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4 Case (B): Time-Varying Scale Process

4 Case (B): Time-Varying Scale Process

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replaced by αˆ p,n since we already know that

:=

σˆ 2q,n

n(αˆ p,n − α ) = O p (1). Therefore,

n2q/αˆ p,n−1 n

∑ |∆i X|2q → p σ2q∗ .

µˆ (2q, 0) i=1

(18)

Once again, let us remind that µˆ (2q, 0) can be easily computed in view of (2) with

σ = 1. By the same token, we could deduce that (still under 4q < α , of course)

σˆ 4q,n

:=

n4q/αˆ p,n−1 n−1

∑ |∆i X|2q|∆i+1X|2q → p σ4q∗ .

µˆ (2q, 2q) i=1

, σˆ 4q,n

) can serve as a desired consistent estimator.

After all, V (ρˆ n , αˆ p,n , σˆ 2q,n

Now we are in a position to complete our main objectives (A) and (B).

3.3 Case (A): Geometric Skewed Stable L´evy Process

When σt ≡ σ > 0, our model reduces to the geometric skewed stable L´evy

process. In this case we can perform a full-joint interval estimation

concerning

the dominating (three-dimensional) parameter (ρ , α , σ ) at rate n.

We keep using the framework of the last subsection. It directly follows from

(15) that

ˆ

ρ

ρ

n

n  αˆ p,n − α  →d N3 (0,V (ρ , α , σ )),

(19)

(σˆ p,n ) p − σ p

where V (ρ , α , σ ) explicitly depends on the three-dimensional parameter

(ρ , α , σ ); recall that p = 2q < α /2. Applying the delta method to (19) in orp

ˆ

ˆ

der

√ to convert (σ p,n ) to σ p,n in (19), we readily get the asymptotic normality of

n(ρˆ n − ρ , αˆ p,n − α , σˆ p,n − σ ); we omit the details. Our first objective (A) is thus

achieved.

In summary, we may proceed with the choice q = 1/4 (so p = 1/2) as follows.

1. Compute the estimate ρˆ n of ρ by (7).

2. Using the ρˆ n , find the root αˆ 1/2,n of (16).

3. Using (ρˆ n , αˆ 1/2,n ) thus obtained, an estimate of σ is provided by, e.g. (recall

(18)),

2

n1/(2αˆ p,n)−1 n

σˆ 1/2,n :=

|∆i X| .

µˆ (1/2, 0) i=1

3.4 Case (B): Time-Varying Scale Process

Now we turn to the case (B). Again by means of the argument give in Section

3.2, it remains to construct an estimator of σα∗ = 01 σsα ds. The point here is that,

different from the case (A), a direct use of (15) is not sufficient to deduce the

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distributional result concerning estimating σα∗ because the dependence of (r, r )

on α is not allowed there. In order to utilize Mn (r) with r depending on α , we

Extracting the second row of (12), we have

n{Mn (r) − µ (r)σr∗+ } →d N1 0, B(r, r)σ2r

.

(20)

+

In view of the condition maxl≤m rl < α /2, we need (at least) a tripower variation

for setting r+ = α . For simplicity, we set m = 3 and

r = r(α ) =

α α α

, ,

.

3 3 3

With this choice, we are going to provide an estimator of σα∗ with specifying its

rate of convergence and limiting distribution.

Let Mn∗ (α ) := Mn (α /3, α /3, α /3). In this case the normalizing factor is

r

/

+

n α −1 ≡ 1, so that

Mn∗ (α ) =

n−2 3

∑ ∏ |∆i+l−1X|α /3,

i=1 l=1

which is computable as soon as we have an estimate of α . We have already obtained the estimator αˆ p,n , hence want to use Mn∗ (αˆ p,n ). For this, we have to look

at the asymptotic behavior of the gap

n{Mn∗ (r(α )) − µ (r(α ))σα∗ } − n{Mn∗ (αˆ p,n ) − µ (r(αˆ p,n))σα∗ },

namely, the effect of “plugging in αˆ p,n ”.

By means of Taylor’s formula

ax = ay + (loga)y (x − y) + (loga)2

1

0

(1 − u)ay+u(x−y)du(x − y)2

applied to the function x → ax (x, y, a > 0), we get

α α α

n Mn∗ (αˆ p,n ) − µ

σα∗

, ,

3 3 3

=

+

n Mn∗ (α ) − µ

n−2

α α α

1√

α /3

σα∗ +

n(αˆ p,n − α ) ∑ xi log xi

, ,

3 3 3

3

i=1

1√

n(αˆ p,n − α )

3

2

1 n−2

√ ∑ (log xi )2

n i=1

1

0

{α +u(αˆ p,n−α )}/3

(1 − u)xi

du,

(21)

where we wrote xi = ∏3l=1 |∆i+l−1 X|. We look at the right-hand side of (21)

termwise. Let yi := ∏3l=1 |n1/α ∆i+l−1 X|.

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• The first term is O p (1), as is evident from (20).

• Concerning the second term, we have

n−2

α /3

∑ xi

log xi =

i=1

1 n−2 α /3

3

1 n−2 α /3

yi log y j − (log n) ∑ yi

n i=1

α

n i=1

1

α α α

3

µ

σα∗ + O p √

, ,

α

3 3 3

n

3

α α α

, ,

.

= O p (1) − (logn) µ

α

3 3 3

= O p (1) − (logn)

• Write the third term as { n(αˆ p,n − α )/3}2 Tn , and let us show that Tn =

o p (1). Fix any ε > 0 and ε0 ∈ (0, α /2) in the sequel. Then

P[|Tn | > ε ] ≤ P[|αˆ p,n − α | > ε0 ] + P |Tn | > ε , |αˆ p,n − α | ≤ ε0 =: pn + pn .

Clearly pn → 0 by the

inf

u∈[0,1]

n-consistency of αˆ p,n . As for pn , we first note that

1

ε0

{α + u(αˆ p,n − α )} ≥ 1 − > 0

α

α

on the event {|αˆ p,n − α | ≤ ε0 }. We estimate pn as follows:

pn = P |αˆ p,n − α | ≤ ε0 ,

1 n−2

√ ∑ (log xi )2

n i=1

1

0

≤ P |αˆ p,n − α | ≤ ε0 , nε0 /α −1/2

≤ P nε0 /α −1/2

{α +u(αˆ p,n −α )}/3 −{α +u(αˆ p,n −α )}/α

(1 − u)yi

1 n−2

∑ (log xi )2

n i=1

n

1

0

du > ε

{α +u(αˆ p,n −α )}/3

(1 − u)yi

du > ε

1 n−2

∑ {(log n)2 + (log yi )2 }(1 + yi )(α +ε0)/3 > Cε

n i=1

1 ε0 /α −1/2

n

(log n)2 → 0

for some constant C > 0. Here we used Markov’s inequality in the last

inequality; note that (α + ε0 )/3 < α /2, hence the moment does exist.

Piecing together these three items and (21), we arrive at the asymptotic relation:

1

n

α α α

1

α α α

Mn∗ (αˆ p,n ) − µ

, ,

, ,

.

σα∗ = − µ

σα∗ n(αˆ p,n − α ) + O p

log n

3 3 3

α

3 3 3

log n

(22)

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Now, recalling (2) we note that the quantity µ (α /3, α /3, α /3) is a continuously

differentiable

function of (ρ , α ). Write µ¯ (ρ , α ) = µ (α /3, α /3, α /3). In view of

the n-consistency of (ρˆ n , αˆ p,n ) and the delta method, we obtain

1

µ¯ (ρ , α ) = µ¯ (ρˆ n , αˆ p,n ) + O p √ .

n

Substituting (23) in (22) we end up with

Mn∗ (αˆ p,n )

1 √

1

n

,

− σα∗ = − σα∗ n(αˆ p,n − α ) + O p

log n µ¯ (ρˆ n , αˆ p,n )

α

log n

(23)

(24)

which implies that

σˆ α∗ ,n :=

Mn∗ (αˆ p,n )

µ¯ (ρˆ n , αˆ p,n )

(25)

serves as ( n/ log n)-consistent estimator of σα∗ . Its asymptotic distribution is the

centered normal scale mixture with limiting variance being

v(ρ , α , σα∗ , σ p∗ , σ2p

) :=

σα∗

α

2

V22 (ρ , α , σ p∗ , σ2p

),

where V22 denotes the (2, 2)th entry of V ; recall that p is a parameter-free con∗ ) can be constant (see Section 3.2). A consistent estimator of v(ρ , α , σα∗ , σ p∗ , σ2p

structed by plugging in the estimators of its arguments.

(24) indicates an asymptotic linear dependence of

√ The stochastic expansion

n(αˆ p,n − α ) and ( n/ log n)(σˆ α∗ ,n − σα∗ ). Of course, this occurs even for constant√σ , if we try to estimate (α , σ α ) instead of (α , σ ). The point is that, plugging

in a n-consistent estimator

of

√α into the index r of the MPV Mn (r) slows down

estimation of σα∗ from n to n/(log n). It is beyond the scope of this article to

explore a better alternative estimator of σα∗ .

4. Simulation Experiments

Based on the discussion above, let us briefly observe finite-sample performance of our estimators. For simplicity, we here focus on nonrandom σ .

4.1 Case (A)

First, let σ is a positive constant, so that X is the geometric skewed stable

L´evy process and the parameter to be estimated is (ρ , α , σ ).

As a simulation design, we set α = 1.3, 1.5, 1.7, and 1.9 with common β =

−0.5 and σ = 1; hence (α , ρ ) = (1.2, 0.7638), (1.5, 0.5984), (1.7, 0.5467), and

(1.9, 0.5132). The sample size are taken as n = 500, 1000, 2000, and 5000. In

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all cases, the tuning parameter q is set to be 1/4, and 1000 independent sample

paths of X are generated. Empirical means and empirical s.d.’s are given with the

1000 independent estimates obtained. The results are reported in Table 4.1. We

see that estimation of (ρ , α ) is, despite of its simplicity, quite reliable. On the

other hand, estimation variance of σ is relatively large compared with those of ρ

and α . Nevertheless, it is clear that the bias is small. Moreover, as α gets close

to 2, the performance of σˆ n becomes better, while that of (ρˆ n , αˆ p,n ) is seemingly

unchanged.

In the unreported simulation results, we have observed that a change of q

within its admissible region does not lead to a drastic change unless it is too small

(see Remark 3.3).

Table 1. Estimation results for the true parameters (ρ , α , σ ) = (0.7638,1.2,1), (0.5984,1.5,1),

(0.5467,1.7,1), and (0.5132,1.9,1) with the geometric stable L´evy processes. In each case, the

empirical mean and standard deviation (in parenthesis) are given.

α = 1.2

n

500

1000

2000

5000

α = 1.5

n

500

1000

2000

5000

α = 1.7

n

500

1000

2000

5000

α = 1.9

n

500

1000

2000

5000

ρ

0.7627

0.7634

0.7645

0.7636

(0.0186)

(0.0137)

(0.0096)

(0.0061)

1.2026

1.2031

1.2031

1.2023

α

(0.0790)

(0.0575)

(0.0437)

(0.0313)

(0.0222)

(0.0162)

(0.0106)

(0.0073)

1.4929

1.5010

1.4986

1.4983

α

(0.1030)

(0.0757)

(0.0564)

(0.0364)

1.6810

1.6830

1.6930

1.6977

α

(0.1103)

(0.0823)

(0.0625)

(0.0375)

1.8553

1.8767

1.8870

1.8971

α

(0.1026)

(0.0808)

(0.0579)

(0.0401)

ρ

0.5988

0.5981

0.5986

0.5984

ρ

0.5476

0.5474

0.5472

0.5466

(0.0219)

(0.0158)

(0.0113)

(0.0070)

ρ

0.5129

0.5133

0.5131

0.5128

(0.0224)

(0.0164)

(0.0109)

(0.0073)

σ

1.1021

1.0450

1.0253

1.0123

1.0751

1.0289

1.0284

1.0169

(0.8717)

(0.4643)

(0.5102)

(0.2854)

σ

(0.4066)

(0.2549)

(0.2355)

(0.1516)

σ

1.0633

1.0567

1.0308

1.0126

(0.2359)

(0.1948)

(0.1611)

(0.1022)

σ

1.0821

1.0535

1.0330

1.0097

(0.1767)

(0.1568)

(0.1111)

(0.0809)

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4.2 Case (B)

Next we observe a case of time-varying but nonrandom scale. We set

σtα =

3

2

cos(2π t) +

,

5

2

(26)

0.6

0.4

0.2

Varying scale

0.8

1.0

so that σα∗ = 0.6.

0.0

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0.0

0.2

0.4

0.6

0.8

1.0

Time

Figure 3. The plot of the function t → σtα given by (26).

With the same choices of (ρ , α ), q, and n as in the previous case, we obtain

the result in Tables 4.2; the estimator of σα∗ here is based on (25). There we can

observe a quite similar tendency as in the previous case.

5. Concluding Remarks

We have studied some statistical aspects in the calibration problem of a geometric skewed stable asset price models. Estimation of stable asset price models with possibly time-varying scale can be done easily by means of the simple empirical-sign statistics and MPVs. Especially, we could estimate integrated

scale, which is a natural quantity as in the integrate variance in the framework of

Brownian semimartingales, with multistep estimating procedure: we estimate ρ ,

α , and σ (or σα∗ ) one by one in this order. Our simulation results say that finitesample performance of our estimators are unexpectedly good despite of their simplicity, except for a relatively bigger variance in estimating σ (or σα∗ ).

We close with mentioning some possible future issues.

• Throughout we supposed the independence between the scale process σ

and the driving skewed stable L´evy process Z. This may be disappointing

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Table 2.

Estimation results for the true parameters (ρ , α ) = (0.7638,1.2), (0.5984,1.5),

(0.5467,1.7), and (0.5132,1.9) with σα∗ = 0.6 in common under (26). In each case, the empirical mean and standard deviation (in parenthesis) are given.

α = 1.2

n

500

1000

2000

5000

α = 1.5

n

500

1000

2000

5000

α = 1.7

n

500

1000

2000

5000

α = 1.9

n

500

1000

2000

5000

ρ

0.7632

0.7636

0.7638

0.7641

1.1951

1.2042

1.2044

1.2025

α

(0.0794)

(0.0619)

(0.0472)

(0.0305)

σα∗

0.6730 (0.3857)

0.6274 (0.3094)

0.6105 (0.2323)

0.6029 (0.1521)

(0.0220)

(0.0159)

(0.0111)

(0.0069)

1.4877

1.4908

1.4960

1.4990

α

(0.1023)

(0.0733)

(0.0573)

(0.0376)

σα∗

0.6697 (0.3031)

0.6551 (0.2488)

0.6349 (0.2033)

0.6151 (0.1414)

(0.0216)

(0.0160)

(0.0113)

(0.0071)

1.6727

1.6801

1.6931

1.6988

α

(0.1038)

(0.0820)

(0.0600)

(0.0393)

0.6832

0.6714

0.6318

0.6116

1.8440

1.8703

1.8851

1.8956

α

(0.1039)

(0.0823)

(0.0588)

(0.0411)

σα∗

0.7196 (0.2233)

0.6762 (0.1897)

0.6412 (0.1349)

0.6168 (0.0998)

(0.0179)

(0.0139)

(0.0098)

(0.0059)

ρ

0.5978

0.5981

0.5985

0.5987

ρ

0.5460

0.5465

0.5468

0.5465

ρ

0.5130

0.5131

0.5138

0.5135

(0.0229)

(0.0159)

(0.0114)

(0.0068)

σα∗

(0.2465)

(0.2280)

(0.1607)

(0.1135)

as it excludes accommodating the leverage effect, however, the simple constructions of our estimators (especially, ρˆ n ) break down if they are allowed

to be dependent. We may be able to deal with correlated σ and Z if we have

an extension of the power-variation results obtained in Corcuera et al. [7]

to the MPV version. To the best of author’s knowledge, such an extension

does not seem to have been explicitly mentioned as yet.

• Assuming that σ is indeed time-varying and possibly random, estimation

of “spot” scales σt is an open problem. Needless to say, this is much more

difficult and delicate to deal with than the integrated scale. We know several results for Brownian-semimartingale cases (see, among others, Fan and

Wang [8] and Malliavin and Mancino [12]), however, yet no general result

for the case of pure-jump Z.

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• Finally, it might be interesting to derive an option-pricing formula for the

case of time-varying scale, which seems more realistic than the mere geometric skewed stable L´evy processes.

References

1. Applebaum, D. (2004), L´evy Processes and Stochastic Calculus. Cambridge University Press, Cambridge.

2. Barndorff-Nielsen, O. E., Graversen, S. E., Jacod, J. and Shephard, N. (2006), Limit

theorems for bipower variation in financial econometrics. Econometric Theory 22,

677–719.

3. Barndorff-Nielsen, O. E., Graversen, S. E., Jacod, J., Podolskij, M. and Shephard,

N. (2006), A central limit theorem for realised power and bipower variations of continuous semimartingales. From Stochastic Calculus to Mathematical Finance, 33–68,

Springer, Berlin.

4. Barndorff-Nielsen, O. E. and Shephard, N. (2005), Power variation and time

change.Teor. Veroyatn. Primen. 50, 115–130; translation in Theory Probab. Appl. 50

(2006), 1–15.

5. Bertoin, J. (1996), L´evy Processes. Cambridge University Press.

6. Borak, S., Hăardle, W. and Weron, R. (2005), Stable distributions. Statistical tools for

finance and insurance, 21–44, Springer.

7. Corcuera, J. M., Nualart, D. and Woerner, J. H. C. (2007), A functional central limit

theorem for the realized power variation of integrated stable processes. Stoch. Anal.

Appl. 25, 169–186.

8. Fan, J. and Wang, Y. (2008), Spot volatility estimation for high-frequency data. Stat.

Interface 1, 279–288.

9. Fujiwara, T. and Miyahara, Y. (2003), The minimal entropy martingale measures for

geometric L´evy processes. Finance Stoch. 7, 509–531.

10. Kallsen, J. and Shiryaev, A. N. (2001), Time change representation of stochastic integrals. Teor. Veroyatnost. i Primenen. 46, 579–585; translation in Theory Probab. Appl.

46 (2003), 522–528.

11. Kuruo˘glu, E. E. (2001), Density parameter estimation of skewed α -stable distributions. IEEE Trans. Signal Process. 49, no. 10, 2192–2201.

12. Malliavin, P. and Mancino, M. E. (2009), A Fourier transform method for nonparametric estimation of multivariate volatility. Ann. Statist. 37, 1983–2010.

13. Masuda, H. (2009), Joint estimation of discretely observed stable L´evy processes with

symmetric L´evy density. J. Japan Statist. Soc. 39, 1–27.

14. Masuda, H. (2009), Estimation of second-characteristic matrix based on realized multipower variations. (Japanese) Proc. Inst. Statist. Math. 57, 17–38.

15. Miyahara, Y. and Moriwaki, N. (2009), Option pricing based on geometric stable processes and minimal entropy martingale measures. In “Recent Advances in Financial

Engineering”, World Sci. Publ., 119–133.

16. Sato, K. (1999), L´evy Processes and Infinitely Divisible Distributions. Cambridge University Press.

17. van der Vaart, A. W. (1998), Asymptotic Statistics. Cambridge University Press, Cambridge.

May 3, 2010

15:41

Proceedings Trim Size: 9in x 6in

007

202

18. Woerner, J. H. C. (2003), Purely discontinuous L´evy processes and power variation:

inference for integrated volatility and the scale parameter. 2003-MF-08 Working Paper

Series in Mathematical Finance, University of Oxford.

19. Woerner, J. H. C. (2004), Estimating the skewness in discretely observed L´evy processes. Econometric Theory 20, 927–942.

20. Woerner, J. H. C. (2007), Inference in L´evy-type stochastic volatility models. Adv. in

Appl. Probab. 39, 531–549.

21. Zolotarev, V. M. (1986), One-Dimensional Stable Distributions. American Mathematical Society, Providence, RI. [Russian original 1983]

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A Note on a Statistical Hypothesis Testing for

Removing Noise by the Random Matrix Theory

and Its Application to Co-Volatility Matrices

Takayuki Morimoto1,∗ and Kanta Tachibana2

1

School of Science and Technology, Kwansei Gakuin University,

2-1 Gakuen, Sanda-shi, Hyogo 669-1337, Japan.

2

Faculty of Informatics, Kogakuin University, 1-24-2 Nishi-shinjuku,

Shinjuku-ku, Tokyo 163-8677, Japan.

Email: morimot@kwansei.ac.jp and kanta@cc.kogakuin.ac.jp

It is well known that the bias called market microstructure noise will arise,

when estimating realized co-volatility matrix which is calculated as a sum

of cross products of intraday high-frequency returns. An existing conventional technique for removing such a market microstructure noise is

to perform eigenvalue decomposition of the sum of cross products matrix and to identify the elements corresponding to the decomposed values

which are smaller than the maximum eigenvalue of the random matrix as

noises. Although the maximum eigenvalue of a random matrix follows

asymptotically Tracy-Widom distribution, the existing technique does not

take this asymptotic nature into consideration, but only the convergence

value is used for it. Therefore, it cannot evaluate quantitatively such a

risk that regards accidentally essential volatility as a noise. In this paper,

we propose a statistical hypothesis test for removing noise in co-volatility

matrix based on the nature in which the maximum eigenvalue of a random

matrix follows Tracy-Widom distribution asymptotically.

Keywords: Realized volatility, market microstructure noise, random matrix theory.

Corresponding author.

203

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1. Introduction

In recent years, we can easily obtain “high frequency data in finance”, so we

may estimate and forecast (co)-volatility more correctly than before using by Realized Volatility (RV) which is a series of the sum of intraday squared log return

and Realized Co-volatility (RC) which is a series of the sum of cross-product of

two log returns, see [2] or [1]. However, it is well known that when forecasting

volatility, RV and RC are contaminated by large biases, so called micro structural

noise which is progressively increased as sampling frequency becomes higher,

see [7]. Thus, the research considers a statistical method of removing such a noise

in RV and RC by using random matrix theory. Doing eigenvalue decomposition

of cross product matrix, we consider that noises in a covolatility matrix are elements corresponding to eigenvalues smaller than the maximum eigenvalue of the

random matrix. It is known that the maximum eigenvalue of a random matrix

will follow Tracy-Widom distribution asymptotically. However, existing methods haven’t taken into consideration a distribution of the maximum eigenvalue of

a random matrix, but have used only the maximum eigenvalue itself, for example, see [9]. Therefore, they cannot evaluate quantitatively a risk of considering

accidentally that essential volatility is a noise.

Therefore, we propose a statistical hypothesis test for removing noise in covolatility matrix based on the nature in which the maximum eigenvalue of a random matrix follows Tracy-Widom distribution asymptotically.

This paper is organized as follows. Section 2 describes theoretical background

of this study and gives brief explanation of random matrix theory and our proposal.

Section 3 investigates empirical analysis. Section 4 concludes.

2. Theoretical Background

In this section, we will introduce theoretical properties of random matrix with

some simulation results.

2.1 Random matrix

Random matrix is a matrix which has random variables as its elements. First,

[16] and [17] developped a eigenvalue distribution of N × N real symmetric matrix A = (ai j ) with elements {ai j } ∼ i.i.d.(0, 1/N). Following [16] and [17], we

introduce N × N real symmetric random matrix A = (ai j ) with elements {ai j |i ≤ j}

which independently follows a distribution with a mean 0 and a variance 1/N. If

eigenvalues of A are λ1 , . . . , λN and an empirical eigenvalue distribution of A is

defined by

N

1

ρA (λ) =

δ(λ − λi ),

N i=1

then

√ 2

 4−λ

lim ρA (λ) = 

0

N→∞

(|λ| ≤ 2)

,

(otherwise)

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4 Case (B): Time-Varying Scale Process

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