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department
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THE DYNAMIC EFFECTS OF
AGGREGATE DEMAND AND SUPPLY DISTURBANCE
Olivier Jean Blanchard
Danny Quah
No.
497
May 1988
The Dynamic
Effects of Aggregate
Demand and Supply
Disturbances.
by
Olivier Jean Blanchard
and Danny Quah
*
February 1988.
*
Both authors are with the Economics Department, MIT, and the NBER. We thank Stanley Fischer,
Rotemberg, and Mark Watson for helpful discussions, and the NSF for financial assistance. We are
grateful for the comments of participants at an NBER Economic Fluctuations meeting, and for the
Julio
also
hospitality of the
MIT
Statistics Center.
The Dynamic
Effects of Aggregate
Demand and Supply
Disturbances.
by
and Danny Quah
EkonomicB Department, MIT.
February 1988.
Olivier Jean Blanchard
Abstract
We decompose
the behavior of
GNP
and unemployment
dynamic
into the
effects of
shocks: shocks that have a permanent effect on output and shocks that do not.
£rst as supply shocks, the second as
We £nd
ment; the
dynamic
that
effect
demand
peaks
after a
and a plateau
Thb
is
bump shaped
effect
on both output and unemploy-
year and vanishes after two to three years.
on unemployment of demand disturbances
effect
interpret the
shocks.
disturbances have a
effect of supply disturbances
ment.
demand
We
two types of
is
a mirror
Up
to a scaJe factor, the
image of that on output. The
on output increases steadily over time, to reach a peak after two years
after five years.
"Favorabie" supply disturbances
may
initially increase
followed by a decline in unemployment, with a slow return over time to
unemployits
original
value.
While
this
dynamic characterization
contributions of
series of
of the
demand and supply
is
fairly sharp, the
data are not as specific as to the relative
disturbances to output fluctuations.
Wis find that the time
demand-determined output Buctuations has peaks and troughs which coincide with most
NBER
troughs and peaks. But variance decompositions of output at various horizons giving
the respective contributions of supply and
demand disturbances
instance, at a forecast horizon of four quarters,
contribution of
demand
we End
that,
are not precisely estimated.
For
under alternative assumptions, the
disturbances ranges from 30 to 80 per cent.
Introduction.
GNP
In response to an innovation in
This fact
horizons.
is
of 1%, one should revise one's forecast
by more Ihan 1% over long
documented by Campbell and Mankiw {1987a), building on
earlier
work by Nelson
and Plosser (1982).
What
in the
does this statistical fact
economy,
its
implications would be clear;
disturbances are, and
But
as
if,
is likely,
mean for macroeconomics?
why
GNP
is
their
dynamic
affected by
effects
all
If
there were only one
we would need
of disturbance
do would be to find what those
have the shape characterized by Campbell and Mankiw.
more than one main type
of disturbance, say
and by aggregate supply disturbances, the interpretation becomes more
moving average representation
of output
the two disturbances.
knowing only the aggregate
Clearly,
to
main type
difficult. In
by aggregate demand
that case, the univariate
a combination of the dynamic response of output to each of
is
Campbell and Mankiw show that
effect
does not allow us to recover those two
GNP
is
equally
consistent with, for example, either white noise productivity growth and small, short-lived, effects of
demand
responses.
GNP,
disturbances on
effects of
We
demand
or instead, with serially correlated productivity growth and larger, serially correlated,
disturbances.
are interested in investigating the
on output
behavior
that this
as well as of disturbances that
in
is
the
their estimated univariate representation for
permanent component are
dynamic
effects of disturbances that
have permanent impact. Studies that impose
necessEirily incapable of addressing this issue
the interesting characterization to obtain).
serial correlation
the components are orthogonal at
permcinent component
is
we
feel
strongly
Allowing the permanent component to have rich
all
decomposed
into
permanent and transitory components where
leads and lags, and such that the variance of the
first
difference of the
either impose additional a priori restrictions on the response of output to each of
the disturbances, or use information from time series other than
first
(and
random walk
arbitrarily close to zero.
To proceed, one must
The
strict
however destroys identification to a profound extent: Quah (1988) shows how any general
difference stationary process can always be
tation.
have only transitory impact
approach has been followed
in
GNP, and
estimate a multivariate represen-
Clark (1987a), and Watson (1986)
adopts the second approach and considers the joint behavior of
GNP
among
others. This paper
and unemployment. This approach has
3
ako been taken
mainly in
its
in
Clark (1987b), Campbell and Mankiw (1987b), and Evans (1986); our approach differs
choice of identifying restrictions. Not surprisingly,
we
find
our restrictions more appealing than
theirs.
Our approach
is
assume that neither has a long run
run
We
conceptually straightforward.
effect
eissume that there are two kinds of disturbances.
We
on unemployment.
assume however that the
on output while the second does not. These assumptions are
effect
first
We
has a long
sufficient to just identify the
two
types of disturbances, their time series and their dynamic effects on output and unemployment.
WhUe
the disturbances are defined by the identification restrictions,
we
believe that they can be given
a simple economic interpretation.
We
think of the disturbances that have a long run effect on output as
being mostly productivity shocks.
We
think of the disturbances that have no such long run effect as being
mostly non-productivity-induced demand disturbances.
as well as
We
find this interpretation useful
supported by the results of the paper. For short, (and we hope, not too misleadingly)
these types of disturbances as aggregate supply and aggregate
Under these
identification restrictions
terization of fluctuations:
ment; the
effect
and reasonable,
effect
demand
and
demand disturbances
disturbances have a
hump shaped
effect
peaks after a year and vanishes after two to three years.
respectively.
is
a mirror
on both output and unemploy-
Up
to a scale factor, the
image of that on output. The
in
may
initially increase
unemployment, with a slow return over time to
While
this
contributions of
dynamic characterization
demand and supply
is
its
unemployment. This
the
we
all
NBER
supply
Eifter five
followed by a decline
original value.
fairly sharp,
the data are not as specific as to the relative
disturbances to output fluctuations.
the time series of demand-determined output fluctuations, that
putting
is
dynamic
effect of
disturbances on output increases steadily over time, to reach a peak after two years and a plateau
years. "Favorable" supply disturbances
refer to
economic interpretation, we obtain the following charac-
this
on unemployment of demand disturbances
we
is
On
the one hand,
we
find that
the time series of output constructed by
supply disturbance realizations equal to zero, has peaks and troughs which coincide with most of
troughs and peaks. But, when
we turn
to variance decompositions of output at various horizons,
find that the respective contributions of supply
and demand disturbances are not precisely estimated. For
instance, at a forecast horizon of four quarters,
we
find that,
under alternative cissumptions, the contribution
4
of
demand
disturbances ranges from 30 to 80%.
The paper has
pretation of the disturbances.
of
demand and supply
contributions of
1.
1
discusses identification. Section 2 discusses our economic inter-
Section 4 characterizes the
Section 3 discusses estimation.
disturbances on output and unemployment.
demand and supply
disturbances to fluctuations
in
dynamic
effects
Section 5 characterizes the relative
output and unemployment.
Identification.
We
assume that there
but
may
two types of disturbances
are
long run effect on either
all
Section
five sections.
have a long run
affecting
unemployment and output. The
unemployment or output. The second has no long run
effect
on output.
We
at length in the
next section, we refer to the
first as
As indicated
no
on unemployment,
assume further that these disturbances are uncorrelated
leads and lags. These restrictions in effect define the two disturbances.
and discussed
effect
first heis
at
in the introduction,
demand disturbances and
to the second
as supply disturbances.
We now
and
U
how
discuss
these restrictions identifj- the two disturbances and their
denote the logarithm of
our assumptions imply that the
GNP
first
and the
level of
difference of
dynamic
unemployment, respectively. As
Y AY,
,
and the
level of
will
effects.
Let
Y
be clear below,
unemployment, U, have a
jointly
stationary bivziriate autoregressive representation:
D{L)Xt
where
with
X
denotes the
2x1
vector
=
where D(0)
vt,
\AY
U]'
uncorrelated, with v
=
[vy
vu]
(1) is
circle.
The white
=n
(l)
in lag operators
D{L)
is
noise bivariate vector sequence
of order n,
t;
is
serially
•
Under our assumption that there
equation
I and E{vtvl)
The matrix polynomial
.
determinantal roots outside the unit
all
=
the reduced form of a
are two types of disturbances in the
model which describes the dynamic
economy, demand and supply,
effects of those disturbances
on
output and unemployment. The vector v comprises reduced form disturbances, which are linear combinations
of these
By
model.
demand and supply
disturbances.
specifying the reduced form as above,
We
we have already imposed
a
number
of restrictions on the
have required that neither of the two reduced form disturbances has a long run
effect
on
5
unemplojTnent; this implies
effect
in
turn that neither of the two types of underlying disturbances has a long run
on unemployment. By contrast, since
in equation (1),
it is
the
first
The only
restriction that
disturbances have no long run effect on output. This
demand and supply
The
it is
is
sides of equation (1)
where
is
Ct
by D[L)~^
the vector of
uncorrelated,
=
C{L)vt,
= D{L)-\
where C(L)
e,t
=
We now
Multiplying both
=
so that C{0)
I.
(2)
demand and supply disturbance
A{L)ct
=
demand
loss of generality take the covziriance
e.t]
[cdt
the supply disturbance. Given the assumption that
,
with
e^t
being the demand
demand and supply disturbances
matrix of
e
to be the identity matrix.
disturbances have no long run effect on output implies that aii(l)
show that our model
demand
=
are
The
in equation
disturbances in the output growth equation
just identified
is
from the estimated reduced form. Combining equations
(3) gives:
Xt
Thus, our model
first
Cf
to zero.
We now
and
(3)
disturbances,
In words, the coefficients on current and lagged
must sum
this
demand
gives:
demand and supply
we can without
condition that
The
that
is:
disturbance and
(2)
is
to have
the key assumption that will allow us to identify the
to obtain the moving average representation of the reduced form.
Xt
(3).
is
we have not yet imposed
the other hand, the moving average representation corresponding to our
model
output which appears
disturbances, and to recover interpretable dynamics from the reduced form.
Xt
On
level of
used.
step
first
and not the
we have allowed both reduced form disturbances, and thus both demand and supply,
a long run effect on the level of output.
show how
difference
is
= A[L)et =
just identified
condition states that
if
C{L)vt,
and only
if
where C(0)
=
/,
and E[vtv't)
there exists a unique matrix
=
CI.
S such
that Set
=
^t
and
SS'
=
5
a square root of the covariance matrix of the reduced form disturbances:
is
n, and A{L)
= C[L)S
and aii(l)
=
0.
imposes three restrictions on the four elements of S. The second imposes a linear restriction on the
first
6
column
of S. This
the fourth restriction needed for identification. That restriction
is
<:ii(l)sii
There always
exists a
Identification
is
S
ci2{l)s2i
that satisGes these
=
given explicitly by:
0.
restrictions.''^
thus achieved by using a long run restriction. This raises a knotty technical issue.
is
well-known that,
unique matrix
+
is
in the
It
absence of precise prior knowledge of lag lengths, inference and restrictions on
asymptotic behavior of the kind we are interested
in here
is
delicate.^
More
precisely,
viewed
in
terms of
Fourier transforms, the restriction immediately above operates on a zero measure interval of the range of
frequencies, and so
to
is
especially suspect.
some 5-neighborhood
It is
of frequency zero,
easy however to generalize this restriction to one that applies
and then to perform estimation
in
the frequency domain.
By
imposing that the restriction holds over a frequency band, we would be using an overidentifying rather than
just identifying restriction.
limit of the results
In
its
Under appropriate
regularity conditions,
from frequency domain estimation,
summary, our procedure
is
as follows.
moving average representation, equation
moving average representation
of the
We
(2).
first
We
we can show that our
results are the
as S tends to zero.
estimate the reduced form, equation
then construct the matrix S, and use
demand and supply disturbance model, equation
it
(l),
and derive
to recover the
(3).
^
The proof is as follows. First obtain the Choleski factorization of Cl. Then any matrix .S is an orthonormal transformation of the Choleski matrix. The restriction on the first column imposed by aii(l) =
determines the orthonormal transformation uniquely, up to column sign chcinges. This is the method we
actually use to construct S.
See for instance Sims (1972). We are extrapolating here from Sims's results which assume strictly
We conjecture that similar problems arise in the VAR case. If anything, one would
expect the difficulties to be compounded. Cochrane (1988) has also recognized this.
exogenous regressors.
2.
Interpretation.
Interpreting residuals in small dimensional systems as "structural" disturbances
interpretation of disturbances as supply and
demand
disturbajices
is
always perilous, and our
is
no exception.
We
discuss various issues
in turn.
Granting our assumption that the economy
is
affected
by two main types of disturbances, productivity
and non-productivity-induced demand shocks, one may reasonably argue that even demand disturbances
have long run
effects
on output: changes
affect the saving rate
in the subjective
and the long run capital stock and output. The presence
learning by doing, also raises the possibility that
all
effects
sample.
effects in the
We
may
that of supply shocks, a proposition
we
demand shocks may
consider reasonable, our decomposition
arbitrarily small relative to that of supply, the correct identifying
is
shown
and
of
Even
if
in the technical appendix. This proposition
well have such long run
demand shocks
effects of
are small
compared
to
"nearly correct" in the
is
following sense: in a sequence of economies where the size of the long run effect of
use. This
effects.
well
be sufficiently long lasting to be indistinguishable
believe that
on output. To the extent however that the long run
may
of increiising returns,
demand shocks may have some long run
they do not, their effects through capital accumulation
from truly permanent
discount rate, or changes in fiscal policy
demand shocks becomes
scheme approaches that which we actually
is
the ajialogue in our context to the "near
identification" proposition in traditional approaches to identification.
Granting our assumption that the economy
permanent
effects
is
affected
on output, the other only with transitory
following Prescott (1987), one might instead view the
disturbances,
will
some with permanent and some with
by two main types of disturbances, one with
effects, alternative
economy
as being affected
transitory effects on output.
then identify those two disturbances. While this
is
interpretations are possible:
Our
by two types
of supply
identifying assumptions
not our prefered interpretation,
we
shall briefly discuss
our results under that alternative interpretation below.
This raises a
final set of issues constituting
low dimensional dynamic system.
having different dynamic
effects
problems inherent
Suppose that, there are
in fact
in
estimation and interpretation of any
many
on output and unemployment. Clearly,
some with permanent and some with
sources of shocks, each of
if
there are
transitory effects on output, as well as
many supply
many demand
them
shocks,
shocks,
some
8
with permanent and some with transitory
effects
in aggregate fluctuations, our decomposition
is
likely to
which aU or most supply shocks have permanent
a transitory effect on output.
the
dynamic
demand
effects of
We
in general.
One might hope
show
on output, and
effects
if
they
be meaningless.
on output, and
all
all
play an equally import2int role
A more
or
interesting case
is
able to distinguish correctly
disturbances from those of supply disturbances. This
From
conditions for this separation to occur.
that in
most demand shocks have only
that, in this case, our procedure
is
unfortunately not true
where we establish necessary and
this explicitly in the technical appendix,
is
we
the reasoning in the technical appendix,
sufficient
also see that
there are important sets of circumstances under which our procedure does correctly identify the different
effects of
demand and
must bear a
stable
supply.
dynamic
Roughly speaking, the
Thus
function confronting the different
demand and supply
components
relation to each other, across the individual
individual supply disturbances.
of
different
if
there
demand
is
output and unemployment
demand disturbances and
only one supply disturbance, and there
is
interpretation of shocks
is
we
find to
subject to various caveats.
be not at
We
across the
a stable production
disturbances, our procedure correctly isolates the
disturbances. This particular set of conditions
To summarize, our
in
believe
dynamic
all
it
effects
unreasonable.
nevertheless to
be reasonable, and to be supported by the results presented below.
We now
briefly discuss the relation of
our paper to others on the same topic.
approach relates to the business-cycle-versus-trend distinction
We
first
examine how our
:
Following estimation, we can construct two output series, a series reflecting only the effects of supply
disturbances, obtained by setting
only the effects of
first series,
is
the supply
stationary.
A
demand
By
is
component
demand
disturbances to zero, and a series reflecting
of output, will be non-stationary while the second, the
By
construction, the
demand component,
construction also, the two series are uncorrelated.
in describing
output movements
is
the "business cycle versus trend" distinction.
no standard definition of these components, the trend
that would realize, were
all
actual output around
trend.
A
realizations of the
disturbances, obtained by setting supply realizations to zero.
standard distinction
While there
all
its
precise definition
is
usually taken to be that part of output
prices perfectly flexible; business cycles are then taken to be the dyncimics of
would obviously be tricky but
is
not needed for our argument.
In
models with
9
It is
tempting to associate the
first series
we
construct with the "trend"
component
second series with the "business cycle" component. In our view, that association
are in fact imperfectly flexible, deviations from trend will arise not only
also
from supply disturbances: business
another way, supply disturbances
separately business cycles and trend
is
unwarranted.
and the
If
prices
from demand disturbances, but
cycles will occur due to both supply
will affect
is
of output
and demand disturbances. Put
both the business cycle and the trend component. Identifying
likely to
be
difficult, as
the two will be correlated through their joint
dependence on current and past supply disturbances.
With
this discussion in
mind, we now review the approaches to identification used by others.
Campbell and Mankiw (1987b) assume the existence
of
two types of disturbances, "trend" and "cycle"
disturbances, which are assumed to be uncorrelated. Their identifying restriction
bances do not
affect
unemployment. The discussion above suggests that
between cycle and trend components
is
unattractive;
if
their
this
is
then that trend distur-
assumption of zero correlation
two disturbances are instead reinterpreted
as
supply and demand disturbances respectively, the identifying restriction that supply disturbances do not
affect
unemployment
is
equally unattractive.
Clark (1987b) also assumes the existence of "trend" and "cycle" disturbances, and also assumes that
"trend" disturbances do not affect unemployment but allows for contemporaneous correlation between trend
and cycle disturbances. While
the
dynamic
this
is
effects of disturbances
The paper
closest to ours
is
an improvement over Campbell and Mankiw,
on output and unemployment
nor
demand
on the
of a
as supply
reduced form identical to equation
are difficult to interpret.
(1)
and demand disturbances respectively. By
above, he also assumes that neither supply
disturbances have a long run effect on unemployment, but that both
level of output.
may have
a long run effect
However, instead of using the long run restriction that we use here, he assumes that
supply disturbances have no contemporaneous
way
ways that
severely constrains
that of Evans (1986). Evans assumes two disturbances, "unemployment"
and "output" disturbances, which can be reinterpreted
assuming the existence
in
it still
of achieving identification;
it
effect
on output.
We
find this restriction less appealing as a
should be clear however that our paper builds on Evans' work.
imperfect information, this would be the path of output, absent imperfect information. In models with
nominal rigidities, this would be the path of output, absent nominal rigidities. In models that assume
market clearing and perfect information, such as Prescott (1987), the distinction between business cycles
and trend is not a useful one.
10
Estimation.
S.
One
problem needs to be confronted before estimation.
final
assumes that both the
around given
level of
possibility
is
in
the US. This raises a
that, in fact,
hysteresiB studied by
We
A
second possibility
bances.
The
first,
if
Summers and coauthor
do not consider
is
it
number
unemployment
disturbances, or by both. This might occur
approach.
difference of the logarithm of
first
The unemployment data suggest however
levels.
over the post war period
One
unemployment and the
The representation we use
for
a small secular increase in
This would require an altogether different modeling
(1986).*
that there are, in addition to
which might include changes
demand
in productivity,
in the
disturbances, two types of supply distur-
has long run effects on output but not on
composition of the labor force, changes in
is
substantially
more
is
what we do below, by
using the residuals to estimate equation
unemployment
of
(1).
first
regressing
of results, those obtained
removing
this
this is the case,
is
than what we want to do.
one strategy
A
short cut,
to allow for a deterministic time trend
unemployment on a
The estimated time trend
3.6% over the post war period. As we
If
from demand and two from supply, and
difficult
equilibrium unemployment drifts slowly through time,
unemployment. This
unemployment
further here.
characterize their dynamic effects. This
in
are stationary
example the economy were characterized by the type of
to postulate the existence of those three disturbances, one
if
(l)
permanently affected by either supply or demand
is
search technology and so on, does have long run effects on unemployment.
acceptable
GNP
equation
of issues.
unemployment. The second, which might include changes
would be
in
and
implies an increase in equilibrium
find this estimate to
be high,
estimated trend from unemployment (which
removed" case below), those obtained removing a trend with slope equal
linear time trend
we
we
report three sets
shall call the "trend
to half of the estimated value ("half
trend removed"), and those obtained not removing any trend ("no trend removed").
We estimate
1950-2 to 1987-2.
twelve lags with
the
*
first five
We
equation
We
little
(l)
using quarterly data for real
GNP
and the aggregate unemployment rate from
present the results of estimation allowing for eight lags.
difference Ln the results.
years, as the
We
also
(We performed estimation with
experimented with estimating the system omitting
Korean War era seems anomalous. Again, the empirical
results
remain practically
understand their argument as applying more to Europe than to the post war US.
11
unchajiged.)
to
Evans
The system
is
the
same
as that estimated
We
for a description of the characteristics of the estimated autoregressive system..
interpreting the dynamic effects of supply cind
4.
by Evans, with a longer sample; we refer the reader
Dynamic
The dynamic
effects of
effects of
demand
demand and supply
disturbances.
disturbances.
demand and supply disturbances
and
are reported in Figures 1
correspond to the case where half of the estimated unemployment trend
The corresponding
before estimation of the joint process.^
directly turn to
is
These
2.
figures
removed from unemployment
figures for the other
two cases are
for the
most
part simil2ir, and are given in the Appendix. Differences between the three sets of results will be pointed out
in the text.
The upper part
of Figure
on output and unemployment.
unemployment on the
now with one standard
The
Y
scale for the logarithm of
unemployment
other;
we
is
is
given on the
of the
the scale for
left,
gives point estimates of the
dynamic
effects
dynamic responses, but
hump shaped
effect
on output and unemployment.
depending on the treatment of the unemployment trend. The
fully
The
size of the effect
removed. The responses
effects are consistent
on output and unemployment,
in
in
on output
is
Their
effects of
largest in the case
effects
peak
demand then
when
the trend
output and unemployment are mirror images of each
with a traditioneJ view of the dynamic effects of aggregate
which movements
in
aggregate
demand
build
up
until the
demand
adjustment of
and wages leads the economy back to equilibrium.
Supply disturbances have an
effect
on the
level of
output which cumulates steadily over time.
We have chosen to present the figures for this case in the text not because we
because the results are -not surprisingly- in between those of the two other cases.
in
demand disturbance
return to this aspect of the results below after discussing the effects of supply disturbances.
These dynamic
prices
1
output
effects of a
deviation bands around those estimates.®
decline to vanish after 8 to 14 quarters.
in
dynamic
and U. Figure 2 again provides point estimates
disturbances have a
after 2 to 4 quarters,
gives point estimates of the
right. Similarly, the lower part of Figure
of a supply disturbance on
Demand
1
like it
The
best but simply
More precisely, these figures give the square root of mean squared deviations from the point estimate
1000 bootstrap replications. Thus, the bands need not be and indeed are not symmetric. In every case,
entire pseudo-histories are created by drawing with replacement from the empirical distribution of the VAR
mnovations. These calculations were performed using a combination of RATS and S on a Sun workstation
running UNIX BSD 4.2.
responses
to
demand
CD
d
CNJ
d
C\J
d
--0.5
to
10
30
20
responses
to
40
supply
m
d
LO
d
LO
10
20
30
40
output response to
demand
output response to supply
o
CM
^
CM
q
LO
CO
o
?\
>
'
CD
d
^
o
/•
•
\'^
V,
'
1-
\
^
/•
v.\
;.
'•.....
/•
q
-
•
'l
^-
/• /
_
•
^
C\J
d
-
,'
w \\
>
IT)
/
/•
C\J
1
/
^
\
1
/
d
•
\
d
/
1
10
1
1
1
20
unempi response
1
30
to
1
1
q
o
40
1
,
1
10
1
1
.
20
demand unempi response
o
d
J
30
1
u
40
to supply
CM
d
o
d
CM
d
CM
d
d
CD
d
d
10
20
30
40
10
20
30
40
12
peak response
is
about two to three times the
initial effect aind tiikes
decreases to stabilize eventually at two thirds of
its
quite different: a positive supply disturbance (that
may
on output)
a few quarters,
initially increase
The response
of
Nominal
real.
rigidities
to maintain constant
new higher
Figure
as
can explain
demand does
employment;
unemployment
is,
why
to
its
Eire
in
original steady state value.
by about
reversed after
The dynamic
effects of
response to a positive supply shock, say an increase in
wage
real
rigidities
can explain
which persists
why
increases in productivity can lead to
until real
wages have caught up with the
1
also sheds interesting light
on the relation between changes in unemployment and output known
yields a coefficient of -2.4, the absolute value of
is
a
mongrel
which
coefficient, as the joint
is
output on annual changes
in
known
as
suggests that there
indeed a tight relation between output and unemployment.
implied coefficient
is
is
roughly equal to
2,
its initial
rise or fall; in
coefficient.
disturbances
exactly what
In the short run, output increases,
unemployment and output deviations
are of opposite
of the coefficient
than Okun's
is
close to 4, higher in absolute value
is
higher for supply disturbances than for
demand
expect. Supply disturbances are likely to affect the relation between output
and employment, and to increase output with
we return
of the
unemployment
of the implied coefficient
we
The absolute value
1
the long run, output remains higher whereas by assumption
That the absolute value
is
Under
lower than Okun's coefficient. In the case of supply disturbances,
value. In the intervening period,
and the absolute value
coefficient.
economy. In the case of demand disturbances. Figure
no such close relation between output and unemployment.
unemployment may
Okun's
in
behavior of output and unemployment
of disturbance affecting the
Finally,
effect
suggestive of the presence of rigidities, both nominal
depends on the type
sign,
run
is
not initially increase enough to match the increase in output needed
after a few quarters,
our interpretation, this coefficient
returns to
unemployment
level of productivity.
unemployment
is
effect
five years.
Okun's Law. In our sample, a regression of annual percentage changes
there
in
The
a supply disturbance that has a positive long
are largely over
unemployment and output
productivity, aggregate
a decrease Ln
unemployment
peak value. The dynamic response
slightly. If this increase occurs, the effect is
and unemployment slowly returns
a supply disturbance on
and
unemployment
place after eight quarters.
little
or
no change
in
employment.
to the issue of alternative interpretations of the data.
Suppose
for
example that the
13
two disturbances we have
effects,
isolated were in fact both supply disturbances, the first having only
the second having permanent effects, and that
all
the behavior of output and
unemployment
in
What
markets cleared under perfect information.
interpretation could be given to the djTiamic responses given in Figure 1?
bances should lead to strong intertemporal substitution
temporary
effects
Temporary productivity
distur-
on labor supply, and this clearly could explain
the upper part of Figure
1.
As permanent disturbances
to
productivity cumulate over time, this could explain the dynamic effects of the disturbances on output in the
lower part of Figure
1.
The
anticipation of higher productivity later
would induce workers,
in reaction to
a positive innovation in productivity, to intertemporally substitute away from current work toward work in
the future; this would explain the initial increase and the subsequent decline in
absence of more information, the dynamic
interpretations.
effects
presented in Figure
1
unemployment. Thus,
in
the
are clearly susceptible to alternative
Output fluctuations absent
Demand
CO
C\J
CO
o
CO
CO
r-*
CO
->;};
o
1950
1960
1980
1970
Output fluctuations due
to
1990
Demand,
CD
O
CM
O
o
CD
O
1950
1960
1970
1980
1990
Unemployment
1950
1960
Unemployment
fluctuations
absent Demand.
1980
1970
fluctuations
due
to
1990
Demand,
CO
OJ
o
C\J
CO
1950
1960
1970
1980
1990
14
Relative contributions of
5.
Having shown the dynamic
demand and supply
effects of
disturbances.
each type of disturbance, the next step
contribution to fluctuations in output and unemployment.
and
entails a
comparison of the
We
historical time series of the
do
this in
is
to assess their relative
two ways. The
demand component
of
first is
informal,
output to the
NBER
chronology of business cycles. The second examines variance decompositions of output and unemployment
in
demand and supply
a.
Demand
disturbances at various horizons,
distwrbcinceB and
NBER
buBiness cycles.
estimation of the joint process for output and unemployment, and our identifying restrictions, we
PYom
can compute the demand components of output and unemployment. These are the time paths of output and
unemployment that would have obtained
innovations to zero,
From
we can generate
demand component
and supply components
The time
series
4 (unemployment).
historical
are
of
absence of supply disturbances. Similarly, by setting
the time series of supply
in the level of
unemployment
output
The upper panel
as vertical lines
is
output and unemployment.
long run effect on output, the resulting
stationary.
corresponding to the "half-trend removed" case
By
the
same token, both the demand
is
presented in Figures 3 (output) and
presents the historical supply component, the lower panel presents the
above the horizontal
All three assumptions about the trend in
component
is
in
demand
are stationary.
demand component. Superimposed on
drawn
components
demand disturbances have no
the identifying restriction that
series of the
in the
quite large, approaching
5%
these time series are the
axis,
NBER
peaks and troughs. Peaks
troughs as vertical lines below the axis.
unemployment
yield qualitatively similar results.
The demand
on each side of the origin. (The demand component
is
larger on
average in the "trend removed" case.)
The peaks and troughs
The two
in
of the
demand component
in
output match closely the
recessions of 1974-1975 and 1979-1980 deserve special mention.
NBER
peaks and troughs.
Our decomposition
about equal proportions to adverse supply and demand disturbances. This
estimated values of the supply and demand innovations over these periods:
is
best
attributes
them
shown by giving the
15
The
73-3
73-4
74-1
74-2
74-3
74-4
75-1
Cd
-0.1
0.2
-0.8
0.9
-1.4
-1.2
-3.0
e.
-1.6
-0.1
-1.0
-1.1
-1.4
0.4
1.2
79-1
79-2
79-3
79-4
80-1
80-2
Cd
-0.7
1.1
0.3
-0.4
-0.4
-3.1
e.
-0.6
-2.0
0.5
-0.9
0.9
-0.9
recession of 1974-75
of negative
demand
is
therefore explained
by an
initial string of
Similarly, the 1979-80 recession
disturbances.
negative supply disturbances, and then
is
first
dominated by a large negative
supply disturbance, and then a large negative demand disturbance. Without appearing to interpret every
single residual,
It
may
we
find these estimated sequences of
also simply be that the supply disturbances of these
supply disturbances, such as changes
short-run contractionary
they
as
may look more
demand
like
effect,
in
output and increases
demand disturbances under our
quite plausible.
two periods were different from typical
which dominate the sample.
in productivity,
with both declines
in
They had a stronger
unemployment. As a
identification restrictions,
and are therefore
result,
classified
disturbances.^
Notice that the supply component
deterministic trend.
to the supply component).
demand component
in
output, presented in the upper panel in Figure
Unemployment
GNP. This
of
demand components
is
oil
unemployment
in
unemployment
fluctuations due to
(the time trend
demand correspond
not a
demand
The model
disturbances.
is
added back
closely to those in the
consistent with our earlier finding on the mirror image
responses of unemployment and output growth to
fluctuation in
3, is clearly
exhibits slower growth in the late 1950's, as well as in the 1970's.
It
Figure 4 gives the supply and
time of the
demand and supply disturbances
moving average
attributes substantial
to supply disturbances, again with increases in the late 1950's
and around the
disturbances of the 1970'e.^
This suggests extending the analysis to incorporate information from another variable. That variable
would be chosen so that it itself is likely to react differently to supply and demand disturbances. Obvious
candidates would be the price level or the relative price of raw materials. In Blanchard and Watson (1986),
evidence from four time series is used to decompose fluctuations into supply and demand disturbances. There
the recession of 1975
demand and supply disturbances,
we reestimated the model, leaving
attributed in roughly equal proportions to adverse
is
demand disturbances.
The estimated dynamic
that of 1980 mostly to
In view of those results,
out 1973-1 to 1976-4.
effects of
both demeind and supply disturbances were nearly
identical to those described above.
By construction, the supply component of unemployment is close to actual unemployment for the first
few observations in the sample. Thus, the large decrease from 1950 to 1952 in the supply component simply
reflects the actual
movement
from 1955-2 through the end
in
of
unemployment
our sample.
We
in this period.
found
little
In light of this,
change
we re-estimated the model
in the empirical results.
16
b.
Variance decompositions.
While the above empirical evidence
computing variance decompositions
Tables
1
through
unemployment
the
is
for
suggestive, a
more formal
output and unemployment at various horizons.
3 give those variance decompositions,
under the three alternative assumptions about
trend. These tables have the following interpretation. Define the k quarter-ahead forecast
error in output as the difference betvifeen the actual value of output and
This forecast error
of k quarters earlier.
the last k quarters.
The number
for
one hundred minus that number.
in
output
at horizon
k,k
=
1,
.
.
,
.
forecast from equation (1) as
in
40 gives the percentage of variance of
The contribution
of supply,
similar interpretation holds for the
not reported,
numbers
parentheses are standard deviations. ° The results are given for
unemployment
Our
A
its
due to both unanticipated demand and supply disturbances
is
the /c-quarter ahead forecast error due to demand.
numbers
assessment can be given by
statistical
all
for
given by
is
unemployment. The
three assumptions about the
trend, as there are important differences across these alternative hypotheses.
identifying restrictions impose only one restriction on the variance decompositions,
namely that
the contribution of supply disturbcinces to the variance of output tends to unity as the horizon increases.
All other aspects are unconstrained.
Two main
First, the
conclusions emerge from these tables.
data do not give a precise answer as to the relative contribution of demand and supply distur-
bances to movements in output at short and medium term horizons. The results vary with the treatment of
the
unemployment
30%
trend: the relative contribution of
(no trend removed) to
reason for this
may
82%
demand
disturbances, four quarters ahead, ranges from
(trend removed); eight quarters ahead,
be seen by comparing Figure
1
in the
demand disturbances
are smaller,
still
ranges from
13%
to
50%. The
with the corresponding Figures in the Appendix, derived
under alternative assumptions about the unemployment trend.
effects of
it
and the
effects of
While they are qualitatively
similar, the
supply disturbances are stronger and quicker,
"no trend removed" case.
Even
for a given
assumption about trend, the standard deviation bands are
the "half trend removed" case, the one standard deviation
band on the
large.
For example, in
relative contribution of
Again, these standard errors are asymmetric, and obtained as described above.
demand
Table
1.
Variance decomposition of output and unemployment
(no unemployment trend removed)
Percentage of variance due to demand:
Output
Horizon
Unemployment
(Quarters)
35.8
100.0
(22.3, 37.6)
(22.3, 0.0)
40.5
96.0
(24.8, 35.3)
(25.4, 2.8)
1
2
35.5
89.0
(22.1, 37.3)
(24.8, 7.8)
3
30.1
80.5
(19.0, 38.4)
(22.4, 13.8)
4
8
(
(
4.0, 31.7)
(
1.5, 14.6)
12
2.
(12.8, 36.7)
37.5
8.4
40
Table
50.1
13.0
6.9, 36.7)
(
6.6, 47.3)
(
3.7, 54.1)
29.5
2.9
Variance decomposition of output and unemployment
(Half trend removed)
Proportion of variance due to demand:
Output
Horizon
nemployment
(Quarters)
57.6
95.5
(21.4, 24.1)
(24.8, 3.2)
1
2
62.1
98.7
(22.3, 21.6)
(27.2, 0.9)
56.7
97.6
(21.2, 23.4)
(26.0, 1.4)
3
4
50.3
93.8
(20.0, 24.4)
(23.4, 3.7)
8
23.9
68.1
(10.4, 25.3)
(13.5, 20.8)
12
54.0
15.4
6.3, 20.2)
(
40
(
9.3, 31.9)
(
7.8, 36.3)
48.6
6.4
(
3.1, 6.8)
Table
3.
Variance decomposition of output and unemployment
(trend removed)
Proportion of variance due to demand:
Horizon
Output
Unemployment
(Quarters)
1
2
3
4
8
12
40
86.4
73.2
(22.8, 8.3)
(20.0, 15.2)
89.8
83.7
(23.5, 6.3)
(27.2, 8.7)
86.5
89.9
(24.2, 7.9)
(30.3, 5.5)
81.8
93.2
(25.7, 9.2)
(30.7, 3.4)
49.5
86.6
(19.6, 14.6)
(23.8, 6.6)
34.0
77.2
(14.7, 12.2)
(19.1, 11.4)
15.9
73.5
[17.6, 13.2)
17
30%
disturbances four quarters ahead ranges from
to
49%.
At longer horizons, the contribution
to
75%. Eight quarters ahead,
demand disturbances
of
it still
14%
ranges from
goes to zero by construction;
it
is
typically small after four years.
The second main conclusion
is
that the contribution of
medium term
substantial and more precisely estimated, at short and
disturbances contribute between
80% and 90%
of the variance of
contribution, eight quarters ahead, varies between
"half trend
removed" case ranges from 60%
ahead. There
is
to
demand disturbances
98%
50%
to
to
unemployment
more
is
demand
horizons. In all three cases,
unemployment four quarters ahead;
87%. The one standard deviation band
four quarters ahead, and from
considerable uncertainty as to the relative contribution of
55%
to
89%
their
in
the
eight quarters
demand and supply
disturbances
at long horizons.
6.
Conclusion.
We
have assumed the existence of two types of disturbances generating unemployment and output dynamics,
the
first
type having permanent effects on output, the second having only transitory
effects.
We have
argued
that these two types of disturbances could usefully be interpreted as productivity and non-productivity
induced demand shocks respectively,
we have concluded
which disappears
demand disturbances have
that
after
or, for short, as
supply and demand shocks. Under that interpretation,
a
hump shaped
make
on output and unemployment
approximately two to three years, and that supply disturbances have an
output which cumulates over time to reach a plateau after
disturbances
effect
five years.
We
effect
on
have also concluded that demand
a substantial contribution to output fluctuations at short and
medium term
horizons;
however, the data do not allow us to quantify this contribution with great precision.
Our
identifying restrictions allow us to ask interesting questions of the
that have permanent effects.
component
to be a pure
While we
two
These questions cannot be addressed by studies that
and
specific extensions.
effects of disturbances
restrict the
permanent
random walk.
find this simple exercise to have
especially to validate
dynamic
refine
The
been worthwhile, we also believe that further work
our identification of shocks as supply and demand shocks.
first is to
We
is
have
needed,
in
mind
examine the co-movements of what we have labeled the demand and
18
supply components of
GNP
with a larger set of macroeconomic vaxiables.
confirm our interpretation of shocks.
We
find in particular the supply
positively correlated with real wages at high to
medium
demand component.^" The second
to enlarge the
extension
is
Preliminary results appear to
component
of
GNP
to be strongly
frequencies, while no such correlation emerges for the
system to one
in four Vciriables,
unemployment,
output, prices and wages. This would re-examine the empirical questions in Blanchard (1986), by using the
long run identification restriction developed in this paper.
This extension appears important; one would
expect that wage and price data will help identify more explicitly supply and
the
The methodology and results
sum of correlations from lags
will
be described
-5 to
in
+5 between
innovations in real wages obtained from univariate
demand
disturbances.
a future paper. The statement in the text refers to
the supply innovation derived in this paper and the
ARIMA
estimation.
19
Technical Appendix
This technical appendix discusses further and establishes the claims
made
in the section
on interpreta-
tion.
First,
asserted in the text that our identification scheme
we
approximately correct even when both
is
disturbances have permanent effects on the level of output, provided that the long-run effect of
output
is
The
small.
first
We now
prove
demand on
this.
element of the model, output growth, has the moving average representation
in
demand and
supply disturbances:
AV; = aii(L)edt
where aii(l)
is
+
ai2(i)e,<
Y
the cumulative effect on the level of output
of the disturbance ej.
representation C{L), together with the innovation covariance matrix
fl, is
The moving average
related to our desired interpretable
representation through some identifying matrix S, such that:
SS'
The model
is
identified
=
demand disturbance be
6 instead,
in the
is
where
implied identifying matrix from that which
thus seen to be a finite-dimensional problem, any matrix
that
In the paper,
implies a different identifying matrix S(S). Let |S'((5)-.S(0)|
measures the deviation
is
C{L)S.
we
selected the unique
0.
Let the long-run effect of the
S, this
=
and A{L)
n,
by choosing a unique identifying matrix S.
matrix S such that aii(l)
For each
=
norm
>
6
=
without
we
maxy,*:
loss of generality.
(5'y*:((5)
use. Since the
will induce the
-
5'yfc(0))
;
this
approximation
same topology, which
is all
needed to study the continuity properties of our identification scheme. All of the empirical results
vary continuously
in
S
relative to this topology. Thus,
|5'(<f)
In words,
if
an economy has long-run
-
effects in
5(0)1
correct point estimates.
—
demand
scheme which incorrectly assumes the long-run
it is
sufRcient to
as
,5
—
show that
0.
that are small but different from zero, our identifying
effects to
be zero nevertheless recovers approximately the
20
We
prove this as follows.
orthogonal matrix
V [6)
Since both 5(0) and S(6) are matrix square roots of
the long-run effect of
demand
is
the
A{1;6)
But
recall that the
and
of supply on the level of output,
for
there exists an
such that:
S(6)=S(0)V(S),
Then
Cl,
elements of the
first
(1, 1)
=
of
=
C{l)S(0)V(6).
C(l)5(0) are respectively, the long-run
when the long-run
any V{S), the new implied long-run
effect of
I.
element in the matrix;
C(l)S{6)
row
V[S)V(Sy =
y/here
demand
effect of
is
demand
identifying assumption that the long-run effect of
demand
is
demand
effects of
Thus
restricted to be zero.
simply the long-run effect of supply (under our
is
zero) multiplied
by the (2,2) element of the
orthogonal matrix V[6). As 6 tends to eero, the (2,2) element of V{6) tends continuously to zero as well.
V [S)
But, up to a column sign change, the unique
This establishes that S[6)
S -* 0.
—
5(0), element by element. Hence,
»
we have shown that
is
the identity matrix.
\S[S)
—
5(0)
—
as
»
|
Q.E.D.
Next, we turn to the effects of multiple
vector of
demand disturbances
fdt,
and a
\Ut
where
with (2,2) element equal to zero
B-jk are
dimension
column vectors
as /,,
and Bii{z]
=
J
p,
demand and supply
-
x
1
disturbances: Suppose that there
vector of supply disturbances
V^2:(L)'
—
z)Pii{z), for
z.
pd
"x
1
so that:
B22(Lyj {f,tj
of analytic functions; Bji has the
(1
f,t,
is
some vector
same dimension
as /j, Bj2 has the
of analytic functions fin.
same
Each disturbance
has a different distributed lag effect on output and unemployment.
Since our
VAR
immediate that we
To
results.
method
allows identification of only as
will not be able to recover the individual
clarify the issues involved,
Suppose that there
Suppose further that the
is
first
we provide an
explicit
many
components
of /
=
it
is
(/^ /^)'.
example where our procedure produces misleading
only one supply disturbance and two
demand disturbance
disturbances as observed variables,
demand
affects only output,
disturbances: fdt
whOe
the second
=
{fdi,t, id2,t]'-
demand disturbance
21
affects only
assume that the true model
An
unrestricted
calculations in
calculations
The supply disturbance
unemployment.
VAR
representation corresponding to this data generating process
be found
Formally,
is:
Rozanov (1965) Theorem
may
both output and unemployment.
affects
in
Essay IV
in
10.1 (pp.
Quah
1
-(^
44-48).
(1986).)
is
found by applying the
(A slightly more readable discussion of those
The implied moving average representation
is:
1
^-'-'-
^'V")"'
straightforward to verify that the matrix covariogram implied by this moving average matches that of
It is
the true underlying model. Further, the unique zero of the determinant
the unit circle.
Therefore this moving average representation
is,
is 2,
and consequently
lies
outside
obtained from the vector
as asserted,
autoregressive representation of the true model.
However,
this
moving average does not
satisfy our identifying
has only transitory effects on the level of output.
This moving average representation
disturbances [fdi, jd2>
/.)•
is
We
assumption that the "demand" disturbance
therefore apply our identifying transformation to obtain:
what we would recover
if
in fact
Notice that while the supply disturbance
the data
/,
affects
is
generated by the three
both output growth and
unemployment equally and only contemporaneously, we would
identify e, to have a larger effect on output
than on unemployment, together with a distributed lag
on output. Further a positive demand
effect
turbance, restricted to have only a transitory effect on output,
is
seen to have a contemporaneous negative
impact on unemployment. In the true model however, no demand disturbances
ment
affect
output and unemploy-
together, either contemporzmeously or at any lag. In conclusion, a researcher following our bivariate
procedure
is likely
to be seriously misled
we
disturbances. Having seen this,
and
dis-
explicitly
We
when
in fact the true
underlying model
is
driven by more than two
ask under what circumstances will this mismatch in the
number
modeled disturbances be benign?
state the necesseiry
and
sufficient conditions for this as a
Theorem which
is
proved below.
of actual
22
Theorem:
(i.)
(ii.)
(iii.)
(v.)
(vi.)
fvii.j
=
X,
ft
=
X
Let
5(L)/t
(fat
I
with /d pd X
.
if
A;
(x.)
and
0,
/. p.
X
otherwise
1
;
;
pii, B21, B12, B22 are column vectors ofanaJytic functions;
55*
ranJc
is fuii
A[z)
—
011(2)
Ct
=
—
(1
=
=
\z\
1,
wiiere
exists a bivariate
^^,
{
on
^^1
I
I,
\
], **'j"ti
\
°ni
^12)
^121) '^22
z)qii(2), with Qii analytic on
=
EtfCt-k
I
if
k
=
Q,
Pn
and B21 Pd x
1,
-^12 aiid
^22
P«
X
1
,'
denotes complex conjugation followed by transposition.
*
moving average representation
022(2)/
y 021(2]
(ix.)
=
1,
= [\-z)p,,[z);
B,,{z)
Then there
(viii.)
;
f',t)'
=
Eftft-k
be a bivariate stochastic sequence generated by
and
\z\
<
X, Xt
for
=
scalar functions, det
yl
A[L)et, such that:
^
for all
\z\
<
1;
and
1;
otherwise.
is
\e.t J
In the bivariate representation, e^
if
and only
if
(i.)
-
(vii.)
and unemployment. There
f,,
and
t, is
orthogonal to
B21
=
li.
622
=
I2
•
and
Jags
B12.
axe pj
demand and
p,
supply disturbances;
tlie
(v.)
disturbances have only transitory effects on the ievei of output.
VAR
fd, at aJi ieads
/i5ii,
describe the true data generating process for
condition that allows the existence of a
by our
orthogonal to
there exists a pair of scalar functions 71, 72 such that:
Conditions
demand
is
procedure
is
described by
VAR mean
(viii.)
-
square approximation.
(x.):
the
observed data
in
output growth
expresses the requirement that
Condition
(vii.)
is
a regularity
The moving average recovered
Theorem guarantees that
there always exists such
a representation.
The second part
of the
Theorem
establishes necessary
and
sufficient conditions
such that the bivariate identification procedure does not inappropriately confuse
turbances. In words, correct identification
is
possible
if
and only
if
on the underlying model
demand and supply
dis-
the individual distributed lag responses
23
in
output growth and unemployment axe
mean
across the different supply disturbances. This does not
and unemployment across demand disturbances must be
up
sets of circumstances
there
in general
a bivariate procedure
dynamic responses
that the
demand component
components
in
in the level of
in
interpretations below.
output growth
identical or proportional, simply that they differ
disturbances. Suppose further that each of the
output has the same distributed lag relation with the corresponding
Then our procedure
is
consistent with our 'production function'-based
correctly distinguishes the
dynamic
(i.)
-
(vs.),
the matrix epectriLl density of
of whose elements are analytic, with det
moving average
in
Roianov
C^
for
\z\
X
is
(1965), there exists a 2
<
1,
and Sx
C
=
CC
M whose second column
length
1,
as a vector.
is
A =
Then
the transpose of the Brst
CM
(1
- zHaiip +
-
z)aiia*i
tiat on
be orthogonal to
\z\
=
1:
(a.)
\a,,\^=P[AP[S;
(b.)
\a,2?
= B[,{B[^y;
/,,
and
e, to
on
=
\z\
from
(ix.)).
row of C evaluated
+
A
This represents
1.
VAR mean
its
Form
at z
has been constructed so that on
lai^r
=
|1
- ^r^li
aisa'j
=
(l
-
=
5^1 (S2i)*
ksiP +
to
matrix function C, each
2
=
the
1,
first
2x2
|a22|2
(^li)'
z)^[^ (B^ J*
be orthogonal to f^ at
+
all
+
B[, {B[,)'
+ B[^ (S^J*
5^2 (522)*
leads
and
\z\
square
(demand)
orthog-onaJ
normalized to have
provides the moving average representation satisfying
For the second part of the Theorem, notice that
|1
x
5(z)5(z)*|, .j.
need not satisfy the condition that the
disturbance have only transitory impact on the level of output (condition
matrix
=
given by Sx{i^)
unit variance orthogonal white noise, obtainable
in
approximation. However such a moving average
e<f
demand and supply
effects of
output and unemployment.
reasoning analogous to that in pp. 44-48
as a
misleading, there are important and reasonable
is
demand
unemployment. This cissumption
Proof of Theorem: By
For
in
under which our technique provides the "correct" answers. For instance, suppose that
only one supply disturbance but multiple
is
demand components
X
disturbances, and
to a scalar lag distribution.
Thus even though
By
demand
sufficiently similcir across the different
=
(viii.)
-
(x.).
1:
;
;
.
lags, it is
necessary and sufficient
24
(c.)
aiiaji
=
fi'ii
(d.)
012022
=
-012 (-^22)
(f.)
\a22?
=
[B2S
;
>
B!,^(B'^^y.
Consider relations
(a.),
(c),
and
Denoting complex conjugation of
(e.).
B
by B, the triangle inequality
implies that:
y=i
j=i
where the inequality
is
B21
strict unless
is
a
complex scalar multiple of
Pn
for each z on
=
\z\
Next, by
1.
the Cauchy-Schwasz inequality,
again y/ith strict inequality except
when B21
is
a
complex scalar multiple of Pn,
for each z on
\z\
=
1.
Therefore:
|aiia2il^
where the inequality
is
strict
A
if
and only
similar argument applied to
there exists
Q.E.D.
some complex
lo^iil^
except when B21
strict inequality is a contradiction as
simultaneously satisSed
^
if
(b.),
an
a
^22^,
on
and
\z\
=
1,
complex scalar multiple of fin on
and 022 axe just scaiar
there exists
(d.),
is
•
/"unctions.
Thus
some complex scalar function
(f.)
|r|
(a.), (c),
')i{z)
=
and
scalar function ')2(^] such that B22
=
72
Bi2-
if
tiie
(e.)
can be
=
7i -^ii-
such that B21
shows that they can hold simultaneously
But
1.
and only
if
This establishes the Theorem.
responses
to
demand
CD
O
d
CM
o
10
20
responses
40
30
to
supply
LO
LO
d
LO
d
LO
10
20
/]o
40
30
UA^twployrv^jii^
tre^nd
^^^lov^J,
responses
demand
to
Ln
*
o
o
o
0.1
0.0
-0.1
•
un
o
_
-0.2
-0.3
/•
\
/
\
-0.4
/
\
S
f
-0.5
-0.6
LO
10
20
responses
30
to
40
supply
in
LO
CJ
o
o
LO
O
LO
10
20
30
^n empf oy^vvOvt
40
^'^^-^
^rvtove^^
25
References
Blanchard, Olivier, "Wages, Prices, and Output:
An
Empirical Investigation,"
mimeo MIT,
1986.
Blanchard, Olivier and Mark Watson, "Are business cycles all alike?" in "The American business cycle;
NBER and University of Chicago Press, 1986, 123-156.
continuity and change", R. Gordon ed,
Cochrane, John, "How big
Economy
is
the
random walk component
in
GNP
?",
forthcoming Journai of Political
1988.
Campbell, J.Y. and N.G. Mankiw, 'Are Output Fluctuations Transitory?"
nomics, November 1987, 857-880.
Quarteriy JournaJ of Eco-
Campbell, John and N. Gregory Mankiw, "Permanent and transitory components
AER Papers and Proceedings, May 1987, 111-117 (b).
in
macroeconomic
fluctuations",
Clark, P.K.
November
"The Cyclical Component
,
in
US Economic
Activity,"
Quarter/y Journai of Economics,
1987, 797-814 (a).
Clark, Peter, "Trend reversion in real output and unemployment" forthcoming Journai of Econometrics,
,
1987
(b).
Evans, George, "Output and unemployment
in
the United States", mimeo, Stanford, 1986.
Nelson, Charles and Charles Plosser, "Trends £ind random walks
of Monetary Economics, September 1982, 10, 139-162.
in
macroeconomic time
Prescott, Edward, "Theory ahead of business cycle measurement". Federal Reserve
Quarterly Review, Fall 1986, 9-22.
Quah, Danny, "Essays
in
Dynamic Macroeconometrics", Ph.D.
dissertation, 1986,
series". Journal
Bank
of Minneapolis
Harvard University.
Quah, Danny, "The Relative Importance of Permanent and Transitory Components: Identification and
Theoretical Bounds," mimeo, March 1988, MIT.
Some
Rozanov, Yu
A., Stationary
Random
Processes, 1965, San Francisco: Holden-Day.
Sims, Christopher, "The role of approximate prior restrictions in distributed lag estimation". Journal
of the American Statistical Association,
March
1972, 169-175.
Summers, Lawrence and Olivier Blanchard, "Hysteresis and European Unemployment", Macroeconomics Annual, 1986, 15-78.
Watson, Mark, "Univariate detrending methods with stochastic trends". Journal of Monetary Economics, July 1986,
18, 1-27.
6 10
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