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Journal of Business & Economic Statistics
ISSN: 0735-0015 (Print) 1537-2707 (Online) Journal homepage: http://www.tandfonline.com/loi/ubes20
True or Spurious Long Memory? A New Test
Arek Ohanissian, Jeffrey R Russell & Ruey S Tsay
To cite this article: Arek Ohanissian, Jeffrey R Russell & Ruey S Tsay (2008) True or Spurious
Long Memory? A New Test, Journal of Business & Economic Statistics, 26:2, 161-175, DOI:
10.1198/073500107000000340
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Published online: 01 Jan 2012.
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Date: 12 January 2016, At: 17:47
True or Spurious Long Memory? A New Test
Arek O HANISSIAN, Jeffrey R. R USSELL, and Ruey S. T SAY
Graduate School of Business, University of Chicago, Chicago, IL 60637 (jeffrey.russell@chicagogsb.edu )
It is well known that long memory characteristics observed in data can be generated by nonstationary
structural-break or slow regime switching models. We propose a statistical test to distinguish between
true long memory and spurious long memory based on invariance of the long memory parameter for
temporal aggregates of the process under the null of true long memory. Geweke Porter-Hudak estimates
of the long memory parameter obtained from different temporal aggregates of the underlying time series
are shown to be asymptotically jointly normal, leading to a test statistic that is constructed as the quadratic
form of a demeaned vector of the estimates. The result is a test statistic that is very simple to implement.
Simulations show the test to have good size and power properties for the classic alternatives to true long
memory that have been suggested in the literature. The asymptotic distribution of the test statistic is also
valid for a stochastic volatility with Gaussian long memory model. The test is applied to foreign exchange
rate data. Based on all the models considered in this article, we conclude that the long memory property
in exchange rate volatility is generated by a true long memory process.
Downloaded by [Universitas Maritim Raja Ali Haji] at 17:47 12 January 2016
KEY WORDS: Regime switching; Structural change; Temporal aggregation.
1. INTRODUCTION
The notion of long-range dependence appears in many different fields, including hydrology, Internet traffic, economics,
and finance. Long-range dependence presents an interesting
case in that it allows for a middle ground between the strict
boundary of I(0) and I(1) processes, requires specialized central limit theorems, and entails statistical analysis quite distinct from the corresponding analysis for short-range dependent
processes. Observing long memory characteristics in a given
dataset, however, does not imply an underlying long-range dependent process as there are spurious causes of this characteristic, such as structural breaks. Unfortunately, detecting the nature of long-range dependence and, in particular, differentiating
between true and spurious generation of this dependence is a
rather difficult task. In this article we provide a test to distinguish between true and spurious long memory that focuses on
temporal aggregation and the differing behavior of the distinct
model classes. Focusing on the temporal aggregation properties
allows us to avoid the problem of specifying or estimating the
number of structural breaks—a problem that plagued the unit
root/structural-break literature (see Perron 1989, 1990; Zivot
and Andrews 1992). This is the approach taken in the context
of testing for long memory by Lobato and Savin (1998). Hence,
our proposed approach is robust. Although our methodology is
potentially applicable to a broad array of time series processes,
the application in this article focuses on the properties of foreign exchange rate volatility.
Long memory of a stationary process can be defined in
either the frequency or time domain. In the frequency domain, a series exhibits long memory if the spectral density
is unbounded at frequency zero. Specifically, any stationary
process Yt with spectral density function f (ω) is a long memory process if f (ω) ∼ cω−α as ω ↓ 0 for some α ∈ (0, 1).
In the time domain, a series exhibits long memory if the absolute values of the autocorrelations are not summable. Specifically, any
stationary process Yt is a long memory process if
limn→∞ nj=0 | Corr(Yt , Yt+j )| is not finite. In practice, this can
be seen in the form of hyperbolic decay of autocorrelations,
that is, much slower decay than the exponential decay of the
ARMA class. For a detailed discussion of these definitions and
conditions under which they are equivalent, see Beran (1994).
Throughout this article we focus on the frequency domain definition. Using these definitions, various authors documented
the existence of long memory in asset return volatilities. Taylor
(1986) demonstrated that the autocorrelations of absolute stock
return series decay very slowly. Such slow decay has also been
shown to exist in the autocorrelations of many other volatility
measures, including various powers of the absolute stock return
series (Ding, Granger, and Engle 1993) and squared returns for
intraday exchange rate data (Dacorogna, Müller, Nagler, Olsen,
and Pictet 1993). Many other studies followed these early studies and provided further evidence of long memory in volatility,
including Robinson (1994) and Baillie (1996).
Early research has predominantly used two models to account for the observed long memory property in volatility, namely, fractionally integrated (exponential) generalized
autoregressive conditional heteroscedasticity (FI(E)GARCH)
in Baillie, Bollerslev, and Mikkelsen (1996) and Bollerslev
and Mikkelsen (1996) and long memory stochastic volatility (LMSV) in Comte and Renault (1996) and Breidt, Crato,
and de Lima (1998). Each model adds a fractionally integrated
process into the well-known GARCH and SV models. Unlike
the case of price dynamics, where the price process must be a
semimartingale to exclude arbitrage possibilities (see Harrison
and Pliska 1981), there is no theory to either suggest or restrict
the long memory specification for volatility dynamics. Thus,
the introduction of fractional components into volatility models
is purely from an empirical perspective without any theoretical
foundation. Furthermore, Granger and Ding (1996) showed that
the parameter estimates of long memory models for volatility
vary substantially between subperiods of a series. This evidence
suggests that the fractionally integrated models may be inconsistent with volatility dynamics.
A structural-break approach, on the other hand, is a completely different approach to representing the empirically documented long memory characteristic of volatilities. In this
approach, locally stationary, short memory models with occasional structural breaks are able to exhibit “spurious long memory.” Early references are Klemeš (1974) and Chen and Tiao
161
© 2008 American Statistical Association
Journal of Business & Economic Statistics
April 2008, Vol. 26, No. 2
DOI 10.1198/073500107000000340
Downloaded by [Universitas Maritim Raja Ali Haji] at 17:47 12 January 2016
162
(1990). Hence, the observed long memory of volatilities may
be due to the nonstationarity of the process as opposed to being generated by an inherently fractionally integrated process.
Granger and Terasvirta (1999), Granger and Hyung (1999), and
Diebold and Inoue (2001) demonstrated this fact using analytic and simulation evidence. The result is that these structuralbreak models may resemble a long memory process in a given
sample size for specific probabilities of shift/break. In general,
the probability of a break must be smaller for larger sample
sizes so that there is only a “small” amount of switching/change
in an observed sample path. Smith (2005) extended this research by deriving (approximations to) the bias of long memory
parameter estimators applied to spurious long memory models.
Differentiating between true and spurious long memory is
relevant from both a statistical as well as an economic perspective. Inference under a stationary long memory model is quite
distinct from inference under structural-break/nonstationary behavior. From an economic perspective, the long-lived impact of
shocks under a long memory model is quite distinct from economic models associated with rare structural breaks. A statistical test to distinguish between true and spurious long memory is
clearly needed. Unfortunately, although there is rather clear understanding of the mechanism generating spurious long memory, we are aware of no formal tests to distinguish between true
and spurious long memory. A particular complication results
from the fact that the usual methods for detecting breaks in a
series whose true data generating process is a fractionally integrated process result in spurious break detection as discussed
by Nunes, Kuan, and Newbold (1995) and Granger and Hyung
(1999), among others. Notably, Hidalgo and Robinson (1996)
and Wright (1998) proposed a test for structural breaks in the
presence of long memory, the former with known potential
break date, the latter with unknown break date. Instead of estimating or identifying such breaks, we provide a methodology
to directly distinguish between true and spurious long memory
using the notion of temporal aggregation.
Mandelbrot and Van Ness (1968) formally introduced the
fractional Brownian motion and fractional Gaussian noise
process and discussed their scaling and self-similarity properties, whereas the relevant notion of self-affinity was discussed in Mandelbrot (1997). A direct implication is that
these processes will have the same long memory at all levels of sampling frequency. More recently, temporal aggregation for long memory processes was considered by various authors in the econometrics literature. Specifically, Beran and Ocker (2000) and Man and Tiao (2001a,b) examined
Gaussian ARFIMA(p, d, q) models and Chambers (1998) presented more general results in which the underlying series is
only assumed to have a Wold representation. The conclusion is
that aggregation acts to modify only the short memory properties and hence the long memory parameter remains constant
across temporal aggregation. Müller et al. (1990) provided evidence of long memory in financial asset prices by empirically
showing that mean absolute price changes are proportional to
the interval size. More recently, Andersen and Bollerslev (1997)
and Andersen, Bollerslev, Diebold, and Labys (2001) used realized volatility and temporal aggregation to provide informal evidence in favor of their finding of long memory. They claimed
that their estimated long memory parameters, using different
Journal of Business & Economic Statistics, April 2008
frequency data (i.e., daily, weekly, monthly), are similar enough
to be consistent with a true long memory process. Hence, although temporal aggregation has been used to provide informal
evidence of long memory, no formal tests have been proposed.
In this article we formalize this approach by providing a specific
test and the corresponding asymptotic distribution for proper
statistical analysis.
The null hypothesis is that the original data series is generated by a stationary, true long memory process. In this case, the
long memory parameter is invariant to temporal aggregation of
the original data. We use the semiparametric long memory parameter estimator of Geweke and Porter-Hudak (1983) (GPH)
which focuses on only the long memory aspect of the series
and hence does not require specifying the short memory dynamics. Furthermore, the GPH estimator allows us to consider
both linear and nonlinear models concurrently. We derive the
joint limiting distributional properties of the GPH estimators
applied to different temporal aggregates of the data. Given this
distribution, we then introduce a Wald test statistic and derive
its asymptotic distribution. Simulations of both true and spurious long memory processes show that both the size and power
of this test are good. Additionally, the quadratic form of the test
statistic is remarkably easy to evaluate. We apply our test to a
high-frequency foreign exchange dataset and cannot reject the
null hypothesis. Based on all of the models we consider in this
article, we conclude that the long memory property in exchange
rate volatility is generated by a true long memory process as opposed to a spurious long memory process.
The remainder of this article is organized as follows. Section 2 contains the distributional properties for both the long
memory parameter estimator as well as our test statistic and also
contains simulation evidence for further clarification. We apply
our analysis to data in Section 3 and conclude in Section 4.
2. PROPERTIES OF GPH ESTIMATES AND
A SPECIFIC TEST
In this section we present the test statistic for distinguishing
between true and spurious long memory. The test is based on
invariance of the long memory parameter over temporal aggregation. The test statistic is formed using the GPH long memory
parameter estimator. In the first subsection, the limiting distribution of the GPH estimator is examined including our extension
to the temporally aggregated case. The second subsection introduces our test statistic as well as its asymptotic distribution.
2.1 GPH Estimates for Temporal Aggregates
The GPH estimator is used because it is semiparametric in
that only periodogram ordinates in a decreasing neighborhood
of the zero frequency are used in the estimation. Thus, we
avoid many potential misspecification issues involving the short
memory and/or linearity aspects of the model. Another reason
we use the GPH estimator is the strong theoretical understanding of its distributional properties (see Robinson 1995; Hurvich,
Deo, and Brodsky 1998; Deo and Hurvich 2001). The former
two studies focus on Gaussian long memory whereas the latter extends the analysis to stochastic volatility with Gaussian
long memory. Similarly, we consider the distributional properties first for Gaussian long memory and then stochastic volatility with Gaussian long memory.
Ohanissian, Russell, and Tsay: True or Spurious Long Memory
Downloaded by [Universitas Maritim Raja Ali Haji] at 17:47 12 January 2016
2.1.1 Gaussian Long Memory. The aforementioned articles show that under proper bandwidth restrictions (i.e., frequencies of the spectral density used), the GPH estimator is
asymptotically normal. In this section we determine the joint
limiting distribution of the GPH estimates obtained using the
temporally aggregated series. Given this joint distribution, we
can then determine the distribution for a test statistic associated
with equality of the long memory parameter across temporal
aggregates of the original series. We begin by stating the formal condition on the long memory processes we consider in
this article.
Condition 1. {Yt } is a stationary mean-zero Gaussian longmemory time series with spectral density f (ω) = |1 −
exp(−iω)|−2d f ∗ (ω), where d ∈ (0, .5), f ∗ (ω) denotes the spectral density of the short memory component of {Yt } and is assumed to be continuous, bounded above, bounded away from
zero, twice differentiable with the second derivative bounded in
a neighborhood of zero.
Note that the assumption of a mean-zero process is without loss of generality as our work will be in the frequency domain and the periodogram of a series is invariant to the mean.
Also note that the symmetry of f ∗ (ω) around the origin implies that f ∗′ (0) = 0. Given a sample Y ≡ (Y1 , Y2 , . . . , Yn )′ ,
we use the notation fj∗ to represent f ∗ (2πj/n) and fj to represent f (2πj/n), which are the spectral densities at specific ordinates for the short memory and original process, respectively.
Now, the GPH estimate of the long memory parameter is calculated from a regression of the log periodogram on the explanatory variable Xj = log |2 sin(πj/n)| and a constant term evaluated at the first l log periodogram ordinates ({log Ij }lj=1 , where
1 n
Ij = 2πn
| t=1 Yt exp(it2π nj )|2 ). Thus,
log Ij = (log f0∗ − C) − 2dXj + log(fj∗ /f0∗ ) + εj ,
∼ .577 is Euler’s constant.
where εj = log(Ij /fj ) + C and C =
Furthermore, we will use the abbreviations that aj = Xj −
l
1 l
2
j=1 Xj and SXX =
j=1 (aj ) . The GPH estimator then is
l
l
1
d = − 2SXX j=1 aj log Ij with an asymptotically normal distribution
√
D
4l(
d − d) → N(0, π 2 /6)
(1)
for l = o(n4/5 ), where →D represents convergence in distribution.
We now introduce additional notation to designate the level
of temporal aggregation (m). Specifically, a superscript (m)
on any statistic represents the corresponding statistic for the
m-temporally aggregated data, where m is a finite positive integer. The m-period nonoverlapping aggregates of Yt are defined
(m)
n
as Zs ≡ m
τ =1 Ym(s−1)+τ for 1 ≤ s ≤ m . Note that the number of ordinates (l) used in the GPH estimator is dependent on
the length of the series, and because temporal aggregation decreases the length of the series, then l(mu ) > l(mv ) for mu < mv .
We next introduce the following condition which details the aggregation levels and number of ordinates used.
Condition 2. (i) For any fixed, but arbitrarily large, aggregation level m, the growth rate of the number of ordinates used
for the GPH estimation is restricted such that l(m) → ∞ and
l(m) = o(n(2−4d)/3 ) as n → ∞.
163
(ii) M denotes the fixed, but arbitrarily large, number of
aggregation levels and (m1 , m2 , . . . , mM ) denote the (ordered)
fixed, but arbitrarily large, aggregation levels for the M series
such that m1 < m2 < · · · < mM .
We note that changing the length of the series results in different frequencies and hence the ωj ’s (which are the frequencies
for the spectral density) are not the same for the same j, that
(1)
(m )
(m)
(mu )
is, ωj = 2π nj , ωj = 2π jm
= ωj v if mu = mv , and
n , ωj
(m)
ωj(km) = ωkj
. For the sake of completeness, we formally intro(m)
duce some of the relevant statistics to be used shortly: Ij represents the periodogram ordinates for the temporally aggregated
(m)
(m)
(m)
1 l(m) (m)
= Xj − l(m)
series, Xj = log |2 sin(π jm
j=1 Xj ,
n )|, aj
l(m) (m)
l(m) (m) 2
(m)
(m)
1
(m)
a log I .
d = − (m)
(a ) , and
S =
XX
j=1
j
2SXX
j=1 j
j
The asymptotical normal distribution in (1) is valid for the
individual GPH estimates obtained using the temporally aggregated series. Our testing approach, based on invariance of the
long memory parameter under temporal aggregation, requires
the joint distribution of these estimates. Specifically, asymptotic
joint normality is established.
We begin by examining the periodograms for series of
differing temporal aggregations. For the general case of an
m-temporally aggregated series, the periodogram is defined by
n/m m
2
m
jm
(m)
Ij =
Ym(t−1)+s exp it2π
,
2πn
n
t=1
s=1
which can be rewritten as
n
2
jm t
m
(m)
Yt exp i2π
Ij =
2πn
n m
t=1
n n
m
s
t
jm
=
−
Ys Yt cos 2π
2πn
n
m
m
s=1 t=1
for 1 ≤ j ≤ mn , where [x] denotes the smallest integer greater
than or equal to x. Note that the ordinates, as well as any statistic based on them, are dependent on the length of the original
series; however, for the sake of notational simplicity we sup(m)
(m)
d(m) is repress this dependence, that is, Ij is really Ij (n),
(m)
ally
d(m) (n), and so forth. Using this representation, Ij can be
(1)
(m)
(1)
written in terms of Ij as Ij = mIj + Y ′ Bj Y for 1 ≤ j ≤ mn ,
where the Bj matrix is defined as Bj ≡ (bj (s, t))1≤s,t≤n and
jm
s
t
m
bj (s, t) ≡
cos 2π
−
2πn
n
m
m
j
− cos 2π (s − t) .
n
The following lemma provides a direct link between the periodogram of an m-temporally aggregated series and the pe(1)
riodogram of the original series. We define Ij as the periodogram of the original series.
Lemma 1. For a process satisfying Condition 1 and an aggregation level (m) with corresponding number of ordinates used
(1)
(m)
(l(m) ) satisfying Condition 2: (Ij − mIj ) →p 0 as n → ∞
for 1 ≤ j ≤ l(m) .
164
Journal of Business & Economic Statistics, April 2008
Corollary 1. Under the assumptions of Lemma 1, for any
(m )
two finite aggregation levels mu and mv with mu < mv : (Ij v −
mv (mu )
) →p
mu Ij
0 as n → ∞ for 1 ≤ j ≤ l(mv ) .
This link between the periodograms of temporally aggregated series provides a framework to derive the asymptotic joint
distribution of the GPH estimates obtained using the temporally
aggregated series, which we state in the following proposition.
Proposition 1. For a process satisfying Condition 1 and aggregation levels with corresponding number of ordinates used
satisfying Condition 2, the joint distribution of the GPH estimates obtained using the temporally aggregated series is asymptotically jointly normal.
Downloaded by [Universitas Maritim Raja Ali Haji] at 17:47 12 January 2016
The following lemma establishes the covariance structure of
the corresponding asymptotically normal joint distribution.
Lemma 2. For a process satisfying Condition 1 and aggregation levels with corresponding number of ordinates used satisfying Condition 2: (4l(mi ) (Cov(
d(mi ) ,
d(mj ) ) − Var(
d(mi ) ))) = o(1)
as n → ∞ for 1 ≤ i < j ≤ M.
This result implies the structure that the covariance between
any two GPH estimates obtained using temporally aggregated
series equals the variance of the lesser aggregated series, asymptotically. This result is confirmed in our simulations. Note
that this structure is consistent with the result in Hausman
(1978) that an efficient estimator must have zero asymptotic
covariance with the difference between the efficient estimator
and any other consistent, asymptotically normal estimator. Although the GPH estimator is not an efficient estimator, the estimate based on the least aggregated data is efficient in the class
of GPH estimates obtained using temporal aggregates or any
linear combination of such estimates. Our results therefore parallel those of Hausman (1978).
2.1.2 Stochastic Volatility With Gaussian Long Memory.
The above derived joint distribution for the GPH estimates obtained using the temporally aggregated series is based on the assumption of a Gaussian time series. To facilitate the use of GPH
estimators for volatility, Deo and Hurvich (2001) extended the
distributional analysis for the GPH approach to a stochastic
volatility model.
The stochastic volatility model they considered is rt =
σ exp(Yt /2)et , where {Yt } is a stationary Gaussian long memory
process independent of the iid mean-zero random variable {et }.
Using the log transformation, this model is linearized as follows:
Zt ≡ log(rt2 ) = μ + Yt + ut ,
(log σ 2
E{log e2t })
(2)
(log e2t
− E{log e2t })
returns {Zt } is a
+
and ut =
where μ =
is iid mean zero. Thus, the log of squared
linear combination of an underlying Gaussian long memory
process plus a noise process. Deo and Hurvich (2001) showed
that adding noise to a long memory process does not change
the asymptotic distribution of the GPH estimator. That is, the
asymptotic distribution of the GPH estimator is the same for
the {Zt } series and the {Yt } series; however, there is a stricter
requirement on the bandwidth choice (i.e., ordinates of the periodogram used for estimation). Specifically, in the Gaussian
long memory case the restriction is l = o(n4/5 ), whereas for the
stochastic volatility with Gaussian long memory case the restriction becomes l = o(n4d/(4d+1) ). Note that for small enough
d this requirement is stricter than the one listed in Condition 2.
Thus, in the stochastic volatility with Gaussian long√ memory
case, the requirement is l = o(n4d/(4d+1) ) for d ∈ (0, 3−1
4 ) and
√
l = o(n(2−4d)/3 ) for d ∈ [ 3−1
4 , .5).
We extend our distribution result from the previous subsection for Gaussian long memory to this stochastic volatility with
Gaussian long memory model. First, we introduce an assumption on the noise process.
Condition 3. {ut } of (2) has a finite eighth moment. (Note
that this condition applies to the iid noise component and does
not restrict the returns process to have a finite eighth moment.)
We now state our next proposition which covers the stochastic volatility with Gaussian long memory model.
Proposition 2. For the {Zt } process of (2) in the stochastic
volatility with Gaussian long memory model where the long
memory process satisfies Condition 1, the iid noise process satisfies Condition 3, and aggregation levels with corresponding
number of ordinates used satisfy Condition 2 along with the
additional restriction that l(mi ) = o(n4d/(4d+1) ), the joint distribution of the GPH estimates obtained using the temporally aggregated series is asymptotically jointly normal with variance–
covariance structure given by Lemma 2.
2.1.3 Simulations. This section provides simulation results. We show that the asymptotic covariance structure of the
previous section provides a very reasonable approximation to
the finite sample behavior of GPH estimates obtained using
temporally aggregated series. In Table 1, we present results
from simulations of an I(.4) process. In theory, the distribution does not depend on the value of the long memory parameter. For the simulations and empirical application in this article,
we use the square root of the number of observations
as our
√
choice for the number of ordinates, that is, l(mi ) = n/mi , as it
is a common rule of thumb used in the literature. Varying the
number of ordinates used in the estimation does not impact the
features of Table 1. Specifically, the number of ordinates used
ranged from n.3 to n.6 . In addition, we considered lower truncation by including a trimming parameter of either 1 or 2. The
ordered aggregation levels are consecutive powers of 2, that is,
(m1 , m2 , . . . , mM ) is defined by mj = 2j−1 for j = 1, 2, . . . , M
for all values of M. Given our simulations and the corresponding GPH estimates obtained using temporally aggregated series,
we compare the theoretical covariance matrix (Table 1a) against
the sample covariance matrix (Table 1b) from simulations for
these estimates.
First, note that the theoretical variance–covariance matrix
does not depend on the value of the long memory parameter.
Simulation results at various degrees of long memory confirm
this fact, and we list only one such set of simulations here which
is representative of the others. Also note that the variance–
covariance structure derived in Lemma 2 (where the covariance
between two estimates equals the variance of the lesser aggregated series) is indeed present in the simulation results. Lastly,
note that the theoretical and sample variances match well for the
longer series, but not as well for the shorter series. This finding
relates directly to a small simulation study in Deo and Hurvich
Ohanissian, Russell, and Tsay: True or Spurious Long Memory
165
Table 1. Variance–covariance matrices for GPH estimators of temporally aggregated series based on (a) theory, (b) simulations,
and (c) an approximation
(a) Theoretical variance–covariance matrix
Downloaded by [Universitas Maritim Raja Ali Haji] at 17:47 12 January 2016
610,304
305,152
152,576
76,288
38,144
19,072
9,536
4,768
2,384
1,192
596
298
610,304
305,152
152,576
76,288
38,144
19,072
9,536
4,768
2,384
1,192
596
298
.0005
.0005
.0007
.0005
.0007
.0011
.0005
.0007
.0011
.0015
.0005
.0007
.0011
.0015
.0021
.0005
.0007
.0011
.0015
.0021
.0030
.0005
.0007
.0011
.0015
.0021
.0030
.0042
.0005
.0007
.0011
.0015
.0021
.0030
.0042
.0060
.0005
.0007
.0011
.0015
.0021
.0030
.0042
.0060
.0086
.0005
.0007
.0011
.0015
.0021
.0030
.0042
.0060
.0086
.0121
.0005
.0007
.0011
.0015
.0021
.0030
.0042
.0060
.0086
.0121
.0171
.0005
.0007
.0011
.0015
.0021
.0030
.0042
.0060
.0086
.0121
.0171
.0242
(b) Sample variance–covariance matrix from 500 simulated series of an ARFIMA(0, .4, 0) model
610,304
305,152
152,576
76,288
38,144
19,072
9,536
4,768
2,384
1,192
596
298
610,304
305,152
152,576
76,288
38,144
19,072
9,536
4,768
2,384
1,192
596
298
.0005
.0005
.0008
.0005
.0008
.0011
.0006
.0008
.0012
.0017
.0006
.0008
.0011
.0017
.0024
.0006
.0008
.0011
.0016
.0024
.0034
.0005
.0007
.0011
.0017
.0024
.0036
.0053
.0004
.0006
.0010
.0015
.0023
.0034
.0050
.0070
.0004
.0006
.0010
.0015
.0023
.0035
.0049
.0071
.0110
.0004
.0005
.0009
.0014
.0020
.0032
.0046
.0069
.0106
.0159
.0003
.0004
.0009
.0014
.0019
.0031
.0046
.0069
.0104
.0158
.0250
.0002
.0004
.0009
.0016
.0022
.0033
.0047
.0073
.0108
.0157
.0254
.0408
(c) Approximated variance–covariance matrix using (3)
610,304
305,152
152,576
76,288
38,144
19,072
9,536
4,768
2,384
1,192
596
298
610,304
305,152
152,576
76,288
38,144
19,072
9,536
4,768
2,384
1,192
596
298
.0005
.0005
.0008
.0005
.0008
.0011
.0005
.0008
.0011
.0016
.0005
.0008
.0011
.0016
.0023
.0005
.0008
.0011
.0016
.0023
.0034
.0005
.0008
.0011
.0016
.0023
.0034
.0050
.0005
.0008
.0011
.0016
.0023
.0034
.0050
.0073
.0005
.0008
.0011
.0016
.0023
.0034
.0050
.0073
.0111
.0005
.0008
.0011
.0016
.0023
.0034
.0050
.0073
.0111
.0168
.0005
.0008
.0011
.0016
.0023
.0034
.0050
.0073
.0111
.0168
.0260
.0005
.0008
.0011
.0016
.0023
.0034
.0050
.0073
.0111
.0168
.0260
.0407
NOTE: The number of observations of the original data series is 610,304 and the aggregated series have aggregation levels of 2j for j = 1, 2, . . . , 11.
(2001). As a result of this finite sample problem, they suggest
using an approximation for the variance in practice. Specifically, the approximation is the variance originally put forth by
Geweke and Porter-Hudak (1983):
−1
l
(m)
2
(m)
Varapprox. d
π 2.
(3)
a
= 24
j
j=1
We find that this approximation matches our simulated variances well, and thus provides a simple alternative to using the variances obtained from simulations. Table 1c lists
the corresponding variance–covariance matrix generated using
this approximation strategy and it indeed closely matches the
variance–covariance matrix from simulations.
2.2 Test Statistic
2.2.1 Statistic and Its Asymptotic Distribution. This section proposes our test statistic. Our null hypothesis is that the
original data are from a stationary, true long memory process
in which case the long memory parameter is the same across
d(m2 ) , . . . ,
all aggregation levels. Specifically, let
d ≡ (
d(m1 ) ,
Downloaded by [Universitas Maritim Raja Ali Haji] at 17:47 12 January 2016
166
Journal of Business & Economic Statistics, April 2008
d(mM ) )′ be the M-dimensional vector of our estimated long
memory parameters, d ≡ (d(m1 ) , d(m2 ) , . . . , d(mM ) )′ be the constant M-dimensional vector of the actual long memory parameters, and be the asymptotic covariance matrix of
d. Our
null hypothesis is that d(m1 ) = d(m2 ) = · · · = d(mM ) = d where
0 < d < .5. This null hypothesis can be tested by considering
the quadratic form (
d − d)′ −1 (
d − d) which is asymptotically
2
distributed as χ (M). Note that the structure of the asymptotic
covariance matrix , as derived in Lemma 2, requires that the
asymptotic variance of each individual GPH estimate be different for the asymptotic covariance matrix to be invertible. Thus,
we require that the number of ordinates used for the estimation
of each temporally aggregated series be different. In practice, d
will not be known so the mean value of the estimates is used
and, hence, the distribution must be adjusted. Specifically, setting the following ((M − 1), M) matrix:
⎛
−1
−1
−1 ⎞
−1
···
1 − M1
M
M
M
M
⎜ −1
−1
−1 ⎟
1
−1
1
−
·
·
·
⎜
M
M
M
M ⎟,
T=⎝ M
···
···
··· ···
···
··· ⎠
−1
M
−1
M
−1
M
··· 1 −
1
M
−1
M
′
the asymptotic distribution for T
d is N(Td, TT ), which is
actually N(0, TT′ ) because the elements of d are all equal
under the null hypothesis. As a result, the test statistic we use is
≡ (T
d),
W
d)′ (TT′ )−1 (T
(4)
which has an asymptotic χ 2 (M − 1) distribution under the null
hypothesis. Alternatively, we could propose a statistic based
on the least and most aggregated series’ estimates only. Simulations (not presented here) show that the long memory estimates using temporally aggregated series need not diverge
monotonically as the level of aggregation increases for all alternative models, thereby supporting the use of multiple GPH estimates. Our definition of the asymptotic covariance matrix for
(
d − d) stated above is inclusive of normalization factors for the
number of ordinates used in the estimation of each temporally
aggregated series. This characterization allows for very simple
application of the test in practice. There is an equivalent characterization, however, in which the normalization for the number
of ordinates is separated from the asymptotic covariance ma
trix. This characterization is written as Dn (
d − d) ∼asy N(0,),
where is independent of the number of ordinates and Dn is
a deterministic matrix that includes that dependence. For this
to be the identity matrix in
characterization, one could set
which case Dn would be the positive definite square root ma is constructed using the
trix of . Note that the test statistic W
(m)
average of all estimates d . It is possible to devise a test statistic that uses a weighted average of
d(m) . The issue of optimal weight for each estimate is not clear and deserves a careful
study.
2.2.2 Size and Power Properties. We now provide simulation results for the proposed test statistic to examine its small
sample properties. We consider the size of the test statistic using
500 simulated series of an I(.4) model and an ARFIMA(p, d, q)
model fitted to Yen/$ log(r2 ) data (which will be introduced in
detail in the following section). We note that these data are best
modeled by an ARFIMA(1, d, 1) specification with the following parameter estimates: long memory parameter of .42, AR
coefficient of .73, and MA coefficient of .87. We present the
size of the test using the corresponding 95% χ 2 (M − 1) critical value for various values of M (i.e., various numbers of aggregated series). We present the empirical size of our test statistic in Table 2 using three sets of variances: theoretical variances, simulated variances, and approximated variances calculated via (3). As expected, the results suggest that there is significant size distortion when using the theoretical variances which
decreases as we exclude the shorter series. The corresponding size distortions essentially disappear, however, when using
the structure of the theoretical variance–covariance matrix [i.e.,
Cov(
d(mi ) ,
d(mj ) ) = Var(
d(mi ) ) for mi < mj ] and using either the
simulated or approximated variances. Given this result, it is fair
to conclude that the main cause of this size distortion is the substantial deviation of the finite sample variances from asymptotic
values. This observation suggests using simulated and approximated variances instead of theoretical variances.
We now consider the power of the statistic for some spurious long memory processes using simulations from five relevant alternative models. Namely, we consider (1) a (nonstationary) random level shift model, (2) a (stationary) random level
shift model, (3) a Markov Switching model with iid regimes,
(4) a Markov Switching model with GARCH regimes, and
(5) a white noise model with a (slow) deterministic trend. Each
of these models has been shown in the literature to be able to
exhibit spurious long memory (in particular, see the articles of
Bhattacharya, Gupta, and Waymire 1983; Diebold and Inoue
2001; Granger and Hyung 1999, among others). The complete
specifications of our five alternative models are listed in the Appendix. Prior to considering the power of the test statistic itself,
Table 2. Empirical size properties of test statistic using 500 series simulation of ARFIMA(0, .4, 0) and ARFIMA(1, .4, 1) models
for 5% theoretical size
12
11
10
9
Size w/theor. var.
Size w/simul. var.
Size w/approx. var.
33%
9%
9%
24%
9%
7%
17%
8%
6%
12%
6%
5%
Size w/theor. var.
Size w/simul. var.
Size w/approx. var.
32%
8%
8%
25%
9%
7%
17%
8%
6%
13%
6%
5%
M
8
7
6
5
4
3
2
ARFIMA(0, .4, 0)
10%
9%
6%
4%
5%
6%
8%
4%
4%
7%
4%
5%
7%
5%
5%
7%
7%
6%
7%
7%
6%
ARFIMA(1, .4, 1)
11%
10%
6%
5%
5%
7%
9%
5%
5%
8%
5%
6%
8%
7%
7%
8%
8%
7%
8%
8%
8%
NOTE: The number of observations of the original data series is 610,304 and the aggregated series have aggregation levels of 2j for j = 1, 2, . . . , M − 1.
Ohanissian, Russell, and Tsay: True or Spurious Long Memory
167
Downloaded by [Universitas Maritim Raja Ali Haji] at 17:47 12 January 2016
Table 3. Mean and empirical 95% confidence interval of estimated long memory parameter from simulations of the five alternative models,
specified in the Appendix, across temporal aggregation of original series
m=1
n = 610,304
m=2
n = 305,152
m=4
n = 152,576
m=8
n = 76,288
m = 16
n = 38,144
m = 32
n = 19,072
m = 64
n = 9,536
m = 128
n = 4,768
m = 256
n = 2,384
m = 512
n = 1,192
m = 1,024
n = 596
m = 2,048
n = 298
RLS–NS
RLS–S
MS–IID
MS–GARCH
Trend
σε2 = 5
ση2 = 1
p = .00001
σε2 = ση2 = 1
p = .003
μ1 = −μ2 = 1
σ2 = 1
p01 = p10 = .001
ω0 = 1, ω1 = 3
α = .4, β = .3
p01 = p10 = .001
c=3
β = −.1
.34
(.14, .58)
.41
(.17, .68)
.50
(.21, .78)
.58
(.27, .85)
.67
(.32, .95)
.76
(.36, .98)
.84
(.42, 1.05)
.89
(.57, 1.06)
.93
(.62, 1.09)
.96
(.73, 1.14)
.99
(.68, 1.26)
1.00
(.59, 1.24)
.30
(.26, .34)
.21
(.16, .27)
.14
(.07, .20)
.09
(.00, .16)
.05
(−.05, .14)
.03
(−.08, .14)
.02
(−.13, .17)
.02
(−.13, .16)
.02
(−.17, .25)
.00
(−.25, .21)
.01
(−.36, .36)
.03
(−.48, .37)
.41
(.35, .46)
.31
(.26, .37)
.23
(.16, .28)
.15
(.07, .24)
.09
(.01, .18)
.05
(−.07, .16)
.02
(−.12, .15)
.01
(−.16, .15)
.00
(−.23, .18)
.01
(−.21, .23)
.01
(−.28, .25)
.01
(−.47, .31)
.36
(.31, .40)
.28
(.23, .34)
.21
(.14, .27)
.14
(.07, .22)
.09
(.00, .18)
.05
(−.07, .17)
.02
(−.14, .17)
.01
(−.18, .16)
.00
(−.22, .19)
.01
(−.23, .22)
.01
(−.35, .27)
.00
(−.47, .32)
.40
(.37, .44)
.47
(.44, .50)
.55
(.51, .59)
.62
(.59, .66)
.68
(.63, .74)
.73
(.67, .79)
.75
(.70, .80)
.76
(.71, .84)
.75
(.69, .81)
.74
(.70, .81)
.73
(.68, .79)
.72
(.68, .78)
we first use simulations to examine the behavior of the GPH estimate of the long memory parameter across different levels of
aggregation. For each model, we select generic parameter values that generate spurious long memory for the original (nonaggregated) data series. The results in Table 3 clearly show that
the long memory parameter estimate does not remain constant
for any of these alternative models across temporal aggregation
of the data. We note that although we do not provide a corresponding table for true long memory processes, simulations for
these models clearly demonstrate the long memory retention
property for true long memory processes. This evidence suggests that our approach should be able to distinguish between
these models and true long memory processes. We now specifically examine the power of our test statistic.
We use 100 simulated series using two sets of parameter values for each of the five models. The first set of parameter values
are the generic ones used in Table 3. The second set of parameter values are determined from simple estimations of these
models using the foreign exchange data we discuss later in the
article. The MS–GARCH model is estimated using deseasonalized returns, and log squared deseasonalized returns are used
for the other models. Our goal here is to ensure that the simulations are valid for relevant parameter values. For some models
the estimated probability of jumps was too large to generate
long memory in the original (disaggregated) series. Hence, the
probabilities were restricted to be small enough to ensure that
the original series has the requisite long memory present. This
is a direct result of the fact mentioned earlier that the key to
the structural-break models exhibiting spurious long memory is
that the number of jumps in a given sample be relatively small.
For a given model, we use 100 realizations in the power study.
The results are listed in Table 4 and are encouraging for our
goal of distinguishing between true and spurious long memory.
The proposed test has excellent power for all of the considered
alternative models.
Our null hypothesis is stated as a stationary, true long memory process that corresponds to a long memory parameter between 0 and .5 and invariant to temporal aggregation. Our
test statistic only considers the invariance to temporal aggregation property and thus, strictly speaking, our statistic only tests
this invariance. To address the rest of the null hypothesis, two
pretests must be performed, namely, the long memory parameter must be tested to be (i) greater than zero and (ii) less than .5.
Specifically, the first pretest involves a null hypothesis of d = 0
with a one-sided alternative hypothesis of d > 0. Similarly, the
second pretest involves a null hypothesis of d = .5 with a onesided alternative hypothesis of d < .5. In each of these pretests,
the null hypothesis would have to be rejected for our test to
be applied. These tests are discussed in the literature (see, e.g.,
Robinson 1995; Velasco 1999). As long as the two pretests have
power that is asymptotically equal to 100%, the asymptotic size
of our test statistic and null hypothesis will not be affected by
these pretests. In the empirical analysis presented here, we ignored any pretests. We did perform simulations that included
168
Journal of Business & Economic Statistics, April 2008
Table 4. Empirical power properties of test statistic using 100 simulated series of the five alternative models given in the Appendix
M = 12
Generic parameter values
RLS–NS
RLS–S
MS–IID
MS–GARCH
Downloaded by [Universitas Maritim Raja Ali Haji] at 17:47 12 January 2016
Trend
Power
Estimated parameter values
Power
σε2 = 5, ση2 = 1
99%
σε2 = 4.4, ση2 = 2.3
93%
σε2 = ση2 = 1
p = .003
100%
σε2 = 2.2, ση2 = 5.7
p = .002
100%
ω0 = 1, ω1 = 3
α = .4, β = .3
p01 = p10 = .001
100%
ω0 = .08, ω1 = .23
α = .28, β = .56
p01 = p10 = .0008
100%
p = .00001
μ1 = −μ2 = 1
σ2 = 1
p01 = p10 = .001
c=3
β = −.1
σ 2 = 1.0
100%
100%
p = .000004
μ1 = −.4, μ2 = −4.0
σ 2 = 2.5
p12 = p21 = .001
c = −4.3
β = −.063
σ 2 = 8.1
100%
100%
NOTE: The number of observations of the original data series is 610,304 and the aggregated series have aggregation levels of 2j for j = 1, 2, . . . , 11.
the two pretests, but do not present those results here in the interest of conserving space. Specifically, we used pretests based
on the long memory parameter estimator in Velasco (1999) for
the models we used in the size and power analysis. The inclusion of the two pretests did not change the size analysis
at all, whereas the power increased to 100% for the RLS–NS
model, which was the only alternative model not to achieve
100% power without the pretests. Alternatively, in some applications the long memory parameter d∗ may be known a priori
and the pretests would not need to be performed. In this case,
d
d ≡ (d∗ , . . . , d∗ )′ with 0 < d∗ < .5 would be subtracted from
(instead of the demeaning performed by T) and the resulting
quadratic form test statistic would then be asymptotically distributed as χ 2 (M).
Throughout this article the null hypothesis has been restricted to stationary, true long memory processes that correspondingly have a constant long memory parameter. Recently,
there has been an interest in time-varying long memory models
(e.g., Jensen and Whitcher 2000; Wang, Cavanaugh, and Song
2001). These models are typically nonstationary and have timevarying spectral representation and are thus locally stationary in
the sense of Dahlhaus (1996, 1997). The properties of our test
statistic in the presence of a time-varying long memory parameter will likely depend on the way the long memory parameter
evolves over time and are beyond the scope of this article.
3. APPLICATION TO FOREIGN EXCHANGE DATA
3.1 Dataset Description
We now apply the proposed test to the intraday foreign exchange quote data. This dataset has been used widely by others,
such as Andersen et al. (2001), in the consideration of the long
memory property of volatility.
Specifically, we use 5-minute returns for DM/$ and Yen/$
from 1986 to 1996 obtained from Olsen and Associates. These
returns are based on interbank FX quotes that appeared on the
Reuters’ FXFX page. Prices at each 5-minute mark are extracted from these quotes by linearly interpolating from the log
bid and the log ask for the two closest ticks. The returns are
then calculated as the change in 5-minute average log bid and
ask prices. Following the approach of Andersen et al. (2001),
weekends and holidays were removed due to the lack of activity
during these periods. In addition, any runs of length 5 or higher
of the same value were removed as these points most likely correspond to problems with the Reuters feed and occur mainly in
the early part of the sample. The interested reader is referred
to Andersen et al. (2001) for a more detailed description of the
dataset.
3.2 Removing Periodicity
An inherent property of the intraday data that we use is
its periodicity. There are clear diurnal patterns in the volatility, as shown in Andersen and Bollerslev (1998). This periodicity manifests itself as periodicity in the autocorrelations
of the absolute, squared, and log squared returns which exists
at rather long lags (288th lag corresponding to the daily frequency). This deterministic behavior could potentially affect
our long memory estimates and results in the same way that
breaks can generate spurious long memory. Furthermore, there
is also the possibility of the long memory being a generalized
long memory process with seasonal long memory (see Gray,
Zhang, and Woodward 1989). Thus, prior to proceeding with
our main analysis, we first remove this periodicity and explore
the seasonal long memory issue.
In considering the periodicity, we first provide a plot of the
autocorrelation function (acf) for the log squared (raw) returns
in Figure 1. Note, all of the figures presented are based on the
DM/$ data as the corresponding plots for the Yen/$ data are
quite similar and are thus not presented in the interest of saving
space. The acf plot shows the telltale periodicity as seen in Andersen and Bollerslev (1998) with the highest peaks at the daily
frequency. As opposed to the more complicated procedure used
in Andersen and Bollerslev (1998) to remove the periodicity,
we begin with a simple adjustment to the returns. Specifically,
we divide each return by the standard deviation of the returns
in that time interval during the day. We find that this simple
standardization is able to remove most of the periodicity and,
although there is still some residual periodicity as seen in the
Downloaded by [Universitas Maritim Raja Ali Haji] at 17:47 12 January 2016
Ohanissian, Russell, and Tsay: True or Spurious Long Memory
169
Figure 1. ACF of log squared returns using raw returns for DM/$
exchange rate data.
Figure 3. ACF of log squared returns using adjusted returns for
DM/$ exchange rate data.
plots, it is essentially as effective as the Andersen and Bollerslev procedure. Figure 3 plots the acf after this standardization
and another adjustment mentioned below.
Holidays, weekends, or other breaks in the temporal continuity of the data introduce specific breaks, which could cause
spurious long memory. To examine this possibility, we plot the
acf of the log squared (standardized) returns for the data as a
whole and also for the case where we only use calendar continuous data. As we see in Figure 2, both of these plots are virtually
identical, and hence, we conclude that these breaks in calendar
time are not affecting our results.
Next, we consider the autocorrelation functions conditional
on the time (hour) of the day. While examining this, we
found large negative autocorrelation spikes corresponding to
the Japanese lunch period mentioned in Andersen and Bollerslev (1998). The Japanese market essentially shuts down during
this period. For this reason, we treat this time period as we do
weekends and simply remove it from the dataset. After doing
so, the conditional acf’s are much more consistent without any
peculiar negative spikes.
Finally, we consider the possibility of seasonal long memory
by plotting the sample periodogram of the log squared (raw)
returns and log squared (adjusted) returns in Figure 4. The raw
series clearly shows a seasonal hump corresponding to the daily
frequency which is almost completely dampened in the adjusted
series, thus providing further evidence that our procedure is effectively removing the periodicity. The other important point to
note is that the only spike in the periodogram is at the zero frequency and not at any other frequency. At the daily frequency,
it is merely a hump and not a spike. Hence, we conclude that
there is no seasonal long memory component in this dataset.
Thus, we sum up the two simple steps of our periodicity removal procedure. We remove the Japanese lunch period, which
corresponds to 3:00GMT to 4:45GMT, as it essentially represents mar
ISSN: 0735-0015 (Print) 1537-2707 (Online) Journal homepage: http://www.tandfonline.com/loi/ubes20
True or Spurious Long Memory? A New Test
Arek Ohanissian, Jeffrey R Russell & Ruey S Tsay
To cite this article: Arek Ohanissian, Jeffrey R Russell & Ruey S Tsay (2008) True or Spurious
Long Memory? A New Test, Journal of Business & Economic Statistics, 26:2, 161-175, DOI:
10.1198/073500107000000340
To link to this article: http://dx.doi.org/10.1198/073500107000000340
Published online: 01 Jan 2012.
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Date: 12 January 2016, At: 17:47
True or Spurious Long Memory? A New Test
Arek O HANISSIAN, Jeffrey R. R USSELL, and Ruey S. T SAY
Graduate School of Business, University of Chicago, Chicago, IL 60637 (jeffrey.russell@chicagogsb.edu )
It is well known that long memory characteristics observed in data can be generated by nonstationary
structural-break or slow regime switching models. We propose a statistical test to distinguish between
true long memory and spurious long memory based on invariance of the long memory parameter for
temporal aggregates of the process under the null of true long memory. Geweke Porter-Hudak estimates
of the long memory parameter obtained from different temporal aggregates of the underlying time series
are shown to be asymptotically jointly normal, leading to a test statistic that is constructed as the quadratic
form of a demeaned vector of the estimates. The result is a test statistic that is very simple to implement.
Simulations show the test to have good size and power properties for the classic alternatives to true long
memory that have been suggested in the literature. The asymptotic distribution of the test statistic is also
valid for a stochastic volatility with Gaussian long memory model. The test is applied to foreign exchange
rate data. Based on all the models considered in this article, we conclude that the long memory property
in exchange rate volatility is generated by a true long memory process.
Downloaded by [Universitas Maritim Raja Ali Haji] at 17:47 12 January 2016
KEY WORDS: Regime switching; Structural change; Temporal aggregation.
1. INTRODUCTION
The notion of long-range dependence appears in many different fields, including hydrology, Internet traffic, economics,
and finance. Long-range dependence presents an interesting
case in that it allows for a middle ground between the strict
boundary of I(0) and I(1) processes, requires specialized central limit theorems, and entails statistical analysis quite distinct from the corresponding analysis for short-range dependent
processes. Observing long memory characteristics in a given
dataset, however, does not imply an underlying long-range dependent process as there are spurious causes of this characteristic, such as structural breaks. Unfortunately, detecting the nature of long-range dependence and, in particular, differentiating
between true and spurious generation of this dependence is a
rather difficult task. In this article we provide a test to distinguish between true and spurious long memory that focuses on
temporal aggregation and the differing behavior of the distinct
model classes. Focusing on the temporal aggregation properties
allows us to avoid the problem of specifying or estimating the
number of structural breaks—a problem that plagued the unit
root/structural-break literature (see Perron 1989, 1990; Zivot
and Andrews 1992). This is the approach taken in the context
of testing for long memory by Lobato and Savin (1998). Hence,
our proposed approach is robust. Although our methodology is
potentially applicable to a broad array of time series processes,
the application in this article focuses on the properties of foreign exchange rate volatility.
Long memory of a stationary process can be defined in
either the frequency or time domain. In the frequency domain, a series exhibits long memory if the spectral density
is unbounded at frequency zero. Specifically, any stationary
process Yt with spectral density function f (ω) is a long memory process if f (ω) ∼ cω−α as ω ↓ 0 for some α ∈ (0, 1).
In the time domain, a series exhibits long memory if the absolute values of the autocorrelations are not summable. Specifically, any
stationary process Yt is a long memory process if
limn→∞ nj=0 | Corr(Yt , Yt+j )| is not finite. In practice, this can
be seen in the form of hyperbolic decay of autocorrelations,
that is, much slower decay than the exponential decay of the
ARMA class. For a detailed discussion of these definitions and
conditions under which they are equivalent, see Beran (1994).
Throughout this article we focus on the frequency domain definition. Using these definitions, various authors documented
the existence of long memory in asset return volatilities. Taylor
(1986) demonstrated that the autocorrelations of absolute stock
return series decay very slowly. Such slow decay has also been
shown to exist in the autocorrelations of many other volatility
measures, including various powers of the absolute stock return
series (Ding, Granger, and Engle 1993) and squared returns for
intraday exchange rate data (Dacorogna, Müller, Nagler, Olsen,
and Pictet 1993). Many other studies followed these early studies and provided further evidence of long memory in volatility,
including Robinson (1994) and Baillie (1996).
Early research has predominantly used two models to account for the observed long memory property in volatility, namely, fractionally integrated (exponential) generalized
autoregressive conditional heteroscedasticity (FI(E)GARCH)
in Baillie, Bollerslev, and Mikkelsen (1996) and Bollerslev
and Mikkelsen (1996) and long memory stochastic volatility (LMSV) in Comte and Renault (1996) and Breidt, Crato,
and de Lima (1998). Each model adds a fractionally integrated
process into the well-known GARCH and SV models. Unlike
the case of price dynamics, where the price process must be a
semimartingale to exclude arbitrage possibilities (see Harrison
and Pliska 1981), there is no theory to either suggest or restrict
the long memory specification for volatility dynamics. Thus,
the introduction of fractional components into volatility models
is purely from an empirical perspective without any theoretical
foundation. Furthermore, Granger and Ding (1996) showed that
the parameter estimates of long memory models for volatility
vary substantially between subperiods of a series. This evidence
suggests that the fractionally integrated models may be inconsistent with volatility dynamics.
A structural-break approach, on the other hand, is a completely different approach to representing the empirically documented long memory characteristic of volatilities. In this
approach, locally stationary, short memory models with occasional structural breaks are able to exhibit “spurious long memory.” Early references are Klemeš (1974) and Chen and Tiao
161
© 2008 American Statistical Association
Journal of Business & Economic Statistics
April 2008, Vol. 26, No. 2
DOI 10.1198/073500107000000340
Downloaded by [Universitas Maritim Raja Ali Haji] at 17:47 12 January 2016
162
(1990). Hence, the observed long memory of volatilities may
be due to the nonstationarity of the process as opposed to being generated by an inherently fractionally integrated process.
Granger and Terasvirta (1999), Granger and Hyung (1999), and
Diebold and Inoue (2001) demonstrated this fact using analytic and simulation evidence. The result is that these structuralbreak models may resemble a long memory process in a given
sample size for specific probabilities of shift/break. In general,
the probability of a break must be smaller for larger sample
sizes so that there is only a “small” amount of switching/change
in an observed sample path. Smith (2005) extended this research by deriving (approximations to) the bias of long memory
parameter estimators applied to spurious long memory models.
Differentiating between true and spurious long memory is
relevant from both a statistical as well as an economic perspective. Inference under a stationary long memory model is quite
distinct from inference under structural-break/nonstationary behavior. From an economic perspective, the long-lived impact of
shocks under a long memory model is quite distinct from economic models associated with rare structural breaks. A statistical test to distinguish between true and spurious long memory is
clearly needed. Unfortunately, although there is rather clear understanding of the mechanism generating spurious long memory, we are aware of no formal tests to distinguish between true
and spurious long memory. A particular complication results
from the fact that the usual methods for detecting breaks in a
series whose true data generating process is a fractionally integrated process result in spurious break detection as discussed
by Nunes, Kuan, and Newbold (1995) and Granger and Hyung
(1999), among others. Notably, Hidalgo and Robinson (1996)
and Wright (1998) proposed a test for structural breaks in the
presence of long memory, the former with known potential
break date, the latter with unknown break date. Instead of estimating or identifying such breaks, we provide a methodology
to directly distinguish between true and spurious long memory
using the notion of temporal aggregation.
Mandelbrot and Van Ness (1968) formally introduced the
fractional Brownian motion and fractional Gaussian noise
process and discussed their scaling and self-similarity properties, whereas the relevant notion of self-affinity was discussed in Mandelbrot (1997). A direct implication is that
these processes will have the same long memory at all levels of sampling frequency. More recently, temporal aggregation for long memory processes was considered by various authors in the econometrics literature. Specifically, Beran and Ocker (2000) and Man and Tiao (2001a,b) examined
Gaussian ARFIMA(p, d, q) models and Chambers (1998) presented more general results in which the underlying series is
only assumed to have a Wold representation. The conclusion is
that aggregation acts to modify only the short memory properties and hence the long memory parameter remains constant
across temporal aggregation. Müller et al. (1990) provided evidence of long memory in financial asset prices by empirically
showing that mean absolute price changes are proportional to
the interval size. More recently, Andersen and Bollerslev (1997)
and Andersen, Bollerslev, Diebold, and Labys (2001) used realized volatility and temporal aggregation to provide informal evidence in favor of their finding of long memory. They claimed
that their estimated long memory parameters, using different
Journal of Business & Economic Statistics, April 2008
frequency data (i.e., daily, weekly, monthly), are similar enough
to be consistent with a true long memory process. Hence, although temporal aggregation has been used to provide informal
evidence of long memory, no formal tests have been proposed.
In this article we formalize this approach by providing a specific
test and the corresponding asymptotic distribution for proper
statistical analysis.
The null hypothesis is that the original data series is generated by a stationary, true long memory process. In this case, the
long memory parameter is invariant to temporal aggregation of
the original data. We use the semiparametric long memory parameter estimator of Geweke and Porter-Hudak (1983) (GPH)
which focuses on only the long memory aspect of the series
and hence does not require specifying the short memory dynamics. Furthermore, the GPH estimator allows us to consider
both linear and nonlinear models concurrently. We derive the
joint limiting distributional properties of the GPH estimators
applied to different temporal aggregates of the data. Given this
distribution, we then introduce a Wald test statistic and derive
its asymptotic distribution. Simulations of both true and spurious long memory processes show that both the size and power
of this test are good. Additionally, the quadratic form of the test
statistic is remarkably easy to evaluate. We apply our test to a
high-frequency foreign exchange dataset and cannot reject the
null hypothesis. Based on all of the models we consider in this
article, we conclude that the long memory property in exchange
rate volatility is generated by a true long memory process as opposed to a spurious long memory process.
The remainder of this article is organized as follows. Section 2 contains the distributional properties for both the long
memory parameter estimator as well as our test statistic and also
contains simulation evidence for further clarification. We apply
our analysis to data in Section 3 and conclude in Section 4.
2. PROPERTIES OF GPH ESTIMATES AND
A SPECIFIC TEST
In this section we present the test statistic for distinguishing
between true and spurious long memory. The test is based on
invariance of the long memory parameter over temporal aggregation. The test statistic is formed using the GPH long memory
parameter estimator. In the first subsection, the limiting distribution of the GPH estimator is examined including our extension
to the temporally aggregated case. The second subsection introduces our test statistic as well as its asymptotic distribution.
2.1 GPH Estimates for Temporal Aggregates
The GPH estimator is used because it is semiparametric in
that only periodogram ordinates in a decreasing neighborhood
of the zero frequency are used in the estimation. Thus, we
avoid many potential misspecification issues involving the short
memory and/or linearity aspects of the model. Another reason
we use the GPH estimator is the strong theoretical understanding of its distributional properties (see Robinson 1995; Hurvich,
Deo, and Brodsky 1998; Deo and Hurvich 2001). The former
two studies focus on Gaussian long memory whereas the latter extends the analysis to stochastic volatility with Gaussian
long memory. Similarly, we consider the distributional properties first for Gaussian long memory and then stochastic volatility with Gaussian long memory.
Ohanissian, Russell, and Tsay: True or Spurious Long Memory
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2.1.1 Gaussian Long Memory. The aforementioned articles show that under proper bandwidth restrictions (i.e., frequencies of the spectral density used), the GPH estimator is
asymptotically normal. In this section we determine the joint
limiting distribution of the GPH estimates obtained using the
temporally aggregated series. Given this joint distribution, we
can then determine the distribution for a test statistic associated
with equality of the long memory parameter across temporal
aggregates of the original series. We begin by stating the formal condition on the long memory processes we consider in
this article.
Condition 1. {Yt } is a stationary mean-zero Gaussian longmemory time series with spectral density f (ω) = |1 −
exp(−iω)|−2d f ∗ (ω), where d ∈ (0, .5), f ∗ (ω) denotes the spectral density of the short memory component of {Yt } and is assumed to be continuous, bounded above, bounded away from
zero, twice differentiable with the second derivative bounded in
a neighborhood of zero.
Note that the assumption of a mean-zero process is without loss of generality as our work will be in the frequency domain and the periodogram of a series is invariant to the mean.
Also note that the symmetry of f ∗ (ω) around the origin implies that f ∗′ (0) = 0. Given a sample Y ≡ (Y1 , Y2 , . . . , Yn )′ ,
we use the notation fj∗ to represent f ∗ (2πj/n) and fj to represent f (2πj/n), which are the spectral densities at specific ordinates for the short memory and original process, respectively.
Now, the GPH estimate of the long memory parameter is calculated from a regression of the log periodogram on the explanatory variable Xj = log |2 sin(πj/n)| and a constant term evaluated at the first l log periodogram ordinates ({log Ij }lj=1 , where
1 n
Ij = 2πn
| t=1 Yt exp(it2π nj )|2 ). Thus,
log Ij = (log f0∗ − C) − 2dXj + log(fj∗ /f0∗ ) + εj ,
∼ .577 is Euler’s constant.
where εj = log(Ij /fj ) + C and C =
Furthermore, we will use the abbreviations that aj = Xj −
l
1 l
2
j=1 Xj and SXX =
j=1 (aj ) . The GPH estimator then is
l
l
1
d = − 2SXX j=1 aj log Ij with an asymptotically normal distribution
√
D
4l(
d − d) → N(0, π 2 /6)
(1)
for l = o(n4/5 ), where →D represents convergence in distribution.
We now introduce additional notation to designate the level
of temporal aggregation (m). Specifically, a superscript (m)
on any statistic represents the corresponding statistic for the
m-temporally aggregated data, where m is a finite positive integer. The m-period nonoverlapping aggregates of Yt are defined
(m)
n
as Zs ≡ m
τ =1 Ym(s−1)+τ for 1 ≤ s ≤ m . Note that the number of ordinates (l) used in the GPH estimator is dependent on
the length of the series, and because temporal aggregation decreases the length of the series, then l(mu ) > l(mv ) for mu < mv .
We next introduce the following condition which details the aggregation levels and number of ordinates used.
Condition 2. (i) For any fixed, but arbitrarily large, aggregation level m, the growth rate of the number of ordinates used
for the GPH estimation is restricted such that l(m) → ∞ and
l(m) = o(n(2−4d)/3 ) as n → ∞.
163
(ii) M denotes the fixed, but arbitrarily large, number of
aggregation levels and (m1 , m2 , . . . , mM ) denote the (ordered)
fixed, but arbitrarily large, aggregation levels for the M series
such that m1 < m2 < · · · < mM .
We note that changing the length of the series results in different frequencies and hence the ωj ’s (which are the frequencies
for the spectral density) are not the same for the same j, that
(1)
(m )
(m)
(mu )
is, ωj = 2π nj , ωj = 2π jm
= ωj v if mu = mv , and
n , ωj
(m)
ωj(km) = ωkj
. For the sake of completeness, we formally intro(m)
duce some of the relevant statistics to be used shortly: Ij represents the periodogram ordinates for the temporally aggregated
(m)
(m)
(m)
1 l(m) (m)
= Xj − l(m)
series, Xj = log |2 sin(π jm
j=1 Xj ,
n )|, aj
l(m) (m)
l(m) (m) 2
(m)
(m)
1
(m)
a log I .
d = − (m)
(a ) , and
S =
XX
j=1
j
2SXX
j=1 j
j
The asymptotical normal distribution in (1) is valid for the
individual GPH estimates obtained using the temporally aggregated series. Our testing approach, based on invariance of the
long memory parameter under temporal aggregation, requires
the joint distribution of these estimates. Specifically, asymptotic
joint normality is established.
We begin by examining the periodograms for series of
differing temporal aggregations. For the general case of an
m-temporally aggregated series, the periodogram is defined by
n/m m
2
m
jm
(m)
Ij =
Ym(t−1)+s exp it2π
,
2πn
n
t=1
s=1
which can be rewritten as
n
2
jm t
m
(m)
Yt exp i2π
Ij =
2πn
n m
t=1
n n
m
s
t
jm
=
−
Ys Yt cos 2π
2πn
n
m
m
s=1 t=1
for 1 ≤ j ≤ mn , where [x] denotes the smallest integer greater
than or equal to x. Note that the ordinates, as well as any statistic based on them, are dependent on the length of the original
series; however, for the sake of notational simplicity we sup(m)
(m)
d(m) is repress this dependence, that is, Ij is really Ij (n),
(m)
ally
d(m) (n), and so forth. Using this representation, Ij can be
(1)
(m)
(1)
written in terms of Ij as Ij = mIj + Y ′ Bj Y for 1 ≤ j ≤ mn ,
where the Bj matrix is defined as Bj ≡ (bj (s, t))1≤s,t≤n and
jm
s
t
m
bj (s, t) ≡
cos 2π
−
2πn
n
m
m
j
− cos 2π (s − t) .
n
The following lemma provides a direct link between the periodogram of an m-temporally aggregated series and the pe(1)
riodogram of the original series. We define Ij as the periodogram of the original series.
Lemma 1. For a process satisfying Condition 1 and an aggregation level (m) with corresponding number of ordinates used
(1)
(m)
(l(m) ) satisfying Condition 2: (Ij − mIj ) →p 0 as n → ∞
for 1 ≤ j ≤ l(m) .
164
Journal of Business & Economic Statistics, April 2008
Corollary 1. Under the assumptions of Lemma 1, for any
(m )
two finite aggregation levels mu and mv with mu < mv : (Ij v −
mv (mu )
) →p
mu Ij
0 as n → ∞ for 1 ≤ j ≤ l(mv ) .
This link between the periodograms of temporally aggregated series provides a framework to derive the asymptotic joint
distribution of the GPH estimates obtained using the temporally
aggregated series, which we state in the following proposition.
Proposition 1. For a process satisfying Condition 1 and aggregation levels with corresponding number of ordinates used
satisfying Condition 2, the joint distribution of the GPH estimates obtained using the temporally aggregated series is asymptotically jointly normal.
Downloaded by [Universitas Maritim Raja Ali Haji] at 17:47 12 January 2016
The following lemma establishes the covariance structure of
the corresponding asymptotically normal joint distribution.
Lemma 2. For a process satisfying Condition 1 and aggregation levels with corresponding number of ordinates used satisfying Condition 2: (4l(mi ) (Cov(
d(mi ) ,
d(mj ) ) − Var(
d(mi ) ))) = o(1)
as n → ∞ for 1 ≤ i < j ≤ M.
This result implies the structure that the covariance between
any two GPH estimates obtained using temporally aggregated
series equals the variance of the lesser aggregated series, asymptotically. This result is confirmed in our simulations. Note
that this structure is consistent with the result in Hausman
(1978) that an efficient estimator must have zero asymptotic
covariance with the difference between the efficient estimator
and any other consistent, asymptotically normal estimator. Although the GPH estimator is not an efficient estimator, the estimate based on the least aggregated data is efficient in the class
of GPH estimates obtained using temporal aggregates or any
linear combination of such estimates. Our results therefore parallel those of Hausman (1978).
2.1.2 Stochastic Volatility With Gaussian Long Memory.
The above derived joint distribution for the GPH estimates obtained using the temporally aggregated series is based on the assumption of a Gaussian time series. To facilitate the use of GPH
estimators for volatility, Deo and Hurvich (2001) extended the
distributional analysis for the GPH approach to a stochastic
volatility model.
The stochastic volatility model they considered is rt =
σ exp(Yt /2)et , where {Yt } is a stationary Gaussian long memory
process independent of the iid mean-zero random variable {et }.
Using the log transformation, this model is linearized as follows:
Zt ≡ log(rt2 ) = μ + Yt + ut ,
(log σ 2
E{log e2t })
(2)
(log e2t
− E{log e2t })
returns {Zt } is a
+
and ut =
where μ =
is iid mean zero. Thus, the log of squared
linear combination of an underlying Gaussian long memory
process plus a noise process. Deo and Hurvich (2001) showed
that adding noise to a long memory process does not change
the asymptotic distribution of the GPH estimator. That is, the
asymptotic distribution of the GPH estimator is the same for
the {Zt } series and the {Yt } series; however, there is a stricter
requirement on the bandwidth choice (i.e., ordinates of the periodogram used for estimation). Specifically, in the Gaussian
long memory case the restriction is l = o(n4/5 ), whereas for the
stochastic volatility with Gaussian long memory case the restriction becomes l = o(n4d/(4d+1) ). Note that for small enough
d this requirement is stricter than the one listed in Condition 2.
Thus, in the stochastic volatility with Gaussian long√ memory
case, the requirement is l = o(n4d/(4d+1) ) for d ∈ (0, 3−1
4 ) and
√
l = o(n(2−4d)/3 ) for d ∈ [ 3−1
4 , .5).
We extend our distribution result from the previous subsection for Gaussian long memory to this stochastic volatility with
Gaussian long memory model. First, we introduce an assumption on the noise process.
Condition 3. {ut } of (2) has a finite eighth moment. (Note
that this condition applies to the iid noise component and does
not restrict the returns process to have a finite eighth moment.)
We now state our next proposition which covers the stochastic volatility with Gaussian long memory model.
Proposition 2. For the {Zt } process of (2) in the stochastic
volatility with Gaussian long memory model where the long
memory process satisfies Condition 1, the iid noise process satisfies Condition 3, and aggregation levels with corresponding
number of ordinates used satisfy Condition 2 along with the
additional restriction that l(mi ) = o(n4d/(4d+1) ), the joint distribution of the GPH estimates obtained using the temporally aggregated series is asymptotically jointly normal with variance–
covariance structure given by Lemma 2.
2.1.3 Simulations. This section provides simulation results. We show that the asymptotic covariance structure of the
previous section provides a very reasonable approximation to
the finite sample behavior of GPH estimates obtained using
temporally aggregated series. In Table 1, we present results
from simulations of an I(.4) process. In theory, the distribution does not depend on the value of the long memory parameter. For the simulations and empirical application in this article,
we use the square root of the number of observations
as our
√
choice for the number of ordinates, that is, l(mi ) = n/mi , as it
is a common rule of thumb used in the literature. Varying the
number of ordinates used in the estimation does not impact the
features of Table 1. Specifically, the number of ordinates used
ranged from n.3 to n.6 . In addition, we considered lower truncation by including a trimming parameter of either 1 or 2. The
ordered aggregation levels are consecutive powers of 2, that is,
(m1 , m2 , . . . , mM ) is defined by mj = 2j−1 for j = 1, 2, . . . , M
for all values of M. Given our simulations and the corresponding GPH estimates obtained using temporally aggregated series,
we compare the theoretical covariance matrix (Table 1a) against
the sample covariance matrix (Table 1b) from simulations for
these estimates.
First, note that the theoretical variance–covariance matrix
does not depend on the value of the long memory parameter.
Simulation results at various degrees of long memory confirm
this fact, and we list only one such set of simulations here which
is representative of the others. Also note that the variance–
covariance structure derived in Lemma 2 (where the covariance
between two estimates equals the variance of the lesser aggregated series) is indeed present in the simulation results. Lastly,
note that the theoretical and sample variances match well for the
longer series, but not as well for the shorter series. This finding
relates directly to a small simulation study in Deo and Hurvich
Ohanissian, Russell, and Tsay: True or Spurious Long Memory
165
Table 1. Variance–covariance matrices for GPH estimators of temporally aggregated series based on (a) theory, (b) simulations,
and (c) an approximation
(a) Theoretical variance–covariance matrix
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610,304
305,152
152,576
76,288
38,144
19,072
9,536
4,768
2,384
1,192
596
298
610,304
305,152
152,576
76,288
38,144
19,072
9,536
4,768
2,384
1,192
596
298
.0005
.0005
.0007
.0005
.0007
.0011
.0005
.0007
.0011
.0015
.0005
.0007
.0011
.0015
.0021
.0005
.0007
.0011
.0015
.0021
.0030
.0005
.0007
.0011
.0015
.0021
.0030
.0042
.0005
.0007
.0011
.0015
.0021
.0030
.0042
.0060
.0005
.0007
.0011
.0015
.0021
.0030
.0042
.0060
.0086
.0005
.0007
.0011
.0015
.0021
.0030
.0042
.0060
.0086
.0121
.0005
.0007
.0011
.0015
.0021
.0030
.0042
.0060
.0086
.0121
.0171
.0005
.0007
.0011
.0015
.0021
.0030
.0042
.0060
.0086
.0121
.0171
.0242
(b) Sample variance–covariance matrix from 500 simulated series of an ARFIMA(0, .4, 0) model
610,304
305,152
152,576
76,288
38,144
19,072
9,536
4,768
2,384
1,192
596
298
610,304
305,152
152,576
76,288
38,144
19,072
9,536
4,768
2,384
1,192
596
298
.0005
.0005
.0008
.0005
.0008
.0011
.0006
.0008
.0012
.0017
.0006
.0008
.0011
.0017
.0024
.0006
.0008
.0011
.0016
.0024
.0034
.0005
.0007
.0011
.0017
.0024
.0036
.0053
.0004
.0006
.0010
.0015
.0023
.0034
.0050
.0070
.0004
.0006
.0010
.0015
.0023
.0035
.0049
.0071
.0110
.0004
.0005
.0009
.0014
.0020
.0032
.0046
.0069
.0106
.0159
.0003
.0004
.0009
.0014
.0019
.0031
.0046
.0069
.0104
.0158
.0250
.0002
.0004
.0009
.0016
.0022
.0033
.0047
.0073
.0108
.0157
.0254
.0408
(c) Approximated variance–covariance matrix using (3)
610,304
305,152
152,576
76,288
38,144
19,072
9,536
4,768
2,384
1,192
596
298
610,304
305,152
152,576
76,288
38,144
19,072
9,536
4,768
2,384
1,192
596
298
.0005
.0005
.0008
.0005
.0008
.0011
.0005
.0008
.0011
.0016
.0005
.0008
.0011
.0016
.0023
.0005
.0008
.0011
.0016
.0023
.0034
.0005
.0008
.0011
.0016
.0023
.0034
.0050
.0005
.0008
.0011
.0016
.0023
.0034
.0050
.0073
.0005
.0008
.0011
.0016
.0023
.0034
.0050
.0073
.0111
.0005
.0008
.0011
.0016
.0023
.0034
.0050
.0073
.0111
.0168
.0005
.0008
.0011
.0016
.0023
.0034
.0050
.0073
.0111
.0168
.0260
.0005
.0008
.0011
.0016
.0023
.0034
.0050
.0073
.0111
.0168
.0260
.0407
NOTE: The number of observations of the original data series is 610,304 and the aggregated series have aggregation levels of 2j for j = 1, 2, . . . , 11.
(2001). As a result of this finite sample problem, they suggest
using an approximation for the variance in practice. Specifically, the approximation is the variance originally put forth by
Geweke and Porter-Hudak (1983):
−1
l
(m)
2
(m)
Varapprox. d
π 2.
(3)
a
= 24
j
j=1
We find that this approximation matches our simulated variances well, and thus provides a simple alternative to using the variances obtained from simulations. Table 1c lists
the corresponding variance–covariance matrix generated using
this approximation strategy and it indeed closely matches the
variance–covariance matrix from simulations.
2.2 Test Statistic
2.2.1 Statistic and Its Asymptotic Distribution. This section proposes our test statistic. Our null hypothesis is that the
original data are from a stationary, true long memory process
in which case the long memory parameter is the same across
d(m2 ) , . . . ,
all aggregation levels. Specifically, let
d ≡ (
d(m1 ) ,
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166
Journal of Business & Economic Statistics, April 2008
d(mM ) )′ be the M-dimensional vector of our estimated long
memory parameters, d ≡ (d(m1 ) , d(m2 ) , . . . , d(mM ) )′ be the constant M-dimensional vector of the actual long memory parameters, and be the asymptotic covariance matrix of
d. Our
null hypothesis is that d(m1 ) = d(m2 ) = · · · = d(mM ) = d where
0 < d < .5. This null hypothesis can be tested by considering
the quadratic form (
d − d)′ −1 (
d − d) which is asymptotically
2
distributed as χ (M). Note that the structure of the asymptotic
covariance matrix , as derived in Lemma 2, requires that the
asymptotic variance of each individual GPH estimate be different for the asymptotic covariance matrix to be invertible. Thus,
we require that the number of ordinates used for the estimation
of each temporally aggregated series be different. In practice, d
will not be known so the mean value of the estimates is used
and, hence, the distribution must be adjusted. Specifically, setting the following ((M − 1), M) matrix:
⎛
−1
−1
−1 ⎞
−1
···
1 − M1
M
M
M
M
⎜ −1
−1
−1 ⎟
1
−1
1
−
·
·
·
⎜
M
M
M
M ⎟,
T=⎝ M
···
···
··· ···
···
··· ⎠
−1
M
−1
M
−1
M
··· 1 −
1
M
−1
M
′
the asymptotic distribution for T
d is N(Td, TT ), which is
actually N(0, TT′ ) because the elements of d are all equal
under the null hypothesis. As a result, the test statistic we use is
≡ (T
d),
W
d)′ (TT′ )−1 (T
(4)
which has an asymptotic χ 2 (M − 1) distribution under the null
hypothesis. Alternatively, we could propose a statistic based
on the least and most aggregated series’ estimates only. Simulations (not presented here) show that the long memory estimates using temporally aggregated series need not diverge
monotonically as the level of aggregation increases for all alternative models, thereby supporting the use of multiple GPH estimates. Our definition of the asymptotic covariance matrix for
(
d − d) stated above is inclusive of normalization factors for the
number of ordinates used in the estimation of each temporally
aggregated series. This characterization allows for very simple
application of the test in practice. There is an equivalent characterization, however, in which the normalization for the number
of ordinates is separated from the asymptotic covariance ma
trix. This characterization is written as Dn (
d − d) ∼asy N(0,),
where is independent of the number of ordinates and Dn is
a deterministic matrix that includes that dependence. For this
to be the identity matrix in
characterization, one could set
which case Dn would be the positive definite square root ma is constructed using the
trix of . Note that the test statistic W
(m)
average of all estimates d . It is possible to devise a test statistic that uses a weighted average of
d(m) . The issue of optimal weight for each estimate is not clear and deserves a careful
study.
2.2.2 Size and Power Properties. We now provide simulation results for the proposed test statistic to examine its small
sample properties. We consider the size of the test statistic using
500 simulated series of an I(.4) model and an ARFIMA(p, d, q)
model fitted to Yen/$ log(r2 ) data (which will be introduced in
detail in the following section). We note that these data are best
modeled by an ARFIMA(1, d, 1) specification with the following parameter estimates: long memory parameter of .42, AR
coefficient of .73, and MA coefficient of .87. We present the
size of the test using the corresponding 95% χ 2 (M − 1) critical value for various values of M (i.e., various numbers of aggregated series). We present the empirical size of our test statistic in Table 2 using three sets of variances: theoretical variances, simulated variances, and approximated variances calculated via (3). As expected, the results suggest that there is significant size distortion when using the theoretical variances which
decreases as we exclude the shorter series. The corresponding size distortions essentially disappear, however, when using
the structure of the theoretical variance–covariance matrix [i.e.,
Cov(
d(mi ) ,
d(mj ) ) = Var(
d(mi ) ) for mi < mj ] and using either the
simulated or approximated variances. Given this result, it is fair
to conclude that the main cause of this size distortion is the substantial deviation of the finite sample variances from asymptotic
values. This observation suggests using simulated and approximated variances instead of theoretical variances.
We now consider the power of the statistic for some spurious long memory processes using simulations from five relevant alternative models. Namely, we consider (1) a (nonstationary) random level shift model, (2) a (stationary) random level
shift model, (3) a Markov Switching model with iid regimes,
(4) a Markov Switching model with GARCH regimes, and
(5) a white noise model with a (slow) deterministic trend. Each
of these models has been shown in the literature to be able to
exhibit spurious long memory (in particular, see the articles of
Bhattacharya, Gupta, and Waymire 1983; Diebold and Inoue
2001; Granger and Hyung 1999, among others). The complete
specifications of our five alternative models are listed in the Appendix. Prior to considering the power of the test statistic itself,
Table 2. Empirical size properties of test statistic using 500 series simulation of ARFIMA(0, .4, 0) and ARFIMA(1, .4, 1) models
for 5% theoretical size
12
11
10
9
Size w/theor. var.
Size w/simul. var.
Size w/approx. var.
33%
9%
9%
24%
9%
7%
17%
8%
6%
12%
6%
5%
Size w/theor. var.
Size w/simul. var.
Size w/approx. var.
32%
8%
8%
25%
9%
7%
17%
8%
6%
13%
6%
5%
M
8
7
6
5
4
3
2
ARFIMA(0, .4, 0)
10%
9%
6%
4%
5%
6%
8%
4%
4%
7%
4%
5%
7%
5%
5%
7%
7%
6%
7%
7%
6%
ARFIMA(1, .4, 1)
11%
10%
6%
5%
5%
7%
9%
5%
5%
8%
5%
6%
8%
7%
7%
8%
8%
7%
8%
8%
8%
NOTE: The number of observations of the original data series is 610,304 and the aggregated series have aggregation levels of 2j for j = 1, 2, . . . , M − 1.
Ohanissian, Russell, and Tsay: True or Spurious Long Memory
167
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Table 3. Mean and empirical 95% confidence interval of estimated long memory parameter from simulations of the five alternative models,
specified in the Appendix, across temporal aggregation of original series
m=1
n = 610,304
m=2
n = 305,152
m=4
n = 152,576
m=8
n = 76,288
m = 16
n = 38,144
m = 32
n = 19,072
m = 64
n = 9,536
m = 128
n = 4,768
m = 256
n = 2,384
m = 512
n = 1,192
m = 1,024
n = 596
m = 2,048
n = 298
RLS–NS
RLS–S
MS–IID
MS–GARCH
Trend
σε2 = 5
ση2 = 1
p = .00001
σε2 = ση2 = 1
p = .003
μ1 = −μ2 = 1
σ2 = 1
p01 = p10 = .001
ω0 = 1, ω1 = 3
α = .4, β = .3
p01 = p10 = .001
c=3
β = −.1
.34
(.14, .58)
.41
(.17, .68)
.50
(.21, .78)
.58
(.27, .85)
.67
(.32, .95)
.76
(.36, .98)
.84
(.42, 1.05)
.89
(.57, 1.06)
.93
(.62, 1.09)
.96
(.73, 1.14)
.99
(.68, 1.26)
1.00
(.59, 1.24)
.30
(.26, .34)
.21
(.16, .27)
.14
(.07, .20)
.09
(.00, .16)
.05
(−.05, .14)
.03
(−.08, .14)
.02
(−.13, .17)
.02
(−.13, .16)
.02
(−.17, .25)
.00
(−.25, .21)
.01
(−.36, .36)
.03
(−.48, .37)
.41
(.35, .46)
.31
(.26, .37)
.23
(.16, .28)
.15
(.07, .24)
.09
(.01, .18)
.05
(−.07, .16)
.02
(−.12, .15)
.01
(−.16, .15)
.00
(−.23, .18)
.01
(−.21, .23)
.01
(−.28, .25)
.01
(−.47, .31)
.36
(.31, .40)
.28
(.23, .34)
.21
(.14, .27)
.14
(.07, .22)
.09
(.00, .18)
.05
(−.07, .17)
.02
(−.14, .17)
.01
(−.18, .16)
.00
(−.22, .19)
.01
(−.23, .22)
.01
(−.35, .27)
.00
(−.47, .32)
.40
(.37, .44)
.47
(.44, .50)
.55
(.51, .59)
.62
(.59, .66)
.68
(.63, .74)
.73
(.67, .79)
.75
(.70, .80)
.76
(.71, .84)
.75
(.69, .81)
.74
(.70, .81)
.73
(.68, .79)
.72
(.68, .78)
we first use simulations to examine the behavior of the GPH estimate of the long memory parameter across different levels of
aggregation. For each model, we select generic parameter values that generate spurious long memory for the original (nonaggregated) data series. The results in Table 3 clearly show that
the long memory parameter estimate does not remain constant
for any of these alternative models across temporal aggregation
of the data. We note that although we do not provide a corresponding table for true long memory processes, simulations for
these models clearly demonstrate the long memory retention
property for true long memory processes. This evidence suggests that our approach should be able to distinguish between
these models and true long memory processes. We now specifically examine the power of our test statistic.
We use 100 simulated series using two sets of parameter values for each of the five models. The first set of parameter values
are the generic ones used in Table 3. The second set of parameter values are determined from simple estimations of these
models using the foreign exchange data we discuss later in the
article. The MS–GARCH model is estimated using deseasonalized returns, and log squared deseasonalized returns are used
for the other models. Our goal here is to ensure that the simulations are valid for relevant parameter values. For some models
the estimated probability of jumps was too large to generate
long memory in the original (disaggregated) series. Hence, the
probabilities were restricted to be small enough to ensure that
the original series has the requisite long memory present. This
is a direct result of the fact mentioned earlier that the key to
the structural-break models exhibiting spurious long memory is
that the number of jumps in a given sample be relatively small.
For a given model, we use 100 realizations in the power study.
The results are listed in Table 4 and are encouraging for our
goal of distinguishing between true and spurious long memory.
The proposed test has excellent power for all of the considered
alternative models.
Our null hypothesis is stated as a stationary, true long memory process that corresponds to a long memory parameter between 0 and .5 and invariant to temporal aggregation. Our
test statistic only considers the invariance to temporal aggregation property and thus, strictly speaking, our statistic only tests
this invariance. To address the rest of the null hypothesis, two
pretests must be performed, namely, the long memory parameter must be tested to be (i) greater than zero and (ii) less than .5.
Specifically, the first pretest involves a null hypothesis of d = 0
with a one-sided alternative hypothesis of d > 0. Similarly, the
second pretest involves a null hypothesis of d = .5 with a onesided alternative hypothesis of d < .5. In each of these pretests,
the null hypothesis would have to be rejected for our test to
be applied. These tests are discussed in the literature (see, e.g.,
Robinson 1995; Velasco 1999). As long as the two pretests have
power that is asymptotically equal to 100%, the asymptotic size
of our test statistic and null hypothesis will not be affected by
these pretests. In the empirical analysis presented here, we ignored any pretests. We did perform simulations that included
168
Journal of Business & Economic Statistics, April 2008
Table 4. Empirical power properties of test statistic using 100 simulated series of the five alternative models given in the Appendix
M = 12
Generic parameter values
RLS–NS
RLS–S
MS–IID
MS–GARCH
Downloaded by [Universitas Maritim Raja Ali Haji] at 17:47 12 January 2016
Trend
Power
Estimated parameter values
Power
σε2 = 5, ση2 = 1
99%
σε2 = 4.4, ση2 = 2.3
93%
σε2 = ση2 = 1
p = .003
100%
σε2 = 2.2, ση2 = 5.7
p = .002
100%
ω0 = 1, ω1 = 3
α = .4, β = .3
p01 = p10 = .001
100%
ω0 = .08, ω1 = .23
α = .28, β = .56
p01 = p10 = .0008
100%
p = .00001
μ1 = −μ2 = 1
σ2 = 1
p01 = p10 = .001
c=3
β = −.1
σ 2 = 1.0
100%
100%
p = .000004
μ1 = −.4, μ2 = −4.0
σ 2 = 2.5
p12 = p21 = .001
c = −4.3
β = −.063
σ 2 = 8.1
100%
100%
NOTE: The number of observations of the original data series is 610,304 and the aggregated series have aggregation levels of 2j for j = 1, 2, . . . , 11.
the two pretests, but do not present those results here in the interest of conserving space. Specifically, we used pretests based
on the long memory parameter estimator in Velasco (1999) for
the models we used in the size and power analysis. The inclusion of the two pretests did not change the size analysis
at all, whereas the power increased to 100% for the RLS–NS
model, which was the only alternative model not to achieve
100% power without the pretests. Alternatively, in some applications the long memory parameter d∗ may be known a priori
and the pretests would not need to be performed. In this case,
d
d ≡ (d∗ , . . . , d∗ )′ with 0 < d∗ < .5 would be subtracted from
(instead of the demeaning performed by T) and the resulting
quadratic form test statistic would then be asymptotically distributed as χ 2 (M).
Throughout this article the null hypothesis has been restricted to stationary, true long memory processes that correspondingly have a constant long memory parameter. Recently,
there has been an interest in time-varying long memory models
(e.g., Jensen and Whitcher 2000; Wang, Cavanaugh, and Song
2001). These models are typically nonstationary and have timevarying spectral representation and are thus locally stationary in
the sense of Dahlhaus (1996, 1997). The properties of our test
statistic in the presence of a time-varying long memory parameter will likely depend on the way the long memory parameter
evolves over time and are beyond the scope of this article.
3. APPLICATION TO FOREIGN EXCHANGE DATA
3.1 Dataset Description
We now apply the proposed test to the intraday foreign exchange quote data. This dataset has been used widely by others,
such as Andersen et al. (2001), in the consideration of the long
memory property of volatility.
Specifically, we use 5-minute returns for DM/$ and Yen/$
from 1986 to 1996 obtained from Olsen and Associates. These
returns are based on interbank FX quotes that appeared on the
Reuters’ FXFX page. Prices at each 5-minute mark are extracted from these quotes by linearly interpolating from the log
bid and the log ask for the two closest ticks. The returns are
then calculated as the change in 5-minute average log bid and
ask prices. Following the approach of Andersen et al. (2001),
weekends and holidays were removed due to the lack of activity
during these periods. In addition, any runs of length 5 or higher
of the same value were removed as these points most likely correspond to problems with the Reuters feed and occur mainly in
the early part of the sample. The interested reader is referred
to Andersen et al. (2001) for a more detailed description of the
dataset.
3.2 Removing Periodicity
An inherent property of the intraday data that we use is
its periodicity. There are clear diurnal patterns in the volatility, as shown in Andersen and Bollerslev (1998). This periodicity manifests itself as periodicity in the autocorrelations
of the absolute, squared, and log squared returns which exists
at rather long lags (288th lag corresponding to the daily frequency). This deterministic behavior could potentially affect
our long memory estimates and results in the same way that
breaks can generate spurious long memory. Furthermore, there
is also the possibility of the long memory being a generalized
long memory process with seasonal long memory (see Gray,
Zhang, and Woodward 1989). Thus, prior to proceeding with
our main analysis, we first remove this periodicity and explore
the seasonal long memory issue.
In considering the periodicity, we first provide a plot of the
autocorrelation function (acf) for the log squared (raw) returns
in Figure 1. Note, all of the figures presented are based on the
DM/$ data as the corresponding plots for the Yen/$ data are
quite similar and are thus not presented in the interest of saving
space. The acf plot shows the telltale periodicity as seen in Andersen and Bollerslev (1998) with the highest peaks at the daily
frequency. As opposed to the more complicated procedure used
in Andersen and Bollerslev (1998) to remove the periodicity,
we begin with a simple adjustment to the returns. Specifically,
we divide each return by the standard deviation of the returns
in that time interval during the day. We find that this simple
standardization is able to remove most of the periodicity and,
although there is still some residual periodicity as seen in the
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Ohanissian, Russell, and Tsay: True or Spurious Long Memory
169
Figure 1. ACF of log squared returns using raw returns for DM/$
exchange rate data.
Figure 3. ACF of log squared returns using adjusted returns for
DM/$ exchange rate data.
plots, it is essentially as effective as the Andersen and Bollerslev procedure. Figure 3 plots the acf after this standardization
and another adjustment mentioned below.
Holidays, weekends, or other breaks in the temporal continuity of the data introduce specific breaks, which could cause
spurious long memory. To examine this possibility, we plot the
acf of the log squared (standardized) returns for the data as a
whole and also for the case where we only use calendar continuous data. As we see in Figure 2, both of these plots are virtually
identical, and hence, we conclude that these breaks in calendar
time are not affecting our results.
Next, we consider the autocorrelation functions conditional
on the time (hour) of the day. While examining this, we
found large negative autocorrelation spikes corresponding to
the Japanese lunch period mentioned in Andersen and Bollerslev (1998). The Japanese market essentially shuts down during
this period. For this reason, we treat this time period as we do
weekends and simply remove it from the dataset. After doing
so, the conditional acf’s are much more consistent without any
peculiar negative spikes.
Finally, we consider the possibility of seasonal long memory
by plotting the sample periodogram of the log squared (raw)
returns and log squared (adjusted) returns in Figure 4. The raw
series clearly shows a seasonal hump corresponding to the daily
frequency which is almost completely dampened in the adjusted
series, thus providing further evidence that our procedure is effectively removing the periodicity. The other important point to
note is that the only spike in the periodogram is at the zero frequency and not at any other frequency. At the daily frequency,
it is merely a hump and not a spike. Hence, we conclude that
there is no seasonal long memory component in this dataset.
Thus, we sum up the two simple steps of our periodicity removal procedure. We remove the Japanese lunch period, which
corresponds to 3:00GMT to 4:45GMT, as it essentially represents mar