each
These
season.
counterparts
to
twelve
equations
In a technical appendix we display the linearized system
and describe how these twelve difference equations are mapped into a state
space representation which can be solved using methods described in King,
Plosser, and Rebelo [1990].
The state space representation essentially has
the same structure as Todd's time-invariant linear-quadratic representation
(TILQ) or Hansen and Sargent's time-varying strictly periodic equilibrium.
The model's
optimal
solution is a series of twelve equations
decision rule
for capital,
hours,
and private
that describe
the
consumption,
one
equation for each season:
[4.1]
Kt+i - A st
where
1
2
3
4 1 2 3 4
1
,2
i3
.4
K t+i - i k t+1 kt+l kt+i kt+i cp t cpt cpt Cpt nt nt nt nt
and
1
St
2
St
3
st
.2
At
4.1
gt
\
,3
At
4 .
At !'•
where the superscripts denote the seasons.
Given these log-linear decision rules for capital, private consumption
and hours,
economy.
it is straightforward to generate time series
First,
a
sequence
of normal
variables
is
drawn
for the model
to mimic
empirical covariance structure of the forcing processes u^_ and
and
the
Once u^_
have been constructed it is straightforward to calculate A^ and g^_.
Then given an initial K q we can construct a sequence of realizations for
the
capital
method.
stock,
hours,
and private
consumption using
following
If this is the jth quarter then use the jth, j+4th and j+8th row
of matrix A along with the current states:
the
22
k£, cp^’]", n^”|, A^ , and g^ to
t
t™
11.
tz* J.
^
^
determine
the
current decisions
consumption and hours.
today's
consumption
for next period's
capital,
and
today's
Given the values of next period's stock of capital,
and
today's
work
effort,
determine the current choices of output,
it
is
straightforward
investment,
real interest rate using equations [2.14],
[2.13],
real wages,
to
and the
[2.17], and [2.18].
In
practice we choose an effective sample length of 88 and provide results
based on 500 draws.
The equilibrium model developed in this paper
across
the entire
moments,
spectrum.
researchers
To
often
focus
decompose
imposes
restrictions
attention on a specific
time
series.
Examples
set
of
of
such
decompositions include first differencing to remove low frequency moments
and seasonal adjustment to remove particular high frequency moments.
Our
objective is to investigate the model's ability to match both the seasonal
patterns uncovered by Barsky and Miron,
moments which Prescott defines
as well as the data's
to be the business
cyclical
cycle phenomena.
To
facilitate these comparisons, we adopt Barsky and Miron's decomposition of
the
stationary,
stochastic
processes
"indeterministic" components.
into
Specifically,
the data to induce stationarity, we regress
dummies:
moments
regressions
Obviously,
seasonal"
and
after log-first differencing
each series on four seasonal
the estimates on the dummy variables define the seasonal patterns
emphasized by Barsky and Miron.
to
"deterministic
calculated
as
using
relating
to
We also adopt the convention of referring
the
indeterministic
cyclical
or
residuals
business
cycle
from
these
phenomena.
the properties of the "seasonal cycle" will vary depending on
the particular decomposition used.
An alternative approach is to simply avoid decompositions.
Hansen and
Sargent [1990] observe that seasonally unadjusted data can be stationary,
23
conditional on a starting season.
moments
approach
for
seasonally
ignores
the
In light of this result, we also report
unadjusted
entire
growth
rates
distinction
in
between
section
business
5.3.
This
cycles
and
First we report results on the seasonal properties of the model.
In
seasonal cycles.
5.2
Seasonal Predictions of the Stochastic Model
summarizing
question:
the
results
particular
attention
is
can a parsimoniously parameterized
paid
to
the
real business
following
cycle
capture the central features of the data at seasonal frequencies?
to facilitate
comparision with Barsky and Miron's work we
model
In order
address
this
question by considering the same set of moments reported in their paper.
The first set of columns in Table 5.1 present the seasonal patterns for the
data set using the log first difference filter.
The seasonal means are
reported in terms of percentage deviations from average growth rates for
the
sample period 1964:1
to
1985:IV.
The
real
interest rate,
which
is
measured by the rental rate on capital, is reported in terms of annualized
rates of return.
variable
which
The table also includes R-square
describe
the
percentage
of
the
statistics
total
variation
for each
in
the
particular time series that is attributable to the deterministic seasonal.
Finally, we report standard errors for each estimate that are based on the
Newey-West [1987] weighting matrix with 12 autocorrelations.
The second and third sets of columns in
results
for
respectively
preference specifications.
the
Table 5.1 contain simulation
time-separable
For each specification,
label the average seasonal means for 500 draws.
the average R-square of the regressions.
and
24
nontime-separable
columns
1 through 4
The fifth column contains
Comparisons of the results
in table 5.1 reveal several
shortcomings of the time-separable model.
seasonal means
in output,
significant
The model sharply overstates the
hours and investment.
The
quarter consumption means are also a poor match.
second and fourth
The predicted R-squares
for these variables also exceeds the respective number in the data in each
instance.
Overall, the time-separable specification does not capture the
seasonal properties of the data.
One way to interpret the time-separable model's shortcomings
focus on hours.
is to
In the fourth quarter, for instance, the model predicts a
large positive seasonal in hours under the first difference filter while
the data indicates that hours are only slightly above their fourth quarter
mean value.
increase
The model's surge in fourth quarter hours
in output which
seasonal.
Since
these
is high
already due
seasonals
are
to a positive
anticipated
wealth effects on consumption are small.
induces a large
and
Instead,
technology
temporary,
their
optimizing households
choose to increase desired savings which shows up as increased equilibrium
investment.
variability
Based on this analysis,
in hours
across
the
it is conceivable
seasons
variability in investment and output.
the
model
that
habit-persistence
act
in
to
smooth
leisure
is
the
cause
Alternatively,
hours
preferences)
across
the
will
also
that the excess
for
the
excess
generalizations of
seasons
smooth
(such
output
as
and
investment.
This proposition is explored in the context of the nontime-separable
preference results in Table 5.1. In order to facilitate comparision of the
NTS
results
with
the
data,
graphically in Figure 5.1.
seasonal
growth
rates
are
also
presented
Recall from Section 3 that estimates
of the
nontime-separabilities produced evidence of local durability in consumption
25
and habit persistence in leisure.
smooth
desired
Figure
5.1
labor
Habit persistence in leisure acts to
supply between
reveals that
the
NTS
adjacent
periods.
Examination
specification captures
many
of
of the
seasonal movements in the data. The model successfully mimics the overall
seasonal patterns
in output,
productivity and capital.
consumption,
For
government purchases,
these variables
the model
average
reproduces
the
sequential pattern of seasonal movements in the data and in most cases the
3
magnitudes .
The
model
is
somewhat
less
successful
with
respect
to
investment, the rental rate on capital, and hours. For the rental rate the
model
consistently
pattern is
overstates
correct.
the
magnitudes,
In the case
although
the
sequential
of investment, the model captures the
sequential pattern of seasons found in the data, but understates the second
quarter rise and overstates the fourth quarter rise in investment growth.
Hoursrepresent
non-time
the
separabilities
model's
in
single
leisure
have
largest failure.
improved
the
Although the
model's
seasonal
predictions, the magnitudes are still off in three out of four quarters and
the model predicts a counter-factual rise in fourth quarter employment. One
strategy that would improve the model's predictions for hours would be to
introduce a more complicated pattern of preference shifters.
see
little
economic
rationale
for
increasing
the
However, we
dimensionality
of
exogenous seasonal shifters that cannot be lined up with calendar events
like
Christmas.
Analternative
strategy
that
we
explore
in
Braun and
Evans(1991) involves modeling time-varying work effort.
A cursory inspection of the graphs might suggest that the fit for capital
not particularly good. Notice however, that the vertical axis is in tenths
of a percent.
26
5.3
Cyclical Predictions of the Stochastic Model
This
model.
subsection examines
the cyclical properties
of the
Two distinct parameterizations are considered:
stochastic
the time-separable
GMM optimum (TS) and the nontime-separable global optimum (NTS).
Table 5.2 contains results relating to relative variability and
cross-correlations
under
three
with
different
output
for both
filters.
The
corresponds
to
moments
differenced
and
regressed
corresponds
to
data
that
calculated
on
four
has
been
parameterizations
heading
using
"one-quarter
data
dummies.
regressed on four seasonal dummies.
and
that
The
data
growth
rate"
been
first
has
heading
Hodrick-Prescott
the
"HP
filtered
filter"
and
then
The heading "Unadjusted one-quarter
rates" corresponds to moments calculated using data that has been first
differenced only.
For
each
filter
we
report
moments
1964:1-1985:4 in the first column.
for
U.S
data
running
from
The second column contains standard
errors for the data's moments reported in column one.
The standard errors
were calculated using a Newey-West weighting matrix with 12 lags. The third
and fourth columns contain results from simulating the model using the TS
and NTS parameterizations. In each case the reported statistics are sample
averages based on 500 draws of length 88.
Looking first at the properties of the data,
filters which incorporate deseasonalization
observe
( one-quarter
that the
two
growth and HP
filter in table 5.5) produce the same general patterns. With respect to
relative variability,
government
purchases
investment
are
about
is about
as
twice
variable
as
as variable
output,
and
as
output,
hours
and
consumption are less variable than output. We do observe some differences
in
moving
from
the
deseasonalized
27
one-quarter
growth
rates
to
the
deseasonalized HP filtered data. In particular, there are significant rises
in the relative variabilities of hours and investment and declines in the
relative variabilities
of consumption and government
expenditures.
With
respect to correlations we observe similar general patterns accross the two
filters. Here the most significant differences occur in the instances of
investment and hours.
More important differences are observed when comparing the first two
filters
data.
with
the
third
filter,
first
differenced
seasonally unadjusted
Output is considerably more variable prior to removal of seasonal
means and
some of the correlations are quite different.
For instance,
government purchases have a correlation with output of about .3 under the
first two filters, but a correlation of
.8 under the third filter.
In
general, the strongest cross-correlations with output occur in data which
have not been seasonally adjusted, but have been first differenced in order
to
induce
stationarity.
This
is
one way
of
interpreting
Barsky
and
Miron's finding that aggregate variables exhibit strong comovement across
seasonal frequencies as well as cyclical frequencies.
Turning to the theory,
properties of the NTS
consider next a comparison of the
cyclical
parameterization with the TS parameterization.
In
performing these comparisons we use the standard errors for the data as a
metric.
The NTS parameterization is successful
in matching many of the
properties of unadjusted 1-quarter growth rates.
It captures most of the
relative variability statistics and cross-correlation patterns found in the
data. The major shortcomings lie in the failure of the model to capture
precisely the relative variabilities of consumption and investment and the
correlations
of
hours, government
expenditures
and
rental
rates
with
output. The TS parameterization has considerably more difficulty matching
28
the unadjusted moments,
missing the relative variability of consumption,
investment, hours and average productivity by wider margins and overstating
the correlation of capital with output. On the basis of these seasonally
unadjusted moments, the NTS
parameterization captures more features of the
data.
If we compare
the predictions
of the NTS
and TS parameterizations
under the two seasonally adjusted filters, we get a different picture. In
many cases the two models1 predictions lie outside a two standard deviation
band around the data.
The largest differences appear to occur under the
1-quarter growth filter.
specification
fails
to
With respect to relative variabilities
capture
the
relative
variabilities
productivity, employment and government expenditures.
fails
to capture
capital
and
the relative variabilities
average
productivity.
In
of
average
The TS specificaton
of consumption,
addition,
the NTS
the
TS
investment,
specification
signficantly overstates the variability of output. Similar patterns emerge
under the HP-filter.
On net,
we would argue
that the NTS
specification
captures more aspects of the relative variabilities in the data with the
failure in hours more than offset by sucesses in consumption,
investment
and overall variability in output.
Comparisons of seasonally adjusted contemporaneous correlations with
output
reveal
1-quarter
growth
contemporaneous
employment
significant
and
filter,
failures
both
correlations
the
rental
rate
of
both
specifications.
specifications
of
investment,
with
output.
fail
to
capture
government
Under
the
Under
the
the
purchases,
HP-filter
both
specifications fail to capture the correlation of output with government
purchases, employment and average productivity.
How do these results compare with the performance of standard business
29
cycle models that ignore seasonality?
Many of the models' predictions are
similar to those of standard real business cycle models. The tendency for
RBC models
using
estimated parameterizations
to
overstate
the
observed
variability of output in the data appears in both Christiano and Eichenbaum
(1990)
and Braun (1990).
benchmark RBC models
Comparison with these other studies finds that
overstate
the contemporaneous
correlation of hours
with output and average productivity with output. This is attributed
to
the absence of important labor supply shifters. Christiano and Eichenbaum
find that when
measurement error in hours
these predicted correlations drop.
is modeled as being
Braun finds that movements
i.i.d.,
in income
taxes shift labor supply thereby reducing both of these correlations.
Overall, we conclude that the NTS specification captures the general
features
of
the
seasonal
cycle
while
continuing
to
capture
the
same
features of the business cycle that have generated so much attention for
this model. The NTS specification successfully captures important aspects
of seasonal fluctuations in output, consumption, average productivity and
investment but
fails
to capture
the
seasonal pattern
in hours.
The TS
specification on the other hand fails to capture many of these moments. At
business cycle frequencies the NTS specification also captures more of the
variability in seasonally adjusted data than the TS specification. Smaller
differences are observed when comparing contemporaneous correlations. Thus,
an additional finding of this analysis is that nontime-separabilities play
an important role in explaining seasonal fluctuations and contribute to the
overall performance performance of business cycle models more generally.6
6.
Conclusions
In
this
paper
we
have
introduced
30
seasonals
shifters
into
an
equilibrium model of the business cycle.
perfect
foresight
seasonal
The model
equilibrium
is
is tractable:
computable
without
the
any
approximations, and an approximate linear solution of the stochastic model
can be
calculated using methods
[1990],
and Todd
[1990],
The
analogous
to
of
structural parameters
estimated using GMM with seasonally unadjusted,
overidentifying restrictions
those
Hansen-Sargent
and seasonals were
postwar U.S.
data.
The
implied by the model cannot be rejected at
conventional significance levels.
Are Barsky and Miron's findings consistent with current equilibrium
business cycle theories as surveyed by Prescott [1986]?
Conditional on our
parameterization, the nontime-separable model predicts most of the seasonal
patterns found in aggregate quantity time series;
the seasonal pattern in labor hours.
a notable exception is
The model also predicts many of the
deseasonalized second moment properties of the data.
question is yes:
Our answer to this
this equilibrium model generally displays the seasonal
patterns discovered by Barsky and Miron
[1989]
as well
as the business
cycle phenomenon.
We view this model as a benchmark in the tradition of Kydland and
Prescott [1982] and Long and Plosser [1983].
While the predictions of the
theory match the data's properties fairly well,
theory require
equilibrium
further
models,
investigation.
seasonal
the assumptions of the
In particular,
variation
in
for
technology
this
is
class
crucial
of
for
delivering seasonal fluctuations in output.
Does the aggregate technology
vary exogenously as much as our estimates
suggest,
or is this
variation due to some misspecification of the technology?
seasonal
Future research
should assess the plausibility of seasonality in aggregate Solow residuals
by examining alternative general equilibrium economies.
31
We conjecture that
this seasonal investigation will provide new macroeconomic insights
into
the importance of labor hoarding (as in Summers [1986]), increasing returns
due to endogenous growth (Romer [1986]), increasing returns due to market
externalities
(Diamond
[1982],
Murphy-Shleifer-Vishny
[1989]),
countercyclical markups of price over cost (as suggested by Hall's
evidence),
the
and propagation mechanisms in general.
above
measured
phenomenon
Solow
nonseasonal.
could
residuals,
produce
even
if
endogenous
true
In principle,
seasonal
technological
[1989]
each of
movements
advances
in
are
The ability of these theories to encompass the results of
this benchmark model and the seasonality in measured Solow residuals is the
topic of our current research.
32
References
Aschauer, D. , 1985, Fiscal Policy and Aggregate Demand, American
Economic Review, 117-127.
Barro, R . , 1990,
Macroeconomics (J. Wiley Company, New York).
Barsky, R. and J. Miron, 1989, The Seasonal Cycle and the Business
Cycle, Journal of Political Economy.
Braun, R . , 1990, The Dynamic Interaction of Distortionary Taxes and
Aggregate Variables in Postwar U.S. Data, unpublished Ph.D.
thesis, Carnegie Mellon University.
Braun, R. , and C. Evans 1991, Seasonal Solow Residuals and Christmas: A
Case for Labor Hoarding and Increasing Returns, manuscript,
Federal Reserve Bank of Chicago.
Chatterjee, S. and B. Ravikumar, 1989, A Neoclassical Growth Model
with Seasonal Perturbations, unpublished manuscript, University
of Iowa.
Christiano, L. and M. Eichenbaum, 1990, Current Real Business
Cycle Theories and Aggregate Labor Market Fluctuations,
unpublished manuscript, Institute for Empirical
Macroeconomics.
Constantinides, G . , 1990, Habit Formation: A Resolution of the Equity
Premium Puzzle, Journal of Political Economy 98, pp. 519-543.
Diamond, P., 1982, Aggregate Demand in Search Equilibrium, Journal of
Political Economy 90, pp. 881-894.
Eichenbaum, M. and L. Hansen, 1990, Estimating Models with
Intertemporal Substitution Using Aggregate Time Series Data,
Journal of Business and Economic Statistics.
Eichenbaum, M . , L. Hansen, and K. Singleton, 1989, A Time Series
Analysis of Representative Agent Models of Consumption and
Leisure Choice Under Uncertainty, Quarterly Journal of
Economics.
Gallant, R . , L. Hansen, and G. Tauchen, 1990, Using Conditional
Moments of Asset Payoffs to infer the Volatility of
Intertemporal Marginal Rates of Substitution,
Journal of Econometrics.
Gallant, R. and G. Tauchen, 1989, Seminonparametric Estimation of
Conditionally Constrained Heterogeneous Processes: Asset
Pricing Applications, Econometrica 57, 1091-1120.
Ghysels, E. 1991, On the Economics and Econometrics of Seasonality,
Discussion Paper C.R.D.E. Universite de Montreal.
33
Hall, R. , 1988, The Relation between Price and Marginal Cost in
U.S. Industry, Journal of Political Economy 96, p p . 921-947.
Hansen, L., 1982, Large Sample Properties of Generalized Method of Moments
Estimators, Econometrica 50, 1029-1054.
Hansen, L. and T. Sargent, 1990, Disguised Periodicity as a Source of
Seasonality, unpublished manuscript, Hoover Institution.
Hansen, L. and K. Singleton, 1982, Generalized Instrumental Variables
Estimation of Nonlinear Rational Expectations Models,
Econometrica 50, 1269-1286.
Heaton, J., 1990, The Interaction between Time-Nonseparable Preferences
and Time Aggregation, unpublished manuscript, Sloan School, MIT.
King, R . , C. Plosser, and S. Rebelo, 1988, Production, Growth, and
Business Cycles, Journal of Monetary Economics 21, 309-342.
King, R . , C. Plosser, and S. Rebelo, 1990, Technical Appendix to
Production, Growth, and Business Cycles, unpublished
manuscript, University of Rochester.
Kydland, F. and E. Prescott, 1982, Time to Build and Aggregate
Fluctuations, Econometrica 50, 1345-1370.
Kydland, F. and E. Prescott, 1989, Hours and Employment Variation
in Business Cycle Theory, manuscript, Federal Reserve Bank
of Minneapolis.
Long, J. and C. Plosser, 1983, Real Business Cycles, Journal of
Political Economy 91, 39-69.
Murphy, K . , A. Shleifer, and R. Vishny, Building Blocks of Market
Clearing Business Cycle Models, NBER Macroeconomics Annual 1989,
pp. 247-287.
Newey, W. and K. West, 1987, A Simple, Positive Definite,
Heteroskedasticity and Autocorrelation Consistent Covariance
Matrix, Econometrica 55, pp. 703-708.
Prescott, E., 1986, Theory Ahead of Business Cycle Measurement,
Carnegie-Rochester Conference Series on Public Policy 27
(Autumn), 11-44.
Summers, L., 1986, Some Skeptical Observations on Real Business Cycle
Theory, Quarterly Review 10 (Federal Reserve Bank of Minneapolis,
Minneapolis), 23-27.
Todd, R . , 1990, Periodic Linear-Quadratic Methods for Modeling
Seasonality, forthcoming in Journal of Economic Dynamics and
Control.
34
T a b le 3.1
GMM Estimates of the Structural Parameters
Time —Separable
Nontime —Separable
Estimate
Std Error
Estimate
Std Error
.2751
.0055
.2803
.0195
a
.3402
.0184
b
-.4956
.0154
0
A1
-.0710
.0027
-.0724
.0031
A2
.0375
.0022
.0385
.0026
A3
-.0190
.0015
-.0193
.0019
A4
.0596
.0019
.0600
.0024
rl
.1819
.0013
.1863
.0011
r4
.1929
.0012
.1929
.0011
dl
.6115
.1747
.6048
.1521
d2
.5807
.1743
.5712
.1518
d3
.6162
.1747
.6072
.1523
d4
.5469
.1748
.5382
.1520
.8756
.0368
.8779
.0319
.0198
.0013
.0195
.0385
.0194
.0014
.0191
.0193
P
an
a e
J—Statistics
20.87
20.72
P-value
(232)
(.146)
/(17)
X2(15)
35
Table 5.1 Seasonal Growth Rates
1
Data
Spring Summer
X-variables
Winter
Output
-7.72
(.59)
4.26
(.42)
Consumption
-7.30
(.48)
Investment
Government
. 2
Winter
Time Separable Estimates
Fall
Spring Summer
R2
.
.
3
Non-Time Separable Estimates
Winter Spring Summer
Fall
R2
Fall
R1
2
-1.04
(.22)
4.16
(.35)
.905
-12.31
5.64
-3.20
9.87
.932
-7.35
3.04
-1.48
5.79
.868
2.59
(.40)
0.04
(.09)
4.34
(.26)
.936
-6.34
.55
-.20
7.08
.958
-7.29
2.29
-1.25
6.25
.914
-14.64
(.68)
11.79
(1.08)
-2.15
(.34)
4.33
(.58)
.903
-35.39
24.79
-14.60
25.20
.946
-10.55
5.41
-4.21
9.35
.830
-4.89
(.37)
2.86
(.79)
0.49
(.46)
1.31
(.32)
.696
-4.13
2.48
.87
.78
.353
-4.10
2.48
.80
.82
.357
.16
(.10)
-.26
(.08)
.09
(.10)
.02
(.10)
.214
.19
-.22
.54
.51
.761
-.082
.19
-.13
.198
Tabor Hours
-3.26
(.16)
2.31
(.21)
.73
(.20)
.07
(.17)
.842
-10.00
4.77
-2.56
7.79
.973
-2.96
.96
-.12
2.12
.898
Avg. Prod.
-4.47
(.52)
1.95
(.40)
-1.78
(.30)
4.09
(.40)
.846
-2.31
8.72
-.64
2.08
.627
-4.39
2.08
-1.37
3.67
.844
4.02
(.48)
3.91
(.44)
4.49
(.49)
3.41
(.42)
.084
4.46
3.95
5.50
3.59
.618
4.28
4.05
4.94
3.83
.336
Capital
Rental Rate
on Capital
.023
1lhe seasonal growth rates for the data were estimated using GMM and a Newey-West weighting matrix.
2The time separable results are sample averages based on simulations with 500 replications of draws of length 88.
3The non-time separable results are sample averages based on simulations with 500 replications of draws of length 88.
Table 5.2 Selected Second Moment Properties, Various Filters
1. Relative Volatility:
(Standard deviation of x)/(Standard deviation of Output)
1-Quarter Growth
X-variables
Std. Err
NTS
HP Filter
TS
.002
.054
.26
.19
.019
.79
1.82
1.70
.023
.43
2.69
1.50
.18
.52
.89
.026
.065
.050
.13
.33
.69
.097
.50
.55
.36
.073
.30
.25
Std. Err
Unadjusted 1-Quarter Rates
NTS
TS
Data
Std. Err
NTS
TS
.026
.60
2.42
.86
.002
.036
.12
.24
.032
.51
2.24
1.31
.031
.55
2.17
1.36
.051
.88
1.96
.66
.003
.033
.069
.063
.054
.97
1.61
.77
.089
.55
3.03
.47
.32
.73
.75
.060
.033
.11
.28
.39
.63
.24
.35
.69
.07
.42
.70
.009
.025
.024
.052
.37
.64
.052
.79
.24
.086
.13
.10
—
—
—
—
•
.016
.73
1.94
1.21
Data
CO
CM
2
Output
Consumption
Investment
Government
3
Capital
labor Hours
Avg. Prod.
3
Real rate
Data
1
2. Contemporaneous correlation: x with Output
1-Quarter Growth
X-variables
Data
Consumption
Investment
Government
.72
.72
.38
Capital2
Labor Hours
Avg. Prod.
Real rate
.26
.46
.82
.25
Std. Err
HP Filter
NTS
TS
Data
.075
.059
.081
.85
.84
.76
.79
.96
.78
.87
.93
.30
.11
.085
.040
.098
.24
.96
.99
-.25
.48
.94
.95
-.25
.28
.67
.59
-
Unadjusted 1-Quarter Rates
NTS
TS
Data
Std. Err
NTS
TS
.030
.024
.11
.90
.97
.85
.92
.96
.86
.96
.94
.83
.010
.010
.030
.98
.97
.69
.89
.98
.64
.070
.10
.080
.36
.97
.99
.41
.93
.98
.50
.81
.93
-.09
.060
.020
.010
.095
.46
.99
.995
-.35
Std. Err
-
-
-
.89
.99
.91
-.50
^The sample period of the data is 1964:11 -1985:IV. The standard errors are for the data's moments; and 12 lags are
employed in the Newey-West estimator. The Stochastic models were simulated 500 times using draws of length 88.
2
The "output” rows reports the standard deviation of output.
3
The capital stock refers to K^_+^, whereas the other variables are X^_.
4
The real rate is not filtered, for comparability with Barsky-Miron [1989].
@FedFRASER