cointegration of stochastic multifractals with application to foreign exchange rates

$: Cointegration of stochastic multifractals with application to foreign exchange rates$
Cointegration of stochastic multifractals with application toforeign exchange rates

V.V. Anha,*, Q.M. Tienga, Y.K. Tseb

aCentre in Statistical Science and Industrial Mathematics, Queensland University of Technology, GPO Box 2434,

Brisbane, Qld 4001, AustraliabDepartment of Economics, National University of Singapore, Kent Ridge, Singapore 119260

Received 15 October 1999; accepted 12 January 2000

Abstract

The existing concept of cointegration applies to integrated processes (in the Box-Jenkins ARIMAframework) or processes with long-range dependence. These processes are assumed to display amonoscaling behaviour (such as that of a fractional Brownian motion). On the other hand, manyturbulent processes are known to be intermittent, hence possess multiscaling characteristics. This paperdevelops a concept of cointegration for these stochastic multifractals. A model is suggested for testingfor cointegration and applied to the exchange rates of three major currencies. 7 2000 IFORS. Publishedby Elsevier Science Ltd. All rights reserved.

1. Introduction

The concept of cointegration plays a key role in studying the causal relationships andprediction of nonstationary processes. The concept was introduced into time series andeconometrics by Granger (1981), and was discussed in a formal setting in Engle and Granger(1987). The underlying idea is based on the concept of stochastic equilibrium, that is, while theprocesses may display nonstationarity individually, a linear combination of them may behaveas a system in equilibrium (such as a stationary process). The processes in the system are thensaid to be cointegrated. It is clear that an equilibrium cointegrated system is easier to handle

Intl. Trans. in Op. Res. 7 (2000) 349±363

0969-6016/00/$20.00 7 2000 IFORS. Published by Elsevier Science Ltd. All rights reserved.PII: S0969-6016(00)00005-8

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* Corresponding author. Tel.: +61-7-3864-2111; fax: +61-7-3864-1508.E-mail address: [email protected] (V.V. Anh).

than the nonstationary individuals; hence the concept is useful in the analysis and prediction ofnonstationary processes.In the literature, cointegration theories have been developed for integrated processes in the

Box-Jenkins ARIMA framework (see Engle and Granger, 1987; Johansen, 1988, 1991;Davidson, 1991; Banerjee et al., 1993; Gonzalo and Granger, 1995), and for processes withlong-range dependence, yielding fractional cointegration (see, for example, Cheung and Lai,1993). These processes are typically monoscaling (i.e., their scaling behaviour can becharacterised by a single parameter). An important example is fractional Brownian motion(fBm) with Hurst index H (see Samorodnitsky and Taqqu, 1994). Their scaling behaviour canbe described by

EjX�t� ÿ X�tÿ r�j q0r qH, qr0, �1�as r40: Here, the symbol0means that the ratio of the left-hand side to the right-hand sidetends to a ®nite constant as r40: It is noted that fBm exhibits long-range dependence (LRD)when 1

2 < H < 1 and the scaling exponent of (1), is a linear function of q with a singleparameter, namely, the Hurst index H.It is commonly accepted in the studies on turbulence that the scaling of turbulent processes

is more complex than (1). In fact, turbulent processes may possess a scaling behaviour

EjX�t� ÿ X�tÿ t�j q0rz�q�, qr0, �2�where the exponent z�q� is a nonlinear function of q, in which case, they are known asmultifractals and display distinct intermittency (i.e. spiky appearance in their sample paths andbursty pattern in their increments) (see Mandelbrot, 1974, Schertzer and Lovejoy, 1985;Meneveau and Sreenivasan, 1991; Davis et al., 1994; Frisch, 1995).This paper will concentrate on a class of nonstationary processes which have the spectrum of

Brownian motion (i.e., H � 1=2), but are not assumed to be monoscaling. In fact, we shallassume that they are multifractal with a multiplicative cascade structure generated from abinomial distribution (de®ned in Section 2). These processes are useful to model ®nancial dataand geophysical data, for example. In particular, we shall demonstrate that major exchangerate series such as the Japanese yen, the Deutsche mark and the British pound possess theabove characteristics. Existing theories of cointegration are not suitable for these processes. Weshall suggest a new de®nition of cointegration and develop a method for its testing. Themethod will be applied to the above exchange rate series.There has been strong empirical evidence in support of a unit root in the exchange rates of

currencies that are free to ¯oat (see, for example, Meese and Singleton, 1982; Meese andRogo�, 1983; Baillie and Bollerslev, 1989). In Baillie and Bollerslev (1989), it was furtherargued that the exchange rates of seven major currencies are cointegrated. That is, they are tiedtogether under a long run relationship. This ®nding, however, was later challenged by Sephtonand Larson (1991) and Diebold et al. (1994). These authors argued that the Baillie±Bollerslevresult is ``fragile` (varies with the sample) and cannot be substantiated. In a rejoinder, Baillieand Bollerslev (1994) put forward the possibility that the exchange rates may be fractionallycointegrated. They maintained that ``a form of cointegration does exist between the exchangerates, so that they do not drift apart in the long run''.

V.V. Anh et al. / Intl. Trans. in Op. Res. 7 (2000) 349±363350

In this paper, we introduce the notion of stochastic multifractals to exchange rate data. Thisconcept enriches the time series structure of the ®nancial series. Furthermore, we extend theconcept of fractional cointegration to multifractal series. Our empirical results show that someexchange rate series demonstrate fractional cointegration under the new de®nition.

2. Intermittency models

Let fX�t�, t 2 Rg be a nonstationary process with stationary increments. These processes havebeen found appropriate to represent data in geophysics and turbulence, for example, Davis etal. (1994). De®ne

Y�t� � jX�t� ÿ X�tÿ 1�jEjX�t� ÿ X�tÿ 1�j �3�

and

Y�t; r� � 1r

�t� r2

tÿ r2

Y�s� ds, r > 0: �4�

It is noted that Y�t�r0 8t and EY�t� � 1: Y�t; r� is a smoothing (coarse graining) of Y�t� withsmoothing window of size r. We shall assume that Y�t; r� is an intermittent process (see Frisch,1995, Chapter 8). The scaling behaviour of Y�t; r� can be described byXÿ

Y�t; r�� q0rt�q�, q 2 R, �5�

as r40, where the sum is taken over all disjoint intervals of length r. The generalised fractaldimension, Dq, of Y�t� is then de®ned as

Dq �t�q�qÿ 1

: �6�

Hentschel and Procaccia (1983) showed that D0 is the fractal dimension of the support of Y�t�,D1 and D2 are the information dimension and the correlation dimension of Y�t�, respectively.An intermittency model for Y�t�, and hence X�t�, is essentially a parametrisation of t�q�:

There have been a large number of intermittency models suggested in the literature ofturbulent processes. Of historical interest, the key models include Kolmogorov's lognormalmodel, the random curdling model (Mandelbrot, 1974), the b-model (Frisch et al., 1978), therandom b-model (Benzi et al., 1984), the a-stable model (Schertzer and Lovejoy, 1985) and thebinomial p-model (Meneveau and Sreenivasan, 1991). In Borgas (1992), a comparison ofintermittency models was undertaken and it was concluded that the binomial p-model was themost satisfactory model and best represented the measurements. Along this line we shallassume that Y�t� is generated by a multiplicative cascade with a binomial generatorcharacterised by a probability p, 0 < pR1=2: In other words, we consider an interval E of unitlength and construct a Cantor set F �T1i�0 Ei on this interval, where E0 � E, Ek contains 2k

subintervals of length 2ÿk obtained by dividing each subinterval of Ekÿ1 into two halves.

V.V. Anh et al. / Intl. Trans. in Op. Res. 7 (2000) 349±363 351

We next de®ne a positive measure m on F as follows. Let 0 < pR1=2 be given and consider aunit mass on E0 (i.e. m0 � 1). Split this unit mass between the two intervals of E1 so that onehas mass p and the other has mass 1ÿ p, the allocation being random. This de®nes m1, whichhas a constant value of 2p on one interval and a constant value of 2�1ÿ p� on the otherinterval. Continue in this way, so that the mass on each interval of Ek is divided randomly intothe proportions p and 1ÿ p between its two subintervals in Ek�1: This de®nes a sequence fmkg,which is a positive martingale; hence it converges almost surely to a limiting mass distributionm on F (Kahane, 1991).Our basic assumption is that Y�t� is generated by such an iterative process (resulting in a

multiplicative cascade, as is known in turbulence), and its scaling behaviour/intermittency isdescribed by (5). Let us now determine the function t�q�: Each generation of the cascade isde®ned by Ek and mk: For each 0RjRk, a number �kj � of the 2k intervals of Ek have masspk�1ÿ p�kÿj, where �kj � � k!

j!�kÿj�! : By the binomial theorem,

XÿYÿk; 2ÿk

�� q�Xkj�0

�k

j

�p qj�1ÿ p� q�kÿj�

� ÿp q � �1ÿ p� q�k: �7�Putting 2ÿk � r (i.e., k � ÿ�log r=log 2�), it follows from Eq. (7) that

logX�Y�k; r�� q� k log

ÿp q � �1ÿ p� q�

� ÿ�log r� logÿp q � �1ÿ p� q�

log 2

� logÿrÿlog2�p q��1ÿp� q��: �8�

From (5) and (8) we get

t�q� � limr40

logX�Y�k; r�� q

log r

� ÿlog2ÿp q � �1ÿ p� q�, �9�

and hence

Dq � ÿlog2

ÿp q � �1ÿ p� q�qÿ 1

: �10�

A related exponent can be introduced by de®ning

EÿY�t; r�� q0r1ÿq�t�q�, qr0 �11�


(Monin and Yaglom, 1975, p. 534). De®ne

K�q� � ÿt�q� � qÿ 1: �12�Then, for the binomial cascade described above,

K�q� � log2ÿp q � �1ÿ p� q�� qÿ 1: �13�

Remark 1. It follows directly from (11) that K�0� � 0, and since EY�t; r� � 1 by de®nition, wealso have K�1� � 0: Writing Y for Y�t; r� and considering r su�ciently small, we get from (11)and (12) that

K�q� � log E�Y q�log r

: �14�

Thus

K 0�q�log r � �E�Yq log Y��logY

E�Y q� ,

K 00�q�log r � E�Y q�ÿEÿY q�log Y�2��

log Yÿ �E�Y qlog Y��2 log Y

�E�Y q��2 :

But ÿEÿY q log Y

��2� ÿEÿY q=2Y q=2 log Y��2

RE�Y q�EÿY q�log Y�2

�by Schwarz's inequality. Consequently, K 00�q�r0, so that K�q� is a convex function (see Fleming,1965, p. 24). It also follows from (14) that K�q� < 0 iff E�Y q� < 1, which holds only if 0 < q < 1:These results are useful when the model (13) is ®tted to empirical data.

Let us now get back to the positive measure Y generated by the binomial cascade algorithm asdescribed above. For each t 2 E0, let

a�t� � limr40�

log YÿB�t; r��

log r

be the local dimension of Y at t, where B�t; r� � fs; jsÿ tj < rg: Let G�a� � ft; a�t� � ag and letf�a� be the Hausdor� dimension of G�a� (see Barnsley, 1988, p. 200). We call f�a� thesingularity spectrum of Y, and we say that Y is a multifractal measure if f�a�6�0 for acontinuum of a: In this de®nition, monofractals therefore consist of singularities all of thesame strength. As a result, the class of monofractals display scale invariance of the type


Y�st��d saY�t�, �15�

where a is the scaling exponent of the monofractal, and �d means equality in distribution. Thatis, the probability density functions on both sides of (15) are the same yielding

E�Y�st�� q� s qaE�Y�t�� q �16�and consequently, the monofractal is characterised by a single scaling exponent (i.e. a�q� � a, aconstant, for all q, and the graph of f�a� consists of one point). An important example of amonofractal is the fractional Brownian motion with Hurst index H, in which case a�q� � H 8q,and a � 1=2 for Brownian motion. On the other hand, multifractals will display heterogeneousscaling, which is characterised by a nonlinear scaling function such as t�q� or K�q� as de®nedabove.

Remark 2. In order to examine directly whether Y�t� is monofractal or multifractal, it is moreconvenient to consider another scaling exponent given by

EjY�t� ÿ Y�tÿ r�j q0rz�q�, qr0,

as r40: It is seen that z�0� � 0, and no other exponent is known a priori (in contrast to K�q�,where there are two a priori exponents: K�0� � 0 and K�1� � 0). Following the same argument asfor K�q�, it can be shown that z 00�q�R0; hence z�q� is concave (see Fleming 1965, p. 26).Furthermore, under the condition that Y�t� is bounded, the function z�q� is monotonicallynondecreasing (Marshak et al., 1994). These results imply that, if Y�t� is a monofractal, itsscaling will be simply given by z�q� � qa, where a is a constant. In particular, for fractionalBrownian motion with Hurst index H, we have z�q� � qH: This result is a convenient tool to testif Y�t� is a monofractal.

There is a relationship between f�a� and t�q�: In fact, let t� denote the concave conjugate of t(also known as the Legendre transform of t), i.e.,

t��a� � infq2R�qaÿ t�q�:

Hentschel and Procaccia (1983) and Halsey et al. (1986) showed heuristically that, if Y isconstructed from a cascade algorithm and if t�q� and f�a� are smooth and concave, thent��a� � f�a� and dually f ��q� � t�q�:

This relationship is called the multifractal formalism, which is also known as thethermodynamic formalism because of the analogue of the Gibbs state, pressure and variationalprinciple in thermodynamics (Bohr and Rand, 1987). The multifractal formalism is a usefultool in applications. In fact, it yields that

t�q� � sup0<a<1

�f�a� ÿ qa

:

Suppose that the supremum is attained at a � a�q� > 0: Then


d

da

ÿf�a� ÿ qa

� � 0,

which implies

q � df

da�a�q��, �17�

and

dtdq� df

dadadqÿ a�q� ÿ q

dadq

� ÿa�q�: �18�Given a sample of X�t�, the mass exponent t�q� can be computed from (3)±(5) (via logregression). The function a�q� is then given by (18) and the singularity spectrum f�a� isobtained from

t�q� � f�a�q�� ÿ qa�q�: �19�For the binomial cascade model, direct computations from Eq. (9) yields

a�q� � ÿpq log2p� �1ÿ p� q log2�1ÿ p�

p q � �1ÿ p� q �20�

f�a� � log2ÿp q � �1ÿ p� q�ÿ q

ÿp q log2p� �1ÿ p� q log2�1ÿ p��

p q � �1ÿ p� q : �21�

The expression for a�q� in Eq. (20) can also be put in the following forms:

a�q� � ÿlog2pp q � �1ÿ p� q log2�1ÿp�

log2p

p q � �1ÿ p� q

� ÿlog2�1ÿ p�log2p

log2�1ÿ p�pq � �1ÿ p� q

p q � �1ÿ p� q

For 0 < pR1=2 we get log2�1ÿ p�=log2pR1: Thus,

ÿlog2�1ÿ p�Ra�q�Rÿ log2p: �22�

Remark 3. The strongest singularity of Y�t� corresponds to q41: It is seen from (20) and (22)that, as q41

amin � ÿlog2�1ÿ p�: �23�


Now,

damin

dp� 1

log 2

1

1ÿ p> 0:

Consequently amin is monotonically increasing as p increases from 0 to 12 : In other words, the

strongest singularity, hence the degree of intermittency, of Y�t� decreases as p increases.Thisresult plays a key role in our de®nition of cointegration of multifractals given in the next section.

3. Cointegration of multifractals

The concept of cointegration was introduced in Granger (1981), and organized in a formalsetting in Engle and Granger (1987), to study the long-run equilibrium relationship between anumber of time series. A time series Xt is said to be integrated of order d, denoted by Xt 2I�d�, if it has an ARIMA �m, d, q� representation given by

�1ÿ y1Bÿ . . .ÿ ymBm��1ÿ B�dXt �ÿ1ÿ c1Bÿ . . .ÿ cqB

q�et, �24�where B is the lag operator BXt � Xtÿ1, et is white noise, and the roots of the AR and MApolynomials are assumed to lie outside the unit circle. Now, consider a vector �X1t, . . . ,Xkt� ofI�d� time series. If a vector a � �a1, . . . ,ak� exists such that Yt � a1X1t � . . .� akXkt 2 I�dÿ b�,where b > 0, the time series are said to be cointegrated, and the resulting cointegrated systemhas an error correction representation. Existing empirical works typically consider d � b � 1;that is, X1, . . . ,Xk 2 I�1� and Y 2 I�0�, which is the class of stationary time series. Thus, thistheory requires that the equilibrium error Yt to be mean reverting, even though X1, . . . ,Xk arenonstationary (e.g. displaying stochastic trends).As noted in Granger and Weiss (1983) and formalised in Cheung and Lai (1993), the above

concept of cointegration can be extended to the case of d and b being fractional. Asinvestigated in Cheung (1993) and the references therein, for example, a number of exchangerate series were found to display LRD, in which case the value of d in (24) varies in the range�1, 3=2� (or d 2 �0, 1=2� if the di�erenced series are considered). For d fractional, the fractionaldi�erencing operator is de®ned as

�1ÿ B�d�X1k�0� ÿ 1�k

�d

k

�Bk �

X1k�0

G�kÿ d�G�k� 1�G� ÿ d�B

k,

where G�� is the Gamma function. A representation of the form (24) (with d fractional) is thencalled an autoregressive fractionally integrated moving average (ARFIMA) process. LetX1, . . . ,Xk be ARFIMA processes with the same LRD exponent d 2 �1, 3=2�: If a vector a ��a1, . . . ,ak� exists for which the resulting linear combination Y � a1X1 � . . .� akXk is anARFIMA process with an LRD exponent d1 strictly smaller than d, we say that X1, . . . ,Xk arefractionally cointegrated.This paper will consider the case in which the processes X1�t�, . . . ,Xk�t� are nonstationary


with uncorrelated increments X�t� ÿ X�tÿ r�, r > 0; but the processes are not assumed to bemonoscaling. In other words, these processes will have a spectral density of the form

f�l� � c

l2, c > 0, l 2 R �25�

(interpreted in a limiting sense (Solo, 1992) or in the sense of time-scale analysis (Flandrin,1989), but are not Brownian motions. In fact we shall allow these processes to be multifractal,and their multifractality/intermittency is characterised by a multiplicative cascade generatedfrom a binomial distribution with probability p (see Eqs. (3±5) and (9)). We shall demonstratein the next section that some major exchange rate series (such as the Japanese yen, theDeutsche mark and the British pound) display such characteristics. For simplicity, we shall saythese processes have generator p. It seems very unlikely that any linear combination of themwould exist as a stationary process. Consequently, both de®nitions of cointegration mentionedabove do not apply to this situation. On the other hand, the development of Section 2 andparticularly Remark 3 suggests the following de®nition:

De®nition 1. Let X1, . . . ,Xk be nonstationary processes with generators 0 < p1, . . . ,pkR1=2,respectively, as described above. If a vector a � �a1, . . . ,ak� exists such that the linear combinationY � a1X1 � . . .� akXk has a generator pY with

pY > maxfp1, . . . ,pkg, �26�we say that the processes X1, . . . ,Xk are cointegrated.

Remark 4. Similar to Brownian motions, the process Y is mean reverting. But this mean reversiondepends on the value of pY: When pY41=2, the process Y becomes monoscaling; hence, a linearcombination of X1, . . . ,Xk (themselves nonstationary and multifractal) may exist as a Brownianmotion. Hence, in a sense, our de®nition of cointegration means convergence to Brownian motion.

Remark 5. Given X1, . . . ,Xk, we may consider the regression

X1 � a2X2 � . . .� akXk � u, �27�where u is a white noise, and the coe�cients a2, . . . ,ak are obtained using least squares. Then, alinear combination for cointegration analysis can be obtained from

Y � X1 ÿ a2X2 ÿ . . .ÿ akXk: �28�

4. Empirical results

We apply the above theory to analyse the exchange rates of three major currencies. Aspointed out by Baillie and Bollerslev (1994), exchange rates may be tied together through along-memory fractionally integrated type of process. The usual analysis of fractionally


integrated time series, however, is restricted to monofractal processes. We shall show in thissection that exchange rate data exhibit intermittency similar to turbulence data. A generalizedconcept of fractional cointegration is then examined.We use the nominal spot exchange rates of the British pound (BP), Japanese yen (JY) and

Deutsche mark (DM). These series are provided by the Commodity Systems Inc. Theyrepresent daily observations from January 1985 through July 1998, totalling 3421 observations.From the plot of the normalised (by dividing by their means) series in Fig. 1, we observe a

fractional Brownian motion appearance in each series. Hence we assume that their spectraldensity takes the form

f�l� � c

jlj2H�1 , c > 0, 0 < H < 1, l 2 R, �29�

near frequency 0. The log periodograms f�l� of BP, DM and JY against log l are computed.The result is shown for DM in Fig. 2 as an example. Their Hurst exponents H, estimated fromthe regression

log f�l� � cÿ �2H� 1�logjlj � u, u0WN, �30�are 0.5241, 0.5190 and 0.5148 for BP, DM and JY, respectively.The ®tted model (the solid line) for DM is also given in Fig. 2. These estimates indicate that

all three time series appear to be Brownian motion �H � 1=2). However, a closer look at theirdi�erenced series, such as that of JY shown in Fig. 3, implies that a scaling structure morecomplex than that of Brownian motion exists; in fact the presence of intermittency is quiteapparent in these di�erenced series. We then compute the exponents z�q� as a function of q 2�0, 10� with steplength 0.1 as de®ned in Remark 2 using log regression. For comparison we also

Fig. 1. The time series JY, DM and BP normalised by dividing by their means, respectively. At the bottom is theresidual series BP� aDM� bJY, which has a Brownian motion appearance.


compute z�q� for Brownian motion (i.e., fBm with H � 0:5). The z�q� functions are plotted inFig. 4. It can be seen that z�q� is a linearfunction of q for fBm as expected (of a monoscalingprocess), but is clearly nonlinear for DM and JY, indicating that these latter series aremultifractal. On the other hand, the z�q� curve for BP appears to be linear. The exponents t�q�are next computed based on (5) with q 2 �0, 10� with steplength 0.1 using log regression. Thecurves K�q� are then obtained as in (13); these are shown in Fig. 5. It is seen that they displaythe theoretical convex shape as demonstrated in Remark 1, with zeroes at q � 0 and q � 1; theK�q� curve for JY is furthest from that of fBm, indicating that JY is more intermittent than

Fig. 2. The log periodogram and ®tted model (continuous line) of the DM series.

Fig. 3. The di�erenced series JY�t� ÿ JY�tÿ 1�:


DM and BP. The model (13) is then ®tted using nonlinear least squares to these K�q� curvesand also to those obtained from the residuals c1 � BPÿ a1DM, c2 � BPÿ a2DMÿ a3JY,c3 � DMÿ a4JY, where the coe�cients a1, . . . ,a4 are least squares estimates. The estimates forthe corresponding p are given in Table 1.Fig. 6 demonstrates for DM that the ®tted model describes the data very well; excellent ®t is

similarly obtained for JY and BP. A low value of p for JY con®rms that this series is moreintermittent than BP and DM. According to the criterion (26), the results of Table 1 indicatethat DM and JY are cointegrated, while BP does not seem to be part of this cointegration. The

Fig. 4. The exponents z�q� of BP, DM, JY, DMÿ aJY (i.e., c3 ) and fBm with H � 0:5:

Fig. 5. The exponents K�q� of BP, DM, JY, DMÿ aJY and fBm.


computed functions z�q� and K�q� in Figs. 4 and 5 support very well this cointegration. In fact,the curves z�q� and K�q� for c3 are closer to that of fBm than the corresponding curves for DMand JY (see Figs. 4 and 5).

5. Conclusions

By assuming that the stochastic processes under study are nonstationary but have aBrownian motion spectrum, the concept of cointegration based on a reduction of long-rangedependence in a linear combination of the processes is not suitable. If any linear combinationstill displays a Brownian motion spectrum, the usual concept of cointegration for integratedprocesses is not applicable either. In this paper, we assume that the processes possess amultiple scaling, and de®ne a concept of cointegration based on a reduction of this multiplescaling towards a monoscaling behaviour. The above empirical results illustrate the practicalrelevance of the concept.The theory of this paper is based on the assumption that the normalised increments of the

stochastic processes behave like a multiplicative cascade generated from a binomialdistribution. Although this model seems to work well in many applications, it would be ofinterest to develop a cointegration theory for a more general class of multiplicative cascades.

Table 1Estimates of the generator p from model (13)

K�q� BP DM JY c1 c2 c3

p 0.325 0.320 0.275 0.320 0.300 0.340

Fig. 6. The exponents K�q� and its ®tted model for DM.


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cointegration of stochastic multifractals with application to foreign exchange rates

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