The full text on this page is automatically extracted from the file linked above and may contain errors and inconsistencies.
Working Paper Series
Seasonal Solow Residuals and
Christmas: A Case for Labor Hoarding
and Increasing Returns
R. Anton Braun and Charles L. Evans
Working Papers Series
Issues in Macroeconomics
Research Department
Federal Reserve Bank of Chicago
October 1991 (W P -91-20)
FEDERAL RESERVE BANK
OF CHICAGO
Seasonal Solov Residuals and Christmas:
A Case for Labor Hoarding and Increasing Returns
*
R. Anton Braun
_
,
**
Charles L. Evans
Revised October 1991
Abstract
In aggregate unadjusted data, measured Solov residuals exhibit large seasonal
variations.
Total Factor Productivity grows rapidly in the fourth quarter at
an annual rate of 24% and regresses sharply in the first quarter at an annual
rate of -30%. This paper
considers two potential explanations for the
measured seasonal variation in the Solow residual: labor hoarding and
increasing returns to scale. Using a specification that allows for no
exogenous seasonal variation in technology and a single seasonal demand shift
in the fourth quarter, we ask the following question: How much of the total
seasonal variation in the measured Solow residual can be explained by
Christmas?
The answer to this question is surprising.
With increasing
returns and time varying labor effort, Christmas is sufficient to explain the
seasonal variation in the Solow residual, consumption, average productivity
and output in all four quarters.
Our analysis of seasonally unadjusted data
uncovers important roles for labor hoarding and increasing returns which are
difficult to identify in adjusted data.
JEL Classification(s):
★
131, 023
Department of Economics
Rouss Hall
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
Ve thank Marty Eichenbaum, Jim Hamilton, Bob Chirinko and seminar participants
at the Federal Reserve Bank of Chicago for helpful comments.
All opinions
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
Prescott [1986] has argued that the variability of Solow's residual is a
reasonable estimate of the variability of exogenous technology shocks.
When
Solow's residual is measured using seasonally unadjusted data for the postwar
U.S. economy,
it exhibits large seasonal variations,
growing rapidly in the
fourth quarter at an annual rate of 24% and falling sharply in the
first
quarter at an annual rate of -30%. This paper starts from the premise that it
is implausible to attribute seasonal variation of this magnitude to changes in
the
state
of
technology.^
We
present
a
model
in
which
fluctuations arise from a single demand shift, Christmas.
all
seasonal
This demand shift
together with misspecification of the traditional production function leads to
large seasonal variation in the Solow residual.
for misspecification,
scale.
Even
when
labor hoarding
technological
growth
and
is
We consider two candidates
external
increasing
aseasonal,
either
returns
to
candidate
in
isolation can induce spurious seasonality in the Solow residual. Our general
equilibrium analysis indicates that:
(1) the economy's seasonal patterns in
all four quarters may be a response simply to a fourth quarter consumption
demand shift, and (2) a combination of labor hoarding and external increasing
returns are important for replicating these patterns
variables
for
the
postwar
U.S.
economy.
Since
in aggregate
our
analysis
quantity
identifies
important roles for labor hoarding and increasing returns, these results have
implications for nonseasonal macroeconomic models.
By focusing on seasonal fluctuations we are able to exploit information
Beaulieu
the first
Argentina
no reason
and Miron [1991] cast doubt on weather explanations by showing that
quarter is a period of negative output growth in Australia and
as well as the United States.
In the Southern Hemisphere, there is
to expect a negative technology seasonal in the first quarter.
2
that
is
typically
ignored
in empirical
macroeconomic
research,
2
This
is
important because the research strategy of selecting a macroeconomic theory
based upon its ability to match certain statistical moments
adjusted data
example,
often
fails
to discriminate
Murphy-Shleifer-Vishny
[1989]
among competing
in seasonally
theories.
survey the many similarities between
technology shock-driven and increasing returns equilibrium theories.
seasonality,
For
Ignoring
they conclude that both theories can be modified in plausible
ways to generate the same implications for aggregate quantity variables.
In a
similar vein, Cooper and Haltiwanger [1989] conclude that production bunching
may
arise
due
to
either
nonconvexities
in production
or
the
arrival
of
3
technology
shocks
in
bunches.
The
literature
on
procyclical productivity also produces mixed results.
labor
hoarding
and
Rotemberg and Summers
[1990] show that the combination of labor hoarding and price inflexibility can
generate procyclical total factor productivity without appealing to technology
shocks. On the other hand, Burnside, Eichenbaum, and Rebelo [1990] find that
labor hoarding, price flexibility, and relatively small exogenous technology
shocks can generate procyclical total factor productivity.
The principal difficulty in selecting one theory, of course, is that each
school of thought interprets the postwar period as being dominated by either
supply shocks
(RBC)
or demand shocks
(Keynesian labor hoarding,
increasing
2
Other researchers have noted this also.
Barsky and Miron [1989] argue that
the seasonal cycle facts for the U.S. contain information for evaluating
alternative macroeconomic theories.
Ghysels [1991] disputes the claim that
the seasonal cycle is like the business cycle, but agrees that seasonally
unadjusted data contain useful information for identifying propagation
mechanisms.
3
Evidence favoring increasing returns is uncovered by Hall [1989], Ramey
[1991], and Caballero and Lyons [1990]. Their conclusions rely on the assumed
exogeneity of their instruments, which is controversial.
Further evidence of
increasing returns which does not rely on these instruments would strengthen
their case.
Chirinko [1991] concludes that increasing returns is more
important than either labor hoarding or technology shocks.
3
returns)-- but not both.
If economists could identify demand shocks, however,
the predicted countercyclical response of labor productivity in RBC models
could
be
compared
with
the
predicted
returns and labor hoarding theories.
is difficult
procyclical
response
in
increasing
Needless to say, such an identification
to achieve unambiguously
in postwar
seasonally
adjusted
time
series data.
Seasonal
fluctuations
offer
valuable
identifying
restrictions.
For
instance, few economists would dispute that Christmas is an important seasonal
event which produces
fourth quarter.
an
increased demand
On the other hand,
seasonal shifters of technology.
seasonal
finding
impulse
that
is weakened
seasonal
patterns in the U.S.
and
that
Christmas
for
it is difficult
in
by
Southern
Beaulieu
Hemisphere
[1990]
in the
identify bonafide
and
Miron's
countries
[1991]
resemble
Together our assumptions that technology is aseasonal
is
an
important
shift
in
demand
restrictions that have strong discriminatory power.
Evans
to
services
The contention that weather is an important
considerably
patterns
consumption
4
provide
identifying
For instance, Braun and
find that seasonal Real Business Cycle models predict totally
aseasonal patterns in output under these identifying assumptions.
transient and anticipated nature of Christmas,
households
Due to the
simply draw down
their savings and increase consumption by equal amounts in the fourth quarter
leaving the level of output unchanged.
RBC models require implausibly large
seasonal variations in technology to explain the seasonal pattern in output.
Thus,
our
identifying
assumptions
offer
strong
explanation for the seasonal pattern in output.
4
evidence
against
one
In this paper we demonstrate4
Our use of seasonal identifying restrictions is similar to Bernanke and
Parkinson's analysis.
Using interwar data, Bernanke and Parkinson [1991]
investigate procyclical productivity in industrial markets.
Under the
plausible identifying assumption that the Great Depression was not caused by a
series of large technology shocks, they find evidence in favor of increasing
returns and labor hoarding.
4
that labor hoarding in conjuction with increasing returns can generate many of
the observed patterns in output, consumption, and the measured Solow residual.
Modeling
seasonal
fluctuations
with
a
single
Christmas
demand
shift
requires us to model economic agents' responses to anticipated and transitory
impulses.
First, the anticipated nature of the Christmas seasonal shift leads
us to model variations in labor effort as driven by convex costs of adjusting
employment.
Burnside, Eichenbaum and Rebelo
[1990] model labor hoarding by
assuming that employment is fixed at the beginning of the period and only
labor effort can respond within a period to shocks.
effort
responds
persistence."*
only
To
to
induce
unanticipated
seasonal
shocks,
In their framework labor
exhibiting
labor hoarding,
quasi-fixed factors must be modeled explicitly.
the
costs
no
of
noticeable
adjusting
Second, the transitory nature
of seasonal shifts leads us to consider convex costs of adjusting capital.
In
an economy with external increasing returns, Baxter and King [1990] found a
negligible response of output to a purely transitory increase in consumption
demand:
consumption rose but investment fell,
leaving output unchanged.
In
the absence of adjustment costs, a similar result is to be expected for the
case of a fourth quarter Christmas seasonal.
If increasing returns is to have
a chance, it must be costly to adjust investment. Third, the nontime-separable
preferences emphasized by Kydland and Prescott [1982], Eichenbaum, Hansen, and
Singleton
[1988],
and others
for business
cycle variability
also play
an
important role in propagating the Christmas demand shock beyond simply the
fourth quarter.
Thus, modeling seasonal fluctuations leads to a specification
that incorporates the same propagation mechanisms that receive wide attention
in models of the business cycle.
In Burnside-Eichenbaum-Rebelo [1990], the impulse response functions of labor
effort to innovations in technology and government purchases appear to be zero
after the initial period's response.
5
Many of the model's parameters governing returns to scale, the magnitude
of adjustment costs and elasticities for work effort are difficult to pin down
on
a
priori
grounds.
A
Generalized
Method
of
Moments
(GMM)
estimation
strategy is used to produce estimates of these parameters. These estimates are
then used to evaluate the seasonal growth rates implied by the model. We find
that both labor hoarding and increasing returns mechanisms are important for
capturing the U.S.
variables
in
the
economy's seasonal fluctuations.
model,
the
hypothesis
that
For each of the real
the
predicted
fluctuations match the data's seasonals cannot be rejected.
seasonal
The estimated
parameterization proves to be remarkably successful at capturing the seasonal
pattern in the measured Solow residual as well as the seasonal pattern in
output, consumption, and
suggest
further
that
average productivity.
labor
hoarding
and
Results reported in section 4
nontime-separabilities
biggest role in propagating the Christmas demand shock.
play
the
Increasing returns
prove to be important for amplifying the seasonal patterns generated by the
other features of the model.
Finally, this model also offers an explanation for the similar seasonal
patterns across countries which Beaulieu and Miron
Christmas-like
celebrations
induce
fourth
quarter
[1991]
have documented.
preference
shifts
consumption demand in many Northern and Southern Hemisphere countries.
extent
that
increasing
returns
to
scale
and
labor hoarding
are
in
To the
important
features of these other economies as well, then our model predicts a seasonal
pattern similar to that found in the U.S.
An outline of the remainder of the paper follows. In section 2 the model
is
described and the seasonal equilibrium
growth path is defined. Section 3
contains a description of the data, the estimation strategy and a summary of
the estimation results. Section 4 evaluates the seasonal implications of the
estimated parameterization and explores the role of the various components of
6
th e
m o d e l.
I n
s e c t io n
5
w e
2.
c o n c lu d e
b y
s u m m a r iz in g
o u r
r e s u lt s .
The Economic Model
In this section we describe the model economy. The presentation of the
economy
leads naturally
competitive equilibrium
to an optimization problem whose
allocation.
to a productive externality.
take account of
solution
is
the
This solution is not Pareto optimal due
As we pose the problem,
the planner does not
the externality. A benevolent social planner could do better
by allowing agents to coordinate. This strategy for calculating competitive
allocations in distorted economies is discussed at length in Romer[1988].
Preferences
The household's period preferences depend upon consumption services c^,
leisure services 1^, and (negatively upon) the intensity of labor effort v .
The period utility function is:
rt log c*
where r
+
a log 1*
-
( v t - v )2
is a seasonal preference shifter.
,
£ > 0, a > 0
[1]
The preference shifter captures
the household's increased desire to consume during the Christmas season:
rt *
where
\
' Q lt + 7 Q 2t + 7 Q3t + '4 Q4 f
> 7 > 0
121
is a quarterly seasonal dummy variable taking on the value of 1 when
period t corresponds to season i and zero otherwise.
Consumption and leisure
services are defined as follows:
•fa
ct -
1* where
cpt + a cpt l ,
T - nt + b ( T -
cpt
allocation.
are
consumption
Ia 1 < 1
nt l
)
|b | < 1
,
expenditures
and
T
[3]
[4 ]
represents
the
total
time
If a > 0 consumption expenditures have a durable quality and are
substitutable across adjacent periods.
If a < 0 consumption expenditures are
complements across adjacent periods, and consumption preferences exhibit habit
7
p e r s is t e n c e .
T he
sam e
in t e r p r e t a t io n s
h o ld
fo r
b
an d
le is u r e
The interpretation of labor effort in the period utility function depends
upon the value of v.
If v>0, v is a bliss point for labor effort.
case, deviations from v provide disutility for the household.
interpreted as
a normal
level
of
labor
effort:
workers
In this
Here v can be
view periods
of
inactivity on the job with the same dissatisfaction as comparable periods of
overactivity.
If v<0, labor effort provides increasing disutility.
In this
case, less work effort is strictly preferred to more.
Production
The representative household has access to a technology which produces
goods (y) using capital (k), labor hours (n), and labor intensity (v):
0 < $ < 1
zt -
z
t
l
[5]
exp( A + ct )
[6]
4t “ *1 < V Zt / 2 ’ *1* *2 > 0
>
Aside from the choice of factor inputs k, n, and v, the level of production is
influenced by three additional factors:
exogenous variation in the state of
technology z^, a productive externality 0 , and convex adjustment costs J
capital and labor (with the specification described below).
on
Each of these
factors will now be discussed separately.
The technology variable z^ is a random walk process in logarithms with
constant drift A.
random variable.
noteworthy.
The impulse
is an independent,
serially uncorrelated
Three observations on the role of z^ in our analysis are
First,
the constant drift term A is nonseasonal-- this is our
identifying restriction that the true technology is aseasonal.
growth
in this economy originates with
z^ since
our
Second,
all
specification of
the
productive externality exhibits local increasing returns
In the balanced growth equilibrium that we analyze,
8
(discussed below).
therefore,
all trending
p r e
variables share the same trend as zf .
c
role
in our analysis
of perfect
Third, the variability of c
foresight seasonal
growth paths,
plays no
but
its
presence satisfies a necessary condition for our econometric relationships to
be well-posed in Section 3.
Increasing
returns
externality variable
capita output.
Murphy,
production
where y
are
captured
by
Marshallian
represents the economy-wide level of per
Shleifer, and Vishny [1989], Caballero and Lyons
influence
the
Marshallian productive externalities have been considered by
and King [1990],
to
in
[1990],
and Baxter
In our framework, the representative household is too small
the
economy-wide
household's control.
output,
so
^
is
taken
to
be
beyond
the
Since the externality is expressed relative to the level
of technology z^, this specification embodies local increasing returns--
as
aggregate economic activity rises relative to trend, the economy becomes more
productive.^
The variable
is a factor which
relates
to
the
cost
of
adjusting
capital and labor hours in terms of lost output:
kt+i - k texp(A+et) \2
exp
{
where ^
-
and ^
)2 -
M
4
-
(
^
[8 ]
n
are positive, and A is the average growth rate of capital as
well as the technology z^. The first term states that it is costly to increase
the capital stock at a rate other than its growth rate.
^Alternatively,
if z
The firm has in place
were deleted from the specification of
in [7], the
externality would grow with economy-wide output, and this would be global
increasing returns. Given our econometric methodology in Section 3, these two
specifications are observationally equivalent. Specifically, for local IR all
growth is exogenous; whereas for global IR the exogenous growth is magnified
by the
process so that some growth is endogenous.
In the global case,
there is a lower value of X which interacts with the same value of ^
as
the local case to produce the same equilibrium as we report in Section 4.
Applying our estimation procedure to the global case would produce this lower
value of A.
9
a technology for assimilating new capital into the production process. This
technology costlessly accepts the normal level of new investment, but other
levels create congestion in the production process.
Likewise, the second term
states that it is costly to increase labor hours at a rate other than its
unconditional
growth
rate,
adjustment cost factor
which
is
zero.
For
this
specification,
the
is in the interval (0,1] and in a nonseasonal steady
state J^-l.^
Period Budget Constraint
The household's period budget constraint is given by:
k t ^ t V V 1 ** J t “ cp t + kt + i * a ' 6) kt
[9]
where S is the rate of capital depreciation per quarter.
be
introduced
into
the
model
and
constraint
[9]
(as
Fiscal policy could
in
Braun
and
Evans
[1991]), but our focus in this paper is the single demand seasonal Christmas
since that is a relative constant across countries.
Planner's Problem
The competitive equilibrium allocations
in a decentralized version of
this
economy are identical to the solution of the following optimization
g
problem.
At time 0, choose a sequence of contingencies { cpt , n^, v , k
t^O } to solve the following:
Due to the inclusion of the growth term in J^, the nonseasonal steady state
of this economy will be the same as an economy which omits adjustment costs.
Besides being plausible, the growth term allows greater comparability with
previous studies.
g
This is a solution strategy previously employed by Romer [1986].
10
max E0£
^t|rtlog(cpt+acpt l ) + alog(T-nt+b(T-nt l )) - |-(vt-v)2
-
t-0
+
where
are
• cpc • V i
] }
+
i9
10i
is a Lagrange multiplier, and the initial values k^, cp
given.
treating
Notice
that
as given:
the
planner
ignores
the
allocations
externality,
while this is suboptimal, it is the analagous problem
to the one faced by small households and firms.
optimal
productive
and n ^
which
solve this
problem
It is well-known that the
are characterized
by
the
first-order conditions for cpt , nt> v , kt+^, and a transversality condition
related to capital (for an example, see Braun and
Evans [1991]).
Furthermore,
assets can be pricedusing intertemporal marginal
rates of substitution in the
usual way.
4
Perfect Foresight Seasonal Equilibrium Growth Path
This
economy
grows
over
time
at
the
progress which is given by A per period.
calendar year, however,
foresight
seasonal
rate
of exogenous
technological
Since preferences shift over the
these growth rates may vary seasonally.
equilibrium growth
path
for this
A perfect
economy
is
a
9
generalization of the standard definition of
a balanced growth path.
The
relevant new feature is that the seasonal growth path is indexed by season.
Thus, consumption in year t and quarter i is linked to consumption in year t+1
quarter
i by
the
following
relationship:
ct+i i “
4A
e ct i*
Along
the
seasonal growth path, consumption will always grow x^% in the winter, Xj% in
9
For example, see King, Plosser, and Rebelo [1988] for a standard definition
of a balanced growth path.
11
th e
s p r in g ,
x^%
in
th e
su m m er,
an d
x^%
in
th e
f a l l . ^
Seasonality 1b Measured golow Res,iduals
Suppose that a researcher attempts to measure Solow residuals for this
economy as Prescott [1986] does.
Armed with the precise knowledge of 0, the
measured Solow residual will b e : ^
St ■ [ A lo g yt - 0 A log kt - (1-0) A log nt ] / (1-0)
- [ (l-0 -^2)
Assuming that
A l o 8
zt +
^ 2
Alog yt *
the deterministic
Alog v t + Alog Jt ^ / C1**)
component of technological
aseasonal, then seasonality in S^ can arise from:
output is seasonal,
seasonal,
[11]
growth
(z^)
is
(1 ) increasing returns if
(2 ) labor hoarding if variations
in labor effort are
and (3) seasonal adjustments in capital and labor hours.
If the
fourth quarter increased desire to consume is strong enough to generate a
seasonal increase in fourth quarter output, then measured Solow residuals will
be proseasonal due to the productive externality.
achieved by a seasonal
increase
in work effort
If the higher output is
(without
a correspondingly
large increase in adjustment costs), then the demand effect is reinforced.
Whether
or
not
a
single
Christmas
seasonal
in
preferences
can
12
explain
seasonality in Solow residuals, in all four quarters, depends upon the model's
ability to generate seasonality in output and labor effort across the entire
calendar year.*
2
1
^ F o r an explicit characterization of this type of seasonal equilibrium path,
see Braun and Evans or Chatterjee and Ravikumar [1989].
X1Ve assume that S^ is an attempt to measure Alog z^ rather than (l-0)Alog z^.
12
Evans [1991] documents that Prescott's measure of the Solow residual is not
exogenous when seasonally adjusted data is used.
The finding that money,
interest rates, and government spending Granger-cause Prescott's residual
could be due to increasing returns or unobserved variations in labor effort of
the form modeled here.
12
3.
Econometric Estimation of the Model's Structural Parameters
The vector of structural parameters $ contains 16 elements:
* - ( 6, 0, T, r, r4 , a, a, b, £, v, 6,
A,
).
In assigning parameter values, there are three categories of parameters:
parameters which can be normalized a priori because
influence upon the analysis;
(2 ) parameters which
their values
are
parameters
Moments.
Since
which
First,
utility
are
econometrically
the parameters
is
ordinal,
we
estimated
(a, T,
a«l
and
estimate
whose
level
and
Method
a
(3)
of
the
consumption
The time allocation is set to T-1369 hours
per quarter (as in Christiano and Eichenbaum [1991]).
index variable
Generalized
set
v) are inherently unidentified.
normalize
preference parameters r and r^.
by
have no
customarily
priori because their values are not well-identified in the data;
(1)
depends
upon v:
we
guarantees that average labor effort will be 1.
13
Labor effort v
set v
at
a
level
is an
which
The parameter < simply
f
>^
defines the units of measure for commodities (thousands of dollars, billions
of yen,
etc.):
analysis.
its value can be selected arbitrarily without affecting the
Second, the discount factor 0 is not well-identified in aggregate
time series data.
We set 0 equal to 1.03
- 25
as in Christiano and Eichenbaum.
Third, the lack of seasonally unadjusted quarterly data on the capital stock
leads
us
to
depreciation
construct
rate
S
is
capital
2.5%
from
per
investment
quarter.
The
flows
assuming
remaining
that
the
parameters
are
econometrically estimated by Generalized Method of Moments.
3.1
GMM estimation
The parameters
a * b,
( 0, r, r^,
13
^ ^ are estimated by1
3
This normalization ensures that the average labor input
corresponds to the average level of labor hours in the data.
13
in
the
model
imposing jointly two sets of moment conditions:
based
upon
the
household's
consumption and leisure;
intratemporal
(1 ) orthogonality conditions
Euler
equation
for
choosing
and (2 ) explicitly equating a set of first moments
in the data with the model's predictions for these moments.
For the first set
of moments, the Euler equation can be written (in terms of observables) as:
where
- ^/(cp^ + a cp^ ^).
a valid
instrument
instrument
set
Xt l^Xt-2 ^
for
includes
Any variable in the time t information set is
estimating
the
time
the
t
parameters
and
t-
in
this
growth
1
equation.
rates
(x^/x^ ^
The
and
^ak°r hours, capital, consumption, and output, as well as four
seasonal dummy variables.
To describe the second set of moment restrictions, let H(x^) refer to the
following transformations of the data:
cpt)qt
(Alog yt/nt)qt
(Alog kt)qt
/
(Alog (l+rt))qt
where
kt/yt
is a 4x1 vector of seasonal dummies, r
]
is a real interest rate, and
the symbol ' denotes transposition, so H(xt) is a 22x1 vector.
the first
rates
of
2 0
elements of the expected value of H(xt> are the seasonal growth
output,
consumption,
labor
productivity,
seasonal change in the real interest rate.
to the average
capital-output ratio
H(xt) -
h(¥) +
ut
14
and
capital,
and
the
The last two elements correspond
and average
definition, the model predicts that
Accordingly,
labor hours.
Given
this
where MU') corresponds to the model's predicted first moments of H(xt) and
is a vector mean zero, serially correlated random variable.
Based upon these
moment restrictions, our estimator of V attempts to set the sample mean of u^
to zero, as well as the sample moments based upon equation [1
2
].
Our choice of moment restrictions is motivated by two concerns.
First,
since labor effort is unobserved, the parameter £ cannot be estimated by Euler
equation methods.
Second, estimating ^
from production function residuals
seems hopeless due to the presence of unobserved variations in labor effort:
no exogenous instruments are available.
14
However,
these parameters can be
estimated by forcing the model to confront the seasonal growth rates in the
data by choosing
Finally,
^
Sims
an<* £, as well as the other parameters.
[1990]
and Hansen and
Sargent
[1991]
have
argued
that
econometricians who use seasonally unadjusted data and misspecify the seasonal
mechanisms may do much worse than econometricians who discard the potential
information content at seasonal frequencies and simply use seasonally adjusted
data.
On the other hand, Ghysels [1991] has pointed out that great efficiency
gains may be possible if seasonally unadjusted data is used.
Thus, there is a
potential trade-off involved in using seasonally unadjusted data, efficiency
gains versus misspecification bias.
Ue
try to address
the bias
issue by
comparing our parameter estimates with other econometric studies which used
seasonally adjusted or annual data.^ 1
4
14
Hall [1988] has noted that his set of instruments would fail to be exogenous
in this setting.
^Another interesting statistical issue is whether the unadjusted time series
data are better characterized by purely indeterministic seasonality, purely
deterministic seasonality, or a mixture of both.
Ve identify Christmas
effects with a fourth quarter mean of r which is larger than the other three
quarters. This single nonzero seasonal mean induces nonzero seasonal means in
other economic aggregates.
Even if r
is stochastic, our economic theory
predicts that economic time series will possess some deterministically
seasonal components.
This argues against purely indeterministic models of
15
3 .2
D a ta
The original data set employed in this study is the Barsky-Miron [1989]
data for the sample period 1964-1985:
adjusted
for
theoretical
seasonality.
constructs
of
U.S. quarterly data which has not been
For
the
empirical
our
model,
analysis
however,
we
to
conform
redefine
to
of
some
the
the
variables as follows (and convert to per capita values).
Output (y) is Gross
National Product per capita.
is nondurables plus
Private consumption (cp)
services consumption expenditures per capita.
Fixed
Investment
plus
Durable
consumption
Investment (i) is the sum of
expenditures,
per
capita.
The
capital stock is computed using the flow 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).
The real interest rate is the ex post return on
three-month Treasury Bills, not seasonally adjusted as reported in Citibase.
The data is converted to per capita values by using the civilian population,
16 years and older.
3.3
Estimation Results
Table
1 presents
moment restrictions
our
estimation results.
in estimating
1 0
The
estimation
structural parameters;
imposes
34
in principle,
there are 24 overidentifying restrictions which are tested by Hansen's [1982]
J-statistic.
The
statistic
is
20.86
with
a
probability
uncovering no evidence against these restrictions.
restrictions
involve matching the model's
seasonality.
16
value
of
0.65,
Recall that 20 of the 34
seasonal predictions
against
the
data's seasonal growth rates.
Informally, this test suggests that the model
successfully captures the data's seasonal properties.
This claim is examined
in more detail in Section 4 where we consider a variety of other tests that
focus
explicitly
Turning
to
on
the
the
model's
individual
seasonal
predictions.
parameters,
our
estimates
using
seasonally
unadjusted data are similar to other estimates in the literature which have
employed seasonally adjusted data.
Our estimate of 0 is .279 which is close
to Prescott's [1986] value of .25 (when output is identified with GNP and does
not include the services of durable consumption goods).
value
of r and
is
.2363.
The
inverse
1/r
The weighted average
corresponds
to
preference parameters estimated by Christiano and Eichenbaum;
4.23
falls
within
the
range
3.92
and
5.15
consumption effect is estimated to be r^/r "
they
1
.0
2 2
report.
;
the
leisure
our value of
The
Christmas
this is the percentage
increase in the marginal utility of consumption services, holding consumption
services fixed.
This value does not seem to be implausibly large.
A
A
The
unusual
precision
of
A
0,
r,
and
is
due
to
the
two
moment
restrictions in H(x^) which are related to the capital-output ratio and the
level of labor hours.
re-estimated,
the
If these two moment conditions are dropped and ¥ is
parameter
estimates
are
essentially
unchanged,
but
the
A
A
standard
errors
respectively.
for
0,
A
r,
and
rise
to
.0352,
.0105,
and
.0107,
Therefore, the unusual precision of these parameter estimates
is due to the inclusion of strong identifying restrictions from the model's
equilibrium predictions.
The
nontime-separability
parameters
a
and
b
are
similar
to
other
researchers' estimates.
The value of a-.445 indicates that consumption goods
have a durable quality:
in seasonally adjusted data, this has been found by
17
The
Eicheribaum, Hansen, and Singleton [1988] and Gallant and Tauchen [1990].^
value of b--.517 indicates that leisure preferences exhibit habit-persistence:
in seasonally adjusted data, this has been found by Eichenbaum, Hansen,
Singleton as well as Braun [1990].
relatively
smooth;
habit-persistence
in
and
This feature makes leisure and labor hours
addition
to
in leisure will
adjustment
smooth
costs
labor hours
for
and
labor
lead
to
hours,
greater
variations in labor effort in response to exogenous shocks.
The adjustment cost parameters are significantly different from zero.^
The capital and labor estimates are 28.45 and 0.258, but these numbers are a
poor indication of their relative effects.
On a quarterly basis, the standard
deviations for the growth rates of capital and labor are 0.35% and 2.21%.
percentage
reduction
in output
due
to
adjusting
capital
(only)
and
(only) by one standard deviation above average is -0.018% and -0.006%.
capital
penalty.
adjustment
Also,
penalty
is
only
about
3
times
larger
than
the
The
labor
So the
labor
these numbers indicate that the direct effect of adjustment
costs on measured Solow residuals is negligible. That is, the effects of Alog
in equation
[11]
are small.
indirect effect may be
large:
As was noted
in response
in Section 2, however,
to a consumption
demand
the
shock,
reducing investment may now be costly enough to induce a large response in
output.
Our estimate of the output elasticity with respect to external increasing
returns is ^
“ .2389.
The elasticity is significantly different from zero.
This value is within the range of estimates reported by Caballero and Lyons
On the other hand, Constantinides and Ferson [1990] find evidence of
habit-persistence in consumption goods preferences (a<0). In simulations of
an equilibrium business cycle model with seasonality, Braun and Evans [1990]
found that durability in consumption (a>0 ) helped the model match key business
cycle moments better than habit-persistence.
^Ghysels [1988] observes that there is a lot of spectral power at seasonal
frequencies for identifying adjustment cost parameters.
18
[1989] and Baxter and King [1990], although Baxter and King use a larger value
of ,33 in their model evaluation.
our
estimate
provides
some
The size and statistical significance of
evidence
that
external
important for explaining seasonal fluctuations;
offered
differ
in Section 4.
from
those
estimate of ^
Nevertheless,
of
provides
are
our
identifying
restrictions
and
[1990]
evidence which
returns
a quantitative assessment is
since
Caballero-Lyons
increasing
Baxter-King
[1990],
our
theirs
and
is both
independent
of
complementary.
Our estimate of £ is 0.0241, so the disutility of labor effort deviations
may in fact be small enough to induce sizable variations.
is .0348, so the estimate is reasonably imprecise.
The standard error
It is important to note
that the hypothesis that variations in labor effort are small would imply that
? is large:
the point estimate and standard error do not support this.
As we
will see in Section 4, estimates of f in the range reported are capable of
yielding
substantial
variations
in
work
habit-persistence
in leisure preferences,
relatively
disutility
effort.
small
associated
costly
with
adjustment
varying
provide evidence for the labor hoarding hypothesis.
implied by the estimates and
Thus,
labor
Finally,
normalization is negative.
the
of
estimated
labor,
effort
and
jointly
the value of v
Thus,
utility
is
strictly decreasing in work effort.
Overall, the estimated parameterization seems reasonable.
The similarity
of many estimates with previous studies suggests that if we had chosen to
"calibrate"
our
parameterization
model
would
using
not
these
have
been
other
very
studies,
different.
the
resulting
Finally,
the
overidentifying restrictions cannot be rejected.
4 .
In
t h is
E v a lu a tin g
s e c t io n
w e
th e
M o d e l's
e v a lu a te
19
th e
S e a so n a l
s e a s o n a l
I m p lic a tio n s
p r o p e r tie s
o f
t
parameterization
and
offer
evidence
on
the
relative
importance
of
hoarding and increasing returns in explaining the seasonal patterns
data. Two criteria
First,
a series
labor
in the
are used to evaluate the model's seasonal predictions.
of hypothesis
tests
are
reported.
These
tests
have
the
benefit of taking into consideration sampling error in the summary statistics
for the data and sampling error in the estimated parameterization. Second, the
seasonal growth rates of the data and the model are simply plotted together.
This latter approach provides a visual summary of the seasonal properties of
the model relative to the data.
Table 2 contains results from a series of hypothesis tests. The results
in
table
questions:
predict
are
2
aimed
providing
information
on
the
Is there evidence of seasonality in the data?
significant
patterns
at
seasonality?
Does
the model predict
following
Does
the
the model
same
found in the data? Column one reports Wald statistics
three
seasonal
that offer
evidence on the first question for each variable individually. The maintained
null underlying the column one results is that the four seasonal dummies for a
particular time series are equal (equation (1), Table 2). These statistics are
constructed from GMM estimates of the average seasonal growth rates in the
data
and
statistic
use
a
Newey-West
indicate
that
covariance
the
null
estimator.
hypothesis
The
of
p-values
no
for
each
seasonality
is
overwhelmingly rejected for each time series. These results are representative
of findings reported by Barsky and Miron [1989]
18
.
Column two reports Wald
statistics that offer evidence on the second question.
The maintained null
hypothesis is that the model's predicted seasonal growth rates are equal.
19
18
Barsky and Miron also find that there is statistically significant
seasonality in the real interest rate although the magnitude of the estimated
seasonals (in levels) is small.
19
The model's predicted seasonal growth rates are a highly nonlinear function
20
The null hypothesis of no seasonality is also sharply rejected for each of the
time-series that the model offers predictions for. On the basis of the results
from these two tests,
we conclude that both the model
and the data offer
strong refutable predictions at seasonal frequencies.
Certainly the most important question is the third one:
predict the seasonal patterns in the data?
Does the model
Column three of Table 2 provides
one metric for evaluating the model 1 s "fit" at seasonal
frequencies.
maintained null hypothesis
model's
in column three
is
that
the
The
predicted
seasonal growth rates for the jth time-series equal the corresponding values
in the data in each of the four seasons. This LaGrange multiplier (or LM) test
is formally a test of particular moment restrictions that were imposed in the
course
of estimation.
statistics were
For hours,
the
Solow residual,
and
calculated using
the
fact
time
that
these
investment,
series
the
can be
expressed as (log) linear combinations of other time series that were included
in the estimation. Eichenbaum, Hansen and Singleton [1984] and Newey and West
[1987b] describe the details of implementing LM tests in the context of GMM
estimation.
Column three contains
surprisingly
null of a common seasonal pattern in all
little
instances.
evidence
As
against
a check we
the
also
calculated a GMM analog to the likelihood ratio statistic and found that the
two statistics were virtually identical.
This collection of statistics provides two important conclusions. First,
the tests reported in columns one and two demonstrate that the statistics have
sufficient power to reject the null hypothesis of no seasonality for the model
and the data. Second, the column three results find no evidence against the
hypothesis that the model correctly predicts the pattern of seasonality found
of the estimated structural parameter vector 4. The asymptotic covariance of
the predicted seasonals is computed using the covariance estimator of ¥ and
the gradient of the nonlinear function.
The Wald statistics are constructed
from these objects in the usual way.
21
in
th e
d a ta .
Turning to the specific predictions of the model, we report plots of the
seasonal
growth rates
for
the data and model
numbers are presented in Table 3).
in figures
These diagrams
1-3
complement
(the
actual
the previous
hypothesis tests in that they offer summary information on the ability of the
estimated parameterization
to
capture
particular
aspects
of
the
seasonal
pattern in the data. We will focus on two aspects of the seasonal pattern: the
magnitude of the model's predicted seasonal in a particular quarter relative
to the data and the ability of the model to mimic the sequential relationship
of
seasons
found
in
the
data.
The
Solow
residuals
labeled
"data"
are
calculated using 0«.28. As was noted in section 3 this number is qualitatively
close to the value of .25 used by Prescott [1986].
One of the principal aims of this paper is to investigate the possibility
that increasing returns and/or time-varying labor effort can explain the large
seasonal variation
reported
in table
in the
2:
Solow Residual.
the model
is
quite
Figure
1 confirms
successful
in
this
predicted Solow residual has the same sequential pattern
magnitudes
found in the data.
These
results
offer
strong
the
results
respect.
The
and captures
the
support
for our
contention that the observed seasonal pattern in the Solow residual is
driven
largely by demand shocks.
In addition to capturing the seasonal pattern in the Solow residual, the
model also mimics important features of seasonality in output, consumption and
average productivity. In all of these instances the model correctly predicts
the sequential seasonal pattern
of the data.
For consumption we do see a
tendency for the model to understate the third quarter deceleration found in
the
data
and
for
output
the
model
understates
the
second
quarter
rise.
However, the hypothesis tests indicate that both of these disparities can be
attributed to sampling error. These
successes across the entire calendar year
22
are particularly striking given that the only seasonal shifter is a fourth
quarter shift in preferences.
Figure 3 displays the seasonal patterns in labor effort.
an unobservable, the data's seasonals cannot be
output
rises
on
the
strength
of
higher
combination with increasing returns,
than
reported.
normal
work
in
the
first
quarter.
Fourth quarter
labor
effort.
In
fourth quarter effort is only 3% above
normal in generating an annualized output growth of 18%.
at
Since this is
These
variations
20
in
Opposing forces are
effort
do
not
seem
implausible.
If we ignore sampling error, the figures suggest that the model fails to
account for some aspects of the seasonal pattern in other variables. For hours
and investment the magnitudes are off in all four quarters,
for the capital
stock they are off in three quarters and for interest rates they are off in
two quarters.
However,
even for these time series
the model
captures many
features of the sequential pattern in the data. The fact that the hypothesis
tests in table
2
fail to reject a common seasonal pattern in individual time
series suggests that there may be considerable sampling error. The most likely
A
sources
for this
sampling error are
A
in £ and
parameters which
govern
respectively the roles of time-varying labor effort and increasing returns.
Both
parameters
are
estimated
with
sizable
standard
errors.
The
case
analyses below demonstrate that variations in these two paramters lead to a
deterioration in the model's seasonal predictions relative to the data.
It is also interesting to examine the role of the various features of the
model
in
explaining
these
seasonal
patterns.
increasing returns can be shut down by setting ^
in work effort can be ruled out by setting
20
Notice
"0 •
and
first,
23
the
Second, time variation
F°r t* 1 6 8 6 values of
The fourth quarter growth rate of labor effort is only 3.8%.
that
£ and ^
adjust
it is never
labor
desirable to vary effort, while
in production.
By
examining
these
two
it is costless to
special
cases
of
the
estimated paramterization we can get some feel for the contributions of labor
hoarding and increasing returns to scale.
The results from these exercises are displayed in Figures 4, 5, and
6
along with the data for purposes of comparison. Consider first output and the
Solow residual. With only increasing returns the Solow residual and output are
essentially flat across all four seasons.
Labor hoarding, on the other hand,
does in isolation produce the correct sequential pattern for these two series
while dramatically understating the magnitudes. On the basis of these diagrams
it would appear that labor hoarding plays a crucial role in generating the
correct seasonal pattern in output and the Solow residual. The primary role of
increasing
returns
then
is
to
magnify
these
patterns.
In
our
estimated
parameterization increasing returns and labor hoarding are clearly interacting
to deliver the seasonal patterns in output and the Solow Residual.
Consumption and investment, on the other hand, continue to display some
seasonality with
properties
only
increasing
of the model.
The
returns.
This
fourth quarter
can
be
seasonal
attributed
demand
to
two
shift helps
explain the fluctuations in the fourth and first quarters. In the second and
third quarters investment and consumption are adjusting endogenously via the
nontime-separabilities in consumption which imply that consumption in adjacent
periods are substitutes.5
5.
This
paper
demonstrates
Conclusion
that
the
seasonal
cycle
contains
valuable
information for uncovering the roles of labor hoarding and increasing returns.
In contrast to business cycles which are arguably induced by both demand and
technology
shocks
of
varying
persistence,
24
seasonal
fluctuations
are
anticipated,
Christmas.
transient,
and
easily
identified with
calendar
events
like
Our findings indicate that increasing returns to scale alone does
not directly explain the seasonality in measured Solow residuals.
However, it
plays an important role in magnifying small variations in work effort. Hall
[1988] has argued that labor hoarding requires implausibly large variations in
work effort to explain cyclical fluctuations in Solow residuals.
fluctuations
this
labor effort
variation of no more
increasing
fluctuations
is not
returns
these
in total
observed in the data.
the
case.
Our
than
variations
estimated parameterization
5.5%
are
factor productivity
on a
labor hoarding--
quarterly basis.
magnified,
that are
of
thereby
the
same
With
producing
magnitude
this
model
offers
an
productive externalities,
explanation
for
the
similar
cross-country seasonal patterns documented by Beaulieu and Miron [1991].
implies
Finally, since our explanation rests on phenomena which
are not country-specific-- Christmas celebrations,
and
For seasonal
25
R e fe r e n c e s
Barsky, R. and J. Miron, 1989, The Seasonal Cycle and the Business
Cycle, Journal of Political Economy.
Baxter, M . , and R. King, 1990, Productive Externalities and Cyclical
Volatility, Rochester Center for Economic Research Working
Paper #245.
Beaulieu, J . 9 and J. Miron, 1991, A Cross Country Comparison of
Seasonal Cycles and Business Cycles, manuscript, Boston University.
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, Seasonality and Equilibrium Business
Cycle Theories, Institute for Empirical Macroeconomics,
Discussion Paper #45.
Burnside, C., M. Eichenbaum, and S. Rebelo, 1990, Labor Hoarding and
the Business Cycle, NBER working paper #3556.
Caballero, R . , and R. Lyons, 1989, The Role of External Economies in U.S.
Manufacturing, manuscript, Columbia University.
Chatterjee, S. and B. Ravikumar, 1989, A Neoclassical Model of
Seasonal Fluctuations, 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, Discussion Paper #24.
Chirinko, R . , 1991, Non-Convexities, Labor Hoarding, Technology Shocks,
Procyclical Productivity: A Structural Econometric Approach,
manuscript, University of Chicago.
Constantinides, G. and W. Ferson, 1990, Habit Persistence and Durability
in Aggregate Consumption: Empirical Tests, unpublished
manuscript, University of Chicago.
Cooper, R . , and J. Haltiwanger, 1989, Macroeconomic Implications of
Production Bunching: Factor and Final Demand Linkages,
NBER Working Paper #2976.
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.
26
Eichenbaum, M . , L. Hansen, and K. Singleton, 1984, Appendix to: A Time
Series Analysis of Representative Agent Models of Consumption and
Leisure Choice Under Uncertainty, manuscript,
Carnegie-Mellon University.
Evans, C., 1991, Productivity Shocks and Real Business Cycles, manuscript,
Federal Reserve Bank of Chicago.
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, Econometrics 57, 1091-1120.
Ghysels, E., 1988, A Study Toward a Dynamic Theory of Seasonality
for Economic Time Series, Journal of American Statistical
Association, 83, 168-172.
Ghysels, E., 1991, On the Economics and Econometrics of Seasonality,
manuscript, University of Montreal.
Hall, R . , 1988, The Relation between Price and Marginal Cost in
U.S. Industry, Journal of Political Economy 96, pp. 921-947.
Hall, R. , 1989, Temporal Agglomeration, NBER Working Paper #3143.
Hansen, L . , 1982, Large Sample Properties of Generalized Method of Moments
Estimators, Econometrics 50, 1029-1054.
Hansen, L., and T. Sargent 1991, Seasonality and Approxiamation Errors
Rational Expectations Models, manuscript, Hoover Institute.
Hansen, L. and K. Singleton, 1982, Generalized Instrumental Variables
Estimation of Nonlinear Rational Expectations Models,
Econometrica 50, 1269-1286.
Heaton, J., 1988, The Interaction between Time-Nonseparable Preferences
and Time Aggregation, unpublished manuscript, University of Chicago.
Kydland, F. and E. Prescott, 1982, Time to Build and Aggregate
Fluctuations, Econometrica 50, 1345-1370.
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, 1987a, A Simple, Positive Definite,
Heteroskedasticity and Autocorrelation Consistent Covariance
Matrix, Econometrica 55, pp. 703-708.
Newey, W. and K. West, 1987b, Hypothesis Testing with Efficient
Method of Moments Estimators, International Economic Review 28,
pp. 777-788.
27
in
Prescott, E., 1986, Theory Ahead of Business Cycle Measurement,
Carnegie-Rochester Conference Series on Public Policy 27
(Autumn), 11-44.
Ramey, V., 1991, Nonconvex Costs and the Behavior of Inventories,
Journal of Political Economy.
Sims, C., 1990, Rational Expectations Modeling With Seasonally Adjusted Data.
Institute for Emprical Macroeconomics Discussion Paper 35.
Singleton, K . , 1988, Econometric Issues in the Analysis of
Equilibrium Business Cycle Models, Journal of Monetary Economics.
28
T a b le
Parameter
1 :
Estimate
GMM P a r a m e t e r
E s tim a te s
Standard Error
6
.2790
0.00091
r
.2350
0.00045
.2402
0.00047
. 0 0 2 1
0.00005
r4
X
28.448
3.64360
.2576
0.05645
a
.4453
0.01123
b
-.5173
0.00854
.2389
0.09240
.0241
0.03481
*1
* 2
CM
J-statistic
Degrees of Freedom
P-value
2 0 . 8 6
24
.6469
21
A Newey-West procedure [1987a] with four lags was used to compute the optimal
GMM weighting matrix.
29
T a b le
(1)
4 x
X
<
f(») - d,
1
>✓
*
CM
(3)
Variable X
f(*)'q
L
+ w
L
T e s t
R e s u lts
f
22
Hn : The model does NOT exhibit deterministic
seasonality . The elements of f(40 are equal.
h 1:
„d
Ho
H y p o th e s is
H q : The data do NOT exhibit deterministic
seasonality
The elements of d are equal.
— d'q t + w t ,
t
2 :
The model's predicted seasonality equals
the data's seasonality.
23
Hj
Ho
4
Solow
303.64
(.000)
437.74
(.000)
0.125
(.998)
Output
413.90
(.000)
348.08
(.000)
0.201
(.995)
Consumption
715.28
(.000)
197.32
(.000)
0.058
(.999)
Investment
1097.45
(.000)
27.35
(.000)
0.137
(.997)
Capital
280.26
(.000)
31.21
(.000)
0.301
(.989)
Labor Hours
831.49
(.000)
112.31
(.000)
24
0.216
(.994)
615.61
(.000)
Labor Prod
227.14
(.000)
391.25
(.000)
0.167
(.996)
Real Rate
31.06
(.000)
54.03
(.000)
0.034
(.999)
22
Equation (2) is predicted by our theoretical model, but our test is not
regression-based. See the text for a description.
23
For both columns 1 and 2, the Wald test statistics are asymptotically
distributed x with four degrees of freedom.
The numbers in parentheses are
probability values of the test statistic.
24
2
The Lagrange Multiplier test statistic is asymptotically distributed x with
four degrees of freedom.
30
T a b le
3 :
S e a s o n a l
25
G r o w th
Variable
Season
Model
Solow Residual
Winter
Spring
Summer
Fall
-7.451
3.512
- .703
5.466
-7.323
3.889
-2.019
5.889
Output
Winter
Spring
Summer
Fall
-6.501
2.924
- .345
4.746
-7.356
4.623
- .674
4.529
Consumption
Winter
Spring
Summer
Fall
-6.744
3.500
-1.322
5.390
-6.876
3.015
.461
4.767
Investment
Winter
Spring
Summer
Fall
-5.637
.854
3.141
2.465
-14.065
12.365
-1.578
4.896
Capital
Winter
Spring
Summer
Fall
.297
.136
.155
.235
.640
.219
.572
.506
Labor Hours
Winter
Spring
Summer
Fall
-1.681
0.491
.165
1.025
-3.126
2.438
.863
.197
Labor Prod.
Winter
Spring
Summer
Fall
-4.821
2.433
- .510
3.721
-4.229
2.186
-1.537
4.332
Real Rate
Winter
Spring
Summer
Fall
2.901
0.812
0.409
-4.122
R a te s
.701
1.247
- .397
-1.162
25
Data
Quarterly rates of growth in percentages, except for the real interest rate
which is the quarterly change in annualized yields (i.e., Alog (l+rt), with r
at annual rates).
31
SOLOW RESIDUAL
OUTPUT
QUARTERLY RATES O F GROWTH
QUARTERLY RATES O F GROWTH
50
.
25
.
PERCENT
PERCENT
0.0
-.
25
-.
50
-.
75
2
3
CONSUMPTION
4
QUARTERLY SEASONS
INVESTMENT
QUARTERLY RATES O F GROWTH
QUARTERLY RATES O F GROWTH
15
10
PERCENT
PERCENT
5
0
5
-10
-5
1
1
2
3
QUARTERLY SEASONS
4
1
4
QUARTERLY SEASONS
Figure
1
QUARTERLY SEASONS
CAPITAL STOCK
QUARTERLY RATES O F GROWTH
PERCENT
LABOR HOURS
Q UARTERLY RATES O F GROWTH
QUARTERLY SEASONS
REAL INTEREST RATE
QUARTERLY RATES O F GROWTH
CHANGE IN ANNUALIZED YIELDS
PERCENT
LABOR PRODUCTIVITY
1
2
3
QUARTERLY SEASONS
4
3
QUARTERLY SEASONS
F ig u r e
OUTPUT
QUARTERLY RATES O F GROWTH
QUARTERLY RATES O F GROWTH
PERCENT
PERCENT
SOLOW RESIDUAL
QUARTERLY SEASONS
INVESTMENT
QUARTERLY RATES O F GROWTH
QUARTERLY RATES O F GROWTH
PERCENT
PERCENT
CONSUMPTION
1
Figure 4
QUARTERLY SEASONS
2
3
QUARTERLY SEASONS
4
1
2
3
QUARTERLY SEASONS
4
CAPITAL STOCK
QUARTERLY RATES O F GROWTH
PERCENT
PERCENT
LABOR HOURS
QUARTERLY RATES O F GROWTH
REAL INTEREST RATE
QUARTERLY RATES O F GROWTH
CHANGE IN ANNUALIZED YIELDS
PERCENT
PERCENTAGE POINTS
QUARTERLY SEASONS
LABOR PRODUCTIVITY
Figure 5
QUARTERLY SEASONS
1
4
QUARTERLY SEASONS
1
2
3
QUARTERLY SEASONS
4
6
QUARTERLY SEASONS
F ig u r e
130333 dO B V 73 0 X30NI
@FedFRASER