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of economics
THE IMPORTANCE OF PERMANENT AND
TRANSITORY COMPONENTS:
IDENTIFICATION AND SOME THEORETICAL BOUNDS
Danny Quah
No.
498
May 1991
Revised
massachusetts
institute of
technology
50 memorial drive
Cambridge, mass. 02139
THE IMPORTANCE OF PERMANENT AND
TRANSITORY COMPONENTS:
IDENTIFICATION AND SOME THEORETICAL BOUNDS
Danny Quah
No.
498
Revised
May 1991
MIT Working paper
The
version
Relative Importance of Permanent and Transitory Components:
Identification
and Some Theoretical Bounds
by
Danny Quah*
08
*
MIT
May
Economics Department and
1991
NBER.
I
thank Olivier Blanchard, Larry
James Stock, Mark Watson, and Jeffrey
Wooldridge for discussions. Victor Solo and three anonymous referees made helpful
comments on ecirher versions of this paper. Finally, I thank Lars Peter Hansen who
through discussions and correspondence helped sharpen the discussion considerably.
All errors and misinterpretations are mine alone.
Christiano, John Cochrane, Steve Durlauf,
The Relative Importance
of Permanent and Transitory Components:
and Some Theoretical Bounds
by
Identification
Danny Quah
MIT
Economics Department and
08
May
NBER.
1991
Abstract
Much macToeconometric
discussion has recently emphasized the economic
significance of the size of the
permanent component
a iarge iiterature has developed that
measured,
tries to
in
GNP. Consequently,
estimate this magnitude
essentially, as the spectral density
of increments in
—
GNP
frequency zero. This paper shows that unless the permanent component
random wedk
at
is
has been misplaced: in general, that quantity
does not identify the magnitude of the permanent component. Further, by
developing bounds on reasonable measures of this magnitude, the paper
shows that a random waJk specification is biased towards estabhshing the
a
this attention
permanent component as important.
1.
A
Introduction
large literature has recently developed that purports to estimate the
GNP,
magnitude
of a time series's
permanent component.
influential papers
by Nelson and Plosser (1982), Watson (1986), Campbell and
Mankiw
For
this literature includes the
(1987), and Cochrane (1988). Using non-parametric reasoning Cochrane
(1988) has, further, developed a measure that allows arbitrary serieJ correlation
in the tramsitory
there
is
components. All
this research,
however, has Jissumed either that
only one disturbance perturbing the time series under study, or that the
—
underlying permanent component has very special structure
for instance, that it
has sericdly uncorrelated increments.
More
recently, other researchers, e.g., Shapiro
and Watson (1988) and Elan-
chard and Quah (1989), have argued that the economic forces underlying
GNP
movements imply that multiple disturbances perturb
permanent component has
rich dynamics.
GNP
In this paper
and that the underlying
I
show that under these
circumstances the measures that had been earlier proposed, in fact, cannot identify
the magnitude of the permanent component. In particular,
derlying permanent
arbitrarily
component
smooth, so that at
dominates that
I
prove that the un-
in every integrated time series
can be taken to be
all finite
series's fluctuations
horizons
can, further, be chosen to be uncorrelated at
smoothness
is
it is
the trajisitory
component that
—these permanent and transitory components
leads and lags.
all
This arbitrary
achievable regardless of the values taken by Campbell and Mankiw's
"long-run effect of a shock" or Cochrane 's or Watson's "size of the
component"
for that integrated time series.
This proposition therefore casts
rious doubt on the usefulness of those measures for cissessing the
time series's permanent component.
random walk
Without
se-
magnitude of a
explicitly identifying the underly-
ing economic disturbances, a researcher cannot quantify the relative importance of
permanent and transitory components.
To
known
see the implications of this arbitrary smoothness result, recall
assertions in the literature: Nelson
Mankiw
(1987) have
serving that
criticized traditional
US GNP might
at
and Plosser (1982) and Campbell and
be better characterized as integrated rather than trend
most transitory
— both
Mankiw
(1987), separately, put forward
claims based on univariate characterization of
ries properties: one, that
that disturbances to
GNP's permanent component
GNP
macroeconomic
effects of disturbances to output. In contrast. Nel-
son and Plosser (1982) and Campbell and
two claims
well-
models of economic fluctuations by ob-
stationary. According to these investigators' reaisoning, traditional
models predict
some
is
GNP's time
se-
highly volatile and, two,
have significant and long-hved
effects.
The accuracy
of the univariate time series models that these investigators used remains controversial (see, e.g.,
Watson
(1986),
Cochrane (1988), Perron (1989), and Christiano
and Eichenbaum (1990)). But, according to the
results in the current
paper the
accuracy of those measurements turns out to be irrelevant for whether permanent
-3disturbances are important for
to
GNP
have long-Uved
integrated and can
still
have
its
GNP fluctuations and for whether most disturbances
The
eflfects.
show
analysis below shows that a time series can be
significant persistence in its innovations,
by
fluctuations dominated
but nevertheless
Thus
treinsitory disturbances.
this
paper
mcikes exphcit an important general message: Because studying the univariate time
series characterizations of a variable leaves unidentified the sources of that variable's
fluctuations, without additional ad hoc restrictions those characterizations are
pletely uninformative for the relative
trcinsitory
components.
To sharpen understanding
explicit lower
bounds
for
of the arbitrary smoothness property,
is
restricted to
derive below
increments
ARIMA
be an
Choosing the permanent component to be a random waUc
serially uncorrelated
I
two natural measures of the importance of a permanent
component when that permanent component
quence.
com-
importance of the underlying permanent and
—
i.e.,
se-
to have
—turns out to maximize both these lower bounds.
This therefore makes precise a sense in which a random walk specification for the
permanent component biases the analysis towards finding the permanent component to be important.
Section 2 provides rigorous statements of our theoretical decomposition results
in a general setting; Section 3
restricted to be
ABIMA
identifiability of arbitrary
does the same for permanent components a priori
processes.
While those two sections consider the
laick
permanent and transitory components, some positive
sults are in fact avaUable. Section 4 considers
of
re-
permanent and transitory components
subject to certain orthogonality and informational restrictions.
I
show that under
those restrictions these components are unique; further, whether such components
exist can
be tested for by a Granger-causality characterizaiton. The paper then
concludes with Section
The reader
5.
will notice that the analysis of Sections 2
through 4 use moment-
matching reasoning to construct permanent and transitory components. Sometimes
a researcher might wish to go beyond
this, i.e., to
construct what
—
in the termi-
nology of stochastic differential equations
—are
called strong sense solutions rather
than only weak sense ones. The Appendix provides calculations that accomplish
The Technical Appendix
this.
contains
all
the proofs.
General Results
2.
This section shows that every integrated sequence admits a decomposition into
permanent and
components with the increments of the permanent com-
trjinsitory
ponent having arbitrarily small variance
—
this
decomposition
the permanent and trcinsitory components are uncorrelated at
First,
estabhsh notation:
=
py
is
{
A random
integrated or difTerence stationary when
AW{t) = W{t) — W{t —
use the
method
in
Doob
1) is
sequence a
f
leads and lags.
t
its first
}
difference or increment
covariance stationary, but
W
itself is not.
We
(1953) pp. 461-463 to always extend definition of covari-
ance stationary sequences over
defined only for integer
all
sequence
W{t), non-negative integer
def
when
possible even
is
>
random walk
1.
all
the integers, even
if
those sequences are initially
For the purposes of this paper, we
if its
ccdl
an integrated
increments are serially uncorrelated (not necessar-
Elements of a sequence, stochastic or otherwise, are denoted by integer
ily iid).
arguments
in parentheses; subscripts indicate either distinct
ments of a matrix. Thus,
for
sequences or the
ele-
example, Yi and Yq are different stochastic sequences
with the <-th element of each written as Yi{t) and Yo{t). Without loss of generality, all
is
some
sity
W
arbitrariness in a 2ir normalization,
we
mean
zero.
Since there
explicitly specify the spectral den-
matrix to be the fourier trsuisform of the covjiriogram matrix sequence:
is
a jointly covau'iance stationary vector sequence,
5w(w)
— TT
covariance stationary sequences are taken to have
=^
Ej~
-oo
^ [Vi^iJWW] e"'"'-'
Finally,
its
spectral density matrix
make
precise the decompositions that
is
aU integrals below are taken from
to X.
Next,
When
we
are investigating:
-5Definition 2.1: Let
Y
(PT) decomposition
Yo
(iii)
leads
and
y;
=
Y{t)
yi(<)
+
lags, then the
Given a
for
for
Y
covariance stationary;
is
and
PT
A permanent-transitory
be an integrated sequence.
is
(ii)
yo(<).
a pair {Yi,Yo) such that:
Yi
Further, if (iv)
is
AYi
is
uncorrected with Yq at
call
Yi a
a transitory component.
permanent component
Permanent
indicates only that disturbances to Yi have long run effects on
y
when
(i)-(iii)
addition (iv)
uncorrelated.
also say that (yi.yo)
decomposes
of 2.1 hold, and that (yi,yo) orthogoneJly decomposes
(ii)
rules out trivial cases. For instance,
might be natural to
set Yi
toY But
.
definition then sensibly asserts that
for
We wiU
in this context
Y, not that the
Y when in
is true.
Condition
it
sericilly
all
said to be orthogonal.
decomposition (yijVo) for Y,
increments of Yi are
integrated and
is
Var(Ayi(<)) and Var(Ayo(t)) are strictly positive;
PT decomposition
similarly, call Yo
(i)
if so,
there
(Yi,Y — Yi)
is
when
y
is
a
random walk,
no transitory component; the
= (y 0)
is
not a
PT
decomposition
y.
From Beveridge and Nelson
(1981),
we know
that every integrated sequence
admits a decomposition into perfectly correlated permanent and trcinsitory components where further the permanent component has serially uncorrelated increments.
Watson (1986) and Cochrane (1988) have considered models where the permanent
component remains a random walk, but has increments that might be imperfectly
correlated with the transitory component. In
all
these, the variance of increments in
the permanent component can be identified from the spectral density of increments
in the original sequence (see, e.g.,
By
contrast, Shapiro
Watson (1986)
or Cochrane (1988)).
and Watson (1988) and Blanchard and Quah (1989) have
considered models where permanent components have richer dynamics than those
in a
random
walk. In these more general specifications, permanent components turn
out to have variances that can no longer be identified from just the second
of the original sequence.
The
moments
extent of this lack of identification can be seen in the
-6following result.
Theorem
I
\\- exp(iw)|-2
^ be an
Let
Fix S, a spectral density satisfying:
2.2:
•
- S^y (0)| dw <
|5Ay(w)
< 5Ay(0) <
oo and
arbitrary non-negative function on [— ir,
tt],
cx>.
0,
and
leads
and
symmetric about
sucii that:
(i)
(ii)
/
<
i/'(w)
|1
-
1
for
w ^
exp(iw)|-2(l
Suppose that
lags,
<
-
and
0;
V'(w))dw
AXi and AXo
and have spectral
Then [X\,Xo)
oo.
are stochastic sequences orthogonal at
densities
an orthogonal
is
<
S^Xi =
and SaXo
i'S
=
(1
~
all
4')S, respectively.
PT decomposition for an integrated sequence whose
increments have the given spectral density S.
Theorem 2.2
asserts that
under regularity conditions the second moments of an
arbitrary integrated sequence are consistent with a wide range of dynamics in the
underlying permanent and transitory components.^
sequences has spectral density equal to the
underlying sequences,
rpS
+ (1
—
rp)S.
with increments
it
is
obvious that
sum
Since the
of orthogonal
of the spectrsJ densities of the
AXi + AXq
Thus, the only subtlety in Theorem 2.2
AXq)
sum
has spectrjJ density
is
S =
whether Xq (a sequence
could be covariance stationary. But this follows from noting
^
The alert reader will notice that Theorem 2.2 only gives a pair {Xi Xo) whose
second moments sum correctly to match a given spectral density 5. The Theorem
does not show how to construct processes {Xi Xq) that will sum to a given process
Y, where the last has increments with spectrjil density S. In the terminology of
,
,
stochastic differential equations,
sense.
The strong
sense solution
Theorem 2.2
is
gives only a solution in the
given below in
Theorem A.l
in the
weak
Appendix.
|
-7that:
Var(Xo)
=
/
11
-
exp(ia;)|-25Ax„(w)da;
=
y
|1
-
exp{iu>)\-^S^Y{u>)il
= SAy(O) y"
~
|1
exp(i«)|-2(l
- i>H) du
-
V(w))
do;
+ /|l-exp(Jw)|-2(SAy(w)-5Ay(0))(l-V(w))dw <
Finiteness results from the
summand's being
first
summand's being bounded from above
sup
-ir<A<ir
Thus,
Xq can
(i),
AXi
^
provides a cleaving of the given spectral density
(1
—
S
%l)S
impose smoothness on
V'
be shown to remain true
is
and the second
by:
and
From
%li)S.
and S
(ii)
=
V'(O)
and
its
random sequence
=
average coefficients, with ^(0)
into
two
else,
S'aXi
analogue for
is
|5(ai)
strictly
—
5(0)
Theorem can
time domain restrictions replace these frequency
from Solo (1989)
conditions. In particular,
S
so that the spectral
1,
everywhere
at frequency zero: the conclusion of the
if certain
the spectral density of a
(ii),
at the origin;
smaller than S. In the Theorem, condition
domain
(ii),
be chosen to be covariance stationary.
coincides with
non-negative pieces
density of
by
|1-V'(A)|- / |l-exp(iw)|-2(5Ay(w)-5Ay(0))(iu'< CO.
J
in fact
Because of
finite
«).
1"
Lemma
1,
we can instead use
that has 1/2-summable
and "5
is
"V*
Wold moving
the spectral density of a
random
sequence that has 1/2-summable Wold moving average coefficients."^
Although we are interested
in obtaining general
discussion considers only the orthogonaJ
first, if
^
we can
find an orthogonal
Recall that a sequence h
summability
Theorem
is
is
PT
Ccise.
The
PT
decompositions, the formal
reaisons for doing so are two-fold:
decomposition satisfying certain properties.
m-summable
if
^.
\j\"^
\h{j)\
preserved under addition and convolution (see
3.8.2.)
<
e.g.
oo,
and that m-
Brilhnger (1981)
-8then we
will
always be able to find a non-orthogonal
satisfying the
that the
PT
same
decomposition otherwise
properties. Second, for certain apphcations (e.g.
decomposition
Theorem 2.2
PT
orthogonal
is
PT
hcis
component
is
arbitrarily
for every positive 6 there exists an ortbogonsil
S = Sax, + ^aXo.
smooth,
^'t^ Var(AA'i)
<
PT
Theorem
in
6.
moment properties
of the integrated sequence of interest. Subject to those conditions,
we can always
find an arbitrarily smooth permanent component. Because we are interested
proposition
—that not
all
rich
2.2.
decomposition (Xi,Xo),
This result requires only -weak regularity assumptions on the
manent components with a
i.e,
increments with arbitrarily small variance.
Corollary 2.3: Fix a spectral density S, satisfying the conditions
where
moments,
decompositions. Significantly, that range always
includes a decomposition where the permanent
where the permanent component
(1990)),
is essential.
provides, for an integrated sequence with given second
a range of feasible orthogonal
Then
Quah
in per-
dynamic structure, Watson's (1986) non-existence
integrated sequences will admit a
component uncorrelated with the transitory component
—
random walk permanent
is
irrelevant here.
Corollary 2.3 implies that without restricting further the dynamics of the per-
manent component Xi the lower bound on Var(AA''i)
is
simply zero.
Therefore,
the results here highlight a similarity between integrated and trend stationary se-
quences: both can have their stochcistic dynamics dominated by transitory components. This observation has been raised elsewhere in the literature but our reasoning
here differs significantly from those other arguments: (1)
weak
The
results require only
regularity assumptions on the univariate dynamics of the integrated sequence
of interest. Thus, contrast the analysis here with,
Rudebusch (1988),
or
West
a trend-stationary sequence
sults here rely neither
(1988),
is
who argue
e.g.,
Clark (1988), Diebold and
that for certain parameter values
close to a difference-stationary one.
(2)
The
re-
on dynamics being only imprecisely estimated, nor on a
researcher's misspecifying a
Wold
representation.
Thus, our analysis
differs
from
-9Cochrane's (1988) and Christiano and Eichenbaum's (1990) criticisms of Campbell
and Mankiw (1987). Arguments about imprecision and possible misspecification
can also be subsumed in the following:
lying population probability
having only
finite
model and
(4)
as
Var(AXi)
in
Quah
The truth
decreases.
The statements here apply
are not conclusions due to
What
component Xq can have
transitory
despite
samples.
(3)
is
of this
is
shown
in the
Xq has
(1990) where the transitory component
Var(AXi) growing
an investigator's
not obvious from the above
autoregressive roots
its
to the under-
is
that the
bounded away from
numerical C8ilculations
fixed autoregressive roots
arbitrarily small.
That the lower bound on the importance of the permanent component
may
at first
1
seem puzzling.
We
is
zero
can get some intuition for this zero lower bound
property by studying more restricted and more explicit decompositions where the
permanent components are constrained to be
ARIMA sequences—less trivial bounds
then result. That analysis forms the content of the next section.
3.
ARIMA
Finite
Components
This section specializes the analysis to
ARIMA
permanent components; doing so
allows explicit formulas for the lower bounds on Var( AYi
W
will
be needed:
i.e.,
the residual in the
its
own
If
is
cova^iance stationary,
minimum mean
let
)
.
Some
additional notation
innov{W) denote
its
square error linear predictor of
innovation,
W bcised on
lagged values.
Theorem
Suppose (yi,yo) decomposes Y, with AYi a moving average sequence of order q. Then (i) Var(in7]ov(Ayi)) > 4-« Say{0); and (ii) Var(Ayi) >
3.1:
•
(g
+
1)"^
•
Say{0). Further, there exist
moving average of order
close to the
bounds
in (i)
The lower bounds
in
q
(different)
FT
decompositions with
AYi
having innovation variances and variances arbitrarily
and
(ii).
Theorem 3.1
are strictly decreasing in the
moving average
order of AYi, and apply regardless of the correlation between the components.
-10Thus, letting AYi be a random walk must maximize the theoreticjJ lower bound
on Yi's contribution to Y.
The
analysis for autoregressive models for
autoregressive
both
its
model
for
AYi
To
variance and innovation variance.
AYi
is
even simpler.
A
first
order
a theoretical lower bound of zero on
suffices to obtain
see this, apply the
arguments
in the
proof of Theorem 3.1 to
S^yM =
|1
- C(l)r' Var(mnov(Ayi)) = S^y{0)
==> Var(innov(Ayi))
=
|1
-
Var(Ayi)
=
1(1
•
C(l)|^
5Ay(0),
and
where now C(l)
simply
let
C(l)
gressive models.
is
t 1-
-
C(l)) X
(1
the projection coefficient in a
The same
+
C{l))-'\
first
S^riO),
order autoregression.
Then
conclusion obviously applies to higher order autore-
But choosing the permanent component
in this
way does make
the transitory component more like an integrated sequence, in that the largest autoregressive root in its
this does not
happen
ARMA representation approaches unity.
for the
moving average
Finally, since a purely autoregressive
model
moving average autoregressive model, the
appUes directly to general
ARMA
models
cases in
is
Theorem
On
the other hand,
3.1.
simply a restriction of a mixed
result for a first order autoregression
for
AYi.
-11under Orthogonality and Informational Restrictions
4. Identification
In certain applications,
Quah
e.g.,
Shapiro and Watson (1988) and Blanchard and
(1989), a researcher wishes to expUcitly construct permanent and transi-
tory components, where these components are
uniquely identified.
somehow
restricted so that they are
For instance, the reseeircher might be interested in those or-
thogoncJ permanent and transitory components contained in the history of output
and unemployment or
in that of
income and consumption. Such an informational
be made precise below.
restriction will
This section answers two questions:
(1)
Can one
such permanent and transitory components, and (2)
how
exist,
is
rich
is
their class?
We will
is
If
whether there
such
PT
available for the existence property
cein
if
a particular orthogond
is
tests in a vector
PT decomposition
PT decomposition
be straightforwardly obtained from the Wold representation of
Quah
This result drives the analysis in Blanchard and
the variables of interest.
(1989), and
decompositions do
—thus, the usual exclusion
Further, under certaun conditions, such an orthogonal
unique and
any
exist
see that a Granger-causality characterization
autoregression can be used to estabhsh
exists.
test
one way of allowing a researcher to uniquely identify permanent
and transitory components. But Blanchard and Quah (1989) never confronted the
existence issue directly and therefore never needed to discuss the Granger-causality
characterization. Finally,
it
we
can be used to construct
in the Technical
arbitrarily
Appendix.
the
PT
PT
let
Y
.
.
.},
that for
PT
decomposition
That construction therefore
explicitly
exists,
—
this is
and
as functions of observable variables.
decomposition. First, following Rozanov (1967),
X
done
produces the
linecir
let
H7{t) denote
combinations of {{(t),f(i
—
l)j^(<
complete under mean square norm. The convolution operator *
b
then
be the integrated sequence of interest for which we seek an
the space spanned by square-summable
2),
such a
decompositions of Theorem 2.2
smooth permanent components
Throughout,
orthogonal
all
will see that if
sequences with
X
integers, the t-th element of 6 * A'
is
covariance stationary and defined over
^.
b{j)X{t
—
j).
is
—
such
all
the
Also, the discussion wiU be
-12more transparent
if
we use the frequency-zero smoothness conditions
form, as discussed above in Section
between existence and uniqueness of a
and the
failure of
Theorem
4.1:
{AY,WY, and
AV
assume:
= C*€,
each entry in
decomposing
Y
C is
full
given
if
a stochastic sequence.
rank at the origin; and
transitory
Y
with
Y
in
(ii)
Call ^
=
and
linearly regular
Wold
representation,
,
W.
If such a decomposition exists, then
Theorem 4.1 can be understood
random sequence and assume
joint history of
W,
with both AYi(t) and Yo(t) contained in H7{t) if and only if
implications of
manent and
fix
decomposition for
1/2-summable. Then there exists {Yi Yq) orthogonally
not Granger causally prior to
The
domain
some system.
^ is jointly covariance stationary
(i)
with spectral density matrix
^
PT
specific orthogonal
be integrated and
in time
following result gives an equivalence
to be Granger-causally prior in
Y
Let
The
2.
it is
as follows: Let
that the researcher
is
AY is
unique.
W be a
interested in the per-
Y
that contain only the information in the
W. The Theorem
asserts that the researcher can first verify
components
in
such permanent and transitory components exist by performing the usual test
Granger causal priority of
for
AY
in
{AY, W)'
.
If
AY
is
found to be not Granger
causally prior, then the algorithm given in the proof (which
the
method
in
Blanchard and Quah (1989))
and transitory components
The
terms of
Y
caui
and
essentially the
same
as
be used to Ccdculate the permanent
W.
reader should note that the construction given here has AYi{t) and Yo{t)
contained in H7{t);
Since the
in
is
first
it
is
this that gives the uniqueness result in
Theorem
4.1.
applications of vector autoregressions (Sims (1980)), such an infor-
mational restriction has been routinely used in going from the Wold decomposition
to
moving average representations that
and Quah (1989)
literature
—for
ichlin (1990),
is
more
readily interpretable
one example where that use Wcis made
instance, Futia (1981),
Quah
are
(1990), and
explicit.
—Blanchard
But a growing
Hansen and Sargent (1991), Lippi and Re-
Townsend (1983)
—questions the appropriateness
of this identifying restriction: explicit economic models can be constructed where
-13the representations that result from this informational restriction bear no relation
to the true underlying dynamics. While this difficulty
is
not central to the current
appUed researcher should note the potential problem
discussion, the
in
more general
contexts.
5.
Conclusion
It is
now well-known that
integrated and trend stationary time series produce differ-
ent implications for classical econometric inference.
to the observable dynamics of economic variables?
How does this difference extend
What are the implications for
economic theorizing?
In this paper,
sum
series as the
I
of
have approached these issues by considering an integrated time
permanent and transitory components.
I
have characterized the
That range turns
range of such decompositions for arbitrary integrated time
series.
out to always include a smooth permanent component,
a permanent component
i.e.,
with increments having arbitrarily small variance. Further, the associated transitory
component need not have high
persistence.
Thus, the observable dynamics of an
arbitrary integrated sequence are similar to those of some trend stationary sequence.
While there are
same long run
increments
is
infinitely
effect of
many
permanent-transitory decompositions,
all
share the
a disturbance in the permanent component in that their
have the same spectral density value at frequency zero. That value
all
therefore uninformative for the importance of
permanent components
in a
time
series.
In
summary, the attention that has been devoted
the permanent
increments,
is
disturbances,
component
component
unwarranted
it
is
in
GNP,
to measuring the size of
as the spectral density at frequency zero of its
—without explicitly identifying the underlying economic
simply not possible to gauge the magnitude of the permanent
in a time series.
Finally,
I
have provided exact lower bounds on the magnitude of permanent
components that are
restricted to be
ARIMA
processes.
Those bounds imply that
-14restricting the
minimum
permanent component to be a random walk maximizes
its
theoretical
importance.
Appendix: Explicit Construction of a Strong Sense Solution
PT decomposition of Section
PT decompositions— decompositions having
the properties specified in Theorem 2.2. We use the notation for a and ^ estabhshed
This appendix shows
4,
how one
can, beginning with the
expUcitly compute other orthogonal
below in the proof of Theorem
4.1.
Theorem
Y
density of
J
A.l: Suppose that
AY
|1
-
is
a given integrated sequence, with the spectral
satisfying:
exp(iu;)|-2|5Ay(w)
- 5Ay(0)| du <
< 5Ay(0) <
oo and
oo,
and that for some given W, {Yi,Yo) is the orthogonal PT decomposition ofY given
by A.l. Suppose further that < S'Ayi(w) < 5Ay(w) for all w in (0,7r]. Let %p be
a non-negative function on [— t, it], symmetric about 0, and such that:
(i)
(ii)
/
<
V(w)
|1
-
Define p
<
1
w ^ 0; and
- V(w))dw <
for
exp(»w)|-2(l
def
= S^Yi/S^y and
oo.
choose sequences a and
p
so that
for
i
w=
and
a =
If AXi =^
PT
a*AY + f3*AYi
decomposition for
The
rem
ip
restrictions
Y
—
^p.
and
with
AXq
SaXi
on S^y and
=^
=
V"
AY-AXi,
then
(XuXq)
is
an orthogonal
ipS^^y
in
Theorem A.l
are
unchanged from Theo-
2.2, but the result here gives an explicit construction for an orthogonjJ
decomposition for Y.
PT
-15Proofs
Proof of Theorem
2.2:
Proof of Corollary 2.3:
J ^^y {w)V'(w) <iw < 2n6.
Proof of Theorem
3.1:
AYi{t)
with C(0)
=
component
Y
i^ 2.1
V
Clearly,
can always be selected so that
Q.E.D.
AYi
Since
is
a moving average process of order
= Y2 C{j)innov{AYi){t - j),
and Y^^j=oC{j)z^
both AYi and
1
for
Q.E.D.
Trivial, following discussion in the text.
,
^
for all
q,
t,
\z\ < 1. But because Yj is a permanent
have the same spectral density at frequency
for
AY
zero:
Var(innov(AYi))
I^C(i)
= SayW-
;=0
Thus the lower bound on Var(innov(Ayi))
«
>
obtained by solving:
«
.
.
is
sup{|x:c(i)f=|(x:cov)l^^j'}
j=0
j=0
subject to C(0)
=
1
^C{j)z^ ^
and
for
\z\
<
1.
;=0
Recsdl that any such polynomial Ylj=o^U)'^^ '-^^ ^^ written as the product of q
monomials: p.^^ C{j)zi = nj=i(l + D{j)z), with \D{j)\ < 1, j = 1, 2,
g, and
D{j) appearing in complex conjugate pairs if not real. Since
.
.
.
,
lE'^^OVI' = \t[i^ +
D{j)z)\'
j=i
its
j
maximization at z
=
1,2,... ,5.
=
I is
j=i
equivalent to the maximization of |1
This occurs at D{j)
=
1
for each j.
is
then 4~*
•
+ D{j)\-^,
for each
Therefore, the solution to
The lower bound on the innovation
Say{0), and results when the moving average representation
the optimization problem attains the value 4'.
variance
= 11\1 + Dij)z\\
-16foi
AYi
is (1
on Var(Ayi)
+ L)^iDDov(AYi),
is
where L
the lag operator. Next, the lower
is
bound
obtained by solving:
9
inf
a^TCiJ)'
2
subject to C(0)
=
^Cij)z^ ^
1,
for
\z\
<
1,
and
Ec(i)
j=0
Substituting out for
a-^
to
j=0
Say{0).
minimize {Yl^j=o^iJ)'^)/\'l2^j=o^iJ)\
the boundary conditions above. First notice that \Yi^j=o^(^)\
Next apply the triangle and Cauchy-Schwarz inequalities:
\tc(j)f <
=
j=0
weneed
,
a^
(trail)'
j=0
<
(Erar)
;=0
•
{±1') =
—
^^^iject to
(Z^l=o 1^0)1)
(E CO?)
-(5+1)'
;=0
j=0
so that:
(Eco?)/|Ec(i)f >(9+i)-^
;=0
Notice that C{j)
=
for j
I
j
=
i=0
0,1, ... ,q, achieves this lower bound,
and because
=
this satisfies the boundary conditions as well. Thus, Var(Ayi) > (5 + l)~^5Ay(0),
and this theoretical lower bound is evidently approached arbitrari/y closely by finite
moving average processes of the form ^j_o ^^ innov{AYi){t — j), where A < 1.
Q.E.D.
is
For the next result, it is convenient to establish a little more notation. If 6
an absolutely summable sequence of numbers, 6 denotes its fourier transform
b(u)
=
Yij Hj)£~"^^
and similarly
,
Proof of Theorem
4.1:
In the
for sequences of matrices.
Wold
representation,
{AY,W)' =
C(0) to be lower triangular with diagonal elements non-negative and
identity covariance matrix. Such a representation exists
has
full
rank, there
is
a unique orthonormal matrix
V
and
is
c
C
*
to
have the
c,
take
unique. Since C(0)
such that C{0)V
is
lower
-17up to column sign changes). Define
triajigular (unique
D
def
= C{j)V and
By construction,
by D{j)
J -if
= iVum)' by T){t) = V'e{t), so that (AY, W)' = C *e = D *t].
H~{t) = H7{t); further, tj has the identity covariance matrix. Also by construction,
V
such a
(£),
Jj)
i.e., no other pair has a lower triangular D{0) and
fundamental for (AY, PV)' having the identity cova^J^lDce
pair is unique,
a disturbance vector
77
* rji and AYq to be D12 * »?2- Since Di2{0) =
and
AYi to be
1/2-sunmiable, Yq can be chosen to be covariance stationary. Thus (11,10)
Du
matrix. Define
D12
is
is
potentially the orthogonal
and
Granger-causally prior to
FT
decomposition satisfying the desired conditions,
the only such decomposition.
if it exists, it is also
W.
This implies that C{j)
is
AY
is
all j,
so
Suppose then that
lower trianguiar for
that V is the identity and D12 is zero. But then Var(yo) = 0; consequently, a FT
decomposition of the desired form does not exist. Conversely, if
is not Granger
causally prior to W, then Ci2{j) ^
for some j, and Cn is not proportional to C\2.
AY
But
Du
then, neither
nor D12 can vanish, thus both AYi and
AYq Lave
strictly
Q.E.D.
positive variance.
Finally, for the last proof, if 6
a complex-valued matrix function, denote
is
point-wise complex conjugate transposed by
6*.
Proof of Theorem A.l:
AXi
Sax^
First, verify that
= |ap5AV +
= S^Y
[|a|2
\P\^SayP
+
|^|V
+
+
its
has the correct spectral density:
2Re[ap'S£,Yp]
2pRe(d^')
Completing the square,
SAX,=S^Y[\a + Pp\^ + m\p-p^)\
= Say [rp^ + ^(1 = ^av V''/')]
Next, check that AA'i and
AXq
are uncorrelated at
calculation, the cross-spectral density
SaXoAXi
Finally, verify that
From
= SayaXi AXq
SaXi = Sayo*
is tiie first
all
leads and lags; by direct
is:
+ SaypP' -
SAYi>
=
0.
difference of a covariance stationary sequence.
the uncorrelatedness just shown, the spectral density
SaXq
= Say — SaXi =
-185Ay(l-V')- Tien,
|l-exp(tw)|~2SAXo(w)dw
/
= ni-exp(tw)|-25Ay(w)(l-V(w))dw
= S^Y{0)J\l-exp{iu)\-^l-i>(u;))dw
+ f\lFrom
condition
that the second
sup
using
(i).
(ii),
the
first
summand
is
exp(fu-)|-2(5Ay(w)
-
5Ay(0))(l
-
V(w)) dw.
summajid is finite. The result then follows from noting
bounded from above by:
|l-t/'(A)|- /
|l-exp(iw)|-2(5Ay(w)-5Ay(0))dw<oo,
Q.E.D.
This completes the proof.
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7367
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