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Working Paper 9116
THE SOURCES AND NATURE OF
LONG-TERM MEMORY IN THE BUSINESS CYCLE
by Joseph G. Haubrich and Andrew W. Lo
Joseph G. Haubrich is an economic advisor
at the Federal Reserve Bank of Cleveland.
Andrew W. Lo is a research associate at
the National Bureau of Economic Research
and an associate professor in the Sloan
School of Management at the Massachusetts
Institute of Technology. For helpful
comments, the authors thank 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.
They also gratefully acknowledge research
support from the National Science Foundation,
the Rodney L. White Fellowship at the Wharton
School of Business, the John M. Olin Fellowship
at the National Bureau of Economic Research,
and the University of Pennsylvania Research
Foundation.
Working papers of the Federal Reserve Bank of
Cleveland are preliminary materials circulated
to stimulate discussion and critical comment.
The views stated herein are those of the authors
and not necessarily those of the Federal Reserve
Bank of Cleveland or of the Board of Governors
of the Federal Reserve System.
November 1991
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ABSTRACT
This paper examines the stochastic properties of aggregate
macroeconomic time series from the standpoint of fractionally integrated
models, focusing on the persistence of economic shocks. We develop a simple
macroeconomic model that exhibits long-range dependence, a consequence of
aggregation in the presence of real business cycles. We then derive the
relation between properties of fractionally integrated macroeconomic time
series and those of microeconomic data and discuss how fiscal policy may alter
the stochastic behavior of the former. To implement these results
empirically, we employ a test for fractionally integrated time series based on
the Hurst-Mandelbrot rescaled range. This test, which is robust to short-term
dependence, is applied to quarterly and annual real GNP to determine the
sources and nature of long-term dependence in the business cycle.
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1. Introduction
Questions about the persistence of economic shocks currently occupy an
important place in macroeconomics. Most of the controversy has centered on
whether aggregate time series are better approximated by fluctuations around a
deterministic trend or by a random walk plus a stationary or temporary
component. The empirical results from these studies are mixed, perhaps
because measuring low-frequency components is difficult. Looking at the class
of fractionally integrated processes, which exhibits an interesting type of
long-range dependence in an elegant and parsimonious way, can help to resolve
the problem. This new approach also accords well with the classic NBER
business cycle program developed by Wesley Claire Mitchell, who urged
examination of trends and cycles at all frequencies.
Economic life does not proceed smoothly: There are good times and bad
times, a rhythmical pattern of prosperity and depression. Recurrent downturns
and crises take place roughly every three to five years and thus seem part of
a nonperiodic cycle. Studying such cycles in detail has been the main
activity of twentieth century macroeconomics. Even so, isolating cycles of
these frequencies has been difficult because the data evince many other cycles
of longer and shorter duration. Mitchell (1927, p. 463) remarks, "Time series
also show that the cyclical fluctuations of most (not all) economic processes
occur in combination with fluctuations of several other sorts: secular
trends, primary and secondary, seasonal variations, and irregular
fluctuations." Properly eliminating these other influences has always been
controversial. No less an authority than Irving Fisher (1925) considered the
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business cycle to be a myth, akin to a run of luck at Monte Carlo. In a
similar vein, Slutzk. (1937) suggested that cycles arise from smoothing
procedures used to create the data.
A similar debate is now taking place. The standard methods of removing a
linear or exponential trend assume implicitly that business cycles are
fluctuations around a trend. Other work (e.g., Nelson and Plosser [1982])
challenges this assumption and posits stochastic trends similar to random
walks, highlighting the distinction between temporary and permanent changes.
Since the cyclical, or temporary, component is small relative to the
fluctuation in the trend component (the random walk part) when viewed
empirically, business cycles look more like Fisher's myth.
This is important
for forecasting purposes, because permanent changes (as in the case of a
random walk) have a large effect many periods later, whereas temporary changes
(as in stationary fluctuations around a trend) have small future effects. The
large random walk component also provides evidence against some theoretical
models of aggregate output. Models that focus on monetary or aggregate demand
disturbances as a source of transitory fluctuations cannot explain much output
variation; supply-side or other models must be invoked (see Nelson and Plosser
[I9821 and Campbell and Mankiw [1987]).
The recent studies posit a misleading dichotomy, however. In stressing
trends versus random walks, they overlook earlier work by Mitchell (1927),
Adelman (1965), and Kuznets (1965), who focused on correlations in the data
that fall between secular trends and transitory fluctuations. In the language
of the early NBER, most recent studies miss Kondratiev, Kuznets, and Juglar
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cycles. The longer-run (lower-frequency) properties can be difficult to
handle with conventional ARMA or ARIMA models because such properties involve
what seem to be an excessive number of free parameters. Of course, an MA(120)
fits the post-Civil War annual data quite well, but most of the relations
would be spurious, and it is doubtful how well such an overfitted
specification could predict. Fractionally differenced processes exhibit
long-run dependence by adding only one free parameter, the degree of
differencing, and show promise in explaining the lower-frequency effects
(i.e., Kuznets' (1965) and Adelman's (1965) "long swings," or the effects that
persist from one business cycle to the next).
Standard methods of fitting
Box-Jenkins models have trouble with the number of free parameters needed for
long-term dependence, especially the sort captured by a fractional process.
We think a better approach is a more 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. We introduce this type of
process in section 2 and review its main properties, its advantages, and its
weaknesses. Section 3 develops a simple macroeconomic model that exhibits
long-term dependence. Section 4 employs a new test for fractional integration
in time series to search for long-term dependence in the data. Though related
to a test developed by Hurst and Mandelbrot, our model is robust to short-term
dependence. Section 5 summarizes and concludes.
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2.
Review of Fractional Techniques in Statistics
A random walk can model time series that look cyclic but nonperiodic. The
first differences of that series (or in continuous time, the derivative)
should then be white noise. This is an example of the common intuition that
differencing (differentiating) a time series makes it "rougher," whereas
summing (integrating) makes it "smoother." Many macroeconomic time series
resemble neither a random walk nor white noise, suggesting that some
compromise or hybrid between the random walk and its integral may be useful.
Such a concept has been given content through the development of the
fractional calculus, i.e., differentiation and integration to non-integer
0rders.l The fractional integral of order between zero and one may be
viewed as a filter that smooths white noise to a lesser degree than the
ordinary integral; it yields a series that is rougher than a random walk but
smoother than white noise. Granger and Joyeux (1980) and Hosking (1981)
develop the time-series implications of fractional differencing in discrete
time. For expositional purposes, we review the more relevant properties in
sections 2.1 and 2.2.
2.1.
Fractional Differencing
Perhaps the most intuitive exposition of fractionally differenced time
series is via their infinite-orderautoregressive (AR) and moving-average (MA)
representations. Let $ satisfy
(l-Lld$
= Et,
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where
e,
is white noise, d is the degree of differencing, and L denotes the
lag operator. If d
=
0, then X, is white noise, whereas if d
=
1, X, is a
random walk. 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
where the binomial coefficient
=
(f)
is defined as
d(d-1) (d-2)- (d-k+l)
k!
for any real number d and non-negative integer k.2 From (2.2), the AR
representation of X, is apparent:
where
4 = (-ilk
(i) .
The AR coefficients are often reexpressed more
directly in terms of the gamma function:
4
k
= (-I)
d
(k) =
r k-d
r(-i)r(i+l)
By manipulating (2.1) mechanically, X, may also be viewed as an
infinite-orderMA process, since
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The particular time-series properties of
X, depend intimately on the value
of the differencing parameter d. For example, Granger and Joyeux (1980) and
Hosking (1981) show that
X, is stationary when d is less than one-half,
and invertible when d is greater than minus one-half. Although the
specification in (2.1) is a fractional integral of pure white noise, the
extension to fractional ARIMA models is clear.
The AR and MA representations of fractionally differenced time series have
many applications and illustrate the central properties of fractional
processes, particularly long-term dependence. The MA coefficients 8, give
the effect of a shock k periods ahead and indicate the extent to which current
levels of the process depend on past values. How fast this dependence decays
furnishes valuable information about the process. Using Stirling's
approximation, we have
for large k. Comparing this with the decay of an AR(1) process highlights a
central feature of fractional processes: They decay hyperbolically, at rate
kd- 1
, rather than at the exponential rate of
pk
for an AR(1).
For example,
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compare in figure 1 the autocorrelation function of the fractionally
differenced series (~-L)'.~'~x,= et with that of the AR(1) X, 0.9%
+
6,.
Although they both have first-order autocorrelations of 0.90, the AR(1)'s
autocorrelation function decays much more rapidly.
Figure 2 plots the impulse-response functions of these two processes. At
lag 1, the MA coefficients of the fractionally differenced series and the
AR(1) are 0.475 and 0.900, respectively. At lag 10, these coefficients
are 0.158 and 0.349, while at lag 100, they fall to 0.048 and 0.000027. The
persistence of the fractionally differenced series is apparent at the longer
lags. Alternatively, we may ask what value of an AR(1)'s
autoregressive
parameter will yield, for a given lag, the same impulse response as the
fractionally differenced series (2.1).
This value, simply the k-th root of
Bk, is plotted in figure 3 for various lags when d
- 0.475.
For large k ,
this autoregressive parameter must be very close to unity.
These representations also show how standard econometric methods can fail
to detect fractional processes, necessitating the methods described in section
4. 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 would give the estimation a poor rating from the usual
Akaike or Schwartz criteria. An explicitly fractional process, however,
captures that pattern with a single parameter, d. Granger and Joyeux (1980)
and Geweke and Porter-Hudak (1983) provide empirical support for this by
showing that fractional models often outpredict fitted ARMA models.
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The lag polynomials A(L) and B(L) provide a metric for the persistence of
5. Suppose 5 represents GNP, which falls unexpectedly
this year. How
much should this alter a forecast of GNP? To address this issue, define
%
as the coefficients of the lag polynomial C(L) that satisfy the relation
(1-L)%
=
C(L)E,,
where the process
5 is given by
(2.1).
One measure
used by Campbell and Mankiw (1987) is
lim
k+m
a,
m
=I
%
k=O
=
C(1).
For large k, the value of 8, measures the response of
5+k
to an
innovation at time t, a natural metric for persistence.
immediate that for 0 < d < 1, C(l)
=
From (2.7), it is
0, and that asymptotically, there is no
persistence in a fractionally differenced series, even though the
autocorrelations die out very ~ l o w l y . This
~
holds not only for d < 1/2 (the
stationary case), but also for 1/2 < d < 1 (the nonstationary case).
From these calculations, it is apparent that the long-run dependence of
fractional processes relates to the slow decay of the autocorrelations, not to
any permanent effect. This distinction is important; an IMA(1,l) can have
small yet positive persistence, but the coefficients will never mimic the slow
decay of a fractional process.
The long-term dependence of fractionally differenced time series forces us
to modify some conclusions about decomposing time series into "permanent" and
"temporary" components. Although Beveridge and Nelson (1981) show that
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nonstationary time series can always be expressed as the sum of a random walk
and a stationary process, the stationary component may exhibit long-range
dependence. This suggests that the temporary component of the business cycle
may be transitory only in the mathematical sense and that it is, for all
practical purposes, closer to what we think of as a long, nonperiodic cycle.
2.2. Spectral Representation
The spectrum, or spectral density (denoted f(o)),
of a time series
specifies the contribution 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 the frequency approaches zero (f(w)
-r
as w
-+
0). Most of the power (variance) seems to be concentrated at low frequencies.
However, prewhitening or differencing the data often leads to
overdifferencing, or "zapping out" the low-frequency component, and frequently
replaces the peak by a dip at zero. Fractional differencing yields an
intermediate result. The spectra of fractional processes exhibit peaks at
zero (unlike the flat spectrum of an ARMA process), but ones not so sharp as
those of a random walk. A fractional series has a spectrum that is richer in
low-frequency terms and that shows more persistence. We illustrate this by
calculating the spectrum of fractionally integrated white noise, and present
several formulas needed in sections 3 and 4. Given %
=
(l-~)-~r,,
the series is clearly the output of a linear system with a white noise input,
so that the spectrum of
% is6
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where z = eiw, and
The identity 1 1-21
f( w )
= C W - ~C
~,
=
= E [ E ~ .]
u2
=
2(1-cos(w))
implies that for small w,
2
2'K '
This approximation encompasses the two extremes of white noise (or a finite
ARMA process) and a random walk.
for a random walk, d
=
For white noise, d
=
0 and f(w)
=
c, while
1 and the spectrum is inversely proportional to &.
A class of processes of current interest in the statistical physics
literature, called l/f noise, matches fractionally integrated noise with d
=
1/2.
3.
A Simple Macroeconomic Model with Long-Term Dependence
Over half a century ago, Wesley Claire Mitchell (1927, p. 230) wrote that
"We stand to learn more about economic oscillations at large and about
business cycles in particular, if we approach the problem of trends as
theorists, than if we confine ourselves to strictly empirical work." Indeed,
gaining insights beyond stylized facts requires guidance from theory.
Theories of long-range dependence may provide organization and discipline in
constructing models of growth and business cycles. They can also guide future
research by predicting policy effects, postulating underlying causes, and
suggesting new ways to analyze and combine data. Ultimately, examining the
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facts serves only as a prelude. Economic understanding requires more than a
consensus on the Wold representation of GNP; it demands a falsifiable model
based on the tastes and technology of ifhe actual economy.
Thus, before testing for long-run dependence, we develop a simple model
in which aggregate output exhibits long-run dependence. The model presents
one reason that macroeconomic data might show the particular stochastic
structure for which we test. It also shows that models can restrict the
fractional differencing properties of time series, thus holding promise for
distinguishing between competing theories. Furthermore, the maximizing model
presented below connects long-term dependence to the central economic concepts
of productivity, aggregation, and the limits of the representative-agent
paradigm.
3.1. A Simple Real Model
One plausible mechanism for generating long-run dependence in output,
which we will mention briefly and not pursue, is that production shocks
themselves follow a fractionally integrated process. 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 many economically
important shocks have long-run correlation properties. Mandelbrot and Wallis
(1969b), for instance, find long-run dependence in rainfall, river flows,
earthquakes, and weather patterns (as measured by tree rings and sediment
deposits) .
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A more satisfactory model explains the time-series properties of data by
producing them despite white noise shocks. This section develops such a model
with long-run dependence, using a linear quadratic version of the real
business cycle model of Long and Plosser (1983) and the aggregation results
of Granger (1980).
In our multisector model, the output of each industry (or
island) follows an AR(1) process, but aggregate output with N sections follows
an ARMA (N,N-1) process, making dynamics with even a moderate number of
sectors unmanageable. Under fairly general conditions, however, a simple
fractional process can closely approximate the true ARMA specification.
Consider a model economy with many goods and a representative agent who
chooses a production and consumption plan. The infinitely lived agent
inhabits a linear quadratic version of the real business cycle model and has a
lifetime utility function of U
=
Cptu(C,),
where C, is an Nxl vector
denoting period t consumption of each of the N goods in our economy. Each
period's utility function u(C,)
u(C,)
where
=
L
C,L
-
is given by
l JBC,,
-C
2 t
is an Nxl vector of ones. In anticipation of the aggregation
considered later, we assume B to be diagonal so that
CLBC,
=
ZbiiCZt.
The agent faces a resource constraint: Total output Y, may be either
consumed or saved. Thus,
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where the i ,j - t h entry Sijt of the NxN matrix St denotes the quantity
of good j invested i n process i a t time t , and where i t i s assumed t h a t any
good Yjt may be consumed or invested.
Output i s determined by the
random l i n e a r technology
Yt =ASt
+
E,,
(3.3)
where et i s a (vector) random production shock whose value i s r e a l i z e d a t
the beginning of period t + l .
The matrix A consists of the input-output
To focus on long-term dependence, we r e s t r i c t A's form.
parameters a i j .
Thus, each sector uses only i t s own output a s input, yielding a diagonal A
matrix and allowing us t o simplify the notation by defining ai = aii.
This diagonal case might occur, f o r example, when a number of d i s t i n c t islands
a r e producing d i f f e r e n t goods.
To f u r t h e r simplify the problem, we assume
t h a t a l l commodities a r e perishable and t h a t c a p i t a l depreciates a t a r a t e of
100 percent.
Since the s t a t e of the economy i n each period i s f u l l y specified
by t h a t period's output and productivity shock, it i s useful t o denote t h a t
v e c t o r Z, = [Y;
EL] ' .
Subject t o the production function (3.3) and the resource constraint
( 3 . 2 ) , the agent maximizes expected lifetime u t i l i t y a s follows:
Max E[UI Zt]
{St}
=
MaxE
{St)
[ 7=t
f /37-tu(~t-
St')
I
Zt
],
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where we have substituted for consumption in (3.4) using the budget equation
(3.2).
This maps naturally into a dynamic programming formulation, with a
value function V(Zt) and optimality equation
With quadratic utility and linear production, it is straightforward to
discover and verify the form of V(Zt):
V(Y,E)
- q'Y + Y'PY + R + E[E'TE],
where q and R ¬e
(3.6)
Nxl vectors and P and T are NxN matrices, with entries
being fixed constants given by the matrix Riccati equation resulting from the
value function's recursive definition.'
Given the value function, the
first-order conditions of the optimality equation (3.5) yield the chosen
quantities of consumption and investment/savings and, for the example
presented here, have the following closed-form solutions:
and
where
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The simple form of the optimal consumption and investment decision rules comes
from the quadratic preferences and the linear production function. Two
qualitative features bear emphasizing. 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. production shocks, the time-to-buildfeature of investment induces
serial correlation. Second, the optimal choices do not depend on the
uncertainty that is present. This certainty equivalence feature is clearly an
artifact of the linear-quadratic combination.
The time series of output can now be calculated from the production
function (3.1) and the decision rule (3.7).
Quantity dynamics then come from
the difference equation
where Ki is some fixed constant. The key qualitative property of quantity
dynamics summarized by (3.11) is that output Yi, follows an AR(1) process.
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Higher output today implies higher output in the future. That effect dies off
at a rate that depends on the parameter ai, which in turn depends on the
underlying preferences and technology.
The simple output dynamics for a single industry or island neither mimics
business cycles nor exhibits long-run dependence. However, aggregate output,
the sum across all sectors, does show such dependence, which we demonstrate
here by applying the aggregation results of Granger (1980, 1988).
It is well known that the sum of two series Xt and Y,, each AR(1) with
independent error, is an ARMA(2,l) process. Simple induction then implies
that the sum of N independent AR(1) processes with distinct parameters has an
ARMA(N,N-1) representation. With more than six million registered businesses
in America (Council of Economic Advisors, 1988), the dynamics can be
incredibly rich
--
and the number of parameters unmanageably huge. The common
response to this problem is to pretend that many different firms (islands)
have the same AR(1) representation for output, which reduces the dimensions of
the aggregate ARMA process. This "canceling of roots" requires identical
autoregressive parameters. An alternative approach reduces the scope of the
problem by showing that the ARMA process approximates a fractionally
integrated process and thus summarizes the many ARMA parameters in a
parsimonious manner.
Though we consider only the case of independent sectors,
dependence is easily handled.
Consider the case of N sectors, with the productivity shock for each
serially uncorrelated and independent across islands. Furthermore, let the
sectors differ according to the productivity coefficient ai. This implies
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differences in ai, the autoregressive parameter for sector i's output
Yi,.
One of our key results is that under some distributional assumptions
about ails aggregate output, 3 follows a fractionally integrated
process, where
To show this, we approach the problem from the frequency domain and apply
spectral methods, which often simplify problems of aggregation.
Let f(w)
denote the spectrum (spectral density function) of a random variable, and let
z = e-iw. From the definition of the spectrum as the Fourier transform
of the autocovariance function, the spectrum of Yit is
Similarly, independence implies that the spectrum of
3
is
The ailsmeasure 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 tit for each
sector. Thus, it makes sense to think of the airsas independently drawn
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from a distribution G(a) and the ai's as drawn from F(a).
the
E,,
Provided that
shocks are independent of the distribution of a,'~,the
spectral density of the sum can be written as
If the distribution F(a) is discrete, so that it takes on m (< N) values,
Y: will be an ARMA (m, m-1) process. A more general distribution leads
to a process that no finite ARMA model can represent. To further specify the
process, take a particular distribution for F, in this case a variant of the
beta distrib~tion.~In particular, let a2 be distributed as beta (p,q),
which yields the following density function for a:
(0
otherwise,
with (p,q) > 0.lo
Obtaining the Wold representation of the resulting process requires a
little more work. First, note that
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where 2 denotes the complex conjugate of z, and the terms in brackets can
be further expanded by long division. Substituting this expansion and the
beta distribution (3.16) into the expression for the spectrwn and simplifying
(using the relation z + 2
=
2 cos(w))
yields
Then, the coefficient of cos(h) is
Since the spectral density is the Fourier transform of the autocovariance
function, (3.19) is the k-th autocovariance of
3.
Furthermore,
because the integral defines a beta function, (3.19) simplifies to /3(p+k/2,
q
- 1)/
/3(p,q).
Dividing by the variance gives the autocorrelation
coefficients, which reduce to
Using the result from Stirling ' s approximation r(a+k)/r(b+k)
= ka-b,
(3.20) is proportional (for large lags) to kl-q. Thus, aggregate output
Y
: follows a fractionally integrated process of the order d
=
1- Q
2'
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Furthermore, as an approximation for long lags, this does not necessarily rule
out interesting correlations at higher, e.g., business cycle, frequencies.
Similarly, comovements can arise as the fractionally integrated income process
induces fractional integration in other observed time series. This phenomenon
has been generated by a maximizing model based on tastes and
technologies.l1
In principle, all of the model's parameters may be estimated, from the
distribution of production functions to the variance of output shocks.
Although to our knowledge no one has explicitly estimated the distribution of
production function parameters, many people have estimated production
functions across industries.12 (One of the better recent studies
disaggregates to 45 industries.13) For our purposes, the quantity
closest to a, is the value-weighted intermediate-product factor share.
Using a translog production function, this gives the factor share of inputs
coming from industries, excluding labor and capital. These inputs range from
a low of 0.07 for radio and television advertising to a high of 0.81 for
petroleum and coal products. Thus, even a small amount of disaggregation
reveals a large dispersion, suggesting the plausibility and significance of
the simple model presented in this section.
Although the original motivation for our real business cycle model was to
illustrate how long-range dependence could arise naturally in an economic
system, our results have broader implications for general macroeconomic
modeling. They show that moving to a multiple-sector real business cycle
model introduces not unmanageable complexity, but qualitatively new behavior
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that in some cases can be quite manageable. Our findings also show that
calibrations aimed at matching only a few first and second moments can
similarly hide major differences between models and the data, missing long-run
dependence properties. While widening the theoretical horizons of the
paradigm, fractional techniques also widen the potential testing of such
theories.
3.2. Fiscal Policy and Welfare Implications
Taking a policy perspective raises two natural questions about the
fractional properties of national income. First, will fiscal or monetary
policy change the degree of long-term dependence? Friedman and Schwartz
(1982), for example, point out that long-run income cycles correlate with
long-run monetary cycles. Second, does long-term dependence have welfare
implications? Do agents care that they live in such a world?
In the basic Ramsey-Solow growth model, as in its stochastic extensions,
taxes affect output and capital levels but not growth rates; thus, tax policy
does not affect fractional properties.l4
However, two alternative
approaches suggest richer possibilities. First, recall that fractional noise
arises through the aggregation of many autoregressive processes. Fiscal
policy may not change the coefficients of each process, but a tax policy can
alter the distribution of total output across individuals, effectively
changing the fractional properties of the aggregate. Second, endogenous
growth models often allow tax policy to affect growth rates by reducing
investment in research, thus depressing future growth.l5
Hence, the
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22
autoregressive parameters of an individual firm's output could change with
policy, in turn affecting aggregate income.
Unfortunately, implementing either approach with even a modicum of realism
would be quite complicated. In the dynamic stochastic growth model, taxation
drives a wedge between private and social returns, resulting in a suboptimal
equilibrium. This eliminates methods that exploit the pareto-optimality of
competitive equilibrium, such as dynamic programming. Characterizing
solutions requires simulation methods, because no closed forms have been
found.16 Thus, it seems clear that fiscal policy can affect fractional
properties.
Explicitly calculating the impact would take this paper too far
afield and is best left for future research.
Those who forecast output or sales will care about the fractional nature
of output, but fractional processes can have normative implications as well.
Following Lucas (1987), this section estimates the welfare costs of economic
instability under different regimes. We can decide if people care whether
their world is fractional. For concreteness, let the typical household
consume C,, evaluating this via a utility function:
Also assume that
m
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23
where
9, =
In r , .
The X term measures compensation for variations
in the process 4(L).
With
9,
normally distributed with mean zero and
variance one, the compensating fraction X between two processes
1
+
X
=
exp
[ $ (1 -
4 and
1/, is
m
0)
1 (1/,: - 43 ] -
(3.23)
k=O
Evaluating (3.23) using a realistic a
=
5, again comparing an AR(1) with
0.9 against a fractional process of order one-fourth,we find that X
p =
=
-0.99996.(This number looks larger than those in Lucas [1987] because the
process is in logs rather than in levels.17) For comparison, this is
the difference between an AR(1) with p of 0.90 and one with p of 0.95. This
calculation provides only a rough comparison. When feasible, welfare
calculations should use the model generating the processes, as only it will
correctly account for important specifics such as labor supply or
distortionary taxation.
4. Rescaled Range Analysis of Real Output
The results in section 3 show that simple aggregation may be one source of
long-term dependence in the business cycle. In this section, we employ a
method for detecting long memory and apply it to real GNP. The technique is
based on a simple generalization of a statistic first proposed by the English
hydrologist Harold Edwin Hurst (1951) and subsequently refined by Mandelbrot
(1972, 1975) and others.18 Our generalization of Mandelbrot's
statistic, called the rescaled range, the range over standard deviation, or
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24
the R/S statistic, enables us to distinguish between short- and long-run
dependence, in a sense that will be made precise below. We define our notions
of short and long memory and present the test statistic in section 4.1.
Section 4.2 gives the empirical results for real GNP. We find long-term
dependence in log-linearly detrended output, but considerably less dependence
in the growth rates. To interpret these findings, we perform several Monte
Carlo experiments under two null and two alternative hypotheses. Results are
reported in section 4.3.
4.1. The R/S Statistic
To develop a method of detecting long memory, we must be precise about the
distinction between long-term and short-term statistical dependence. One of
the most widely used concepts of short-term dependence is the notion of
"strong-mixing" (based on Rosenblatt [1956]), a measure of the decline in
statistical dependence of two events separated by successively longer time
spans. Heuristically, a time series is strong-mixing if the maximal
dependence between any two events becomes trivial as more time elapses between
them. By controlling the rate at which the dependence between future events
and those of the distant past declines, it is possible to extend the usual
laws of large numbers and central-limit theorems to dependent sequences of
random variables. Such mixing conditions have been used extensively by White
(1980), 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
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25
of our null hypothesis. As Phillips (1987) observes, these conditions are
satisfied by a great many stochastic processes, including all Gaussian
finite-order stationary ARMA models. Moreover, the inclusion of a moment
condition allows for heterogeneously distributed sequences (such as those
exhibiting heteroscedasticity) , an especially important extension in view of
the nonstationarities of real GNP.
In contrast to the "short memory" of weakly dependent (i.e.,
strong-mixing) processes, natural phenomena often display long-term memory in
the form of nonperiodic cycles. This has led several authors, most notably
Mandelbrot, to develop stochastic models that exhibit dependence even over
very long time spans. The fractionally integrated time-series 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, violating the conditions of strong-mixing. To detect long-term
dependence (also called strong dependence), Mandelbrot suggests using the R/S
statistic, which is the range of partial sums of deviations of a time series
from its mean, rescaled by its standard deviation. In several seminal papers,
Mandelbrot demonstrates the superiority of the R/S statistic over more
conventional methods of determining long-run dependence, such as
autocorrelation analysis and spectral analysis.l9
In testing for long memory in output, we employ a modification of the R/S
statistic that is robust to weak dependence. In Lo (1991), a formal sampling
theory for the statistic is obtained by deriving its limiting distribution
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analytically using a functional central-limit theorem.
20
We use this
statistic and its asymptotic distribution for inference below. Let Xt
denote the first difference of log-GNP;we assume that
where p is an arbitrary but fixed parameter. Whether or not
long-term memory depends on the properties of E,.
H, the sequence of disturbances
(Al)
E[et]
(A2)
suptE[
=
E,
( E ~ ) is
For the null hypothesis
satisfies the following conditions:
0 for all t.
JE,~']
<
a
for some p > 2.
exists, and
(A4)
X, exhibits
u2
>0
strong-mixing,with mixing coefficients % that
satisfy21
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 numbers and the (functional)
central-limit theorem to obtain. Note that we have not assumed stationarity.
Although condition (A2) rules out infinite-variance marginal distributions of
E ~ ,such
as those in the stable family with characteristic exponent less
than two, the disturbances may still exhibit leptokurtosis via time-varying
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27
conditional moments (e.g., conditional heteroscedasticity).
Moreover, since
there is a trade-off between conditions (A2) and (A4), the uniform bound on
the moments may be relaxed if the mixing coefficients decline faster than (A4)
requires.22 For example, if we require
6,
to have finite absolute
moments of all orders (corresponding to /3
+ co),
faster than l/k. However, if we restrict
6,
then % must decline
to have finite moments only up
to order four, then % must decline faster than l/k2. These conditions
are discussed at greater length in Phillips (1987), to which we refer
interested readers.
Conditions (Al) through (A4) are satisfied by many of the recently
proposed stochastic models of persistence, such as the stationary AR(1) with a
near-unit root. Although the distinction between dependence in the short
versus the long run may appear to be a matter of degree, strongly dependent
processes behave so differently from weakly dependent ones that our dichotomy
seems quite natural. For example, the spectral densities of strongly
dependent processes are either unbounded or zero at frequency zero. Their
partial sums do not converge in distribution at the same rate as weakly
dependent series, and graphically, their behavior is marked by cyclic patterns
of all kinds, some that are virtually indistinguishable from trends.23
To construct the modified R/S statistic, consider a sample XI, 3 ,
...,
X,, and let
En denote the sample mean
1
lj X,.
Then, the modified R/S statistic, which we shall call
a, is given by
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a=-
k
-
[x, - gn)
- Min
1k [xj - a,)],
j=l
where
2
and 6,
and 7 are the usual sample variance and autocovariance estimators
j
of X. Q,, is the range of partial sums of deviations of Xj from its mean,
k, normalized by
an estimator of the partial sum's standard deviation
divided by n. The estimator 3,(q)
involves not only sums of squared
deviations of Xj, but also its weighted autocovariances up to lag q; the
weights wj(q) are those suggested by Newey and West (1987), and they always
2
yield a positive estimator 6,(
q)
.24
Theorem 4 . 2 in Phillips
(1987) demonstrates the consistency of 3,(q)
under the following
conditions:
<
for some B > 2 .
(A2')
supt ~[lr,~~']
(A5)
As n increases without bound, q also rises without bound, such that
- o(n1/4 .
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29
The choice of the truncation lag q is a delicate matter. Although q must
increase with the sample size
(although at a slower rate), Monte Carlo
evidence suggests that when q becomes large relative to the number of
observations, asymptotic approximations may fail dramatically.
If the
chosen q is too small, however, the effects of higher-order autocorrelations
may not be captured. Clearly, the choice of q is an empirical issue that muust
take into account the data at hand.
Under conditions (Al), (A2'), (A3)
statistic V,, =
...
A(5),
Lo (1991) shows that the
has a well-defined asymptotic distribution given by
the random variable V, whose distribution function Fv (v) isz6
Using F,, critical values may be readily calculated for tests of any
significance level. The most commonly used values are reported in tables la
and lb. Table la reports the fractiles of the distribution, while table lb
reports the symmetric confidence intervals about the mean. The moments of V
are also easily computed using the density function fv; it is
straightforward to show that E[V]
=
$ and E[V 2 ]
=
7r2
-.
6
Thus,
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30
the mean and standard deviation of V are approximately 1.25 and 0.27,
respectively. The distribution and density functions are plotted in figure 4.
Note that the distribution is positively skewed and that most of its mass
falls between three-fourths and two.
If the obsemations are independently and identically distributed with
variance a:, our normalization by 3,(q)
is asymptotically equivalent to
1 lj(xj
normalizing by the usual standard deviation estimator sn = [ii
The resulting statistic, which we call
-
En)2]1'2.
on, is precisely the one proposed
by Hurst (1951) and Mandelbrot (1972):
O n n' 2 [
Max
k
1
j=l
Xj -
En -
Min
19511
k
1
(xj - En}] .
j=l
Under the more restrictive null hypothesis of i.i.d. observations, the
vn = Gn/fi can be shown to converge to V as well. However, in
the presence of short-range dependence, vn does not converge to V,
statistic
whereas Vn still does. Of course, if the particular form of short-range
dependence is known, it can be accounted for in deriving the limiting
distribution of vn. For example, if Xt is a stationary AR(1) with
autoregressive parameter p , Lo (1991) shows that
where ( = *j(l+p)/(l-p).
vn converges to (V,
But since we would like our limiting
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31
distribution to be robust to general forms of short-range dependence, we
use the modified R/S statistic Vn below.
4.2. Empirical Results for Real Output
We apply our test to two time series of real output: quarterly postwar
real GNP from 1947:IQ to 1987:IVQ, and the annual Friedman and Schwartz (1982)
series from 1869 to 1972. These results are reported in table 2. Entries in
the first numerical row are estimates of the classical R/S statistic
f,,
which is not robust to short-term dependence. The next eight rows are
estimates of the modified R/S statistic Vn(q) for values of q from one to
eight. Recall that q is the truncation lag of the spectral density estimator
at frequency zero. Reported in parentheses below the entries for Vn(q) are
estimates of the percentage bias of the statistic qn, computed as
100
[fn/vn(q>
-
11.
The first column of numerical entries in table 2 indicates that the null
hypothesis of short-term dependence for the first difference of log-GNP cannot
be rejected for any value of q. The classical R/S statistic also supports the
null hypothesis, as do the results for the Friedman and Schwartz series. On
the other hand, when we log-linearly detrend real GNP, the results are
considerably different. The third column of numerical entries in table 2
shows that short-term dependence may be rejected for log-linearly detrended
quarterly output with values of q from one to four. That the rejections are
weaker for larger q is not surprising, since additional noise arises from
estimating higher-order autocorrelations. When values of q beyond four are
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32
used, we no longer reject the null hypothesis at the 5 percent level of
significance. Finally, using the Friedman and Schwartz time series, we only
reject with the classical R/S statistic and with V,(l).
The values reported in table 2 are qualitatively consistent with the
results of other empirical studies of fractional processes in GNP, such as
Diebold and Rudebusch (1989) and Sowell (1989).
For first differences, the
R/S statistic falls below the mean, suggesting a negative fractional exponent,
or in level terms, an exponent between zero and one. Furthermore, though the
earlier papers produce point estimates, the imprecision of these estimates
means that they do not reject the hypothesis of short-term dependence. For
example, the standard-deviation error bounds for Diebold and Rudebusch's
two point estimates, d
=
0.9 and d
=
0.52, are (0.42, 1.38) and (0.06, 1.10),
respectively.
Taken together, our results confirm the unit-root findings of Campbell and
Mankiw (1987), Nelson and Plosser (1982), Perron and Phillips (1987), and
Stock and Watson (1986).
That there are more significant autocorrelations in
log-linearly detrended GNP is precisely the spurious periodicity suggested by
Nelson and Kang (1981).
Moreover, the trend plus stationary noise model of
GNP is not contained in our null hypothesis; hence, our failure to reject the
null hypothesis is also consistent with the unit-root model . 2 7
To see
this, observe that if log-GNP y were trend stationary, i.e., if
t
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33
where
r,~,
is stationary white noise, then its first difference X, would
simply be X,
=
/3
+
r , , where r , = 'It
-
But this innovations
process violates our assumption (A3) and is therefore not contained in our
null hypothesis.
Sowell (1989) has used estimates of d to argue that the trend-stationary
model is correct. Following the lead of Nelson and Plosser (1982), he
investigates whether the d parameter for the first-differenced series is close
to zero, as the unit-root specification suggests, or close to minus one, as
the trend-stationary specification suggests. His estimate of d is in the
general range of -0.9 to -0.5, providing some evidence that the
trend-stationary interpretation is correct. Even in this case, however, the
standard errors tend to be large, on the order of 0.36. Although our
procedure yields no point estimate of d, it does seem to rule out the
trend-stationary case.
To conclude that the data support the null hypothesis because our
statistic fails to reject it is premature, of course, since the size and power
of our test in finite samples is yet to be determined.
4.3. Size and Power of the Test
To evaluate the size and power of our test in finite samples, we perform
several illustrative Monte Carlo experiments for a sample of 163 observations,
which corresponds to the number of quarterly observations of real GNP growth
~ simulate two null hypotheses:
from 1947:IQ to 1987:I V Q . ~We
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independently and identically distributed increments, and increments that
follow an ARMA(2,2) process. Under the i.d.d. null hypothesis, we fix the
mean and standard deviation of our random deviates to match the sample mean
and standard deviation of our quarterly data set: 7.9775 x
1.0937 x
and
respectively. To choose parameter values for the
ARMA(2,2) simulation, we estimate the model
using nonlinear least squares. The parameter estimates are as follows
(standard errors are in parentheses):
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 R/S statistic rejects too frequently. The 5
percent test using the modified R/S statistic with q
=
3 rejects 4 . 6 percent
of the time, closer to the nominal size. As the number of lags increases to
eight, the test becomes more conservative. Under the ARMA(2,2) null
hypothesis, it is apparent that modifying the R/S statistic by the spectral
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2
density estimator &,(q)
is critical. The size of a 5 percent test based on the
classical R/S statistic is 34 percent, whereas the corresponding size using
the modified R/S statistic with q
=
5 is 4.8 percent. As before, the test
becomes more conservative when q is increased.
Table 3 also reports the size of tests using the modified R/S statistic
when the lag length q is optimally chosen using Andrews' (1987) procedure.
This data-dependent procedure entails computing the first-order
autocorrelation coefficient j(1) and then setting the lag length as the
integer value of
fin,
wherez9
Under the i.i.d. null hypothesis, Andrews' formula yields a 5 percent test
with empirical size 6.9 percent; under the ARMA(2,2) alternative, the
corresponding figure is 4.1 percent. Although significantly different from
the nominal value, the empirical size of tests based on Andrews' formula may
not be economically important. In addition to its optimality properties, the
procedure has the advantage of eliminating a dimension of arbitrariness from
the test. Table 4 reports power simulations under two fractionally differenced
alternatives: (1
- L)d et
=
qt, where d
=
(1/3, -1/3).
has shown that the autocovariance function yc(k) equals
Hosking (1981)
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Realizations of fractionally differenced time series of length 163 are
simulated by pre-multiplying vectors of independent standard normal random
variates by the Cholesky factorization of the 163 x 163 covariance matrix,
whose entries are given by (3.11).
is chosen to yield unit variance
E,.
To calibrate the simulations, a:
We then multiply the
e,
series by
the sample standard deviation of real GNP growth from 1947:IQ to 1987:IVQ and
add the sample mean of real GNP growth over the same period. The resulting
time series is used to compute the power of the R/S statistic (see table 4).
For small values of q, tests based on the modified R/S statistic have
reasonable power against both of the fractionally differenced alternatives.
For example, using one lag, the 5 percent test has 58.7 percent power against
the d
=
1/3 alternative and 81.1 percent power against the d
=
-1/3
alternative. As the lag is increased, the test's power declines.
Note that tests based on the classical R/S statistic are significantly
more powerful than those using the modified R/S statistic. This, however, is
of little value when distinguishing between long-term and short-term
dependence, since the test using the classical statistic also has power
against some stationary finite-order ARMA processes. Finally, note that tests
using Andrews' truncation lag formula have reasonable power against the d
=
-1/3 alternative, but are considerably weaker against the more relevant d
=
1/3 alternative.
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37
The simulation evidence in tables 3 and 4 suggests that our empirical
results do indeed support the short-term dependence of GNP with a unit root.
Our failure to reject the null hypothesis does not seem to be explicable by a
lack of power against long-memory alternatives. Of course, our simulations
are illustrative and by no means exhaustive; additional Monte Carlo
experiments will be required before a full assessment of the test's size and
power is complete. Nevertheless, our modest simulations indicate that there
is little empirical evidence of long-term memory in GNP growth rates. Perhaps
a direct estimation of long-memory models would yield stronger results, an
issue that has recently been investigated by several authors.30
5
.
Conclusion
This paper has suggested a new approach for investigating the stochastic
structure of aggregate output. Traditional dissatisfaction with conventional
methods
--
from observations about the typical spectral shape of economic time
series to the discovery of cycles at all periods - - calls for such a
reformation. Indeed, recent controversy about deterministic versus stochastic
trends and the persistence of shocks underscores the difficulties even modem
methods have in identifying the long-run properties of the data.
Fractionally integrated random processes provide one explicit approach to
the problem of long-term dependence; naming and characterizing this aspect is
the first step in studying the problem scientifically. Controlling for
long-term dependence improves our ability to isolate business cycles from
trends and to assess the propriety of that decomposition. To the extent that
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38
long-term dependence explains output, it deserves study in its own right.
Furthermore, Singleton (1988) has pointed out that dynamic macroeconomic
models often inextricably link predictions about business cycles, trends, and
seasonal effects. So, too, is long-term dependence linked: A fractionally
integrated process arises quite naturally in a dynamic linear model via
aggregation. Our model not only predicts the existence of fractional noise,
but suggests the character of its parameters. This class of models leads to
testable restrictions on the nature of long-term dependence in aggregate data,
and also holds the promise of enhancing policy evaluation.
Advocating a new class of stochastic processes would be a fruitless task
if its members were intractable. But in fact, manipulating such processes
causes few problems. We construct an optimizing linear dynamic model that
exhibits fractionally integrated noise, and provide an explicit test for such
long-term dependence. Modifying a statistic developed by Hurst and Mandelbrot
gives us a statistic robust to short-term dependence. This modified R/S
statistic possesses a well-defined limiting distribution, which we have
tabulated. Illustrative computer simulations indicate that this test has
power against at least two specific alternative hypotheses of long-term
memory.
Two main conclusions arise from our empirical work and from Monte Carlo
experiments. First, the evidence does not support long-term dependence in
GNP. Rejections of the short-term-dependence null hypothesis occur only with
detrended data and are consistent with the well-known problem of spurious
periodicities induced by log-linear detrending. Second, since a
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39
trend-stationary model is not contained in our null hypothesis, our failure to
reject may also be viewed as supporting the first-difference stationary model
of GNP, with the additional result that the stationary process is at best
weakly dependent. This supports and extends Adelman's conclusion that, at
least within the confines of the available data, there is little evidence of
long-term dependence in the business cycle.
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Footnotes
1.
The idea of fractional differentiation is an old one (dating back to an
oblique reference by Leibniz in 1695), but the subject lay dormant until
the nineteenth century, when Abel, Liouville, and Riemann developed it
more fully. Extensive applications have only arisen in this century;
see, for example, Oldham and Spanier (1974). Kolmogorov (1940) was
apparently the first to notice its applicability in probability and
statistics.
2.
When d is an integer, (2.3) reduces to the better-known formula for the
d!
We follow the convention that
binomial coefficient, k! (d-k)
!'
= 1 and
= 0.
(8)
(8)
3.
See Hosking (1981) for further details.
4.
See Cochrane (1988) andQuah (1987) for opposing views.
5.
There has been some confusion about this point in the literature. Geweke
and Porter-Hudak (1983) argue that C(l) > 0. They correctly point out
that Granger and Joyeux (1980) erred, but then incorrectly claim that
(1) = 1
) If our equation (2.7) is correct, then it is apparent
that C(l) = 0 (which agrees with Granger [I9801 and Hosking [1981]).
Therefore, the focus of the conflict lies in the approximation of the
ratio r(k+d)/r(k+l)
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 the solution to the following
recursive relation (see, for example, Iyanaga and Kawada [1980, section
179.A]) :
r(x+i) = x~(x)
and the conditions
r(1)
=
r(x+n) 1 lim - 1.
n - . ~ nxI'(n)
6.
See Chatfield (1984, chapters 6 and 9).
7.
See Sargent (1987, chapter 1) for an excellent exposition.
8.
See Theil (1954).
9.
Granger (1980) conjectures that this particular distribution is not
essential.
10. For a discussion of the variety of shapes the beta distribution can take
as p and q vary, see Johnson and Kotz (1970).
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Two additional points are worth emphasizing. First, the beta
distribution need not be over (0,l) to obtain these results, only over
( 1 ) Second, it is indeed possible to vary the aifsso that ai
has a beta distribution.
Leontief, in his classic (1976) study, reports own-industry output
coefficients for 10 sectors, investigating how much an extra unit of food
will increase food production. Results vary from 0.06 (fuel) to 1.24
(other industries).
See Jorgenson, Gollop, and Fraumeni (1987).
See Atkinson and Stiglitz (1980).
For example, see Romer (1986) and King, Plosser, and Rebelo (1987).
See King, Plosser, and Rebelo (1987), Baxter (1988), and Greenwood and
Huffman (1991).
We calculate this using (2.7) and the Hardy-Littlewood approximation for
the resulting Rieman Zeta Function, following Titchmarsh (1951, section
4.11).
See Mandelbrot and Taqqu (1979) and Mandelbrot and Wallis (1968,
1969a-c) .
See Mandelbrot (1972, 1975), Mandelbrot and Taqqu (1979), and Mandelbrot
and Wallis (1968, 1969a-c).
This statistic is asymptotically equivalent to Mandelbrot's under
independently and identically distributed observations. However, Lo
(1991) 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 size, which involves no statistical inference;
such a relation clearly is unaffected by short-term dependence.
)
a stochastic process on the probability space (fl,
Let ( E ~ ( w ) be
F, P) and define
sup IP(AnB) - P(A)P(B)I
AcF,BcF
(A-l,Wl
The quantity a(A,B) is a measure of the dependence between the two
a(A,B) =
a fields A and B in F.
Denote by B: the Bore1 a field generated
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t
by [E,(w), . . . , E~(w)], i.e., B, = u[E,(w),
the coefficients cr, as
cr, = sup
a
...,
E~(w)] c F. Define
(B'-~, Bj+k" ) .
j
Then, (E,(w))
is said to be strong-mixing if lim cr,
=
0.
0.00
For further details, see Rosenblatt (1956), White (1984), and the papers
in Eberlein and Taqqu (1986).
See Herndorf (1985). Note that one of Mandelbrot's (1972) arguments in
favor of R/S analysis is that finite second moments are not required.
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 for purposes of inference, a stronger moment
condition is needed.
See Mandelbrot (1972) for further details.
ui(q) is also an estimator of the spectral density function of
Xt at frequency zero, using a Bartlett window.
See, for example, Lo and MacKinlay (1988).
V may be shown to be the range of a Brownian bridge on the unit interval.
See Lo (1991) for further details.
Of course, this may be the result of low power against stationary but
near-integrated processes, an issue that must be addressed by Monte Carlo
experiments.
All simulations were performed in double precision on a VAX 8700 using
the IMSL 10.0 random number generator DRNNOA. Each experiment consisted
of 10,000 replications.
In addition, Andrews' procedure requires weighting the autocovariances by
j (j = 1,
1
-
1
(j
q+l
=
...,
1,
[%I ) , in contrast to Newey and West's (1987)
. . . , q),
where q is an integer but
(4) need not be.
See, for example, Diebold and Rudebusch (1989), Sowell (1987), and Yajima
(1985, 1988).
www.clevelandfed.org/research/workpaper/index.cfm
43
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www.clevelandfed.org/research/workpaper/index.cfm
pj f o r
(I-L)
0
-
-
475
Et
7
m
z
0
0
-
=
t
6
0
DL
x
0*
0
g
t
3
Q
h]
0
0
00
30
60
90
120
LAG
Source : Authors
Figure 1
Autocorrelation functions of an AR(1) with coefficient 0.90 (dashed line) and
a fractionally differenced series X, = (1 - L ) - ~ c , with differencing parameter
d = 0.475 (solid line). Although both processes have a first-order
autocorrelation of 0.90, the fractionally differenced process decays much more
slowly.
www.clevelandfed.org/research/workpaper/index.cfm
Source:
Authors
Figure 2
Impulse-response function (solid line) of the fractionally differenced time
series X,
- (1 -
L ) - ~ Efor
~ differencing parameter d
- 0.475.
For comparison,
the impulse-response function of an AR(1) with autoregressive parameter 0.90
is also plotted (dashed line).
www.clevelandfed.org/research/workpaper/index.cfm
Equivalent p of AR(I)
w
0
- . \ . : . .. . . . . ... . . . ... .
.
- . . :.
.
:
. . ......
.
.
-
.
..r
.
.
.
.
:
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
..
.
.
.
- . . . . . . ... . . . .. . . . .. . . . .. . . . ... . . . ... . . . .. . . ... . . . ... . . . .. . .
-
.
---
.
.:
.
.
.
.
.
.
.
......
.
.
.
.
.
.
.
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.
0
~
"
"
. . ,.
.
13 0 "
..
.
.
.
.
.
.
.
.
. . . .. . . . . . . . . . . . . . . . . . . ..
.
.
.
-
.
.
.
.
.
.
. i . . . . . . . . . . . . . . . . . . . . . . I. . . . . : . . , . . . . ..
.
.
.
.
.
.
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.
.
.
.
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.
.
.
.
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.
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- . _ I _
................................................... -.
.,-..
.-.
.&.
..,. .
.
.
.
.
.
.
.
.
.
.
.
.
.
. .
.
.
:
.
.
.
.
.
.
.
.
.
.
0
..........
.<
.
.
. . :. . . . ... . . . . . . . . . . . ..I . . . -. - . . . . : . . .
. . .: . . . .. . .
.
.
.
.
.
.
.
.
.
.
. . . . . .. . ." . . . . . . .. . . . . . .
.. ' . ". ........................
;
. . . . . . . . . . .
"
~
"60 "
_
;
-
1
"
"90
120
LAG
Source: Authors
Figure 3
Values of an AR(1)'s autoregressive parameter required to generate the same
k-th order autocorrelation as the fractionally differenced series X, =
(1 - ~ ) - ~ e for
,
differencing parameter d = 0.475 (solid line). Formally,
this is simply the k-th root of the fractionally differenced series'
impulse-response function (dashed line). For large k , the autoregressive
parameter must be very close to unity.
www.clevelandfed.org/research/workpaper/index.cfm
Source : Authors
Figure 4
Distribution and density function of the range V of a Brownian bridge. Dashed
curves are the normal distribution and density functions with mean and
variance equal to those of V.
www.clevelandfed.org/research/workpaper/index.cfm
Table la.
Table Ib.
Fractiles of the Distribution Fv(v)
Symmetric Confidence Intervals about the Mean
Source: Authors
www.clevelandfed.org/research/workpaper/index.cfm
R/S analysis of real GNP; .
Table 2
indicates log-linearly detrended
quarterly real GNP from 1947:IQ to 1987:IVQ, and
indicates the first
differences of the logarithm of real GNP.
and AfS are defined
g:.
similarly for the Friedman and Schwartz series. The classical R/S statistic
9, and the modified R/S statistic Vn(q) are reported.l
-
Under the null hypothesis H (conditions [All , [A2' ], and [A31 [AS]) ,
the limiting distribution of V,(q) is the range of a Brownian bridge, which
m.
has a mean of
Fractiles are given in table la; the 95 percent
confidence interval with equal probabilities in both tails is (0.809, 1.862).
Entries in the %-Bias rows are computed as ([~,/v,(~)]"~
- 1) 100 and are
estimates of the bias of the classical R/S statistic in the presence of
short-term dependence. Asterisks indicate significance at the 5 percent
level.
Source: Authors
www.clevelandfed.org/research/workpaper/index.cfm
Table 3
Finite sample distribution of the modified R/S statistic under i.i.d. and
ARMA(2,2) null hypotheses for the first difference of real 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:IQ to 1987:IVQ; parameters of the ARMA(2,2) were chosen to
match point estimates of an ARMA(2,2) model fitted to the same data set.
Entries in the column labeled "q" indicate the number of lags used to compute
the R/S statistic. A lag of zero corresponds to Mandelbrot's classical R/S
statistic, and a non-integer lag value corresponds to the average (across
replications) lag value used according to Andrews' (1991) optimal lag formula.
Standard errors for the empirical size may be computed using the usual normal
approximation; they are 9.95 x
2.18 x
5, and 10 percent tests, respectively.
and 3.00 x
for the 1,
i.i.d. Null Hypothesis:
n
Q
Min
Max
Mean
S. D.
Size 1%-Test
Siee 5%-Test
Size 10%-Test
163
0
0.522
2.457
1.167
0.264
0.022
0.081
0.138
163
1.5
0.525
2.457
1.171
0.253
0.015
0.069
0.121
163
163
163
163
163
163
163
163
1
2
3
4
5
6
7
8
0.533
0.564
0.602
0.641
0.645
0.636
0.648
0.657
2.423
2.326
2.221
2.136
2.087
2.039
1.989
1.960
1.170
1.174
1.179
1
.
1.189
1.193
1.198
1.203
0.254
0.246
0.239
0.232
0.225
0.219
0.213
0.207
0.016
0.011
0.009
0.006
0.004
0.002
0.000
0.000
0.069
0.058
0.046
0.036
0.030
0.024
0.018
0.015
0.125
0.111
0.097
0.082
0.071
0.061
0.050
0.040
ARMA(2,2) Null Hypothesis:
n
q
Min
Max
Mean
S. D .
Size 1%-Test
Size 5%-Test
Size 10%-Test
163
0
0.746
3.649
1.730
0.396
0.175
0.340
0.442
163
6.8
0.610
2.200
1.177
0.229
0.009
0.041
0.OM
163
163
163
163
163
163
163
163
1
2
3
4
5
6
7
8
0.626
0.564
0.550
0.569
0.609
0.616
0.629
0.644
3.027
2.625
2.412
2.294
2.241
2.181
2.109
2.035
1.439
1.273
1.202
1.180
1.178
1.180
1.180
1.179
0.321
0.279
0.257
0.244
0.236
0.229
0.222
0.215
0.034
0.010
0.012
0.012
0.010
0.008
0.006
0.005
0.110
0.054
0.055
0.054
0.048
0.040
0.034
0.030
0.182
0.111
0.108
0.102
0.093
0.082
0.079
0.066
Source: Authors
www.clevelandfed.org/research/workpaper/index.cfm
Table 4
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 chosen to
match the sample mean and variance of quarterly real GNP growth rates from
1947:IQ to 1987:IVQ. Entries in the column labeled "q" indicate the number of
lags used to compute the R/S statistic; a lag of zero corresponds to
Mandelbrot's classical R/S statistic, and a non-integer lag value corresponds
to the average (across replications) lag value used according to Andrews'
(1991) optimal lag formula.
n
q
Min
Max
Mean
S.
163
0
0.824
4.659
2.370
163
6.0
0.702
2.513
163
163
163
163
163
163
163
163
1
2
3
4
5
6
7
8
0.751
0.721
0.708
0.696
0.700
0.700
0.699
0.694
3.657
3.140
2.820
2.589
2.417
2.297
2.195
2.107
D.
Power 1%-Test
Power 5%-Test
Power 10%-Test
0.612
0.637
0.778
0.839
1.524
0.286
0.017
0.126
0.240
2.004
1.811
1.688
1.600
1.534
1.482
1.440
1.405
0.478
0.409
0.363
0.330
0.304
0.282
0.264
0.249
0.416
0.254
0.141
0.068
0.027
0.008
0.001
0.000
0.587
0.448
0.331
0.234
0.158
0.096
0.056
0.027
0.680
0.545
0.440
0.350
O.27l
0.201
0.141
0.097
Power 1%-Teat
Power 5%-Test
Power 10%-Test
D.
n
q
Min
Max
Mean
S.
163
0
0.352
1.080
0.614
0.103
0.849
0.956
0.981
163
4.1
0.449
1.626
0.838
0.142
0.211
0.456
0.600
163
163
163
163
163
163
163
163
1
2
3
4
5
6
7
8
0.416
0.466
0.512
0.546
0.564
0.600
0.658
0.652
1.251
1.344
1.467
1.545
1.667
1.664
1.731
1.775
0.708
0.779
0.837
0.887
0.931
0.970
1.007
1.041
0.116
0.125
0.132
0.137
0.141
0.144
0.147
0.149
0.587
0.350
0.194
0.100
0.046
0.019
0.008
0.004
0.811
0.631
0.458
0.309
0.200
0.124
0.074
0.041
0.895
0.758
0.612
0.47l
0.334
0.236
0.158
0.10s
Source: Authors
@FedFRASER