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ALFRED
P.
WORKING PAPER
SLOAN SCHOOL OF MANAGEMENT
THE SOURCES AND NATURE OF LONG-TERM MEMORY
IN
THE BUSINESS CYCLE
by
Joseph G. Haubrich and Andrew W. Lo
Latest Revision: August 1989
Working Paper No. 3062-89-EFA
MASSACHUSETTS
INSTITUTE OF TECHNOLOGY
50 MEMORIAL DRIVE
CAMBRIDGE, MASSACHUSETTS 02139
lAUG
23 1989
THE SOURCES AND NATURE OF LONG-TERM MEMORY
IN
THE BUSINESS CYCLE
by
Joseph G. Haubrich and Andrew
W. Lo
Latest Revision: August 1989
Working Paper No. 3062-89-EFA
AU6 2 5ig89
RECaVED
THE SOURCES AND NATURE OF LONG-TERM MEMORY
IN
THE BUSINESS CYCLE
Joseph G. Haubrich* and Andrew
First Draft:
Latest Revision
We
May
:
W. Lo **
1988
August 1989
examine the stochastic properties of aggregate macroeconomic time
series
from the
standpoint of fractionally integrated models, and focus on the persistence of economic
shocks.
We
develop a simple macroeconomic model that exhibits long-term dependence,
a consequence of aggregation in the presence of real business cycles.
We
derive the re-
between properties of fractionally integrated macroeconomic time series and those
of microeconomic data, and discuss how fiscal policy may alter their stochastic behavior.
To implement these results empirically, we employ a test for fractionally integrated time
series based on the Hurst-Mandelbrot rescaled range. This test is robust to short-term
dependence, and is applied to quarterly and annual real GNP to determine the sources
and nature of long-term dependence in the business cycle.
lation
'Department of Finance, Wharton School, University of Pennsylvania.
** Sloan School of Management, Massachusetts Institute of Technology and
NBER.
We thank the editors and the referee for comments and we are grateful to Don Andrews, Pierre Perron, Fallaw Sowell, and
seminar participants at Columbia University, the NBER Summer Institute, the Penn Macro Lunch Group, and the University
of Rochester for useful discussions Research support from the National Science Foundation (Grant No. SES-8520054), the
Rodney L. White Fellowship at the Wharton School (Lo), the John M. Olin Fellowship at the NBER (Lo), the Batterymarch
Fellowship (Lo), and the University of Pennsylvania Research Foundation (Haubrich) is gratefully acknowledged.
1.
Introduction.
Questions about the persistence of economic shocks currently occupy an important
Most controversy has centered on whether aggregate time
place in macroeconomics.
are better approximated by fluctuations around a deterministic trend, or by a
series
random
walk plus a stationary or temporary component. The empirical results from these studies
are mixed, perhaps because the time series representation of output
wider
clciss
of processes than previously considered.
may
belong to a
much
In particular, earlier studies
have
ignored the class of fractionally integrated processes, which serves as a compromise between
the two stochastic models usually considered, and which also exhibits an interesting type
of long-range dependence. This
new approach
also accords well with the classical
business cycle program exemplified by Wesley Claire Mitchell,
trends and cycles at
Economic
all
and depression, a "rhythmical fluctuation
crises take place
Even
because the data evince
in activity."
Recurrent
roughly every 3 to 5 years and thus seem part of a non-
Studying such cycles
century macroeconomics.
cult
of
frequencies.
ing periods of prosperity
periodic cycle.
who urged examination
does not proceed smoothly: there are good times, bad times, alternat-
life
downturns and
NBER
in detail
has been the main activity of twentieth
so, isolating cycles of these frequencies
many
has been
other cycles of longer and shorter duration.
diffi-
Wesley
Mitchell (1927, p. 463) remarks "Time series also show that the cyclical fluctuations of
most
[not
sorts:
tions."
all]
economic processes occur
secular trends, primary
combination with fluctuations of several other
in
and secondary, seasonal variations, and
irregular fluctua-
Properly removing these other influences has always been controversial.
No
less
an authority than Irving Fisher (1925) considered the business cycle a myth, akin to runs
of luck at
Monte
Carlo. In a similar vein, Slutzky (1937) suggested that cycles arose from
smoothing procedures used to create the data.
A
similar debate
is
now taking
The standard methods
place.
of removing a linear or
exponential trend assume implicitly that business cycles are fluctuations around a trend.
Other work
similar to
changes.
[e.g.
Nelson and Plosser (1982)] challenges this and posits stochastic trends
random
walks, highlighting the distinction between temporary and permanent
Since empirically the cyclical or temporary component
fluctuation in the trend
like Fisher's
[as in
2.7
myth. This
the case of a
component
is
important
random
[the
random walk
for forecasting
1
-
small relative to the
part], business cycles look
more
purposes because permanent changes
walk] today have a large effect
-
is
many
periods later, whereas
8.89
temporary changes
The
large
[as in
stationary fluctuations around a trend] have small future effects.
random walk component
of aggregate output.
also provides evidence against
theoretical models
Models that focus on monetary or aggregate demand disturbances
cannot explain much output variation; supply side
as a source of transitory fluctuations
or other models
some
must be invoked
[e.g.
Nelson and Plosser (1982), Campbell and
Mankiw
(1987);.
However, the recent studies posit a misleading dichotomy.
sus
random walks, they overlook
who have
Kuznets,
earlier
In stressing trends ver-
work by Mitchell [quoted above], Adelman, and
stressed correlations in the data intermediate between secular trends
and transitory fluctuations.
In the language of the interwar
Kondratiev, Kuznets and Juglar cycles.
NBER,
they are missing
In particular, in the language of
modern time
has ignored
series analysis, the search for the stochastic properties of aggregate variables
the
clciss
of fractionally integrated processes.
These stochastic processes, midway between
white noise and a random walk, exhibit a long-range dependence no
can mimic, yet lack the permanent
effects of
an
ARIMA
of explaining the lower frequency effects, the long swings
Adelman
finite
ARMA
model
They show promise
process.
emphasized by Kuznets (1930),
(1965) or the effects which persist from one business cycle to the next. Since long
memory models
they
may
for a
more
exhibit dependence without the permanent effects of an
ARIMA
process,
not be detected by standard methods of fitting Box-Jenkins models. This calls
direct investigation of this alternative class of stochastic processes.
This paper examines the stochastic properties of aggregate output from the standpoint of fractionally integrated models.
reviewing
its
main
We
properties, advantages,
introduce this type of process in Section
and weaknesses. Section
macroeconomic model that exhibits long-term dependence. Section
for fractional integration in
Though
dence.
2.7
conclude
3 develops a simple
employs a new
time series to search for long-term dependence
related to a test of Hurst's and Mandelbrot's,
We
4
it
is
2,
test
in the data.
robust to short-term depen-
in Section 5.
-2-
8.89
Review of Fractional Techniques
2.
A random
walk can model time
in Statistics.
The
but non-periodic.
series that look cyclic
first
differences of that series [or in continuous time, the derivative] should then be white noise.
This
ries
an example of the
is
makes
it
"rougher," whereas
macroeconomic time
that
common
intuition that differencing [differentiating] a time se-
summing
[integrating]
series look like neither a
some compromise
or hybrid between the
it
makes
it
"smoother."
random walk nor white
random walk and
noise, suggesting
integral
its
Many
may
be useful.
Such a concept has been given content through the development of the fractional calculus,
i.e.,
differentiation
between
and
1
and integration
may
the ordinary integral;
than white noise.
to non-integer orders.
be viewed as a
it
filter
The
that smooths white noise to a lesser degree than
yields a series that
rougher than a random walk but smoother
is
Granger and Joyeux (1980) and Hosking (1981) develop the time
implications of fractional differencing in discrete time.
ries
fractional integral of order
have employed fractionally difference time
series,
se-
Since several recent studies
we provide only
a brief review of their
properties in Sections 2.1 and 2.2.
2.1.
Fractional Differencing.
Perhaps the most intuitive exposition of fractionally differenced time
infinite-order autoregressive
and moving-average representations. Let Xt
(l-I)'^X,
where
d
=
tf is
white noise, d
then Xt
is
is
=
series
is
via their
satisfy:
(2.1)
et
the degree of differencing, and
L
denotes the lag operator.
random walk
if
d
white noise, whereas Xt
is
a
=
1.
If
However, as Granger
and Joyeux (1980) and Hosking (1981) have shown, d need not be an
integer.
From
the
binomial theorem, we have the relation:
The idea
'
lay
dormant
only arisen
its
an old one, dating back to an oblique reference by Leibnit in 1695, but the subject
Liouville, and Riemann, developed it more fully Extensive applications have
this century; see, for example, Oldham and Spanier (1974). Kolmogorov (1940) was apparently the first to notice
of fractional differentiation
until the 19th century
in
ie
when Abel,
applications in probability and statistics.
^See, for example, Diebold and Rudebusch (1989) and Sowell (1989a, b) for further details.
2.7
- 3 -
8.89
—
(1-^)'
where
(^)
is
=
E(-l)'(^)^'
(2-2)
the binomial coefficient extended to non-integer values of d in the natural way.
By manipulating
(2.1) mechanically,
Xt also
ha.s
an infinite-order
particular, expressing the binomial coefficients in terms of the
=
X.
The
when d
is
d.
have:
(2.3)
less
For example, Granger and Joyeux (1980) and Hosking (1981) show
than
^,
Although the specification
to fractional
The
ARIMA
AR
applications,
and
and
Xt
MA
is
in (2.1)
models
is
stationary;
is
when d
is
greater than
— j, Xt
is
invertible.
a fractional integral of pure white noise, the extension
clear.
representations of fractionally differenced time series have
many
illustrate the central properties of fractional processes, particularly long-
term dependence. The
MA
coefficients Bj^ tell the effect of a
indicate the extent to which current levels of the process
this
we
function,
pl|^.
B, =
B(L)..
gamma
representation. In
particular time series properties of Xt depend intimately on the value of the differ-
encing parameter
that
=
(.-L)-^<,
MA
shock k periods ahead, and
How
depend on past values.
dependence decays furnishes valuable information about the process. Using
fast
Stirling's
approximation, we have:
Bu
for large k.
Comparing
this
-—
«
(2.4)
with the decay of an AR(l) process highlights a central
feature of fractional processes: they decay hyperbolically, at rate k
exponential rate of p
for
an AR(l). For example, compare
function of the fractionally differenced series
0.9X(_i
+
£(.
Although they both have
autocorrelation function decays
^See Hosking (1981)
2.7
1
—
L)^^^'^^Xt
Figure
=
Cf
1
rather than at the
the autocorrelation
with the AR(l) Xt
first-order autocorrelations of 0.90, the
much more
functions of these two processes. At lag
(1
in
,
rapidly. Figure
=
AR(l)'s
2a plots the impulse-response
the MA-coefficients of the fractionally differenced
for further details.
- 4 -
8.89
series
and
and the AR(l) are 0.475 and 0.900 respectively;
at lag
series
lags. Alternatively, v/e
may
autoregressive parameter will, for a given lag, yield the
fractionally differenced series (2.1). This value
in
and 0.349,
100 they are 0.048 and 0.000027. The persistence of the fractionally differenced
apparent at the longer
is
at lag 10 they are 0.158
when d —
Figure 2b for various lags
must be very
ask
of an AR(l)'s
same impulse-response
simply the
is
what value
A;-th
root of Bf^ and
is
as the
plotted
For large k, this autoregressive parameter
0.475.
close to unity.
These representations also show how standard econometric methods can
methods
fractional processes, necessitating the
of Section 4.
fail
to detect
Although a high order
ARMA
process can mimic the hyperbolic decay of a fractionally differenced series in finite samples,
the large
number
of parameters required
usual Akaike or Schwartz Criteria.
pattern with a single parameter
Hudak
explicitly fractional process, however, captures that
Granger and Joyeux (1980) and Geweke and Porter-
d.
(1983) provide empirical support for this by showing that fractional models often
out-predict fitted
The
lag
represents
ARMA
models.
polynomial B{L) provides a metric for the persistence of Xf.
GNP, which
forecast of future
given by (2.1).
Rudebusch (1989)
unexpectedly this year.
falls
GNP? To
polynomial C{L) that
is
An
would give the estimation a poor rating from the
address this issue, define
satisfies the relation (1
How much
should this change a
as the coefficients of the lag
Cj^
— L)Xt =
Suppose Xt
C{L)et, where the process Xt
One measure used by Campbell and Mankiw
(1987) and Diebold and
is:
lim B,,
=
J^C,,
=
C(l)
(2.5)
.
k=0
For large k, the value of Bj^ measures the response of Xt^k ^o ^" innovation at time
natural metric for persistence."*
and asymptotically there
is
From
(2.4),
no persistence
the autocorrelations die out very slowly.
case), but also for j
<
d
<
1,
when
it is
immediate that
for
<
<i
<
in a fractionally differenced series,
This holds true not only for d
the process
is
<
1,
C(l)
t,
=
a
0,
even though
^ (the stationary
nonstationary.
*But sec Cochrane (1988), Quah (1988), and Sowell (1989c) for opposing views.
'There has been some confusion in the hterature on this point. Geweke and Porter-Hudak (1983) argue that C(l) > 0.
They correctly point out that Granger and Joyeux (1980) have made an error, but then incorrectly claim that C(l) ;= l/r{d).
If our equation (2 4) is correct, then it is apparent that C(l) =
(which agrees with Granger (1980) and Hosking (1981)].
Therefore, the focus of the conflict lies in the approximation of the ratio r(/c + d]/r{k + 1) for large k. We have used Stirling's
approximation. However, a more elegant derivation follows from the functional analytic definition of the gamma function as
2.7
- 5 -
8.89
From
these calculations,
apparent that the long-range dependence of fractional
is
it
processes relates to the slow decay of the autocorrelations, not to any permanent effect.
This distinction
is
important; an IMA(l,l) can have small but positive persistence, but
the coefficients will never mimic the slow decay of a fractional process.
The long-range dependence
of fractionally differenced time series forces us to
some conclusions about decomposing time
modify
"permanent" and "temporary" com-
series into
ponents. Although Beveridge and Nelson (1981) show that non-stationary time series
always be expressed as the
sum
of a
random walk and a
may
stationary process, the stationary
component may exhibit long-range dependence. This suggests that the temporary compo-
may be
nent of the business cycle
practical purposes, closer to
The presence
what we think
if
Nelson and Plosser (1982) argue that deterministic detrending
the cylical component
(1986) shows that the
or trend part
for all
of as a long non-periodic cycle.
overstate the importance of the business cycle
walk. However,
is,
of fractional differencing has yet another implication for the Beveridge-
Nelson (BN) approach.
may
transitory only in the mathematical sense and
BN
decomposition
if
secular
movements
follow a
an AR(l) with a nearly unit root, McCallum
is
will assign too
and understate the contribution
of the
much
variance to the permanent
temporary component.^ But suppose
A
decom-
differencing the series the appropriate [fractional]
number
the temporary component follows the fractionally differenced time series (2.1).
position in the spirit of
BN,
i.e.,
random
of times, leaves no temporary component. However, taking first-differences of the series as
Plosser and Schwert (1978) suggest will obviously overstate the importance of the cyclical
component. For example,
order, the temporary
may
reverse
we
difference (2.1) once, instead of the appropriate fractional
component becomes
(fractional) order leaves
noise
if
— L)^~
(1
e^.
Differencing by the appropriate
no temporary component. Therefore, the presence of fractional
McCallum's
result.
the Bolution to the following recursive relation
[see, for
example, lyanaga and Kawada (1980 Section 179.A)):
r(i+l)
=
xT{x)
and the conditions;
r(i)
=
1
lin,
n
That an AR(1) with
2.7
coefficient 0.99
is in
— ao
ii^^
any sense "temporary"
- 6 -
=
1
n'Tln)
is
arguable.
8.89
Spectral Representation.
2.2.
The spectrum,
or spectral density [denoted /(w)] of a time series specifies the contri-
bution each frequency makes to the total variance. Granger (1966) and Adelman (1965)
have pointed out that most aggregate economic time series have a "typical spectral shape"
where the spectrum increases dramatically
as
w —
>
0].
Most
as the frequency approaches zero [/(cj)
—
>
oo
of the power or variance seems concentrated at low frequencies, or long
periods. However, pre-whitening or differencing the data often leads to "over-differencing"
component", and often replaces the peak by a dip at
or "zapping out the low frequency
The spectra
Fractional differencing yields an intermediate result.
exhibit peaks at
the
random
more
[unlike the flat
walk's.
A
We
persistence.
spectrum of an
fractional series hais a
0.
of fractional processes
ARM A process], but ones not so sharp as
spectrum richer
low frequency terms, and
in
by calculating the spectrum of fractionally integrated
illustrate this
white noise, and also present several formulas needed later on.
Given Xt
=
(1
— L)~
if,
the series
is
noise input, so that the spectrum of Xt
=
T.
1
The
identity
|l
— zf =
2[l
^^"^
^dT
— 2r
^TT
—
cos(t<;)]
fiu)
output of a linear system with a white
is:
a2
^(^)
clearly the
2
=
e-*",
implies that for small
=
u)
=
o'
E{ei].
we have:
c^^.
cu;-2^
(2.7)
This approximation encompasses the two extremes of a white noise
process]
and a random walk. For white
walk, d
=
1
and the spectrum
is
noise, d
=
0,
and
f{u>)
inversely proportional to
(2.6)
o;'^.
=
c,
A
a finite
ARMA
while for a
random
[or
class of processes of
current interest in the statistical physics literature, called l// noise, matches fractionally
integrated noise with d
''See Chatfield (1984,
2.7
=
I.
Chapters 6 and
9).
- 7 -
8.89
A
3.
Simple Macroeconomic Model with Long-Term Dependence.
Over half
to learn
if
a century ago,
more about economic
we approach
Models
oscillations at large
p. 230)
wrote that "We stand
and about business cycles
the problem of trends as theorists, than
empirical work."
theory.
Wesley Claire Mitchell (1927
if
we
in particular,
confine ourselves to strictly
Indeed, gaining insights beyond stylized facts requires guidance from
of long-range
dependence may provide organization and discipline
construction of economic paradigms of growth and business cycles.
They can guide
in
future
research by predicting policy effects, postulating underlying causes, and suggesting
ways to analyze and combine data. Ultimately, examining the
the
new
facts serves only as a prelude.
Economic understanding requires more than a consensus on the Wold representation
GNP;
demands a
it
falsifiable
model based on the
tastes
of
and technology of the actual
economy.
Thus, before testing
for long-range
dependence, we develop a simple model of economic
equilibrium in which aggregate output exhibits such behavior.
It
presents one reason that
macroeconomic data might show the particular stochastic structure
shows that models can
also
for
which we
restrict the fractional differencing properties of
time
test.
It
series, so
that our test holds promise for distinguishing between competing theories. Furthermore,
the majcimizing model presented below connects long-rajige dependence to central economic
concepts of productivity, aggregation, and the limits of the representative agent paradigm.
3.1.
A
Simple Real Model.
One
will
plausible
mechanism
for generating long-range
mention here and not pursue,
integrated process.
is
dependence
in
output, which we
that production shocks themselves follow a fractionally
This explanation for persistence follows that used by Kydland and
Prescott (1982). In general, such an approach begs the question, but in the present case
evidence from geophysical and meteorological records suggests that
important shocks have long run correlation properties.
for
example, find long-range dependence
many
economically
Mandelbrot and Wallis (1969),
in rainfall, riverflows,
earthquakes and weather
[measured by tree rings and sediment deposits].
A more
them
satisfactory
model explains the time
despite white noise shocks.
data by producing
This section develops such a model with long-range
dependence, using a linear quadratic version of the
2.7
series properties of
- 8 -
real business cycle
model of Long and
8.89
Plosser (1983) and aggregation results due to Granger (1980). In our multi-sector model
the output of each industry (or island) will follow an
N
sectors will not follow an
AR(l) process. Aggregate output with
AR(l) but rather an ARMA(A'',A'^-1). This makes dynamics
with even a moderate number of sectors unmanageable. Under
fairly general conditions,
however, a simple fractional process will closely approximate the true
Consider a model economy with
many goods and
and consumption plan. The
a production
version of the real business cycle model.
U =
A'^
^/9'u(C() where Ct
goods
is
u{Ct)
where
l
is
assume
B
total
may be
process
either
is
j-th entry S^^t of the
et is
t
+
i
at time
a lifetime utility function of
function u{Ct)
is
given by:
- \c[BCt
(3.1)
and
it is
Sti
=
later,
we
face a resource constraint:
Yt
(3.2)
matrix St denotes the quantity of good j invested
assumed that any good
may be consumed
Yjt
or invested.
determined by the random linear technology:
a (vector)
1.
yielding a diagonal
might occur,
for
A
=
ASt
+
(3.3)
et
random production shock whose
The matrix A
term dependence we
value
is
realized at the beginning of
consists of the input-output parameters a^,.
restrict A's form.
Thus, each sector uses only
its
To focus on
own output
matrix and allowing us to simplify notation by defining
Cj
=
long-
as input,
a^^.
This
example, with a number of distinct islands producing different goods. To
further simplify the problem,
2.7
C[i
+
NxN
Yt
period
The agent has
C[BCt = Yi ^ti^ff "^^^ agents
consumed or saved, thus:
z,
t,
chooses
agent inhabits a linear quadratic
to be diagonal so that
where the
where
who
an A^xl vector of ones. In anticipation of the aggregation considered
output Yt
Output
infinitely lived
utility
=
Ct
in
a representative agent
an A^xl vector denoting period-t consumption of each of the
our economy. Each period's
in
ARMA specification.
all
commodities are perishable and capital depreciates
- 9 -
at a
8.89
rate of 100 percent. Since the state of the
period's output and productivity shock,
economy
it is
in
each period
is
fully specified
useful to denote that vector Zt
=
by that
[V/
e[]'.
Subject to the production function and the resource constraints (3.2) and (3.3), the
agent maximizes expected lifetime
utility:
Y,0'-\{Yt-Sti)
{5,}
'
where we have substituted
maps
{St)
'
'
"-
consumption
for
naturally into a dynamic
T
=
(3.4)
t
in (3.4)
using the budget equation (3.2). This
programming formulation, with a value function
V [Zt]
and
optimality equation:
V[Zt)
=
U^.^[u[Yt-Sti) + 0E\V{Zt+^)\Zt]\
\St]
With quadratic
utility
and
K
(3.5)
.
)
linear production,
it
is
straightforward to show that V[Zi)
is
quadratic, with fixed constants given by the matrix Riccati equation that results from
Given the value function, the
the recursive definition of the value function.
first
order
conditions of the optimality equation (3.5) yield the chosen quantities of consumption and
investment/savings and have closed-form solutions.
The simple form
of the optimal consumption
and investment decision
the quadratic preferences and the linear production function.
Two
rules
comes from
qualitative features bear
emphasis. First, higher output today will increase both current consumption and current
investment, thus increasing future output. Even with 100 percent depreciation, no durable
commodities, and
i.i.d.
serial correlation.
Second, the optimal choices do not depend on the uncertainty present.
production shocks, the time-to-build feature of investment induces
This certainty equivalence feature
The time
series of
is
clearly
an artifact of the linear-quadratic combination.
output can now be calculated from the production function
and the optimal consumption/investment decision
rules.
(3.1)
Quantity dynamics then come
from the difference equation:
y,(+l
*See Sargent (1987, Chapter
2.7
1) for
-
^^-^
—
^y,t
+ K, +
e,t+i
(3.6)
an excellent exposition.
- 10 -
8.89
or
Yu^l
where
Pj
and X, are
summarized by
=
fixed constants.
(3.7)
is
a^,
which
in
The key
That
effect dies off at a rate that
mimics business
However, aggregate output, the sum across
It
sum
has an ARMA(A^,A'^
—
of
A'^
all
is
two
series,
Xt
an ARMA(2,1) process. Simple induction
With over
six million registered businesses in
America (CEA, 1988), the dynamics can be incredibly
unmanageably huge. The common response
different firms (islands)
of
independent AR(l) processes with distinct parameters
representation.
1)
sum
well-known that the
is
each AR(l) with independent error,
then implies that the
ters
depends on the
show such dependence, which we now demonstrate by applying the aggre-
gation results of Granger (1980,1988).
Yt,
dynamics
Higher output today
for a single industry or island neither
cycles nor exhibits long-run dependence.
and
qualitative property of quantity
turn depends on the underlying preferences and technology.
The simple output dynamics
sectors, will
(3.7)
£,(^1
that output Y^t follows an AR(l) process.
implies higher output in the future.
parameter
+ K, +
<^^Y^t
rich,
to this
and the number
problem
is
of
parame-
to pretend that
many
have the same AR(l) representation for output, which reduces the
dimensions of the aggregate
ARMA
An
autoregressive parameters.
This "canceling of roots" requires identical
process.
alternative approach reduces the scope of the problem by
ARMA process approximates a fractionally integrated process, and thus
the many ARMA parameters in a parsimonious manner. Though we consider
showing that the
summarizes
the ca^e of independent sectors, dependence
Consider the case of
related
A'^
for sector I's
output
sumptions on the
Y^°'
=
X^j_i Vit
.
ttj's
Y^^.
easily handled.
sectors, with the productivity shock for each serially uncor-
and independent across
the productivity coefficient
is
a^.
islands.
Furthermore,
This implies differences
One
of our key results
is
let
the sectors differ according to
in Qj,
the autoregressive parameter
that under
some
distributional as-
aggregate output Vj" follows a fractionally integrated process, where
To show
this,
we approach
this
problem from the frequency domain and
apply spectral methods which often simplify problems of aggregation.^ Let /(w) denote
the spectrum [spectral density function] of a
random
variable,
and
let z
=
e~*'^.
From
the definition of the spectrum as the Fourier transform of the autocovariance function, the
*See Theil (1954).
2.7
- 11 -
8.89
spectrum of
yj^
is:
a'
^•(")
Similarly,
=
u^^^-
''•''
independence implies that /(w), the spectrum of
The
of the individual Fit's.
q:,'s
the
Y^°', is
sum
of the spectra
mezisure an industry's average output for given input.
This attribute of the production function can be thought of as a drawing from nature, as
can the variance of the productivity shocks
for
e^t
each sector. Thus,
it
makes sense
to
think of the Oj's as independently drawn from a distribution G[a) and the Oj's drawn from
F{a). Provided that the
density of the
£,j
sum can be
shocks are independent of the distribution of
the distribution F{a)
ARMA(m,m —
1)
the spectral
written as:
/H
If
Qj's,
is
process.
=
^£[^2]
J
discrete, so that
A more
/
.
2n
--J-^dF(a)
\l
takes on
it
(3.9)
— azr
m
(<
values, F^" will be an
A'^)
general distribution leads to a process no finite
ARMA
model can represent. To further specify the process, take a particular distribution
Beta distribution.
in this case a variant of the
In particular, let
a
for
F,
be distributed as
Beta(p,g), which yields the following density function for a:
dF{a)
=
~
^(^''^
{
~
(3.10)
otherwise
with p
>
requires a
and
little
q
>
0.^^
Obtaining the Wold representation of the resulting process
more work. As a
first
step,
expand the spectrum
(3.10) by long division.
Substituting this expansion and the Beta distribution (3.10) into the expression for the
spectrum and simplifying [using the relation
z
+
z
=
2cos(a;)] yields:
'"
Granger (1980) conjectures that the particular distribution
"
For a discussion of the variety of shapes the Beta distribution takes as p and
2.7
is
not essential but only proves the result for Beta distributions.
- 12 -
q vary, see
Johnson and Kott (1970)
8.89
/M
Then
=
y^
^2
+ 2^a^os(Mj^(^c.2p-l(l-a2)9-2da.
the coefficient of cos(A;u;)
(3.11)
is:
2a^
c,2p+^-l(i
_ ^2)9-2^^
(3.12)
_
/:
Since the spectral density
the
A:-th
autocovariance of
simplifies to
coefficients
0{p
+
is
the Fourier transform of the autocovariance function, (3.12)
Yf'^.
k/2,q —
is
Furthermore, the integral defines a Beta function, so (3.12)
Dividing by the variance gives the autocorrelation
l)//?(p, g).
which reduce to
*'
r(p)
r(p
+
1
+ 0+1)
which, again using the result from Stirling's approximation T{a
proportional
(for large lags) to
integrated process of order d
—
Thus, aggregate output
k^~^.
1
—
f"
+
k)/T{b
+
A;)
«
k°'~",
is
Yf^ follows a fractionally
Furthermore, as an approximation for long
this does not necessarily rule out interesting correlations at higher, e.g.
lags,
business cycle,
frequencies. Similarly, co-movements can arise as the fractionally integrated income process
may
induce fractional integration
in
maximizing model given tastes and
In principle,
all
other observed time series.
This has arisen from a
technologies.-^
parameters of the model may be estimated, from the distribution of
production function parameters to the variance of output shocks. Though to our knowledge
no one has
explicitly estimated the distribution of production function parameters,
people have estimated production functions across industries. ^^
studies disaggregates to 45 industries.
One
many
of the better recent
For our purposes, the quantity closest to a,
is
the
''Two additional points are worth emphasizing First, the Beta distribution need not be over (0,1) to obtain these results,
only over (a,l) Second, it is indeed possible to vary the a.'s so that a, has a Beta distribution.
'^Leontief, in his classic study (1976) reports own-industry output coefficients for 10 sectors:
how much an extra
unit of food
food production These vary from 0.06 (fuel) to 1.24 (other industries).
'* Jorgenaon, GoUop and Fraumeni
(1987).
will increase
2.7
- 13 -
8.89
value-weighted intermediate product factor share. Using a translog production function,
this gives the factor share of inputs
These range from a low of 0.07
in
coming from
TV
radio and
industries, excluding labor
and
capital.
advertising to a high of 0.811 in petroleum
and coal products. Thus, even a small amount of disaggregation reveals a large dispersion
and suggests the
3.2.
plausibility
and significance of the simple model presented
in this section.
Welfare Implications.
Taking a policy perspective
of national income.
that they
live in
People
raises a natural question
about the fractional properties
Does long-term dependence have welfare implications? Do agents care
such a world?
who must
predict output or forecast sales will care about the fractional nature
of output, but fractional processes can have normative implications as well.
Lucas (1987),
regimes.
let
We
this section estimates the welfare costs of
can decide
if
the typical household
economic
people care whether their world
consume
under
different
fractional. For concreteness,
C(, evaluating this via a utility function:
oo
.l-CT
U
is
instability
Following
^
1
0'
l-ai
-E\Cl
-c
(3.14)
t=0
Also assume:
InCt
=
(l
+
A)^^^LSt
(3.15)
fc=0
where
With
rjj
Tji
=
Ine^.
The
A term measures compensation for variations in the process (^{L).
normally distributed with
between two processes
4>
and
xJj
mean
and variance
1,
the compensating fraction A
is:
oo
1
+
exp
A
Ln-o)^(4-4)
(3.16)
k=0
Evaluating this using a
2.7
realistic
(7
=
5,
again comparing an AR(l) with p
- 14 -
=
0.9
and
8.89
fractional process of order 1/4,
those
is
in
we
Lucas because the process
find that A
is
in logs
=
—0.99996
[this
number
looks larger than
rather than in levels]. ^^ For comparison, this
the difference between an AR(l) with p of 0.90 and one with p of 0.95. This calculation
provides a rough comparison only.
model generating the processes,
When
as only
it
feasible, welfare calculations
will correctly
should use the
account for important specifics,
such as labor supply or distortionary taxation.
'^
We
calculate this using (2.4) and the Hardy-Littlewood approximation for the resulting
Titchmarsh, 1951, sec
2.7
Riemann Zeta Function,
following
4 11
- 15 -
8.89
R/S
4.
The
Analysis of Real Output,
results of Section 3
dependence
show that simple aggregation may be one source of long-term
in the business cycle.
memory and apply
statistic first
it
to real
In this section
GNP. The technique
we employ a method
is
for detecting long
based on a simple generalization of a
proposed by the English hydrologist Harold Edwin Hurst (1951), which has
subsequently been refined by Mandelbrot (1972, 1975) and others.^
Our
generalization
of Mandelbrot's statistic [called the "rescaled range" or "range over standard deviation"
or R/S] enables us to distinguish between short
made
in
a sense to be
precise below.
We
define our notions of short
4.1. In Section 4.2
term dependence
and long memory and present the
we present the empirical
in log-linearly
results for real
under two
null
To develop
a
method
of detecting long
used concepts of short-term dependence
is
Section 4.3;
several
less
we
find long-
dependence
in
Monte Carlo experiments
results in Section 4.3.
Statistic.
between long-term and short-term
(1956), which
we perform
and two alternative hypotheses and report these
The Rescaled Range
tinction
GNP
test statistic in Section
detrended output, but considerably
the growth rates. To interpret these results,
4.1.
and long run dependence,
is
a measure of the decline
by successively longer spans of time.
memory, we must be
statistical
precise about the dis-
dependence. One of the most widely
the notion of "strong-mixing" due to Rosenblatt
in statistical
dependence of two events separated
Heuristically, a time series
maximal dependence between any two events becomes
trivial as
is
strong-mixing
if
the
more time elapses between
them. By controlling the rate at which the dependence between future events and those of
the distant past declines,
limit
it is
possible to extend the usual laws of large
numbers and central
theorems to dependent sequences of random variables. Such mixing conditions have
been used extensively by White (1982), White and Domowitz (1984), and Phillips (1987)
for
example, to relax the assumptions that ensure consistency and asymptotic normality of
various econometric estimators.
We
adopt this notion of short-term dependence as part of
our null hypothesis. As Phillips (1987) observes, these conditions are satisfied by a great
many
stochastic processes, including
Moreover, the inclusion of a
moment
all
Gaussian finite-order stationary
ARMA
models.
condition also allows for heterogeneously distributed
'*See Mandelbrot and Taqqu (1979) and Mandelbrot and Wallis (1968, 1969a-c).
2.7
- 16 -
8.89
sequences [such as those exhibiting heteroscedasticity], an especially important extension
in
GNP.
view of the non-stationarities of real
In contract to the "short
phenomena
natural
memory"
of
weakly dependent
often display long-term
memory
in
[i.e.,
strong-mixing] processes,
the form of non-periodic cycles.
This has lead several authors, most notably Mandelbrot, to develop stochastic models
that exhibit dependence even over very long time spans.
series
The
fractionally integrated time
models of Mandelbrot and Van Ness (1968), Granger and Joyeux (1980), and Hosking
(1981) are examples of these. Operationally, such models possess autocorrelation functions
that decay at
much
slower rates than those of weakly dependent processes, and violate
To
the conditions of strong-mixing.
detect long-term dependence [also called "strong
dependence"], Mandelbrot suggests using the range over standard deviation (R/S)
also called the "rescaled range,"
The R/S
river discharges.
from
statistic
mean, rescaled by
its
which was developed by Hurst (1951)
its
is
statistic,
in his studies of
the range of partial suras of deviations of a time series
standard deviation. In several seminal papers, Mandelbrot
demonstrates the superiority of R/S to more conventional methods of determining long-run
dependence [such as autocorrelation analysis and spectral
In testing for long
that
is
is
memory
in
analysis].
output, we employ a modification of the
R/S
statistic
robust to weak dependence. In Lo (1989), a formal sampling theory for the statistic
obtained by deriving
theorem.
We
its
limiting distribution analytically using a functional central limit
use this statistic and
its
asymptotic distribution for inference below.
Let Xt denote the first-difference of log-GNP; we assume that:
Xt
where n
is
= ^ + ^
(4.1)
an arbitrary but fixed parameter. Whether or not Xt exhibits long-term
depends on the properties of
{ft}-
As our
null hypothesis
memory
H, we assume that the sequence
of disturbances {e^} satisfies the following conditions:
{A\)
E\et\
=
for all
t.
*'See Mandelbrot {1972, 1975), Mandelbrot and Taqqu (1979), and Mandelbrot and Wallis (1968, 1969a-c).
'*Thi8 Btatietic is asymptotically equivalent to Mandelbrot's under independently and identically distributed observations,
however Lo (1989) shows that the original R/S statistic may be significantly biased toward rejection when the time series is
short-term dependent. Although aware of this bias, Mandelbrot (1972, 1975) did not correct for it since his focus was on the
relation of the R/S statistic's logarithm to the logarithm of the sample site, which involves no statistical inference; such a
relation clearly is unaffected by short-term dependence
2.7
- 17 -
8.89
{A2)
suptE\\et\^]
{A3)
a^
{A4)
{et}
=
<
oo for
limn-
is
some
/?
>
2.
strong-mixing with mixing coefficients
°°
>
and a
exists
-ni^U^if
a;,
0.
that satisfy 19
l_i
k=l
Condition [Al)
is
standard. Conditions {A2) through {A4) are restrictions on the maximal
degree of dependence and heterogeneity allowable while
still
permitting some form of the
law of large nimibers and the [functional] central limit theorem to obtain.
we have not assumed
marginal distributions of
less
than
moments
et
such as those
may
the disturbances
2,
[e.g.
Although condition
stationarity.
still
coefficients decline faster
moments
of
than l/k. However,
must decline
rules out infinite variance
(.42)
family with characteristic exponent
exhibit leptokurtosis via time-varying conditional
conditional heteroscedasticity]. Moreover, since there
conditions {A2) and {A4), the uniform
absolute
in the stable
all
if
For example,
orders [corresponding to
restrict
et
is
a trade-off between
bound on the moments may be relaxed
than {A4) requires.
we
Note that
/?
to have finite
—
>
oo],
if
we
require to
if
the mixing
et
have
then a^ must decline
moments only up
finite
feister
to order 4, then aj^
faster than l/k"^.
Conditions {Al)
—
{A4) are satisfied by
many
of the recently proposed stochastic
models of persistence, such as the stationary AR(l) with a near-unit root. Although the
distinction
between dependence
of degree, strongly
series that
in the short versus long
dependent processes behave so
diff"erently
our dichotomy seems most natural.
strongly dependent processes are either
sums do not converge
graphically, their behavior
is
marked by
may appear
to be a matter
from weakly dependent time
For example, the spectral densities of
unbounded
in distribution at the
runs
same
or zero at frequency zero. Their partial
rate as weakly dependent series.
cyclic patterns of all kinds,
some that
And
are virtually
indistinguishable from trends. '^^
Consider a sample Xi, X2, ...,Xn and
let
the modified re-scaled range statistic, which
Xn
we
denote the sample
shall call
Qn,
is
mean ^
^
X,. Then
given by:
'*For the precise definition of strong-mixing, and for further details, see Rosenblatt (1956), White (1984), and the papers
Eberlein and Taqqu (1986).
'°See Herrndorf (1985).
moments
are not required.
Note that one of Mandelbrot's (1972) arguments in favor of R/S analysis is that finite second
This is indeed the case if we are interested only in the almost sure convergence of the statistic.
However, since we wish to derive its limiting distribution
^'See Mandelbrot (1972) for further details
2.7
in
for
purposes of inference, a stronger
- 18 -
moment
condition
is
needed.
8.89
Qn
(4.2)
^n{g)
- -
- -
;=1
3=1
where
^E(^;-^n)' + ^E'^;(9){ i: (X.-^n)(X._,-X,)
=
lig)
-
and a^ and
^2
+ 2^u;j(9)7y
^. are the usual
w,(g)
Xj from
its
of squared deviations of X., but also
those suggested by
always positive.
q
9+1
<
n.
its
Theorem
^,1(9) involves
weighted autocovariances up to lag
Newey and West
Qn
mean, normalized by an estimator
The estimator
partial sum's standard deviation divided by n.
is
1
sample variance and autocovariance estimators of X.
range of partial sums of deviations of
u)j{q) are
=
(4.3)
I
(1987),
and
is
the
of the
not only sums
q;
the weights
yields an estimator o^iq)- that
4.2 of Phillips (1987) demonstrates the consistency o( an{q)
under the following conditions:
<
00 for
some /?>
{A2')
supt
{A5)
As n increases without bound, q
E\\et\'^^]
2.
also increases without
bound such that
q
~
o(nV4).
The
choice of the truncation lag g
[but at a slower rate than] the
becomes large
dramatically.
relative to the
However,
autocorrelations
may
q
is
a delicate matter.
sample
number
size,
d„(9)
ie
2.7
q
must increase with
Monte Carlo evidence suggests that when
of observations, asymptotic approximations
cannot be chosen too small otherwise the
not be captured. The choice of q
must therefore be chosen with some consideration
^^ See, for
Although
also an estimator of the spectral density function of Xi
example, Lo sind MacKinlay (1989)
-19-
at
is
clearly
may
q
fail
effects of higher-order
an empirical issue and
of the data at hand.
frequency lero, using a Bartlett window.
8.89
If
the observations are independently and identically distributed with variance o^, our
normalization by dn{^)
deviation estimator Sn
asymptotically equivalent to normalizing by the usual standard
is
=
~
[^ IZi(-^j
p
-X'n)
resulting statistic, which
we
Max
Qr
Qn,
call
and Mandelbrot (1972):
precisely the one proposed by Hurst (1951)
is
The
•
(4.4)
l</t<
To perform
we
require
its
with the standardized re-scaled range F^
statistical inference
distribution.
Although
finite-sample distribution
its
sample approximation has been derived
(A3)
(-45). In particular,
variable
we
call
V
,
in
is
=
not apparent, a large-
Lo (1989) under assumptions {Al),
(^2'),
the limiting distribution of V^, which corresponds to a
has the following
and
c.d.f.
Qn/y/^,
and
random
p.d.f.:
oo
=
Fv{v)
+ 2^(l-4fc2v2)e-2(H^
1
(4.5)
Jb=l
(4.6)
k=l
Using Fy
,
critical values
may
readily be calculated for tests of any significance level.
most commonly used values are reported
computed using fy\
the
it
is
mean and standard
in
Table
1.
The moments
V
V
are also easily
= ^,
thus
are approximately 1.25 and 0.27 respectively.
The
straightforward to show that E\V]
deviation of
of
The
= y/^ and
^^[V^]
distribution and density functions are plotted in Figure 3. Observe that the distribution
positively
is
skewed and most of
its
mass
falls
between | and
Although Fy completely characterizes the rescaled range
pothesis of short-range dependence,
pendence
is
considerably different.
its
2.
statistic
under the
null hy-
behavior under the alternative of long-range de-
Lo (1989) shows that
for the fractionally differenced
alternative (2.1), the normalized rescaled range V^ diverges to infinity in probability
d G (0,2) and converges to zero
against (2.1), a test for long-term
'*In fact, Lo (1989) shows that such a test
2.7
is
in probability
when d G (— j,0).
memory based on Vn
is
consistent against a considerably
-20
when
Therefore, at least
consistent.
more general
class of alternatives.
8.89
4.2.
Empirical Results for Real Output.
We
GNP
apply our test to two time series of real output: quarterly postwar real
from
1947:1 to 1987:4, and the annual Friedman and Schwartz (1982) series from 1869 to 1972.
The
results are reported in Table 2.
These
the classical rescaled range V^ which
row of numerical entries are estimates
first
not robust to short-term dependence.
is
eight rows are estimates of the modified rescaled range Vn{q) for values of q
Recall that q
zero.
bieis
first
and
computed
is
column of nxmierical
short-term dependence for the
of
q.
The
for the
Friedman and Schwartz
with values of q from
beyond
4 are used,
to
8.
l]-
log-GNP cannot be
third
rejected for any value
supports the null hypothesis.
When we
we no
The
results
GNP,
log-linearly detrend real
column of numerical
entries in Table 2
show
be rejected for log-linearly detrended quarterly output
rejections are
weaker
for larger q
not surprising
is
When
values
longer reject the null hypothesis at the 5 percent level of
Friedman and Schwartz time
series,
we only
reject
with the
range and with Vn{l).
classical rescaled
tests
~
[(^n/^n(9))
from estimating higher-order autocorrelations.
significance. Finally, using the
The
statistic also
That the
to 4.
1
since additional noise arises
of q
1
entries in Table 2 indicate that the null hypothesis of
The
may
that short-term dependence
•
series are similar.
considerably.
diff"er
as 100
first-diff"erence of
range
classical rescaled
the results
from
parentheses below the entries for Vn{q) are estimates of the percentage
in
of the statistic Vn,
The
The next
the truncation lag of the estimator of the spectral density at frequency
is
Reported
of
on the
diff"erenced series
ought to be particularly striking, especially since
the classical version [Vn(0)] rejects too often, on evidence of merely short term dependence.
Even over-differencing
its first diff"erence,
not a problem:
and the
The detrended
cance disappears
is
series
when
test
more
may
2.7
not adequately control for short term dependence.
suggested by Nelson and
Kang
(1981).
GNP
is
In
pre-
Their results are
show that inappropriate detrending introduces a great deal
into low frequencies.
Taken
dence.
of dependence, but even there the signifi-
significant autocorrelations in log-linearly detrended
particularly cogent since they
power
ought to pick that up as well.
only a year of lags are included. Partly due to decreasing size and
cisely the spurious periodicity
of
the series has a fractional component, so will
show more evidence
power, at these lags the test
addition, finding
if
as a
whole then, our results accept the
null hypothesis of short
Equivalently, they reject the alternative of long term dependence;
- 21 -
term depen-
GNP
has no
8.89
fractional
component. This finding deserves some notice simply because of the many pre-
From the path-breaking
vious attempts (direct or indirect) to resolve the issue.
work
of
early
Mandelbrot and Wallis (1969b) and Adelman (1963) to the sophisticated recent
estimates of Diebold and Rudebusch (1989), econometricians have not conclusively established the existence or non-existence of long
term dependence.
and Rudebusch have rather large standard
of Diebold
In particular the estimates
errors, leaving
precise conclusions about the existence of fractional differencing.
tighter distribution,
allow
more
and
in
Though by no means the
economy, whether
test statistic has a
how
final
work on the subject, our
term dependence.
tests present strong evidence against long
is
Our
to reach
conjunction with the size and power results reported below,
definite conclusions.
Our main concern
them unable
these results improve our ability to model the aggregate
purely statistical sense of estimating the right stochcistic process
in the
or in the theoretical sense of developing the appropriate equilibrium structure.
perspective, finding only short term dependence simplifies
From
either
some matters while complicating
others.
From
the standpoint of obtaining a correct statistical representation, our results sub-
stantially circumscribe the class of stochastic processes
simplifies the problem,
whether the goal be to measure the persistence of economic shocks,
assess the welfare costs of
economic
instability, or derive the correct theory.
models of trend (perhaps stochastic) and
tic
structure of
needed to characterize GNP. This
GNP. On
finite
ARMA
The standard
adequately capture the stochas-
the other hand, rejecting fractional processes removes a simple
explanation to the problems plaguing this area.
The
intriguing possibility that the con-
tending sides had each misread the true, fractional, nature of
GNP
unfortunately
is
false,
and the disagreement must have another, perhaps deeper source.
These
results also confirm the unit root findings of
Campbell and Mankiw (1987),
Nelson and Plosser (1982), Perron and Phillips (1987) and Stock and Watson (1986).
cannot reject the null hypothesis of a
As mentioned
diff"erence stationary short
stationary noise model of
GNP
to reject the null hypothesis
observe that
if
log-GNP
* Of courBe, this may be
Monte Carlo experiments
2.7
term dependent process.
before, the significant autocorrelations appearing in detrended
the spurious periodicity suggested by Nelson and
yt
is
is
Kang
We
GNP
indicate
(1981). Moreover, the trend plus
not contained in our null hypothesis, hence our failure
also consistent with the unit root model. ^^
were trend stationary,
To
see this,
i.e.:
the result of low power against stationary but near-integrated processes, and must be addressed by
- 22 -
8.89
=
yt
where
€(
=
rjt
-
T]t
is
stationary white noise, then
T]i^i.
But
Q +
/3<
+
its first-difference
this innovations process violates
(4.7)
rjt
Xt
is
simply Xt
=
+
our assumption (A3) and
et
where
therefore
is
not contained in our null hypothesis.
From
the perspective of a theorist, the tests in Table 2 provide
more information than
usual because they also serve to reject a particular model. Recall the section's main point;
moving
to a multi-sector
test provides
no evidence
model can produce qualitatively
for those
does not qualitatively affect
its
different
output dynamics. Our
dynamics. The American economy's multi-sector nature
output dynamics. For
many
questions, single sector models
provide a rich enough environment. Once again this represents a simplification;
us to avoid a broad and
As
difficult class of
sectors, rejecting the predictions of a multi-sector
tential puzzle. Currently the strong
allows
models.
before, the results also complicate the theoretical picture. Because the
economy has many
it
American
model presents a po-
assumptions needed to produce long term dependence
take the edge off any puzzle, but these are only sufficient conditions. Necessary conditions,
unknown.
or broader sufficient conditions, are
We
don't
know enough about what
pro-
duces fractional differencing or how closely actual conditions match those requirements to
worry about not finding
it.
If
broader conditions produce long term dependence - and
if
actual sector dynamics meet those conditions - then a genuine puzzle arises.
Related research
GNP
may
also turn
up puzzles, making the
lack of fractional processes in
anomalous. For example Haubrich (1989) finds fractional processes
across several countries.
The output
in
consumption
results here therefore rule out several natural expla-
nations, such as a constant marginal propensity to
consume out
of fractionally differenced
income.
From another
tempts.
many
perspective, our results impose a discipline on future modelling at-
As Singleton (1988) points out, dynamic macro models make predictions over
frequencies: seasonal, cyclical,
the facts on long term dependence.
and longer. Their predictions must now conform to
Future dynamic models - multi-sector or otherwise
- must respect this constraint, or bear the defect of having already been rejected.
restrictive this will be
is
unknown.
To conclude that the data support the
reject
2.7
it
is,
How
null hypothesis
of course, premature since the size
- 23 -
and power
because our statistic
fails to
of our test in finite samples
is
8.89
yet to be determined.
We
perform
Monte Carlo experiments and report the
illustrative
results in the next section.
4.3.
The
Size
and Power of the
To evaluate the
trative
the
size
and power
Monte Carlo experiments
number
Test.
of our test in finite samples,
for a
sample
of quarterly observations of real
we perform
size of 163 observations,
GNP
several illus-
corresponding to
growth from 1947:2 to
We
1987:4.'^^
simulate two null hypotheses: independently and identically distributed increments, and
increments that follow an
the
mean and standard
ARMA(2,2)
process.
deviation of our
standard deviation of our quarterly data
To choose parameter values
{\
-
4>iL
-
using nonlinear least squares.
=
set:
^l
+
{\
i.i.d.
deviates to
match the sample mean and
null hypothesis,
fix
7.9775 x 10~^ and 1.0937 x 10~^ respectively.
ARMA(2,2)
for the
4>2L^)yt
random
we
Under the
simulation,
+ eiL +
we estimate
tt~WN[0,ol)
d2L^)it
The parameter estimates
the model:
(4.8)
are [with standard errors in paren-
theses]:
4>i
=
0.5837
Oi
=
(0.1949)
4>2
=^
(0.1736)
- 0.4844
$2
=
(0.1623)
ti
=
- 0.2825
0.6518
(0.1162)
0.0072
(0.0016)
a1
=
0.0102
Table 3 reports the results of both null simulations.
It is
apparent from the
"I.I.D. Null"
Panel of Table 3 that the 5 percent test based on
the classical rescaled range rejects too frequently.
^* All simulations were performed in double precision on a
each experiment was comprised of 10,000 replications.
VAX
2.7
- 24 -
The
8700 using the
5 percent test using the modified
IMSL
10.0
random number generator DRNNOA;
8.89
=
rescaled range with q
number
estimator d^{q)
it
the size of a 5 percent test based on the classical rescaled range
is
34 percent, whereas the corresponding size using the modified
is
4.8 percent.
As
As the
apparent that modifying the rescaled range by the spectral density
is
critical;
is
size.
becomes more conservative. Under the ARMA(2,2)
of lags increcises to 8, the test
null hypothesis,
nominal
3 rejects 4.6 percent of the time, closer to its
before, the test
R/S
statistic
becomes more conservative when q
is
=
with q
5
increased.
Table 3 also reports the size of tests using the modified rescaled range when the
lag length q
is
chosen optimally using Andrews' (1987) procedure. This data-dependent
procedure entails computing the first-order autocorrelation coefficient p(l) and then setting
the lag length to be the integer- value of
Under the
null,
i.i.d.
percent; under the
may
77-^(l-p2)2
=
&
Andrews' formula
ARMA(2,2)
significantly different
formula
K^
^
A^n
Mn, where.
(4-9)
yields a 5 percent test with empirical size 6.9
alternative, the corresponding size
from the nominal value, the empirical
is
4.1 percent.
size of tests
not be economically important. In addition to
Although
based on Andrews'
optimality properties, the
its
procedure has the advantage of eliminating a dimension of arbitrariness
in
performing the
test.
Table 4 reports power simulations under two fractionally differenced alternatives:
L)
£(
=
function
rji
where d
7e(fc)
of
e^
is
=
1/3,-1/3.
given by:
k)
2
Realizations of fractionally differenced time series
[of
multiplying vectors of independent standard normal
factorization of the 163 x 163 covariance matrix
calibrate the simulations,
a"^
is
2.7
-^
[j
=
l,...,fl),
random
q is
an integer and
- 25 -
Mn
^
variates by the Cholesky-
entries are given
chosen to yield unit variance
where
,
length 163] are simulated by pre-
GNP
'^In addition, Andrews' procedure requires weighting the autocovariances by
-
l^
1
/
,
whose
multiplied by the sample standard deviation of real
1
—
Hosking (1981) has shown that the autocovariance
T(l-2d)T{d +
and West's (1987)
(1
1
-
€t's,
by
(4.10).
the {e^} series
is
To
then
growth from 1947:1 to 1987:4,
j^
(j
=
1,
.
.
.
,
[A/„]) in contrast to
Newey
need not be.
8.89
and to
this series
period.
The
added the sample mean of
is
resulting time series
is
real
GNP
growth over the same sample
used to compute the power of the rescaled range; Table
4 reports the results.
For small values of
power against both
tests
q,
based on the modified rescaled range have reasonable
fractionally differenced alternatives. For example, using one lag the 5
=
percent test against the d
alternative this test hcis 81.1 percent power.
power
declines.
Note that
d=
1/3 alternative has 58.7 percent power; against the
tests baised
on the
powerful than those using the modified
R/S
As the
lag length
classical rescaled
increased, the test's
is
range
however,
statistic. This,
—1/3
is
is
significantly
more
value
when
of
little
distinguishing between long-term versus short-term dependence since the test using the
classical statistic also has
power against some stationary
Finally, note that tests using
against the d
d
—
= —1/3
ARMA
processes.
Andrews' truncation lag formula have reasonable power
alternative but are considerably weaker against the
more
relevant
1/3 alternative.
The simulation evidence
in
Tables 3 and 4 suggest that our empirical results do indeed
support the short-term dependence of
null hypothesis does not
alternatives.
additional
test's size
little
is
finite-order
Of
seem
GNP
with a unit root.
to be explicable by a lack of
course, our simulations were illustrative
Monte Carlo experiments must be performed
and power
is
Our
failure to reject the
power against long-memory
and by no means exhaustive;
before a
full
assessment of the
complete. Nevertheless our modest simulations indicate that there
empirical evidence in favor of long-term
memory
in
GNP
growth
rates.
the direct estimation of long-memory models would yield stronger results and
is
Perhaps
currently
being investigated by several authors.'^®
5.
Conclusion.
This paper hzs suggested a new approach to the stochastic structure of aggregate
output.
Traditional dissatisfaction with the conventional methods - from observations
about the typical spectral shape of economic time
periods - calls for such a reformulation.
series, to
the discovery of cycles at
all
Indeed, recent controversy over deterministic
versus stochastic trends and the persistence of shocks underscores the difficulties even
modern methods have
of identifying the long
run properties of the data.
^'See, for example, Diebold and Rudebusch (1989), Sowell (I989a,b), and Yajima (1985,1988).
2.7
- 26 -
8.89
Fractionally integrated
random
processes provide one explicit approach to the prob-
lem of long-term dependence; naming and characterizing
studying the problem
it
Controlling for
from trends and to
late business cycles
extent that
scientifically.
Jissess
its
this aspect
the
is
first
step in
presence improves our ability to
the propriety of that decomposition.
explains output, long-term dependence deserves study in
own
its
iso-
To the
right. Fur-
thermore, Singleton (1988) has recently pointed out that dynamic macroeconomic models
often link inextricably predictions about business cycles, trends,
too linked
in a
is
and seasonal
effects.
So
long-term dependence: a fractionally integrated process arises quite naturally
dynamic
linear
model via aggregation. This model not only predicts the existence of
fractional noise, but also suggests the character of
its
parameters. This
on the nature of long-term dependence
leads to testable restrictions
in
clciss
of models
aggregate data, and
also holds the promise of policy evaluation.
Advocating a new
were intractable.
class of stochastic processes
would be a
fruitless task
if its
In fact, manipulating such processes causes few problems.
members
We
con-
structed an optimizing linear dynamic model that exhibits fractionally integrated noise,
and provided an
explicit test for such long-term
and Mandelbrot gives us a
R/S
statistic possesses
trative
statistic
dependence. Modifying a
statistic of
Hurst
robust to short-term dependence, and this modified
a well-defined limiting distribution which
computer simulations indicate that
this test
we have
tabulated. Illus-
has power against at least two specific
alternative hypotheses of long-memory.
Two main
conclusions arise from the empirical work and
First, the evidence does not
term dependence
support long-term dependence
null hypothesis occur only
in
Monte Carlo experiments.
GNP.
Rejections of the short-
with detrended data, and
is
consistent with
the well-known problem of spurious periodicities induced by log-linear detrending. Second,
since a trend-stationary
may
model
is
not contained in our null hypothesis, our failure to reject
also be viewed as supporting the first-difference stationary
additional result that the resulting stationary process
is
model of GNP, with the
weakly dependent at most. This
supports and extends the conclusion of Adelman that, at least within the confines of the
available data, there
2.7
is little
evidence of long-term dependence in the business cycle.
- 27 -
8.89
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2.7
- 31 -
Model with Long-Memory Stationary
8.89
C-t:l
Creoteo b> Aoo'e**
o
w
C'eoteo b* Anorcf w
_C
Ap*
CRrt-Ej g.
Table
P{V <
v)
la. Fractiles of the Distribution
Fv{v).
Table
R
GNP;
2.
GNP
from 1947:1 to
1987:4, and Ay*^ indicates the first-differences of the logarithm of real GNP. y^^^ and Ay^^ are
defined similairy for the Friedman and Schwartz series. The classical rescaled range V„, and the
'S analysis of real
y^^^
indicates log-linearly detrended quarterly real
modified rescaled range Vn{q) are reported.
Table
3.
R/S statistic under i.i.d and ARMA(2,2) null hypotheses for the first-difference of
log-GNP The Monte Carlo experiments under the two null hypotheses are independent and consist of 10,000 replications
each Parameters of the i.i.d. simulations were chosen to match the sample mean and variance of quarterly real GNP growth
rates from 1947 1 to 1987:4, parameters of the ARMA(2,2) were chosen to match point estimate* of an ARMA(2,2) model
fitted to the same data set Entries m the column labelled "q' indicate the number of lags used to compute the R/S statistic;
corresponds to Mandelbrot's classical rescaled range, and a non-integer lag value corresponds to the average (across
a lag of
replications) lag value used according to Andrew's (1987) optimal lag formula Standard errors for the empirical sire may be
computed using the usual normal approximation; they are 9.95 x 10~*, 2.18 x 10~^, and 3.00 x 10"^ for the 1, 5, and 10
Finite sample distribution of the modified
real
percent tests respectively.
I.I.D. Null
Hypothesis:
n
Table*.
Power of the modified R/S statistic under a Gaussian fractionally differenced alternative with differencing parameters d =
1/3, —1/3 The Monte Carlo experiments under the two alternative hypotheses are independent and consist of 10,000 replications
each Parameters of the simulations were choeen to match the sample mean and variance of quarterly real GNP growth rates
from 19471 to 1987:4 Entries in the column labelled 'q" indicate the number of lags used to compute the R/S statistic; a
lag of
corresponds to Mandelbrot's classical rescaled range, and a non-integer lag value corresponds to the average (across
Andrew's (1987) optimal lag formula.
replications) lag value used according to
1/3:
n
59^0
1)31
\
Date Due
DEC
kiV
1990
16 1993
WP''''
Lib-26-67
MIT
3
iiBRARiE"; nnpL
i
TDSD DD57DM35
5