Carnegie-Rochester
Conference
Series on Public
Policy
22 (1985J
g-54
North-Holland
MACROECONOMIC TIME-SERIES,
BUSINESS CYCLES AND WXROECON@J~IC POLICIES
Rene M. Stulz
Ohio State
University
and
Walter
Wasserfallen*
Universitk
I.
The
decomposition
and
a seasonal
cal,
cyclical
is
part
is
believed
respond
the
cyclical
to
order
to
is
walk,
for
States.
as
the
the
for
better
approximated
a large
cycle
the
number
growth
annually
component
component
special
of
real
in
real
(1976).
See, for
instance,
and Taylor
(1979).
0 167-2231/85/$03.30
01985
Barre
(1978),
Elsavier
Science
Publishers
and
to
not
directly
observable.
In
it
has
become
common practice
in
growth
Unfortunately,
and seasonals
seasonal
(1982)
(1983).
show
that
model,
the
by
out
(1973),
B.V. (North-Holland)
growth
com-
a random
in
a random
to
a
in the
specifically
time-series
turns
Lucas
by including
dummy variables
approximated
variables
Lawrence
it
activity
*The excellent
research
assistance
of Peter
F. Wyss is gratefully
&lit.?
and Manfred
Neumann have kindly
provided
part
of the data.
We
of the Macroeconomics
Seminar
of the Economics
Department
at the Ohio
participants
of
the
Carnegie-Rochester
Conference
on Public
Policy,
McCulloch,
Charles
Nelson,
Manfred
Neumann,
Jiirg
Niehans,
Charles
Plosser,
Brunner
for
helpful
comments.
We thank
the Center
for
Research
in
Business
at the University
of Rochester
for partial
financial
support
of
I
The
because
is
measured
is
interest,
measures.
by a stochastic
of
a cycli-
macroeconomics.
policy
cycle,
long-run
or growth,
in
fluctuations
fiscal
others,
business
correct
a trend
history
be of
and where appropriate,
1 Ne,lson/Plosser
regressions.
If
business
to
short-run
and
into
a long
trend,
respective
ponent
to
variables
has
considered
monetary
approximate
time
real
dominate
to
work
linear
of
component
component,
empirical
INTRODUCTION AND SUMMARY
generally
directly
Bern
the
United
walk,
be much
the
smaller
acknowledged.
Jacques
also
thank
the members
State
University,
the
April
1984,
Huston
and especially
Karl
Government
Policy
and
this
research.
Mishkin
(1982),
Sargent
than
if
the
This
follows
trend,
growth
component
from
the
the
fact
residuals
correlations,
of
which
by sample
size.
sition
time-series
of
rather
than
in the
In
first
part
in
results
and
moving
does
not
persist
plaining
the
business
cycle
trend.
of
which
is
fact
that
of
invest
the
paper
output
has
work
of
has
shows
that,
monetary
our
can
policy
and,
model
is,
therefore,
economic
which
changes
in
theories,
cluding
remarks.
2This
research
Black
can
only
monthly
and
in general
our
the
by a low-
business
cycle
output
trend
be
both
In
is
shown
the
that
in
of
the
both
are
how
by
which
focus
King/Plosser
10
focuses
when
output
changes
in
paper
the
(1982),
ends
real
and
business
trend
of
of
the
on the
they
can
technologies
invest
their
lesson
of
hence,
macro-
real
technologies,
considered
on
in
households
and,
its
in output
The model
i.e.,
The
part
of
output
usually
policy.
the
changes
The major
of
theory,
in
while
from
a theory
second
a change
output.
trend
output
relevant,
for
2
ex-
theory,
of
changes
trend.
analysis.
caused
models
growth
be empirically
distribution
in
(1981),
of
deviations
an explanation
by growth
recent
Long/Plosser
Traditionally,
domain
a change
monetary
on
the
its
the
which
i.e.,
builds
in the
explain
It
considered
(1979).
ro-
We show that
changes
empirical
changes
variables
cycle
the
approximated
shows that
distribution
that
usually
(1982).
the
provides
about
the
fluctuations,
are
instance,
in
that
results,
to
from
choose
brings
hence,
perma-
used
We use
We find
explain
must
among many technologies.
wealth
of
study
to our
been
to
which
with
households
implies
component.
Their
can be well
decompo-
in the
some evidence
fluctuations.
tried
output
a model
consistent
all,
macroeconomic
fluctuations
we offer
by changes
(1982).
time-series.
According
of
theory
deviations
the
Nelson/Plosser
at
determined
trend
cycle,
auto-
one year.
in
trend
macroeconomic
paper,
or
role
Our empirical
output,and
of
on a time
empirically,
business
trend.
positive
a stochastic
dominated
countries.
exists
process.
part
a crucial
several
walk
and
exclusively
we provide
paper,
results
it
beyond
The empirical
play
at
if
average
using
of Nelson/Plosser,
only
on U.S.
the
component,
show that,
i.e.,
this
strong
artifact
are
by a time
a random
exhibit
(1982)
transitory,
of
regresses
variables
variables
look
corroborate
order
real
one
statistical
real
the results
and focused
data
cyclical
a pure
of
nent
quarterly
if
approximated
regression
Nelson/Plosser
fluctuations
bustness
of
annual
data
erroneously
that
the
are
that
the
is
by
with
our
variables
and
by
business
some con-
cycle;
Kydland/Prescott
see,
(1981).
for
EMPIRICAL FINDINGS
II.
In
this
important
under
a number
part,
insights
into
the
frameworks
and
cal,
techniques,
seasonal
unit
to
roots
are
are
States,
various
the
time-series
analysis
lationships
cations
of
our
section
1.
Statistical
It
cycles
in
to
correct
the
time-series
components
in
components
z is
trend,
the
variable
data,
for
growth
population
readily
to
isolate
is of
the
Germany.
The
allow
re-
next.
are
empirical
The impli-
outlined
work
investigation
in the
the
on business
for
cyclical
part.
T,
C, and S are
growth
The
and
underlying
form
studies
used
of
the
which
included
153)
components
seem
in
however
to
investigation.
parts.
(1).
which
are
variables,
pg.
the
the
international
presented
and
under
seasonal
equation
of
explicitly
(1982,
from
namely
and therefore
the
measurement
theoretical
under
and
of
logarithms
are
come
empiri-
= Tt + Ct + St
version
observable
order
model
cyclical,
empirical
cycle
the
and
part.
practice
Two approaches
in
business
are
informal
normal
and West
of
cycli-
more
contain
and quarterly
the
Frameworks
zt
the
for
this
to
countries,
Estimates
variables
common
unobserved
cative
activity
magnitudes
trend,
analysis
France,
allow
First,
first-order
empirical
monthly
is
seasonal
its
real
addition
for
industrialized
movements.
findings
of
the
which
the
into
two sections
Britain,
measured
seasonal
between
final
where
are
of
of
time-series
tests
The next
Great
out
as follows:
In
formal
applied.
Switzerland,
the
discussed.
developed
carried
organized
decompose
components
are
characteristics
is
The data
underlying
cal results.
post-war
period
and include
five
United
tests
stochastic
used
some recently
seasonal
empirical
The presentation
investigation.
statistical
of
note
Empirically,
may justify
to
business
for
cycle.
are
considered
the
respective
that
"using
and
direct
secular
With
respectively
prefer
however,
a multipli-
we mainly
an additive
account
seems unsatisfactory,
suffice
Some authors
work
with
formulation.
and seasonal
the
relevant
first
for
approach,
economic
regressions.
observable
since
neither
of
the
growth,
Nelson/Plosser
variables
measures
movements
to
factor
account
inputs
nor
are
not
technology
available."
The second
strategy
relies
on the
11
stochastic
properties
of
the
series
itself.
The
about
of
the
this
emerging
unobserved
part,
ture
basic
But
processes
to
is
we consider
linear
time
the
first
which
two
trend
following
variable
and
t
denotes
dummies
are
quarterly
and
lar
from
the
of
the
estimated
seasonal
operator
order.
is
3 This
of
This
the
litera-
a deterministic
plus
for
are
a stationary
It
is
given
is
by
would
be the
as Lkzt
white
case
noise
if
the
important
to
an
measure
is
trend
term
proxy
to note
that
the
nature,
equation
to allow
for
= zt-k'
rather
for
secu-
and only
reason
the
to
treat
(3)
(2)
(1976).
is the
multipli-
A normal
non-stationarities,
and
leading
a
to
(3)
and seasonal
Both
random
process
with
structure.
procedure
and the
Box/Jenkins
normal
a
term
s-l
4 for
as an adequate
This
by
respectively
is
of
way.
alternative
proposed
term,
that
We know of no compelling
r(LS)
is
taken
zt.
is
constant
an error
a deterministic
= fit
n
often
It
of
e(L)r(LS)(l-L)(l-LS)Zt
are
to the
set
autocorrelation
constant
St.
appropriate
periodicity,
series
an asymmetric
included
an
P symbolizes
the
z are
developed
process
root
seasonal
P'S
stochastic.
in such
L, defined
3
belongs.
zero.
form
Oj’S
in addition
the
because
dummies
recently
unit
and
stochastic
with
effects
an unspecified
in
(2)
parts
seasonal
$(L)
of
is
(2)
the
data.
but
component
terms
seasonal
class
contains
of
section
+ Ut
the
monthly
equation
A more
cative
model
final
component
actually
mean
that,
is
zero
component
three
s
of
seasonal
cyclical
first
and
Note
12 for
cyclical
and
the
seasonal
y.D.
J Jt
mean
The time-series
Tt and'the
s-l
L
j=l
+
included.
unconditional
obvious
the
cyclical
In accordance
an unconditional
a time-trend,
dummy variables.
the
In
the
identify
testing.
The
some assumptions
equation:
seasonal
of
series.
be that
to
deterministic
with
the
requires
investigated
empirical
Zt = “0 + a$
where
is
models.
component
of
will
task
the
to
problem
parts
restriction
the
amenable
stochastic
extraction
individual
our
stationary.
question
signal
shock.
contains
12
b(L)
polynomials
and r(LS)
may be of
The differenced
moving
average
in
variable
terms.
the
lag
infinite
wt =
(1-L)u-LS)zt = I (Zt-zt-l)-(zt-s-zt-s-l)
and
invertible
what
the
ARMA-process.
cyclical
treat
all
parts
This
topic
will
In
the
information
First,
the
(3)
are
roots
part
of
z should
with
two
sections,
about
the
empirical
autocorrelation
the
in
this
it
It
way,
namely
in the
evidence
as
will
be
the
implied
respect.
Second,
more
of
of
two
zt
representation
be possible
part.
which
yields
competing
models.
by equations
formal
to
processes.
this
presented
of
obvious
stochastic
section
of
a stationary
immediately
however
structures
autoregressive
follow
not
will
last
relevance
to
is
be.
further
next
assumed
case,
a symmetric
be dealt
exploited
in
of
z in
1 is
In this
(2)
tests
a time-series
for
are
and
unit
carried
out.
2.
Autocorrelation
As
first,
trends
too,
used.
enced,
are
However
the
characterized
0.25.
If,
not
correct
zt
is
of
produce
white-noise
function,
starting
from
in
the
data
and
and
all
lags
stochastic
the
lags
illustrate
the
residual
pt
samples
would
have
procedure
seasonally
differprocess
except
and s+l,
trend
model
stationary
the
and the
s-l
z on a linear
trend
but a spurious
residuals,
of
to
from
moving-average
of
a value
large
first
and
structures
residuals
used
a
and sto-
and deseasonalizing
at
-0.5,
hand,
the
is
a non-invertible
of
deterministic
be correct
detrending
autocorrelations
other
a regression
to
1984),
autocorrelation
case
(2)
(1981,
estimated
inappropriately
a value
representation
the
counterpart
follow
zero
take
on the
white-noise,
estimated
w's
by
they
model
the
if
resulting
and s, where
quate
if
the
of
A simple
(2).
Assume
Its
and
Nelson/Kang
between
by comparing
z's
model
and
distinguish
be obtained
implications.
property
(1977)
to
differenced
be white-noise.
this
of
can
test
deterministic
relevant
is
Chan/Hayya/Ord
informal
appropriately
partly
to
by
rather
chastic
of
Structures
shown
for
with
(3)
and
lags
1
a value
is
an ade-
differences
w are
and seasonal
dummies would
estimated
autocorrelation
almost
one
the
residuals
and
dying
off
only
very
slowly.
Issues
or
U'S,
spectively
case
are
where
highly
become
the
more
stationary
serially
the
autocorrelated
(l-l$IL)(l-rILs)!Jt
if
differences
correlated.
residuals
the multiplicative
complicated
in
in
Given
the
is
deterministic
especially
equation
the
= e(L)A(Ls)e,
13
(3),
equation
the
(2),
w's,
results
presented
trend
and seasonal
interesting.
ARIMA-process
in
Suppose
are
below,
model
pt
the
rethe
(2)
follows
with
et white
noise.
wt would
then
be
Wt = ( 1-L)(1-LS)f3(L)A(LS)
(l-@lL)
(l-r$)
With
$1
from
and
a pure
low
wt would
pg.
to
stochastic
properties
point
one,
If
process.
little
in
wt
e(L)
be
the
States,
work,
Switzerland,
economy,
almost
samples
rate.
In
are
series
addition,
are
post
World
cept
for
real
price
impossible
under
these
deterministic
and
of the autocorrelation
the stationary
differ-
With
War II
to
seasonally
unadjusted.
production,
and the
which
and
the
real
wage
rate
throughout
and
rates
of
either
of
and real
quarterly
are
exchange
for
the
are
available
United
is
monthly,
ex-
The
are
data
GNP, industrial
The data are
States.
from
all
period
or
real
real
rates,
annually.
respect
used.
the
unemployment
unemployment
only
United
- are
The observation
available
in that
unemployment
Germany
production,
is
is
- the
state
logarithms.
Exceptions
sources
and West
exception
periodicity
GNP,
countries
the
like
the
natural
and the
real
five
representing
industrial
variables
included.
Swiss
official
of
As Nelson/Plosser
between
for
France,
variables
GNP,
transformed
data
Britain,
on quantity
as
also
relevant
Great
lies
such
rates
in addition,
(3).
empirical
emphasis
indistinguishable
are,
dependence.
becomes
finite
would
and A(L’)
serial
it
out,
distinguish
in equation
from
to
trend
and seasonal
effects
on the basis
of the residuals
in equation
(2) and of
In
The
close
show only
147-149)
circumstances
ences
sufficiently
moving-average
order,
(1982,
rl
(5)
et
the
authors
into
two
upon
request.
The
results,
Regressing
the
dummies
cal
- shown
outcome:
serial
of
under
A very
for
and
with
the
that
the
parently
the
level
all
decay
random
time
seasonal
elements.
the
the
in
1,
respective
that
trend
"Deterministic
Trends"
statistic
indicating
dummies,
residual
for
cases.
about
of
residuals.
generally
most
the
half
serial
divided
on a time
with
hypothesis
is
in
the
slowly
are
series
estimated
very
walk
Inspection
Table
heading
the
series
only
excellent
in
low Durbin-Watson
correlation
reveals
0.9
presented
original
and
Judged
according
of
the
correlation
14
high
first-order
the
is
variables
in the
the
at
about
consistent
Note
explanatory
to
analysis
start
which
variables.
positive
one typi-
complete
autocorrelations
lag,
and seasonal
- yields
A more
increasing
the
trend
parts.
further
power
ap-
significance
contain
estimated
seasonal
residuals,
of
however,
cies
shows
at
seasonal
are not
able
The
Based
get
lags
are
be quite
over
because
the
nominal
rates.
The
that
examined
the
random
these
findings
walk
are
4The
that
5
The
most
the
autocorrelation
real
especially
with
results
all
(1982)
from
are
for
one
in
are
however,
as
the
case
noted
measured
the
averaging
of
of
the
first
differenced
residuals
with
the
They
which
interval
to
be
obser-
very
purpose.
subperiods
have
been
formed
according
to
15
the
graphical
picture
of
the
cycliis
annually.
proved
by
trend,
differences,
shorter
of
above,
the
consistent
to
ob-
generated
that
in first
seems
rates
a deterministic
due
function
as
hypothesis
data
lag
which
interesting,
series
furthermore
U.S.
at
exchange
model
the
deviation
correlation
likely
for
as
appear
for
dependence
are
Remember,
consistent
as
serial
period
be that
processes.
Our
serial
to
lag,
coefficients
results
an appropriate
Nelson/Plosser
positive
The
provides
also
order
the
in
autocorrelations
seasonal
The variables
that
Table
differenced
first
Higher
the
of
by Box/Jenkins
significant
the
alone
for
part
proposed
patterns.
reveal
seems
measured
is, however,
vations.
criterion.
second
seasonally
and/or
rate
average
results
of
remaining
lags
dummies
function
the
partly
few
time.5
autocorrelated.
observe
and
flexible
conclusion
moving
component,
highly
over
in
dependen-
seasonal
procedure
furthermore
recent
general
order
autocorrelation
the
summarized
The
error
that
of
be first
first
significant
correctly.
and show no consistent
unstable
served
for
the
by a two-standard
subperiods
cal
at
indicates
identification
must
exhibit
effects
are
stationary.
irregular
low
informal
still
finding
themselves
occur
judged
This
series
seasonal
variables
to
generally
all
characteristics
on the
all
order
to
capture
time-series
(1976),
at
almost
lags. 4
to
main
various
1.
that
series.
useful
TABLE
Statistical
Variable
Obs.
Per.
I
Characteristics
Deterministic
N
Trend
Trends
Seas.
R2
Time-Series
DW
Dumm i es
United
stat.
Sign.
Analysis
Autocorr.
Diff.
States
Real GNP
Industrial
l/47-IV/81
Prod.
I /47-2/82
‘140
422
+
+
0
0.99
0.12
1:0.35,
2:0:21
0
0.98
0.04
1:0.48,
2:0.29,
12:
14:
Unemployment
Rate
i /48-2ja2
410
+
0
0.27
0.04
-0.22,
-0.23,
13:
15:
3:0.18,
-0.20,
-0.16
1:0.16,
2:0.27.
3:0.19.
4:0.15,
5:0.17,
lO:-0.14,
12:-0.22,
15:-0.12.
24:-0.13
2/40-l
/63
180
+
0
0.16
0.07
2:0.32,
lo:-0.20,
2/63-2/70
85
-
0
0.80
0.18
3:-0.24,
3/70-2/82
144
l
0
0.22
0.05
1:0.22,
3:0.21,
5:0.24,
12:-0.27
4:0.35
2:0.34,
3:0.17,
4:0.26
Real
Wage
I /64-2182
218
+
0
0.17
0.01
3:0.21,
12:-0.22,
9:0.14,
11:0.16,
24:-0.29
5/64-0/72
9/72-2/02
100
114
+
-
0
0.96
0.18
0
0.56
0.03
l2:-0.31
2:0.24,
3:0.30,
1948-1981
34
+
0.95
0.18
l/61-1/83
09
+
J
0.75
0.17
l/61-1/83
89
-
J
0.21
0.21
I /70-l/83
I /70-6/83
53
162
+
+
0
0.95
0.10
J
0.91
0.40
l/68-11/83
62
+
0
0.97
0.15
I /60-l
/83
l/60-1/83
93
93
+
+
J
0.92
0.29
J
0.76
0.14
330
+
0
0.66
24:-0.24
Switzerland
Real
GNP
industrial
Prod.
Employment
4:-0.42
France
Real
GNP
Industrial
Real
Wage
Great
Real
Prod.
12:0.27
4:-0.46,
3:0.25,
6:0.38.,
4:-0.49
0.01
1:0.35.
2:0.15,
4:0.18,
1:0.34,
l2:-0.32
12:-0.44
Britain
GNP
Industrial
Unemployment
Real
2:0.35
1:0.42.
Wage
Prod.
Rate
l/56-6/83
l/56-12/70
180
+
/
0.31
0.09
l/76-12/79
48
0
0
0.30
0.12
l/63-5/83
245
+
0
0.92
0.15
I:-0.22,
12:-0.42
l/63-12/72
120
+
0
0.96
0.47
l:-0.49,
2:0.22,
l/73-12/82
120
+
0
0.51
0.22
13:0.41,
l2:-0.39
16
lo:-0.34
3:0.14,
13:-0.28
13:0.24
12:-0.49
Table
1 Continued
Statistical
Variable
Obs.
Per.
N
Characteristics
Deterministic
Trend
Trends
Seas.
R*
Time-Series
DW
Dummies
West
Stat.
Sign.
Analysis
Autocorr.
Diff.
Germany
Real GNP
Industrial
Prod.
l/60-IV/82
l/63-12/82
92
240
+
+
J
/
0.96
0.42
1.1
0.85
0.37
1.1
I:-0.60,
3:0.19,
7:-0.14,
11:0.20,12:-0.26
23:0.27,
Unemployment
Rate
l/68-1/83
181
+
J
0.77
0.03
l/68-8/74
80
+
4
0.36
0.09
l/75-8/81
80
-
J
0.45
0.08
1.1
1.1
1.1
4:-0.21,
24:-0.21
1:0.35,
2:0.31,
3:0.35,
4:0.22,
7:0.16,
12:-0.20
1:0.35,
2:0.24,
3:0.37,
12:-0.28
Real
Exchange
Rates
S.Fr./S
I /73-2/82
110
-
0.25
0.12
S.Fr./E
l/73-6/83
126
f
0.1 I
0.09
1.0
S.Fr./FF
l/73-6/83
126
-
0.46
0.15
1.0
S.Fr./IJM
l/73-2/83
121
-
0.71
0.16
1.0
Notes
:
Obs.
Per.:
N:
0:
+,
Observation
Number
Not
-:
Stat.
significantly
different
Significantly
Cliff.:
Autocorr.:
serial
zero
negative
different
from
zero
of
freedom
degrees
statistic
differences.
by the
Autocorrelations
standard
from
positive,
for
Stationary
followed
Sign.
observations
R2, adjusted
Durbin-Watson
DW:
1:0.19
period
of
Significantly
4:
rT*:
1.0
error
correlation
Given
order
of
seasonal
significantly
criterion.
is
the
different
Given
coefficient.
17
order
of
normal
differencing,
differencing.
is
from
the
lag,
zero,
followed
judged
by
by
the
a
two-
estimated
3.
Unit
Root
More
this
formal
study,
(1982a,b)
test
have
equation
the
between
which
do not.
that
correlation
The
based
(1976)
vt
t-value
to
of
seasonality,
characteristics
are
included
lag
in all
presented
unit
comes
insignificant
usual
t-statistic.
mates
is
reported
6
In
from
addition
in
in
aI
regression
elements
to
of
the
autodis-
and
results
seasonal
series
presented
dummies
+ "0 + 51t
unit
and
auto-
is
taken
(1976,
except
the
that
isolated
8.5.2).
Therefore,
for
Note
is
(6)
through
by comparing
Table
examined.
+ Et
root
= 1 is tested
Fuller
are
on estimating
carrying
2.
cannot
the
All
p of
p extends
to
seasonal
lagged
the
usual
series,
possible
number
For almost
the
The
Only
this
and
West
outcome.
containing
the
the
majority
i2
Hasza/Fuller
out
all
literature,
(6)
series,
of
the
These
the
by ordinary
the
The trend
the
by Nelson/Plosser
to
equation
be rejected.
for
time-series
chosen
helpful
root
from
For
according
possible
regression.
based
problems.
siderably
relevant
;,
in Table
order
further
a
differ-
first
season-
least
squares
cases.
The results
are
is
(3),
assuming
purely
The following
discussion
+ . . . + gpvtmp
neglected,
in the
in
and Hasza/Fuller
lags respectively.
no seasonal
characteristics
The hypothesis
irrespective
(1976)
seasonal
drawn
investigated
by
The
al.
and
significance
+ glvt-l
= zt-zt-l.
coefficient
models
procedures
containing
on the
regression
al
(2)
order.
is
and given
these
finite
coefficients
at seasonal
model for
time-series
with
Zt = alzt-l
ences
for
time-series
the
by Fuller
models
of
The distinction
is,
Fuller
where
two
to
developed
point
representations
above,
the
been
starting
combining
tinguishes
relevant
procedures,
recently
The
.6
regressive
from
Tests
as
are
a first
judged
statistic
unemployment
findings
of
a1 moreover
variables
Durbin-Watson
German
hypothesis
parameter
rate
consistent
beby
the
indicate
no
deviates
with
con-
the
esti-
(1982).
seasonal
(1982b).
private
It
elements,
is given
correspondence
tests.
18
the
most
general
model
by
with
Professor
Fuller
was
very
zt = “1Zt-1
+ a*(zt-s-zt+l)
+ kIwt-I
+ . . . + khwt-h
o + Bit
+f3
with
ly
s-l
z
+
roots
order
are
again
to
present.
are
zero,
normal
equal
normal
isolated
root
in the
the
if
and
following
nary
least
(7)
t
to p+sP where
seasonal
as coefficients.
a first-order
Under
normal
these
and seasonal
differences
are
respective-
part.
CII and a2 are
equal
and a first-order
The
to
(7)
unit
one and
seasonal
equation
unit
can
be written
(7)
with
as
= 50 + 6It
hypotheses
p and P denote
autoregressive
circumstances,
a(L)r(LS)(l-L)(l-LS)Zt
The
+ E
J Jt
h is
of
a3 equal
y.0.
j=l
wt=(l-L)(l-LS)zt.
the
+ a3(zt-l-zt-s-l)
tested
by
+
s-l
z
j=l
y.0.
+ Et
J Jt
estimating
equation
ordi-
squares:
Hl:
CXI = 1,
assuming
test-statistic
1.0.
is
;,
from
that
the
calculated
Fuller
seasonal
model
is
as a conventional
(1976,
table
8.5.2)
stationary.
t-value
is
used
to
The
relative
to
perform
the
test.
H2:
aI
lated
and
Fuller
H3:
The
HI,
compared
(1982,
'LI
entry
in Hasza/Fuller
based
cannot
is
the
relevant
F-statistic
value
03 = 60 = aI
In
taken.
(1982,
on equation
be
A conventional
is
in
of
calcuHasza/-
5.1).
= a2 = 1 and
F-value
= 1,
to
table
usual
findings
a1
= CX~ = 1 and a3 = 0.
rejected.
(7)
this
table
are
The
19
= rj(al1
j)
= 0.
case,
is
Again,
the
the
relevant
5.1).
shown
in
seasonal
Table
model
3.
The
is
hypothesis
assumed
to be
TABLE
Tests
Estimated
Variable
United
Real
equation:
Obs.
Per.
l/47-IV/81
Prod.
Unemployment
Rate
t/47-2/82
l/48-2/82
2/48-l/63
2/63-2/70
3/70-2/82
Wage
l/64-2/82
5/64-8/72
9/72-2/82
Great
Real
N
zt-,
P
Unit
+ g,vt-,
al
Roots
+ ...
+ gpvt-p
+ 60
BO
+ f3,t
5,
+ E+
R2
DW
140
8
0.92
0.53
0.0007
0.99
2.02
12
(-2.33)
0.98
(2.38)
0.08
(2.21)
0.00007
0.99
1.98
(-2.11)
(2.20)
0.10
(1.99)
0.0002
0.98
2.00
(2.05)
180
t-2.60)
0.94
0.20
(2.01)
0.0009
0.96
2.01
85
(-2.46)
0.98
(1.91)
-0.10
(1.64)
0.0007
0.97
2.02
144
C-0.25)
0.95
C-0.12)
0.00
(0.31)
0.0009
0.97
I .98
218
12
C-2.30)
0.98
C-0.01)
0.02
(1.64)
0.000003
0.99
I .97
12
(-2.51)
0.75
(2.70)
0.16
(0.38)
0.0003
0.99
2.11
12
(-2.83)
0.92x
(2.74)
0.10
(2.98)
-0.00005
0.98
2.06
(-3.73)
(3.56)
0.98
2.01
422
410
100
114
12
0.97
(-2.38)
Britain
GNP
Industrial
Unemployment
l/60-1/83
Prod.
Rate
l/60-1/83
l/56-6/83
l/56-12/70
l/76-12/79
Real
= a,
First-Order
States
GNP
Industrial
Real
zt
for
2
Wage
l/63-5/83
l/63-12/72
l/73-12/82
93
0.87
1.18
(-1.92)
(1.32)
0.0005
93
0.89
(1.96)
0.49
0.0003
0.95
1.84
330
(-1.90)
0.98
(1.96)
-0.02
(0.97)
0.0007
0.99
1.77
180
(-3.28)
0.88X
C-0.93)
0.17
(3.35)
0.0006
0.92
1.87
0.86
I .90
0.99
2.03
48
(2.15)
(-3.63)
0.80
(3.10)
2.54
C-2.50)
C-0.99)
-0.005
245
0.95
(1.54)
0.22
(1.76)
0.48
(1.47)
120
C-1.71)
0.89
0.0003
0.98
I .99
C-0.97)
0.84
(0.97)
0.73
(1.15)
0.0001
0.87
2.03
(-2.36)
(2.36)
(1.98)
120
20
0.00008
TABLE
Tests
Obs.
Variable
Per.
for
N
2 Continued
First-Order
P
Unit
Roots
Ql
60
R2
01
DW
Switzerland
Real
1948-1981
GNP
Industrial
Prod.
34
l/61-1/83
89
5
0.95
8
(-0.30
0.91
(-I
l/61-1/83
Employment
89
8
1
.59)
0.92
(-2.38)
0.56
0.0001
0.99
2.02
(0.36)
0.47
(0.02)
0.0002
0.93
1.86
(1.68)
0.38
(0.53)
0.96
1.95
-0.00009
(2.39)
(-1.94)
France
Real
GNP
I /70-t
/83
53
8
0.91
(-I
Industrial
Prod.
l/70-6/83
162
I2
.07)
0.85
l-1.97)
Real
Wage
West
Germany
Real
GNP
I /68-I
I /83
I /60-I
industrial
Prod.
Unemployment
62
V/82
92
/63-12/82
Rate
/68-l
8
8
240
/83
12
181
/68-8/74
12
80
/75-8/8,
80
Real
Exchange
“t
Obs.
126
,/73-2/83
0.96
2.58
(2.05)
(0.26)
0.99
2.04
0.99
2.00
0.95
2.21
0.94
0.27
(1.14)
0.96
0.21
C-0.58)
0.97
(0.64)
0.16
C-0.58)
(0.69
2.01**
0.0003
(0.60)
0 .ooo I
(0.25,
-0.00004
1
C-0.33)
0.28
0.008
0.88
0.14
12
(0.87)
1.07
(2.86)
0.0008
0.87
0.82
(3.20)
-1.82
(0.25)
12
(I .27)
3.93”
0.95
I .75
0.002
(2.88)
C-6.00)
I2
0.94
0.24
0.00005
0.91
I .99
I2
(-1.42)
0.94
(0.90)
0.22
(0.18)
0.0001
0.93
I .98
I2
C-2.30)
0.89
(I .96)
0.50
(I .49)
-0.0002
0.92
2.00
12
(-2.57)
0.84
(2.52)
0.78
(-I .75)
-0.0004
121
0.96
I .98
C-3.12)
(3.1,)
c-2.80)
:
= z
- =t-1
P&.:
Observation
N:
Number
P:
Number
period
of
of
estimated
R2:
DW:
t-values
are
given
relative
*
l *
2.01
(5.91)
1.17
126
I /73-6/83
S.Fr./DM
Notes
110
I /73-6/83
S.Fr./FF
0.99
Rates
I /73-2/82
S.Fr./f.
0.0004
(0.60)
0.00002
(-1.05)
(18.38)
S.Fr./%
I.10
(1.09)
0.69
observations
lagged
differences
regressions
are
R2, adjusted
Ourbin-Watson
shown
in
to 1.0.
a,
a,
for
parentheses
is
is
significantly
significantly
degrees
included.
The
degrees
of
freedom
of
the
N-p-3.
of
freedom
statistic
below
the
different
different
estimated
from
from
21
coefficients.
one
one
at
at
the
the
The
5%-level
l&level
t-value
for
cx,
is
GNP
Real
Industrial
Germany
West
Industrial
France
Employment
industrial
Prod.
Prod.
V/82
l/63-12/82
I /60-I
I /70-6/83
l/61-1/83
l/61-1/83
240
92
162
09
89
114
Q/72-2/02
Prod.
100
5/64-a/72
Switzerland
218
I /64-2/02
N
United
States
Real Wage
Per.
Obs.
Variable
It
= a,~~-,
-0.18
24
8
24
0.59
0.32
(-0.96)
0.82
0.94
(-1 .oo)
0.95
(-3.05)
0.07
0.76
8
(-1.77)
0.55
0.93
C-1.15)
0.93
8
C-0.48)
-0.77
(-3.11)
0.98
24
24
0.07
0.99
o2
FM(SD)
+ a3(~t-,-~t-5-,l
Estimated
First-Order’and
(-0.1 I )
-0.28
o1
for
24
h
+ a2(zt-zte5-,)
Tests
3
-0.02
1.45
-0.20
-0.29
0.36
-0.04
-0.22
(tul
+ k,wt-,
-0.05
equation
First-Order
TABLE
(-0.88)
C-0.42)
0.96
0.98
(0.64)
I .Ol
C-1.54)
0.94
(-1.12)
0.93
(-0.31)
(-2.28)
0.99
1 .QQ
(0.09 1
0.62
o1
FM
0.93
0.91
1.03
0.90
0.95
0.81
0.70
0.81
o2
Unit
+ . . . + khw+h
I model ):
Seasonal
+ f3,
Roots
-0.33
-0.43
-0.37
-0.09
-0.33
-0.11
0.02
-0.12
a3
+ B,t
yjDjt
17.62**
5.48
15.49*
6.67
5.42
13.70’
6.68
16.18*
3.58
3.47
3.60”
2.61
2.18
2.93
I .78
3.29
@(d+4)
n-d-4
Tests
+ ~~
Roots
,#,(3)
n-d-4
Unit
+ L
3.46*
1.43
4.04
2.25
2.49
2.56s”
1.41
2.63**
FM (SD)
Seasonal
0.30
2.53
0.64
0.10
0.25
0.29
0.39
0.24
RM
Dummies
Britain
GNP
120
l/70-1
t-value
of
The tested
The
**
tested
degrees
The
The
*
for
the
= o2
model
hypothesis
is
rejected
=
=
(a,
is
estimated
CL, = a2
u,
differences
)
0.94
0.44
0.56
-0.38
0.36
C-1
)
I .02
(1.67)
0.91
(-1.33)
0.95
o3
“3
a3
at
the
1% level.
given
relative
at the 5% level.
= f3,
= 0
= I,
regressions
1 and
I and
= a2
= 0)
to
are
least
= yj
with
1.0.
at
= 8,
included
)
-0.26
(-2.31
0.26
0.83
C-1 .Ol )
0.78
t-1.52)
21
0.74
0.90
0.13
0.57
(-1.32)
0.92
(-1.08)
0.96
(-2.38)
with
seasonal
dummies
without
seasonal
dummies
lagged
observations
aI,
in parentheses,
hypothesis
is rejected
of
Test
rp (d+4),
n-d-4’
freedom
Test
,#(3)
n-d-4’
for
Restricted
FM:
RN:
of
Number
model
model
of
Number
$-It-l
) - (z+-s-Zt.+,
Observation
period
24
24
120
l/63-12/72
2/82
24
245
24
24
8
8
24
24
l/63-5/83
180
l/56-12/70
93
330
/83
I /60-l
93
80
181
I /56-6/83
/83
/83
I /60-l
l/75-8/81
1/68-l
Full
Full
of
Rate
Prod.
Rate
FM(SD):
N:
h:
Obs.
Per.:
:
Notes
W+ =
Wage
Real
Unemployment
Industrial
Real
Great
Unemployment
j)
N-h-s-4.
(all
seasonal
dummies
= 0
0.22
-0.08
-0.01
0.08
-0.12
-0.31
-0.67
0.12
0.10
0.96
.47)
0.92
I .02
.42)
0.98
(-I
.64)
C-0.21)
0.84
(-I
C-1.59)
0.95
(-I
C-1.34)
(-0.62
0.92
0.96
C-0.86)
0.98
(-2.37)
1
0.62
0.79
0.68
0.72
0.84
0.91
0.93
I .@J
1.02
0.07
-0.45
-0.04
-0.02
-0.13
-0.40
-0.85
0.03
0.03
19.05**
6.50
13.41
25.36**
16.01
18.22+*
10.19’
10.21
6.76
l
4.03”
2.02
2.84
5.02**
3.31
4.31+
4.30’
4.38’
1.92
4.09**
I .38
1.78
4.40*+
2.10”
3.02
2.10
5.00”
1.60
0.36
0.70
0.37
0.17
0.34
1.16
0.90
I .44
0.71
stationary
in
strates,
of
hypothesis
creases
in
to
the
that
11 if
Out of
their
model
mentioned
above,
a comparison
this
of
siderably
used
first-order
one.
unit
However,
two
whereas
would
the
in
the
therefore
rameter
ever,
is
given
be
inversely
the
time
the
required
economic
grounds
of
of
the
unit
speed
The
therefore
In
the
to
across
in
no case
can
component
is
to
are
get
this
the
it
is
the
highly
present.
First
a stationary
better
of
through
24
mean and
of
the
pa-
interval.
period,
HowaI
should
Second,
is
long
implausibly
.9.
is
the
Third,
also
a first-order
that
are
easily
unit
a stochastic
of
Seasonal
a seasonal
the
and countries.
sections
differencing
on
in favor
proposition
markets
that
time-series.
captured
in the
non-stationary,
observations.
preceding
likely
a
below
First,
accepts
and
pro-
characteristics
around
same in all
hypothesis
test
long-run
and time-series
the
con-
model.
different
is
adjustment
one
Second,
between
measurement
mean
As
slightly
Three
actually
unless
is virtually
that
present,
is
well.
The
a constant
between
countries
interpretation,
series
long-run
IXI coefficient
results
completely
the
interval
the
the
point.
is
one
time
a2 increases
interpretation.
of
to
discriminating
the
cyclical
time
return
means
required
are
if
root
dummies
alone.
coefficient
to
hypothesis.
a seasonal
from
this
in
swings.
unit
a possible
the
value
at
return
irrespective
presented
This
walk
to
adjustment
results
summarized.
they
the
root
of
rejected.
random
of
to
of
power
cyclical
the
one
length
related
uniformity
the
to
the
are excluded
present,
would
long-run
favor
close
dummies
in-
time-trend
quite
seasonal
that
criti-
restricted
that
are
number
seasonal
seasonality
interpretation
it
appears
autocorrelation
is
this
null
It
for
the
the
aI and a2 are
case
below
the
both
no
root
case,
under
or
joint
root
on the
3).
dummies
an
parameters
describes
unit
that
First,
be mentioned
economic
a unit
results
QI
zero
virtually
and
exhibit
empirical
the
must
have
other
show
and FM reveals
root
If
not
is
to
root
inclusion
on the
seasonal
be noted.
one unit
caveat
the
cases.
on the
if
and FM demon-
The evidence
equal
Table
seasonal
above
depend
(7).
the
in
one if
One possible
not
if
must
is
FM(SD)
towards
cedures
set
column
including
does
is,
completely
(last
FM(SD)
for
that
are
significance
zero
columns
examined,
values
characteristics
and a3 to
the
and a first-order
tested,
dummies
of
in equation
17 series
The
H3 is
seasonal
series
outcome
a first-order
8 cases.
Two additional
lose
the
dummies
is mixed.
value
A comparison
that
seasonal
H2,
present,
and
case.
moreover,
exclusion
cal
this
root
be
trend
or
data
is
the
patterns,
ARIMA-model
if
than
through
seasonal
volved,
meaning
pirical
procedure.
follows
dummy variables.
that
The
a pure
these
moving
findings,
considered
an
would
q
Q
would
4.
q would
various
The
quarterly
countries.
lags
are
in
shown
s with
and
Based
all
on
variables
a-
polynomials
Q would
possibly
some
series
the
generally
intermediate
(1-L')
and
are
investigated
(1979)
of
production
work
and
First,
in
the
the
n(LS)
the
and distributed
the
ARIMA-residuals.
series
and seasonal
influences
from
the
than
vice
effects
States
occur
by
and
seasonal,
differences
indices
that
follow
over
the
endogenous
stochastic
and
the
non-seasonal
processes
period
variables
where
polynomials.
here.
25
empirical
1965-1981.
of
a dynamic
the
results
ordinary
A constant
simultaneous
the moving-average
This
issue
is
on the
contempora-
estimated
first
the
most
Third,
example
is
bivariate
versa.
are
that
of
not
be
differences
The
regressions
can
among
United
even
The
the
relationships
be stronger
lag
examined
Second,
feedback
alternaestimation
all
normal
periods.
are
includes
parameter.
The strongest
time
A representative
4.
seasonal
different
cross-correlograms
.positive
short.
observes
two
industrial
differences
seem to
cases.
magnitudes
for
conclusions.
stationary,
necessarily
product
GNP and
unexpectedly,
Table
real
and
moving-average
countries
most
Plosser
the
inem-
generally
order.
of
e-
non-seasonal
as bivariate
indicate
production
not
the
The statistical
by an ARIMA-process
in
in
real
stationary
surprisingly
industrial
into
for
seasonal
Not
European
do
low
seasonality,
between
purpose.
the
one
West
model
For
techniques
some general
cross-correlations
7
of
than
zero.
as well
described
squares
orders
connections
that
for
allow
including
is
to
series
for
regressions
neously
is
Connections
of ARIMA-models
well
series
representation
containing
smaller
econometric
used
results
the
be
international
with
tively
the
relatively
root
an adequate
be one.
International
The
of
of
is
of
variables
restricted
both
process
unit
data
part
statistical
denote
For
and
the
= eq(L)AQ(LS)qt
respectively.7
equal
1,
parameters
of
stationary
average
adequate
be a variant
and
a seasonal
differencing
remaining
(I-L)(d)z,
where
Generally,
seasonal
portion
pursued
least
of
the
term
equation
factors
further
TABLE
International
4
Relationships
- An Example
Endogenous:
Exogenous:
Industrial
Industrial
Production
Production
Switzerland
United
States
United
Lag:
Great
West
Britain
Germany
States
0
4
0.33
0.41
(1.89)
(4.10)
0’.29
(1.52)
Switzerland
Lag:
0.19
0
(3.02)
Great
Lag:
Britain
0.27
0
(2.80)
3
0.32
(2.16)
West
Germany
Lag:
0
0.51
(5.41
)
0.48
0.38
(2.69)
(4.35)
0.41
4
(2.28)
.
R2
DW
Notes
The
shown
DW:
0.50
0.46
0.53
1.62
2.05
2.06
2.00
:
logarithms
R2:
0.55
in
R2,
regressions
of
the
parentheses
adjusted
Durbin-Watson
are
estimated
Four
variables.
below
for
degrees
the
estimated
of
in
the
lagged
first
dependent
coefficients.
freedom
statistic
26
and
seasonal
variables
differences
are
included.
of
the
natural
t-values
are
and four
lagged
that
significantly
time
lag.
These
tries
the
stress
such
as the
the
the
The
model
given
the
time-series
tation
of
to the
qualitative
assumed
correlation
Business
of
the
business
allow
the
limited
to
cycle,
maximum
to
ones
import
all
countries
the
last
Cycle
the
used
consider
and (9)
and taking
primary
degree
drift,
part.
(l-L)Tt
= nit
(l-LS)st
=
to
shocks
in
to
rest
the
a
explanation
the
be
influences,
implied
measure,
in
all
(1981)
non-seasonal
model
then
and
with
respect
component,
mean of
zero.
seasonal
followed
trend
In order
are
Based
component
is
regularities
Nelson/Plosser
models.
persistence
context.
the
which
The auto-
the
that
persistence,
whereas
represen-
cyclical
describing
importance
of
an adequate
is now explored
The procedures
Beveridge/Nelson
Q = 1, the
real
to worldwide
an unconditional
of
exclusively
found
of
with
or
presented
Measurement
(9),
cyclical
seasonal
by
however,
walk
dominant
evidence
transmitted
no
decade.
investigation,
possible
seen
or
coun-
the
An alternative
of
the
is
goods.
easily
a small
industrialized
the
are
over
with
of
a random
attributed
actions
under
structure
to
policy
by equation
be stationary
of
States,
for
is
only
concerning
United
characteristics
to
question
be that
increases
for
economies
The
from
common reaction
price
with
real
no
It
included.
answer
the
demand
oil
Implications
5.
the
receives
example
via
observed
the
would
for
world
would
that
however,
country,
also
are
integrated.
One possibility
large
is
well
are
effects
suggest
fairly
forces,
here.
of
variables
positive
findings
are
impulse
dependent
are
equivalent
(1982),
to
who,
on equations
(1)
becomes
+ K
(l-Al?)
(10)
nzt
ct = f(L) q’3t
and therefore
(1-L)(l-Ls)zt
= wt = (1-L')
nit
+ (1-L) (l-LS)f
Qlts n2t
and n3t
constant
term,
are
which
mutually
vanishes
uncorrelated
in
equation
27
+ (1-L)(1-A1LS)n2t
(11)
(L)r13t
white
(11)
noise
sequences
because
seasonal
and K is
differ-
a
encing
is
real
applied
exchange
clusively
to
it.
rates,
It
since
if
the
side,
of
associated
lag
trend
given
lag
gation.
hit,
For
the
above.
therefore
obviously
(=s+l)
is
and
cyclical
section,
on
it
s
the
low
must
denote
the
is
pure
cyclical
shocks
create
This
does,
course,
not
are
created
conclusions
‘For
by a series
remain
an
alternative
of
unchanged
approach
a moving
therefore
of
ex-
in
take
white
if
the
if
which
shocks
a business
yields
similar
28
at
2 is
the
variance
zeros
at
analysis
lags
in
that
index
in
it
see
real
at
appears
magnitudes.
"business
considered
McCulloch
this
exists
it
same direction.
is
the
2 to s-2.
Therefore,
in the
results,
if
If
njt,
presented
component,
1,
+Yo,
to
cycle
lags
-( l+$
equal
possibility
must
no cyclical
is
no persistence
independent
investif(L)
exists
)"2
for
findings
that
virtually
the
under
respectively.
momentum.8
the
empirical
and n2
cyclical
1 up
correlation
nI
forecastable
exclude
the
+ 2(1+*i
the
lag
F is
an uninter-
serial
there
intermediate
s+l
process
imply
values
of
s and
where
coefficients
= 20;
and
s-l,
average
hand
seasonal
structures,
the
noise,
with
the
variable
to
Even
variances
that
the
finite
right
from
non-zero
closely
yO
1,
s-F-2,
autocorrelation
results
be concluded
that
F+2 to
time-series
and
to s+l,
empirical
little
that
is
the
at
lags
data,
order.
the
at
would
quite
(11)
on
autocorrelations
series
, where
very
zero
lags
1 (5)
non-zero
contains
point
be governed
dependencies
yields
at
observed
= 0,
are
must
component
differences
f(L)
all,
of
part
zeros
the
of w extends
Based
last
be of
if
component
correlogram
from
correspond
and 0;
wt.
different
order
and -(o:+~AIcJ~)/Y~
"&Y.
serial
stationary
Given
that
generates
irregular
features
this
by equation
first
(monthly)
of
at
walks,
w implied
The
quarterly
likely
for
component,
The
component
These
random
of
intermediate
most
follow
finite.
n2t.
presented
the
with
with
but
rupted,
coefficients
of
is
through
F+s+l
order of f(L).
the cyclical
s-l
f(L)
obvious
component.
structure
Autocorrelations
only.
are
to
order
already
they
by a stochastic
The autocorrelation
becomes
(1975).
cycles"
These
which
is defined
as a weighted
linear
The characterization
series
presented
business
index
1, the
industrial
logarithm
of
to
industrial
The
whereas
series.
In
ponent
is
this
the
persistence
in
depending
shows
exclusively
on
chosen
results
As
The
variable
lag
is
is
not
certainly
regressions
than
the
natural
is
this
exhibit
the
large
com-
findings,
high
degree
totally
of
spurious,
(1985)
furthermore
may crucially
depend
surprising
because
"explained"
more
almost
random
trend
on our
the
an
com-
of
a highly
is
models
Swiss
Figure
dummies
Based
Wasserfallen
This
of
cycles,
indicator
cycle
In
in
trend.
size.
point.
residuals
mentioned,
cycle
business
the
logarithm
results
already
example,
the
on business
procedures.
distributed
a deterministic
natural
obvious.
sample
from
and seasonal
the
operation
business
dependent
usual
rather
for
measurement
autocorrelated
the
in
by a deterministic
resulting
autocorrelated
a regression
on a trend
misleading.
test
clarify
from
literature
the
that
residuals
are
proxied
is
to
time-
highly
deviation
useful
differencing
empirical
usually
approach
is
differences
differences
the
the
in macroeconomic
the
A representative
production
stationary
index.
as
to
9
series.
component
effects.
estimated
individual
contrast
measured
of
of
cyclical
sharp
production,
the
swings,
the
in
seasonal
time-series
pared
is
indicators
and fixed
of
the
of
above
cycle
time-trend
combination
on
a highly
easily
purely
by
random
series.I'
MACROECONOMIC POLICIES
III.
Macroeconomists
traditionally
stochastic
process,
deviations
from
of
the
empirical
‘The
I
recessions,
IO
use
of
Nelson/Kang
and
(1984)
detrended
state
to
on
‘many ’
at
the
page
output
mean
traditional
this
is
OF THE OUTPUT RATE
the
the
suggests
least
as plausible
classical
“Our
economic
activities
deviations
business
cycle.
of
followed
The
a new view
as the
the
presents
of
the
traditional
business
business
by
a
correlated
the
that
definition
definition
follows
and serially
paper
6:
rate
approach,
mean characterize
in
rate
in
this
its
conforms
They
expansions
contractions,
wrongly
output
approach
(1946).
among
from
that
a constant
With
presented
of the
indicator
assume
exhibits
mean.
rate
evidence
BurnsAlitchel
consensus
which
the
output
distribution
AND THE DISTRIBUTION
cycle
cycles
by
as
rsimilarly
general’
associated
with
revival.”
note
a
number
of
important
variables.
29
additional
pitfalls
the
a
A
Residuals
time-trend
Stationary
(1 - L)(l
of a regression
and seasonal
on
dummies
differences
- L’)
Mean
I
Figure 1
Quarterly
index of industrial production
Switzerland 196 l-l 983
30
This
view.
over
new view
time
the
and that
output
rate
With
the
growth
if
the
try
on
to
one
to
In
this
reflects
part
the
model
of
the
of
factors
II
the
the
with
cycle
variance
This
part
an
of
informal
Section
111.2.
111.3.
nominal
discusses
rate of
interest
rate
mean
households
choose
their
lifetime
expected
with
depend
the
only
in
variance
the
output
part
the
of
the
factors
of
which
associated
changes
Section
of
some
of
the
paper
rate
and
in
111.1.
of
the
the
pro-
results.
households.
are
Section
derives
the
real rate of
solved
is
and
their
the
for
policies
there
mean
allocate
emphasizes
investment
Unless
technology,
because
shocks
rate.
output
utility.
households
rate
stock.
as follows.
and
to
mean and the
both
money
problem
output
real
unanticipated
model
the
the
and
has
mean.
model which
derived
in the
of
-
one
the
case,
output
new view,
rate,
that
changes
the
the
its
Model
this
of
of
from
this
from
that
problem.
Section
111.4.
Section
111.5.
derives
the
the
consumption
one
on how
that
rate
with
output
shows
of the
optimization
of
developed
in
organized
of
the
case,
the
while
that
output
rate
of
the
Discussion
the
rate
is
distribution
The
duced
paper
the solution
of
interest.
Finally,
An Informal
and
this
growth
the
discussion
and the
The model
in
-
of
rate,
rate,
stochastically,
-
produce
output
distribution
change
to
the
a general
equilibrium
of the output
rate
theory
theory
presents
of
of
output
suggests
deviations
The model
can
used
the
follows
empirical
growth
technologies
business
fluctuations
the
It
the
output
of
as its
paper.
the
paper
paper,
we build
The distribution
this
with
mean and the
vides
of
deviations
in this
mean.
the
in
the
of
fluctuations
about
output
associated
affect
the
facts
in Part
variance
the
stochastically
of
of
deviations
mean as well
new view.
mimics
presented
in its
mean
fraction
the
changes
distribution
the
presented
from
rate
correlation
the
the
a larger
explain
changes
of
explain
deviations
wants
explain
serial
explaining
evidence
explain
the
mean output
little
view
focuses
mean
than
1.
is
mean.
The empirical
rate
the
its
traditional
theory
mean.
that
there
from
macroeconomists
in
is
only
the
fact
that
households.
which
maximize
one commodity
variance
resources
of
over
the
prooutput
technologies
and commodities.
The
model
households
perfect
here,
developed
choose
markets.
and
it
can
the
Only
in
this
part
of
of
the
distribution
one
be produced
commodity
is
by a variety
31
the
paper
output
produced
of
lets
rate
in
constant
infinitely-lived
in
the
an economy
economy
stochastic
with
considered
returns
to
scale
technologies
quantity
of
lessly
at
point
the
in
and the
Households
time.
of
various
choose
of
technologies,
and
to
hedge
unanticipated
if
households
nologies
used
nologies
change
choose
a different
absence
of
the
in
In
that
the
model
government
a target
change
for
account
the
the
the
perfect,
so that
to
price
of
plus
price
of
rate
the
in
the
in
rate
in the
wealth
among
portfolio
they
technologies.
over
money,
a diversified
tolerance,
change
of
output
their
risk-averse,
risk
the
time
only
mix
tech-
of
because
and nominal
assets
expands
here,
monetary
control
of
money
follows
the
itself
the
price
of
the
stock
of
the
of
the
variable.
its
to
role
assumed
to
try
to
rate
of
take
assumed
is
rate
to
be imgrowth
the
growth
deviations
growth
into
rate
time.
the
for
actual
The
target
the
over
stock
growth
Consequently,
from
is
money
target
a random
money
process
stock
the
to
is
The target
unexpectedly
money
rate
It
money
money.
changes
government's
variable.
rate.
of
households
an important
output
a stochastic
of
growth
plays
the
price
policy
control
the
policy
of
lead
tech-
money
of
households.
can
try
case,
in
of
money
would
In this
choose
monetary
the
aggregate
can
invest
default-free
a safe
bonds.
real
in
rate
demand
nominal
rate
change
of
is
an increasing
in
the
of
of
for
nominal
interest
price
of
expected
is
bonds
the
rate
of
and
There
nominal
is
rate
is
As the
nominal
of
change
32
in
the
are.
price
rate
of
of
the
assumed
commodity,
default-free
no outside
of
to
must
zero.
function
opportunity
In
of
the
cost
of
rate
of
interest,
of
the
price
in
bonds
supply
interest
equal
a decreasing
money.
of
production
bonds,
return.
the
function
the
the
nominal
Consequently,
of
a decrease
of
absence
uncorrelated.
free
the
the
would
is
Households
offer
both
they
a random
of
In
invest
invest
relative
for
achieve
how to
in the
change
actual
consume
than
money
be serially
balances,
that
to
money
growth
of
by households
technologies
its
government's
required
the
rate
variance
distribution
uses
fact
However,
the
the
cost-
of
set
of
used
Households
sufficiently
introduction
developed
determination
achieve
of
as the
that
The existence
of
opportunity
rate.
changes
would
time.
mix
money,
the
rate
production
assumed
be changed
balances.
would
were
is
can
technologies
choose
constant
over
investment
in the
they
exhibit
of
mean and the
Households
if
It
be risk-averse.
investors
assets.
time.
technology
output
real
to
the
risky
against
the
of
assumed
same way as risk-averse
over
in each
The mix
services
are
would
randomly
invested
distribution
commodity
households
change
commodity
each
determines
the
which
the
which
default-
be such
that
equilibrium,
expected
real
it
of
of
real
follows
money
rate
balances
(which
that
is
equivalent
to
households'
wealth
an increase
holdings
is
the
sum of
an
consequently,
households'
the
risk
aversion
real
assets
about
real
of
the
stock
of
real
returns
and
makes
cannot
attractive
to
must
interest
interest
also
the
interest
This
less
crease
economy.
about
and Tobin
which
households
between
to
risk
a fall
in
of
the
the
nominal
be
in
the
real
rate
an increase
real
in
rate
explicitly
and
in
the
of
interest
real
nominal
interest
is
their
of
bonds
the
of
result
techrate
in
a fall
this
The
by the
invest
nominal
in the
(1965),
of
stock
given
to
While
maximize
willing
the
this
risk
supply
the
bear
of
bear
the
premium
with
risk
a given
In
premium
amount
this
rate
implicitly
for
in
derived
here
expected
A fall
of
more
risk
33
the
to
utility
in
the
the
in the
that
the
households
safe
real
of
fact
following
however,
rate
of
households
distribution
reflects
Consequently,
asset.
and
paid
a given
model,
by investing
safe
output
production.
premium
interest.
they
expected
risk
risk
in
to
of
the
the
associated
change
rate
no outside
the
are
in the
the
equilibrium.
Therefore,
an increase
affect
in production
a decrease
production.
(1963)
increases
nominal
in
the
investments
households
in
distribution
of
production
for
interest
production
not
joint
re-establish
However,
brings
corresponds
rate.
to
risky
on investments
an increase
that
in
in the
all
following
of
does
a given
the
of
rate
returns
not
decreases
portfolio
nominal
as-
distribution
wealth
interest
for
necessary
accompany
in
difference
bearing
are
invest
of
that,
households
joint
investments
rate
production
is
the
nomithese
consumption.
The
terest
the
constant
With
a given
of real
Mundell
in
lifetime
decreases
Furthermore,
real
production,
attractive
equilibrium
in a model
real
balances.
exhibit
and the
in
in
nominal
the
less
of
of
real
interest
shares.
households'
be altered.
it
rate
of
the
substitutes.
invest
on investments
more
models
the
households'
households
households'
increase
means
it
to
that
function
to
in
in
returns
maintain
of
be gross
the
the
This
attractive
interest
decreases
the
and of
rate
distribution
the
Hence,
nologies
of
of
in
invested
real
the
in
want
investments
interest.
expected
inflation)
model,
expenditure
utility
a given
commodity.
is
of
commodity
assumed
to
an
value
for
on
made more
rate
is
they
total
the
commodity
to
it
the
nominal
assumed
The change
of
the
households'
wealth
commodity
production.
rate
in
Consequently,
decreases
of
a decrease
real
investments.
of the
rate
In this
and constant
are
the
returns,
amount
stock
paper,
nal
of
the
expected
balances.
wealth.
relative
sumptions
the
real
increase
real
Throughout
and
in
of
asset,
households
infor
rate
the
of
output
households
change
in the
cannot
de-
as there
must
is
change
the
distribution
Whenever
the
ance
of
risk
they
output,
nominal
rate
must
rate
an
and
an
change
of the
The
technologies
by a decrease
in
the
nominal
of
the
output
in
the
nominal
the
same
rate.
of
in
effect
of
can reduce
the
output
rate.
the
the
variance
interest
of
interest
obtain
the
growth
through
the
variance
as
a decrease
increase
in
the
the
in
Conseboth
the
households
As in
of
the
money
premium
growth
the
output
result.
rate
of
if
that
a risk
of
amount
rate.
this
vari-
However,
decreases
to
bear.
the
of
output
The assumption
crucial
they
function
expected
of
risk
before
in
rate
variance
rate
the
in the
is
an increase
price
model
which
we want
for
effect,
rate
expected
of
the
rate
of
of money.
to
investment
by
(b)
at
stock
input
is
dKi
a discrete-time
in
the
of
which
Cox/Ingersoll/Ross
built
of
given
here
(a)
sector
whose
is
the
only
from
policies
the
commodity
output
of
is
the
invested
the
ith
in-
the
model
hold
real
change
sto-
The ex-
processes.
commodity
in
rate
the
commodity.
in n production
commodity
of
households
one
because
output
dynamics
differs
because
there
model
the
production
The
itself.
ith
Ki.
production
process
is
a
by:
= 'Ki K.dt
1
Eaton
for
be invested
produce
quantity
given
which
of
to a finance
distribution
and
(1978)
instantaneous
models
(1980),
model
k, can
to
the
variable
“For
an economy
of capital,
The
the
set
is a government
time. 11
required
i=l ,...,n,
process.
choose
The
model
We turn
there
through
We look
random
set.
the
finance.
Cox/Ingersoll/Ross
and
chastically
only
extends
in
opportunity
opportunity
balances
here
used
show how households
developed
isting
developed
is widely
a given
vestment
for
of
decrease
the
has
households
variance
the
interest,
of
The Economy
(1978)
Jones
model,
the
amount
an increasing
of
that
stock
is
of
increase
out
money
2.
in this
the
mix
variance
increases
turns
wealth
among many technologies
model
it
decrease
an efficient
increase
the
can choose
stock
invested
be accompanied
quently,
this
of
is
to
by decreasing
chose
the
output
as it
bear
households
mean
risk
of
+ aKiKidzKi
incorporate
(1981).
alternative
real
GertIer/Grinols
Ui
balances
(1982),
model.
34
(12)
in
the
and
Cox/lngersolI/Ross
Stulz
(1984).
(1978)
See
also
model
Lucas
,
see
(1984)
where
is
d+.
process.
the output
12’
changes.
vector
s of
from
(12)
the
unit
of
that
the
implies
world
each
that
for
a
standard
and the
change
as
households
Wiener
variance
the
state
is
also
assumed
processes
exhibit
which
government
the
by a lxs
to
follow
an Ito
follow
of
of
is described
variable
variables
production
of
rate
can
state
endogenous
time
output
process
the
variables;
of
expected
a production
The state
so
per
economy,
state
process
Equation
increment
this
rate
world
Ito
the
In
constant
an
process.
returns
to
scale.
We want
the
proceeds
the
money
money
the
to
from
stock
of
households'
lead
whole
nously
the
money stock
transfers,
we take
of
M=M($,t).
tained
by differentiating
sis,
we study
only
tries
make
by the
growth
of
rate
of
effects
time.
of
monetary
readers
who
of
as
not
in
Ito’s
the
at
of
with
the
price
would
provide
monetary
instantaneous
on the
of
derive
references
these
is
on
techniques,
introduction.
35
as
In
exogechanges
by
assumed
that
involving
the
To simplify
that
the
the
the
relevant
over
the
dynamics
Fischer
the
analy-
This
governis
the
and
(1975)
of
capthe
considered
here
distribution
of
information
money
the
be ob-
The variance
policy
processes
of
can
the
P,.
conditional
Ito
state
supply
a rate
The
all
paper
which
We assume
TI grow
type
stock
commodity
of
an Ito process.
the
money
as being
money
Lemma.
policy.
money
the
this
operations
II follows
analysis
familiar
It
loga-
affect
of transfers.
the
be a function
using
71 captures
(1978)
are
of
the
in
bonds.
change
of
that
policy
complete
to
of
commodity,
open-market
of monetary
2
analysis'
change
A more
assumed
the
net
in
from
obtained
for
government
the
consumption
of
instantaneous
price
the
of
no government
M(s,t)
stock
operations.
because
M is
the
of
changes
not
in
rebate
proceeds
least
will
analysis
money
a stock
and
are
one type
‘2Cox/ln9ersoll/Ross
the
only
rise.
the
actions
owns
at
changes
the
the
results
policy
that
view
open-market
There
assumption
rate
simplifies
the
of
The
ment
the
because
not
so that
If
the
monetary
in the
production
supply
world,
to
hold,
can
changes
government
change
money
one
of
and money.
The
not
extent
because
can
commodity
will
does
sector,
households.
households,
To the
because
and
to
wealth.
The
government
private
in
analysis,
time
tured
paper
effects
given.
over
rebated
the
of
changes
to
to
the
wealth
function,
real
partially
applying
to
real
wholly
this
utility
in
creation
the
are
remainder
a model
money
affect
creation
rithmic
the
consider
next
for
about
the
instant
of
M(_s,t)
optimal
which,
control.
provides
For
a
useful
given
the
dynamics
the
assumed
not
discussed
for
dynamics
for
here,
To capture
the
the
assumed
3.
The Households'
the
as
effect
are
quantity
it
price
adds
of
of
all
demanded
money.
to
in the
the
Optimization
that
of
money
nothing
changes
to be functions
We assume
of
However,
the
of
such
main
monetary
vector
by households,
a derivation
results
policy
of
variables.
same,
are
is
this
regime,
state
yield
paper.
!.I,, and
IJ~
Problem
households
are
the
infinitely-lived
and
maximize:
m
Et I e-pT
t
where
the
C(T)
is
holdings
tive
risk
tolerance
households
is
asset
risky
is
processes
real
terms
of
invest
in
one
equal
the
given
corresponds
m(T)
to
r.
instantaneous
nominal
risky
of
expected
rate
asset
the
+ a,dz
optimal
to
n risky
assets
assume
that
return
of
(12).
equal
have
a real
bond,
which
rate
of
invested
in
whose
return
real
rate
consist
of
all
markets
investments
One risky
rela-
wealth
asset
inare
in
is
of
pro-
a bond
to R.
of
return
equal
to
is:
(13)
71
rate
of
II.
consumption
that
assets
instantaneous
of return
nominal
growth
we
We
real
wealth
different
whose
constant
The
households'
n+l
asset
to
implies
shares.
the
in
by equation
= Rdt + Vndt
household's
per feet
and
function
processes.
first
’ 3Morespecifically,
is
fraction
and
of return
U, is the
commodity
expenditure
can
and
is
the
rate
The
there
the
production
-d*l
I1
where
is
instantaneous
We take
the
ni
Consequently,
a safe
the
utility
constant
nonstochastic
perfect.13
duction
of
balances.
and
real
in
with
rate
Households
vestments
IdT
instantaneous
w.
i.
in
return
consumption
of real
households'
risky
is
the
The
l-a
a [c(T)‘m(T)
there
and
are
competition.
36
no
portfolio
transaction
policies
costs,
no
taxes,
must
and
be
that
such that
the
following
n+l
z ni
1=1
dw =
where
is
dIi/Ii
flow
budget
d1.
( $i
the
- rdt
nominal
bond
and
the
and consequently,
variance-covariance
of holding
this
vestments
to
balances
of
- Rmdt - cdt
of
the
plus
knowing
fraction
have
perfectly
T (TR)
the
true
this
solve
is
the
to
1
the
first
!,,L*+(,
1-l
Asset
In
the
price
See
of
the
is
-cs
wealth
fact
that
real
returns,
one forms
opportunity
the
the
cost
of
output
are
the
(1979)
returns
and
of
state
in-
varia-
m(l-TR)l]
w
wO
function
c(w,l,t)
(n+l)x(n+l)
of
of
zeros.
matrix
respect
c(w,_s,t)
to
in
state
opportunity
w.
with
the
of
of
(n+l)xs
variables.
cost
The
respect
matrix
U and Uas is
changes
of
and cw
variance-covariance
the
with
of
tolerance
expenditures.
c(w,l,t)
with
derivatives
vector
rate.
risk
by the
be the
Policy
(relative)
given
to
portfolio
(15)
consumption
taken
for
portfolio
of
function
returns
and Monetary
the
optimal
asset
absolute
partial
inverse
asset
model,
Fama/Farber
level
the
of
the
a lx(s-1)
Demands
this
when
function
are
U is the
variable
Cj is
of
vector
is
matrix
distribution
14
c=q+Rm
lxs
variables.
of
utility
derivative
state
balances.
In
households'
the
Rm is
the
of
-1
coefficient
indirect
returns,
state
asset.
correlated
W
the
is the
covariance
ith
we get:
(u-r.l)+y
-
expenditures
gs
asset
4.
is
partial
vector
the
assets
l4
for
distribution
portfolio,
T
households'
Consumption
the
reflects
as distinct
returns.
(14)
of
of
This
W
the
return
m/w.
asset
households
for
n=(G)Y-
where
satisfied:
balances.
model,
Solving
bles.
rate
be treated
matrix
real
In
real
cannot
is
]w + rwdt
instantaneous
(la),
nl corresponds
in the nominal
bond
equation
invested
equation
The
holding
real
Uncertainty
households
With
a discussion
37
to
a non-stochastic
of
stochastic.
choose
holdings
of
real
hold
determines
investment
balances
when
the
oppor-
changes
in
tunity
set,
i.e.,
the
stochastically
folio
whose
choose
and
If
hedge
the
against
of
relative
to
hold
their
hold
able
such
lifetime
a mean-variance
to hedge,
the
in
state
myopic
function,
policy,
monetary
intuition
policy
behind
households
know
way
decreases
which
against
risk
with
households
are
otherwise,
their
changes
on their
As this
asset
households
try
It
lifetime
the
useful
changes
in
utility
protect
the
policy
of
expected
this
dynamics
i,
sufficiently
asset
way.
to
is
affect
hedge
of
re-
positively
they
would
offset
the
state
process,
adverse
the
a
risk-averse
partly
in the 1.th
the
If
in
degree
than
in a production
can
policies.
wealth
against
monetary
try
their
this
their
implies
policy
return
a
vari-
the
fact
effects
households'
of
choice
commodity.
this
example,
rate
if
of changes
themselves
illustrate
the
Breeden
in
in monetary
to
the
variable
less
have
in
they
be
changes
following
utility,
they
to
they
investment
some asset's
be an investment
to produce
unrealistically,
15See
to
purpose
variable,
state
if
with
if
monetary
i
variable
than
function
in the
variable
choose
households
hedge
affects
If
invest
expected
can
changes
is
in
to
state
enough.
unanticipated
able.
technologies
in
for
the
be explained
expected
coefficient
changes
affect
households
a state
utility
variable
lifetime
high
likely
so that
unanticipated
can
state
changes
is
effect
that
ith
in
unanticipated
can
result
the
aversion
correlated
For
this
that
against
mean--
households
be correlated
logarithmic
15 .
policies
uncertainty
unanticipated
lative
of
hedge
the
extent
do not
the
and consumption
households
must
they
in
their
that
a smaller
households
as
if
means
However,
portfolio.
risky
assets
Furthermore,
portfolio
When
The
efficient
returns
of
utility
to hold.
time,
that
technologies
choose
change
to
of
dominates
an unanticipated
utility
no mix
variables
This
change
port-
means
return
over
state
one.
expected
variables.
logarithmic
that
they
in
exceeds
This
a way that
stochastically
changes
aversion
5 do not
efficient
- r.1).
whose
portfolio
change
unanticipated
risk
in such
the
vector
a mean-variance
U-I(k
portfolio
of
variables
a portfolio
affects
a
return
state
to
in usage
yields
by the
hold
proportional
technologies
the
given
households
are
balances
space
variables
time,
weights
the
real
variance
state
over
of
discussion
we assume
change
of
u'rr
(1983)
38
of
are
with
that
the
a concrete
there
price
assumed
is
of
to
example.
only
one
money,
P,.
be
such
that
state
Not
an
unexpected
increase
dl’, =
c
is
assumed
example
even
changes
in
with
stylized
facts
state
is
useful
variables
for
other
note
an increase
finance.
index
assume
than
one.
in p,:
that
simplify
(A.l.)
our
with
example
in this
their
shares. 16
unanticipated
economy,
according
in R.
two
important
we can define
instantaneous
utility
the
Consequently,
commodity
as
folio
of
with
(1981),
among
has
Index,
negatively
others
have
stocks
show 'that
return,
i.e.,
expected
with
evidence
R are
the
to:
that
inflation,
positively
commodity
using
Nelson
correlated
provided
of
deflated
inflation.
a real
of
in
in terms
others,
stocks
function
changes
returns
common
correlated
the
(17)
because
the
an
function
$P = (&)g!
flation,
and
unanticipated
changes
embodies
Using
this
processes
u'n and R and (A.2)
as
p evolves
To
of production
correlated
First,
households,
than
returns
that
expenditure
price
to
the
negatively
of modern
the
implies
smaller
with
to
constant
numeraire,
but
perfectly
index
exhibits
it
interesting
price
money
(16)
uncorrelated
in
is
of
positive,
further,
II are
price
- p,dt).
be
in v,, are
It
exact
Q.($
to
changes
changes
in the
the
(1976),
of
correlated
price
index
Fama/Schwert
a well-diversified
a return
the
nominal
which
of
by the
inflation.
interest
is
p are
(1977),
the
in-
negatively
of
common
Consumer
Price
Fama (1976)
rates
case
portand Schwert
portfolio
deflated
rate
with
a well-diversified
when
and
are an increasing
assumption
(A.2)
holds.
With
nominal
‘%ee
the
assets
assumptions
can be written
Samuelson/Swamy
we have
made
as:
(1974).
39
for
this
example,
the
demand
for
(18)
"1 =
*
where
the
(A.21,
changes
state
in
in the
log
With
fall
variable
changes
the
is
the
log
taken
of
type
of
in 71, i.e.,
the
of
the
policy
the
uncertainty
price
services
adverse
relative
risk
a short
household
assets
which,
because
of
an
if
in
in
opportunity
balances.
expects
cI,,R
to
This
policy
position
about
with
future
results
of
on
to
is
nominal
assets
investment
Rates
and Real
nominal
it
makes
Finally,
worsening
expected
hedge
the
loss
TI.
an
of
return
that
also
against
the
from
TR < 1,
a household
to
of
do so 'by
in
the
in
believed
policy
and consumption
the
example
coefficient
position
offsets
the
the
against
unexpectedly,
a fall
monetary
assets
asset
hedge
one
takes
an
an unexpected
set.17
because
this
short
widely
increases
to
TR < 1 and can
an unanticipated
consequently,
While
its
in
condition
it
its
TI falls
decreases
it
opportunity
can affect
If
and
nominal
i.e.,
a decrease
in worse
wants
hedged,
it
about
is
because
accompanies
as
that,
demand for
of interest.
one,
fully
as
in
investment
uncertainty
specific
is
be negative,
implies
the
rate
assumption
an unanticipated
in R if
assets.
71 corresponds
example
uncertainty
creases
nominal
the
interest
than
R which
Hence,
brings
A household
gain
set,
real
of
nominal
household
short
From
correlated
here,
increase
smaller
in
the
investment
rate
unanticipated
holding
of
R.
A household
balances.
increase
fall
additional
is
position
the
of
negatively
of money,
in R.
an unanticipated
tolerance
unanticipated
worsening
real
of
makes
log
postulated
of one unit
of
effects
taking
be the
perfectly
of P,,.
an increase
pi? and, consequently,
from an increase
in the nominal
cost
to
R are
and,
example
demands,
follow
from
uncertainty
opportunities,
consequently,
just
creates
discussed
one should
a number
it
increases
points
dethe
out
not forget
that
of restrictive
how
the
as-
sumptions.
5.
Naminal
In this
rate
of
these
model,
interest
two
“For
Interest
real
R are
variables
a discussion
balances
of
held
endogenous
are
what
Balances
determined.
hedging
by households
variables.
means
in
40
In
as well
this
section,
To
be
able
to
a model
like
this,
see
as the
characterize
Breeden
nominal
we show how
(1984).
our
solution
more
utility
precisely,
function
real
returns
With
a
quently,
we
(i.e.,
6=0).
on production
processes
logarithmic
utility
our
do not
results
The demand
variables.
that
of
assumptions
the
hT'
the
as
the
for
long
demand
logarithmic
easier
to
general
assets
nominal
the
state
variables
function
the
now given
simplifies
not
it
be true
by households
must
of the
be equal
follows
Ito
asset
of
irrespectively
the
dynamics
processes.
demands,
endogenous
it
makes
the
capital,
k,
variables
in
the
the
real
real
value
nominal
assets
of
a
balances
they
hold,
as there
we have:
expenditures
are
equal
to
PW in
this
case,
I8 The
Rm.
=
real
and
it
(21)
wealth
of
can be rewritten
of
households
monetary
wealth,
utility
function
expected
instance,
of
wealth
is equal
m,
i.e.,
to
the
w=m+k.
sum of
Hence,
the
stock
equation
of
(21)
as:
indirect
J(w,z,t)
for
it
that:
However,
function
of
Because
(20)
consumption
p(l-a)W
See,
state
model.
that
to
Consequently,
holds
n1w = m.
as
S.
Conse-
hedge.
nature
or about
the
value
equilibrium,
Furthermore,
the
in
by:
here
P,, follow
equilibrium
must
given
variables
and
expectations
bonds.
is
assets
rational
are no nominal
do
on the
(A.1')
changes
(19)
equilibrium
In
held
state
characterize
households
with
logarithmic
.
for
utility
uncorrelated
restrictions
nominal
made about
as the
are
function,
require
nIw = (G)(R+pn-r)w
0
ll
Notice
assume that
households
have a
18
Furthermore,
we assume that
Cox/lngersolI/Ross
in
a
general
=
(1978)
equilibrium
of
wealth
is
(I/p)e-Ptlnw+G(q,t)
for
setting.
41
a derivation
given
in
this
case
by:
.
of
the
expected
indirect
utility
R = p(l-a)
Equations
(19)
unknowns,
R and m.
ii
+ l1
and (21')
(21’)
can be viewed
Rewriting
as a system
equation
(19),
of
two equations
in two
we have:
(19’)
Consequently,
yield
for
a given
R as a function
(20')
shows
equilibrium.
the
the
to
R times
Figure
of
in the
expected
(a)
rate
in
variance
of
vector
A
policy.
Such
next
the
The real
and
constant
of
the
rate
the
price
change
to
of
on the
of
portfolio
to
satisfy
balances
must
On Figure
TI.
they
2, m*
As r,
is
real
can also
carried
highlight
an increase
and
the
out
all
depend
continuously
change
rate
of
in
in
in
of
of
the
the
changes
remainder
under
this
monetary
interest
because
the
effects
05
a
of
in-
an increase
change
analysis
of
(b)
+,
to
the
interest
section
r,
rate
TI, and (c)
may change
affect
the
nominal
interest
corresponds
However,
to
(21')
Equation
real
wealth.
of money
of
2,
shown
next
technologies
real
effect
real
of
of
rate
technologies.
section
of
utility
of
and
m for
marginal
u,, or 0:
is
R and
be related
the
variables
in
a change
section.
in
the
rate
(19')
two functions.
of m and R.
change
state
change
between
study
in
of
the
of
model.
to
an increase
exist'
utility
values
equations
these
how R and m must
the
marginal
interest,
2 plots
must
that
the
of
shows
be used
terest
on the
which
equilibrium
2 can
rate
Figure
(21')
condition
and R* are the
the
m.
relation
Equation
first-order
be equal
real
of
of
this
assumption
changes
in
in
r is
opposite
of
real
of
monetary
policy.
In
the
equation
effect
(21').
in
a vertical
affect
the
crease
r
the
in
decreases
same
in
for
Figure
in
by
u,, to
token,
given
in
P,
the
~5
of
rate
risk
real
interest
equation
in
and does
that
an
balances
1-1,
not
in-
held
premium
follow
at
the
R, an increase
paid
R must
decreases
42
the
of
a decrease
(19')
follows
premium
in
r or
of
Given
by
interest.
of
risk
An increase
the
It
decreases
rate
nominal
in
in
the
balances.
side
by equation
(21').
nominal
:eestablish
an increase
right-hand
an increase
curve
decreases
level
on the
equation
the
R+~J -1".
an increase
a given
2,
the
a given
U,
of
no effect
a decrease
that,
a decrease
a fall
in
and increases
Notice
or
r has
given
or
effect
in P,, for
shift
curve
in
households
i.e.,
u,, or
Consequently,
creates
the
an increase
m, a change
or
(19'),
of
its
risk
on nominal
in
assets,
an increase
earlier
premium
r
in
level.
on nominal
r
By
R’
-Pn
B
+I
I
II
*
m*
m
Figure 2
Equation (19’) characterizes portfolio equilibrium
for households
Equation (2 1’) relates holdings of real balances to consumption expenditures
R* corresponds to the equilibrium
interest rate
m* corresponds to the equilibrium
holdings of real balances for a given
amount of real wealth
43
assets
must
sis
per
unit
rise
of
to
the
on nominal
tively
of
variance,
bring
the
variance
the
risk
6.
The Real
an increase
in
05
negative.
If
nominal
on
uncorrelated
of
r and
the
the
i.e.,
row
(A.l'),
processes
can be written:
duction
=
negative.
the
V-l
=e
(!e
requires
of
of
-
nominal
the
risk
premium
returns
asset
wealth.
may decrease
affect
case,
rate.
the
real
We focus
changes
the
in
TI are
in monetary
rate
Let
out
taken
holdings
of
of
policy
return
on the
real
on an asset
the subscript
of a matrix.
investments
e denote
With
in
as-
production
(22)
r.1)
that
nega-
In this
policy
output
real
been
the
analy-
processes.
a change
have
interest
Fluctuations
payoff.
households'
of
This
real
unanticipated
instantaneous
instantaneous
if
have
in monetary
which
rate
level.
hold
invested
production
effect
and column
sumption
n-e
is
in
of
the
a safe,
first
holding
of
distribution
case
returns
we derive
Equilibrium
the
earlier
not
and Macroeconomic
simple
nominal
assets
return
assets
Interest
the
promises
rate,
of
its
does
we show how changes
interest,
the
output
rate
on nominal
with
First,
the
real
section
interest
that
the
Rate
of
which
with
of
the
to
is
premium
of
back
of
discussion
rate
premium
assets
In this
rate
risk
effect
correlated
the
and consequently,
the
whole
stock
of
capital
be invested
in
pro-
processes:
l'n
w = k
- -e
Using
equations
Using
rate
the
of
(22)
and (23),
k = i'$I
(Ee-r.A)w
fact
w=m+k,
that
interest
as a function
we get:
(24)
we can
of
rewrite
equation
(24)
to obtain
the
real
m and k:
(25)
where
a = A'l$l~e
and b = A'y!&'A.
a and b are
44
exogenously
given.
They
change
stochastically
some production
over
process
Differentiating
Equation
k
rate
of
rate
the
light
growth
of
real
rate
relative
of
of
crease
in
real
more
in
households
falls
by
librium,
real
of
interest
interest
wants
to
than
can
to
is
expected
given
by the
n,,
the
variables
returns
result
are
invest
interest
increase
the
nominal
in
the
of
fall
nominal
real
in the
rate
of
their
interest.
An in-
households'
bond,
holdings
a constant
real
in-
processes
and
To understand
why
notice
for
that,
for
nominal
However,
bond.
attractive,
constant
fraction
demand
bond
effect
of
a fall
the
in
equithe
nominal
so that
a
assets
Consequently,
in
pk=E(dk/k).
in the
real
that
the
Notice
which
stock
of
commodity
k.
Using
equation
output
rate
can
be obtained
rate
of
_l'V-l(~
=e
With
of
R.
an increase
Ee(w/k),
p’k=
a simple
rate
no household
asset.
the
the
in the
expla-
has
balances.
$
and of
lifetime
seems paradoxical
following
less
the
production
nominal
of
a fall
it
in
households'
holdings
must
in
For
an
their
a fall
the
following
more
of growth
model
rate
in
variance
this
wealth.
less
of
nominal
rate
decreases
real
states
the
endogenous.
real
by
expected
of
in
a constant
a given
invest
interest,
in that
OUtpUt
this
decisions
their
in
in the
in
expected
with
to
invest
higher
an increase
literature,
be no investment
We now consider
the
of
want
make the
invest
that
associated
for
hence,
is
in the
While
rate
of
there
rate
the
As an increase
Consequently,
households
bond
to
rate
less
is
production
and,
interest
follows
interest.
processes
nominal
of
higher.
macroeconomic
nominal
want
real
of
households
the
given
m, it
aversion,
rate,
rate
are
households.
production
the
real
interest
the
balances
terest
of
we get:
II or a decrease
model,
risk
in
of
of
In this
wealth
of
rate
much
nation.
the
decreases
utility
of
(25),
balances
the
real
expected
distribution
(26)
that
real
of
TI decreases
in
the
>o
means
in which
interest
of the
if
time.
b(m+k)2
(26)
world
only
over
equation
dr
-=
dm
the
time
changes
real
corresponds
interest
to
the
(23),
as
rate
interest
expected
expected
which
of
output
output
defines
a function
on
the
of
rate
divided
vector
exogenous
r:
(27)
+ -r.l) _
45
Differentiating
the
expected
output
rate
with
respect
to
r,
we obtain:
duk (~‘!!,l~)(~;v,l~e)- (~;!!,ll)Q’v,‘re~
-=
dr
(28)
(l’V-l(~
- -e -e -r.A))’
Merton
(1972)
side
is
of
equation
related
expected
models
change
output
in
the
the
real
willing
would
stock
of
must
shift
stock
is
bear
the
a less
left
obtain
rate.
variance
As the
variance
has
implies
risk
rate
as,
of
in the
for
the
output
and
change
of
the
that
households
risk.
in
is
the
of
as a decrease
unchanged
households
of
the
capital
the
would
variance
in the
in
commodity
the whole
expected
pro-
have
by a de-
of
the
output
function
of
an increase
expected
been
a decrease
accompanied
money,
longer
previous
an increasing
price
change
Therefore,
money
is
the
and
falls,
households'
of
interest
rate
the
less
price
rate
of
output
sense
output
of
the
otherwise,
no
before
none
causes
are
interest
so that
equi-
which
Consequently,
plan,
conse-
to maintain
part
of the
In equilibrium,
in production.
variance
rate
of
technologies,
an adverse
and,
they
did
of
between
well-understood
policy,
that
as they
real
production
same effects
model,
the
of
of
the
such
is
the
the
only
of
of
technologies
of
the
expected
monetary
distribution
time-series
output
and of monetary
By a proper
(26).
and
that
of
dynamics
equation
technologies
price
monetary
rate
rate
in
of
the
that
change
of money.
dynamics
the
of
unchanged
rate
rate
the
distribution
of
fall
expected
expected
the
price
there
must
in
efficient
nominal
by inspection
that
asset
fall
growth
the
In this
examples
on that
a change
As the
higher
is
models,
return
risky
was not
interest
In these
model,
fall,
output
relation
attractive
an
also
This
more
to
right-hand
expected
asset
be invested
idle.
must
in
the
of
in production.
crease
of
rate
(1984)).
be invested
expected
the
interest.
no longer
to
on the
the
riskless
of
the
same amount
must
expression
of
in
plan
of the
real
Breeden
households
capital
of
rate
and the
For
the
investments
rate
to
real
(see
of
Consequently,
for
in the
of
present
shares
stock
able
rate
the
of
portfolio
duction
the
makes
policy.
output
to
money
wealth
to
monetary
positive.
real
In
numerator
is
technologies
librium.
the
rate
without
quently,
that
(28)
positively
the
in
proves
one
state
variable
money,
LI.,,.
If
the
it
the
output
output
which
P,, follows
46
policy,
is
is
are
of
possible
rate
is
rate.
For
the
expected
a martingale,
a function
as can be noted
specification
policy,
of
of
rate
the
to
the
dynamics
construct
same as the
instance,
rate
then
the
suppose
of
change
expected
output
rate
changes
follows
in
for
a larger
the
output
all
the
in
a martingale
~1, is
high
also.
enough,
fraction
of
changes
its
mean.
One can
in
the
expected
output
or
expected
output
examples
monetary
in
policy
which
contribute
The empirical
analysis
decompose
cyclical,
and
the
a number
pirical
work
stationary
of
this
in which
to
changes
in
technologies
the
changes
If
shows that,
into
and
in
the
countries
and
are
An important
unable
of
that
the
these
data
is
present
in
ex-
quarterly
reject
of
the
is
time-series
a
trend,
time-series
and
to
these
implication
differencing
trend
of
monthly
of
trend,
a deterministic
a stochastic
We examine
in general,
a stochastic
into
representation
root.
first
paper
than
representation
that
if
the
the
decomposition
business
cycle
changes
in
with
changes
in
in
the
the
macroeconomic
dynamics
series
in
time-
hypothesis
exhibits
a
for
em-
results
required
trend
the
output
can
theory
be
caused
model,
macroeconomic
as
to
get
a
is
used,
by
on the
portfolio
the
variance
of
growth
the
nominal
rate
When the
households'
changes
as
by
of
interest
real
wealth
is
in
shown
is
if
real
variables
affect
the
distribution
and
the
held
decreases
falls,
the
households
47
one wants
to
explain
in
which
macroeconomic
in
trend
of
by
In
output
the
through
An increase
instance,
households'
choose
the
time--
considered
of
for
the
policies.
by households.
money stock,
Conseexplain
usually
macroeconomic
assets
vari-
to
changes
in
of
more
crucial
model
the
that
for
In
component.
it
changes
of
rate
cyclical
as
artifact.
account
a theoretical
properties
It
policies
the
the
time-series
same
study.
well
seem to
decomposition,
We present
have
trend
be a statistical
than
trend
macroeconomic
empirical
growth
effect
of
a stochastic
to
component
time-series
stochastic
output
our
out
trend
fluctuations.
of
involving
turns
the
macroeconomic
quently,
their
of
root.
is
particular,
ation
changes
of
time-series.
However,
much
deviations
examples
explaining
rather
component.
unit
unit
in
component
autoregressive
first-order
the
construct
time-series
autoregressive
a first-order
for
than
of
account
CONCLUDING REMARKS
presented
a seasonal
a time-series,
the
variance
rate
can be attributed
the
to
marcroeconomic
and a seasonal
a cyclical,
also
the
output
rate.
IV.
one should
rate
rate
both
if
expected
output
from
in
series
in the
rate
changes
that
in the
variation
technologies
hibits
Furthermore,
changes
increases
real
to bear
wealth.
a smaller
in
amount
the
output
of
growth
production
rate
and in the
Consequently,
risk.
of
the
expected
money
value
stock
of
causes
output.
48
an increase
in
the
variance
of
a decrease
in
the
variance
of
References
Barro,
R.J.
(1978)
Beveridge,
Unanticipated
Money,
United
Journal
States,
S. and Nelson,
(1981)
Permanent
Measurement
Level
86;
549-580.
In
the
of
Economic
Components
the
with
Time
Series
Particular
"Business
Cycle,"
Journal
and Business
Cycles,
Working
into
Attention
of
Monetary
151-174.
Equilibrium
of
Management,
Cambridge,
G.E.P.
Time
Institute
of
Sloan
Technology,
G.M.
Series
Francisco:
Breeden,
Massachusetts
Paper.
MA.
and Jenkins,
(1976)
Analysis,
Forecasting
and Control,
rev.
ed.
San
Holden-Day.
D.T.
(1984)
(1984)
Futures
Markets
A.F.
(1946)
Markets,
Consumption,
Production
(1977)
ng , Journal
and Mitchell,
Hayya,
Journal
of
and
Financial
Options:
Hedging
Economic
of
Interest
and Optimality
Theory,
32,
Rates:
A
275-300.
Synthesis,
Economics.
W.C.
Measuring
search,
H.K.,
and Commodity
in Incomplete
f orthcomi
Chan,
Price
Decomposition
of
7:
General
School
Burns,
the
Economy,
F.
(1979)
Box,
to
and Transitory
Economics,
Black,
and
Political
C.R.
A New Approach
to
Output,
of
Business
Cycles,
National
Bureau
of
Economic
Re-
New York.
J.C.,
and Ord,
A Note
on Trend
gression
Versus
J.K.
Removal
Variate
Methods:
Differencing,
744.
49
The Case
of
Polynomial
Econometrica,
45:
Re737-
Cox,
Ingersoll,
J.,
(1978)
J.,
A
and Ross,
Theory
Eaton,
the
of
unpublished
S.A.
Term
Structure
of
Interest
Rates,
paper.
J.E.
Fiscal
(1981)
Policy,
Capital
Inflation
, The Review
of
and
Economic
the
Accumulation
Studies,
48:
of
Risky
435-447.
Fama, E.F.
(1976)
Fama,
E.F.
Foundations
and Farber,
(1979)
Fama, E.F.
Bonds
New York:
Basic
Books.
and Foreign
Exchange,
American
Economic
Review,
639-649.
and Schwert,
(1977)
Finance.
A.
Money,
69:
of
Asset
G.W.
Returns
and Inflation,
Journal
of
Financial
Economics,
Journal
of
Political
Economy,
5:
115-146.
Fischer,
S.
(1975)
The
Demand
for
Index-Bonds,
83:
509-534.
Fuller,
W.A.
(1976)
Gertler,
Introduction
M. and Grinols,
(1982)
D.P.
(1982a)
Randomness
Testing
for
al Time
Series
the
Wiley.
and
Investment,
Journal
of
Monetary
Eco-
W.A.
for
al Time Series
(1982b)
New York:
239-258.
and Fuller,
Testing
Time Series.
E.
Monetary
nomics,
Hasza,
to Statistical
Nonstationary
Models,
Parameter
hr?QlSofStQtiStiCS,
Nonstationary
Models,
Specifications
Parameter
Report
Census.
50
prepared
10:
in
1209-1216.
Specifications
for
the
Season-
in
U.S.
Season-
Bureau
of
Jones,
E.P
(1980)
Intertemporal
Ph.D.
Financial
King,
R.G.
The
Kydland,
of
American
Money,
F. and Prescott,
E.C.
Time to Build
and
Lawrence,
Lucas,
and
Review,
the
Prices
74:
Persistence
Carnegie-Mellon
Rational
Technology,
in
a Real
Business
363-380.
of
University,
Expectations,
the
Inflation-Output
the
United
11:
225-245.
J.B.
and Plosser,
(1983)
Real
R-E.,
Unemployment.
Pittsburgh,
Working
PA.
Shocks,
Tradeoff,
Kingdom
and
Some Time
1957-1977.
Journal
of
the
Series
Stability
of
Evidence
Monetary
for
Economics,
C.I.
Business
Some
Cycles,
Journal
International
American
(1984)
Money
Series
McCulloch,
Supply
of Political
Economy,
91:
39-69.
Jr.
(1973)
Evidence
Economic
in
on
Review,
a Theory
of
Public
Policy,
Meltzer.
Amsterdam:
The Monte
Carlo
on
63:
Output-Inflation
Tradeoffs,
326-334.
Finance,
21:
Carnegie-Rochester
(eds.)
K.
Brunner
Conference
and
A.H.
North-Holland.
J.H.
(1975)
13:
Merton,
of
C.
(1983)
Long,
Credit
Economic
(1981)
Paper,
Institute
C.I.
Behavior
Cycle,
Unpublished
MA.
and Plosser,
(1984)
Massachusetts
Thesis.
Cambridge,
and Monetary,Equilibrium,
Cycle
in Business
Activity,
Economic
Inquiry,
303-321.
R.
(1972)
An Analytic
Journal
Derivation
of Financial
of
the
and Quantitative
51
Efficient
Analysis,
Portfolio
5:
155-178.
Frontier,
Mishkin,
F.S.
(1982)
Mundell,
Does
Anticipated
gation,
Journal
of
Policy
Matter?
Political
Economy,
An
Econometric
90:
Investi-
22-51.
R.A.
Inflation
(1963)
and Real
Interest,
Journal
Political
of
Economy,
71:
280-282.
Nelson,
C.R.
Inflation
(1976)
Nelson,
C.R.
and
Finance,
31:
and Kang,
H.
(1981)
Spurious
Pitfalls
C.R.
in
Trends
of
Plosser,
49:
Use
Journal
Common Stocks,
Journal
of
Inappropriately
Oetrended
Time
741-751.
of
Time
as
an Explanatory
of Business and Economic
Variable
Statistics,
2:
in
73-82.
C.I.
and
Monetary
The
Random Walks
Economics,
Analysis
metrics,
Samuelson,
P.A.
(1974)
in Macroeconomic
10:
Time
Series,
Journal
139-162.
8:
of
Seasonal
Economic
Models,
Journal
Econo-
of
147-163.
and Swamy, S.
Invariant
Index
Synthesis,
American
Numbers
and
Economic
Canonical
Review,
Duality:
Survey
and
566-593.
T.J.
(1976)
A Classical
Journal
Schwert,
on
C.I.
(1979)
Sargent,
the
and Plosser,
(1982)
Return
in
Econometrica,
Regression,
Nelson,
of
Periodicity
Series,
(1984)
Rates
471-483.
of
Macroeconometric
Political
Economy,
Model
84:
for
the
United
States,
207-237.
G.W.
(1981)
The
flation,
Adjustment
Journal
of
of
Stock
Finance,
52
Prices
36:
to
15-29.
Information
About
In-
stulz,
R.M.
(1984)
Taylor,
Currency
Preferences,
termination
of
of Money,
Credit
Power
Rates
in an Optimizing
and Banking,
16:
Risks,
and
the
Model,
De-
Journal
302-316.
J.B.
(1979)
Estimation
Expectations,
Tobin,
Purchasing
Exchange
and Control
of
Econometrica,
a Macroeconomic
47:
Model
with
Rational
1267-1286.
J.
(1965)
Wasserfallen,
(1985)
Money
and Economic
Growth,
Econometrica,
33:
671-684.
the
Phillips-Curve
W.
Forecasting,
An
Empirical
Rational
Expectations,
Investigation,
forthcoming.
53
Journal
and
of
Monetary
Economics,
-