Time series can have a long memory. Those systems are not independently identically distributed. This phenomenon is often referred asburstinessin the literature.18The under- lying stochastic processes for such burstiness are calledfractal. Fractal processes with a long memory are called persistent. A common characteristic of those fractal processes is that their space time is governed parsimoniously by power law distributions. This effect is called the “Noah-Effect”, explaining the occurrence of heavy tails and infinite variance.
It can be observed as the tendency of time series for abrupt and discontinuous changes.
Another property of fractal processes is hyperbolically decaying autocorrelations, which is known as the “Joseph-Effect”. It is the tendency of a persistent time series to have trends and cycles.
The examination of fractal processes in finance has become a popular topic over the years.19 For a long-memory process, we observe that larger-than-average representations are more likely followed by larger-than-average representations instead of lower-than- average representations. Hurst developed a statistic to examine the long memory of a sto- chastic process. As significant autocorrelations are often not visible, he came up with a new methodology to provide a measure (the Hurst-Exponent) for long-range dependence within a time series.
Due to the failures of traditional capital market theory which is largely based on the theory of martingales, researchers experienced that markets do not follow a purely random walk. The fractal market hypothesis was developed. The existence of self-similar struc- tures is a major component of it. For self-similar processes, small increments of time are statistically similar to larger increments of time.
Self-Similarity is defined as follows:20LetXt be a stochastic process with a continuous timet.Xt is self-similar with self-similarity parameterH (H-ss), if the re-scaled process with time scalect,c−HXct, is equal in distribution to the original processXt,
Xt=d c−HXct. (40)
This means, for a sequence of time points t1, . . . , tk and a positive stretch factor c, the distribution ofc−H(Xct1, . . . , Xctk)is identical with the one ofXt1, . . . , Xtk. In other words, the path covered by a self-similar process always looks the same, regardless of the scale it is observed with. In terms of financial data this means: no matter if we have intra- day, daily, weekly, or monthly data, the plots of the resulting processes have similar looks.
For further information on self-similarity we refer to Samorodnitsky and Taqqu (1994), or Beran (1994).
18Willinger, Taqqu and Erramilli (1996).
19For example, we refer to Mandelbrot (1997a, b, 1999), and Peters (1994).
20Beran (1994).
Ch. 10: Stable Non-Gaussian Models 423
4.1. Fractal processes and the Hurst-Exponent
First, we consider a process without a long memory. A perfect example is Standard Brown- ian Motion, which is characterized as a standard random walk. 21 Commonly known is Einstein’s “to the one-half” – rule, describing the distance covered by a particle driven by Standard Brownian Motion. It states that the distance between consecutive values of the observed time series of this particle is proportional to the square root of time:22
R∼T0.5. (41)
The power of 0.5 refers to the Hurst-Exponent which is already known as the self- similarity parameter. For Standard Brownian Motion, the Hurst-ExponentH is equal to 0.5 which means that we have an unbiased random walk. A process with a Gaussian lim- iting distribution but a Hurst-ExponentHdifferent from 0.5 is called Fractional Brownian Motion. Fractional Brownian Motion differs from Standard Brownian Motion by the fact that it is a biased random walk. The odds are biased in one direction or the other.
Definition of Fractional Brownian Motion.23 Let us assume a self-similar Gaussian process withXt,t∈R, having mean zero and the autocovariance function
Cov(Xt1, Xt2)=1
2 |t1|2H+ |t2|2H− |t1−t2|2H
VarX(1), (42)
whereHis the self-similarity parameter andH∈(0,1).
Such a process is called a Fractional Brownian Motion. ForH=1/2 it becomes a Stan- dard Brownian Motion.
The increments of Fractional Brownian Motion,Yj=BH(j+1)−BH(j ),j∈Z, form a stationary sequenceYj which is called Fractional Gaussian Noise.
Fractional Gaussian Noise. A sequence of Fractional Gaussian Noise has the following properties:
(i) its mean is zero,
(ii) its varianceEYj2=EBH2(1)=σ02, and (iii) its autocovariance function is
r(j )=σ02 2
(j+1)2H−2j2H+(j−1)2H ,
wherej ∈Z,j0, andr(j )=r(−j )forj <0.
21See Campbell, Lo and McKinlay (1997).
22Peters (1994).
23Samorodnitsky and Taqqu (1994).
424 B. Martin et al.
Forj→ ∞,r(j )behaves like a power function.
jlim→∞r(j )→0. (43)
The autocorrelations are given by ρ(j )=1
2
(j+1)2H−2j2H+(j−1)2H
, (44)
wherej 0 andρ(j )=ρ(−j )forj <0. As j tends to infinity,ρ(j )is equivalent to H (2H−1)j2H−2.24
In the presence of long memory, 1/2< H <1, the correlations decay to zero so slowly that they are no more summable:
∞ j=−∞
ρ(j )= ∞. (45)
ForH =1/2, i.e., a Gaussian i.i.d. process, all correlations at non-zero lags are zero.
For 0< H <1/2, the correlations are summable, and it holds:
∞ j=−∞
ρ(j )=0. (46)
H=1 impliesρ(j )=1. ForH >1, the condition−1ρ(j )1 is violated.
For 0< H <1, a Gaussian process with mean zero and the given autocovariance func- tion is self-similar and has stationary increments (H-sssi). The above autocovariance func- tion is shared by all GaussianH-sssi processes.
Fractional processes with stable innovations. There are many different extensions of the Fractional Brownian motion to theα-stable case withα <2. Most common is the so called Linear Fractional Stable Motion or, Linear Fractional Lévy Motion.
In an analogy to the Gaussian case withα=2, the increments of Linear Fractional Stable Motion25show long-range dependence forH >1/α. LRD forα <1 does not exist, asH must lie in(0,1). Processes withH=1/αare calledα-stable Lévy Motion whose incrementsX(tj+1)−X(tj)are all mutually independent.
Forα-stable Lévy processes with infinite variance, we carefully have to interpret the value obtained forH and how it is related to the parameterd measuring the degree of long-range dependence.
24Beran (1994).
25Samorodnitsky and Taqqu (1994).
Ch. 10: Stable Non-Gaussian Models 425
H, the Hurst-Exponent, is the scaling parameter and describes asymptotical self-simi- larity:
For finite variance processes, the relation betweenHanddis H=d+1
2. (47)
For processes with infinite variance (α <2), the relation is H=d+1
α. (48)
Ifd >0, the time series is governed by a long-memory process.
There is a number of methods to distinguish a purely random time series from a fractal one. For example, the classical R/S analysis26determines the parameterHof a time series.
The resulting graph is called pox-plot of R/S or rescaled-adjusted-range plot.
Before the classical R/S method will be described, we briefly explain two other methods to derive the Hurst-ExponentH, the Aggregated Variance Method and the similar method Absolute Values of Aggregated Series.27
4.2. The Aggregated Variance Method
The original time seriesX=(Xi, i=1, . . . , N )is divided into blocks. Each block has the sizemelements. The indexk labels the block. The aggregated series is calculated as the mean of each block:
X(m)(k)= 1 m
km
i=(k−1)m+1
Xi withk=1,2, . . . , N
m
. (49)
After building the aggregated series, we get the sample variance ofX(m)(k)as VarX (m)= 1
N/m
N/m
k=1
X(m)(k)2
−
1 N/m
N/m
k=1
X(m)(k) 2
. (50)
The procedure is repeated for different values ofm{mi, i1}. The chosen values for mshould be equidistant on a log-scale, i.e.,mi+1/mi=C.
AsX(m)scales likem(H−1), the sample varianceVarX (m)behaves likem(2H−2). Thus, plotting a log-log representation ofmandVarX (m), the plots form a straight line with slope 2H−2.
26Mandelbrot and Wallis (1968).
27Teverovsky, Taqqu and Willinger (1995) as well as Teverovsky, Taqqu and Willinger (1998).
426 B. Martin et al.
4.3. Absolute Values of the Aggregated Series
This method is similar to the Method of Aggregated Variance explained above. Starting again with the aggregated series, we calculate the sum of the absolute values of the aggre- gated series.
1 (N/m)
N/m
k=1
X(m)(k). (51) If the original series has a long-range dependence parameterH, the log–log-plot ofm versus the corresponding values of the statistic provides us with a line of slopeH−1.
4.4. Classical R/S analysis
Let us assume we have a time series ofN consecutive values.Y (n)=n
i=1Xi,n1, is the partial sum andS2(n)=1nn
i=1[Xi−n−1Y (n)]2,n1, is the corresponding sample variance.
We defineZ(t)=Y (t)−ntY (n). Therescaled-adjusted-rangestatistic or R/S statistic is defined by
R
S(n)= 1 S(n)
0maxtnZ(t)− min
0tnZ(t)
. (52)
R/S is called the rescaled adjusted range as its mean is zero, and it is expressed in terms of the local standard deviation. For largen, the expected value of the statistic approaches c1nH:
E
R/S(n)
∼c1nH, (53)
wherec1 is a positive, finite constant and does not depend onn. In case of long-range- dependence in a Gaussian process, the values forH range in the interval(0.5,1.0). For an i.i.d. Gaussian process (i.e., pure random walk) or a short-range dependent process, the value ofR/S(n)approachesc2n0.5.c2is independent ofn, finite, and positive.
E R/S(n)
∼c2n0.5. (54)
The practical application of the R/S analysis is performed graphically. It is described in Mandelbrot and Wallis (1968).
With this procedureKdifferent estimates ofR/S(n)are obtained. It starts with dividing the total sample ofN consecutive values intoKblocks, each of sizeN/K. We define
k(m)=(m−1)N
K +1 (55)
Ch. 10: Stable Non-Gaussian Models 427
as the starting points of each block, where K is the total number of blocks and m= 1, . . . , Kis the current block number. Now we compute theR(n, k(m))/S(n, k(m))for each lagnsuch thatk(m)+n < N. All data points beforek(m)are ignored in order to avoid the influence of particular short-range dependence in the data.
Plotting the log(R(n, k(m))/S(n, k(m)))for each block versus log(n), we can estimate the slope of the fitted straight line. The classical R/S analysis is quite robust against variations in the marginal distribution of the data. This is also true for data with infinite variance.
Calculating the Hurst-ExponentH and the stability indexαof the process innovations, the long-range dependence parameterdis obtained by
d=H−1
2, (56)
for finite variance (α=2), and b d=H−1
α, (57)
for infinite variance (α <2).
Long-range dependence occurs ifdis greater than 0.
The R/S analysis is a nonparametric tool for examining long-memory effects. There is no requirement for the time series’ underlying limiting distribution. In case of an underlying Gaussian process (α=2), a Hurst-Exponent ofH=0.5 implies that there is no long-range dependence among the elements of the time series.
For 0.5< H <1, a Gaussian time series is calledpersistent.28A persistent time series is characterized by long-memory effects. If long memory is present, the effects occur regard- less of the scale of the time series. All daily changes are correlated with all future daily changes, and all weekly changes are correlated with all future weekly changes. The fact that there is no characteristic time scale is an important property of fractal time series.
0< H <0.5 signals anantipersistentsystem for finite variance. Such a system reverses itself more frequently than a purely random one. At the first glance, it looks like a mean- reverting process. But this would actually require a stable mean, which is not the case in such systems.
4.5. The modified approach by Lo
Hurst’s R/S statistic turned out to react sensitively towards short-memory processes. Thus, Lo (1991) modified the classical R/S statistic, now showing robustness towards short-range dependence.29Lo’s statistic only focuses on lagn=N, the length of the series.30Multiple lags are not analyzed, the statistic does not varynover several lags< N.
28Peters (1994).
29Lo (1991).
30Teverovsky, Taqqu and Willinger (1998).
428 B. Martin et al.
Compared to the graphical R/S method, which delivers an estimate of the parameterH, Lo’s modified statistic just indicates the presence of long-range dependence, but does not deliver an estimate of the Hurst-Exponent. The statistic performs a test of the hypotheses
• H0: no long-range dependence.
Instead of the ordinary sample standard deviationS for normalization, there is an ad- justed standard deviationSqin the denominator.Sqconsiders the elimination of short term memory to the statistic. As it is known that the R/S statistic responds very sensitively to- wards short range dependence, the influence of short range dependence can be offset by normalizingR with a weighted sum of short-lag autocovariances. To the varianceS2Lo added weighted autocovariances up to orderq.31 His modified statisticVq(N )is defined by
Vq(N )=N−1/2R(N )
Sq(N ), (58)
with
Sq(N )= S2+2
q
j=1
wj(q)γˆj, (59)
whereγˆjis the autocovariance of orderj for the observed time series.wj(q)is defined as wj(q)=1− j
q+1 withq < N. (60)
The statisticVq(N )is applied for a hypothesis test. It checks if the null hypothesis of the test can be rejected or not, given a certain confidence level. The two hypotheses are:
• H0: no long-range dependence present in the observed data, 0< H 0.5.
• H1: long-range dependence is present in the data, 0.5< H <1.
The statistic assumes a Gaussian process (α=2). In cases where the value ofVq(N )lies inside the interval[0.809,1.862],H0is accepted as the statistic is in the 95% acceptance region. ForVq(N )outside the interval[0.809,1.862],H0is rejected.
Lo’s results are asymptotic assumingN→ ∞andq=q(N )→ ∞.32However, in prac- tice the sample size is finite and the value of the statistic depends on the chosenq. Thus, the question arises, what would be the proper value forqin order to perform the hypothesis test? Andrews (1991) has developed a data driven method for choosingq:33
qopt= 3N
2 1/3
2ρˆ 1− ˆρ2
2/3
, (61)
31Peters (1994).
32Teverowsky, Taqqu and Willinger (1998).
33See Lo (1991).
Ch. 10: Stable Non-Gaussian Models 429
here[ã]stands for the greatest integer smaller than the value between.ρˆis the first order au- tocorrelation coefficient. Therefore, choosing Andrews’qassumes that the true underlying process is AR(1).
Critique of Lo’s statistic. Lo’s statistic is applied by calculatingVqfor a number of lags q, plotting those values againstq. The confidence interval for acceptingH0 at the 95%
confidence level is plotted as well.
Simulations have shown that the acceptance ofH0(and therefore the value ofVq(N )) varies significantly withq. Taqqu, Willinger and Teverowsky (1998) found that the larger the time series and the larger the value forq, the less likelyH0is rejected.
Whereas, Lo’s statistic just checks for the significance of long-range dependence, the graphical method of the classical R/S provides relatively good estimates ofH.
For small q the results ofVq usually vary strongly. Then a range of stability follows after the so called “extra” short-range dependence has been eliminated, and the only effect measurable for the statistic would be long-range dependence.
Applying the statistic to Fractional Brownian Motion withH >0.5, which is a purely long-range dependent process without short memory effects,Vq is expected to stabilize at very low values ofq. Unfortunately this could not be confirmed by the testing of Taqqu, Willinger and Teverowsky (1998). Moreover, they demonstrate that – ifq is large enough – the following holds forVq(N )andq1/2−H:
Vq(N )q1/2−H. (62)
ForH >0.5, Vq decreases with increasing q. Even for strongly fractional processes with time series containing 10000 samples, Taqqu, Willinger and Teverowsky found that, with increasing values for q, the probability ofVq lying inside theH0 95% confi- dence interval and accepting the null-hypothesis grows. To mention three cases only: for q =500 andH =0.9 the null-hypothesis (no long-range dependence) is accepted with 90% for Fractional Brownian Motion, with 92% for FARIMA(0.5, d,0), and with 94% for FARIMA(0.9, d,0).34
Lo’s test is very conservative in rejecting the null-hypothesis. It works for short-range dependence, but in cases of long-range dependence it mostly accepts the null-hypothesis.
The statistic of Lo is certainly an improvement compared to the short-range sensitive clas- sical R/S, but should not be used isolated without comparing its results with other tests for LRD.
In practical applications, the question for a proper choice ofq remains. The value of Andrews’ data driven qopt depends on the econometric model underlying the observed time series, but, the appropriate model is not known in advance. Andrews’ choice bears the assumption that the time series obeys an AR(1)process.
34FARIMA(0.5, d,0)means a fractional ARIMA process with an AR(1)coefficient of 0.5 and an MA(1)coef- ficient of 0.
430 B. Martin et al.
It used to be a common way to asses long-range dependence by looking at the rate at which the autocorrelations decay. With a Hurst-ExponentH different from 0.5 the corre- lations are no longer summable. Such non-summability of autocorrelations used to be seen as a comfortable way of assuming long-range dependence. But there are pitfalls: if the un- derlying process is considered to follow a stable law withα <2, a second moment does not exist and therefore autocorrelations do not exist either.
It can be concluded that – if testing for long-range dependence – the application of a single technique is insufficient.
4.6. The statistic of Mansfield, Rachev and Samorodnitsky (MRS)
Long-range dependence means that a time series exhibits a certain kind of order over a long comprehensive period. Instead of pure chaos with no rule in the price movements of an asset, we can find periods of time with its sample mean significantly different from the theoretical mean. The stronger the long-memory effects in the time series, the longer an interval of the series whose mean deviates from the expected value.
Mansfield, Rachev and Samorodnitsky (1999) concentrate on this property of LRD- exhibiting time series. This property of LRD is valid regardless of the assumed underlying stochastic model.
The authors define a statistic that delivers the length of the longest interval within the time series, where the sample mean lies beyond a certain threshold. The threshold is set greater than the finite meanEXi of the whole time series. Furthermore, the time series is assumed to follow a stationary ergodic process.
Expressed in mathematical terms, the statistic is defined as Rn(A)=sup
j−i: 0i < jn, Xi+1+ ã ã ã +Xj
j−i ∈A
, (63)
which is defined for everyn=1,2, . . .. If the supremum is taken over the empty set, the statistic is defined to be equal to zero.
The set A is defined either as
A=(θ ,∞) withθ > à, (64)
or as
A=(−∞, θ ) withθ < à, (65)
whereàis the theoretical mean of the time series.
Rn(−∞, θ )andRn(θ ,∞)are interpreted as “greatest lengths of time intervals when the system runs under effective load that is different from the nominal load”.35 In the following, the examination is restricted toRn(θ ,∞).
35Mansfield, Rachev and Samorodnitsky (1999).
Ch. 10: Stable Non-Gaussian Models 431
A theoretical way to examine a time series for long-range dependence would be the log–log plot ofRn(θ ,∞)versusn. In the case of long-range dependence, the slope of the plot would be expected to be greater than 1/αwithαas the tail index. However,αis not known in advance. Therefore, Mansfield, Rachev and Samorodnitsky developed a statistic that does not rely on an a-priori tail index. They defined
Wn(θ )=Rn(θ ,∞)
Mn , (66)
whereMn=max(X1, . . . , Xn)is the largest of the firstnobservations,n1. This sta- tistic has a self-normalizing nature, and because of the denominator it has the ability to compensate for the effects of the tail-indexα.
In case of short-range dependence, the ratioWn(θ )approaches a weak limit asn→ ∞. In case of long-range dependence,Rngrows faster thanMnand the statistic diverges.
For visualization, the statisticθ Wn(θ ) is plotted againstθ. Its limiting distribution is independent ofθ. A difficult task is the selection of the proper range ofθ. It has to be determined empirically by looking where the values forθ Wn(θ )stabilize.
Once the value of the statistic is at least 19 for a certainθ then long-range dependence is present at a significance level of 0.05.
4.7. Empirical results for long-range dependence in credit data
For our empirical examination of long-memory effects in daily credit return data, we use the returns of bond indices provided by Merill Lynch.36We have selected four indices with time series of daily index prices from January 1988 to April 2000. Each index represents a number of bonds with similar properties (see explanation in Table 15). As the analysis of long-memory effects requires large data samples, an important criterion for the selection of an index was the available sample size. The sample sizes are listed in Table 16.
We apply three different methods for estimating the self-similarity parameterHand two methods performing a hypothesis test regarding the presence of LRD. As explained before, we have chosen
(i) the “Aggregated Variance Method”,
(ii) the method “Absolute Values of Aggregated Series”,
(iii) the classical R/S analysis developed by Mandelbrot and Wallis, (iv) Lo’s modified R/S statistic, and
(v) the statistic of Mansfield, Rachev and Samorodnitsky (MRS).
All these methods have been implemented with Matlab 5.3. Methods (i)–(iii) provide an estimate of the Hurst-ExponentH. Method (iv) tests if the null hypothesis “no long-range dependence” has to be accepted or rejected at a given confidence level. Method (v) is also a hypothesis test, however, contrary to Lo’s test it works independently of the tail index.
36The time series were obtained via Bloomberg’s Index Section.