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THE RELATIVE IMPORTANCE OF PERMANENT AND
TRANSITORY COMPONENTS;
IDENTIFICATION AND SOME THEORETICAL BOUNDS
Danny Quah
No. 498
June 1988
massachusetts
institute of
technology
50 memorial drive
Cambridge, mass. 02139
THE RELATIVE IMPORTANCE OF PERMANENT AND
TRANSITORY COMPONENTS;
IDENTIFICATION AND SOME THEORETICAL BOUNDS
Danny Quah
No. 498
June 1988
The
Relative Importance of Permanent and Transitory Components:
Identification
and Some Theoretical Bounds.
by
Danny Quah
*
June 1988.
Department of Economics, MIT and NBER. I am grateful to Olivier Blanchard and Jeffrey Wooldridge
ongoing discussions that have helped sharpen my understanding of the issues here. Conversations with
Robert Engle and Mark Watson have also been useful. I thank the MIT Statistics Center for its hospitality.
*
for
All errors and misinterpretations are mine.
The
Relative Importance of Permanent and Transitory Components:
Identification
and Some Theoretical Bounds.
by
Danny Quah
Ekonomics Department, MIT.
June 1988.
Abstra.ct
The
relative contribution of permiLnent
importance
in the
and transitory disturbances
A number
study of economic fluctuations.
is
a question of considerable
of alternative empirical models have
been proposed and used to estimate the relative sizes of these different components. These empirical
models are typically just-identified or over- identified by assumptions that are open
to dispute. This
paper develops exact theoretical bounds on the relative importance of the permanent and transitory
components focusing only on assumptions of orthogonality and lag
the orthogonality restriction
minimum importance
is
inessential
and that there
is
maximized by setting
proves that for any given difference stationary time
sum
of a series that
residual series.
is
arbitrari/y
The "long run
smooth
(i.e.
series, there
same regardless of the researcher's assumptions
the
permanent and transitory components. The
permanent and transitory components
in
it
to a
its
lag length.
random
walk.
The paper
is
shown
to be
and orthogonality between
theoretical results are applied to
output.
Thus the
and a stationed
permanent component
regcirding lag lengths
US aggregate
tiiat
aJways exists a decomposition into
"close" to being deterministic)
effect" of a disturbance in the
the
The paper shows
a direct relation between the theoretical
of the permanent component and assumptions on
importance of the permanent component
the
is
lengths.
examine possible
1.
Introduction.
What
the relative importance of disturbances that
is
economic fluctuations? What are the dynamic
papers have recently emphasized that these
A
analysis.
partial
tire
transitory versus disturbances that are
effects of
Eire
in
such disturbances? Both empirical and theoretical
questions of considerable importance in macroeconomic
of such papers includes Beveridge
list
permanent
and Nelson (1981), Campbell and Mankiw (1987),
Clark (1987), Cochrane (1988), Diebold and Rudebusch (1988), Harvey (1985), King, Plosser, Stock and
Watson
and Plosser (1982), Watson (1986) and West (1988).
(1987), Nelson
Suppose that aggregate output does contain a unit root.
Under
this
may
maintained hypothesis, one
the fundamental disturbance to an
choose to identify the innovation in aggregate output as
economy (where innovation
projection residual on lagged values of the variable
itself). If
is
used
in
the technical time series sense of
one adopts this view, there
is
then
little left
be said or done: estimating the dynamic response of the economy to this one fundamental shock
an exercise
in
is
in fact
simply
parametrizing the Wold representation.
Alternatively
there
is
to
still
may
maintaining the hypothesized unit root property for output, one
more than one fundamental disturbance that
question to disentangle the dynamic
decompose a unit root time
eff'ects
drives aggregate output.
of the different disturbances.
series (aggregate
output say)
A
It is
conjecture that
then an interesting
convenient place to begin
is
to
terms of two components, one permanent and
in
one transitory.
Some well-known such decompositions
(1981)),
models
is
are the Beveridge-Nelson decomposition (Beveridge
and the unobserved components representation, as
is
that the permanent
component
is
restricted to
in
Watson
(1986).
A
and Nelson
key characteristic in these
be a pure random walk, that
is,
its first
difference
serially uncorrelated.
King, Plosser, Stock and Watson (1987), Blanchard and
Quah
(1988) and Shapiro
and Watson (1988)
have recently considered models of aggregate output where the
serial correlation properties of the
component
to study the
are unrestricted.
The motivation
for
doing this
is
dynamic
eff'ects
bances that wiU turn out eventually to have permanent impact. Thus this breaks the
permanent
of those distur-
artificial distinction
between short run and long run fluctuations. King, Plosser, Stock and Watson use a common trends rep-
3
resentation whereas Blanchard and
Quah and Shapiro and Watson
permanent and transitory components.
of output into "interpret able"
in the
and Nelson), the innovations
construct an orthogonal decomposition
common
In the first
written as a
is
random
speaking therefore the
walk,
common
its first
trend
difference
itself is
(as in
Beveridge
trends are allowed to be correlated with the innovations in
the residual terms so that short run and long run fluctuations are not distinct.
trend
paper
is
Thus while each common
correlated with the stationary component; strictly
not the entire permanent component.
On
the other hand, the
second and third papers explicitly construct the moving average representation of the permanent component.
In both of these cases therefore
Shapiro and Watson, and
in the
(i.e.
both the orthogonal decompositions
common
Blanchard and Quah and
in
trends representations in King, Plosser, Stock and Watson) one can
meaningfully discuss the dynamic effects of both permanent disturbances and transitory disturbances.
contrast, restricting the
in
permanent component
to be a
random walk orthogonal
to the transitory
By
component
simply assumes away any interesting answer to this question.
This paper considers the
relative
effects of certain
kinds of identifying assumptions for the question of the
importance of permanent and transitory components. More
specifically, this
paper uses only orthog-
onality and lag length restrictions to develop theoretical bounds on the relative magnitudes of
permanent
versus transitory components. Thus formally the results below relate specifically to the issue of econometric
identification.
It is
importcint to carry out the kind of "sensitivity analysis" exercises as in this paper. For example,
identifying
permanent and transitory components from a bivariate
VAR
(as in
other researchers to "try a different variable, and see what happens".
(i.e.
achievable)
bounds on how much these
results
similar concerns
The remainder
sition for the
is
The
results
below provide tight
on the relative importance of permanent and transitory
components can change should those other researchers work
somewhat
Blanchard and Quah) tempts
sufficiently hard.
An
interesting paper with
Hasbrouck (1988).
of this paper
is
organized as follows: Section 2
first
provides a general existence propo-
decomposition of a unit root process into (orthogonal) permanent and transitory components.
This section then presents a general approximation result: given an (almost) arbitrary pattern of serial correlation
and an (almost) arbitrary observed unit root process, there
exists a
permanent component
for that
4
process with exactly that pattern of serial correlation. However regardless of the orthogonality cissumption
between the permanent and transitory components or the hypothesized
permanent component
always the same. The results
an innovation
in the
imply that (except
in unrealistic
degenerate cases) economic examples of difference stationary
sequences can be regarded as
made up
of a stationary part
is
and a "permanent" part that
is
close to a deter-
ministic time trend. Section 3 specializes the general theory and provides exact calculations for
components that are
first
is
finite
ARIMA
processes.
Then
hypothesized to be a g-th order moving average,
while the variance of
its
preceding sections to
all
is
trivial
and equal to
US GNP. The paper
the results.
its
if
the
first
difference of its
variance has greatest lower
innovation has greatest lower bound given by 4~'5(0).
in the autoregressive case
permanent
Let 5(0) be the spectral density at frequency zero of the
difference of the observable original process.
proofs of
permanent
effect of
component, the long run
in this section
serial correlation in the
permanent component
bound
The
[q
+
l)~''^5(0),
greatest lower
bound
zero. Section 4 presents the application of the ideas of the
concludes with a brief Section
5;
the Technical
Appendix contains
General Results.
2.
Consider a representative random process Y, assumed to be difference stationary.
being comprised of two kinds of disturbances, one that has permanent
effects,
We
wish to view this as
and the other having only
transitory effects.
Let
W{t —
W
he
a.
stochastic sequence and
The elements
1).
let
AW
denote the
^(*)
—
of a sequence (stochastic or otherwise) will be denoted
by integer arguments
in
parentheses; subscripts will indicate distinct sequences or the elements of a matrix.
and Yi are
there
is
arbitrariness in a 27r normalization,
we
= 2,=_oo
W
Unless stated otherwise,
E' [W^(y)W^(0)') e"'"-'.
for
and
example Yoo
yi(t).
Since
specify explicitly the spectral density matrix to be
the fourier transform of the covariogram matrix sequence: for
Sw{<jj)
Thus
different stochastic processes, with the t-th element of each written as Foo(t)
some
=
Aiy(f)
difference sequence:
first
a covariance stationary vector process,
all
integrals are taken
from
—n
to w. All
proofs are in the Technical Appendix.
Y
Definition 2.1: Let
for
Y
(i)
(ii)
(iU)
is
Yea
A
be a difference stationary sequence.
permanent-transitory (PT) decomposition
a pair of stochastic processes Yoo, Yi such that:
difference stationary,
is
Var(Ay«,), Var(Ari)
=
AY{t)
>
is
covariance stationary;
0;
+ AYi(t)
AYoo{t)
and Yi
in
mean
square,
i.e.,
E
\AY{t)
-
AYoo{t)
-
AYi(t)\^'\
= 0.
Further;
(iv)
If
AYoo
is
orthogonid to Y^ at
all
ieads
and
Notice that the decomposition of interest
stochastic sequences
AV
AY{t) — AYoo{t) — AYi(t)
To
see
why
is
the
If
further (b.)
sum
first
this is
and AY^o
is
a
+ AYi
random
lags, then the
is
PT decomposition
in the sense of
mean
is
said to be orthogonal.
square (condition
should be indistinguishable in that for each
variable with zero
mean and
(Hi)):
t
the difference
variance.
important consider the following (incorrect but frequently used) argument:
difference of a covariance stationary sequence, thus
AY^o and AYi
are orthogonal at all leads
of the individual spectral densities.
Then
(c.)
and
its
lags,
the two
(a.)
AYi
spectral density vanishes at frequency zero.
then the spectral density of their
sum is
the
one can always construct an orthogonal decomposition
6
of
Ay
into
AYoo and AYi: simply
Let the spectral density of Ayoo nowhere exceed that of
Ay, and
spectral density equal to the difference in spectral densities of
Ayoo
is
AY
the spectral densities of
let
of course then arbitrary (so the argument goes)
up to
and Ayoo be equal
choose Yi so that
Ay
at frequency zero.
its first
difference has
and Ayoo- The permanent component
satisfying these
two conditions, and therefore
one can choose Ayoo to have as small a variance as one wishes.
Why
is
the preceding argument incorrect?
done by the above
there
is
no sense
is
in
The key observation here
simply to write the spectral density of
which by doing
so,
rate in consumption of durables
as a percentage of the
in
which US
GNP
is
a decomposition of a
is 1.0.
sum
of durables
random process
Y
all
in
has actually been constructed.
US GNP
is
that one has
However
To
1.5 while the variance of the
Suppose further that the variance of the range
mean temperature on
the
that technically
Ay as the sum of two spectral densities.
a decomposition of
suppose for instance that the variance of the growth rate
is
a representative Malaysian beach
is
in daily
0.5.
Is
see this
growth
temperature
there any sense
consumption and daily temperature? One has not constructed
(say of
GNP
into
permanent and transitory components)
if
one has
simply shown an appropriate "adding up" property for second moments.
Notice that this
is
also
why
the
Wold decomposition theorem
is
proven by constructing a sequence of
projections, cind not simply as a result of factoring spectral densities.
This discussion suggests that one should be suspicious of the
the decomposition Y(t)
=
Nelson have shown that
if
always
well-known however that
exists.
It is
also
Yoo(t)
is
yoo
+ Yi(t)
without
first
a random walk and
Yi at aU leads and lags, such a decomposition
is
if
may
common
practice of simply writing
down
establishing a general existence result. Beveridge
and
perfectly correlated with Yi, then such a decomposition
yoo
is
required to be a
not exist.
We
random walk and orthogonal
to
therefore provide here a general existence
proposition.
To
rule out trivial degenerate Ccises, both
strictly positive variances. Notice that the
permanent and transitory components
permanent component yoo
Nelson and Plosser (1982, pp.155-158) briefly treat a
first
is
are required to have
not required to be a
random
order moving average model for Ayoo in
a discussion to bound the relative importajice of permanent and transitory components in output.
concluded that for
US GNP
walk.
They
the standard deviation of innovations in the permanent component relative to
7
that of the transitory component
is in
the neighborhood of five or six.
The
below
results
may be viewed
as
generalizing their calculations.
Sometimes
it
of interest to
is
impose additional conditions on a
conditions has been proposed by Blanchcird and
is
set of
W
if
each entry
in
77
r]
can be recovered
combination of square summable linear combinations of the components of current and lagged
as a linear
values of
One such
decomposition.
Recall that a white noise vector process
(1988).
for a covariance stationary vector process
fundamental
said to be
Quah
PT
W.
Rozanov
(See for example
For
invertibility.)
Y
(1965); this
a difference stationary sequence,
is
simply the multivariate analogue of Box- Jenkins'
we wiU
identify its innovation with the innovation of
its first difference.
Proposition 2.2 (Blanchard-Quah): Let
such that (AY, X)'
orthogonal
for
if
decomposition
it
and only
exists,
AY
if
then
(Vooi ^1)
it is
is
forY such
and of
full
is
Y
to be the log of
GNP
selecting an
and
Further,
if
demand
it is
minimum importance on
X
is
wUl serve
as a useful
the permanent cind transitory components in Y. However there
benchmark
is
all
PT
actually
Appendix
for the
may
be no
X
loss in
of their paper.)
but
in
doing so
Y
if
The
subsequent discussion.
is
crucial for identifying
in isolating the
arise out of a suspicion
some well-defined permanent component
indicates that there
other series.
may
AY,
the transitory component.
sometimes interest
nent component without using such multivariate information. This
causally prior to
such an orthogonal
well-justified. (See in particular
disturbances, aa well as the results in the
in this Proposition
The above Proposition
fundamental
Granger caused by
Notice that in the decomposition above, information on the second Vciriable
series.
is
X to be the measured unemployment
X such that
Blanchard and Quah argue persuasively however that their choice for
is
and Yi
that the vector of innovations in Y^o
not vice versa will result in a decomposition that places
of a researcher that there
be
an interesting relation between Granger causality and the decomposition
by the proof of the Proposition,
decomposition asserted
X
rank spectral density. Then there exists an
not Granger causally prior to X.
proposed by Blanchard and Quah who take
their discussion of multiple
let
unique.
turns out that there
rate. In particular
be a difference stationary stochastic process, and
jointly covariance stationary,
PT decomposition
(AY, X)
Thus
is
Y
on the part
independent of
and only
if
F
perma-
is
all
other
Granger
8
The next proposition provides necessary and
Proposition
Suppose that
2.3:
Y
is
for all
t.
(ii)
(Yoo,Yi)
is
PT decomposition
a
= Say(^) =
faj
5Ay„(w)
(b)
J S^Y^{w)dw>0,
{Yoo,Yi)
is
if
and only
5'ArAr„(w)
at
Ycx,
and Yi are
PT
„
(
"
_
1.
decompositions.
such
stoc/iastic processes
Suppose further that (AYoo, Ay)'
S=
stationary wit A bounded spectral density matrix
(i)
and that
difference stationary,
E\AY{t) - AY^{t) - AYi[t)f =
that
sufficient conditions characterizing
is
jointly covariance
Then:
if
w =
0;
and
j{SAY„H + S^Y(w)\dw>2ReJS^YAY^Hdu}.
an orthogonal
PT decomposition
(a)
as in
(i);
(b)
as in
(i);
(c)
Say > Say„ = Sayay„
if
and only
if
and
In the following,
we
at all w.
will repeatedly use the
tain proposed candidates are in fact
above alternative representation result to establish that
PT decompositions. We also immediately
cer-
have the following convenient
implication:
Corollary 2.4:
AYco
is
If (Yooi ^i) is
an orthogonal
Granger causally prior
correlation pattern, then
Thus except
AY
is
to
AY.
PT decomposition
If further
AYoo and
for a difference stationary sequence Y, then
Ay
do not have precisely the same
serial
not Granger causally prior to Ayoo.
in degenerate cases the original series
and a permanent component are distinguished by
the Granger causality patterns between them.
We now use these characterizations to derive restrictions on the dynamics of possible PT decompositions.
The
following
Theorem
is
the principal result of this paper.
2.5: Let
Y
be a difference stationary stochastic process, and
let {Yoo, Yi)
be a
PT decomposition
for Y.
(i)
Suppose
definite
[Yoo jYiY
hasa full rank variance covariance matrix, and spectral density matrix strictly positiire
and bounded from above. Let S be a spectral density such that
at
w =
S(uj)
=
Say{^),
^^^^
9
f S{uj)doj >
that
(ii)
AXoo has
Suppose that
that
PT
but
0,
is
otherwise arbitrary. Then there exists a
(Xqo)
-X^i)
for
Y
such
spectral density equal to S.
{Yoo, Yi) is an orthogonal
Say — S >
0,
with equality at
decomposition (Xqo,
This Theorem
PT decomposition
is
-X^i)
for
Y
w =
PT
0,
Y
decomposition for
but
such that
is
,
and
S
Jet
be a spectral density such
otherwise arbitrary. Then there exists an orthogonal
AXqo
hiLS
a general possibility result. In words,
it
spectral density equal to S.
says that for any hypothesized serial correlation
behavior, there exists a permanent component that has exactly that dynamic pattern of correlations, and
such that the deviation from
it
in the observed
Notice that the existence claim
and therefore circumvents the
The Theorem
is
Y
is
covariance stationary.
proven by explicitly constructing the
PT
(meem square) decomposition,
criticisms above.
(together with the preceding Propositions) also makes the following strong identification
statement: regardless of the dynamic structure of the hypothesized permanent component, the long run
effect of
an innovation in the permanent component
is
always the same and equal to the square root of the
spectral density at frequency zero of the observed data itself. This extends Watson's (1986) statement for
unobserved components models (where the permanent component
is
restricted to be a
random walk and
orthogonal to the transitory component) to the case of general permanent-transitory decomposition models,
where neither orthogonality nor random walk behavior
The
results
imply that there are many
imply identical long run conclusions (and
any one
PT
PT
is
assumed.
decompositions
all
in fact conclusions that
whom
of
fit
equally well. While they
all
can be adduced without ever estimating
decomposition), each also provides a different picture of the short run dynamics implied by a
permanent disturbance.
We
have the following immediate corollary:
Proposition
2.6:
position for Y.
Var(AXoo) <
The
Let
Then
Y
for
be a difference stationary stochastic process, and
any real number
6
>
0,
there exists a
PT
let
(Yoo,Yi) be a
PT
decomposition (Xco,.X^i) for
decom-
Y
where
<5.
implications of this Proposition wUl be meide
more concrete
in the
next section. Notice that this
10
result applies
whether or not one seeks only orthogonal
any nonstationary process, one can always imagine
component, where the permanent component
is
it
PT
to
decompositions.
In words,
says that given
it
be comprised of a permanent and a transitory
smooth
arbitrarily
have the variance of
(i.e.
changes
its
arbitrarily small).
But the limiting
ceise
a deterministic trend.
(which
Thus
is
never attained, but can be approached arbitrarily closely)
this result provides
a sense in which those
who have argued
significantly
(1988), Diebold
from other arguments that have been offered
and Rudebusch (1988) or West (1988))
the observed process
Y
itself,
Cochrane (1987) has
result here. Notice
number
also presented
of covariogram terms,
result here
is
it
However
example Clark
imposes no requirements on the dynamics of
but rather applies regardless of those dynamics.
however that
trend stationary models
as
in the literature (see for
simply
that difference
stationary stochastic processes aren't that different from trend stationary processes are correct.
it differs
is
(as
an approximation argument that
may
he correctly emphasizes) his argument
is
at first
appear identical to the
actually one of matching a finite
and speaks to the problem of econometrically distinguishing difference and
when one only has
available a finite data segment.
one that applies to the underlying probability model, and
By
is
contrast the representation
not a problem of statistical
inference.
The Proposition on
relation permitted.
be a
finite
ARIMA
arbitrarily small variance
The next
process.
difference than the trivial
well.
Yoo
is
The
makes no assumptions about the form
section provides exact results
It
bound
when
the permanent component
turns out this imposes a stricter lower
of zero:
bound on the
random walk.
is
required to
Vciriance of the first
however the flavor of the current result carries over to that case as
results of the next section will also put in perspective the results that
restricted to be a
of the serial cor-
have been obtained when
11
3.
Finite
We
for
ARIMA
Components.
consider finite moving average models for AYao, and then finite autoregressive models.
first
mixed moving average autoregressive models follow from the
finite
Let injiov(W^) denote the innovation in the stochastic process
we
identify
(i)
(ii)
Suppose
3.1:
(Vrc, Yi) is a
a moving average process of order
is
results
autoregressive case.
W. As
above,
if
V7
difference stationary,
is
mnov[W) with innov(AW^).
Proposition
AYoo
The
PT
q.
decomposition for the difference stationary sequence Y, and
Then
for
S^y
the spectral density of
AY:
> 4-« Say (0); and
Var(innov(Ayoo))
•
Var(An«) > (g+ 1)-' Say{0).
•
PT
Further, there always exist (different)
differences are moving' average process of order q,
first
arbitrarily close to the theoretical lower
The lower bounds
permitted on AY^o-
bounds
in
(i)
its
permanent components whose
and
(ii).
in Proposition 3.1 are strictly decreasing in the order of the
moving average process
Thus, we can also immediately conclude that letting AFqo be a pure random walk
specification sets g
component with
that have
and whose innovation variances and variances are
maximizes the contribution of the permanent component to
random walk
Y
decompositions for
=
1,
and consequently
Y
,
in the sense of variance
identifies the variance of the
decomposition. The
change
in the
permanent
innovation variance with the square of the long-run impact of a unit innovation (the
sum
of the coefficients).
With the
AYoo
is
result for finite
simple.
A
first
able) theoretical lower
arguments
=
Then simply
to the trivial lower
in
hand, the situation for autoregressive models for
order autoregressive model for AFoo suffices to obtain an (arbitreirily closely achiev-
bound
of zero
on both
as in the proof of the Proposition
and to Var(Ayoo)
gression.
moving average models
^]j!gn
let
bound
C(l)
-^Ay
| 1.
(0),
innovation variance and variance.
above to 5Ay„(0)
where now C(l)
Since a
of zero, the
its
same
first
will
Next, since a purely autoregressive model
is
is
=
|l
—
C(l)|~
To see
this,
apply the same
Var(innov(AFoo))
the projection coefficient in a
first
=
'5Ar(0),
order autore-
order autoregressive model already comes arbitrarily close
be true of higher order autoregressive models.
simply a restriction of a mixed moving average autoregressive
12
model, the result for a
4.
An
first
order autoregression applies directly to general
ARMA
models
for
AVoo-
GNP.
Empirical Application:
This section describes the results of applying the ideas of the preceding sections to examining permanent
and transitory components
in
US GNP.
From Proposition
for aggregate output.
rate of aggregate output
is
First
2.2,
we
establish that there exists an orthogonal
suffices to find
it
some stationary
series
PT
decomposition
X such that the growth
not Granger causally prior to X.
Blanchard and Quah (1988) used the measured rate of aggregate unemployment
in their study;
it
is
convenient to do so here as well (although any such stationary series will do). Marginal significance levels
for testing the coefficients
unemployment lagged
are
on unemployment to be zero
0.86%
,
1.71%
,
and 4.57%
in the projection of
RATS
The
itself
and
and 12-lag bivariate projections. The
for the 4-, 8-
data are quarterly from 1948:1 to 1987:2 and the marginal significance
by the
output growth on
levels are for the F-statistics
reported
econometric package.
discussion following Definition 2.1 warns against simply examining the spectral density of output
growth
for evidence
results
above together with Propositions
on a permanent-transitory decomposition
not misled by doing
2.2
and
Figure 4.1 graphs two
so.
2.3
for output.
and Theorem
diff'erent
However the Granger causality
2.5 assure us that in this case,
we
are
estimates of the spectral density function for
output growth.
One
17), the other
is
an autoregressive spectral density estimate. For the second, the eight- lag autoregressive
representation
is
estimated by least squares, then the reciprocal of the square of the fourier transform of the
is
projection representation
a smoothed periodogram estimate (using a rectcmgular two-sided
is
filter of
length
graphed here. Under standard regularity conditions, both of these are pointwise
consistent for the true spectral density function (see for example BriUinger
the surrounding discussion [1981, pp. 147-9] and Berk
Theorem
1 [1974]).
Theorems
5.6.1
and
5.6.2 cind
Notice that the overall shape of
the spectral density estimates in Figure 4.1 are roughly the same, although differing in details.
Our focus here
PT
is
not the value of the spectral density at any particular
decompositions such as those described
in
fijced
frequency, but whether
Propositions 2.6 and 3.1 can be found for aggregate output.
Recall that these decompositions are such that the permanent
component
is
smooth
in the sense of
having
Figore 4.1
GNPGrovth
atrtoregressive estimate
0.00
0.00
0.25
0.50
V
0.75
a irvction of PI
1.00
Fi^» 4.2
GUP «ad Supply DistarbanoM
—
—
0.00
0.00
0.23
0.50
Frgqgency as « fraction of PI
0.75
1.00
GHPgrovth
Supply
Figore 4.3
GUP
Grovtlt, Hill ItutoTBtiott Varitiwe
—
—
-
Artaal GUP
Order
1
Order 3
Order 5
0.00
0.00
0.25
0.50
Txtfptcucf tB a fraction of PI
0.75
1.00
—
Order 10
--
Order 15
Figore 4.4
GNP GrovUc Uia Vuiafioe
3.00
T
—
—
2.50-
-
ActaalGNP
Order
1
Orders
Order 5
0.00
0.00
0.23
0.30
Freqaeacy as a fraction of PI
0.73
1.00
—
Ord«rlD
-
Order 13
13
"small" variance.
Figure 4.2 presents the estimated spectral densities for the growth
of output (in the terminology of Blanchard and Quah). These are
in
output and in the supply component
smoothed periodogram estimates
specified otherwise, all spectral density estimates hereafter are obtained
window
using a rectangular two-sided
realization"
of length 17).
Due
by Blanchard and Quah.
and
for the
by smoothing the periodogram
The supply component
is
calculated from "historical
to sampling error, the estimates of the spectral densities at
they are 1.83 and 1.76 for the
frequency zero of these two stochastic sequences are not exactly equal:
original data
(unless
supply component respectively. For the purposes of graphing the spectral densities
in Figure 4.2, that for the
supply component
is
scaled
upwards so that the spectral
densities coincide at zero.
Notice that even in the presence of sampling error and after upwards scaling, except for one or two ordinates
the estimate for the supply component never exceeds that for the original data. This
is
an implication of
the orthogonal nature of the supply-demand decomposition in Blanchard and Quah.
The supply component
as calculated
by those authors
not "trivially implied by their assumptions"
(I
is
evidently not special in any way, and
certainly
have heard this assertion made a number of times). The results
that they actually obtain derive precisely from their use of the
in the
is
unemployment
system that they estimate. The use of the unemployment rate
series
rate as the additional indicator
is
sensible for reasons that are
described in their paper.
Figures 4.3 and 4.4 again graph the estimated spectral density function for output growth. Superimposed
on
this
and scaled to coincide
at
frequency zero are the theoretical spectral densities of moving average
permanent components attaining the lower bounds described
spectral densities of
minimum
Figure 4.4 graphs those of
Without
in Proposition 3.1.
innovation variance permanent components (part
minimum
variance permanent components (part
explicitly developing precision properties for these estimates,
PT decomposition where
the permanent
component
is
a pure
random walk
(ii)
it is
(i)
Figure 4.3 graphs the
of the Proposition), and
of the Proposition).
difficult to
say
if
an orthogonal
(say) "fits" aggregate output.
appropriate condition in these graphs would be that the spectral density at zero must also be the
value for the spectral density everywhere.
exists,
However we note that
if
some orthogonal
PT
The
minimum
decomposition
then there necessarily also exists another such decomposition with a permanent component that
is
14
even smoother
in
5.
(in the sense of
having a smaller innovation variance or Vtiriance): this follows from the way
which richer moving average structures collapse
in
towards the horizontal axis
in
Figures 4.3 and 4.4.
Conclusion.
This paper has considered the general problem of decomposing a difference stationary process into the
of a
sum
permanent and a transitory component.
It is
by now well-known that unit root and trend stationary time
How
implications for classical econometric inference.
dynamics
We
smooth
of
series
data generate drastically different
does this difference carry over onto the observable
economic variables?
have shown that without lag length restrictions, the permanent component
in the sense of
having
its
changes be of arbitrarily small variance. Thus there
the observable dynamics in a unit root sequence
is
may
is
be arbitrarily
a sense in which
close to that in a trend stationary sequence.
"long run effect" of a disturbance in the permanent component
is
The
precise
always identified and identical, regardless
of lag length and orthogonaUty assumptions.
We
have also derived exact lower bounds on the variability in the permanent component when that
permanent component
the permanent
is
restricted to be a finite
component
is
a
ARIMA
random walk maximizes
process.
We
have shown that the case when
the importance of that permanent
component
for
explaining the observed data.
In application to
preted as the
sum
US
of a stationeiry
supply component that
components:
it is
is
GNP
can be inter-
arbitrarily
smooth. The
aggregate output, the theoretical results here indicate that
component and a permanent component that
calculated by Blanchcird and
Quah
is
neither the smoothest nor the most volatile.
is
seen to be one of
many
possible
permanent
15
Technical Appendix.
Proof of Proposition
representation
= C
I
I
By
2.2:
*
the
1,
I
the non-negative integers; C(0)
Wold Decomposition Theorem,
is
wiiere
C
is
V
lower triangular;
D = CV
such that
has
see that
r)
its (1,2)
fundamental for [AY, X)' and
is
,
By
to the identity.
is
a unique moving average
iias
denotes convolution; and
*
=
e
(£i,£2)' is serially
fundamental for {AY, X)'. There exists a unique
is
entry
sum
(^^\=D*V'e=D*r),
we
j
an array of square-summable sequences, zero except on
uncorrelated with the identity covaxiance matrix, and
orthogonal matrix
„
I
to zero. Writing
where
=V'e,
r?
serially uncorrelated with variance covariance
the construction, such a {D,r]) pair
is
unique,
matrix equal
no other pair admits simultaneously
i.e.,
a (1,2) entry in the array of moving average coefficients that sums to zero, and a fundamental disturbance
vector that
2,-Ci2{y)
is
=
contemporaneously uncorrelated. Identify AYoo
0,
AYi has
be
Dn
AYi
* rji,
AY is Granger causally prior to X.
moving average representation [AY, X)' =
5
*
wiiere the (1,2) entry in
i/,
be identically zero; the variance covariance matrix of the serially uncorrelated v
that the (1,2) entry in
But then AYi
is
C
prior to
is
is
identically zero, so that the orthogonal matrix
identically zero.
Proof of Proposition
at
(i)
w =
0.
SaYiAY„ = S^YAY„ - S^Y^ =
SaYoc
Var(Ayi)
Since
* ^2-
identically zero
AV
if
is
B
can be taken to
arbitrary. It then follows
above
is
simply the
=
=
0-
Suppose
and thus no nontrivial
By
0.
PT
Dn
when
AY
is
* rji)
Thus we have established
=
J
[5Ay„(w)
+
=
^
\
S/\Yi
Say^c
(a).
•
and AYi
Say„,
+ ^ay -
(or
Cn
D12
and C12
* r/sj
have
Q.E.D.
Since Yi
is
covariance
this implies that at
2ReS'ArAyco- At w
Further since VarfAFi)
SayH
PT
not Granger causally
linear combination of
decomposition for Y.
the inequality \S^YiAYa^
Next, recall that S^Yi
S^Y.dij
J
(1^00,^1) is a
identity.
Granger causally prior to X, a
consider
i.e.,
V
is
This implies that
and thus are uniquely determined by the above construction.
2.S:
=
S^y —
Thus
then follows that both AY^o (or equivalently
It
strictly positive variances,
stationary, S^Yi (w)
0.
Next suppose the opposite,
exist.
X. Then C12 cannot be
then becomes
=
identically zero so that Var(y"i)
decomposition does not
D12
to be
spectral density that vanishes at frequency zero, so that then Yi itself can be
chosen to be covariance stationary. Suppose then that
there exists a
to
>
- 2Re5ArAr„(c^)] dw >
0,
0.
=
w =
0,
0, this
16
=
Next Var(Ayoo)
at
w =
0,
S^Yi
stationary-
for Y. (a)
By
/ S^^y^ (w) >
= 2S^Y„ —
both
(b),
Suppose
orthogonality, for
since for all u,
S^n =
PT decomposition.
(yooi Yi) is
w,
=
2Re5'Ar A^co
2
[5Ay„
^
^'i)
=
S'Ar^]
Thus Yi can be chosen
0.
PT decomposition
an orthogonal
S^YiAY^ — Sayay„ - Say„ =
-^Ay
- Say„ >
In addition, if (c)
By
2.4:
AYi
then
is true,
Y.
for
is
of
AY
This
is
orthogonal to AVoo at
By
(b).
all
AY. Next
(i)
(1^00,^1) is a
Jeads and iags, so that
Proposition 2.3, (Yoo,Yi) being an orthogonal decomposition for
(11
S^y =
(1
+
V')^Aroo
/
-^"^
)
some
— ^ay„- But
the projection of AYoo on lead, current and lag values of
+ if>)~^. When
AYoo
^Jid
AY
do not have exactly the same
thus the coefEcient sequence itself
is
two-sided and symmetric.
Consequently
Y
implies
reai symmetric
then the projection
•S'^y^ /•^'aVoo
is
is
~
AY has coefficients with
nontrivial
is
AY
^^
Granger
serial correlation patterns,
But then the coefEcient sequence has a fourier transform that
varies over (— t, +7r].
By
(c).
-^Arco- Farther
simply the identity however and therefore places zero weight on lead values. Thus AYoo
fourier transform {l
is
ip
and real, and
not Granger-causally
Q.E.D.
prior to AYoo-
Proof of Theorem
2.5:
(i)
Choose
be the (one-sided) sequence of Wold moving average coefEcients
to
b
for the spectral density
S/S^y^,
i-^-,
\b\^
w =
=
Xoo
=
0,
PT decomposition
Sayay„ =
and
Then
to be covariance
only remains to establish
It
on lead, current and lag values of AYoo has coefHcients whose fourier transform
causcdly prior to
(a).
Q.E.D.
zero at frequency zero,
ip,
.
which implies that
0,
that the joint spectral density of (^00,^1)' can be written as S^^y^,
positive definite function
(Foo ^i ) is a
Conversely, suppose (a)
0, (c.) follows.
an orthogonid decomposition.
Proof of Corollary
—
To prove the converse, suppose
(b).
AYi and AFoo Aave strictly positive variances. Thus
(yooi
all
and so we have established
0,
we have J^y
^[j]
1-
S'et
=
b *
jointly covariance stationary vector sequence
5ax„
\Sayax„
•S'^y
Yoo,
and Xi
[AXoojAY)'
\^(b 0\(
(
Say
J
Notice that since
S-
= Y—
S{(jj)
=
5'Ay(w)
= 5Ay„(w)
at
The spectral density matrix of the
Xoo-
is:
S^Y^
\ (l'
.
iJ\Sayay^
^(\b?SAY^
S^y
J
\0
\^f,
S
\0
0\
l)
\
'
Vi5AyAy„
Thus AXoo has the required dynamics in
for Y, this matrix
is
seen to have
all its
Say„ J
S. Further since
b
elements equal at
=
w
1
Vi-^AyAy^
atw
=
0.
=
0,
and
S^Yo. )
(Yoo, ^i) is
Finally, the
a
PT decomposition
determinant of this spectral
17
density matrix
is
S S^Y -
\b\''
\S^Y ^yJ^
=
-
det
\b\^
(
Z^'"'X'^AYAYo^
)
JAY J
which implies:
Because
rank.
in addition
By
Proposition
decomposition for
as SaYo,
/ S{w)
I
b,c square
1
1
Y
/
,
doj
>
0,
is
is
))
fo^
some
V"
the requirement that its
exist
Since (Foo, Yi)
(ii)
whose
(I
c\ „
lj*^^^"l,l
first
symmetric positive
Ay)
difference be b*
definite functions
AYoo
=
+
'^x^^^Ya^i^
Sx,
+
b, c
—
mean
+
AF
,
i>(tL))
c *
)'
=
PT
can be written
at
Ay. By
w =
0.
For
construction,
4>x, with V'x(w)
V")
—
1
[
=
-
..
square, Xoo{b,c)
0.
it is
1
•
is
uniquely defined by
= Y — X^o (&) c) By Proposition
.
necessary and sufficient that there
w =
at
,
= S~^{Say —
=
1
Let Xi(b,c)
PT decomposition,
symmetric positive deSnite function, vanishing at w
that their fourier transforms
AFoo
an orthogonal
f'b'
ip) [c'
+ c * AY.
above can be represented as Sx
Theorem, and deBne V'x
\
I
1
difference vanishes in
first
b *
is
is full
and has spectral density m.atrix given by:
{Xoo{b,c),Xi[b,c)) to be an orthogonal
matrix of (AXqo,
real
decomposition for Y.
define AXoo{b,c) to be the process
jointly covariance stationary,
to a stochastic process
2.3, for
PT
positive definite real symmetric function with
b
Up
a
Proposition 2.3 implies that the spectral density matrix of ( AVoo
summable sequences,
(AXqo, Ay)'
/
we then have that the variance covariance matrix of (AXoo, Ay)'
(Xoo,-^i)
2.3,
^AY
\-^AYAX„
J
0,
such that the spectral density
Sx
Set
S), which
is
Select square
to
S
in
the statement of the
therefore guaranteed to be a
summable sequences
b,c such
satisfy:
{S/S^Ya,)
'
for
V'x/V'
w ^
0,
and
w =
at
0;
c=(l + rP)-'[{S/SAYj-b].
Notice that
b is
restricted only to the extent that its
Since the right hand side
is
real
modulus on each frequency
symmetric and positive
definite,
it
is
satisfies the
a spectral density.
above
equality.
Thus
b
can be
chosen to be the (one-sided) Wold moving average coefficients corresponding to that spectral density.
straightforweird to verify that these sequences b,c imply that:
VO
1 ;
U
l
+ rpj\c*
l)
\Say^J\1
1
+ rPxJ
It is
18
Thus {Xoo(b,c),Xi{b,c))
Proof of Proposition
is
no greater than
PT decomposition.
an orthogona.1
is
2.6: It
always possible to choose
is
Q.E.D.
S
in the
Theorem
to be such that
^ J S[(j)da}
Q.E.D.
S.
Proof of Proposition
3.1: Since
AYoo
AY„ (f) =
a moving average process of order
is
^
C{j)innov{AY^){t -
for
j),
aU
q,
t,
3=0
with C(0)
=
1,
X^y=o ^(j)^''
^ ^"^ Nl
7^
^
1-
Since Yoo
a permanent
is
Var(innov(AF<„))
=
component
for Y,
Say{0).
y=o
Thus the lower bound on Var(innov(Ayoo))
is
obtained by solving:
sup
c
y=o
1
=
subject to C(0)
j=0
^
1,
may
Recall that any such polynomial Yl'i=o ^(j)^^ above
Z)'=o^(j)^^
= ny=i(l +
C'(y)'=^).
=
^'^^ 1-^0)1
if
not
is
equivalent to the maximization of
each
real.
j.
Since \'^C{j)z'\'^
+
\1
+
l.y
=
l.
^
for
|2r|
<
1.
be written as the product of q monomials:
2,...,g appearing in complex conjugate pairs
+ ^U)^)? = n'=i
D(y)p, for each j
=
|1
+
1,
2,...,g.
D{j)z\^,
its
is
L)'' innov{AYoo) ,
then 4~'
where L
•
maximization
1
for
5Ar(0), and results when the moving average representation for AYoo
is
the lag operator.
inf
(Co-'
=
=
1,
E
3=0
<^0>''
^
Next, the lower bound on Var(Ayoo)
cy^Tcijf
3=0
for
\z\
<
1,
and
|
J2 C(:?')l^<^^ =
3=0
1
The lower bound on the
solving:
subject to C(0)
=
at z
This occurs at D{j)
Therefore the solution to the optimization problem attains the value 4'.
innovation variance
(l
|n'=i(l
^
C(j)z'
Say{0).
is
is
obtained by
19
Substituting out for
cr^,
we need
tions above. First notice that
minimize iYl'i=o ^U)'') I
to
^^_Q
< (Sy=o
C(y)
< it\^u)i\\
<
I^O)!)
Sv=o
•
^iJ)
subject to the boundary condi-
Next apply the Cauchy-Schwarz inequality:
l±\cu)A lE^M =
•
lE^wl
•(^+1)-
3=0
Therefore we have:
E^ur]/
Notice that C{j)
this satisfies the
cal lower
Ey=o
bound
=
1
for j
=
0,1,
.
.
.
,q,
boundary conditions
is
>(g + i)-^
3=0
iy=o
achieves this lower bound, and because Yl'',=o ^^
as well.
Thus Var(Ayoo) >
evidently approached arbitrarily closely by finite
A^innov(Ayoo)(f
-
j),
where A
<
1.
Q.E.D.
(?
+
~
limATi
1)~^S^y{0), and
^~
i-xl
>
this theoreti-
moving average processes of the form
20
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M/<R
8?991
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