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Time varying long memory parameter.

Self-similarity and related fields, Le Touquet, 2010.

June 7, 2011

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Outline

Long memory processes

Increment stationary processes Examples

Semi-parametric estimation Fourier and wavelet analysis

Fourier and wavelet semiparametric estimation Time varying long memory

The model

Local long memory estimation Bibliography

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Increment stationary processes

Short memory processes

Short memory

A weakly stationary processX ={Xk}k∈Z with spectral densityf hasshort memoryiff(λ) is bounded away from 0 and ∞asλ→0.

A standard stronger condition is the so calledshort range dependencecondition :

X

k∈Z

γ(k)∈R\ {0},

whereγ is the autocovariance function ofX.

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Increment stationary processes

Generalized processes indexed by sequences

Given a processX ={Xk}kZ, define a generalized process indexed by sequencesh= (hk)∈RZ as follows:

X(h) =X

k

hkXk .

LetVd denote the set of real valued sequences (hk)∈RZ with finite support, such that

Z π

−π

|h(λ)|2 |λ|−2ddλ <∞, whereh(λ) =X

k

hke−ikλ.

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Increment stationary processes

Generalized long memory processes

Definition ford ∈R

The processX has memory parameter d and short-range spectral densityf if f is a short memory spectral density and, for any h∈Vd, the r.v. X(h) has finite variance given by

var(X(h)) = Z π

−π

|h(λ)|2f(λ)dλ

withf satisfying

f(λ) =|1−e−iλ|−2d f(λ). (1)

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Increment stationary processes

Equivalent definition

Ifd <1/2 thenVd contains all finitely supported sequences and (Xt) is a weakly stationary process andf is usual spectral density.

For alld ∈Ran equivalent definition is the following.

Definition

The processX is said to have memory parameterd ∈Rand short-range spectral densityf if for any non-negative integer k>d−1/2, its k-order difference∆kX is covariance stationary with spectral density satisfying

f(λ) =|1−e−iλ|−2(d−k)f(λ).

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Examples

Examples

1. theFractional Gaussian noise (fGn) with Hurst parameterH has long memory parameter d =H−1/2∈(−1/2,1/2).

2. thegeneralized fractional Brownian motion (fBm) with Hurst parameter H sampled in discrete time has long memory parameter d =H+ 1/2∈R.

3. theFARIMA(p,d,q) has long memory parameterd (provided the AR and MA operators have no zeros on the unit circle).

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Examples

Fractional Gaussian noise

Definition

Given aHurst indexH∈(0,1), the fGn is defined as the discrete time process made of fixed lag increments of thefBm{B(H)t }t∈R,

Y(H)t =B(H)t −B(H)t−1, t∈Z

The covariance function reads for alls,t∈Z,

cov

Y(H)s ,Y(H)t

= σ2 2

n||t−s| −1|2H+||t−s|+ 1|2H−2|t−s|2Ho ,

∼σ2H(H−1/2)|t−s|2(H−1) as |t−s| → ∞.

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Examples

Fractional Gaussian noise

HenceY(H) is stationary time series. Its spectral density is given by

fH(λ)∝ |1−e−iλ|2

X

k=−∞

|λ+ 2kπ|−2H−1, λ∈(−π, π). (2)

(We will come back later to this Formula).

Observe that, asλ→0,

fH(λ)∼C|λ|−2H+1

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Examples

Generalized fBm

Thegeneralized fractional Brownian motion B(H), where H∈Ris parameterized by a family Θ(d) of “test” functionsθdefined on R and is defined as follows: {B(H)(θ), θ∈Θ(d)} is a mean zero Gaussian process with covariance

cov

B(H)1),B(H)2)

= Z

R

|ξ|−2dθ1(ξ)θ2(ξ)dξ , (3) whered =H+ 1/2,θ is the Fourier transform ofθ,

θ(ξ) = Z

θ(t)e−iξtdt, and Θ(d) is a set of test functions θ satisfying

Z

R

|ξ|−2d(ξ)|2dξ <∞.

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Examples

Generalized fBm: the case 0 < H < 1

Clearly, Θ(d) can be taken exactly as the class of tempered distributionsθsuch that

Z

R

|ξ|−2d(ξ)|2dξ <∞.

Let 0<H<1 i.e. 1/2<d <3/2. Then δt−δ0 ∈Θ(d) for all t∈R. Moreover the process B(H)t−δ0) is a H-self-similar Gaussian process, with stationary increments; thus

n

B(H)t ,t ∈Ro d

∝n

B(H)t−δ0),t ∈Ro .

It follows that cov

B(H)t ,B(H)u

∝ Z

R

|ξ|−2d(1−e−iξt) (1−eiξu)dξ .

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Examples

Generalized fBm: the case 0 < H < 1

By continuity of the sample paths ofB(H)t , we further get for a wide class of functionsθ(e.g. continuous, compactly supported and with vanishing integral)

Z

B(H)t θ(t)dt∝d B(H)(θ).

It also follows that cov

Y(H)t ,Y(H)u

∝ Z

R

|ξ|−2d|1−e−iξ|2ei(u−t)ξdξ .

The formula for the spectral density (2) follows.

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Examples

Fractional integration of a stationary process

Stationary processes

Letx ∈RandY ={Yt, t ∈Z}be a weakly stationary process with spectral measureν such that

Z π

−π

1−e−iλ

−2x

dν(λ)<∞.

Thex-order fractionally integrated process ofY is defined by

−xY

t= Z π

−π

1−e−iλ −x

eiλtdYbλ

whereYb denotes the spectral representation of Y.

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Examples

Fractional integration of a long memory process

The previous definition can be extended to generalized processes so that, ifY have long memory parameterd,∆−xY has long memory parameterd +x and same short-range spectral density asY.

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Examples

FARIMA processes

Definition

The FARIMA(p,d,q) process is defined as the process∆−dY with Y taken as an ARMA(p,q) process.

Hence it admits a generalized spectral density of the form

f(λ) =σ2|1−e−iλ|−2d

1 +Pq

k=1θke−iλk 1−Pp

k=1φke−iλk

2

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Examples

Less standard examples

1. FARIMA stable processes. See Stoev and Taqqu (2004).

2. Stochastic volatility models. See Hurvich, Moulines, Soulier (2005).

3. Infinite source Poisson model. See Fa¨y, Roueff, Soulier (2007) 4. Extension to locally stationaryprocesses, Roueff and von

Sachs (2011), see below.

5. Gaussian subordinator(in progress), X(t) =g(Yt) with Yt Gaussian. The long memory parameter of X depends on the one ofY and the Hermite rank of g. See Clausel, Roueff, Taqqu, Tudor (2010).

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Fourier and wavelet analysis

Fourier and wavelet analysis

Fourier and wavelet transforms arelinear orthogonal transforms used to performsecond order analysis.

Data

DFT Periodogram

− − − − − − − − − − −− − − − − − − − − − − −−

DWT Scalogram

X1 ... Xn

dn(2πk/n),1≤k≤n In(2πk/n) =|dn(2πk/n)|2

− − − − − − − − − − −− − − − − − − − − − − −−

Wj,k,1≤k ≤nj,0≤j ≤J bσ2j = 1 nj

nj

X

k=1

W2j,k Herenj ∼n2−j andJ ∼log2(n).

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Fourier and wavelet analysis

Fourier analysis

The discrete Fourier transform (DFT) of a sampleX1, . . . ,Xn is defined as the Fourier coefficients

dh,nX (ω) = (2πn)−1/2

n

X

t=1

(Xt−X¯n)eitω ,

Where

n= 1 n

n

X

k=1

Xk .

The Fourier coefficients are usually computed at frequencies 2πk/n, 0<k <n, via

FFT(X1, . . . ,Xn).

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Fourier and wavelet analysis

Periodogram: definition

The periodogram is defined as

In(ω) =|dn(ω)|2

It can be interpreted as thespectral densityof the MA(n) process having autocovariance function

γbn(τ) = 1 n

X

1≤k,k+τ≤n

(Xk −X¯n)(Xk+τ−X¯n), τ ∈Z.

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Fourier and wavelet analysis

Periodogram of a long memory process

FARIMA(0, 0.25 ,0)

Time

0 100 200 300 400 500 600

−1012

● ●

● ●

●●

●●●●

1e−051e−031e−01

I(λλ)

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Fourier and wavelet analysis

Wavelet assumptions

(W-1) φandψ are compactly-supported, integrable, and φ(0) =R

−∞φ(t)dt = 1 andR

−∞ψ2(t)dt = 1.

(W-2) There existsα >1 such that supξ∈R(ξ)|(1 +|ξ|)α<∞.

(W-3) The functionψ hasM vanishing moments, R

−∞tmψ(t)dt = 0 for all m= 0, . . . ,M−1 (W-4) The functionP

k∈Zkmφ(· −k) is a polynomial of degreem for allm= 0, . . . ,M−1.

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Fourier and wavelet analysis

Strang Fix conditions

Under (W-1), (W-3) is equivalent to

(ξ)|=O(|ξ|M) as ξ →0. (4) Under (W-1), (W-4) is equivalent to

sup

k6=0

(ξ+ 2kπ)|=O(|ξ|M) as ξ→0. (5)

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Fourier and wavelet analysis

Discrete wavelet transform (DWT)

Letψj,k(t) = 2−j/2ψ(2−jt−k) whereψ is a wavelet. For x(t) defined on continuous timet ∈R, the DWT is defined by

Wj,k = Z

−∞

x(t)ψj,k(t)dt, j ∈Z, k ∈Z.

Forxk defined on discrete timek ∈Z, we use the interpolated version

x(t) =X

l∈Z

xlφ(t−l), t ∈R,

resulting inWj,k =X

l∈Z

hj,2jk−lxl = [↓j (hj∗x)]k.

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Fourier and wavelet analysis

Conditions on the filters

1. Finite support: For each j,{hj(τ)}τ∈Z has finite support.

2. Uniform smoothness: There exists M ≥K,α >1/2 and C >0 such that for all j ≥0 and λ∈[−π, π],

|bhj(λ)| ≤ C2j/2|2jλ|M

(1 + 2j|λ|)M+α . (6) 3. Asymptotic behavior: There exists some non identically zero

functionbh such that for anyλ∈R,

j→+∞lim (2−j/2bhj(2−jλ)) =bh(λ). (7)

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Fourier and wavelet analysis

Daubechies Filters at different scales j

We plot 2−j hj(λ)

2

when φandψ are the Daubechies wavelets withM = 2.

10−3 10−2 10−1 100

0 0.2 0.4 0.6 0.8 1 1.2 1.4

j=1 j=2 j=3 j=4 j=5 j=6 j=7 j=8

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Fourier and wavelet analysis

MRA

In practice, waveletsφand ψare associated with a mutliresolution analysis(MRA), in which case, Wj,k are obtained iteratively using two finite filterh andg as follows, for allj ≥1,

x0,k =xk xj,k =X

t

xj−1,2k−tgt = (↓g?xj−1,·)k

Wj,k =X

t

xj−1,2k−tht = (↓h?xj−1,·)k

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Fourier and wavelet analysis

Teletraffic data

Around 2 hours IP traffic record aggregated every second.

0 1000 2000 3000 4000 5000 6000 7000

0 200 400 600 800 1000

s

nb. paquets

0 200 400 600 800 1000 1200

−0.2 0 0.2 0.4 0.6 0.8 1 1.2

s

Correllation

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Fourier and wavelet analysis

DWT of Teletraffic data

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Fourier and wavelet analysis

And its scalogram

1 2 3 4 5 6 7 8 9 10

102 103 104 105

0 0.2 0.4 0.6 0.8 1 1.2 1.4

Time varying long memory parameter. 29/53

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Fourier and wavelet semiparametric estimation

Semi-parametric assumption

X has long memory parameter d and short-range spectral density f satisfying

|f(λ)−f(0)| ≤Cf(0)|λ|β , (8) for someC, β >0. (semiparametric assumption)

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Fourier and wavelet semiparametric estimation

Fourier VS wavelet : a quick view

Fourier methods Wavelet methods Resolution Prediff. of orderδ M vanishing moments param. Taper of orderτ Fourier decay α

d Range δ−1/2−τ <d <δ+ 1/2 1/2−α<d <M+ 1/2

Estimators GPH, LWF LWW, LWR

Asymp. var. %as τ % depends on d

See Fa¨y et al. [2009] for details.

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Fourier and wavelet semiparametric estimation

Wavelet spectrum

One has the following asymptotic equivalence for thewavelet spectrum,

σ2j =var(Wj,k)∼σ2 22dj, as j → ∞.

Under the semiparametric assumption (8), the∼ can be made more precise,

σ2j −σ222dj

≤C 2−βj 22dj . (9)

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Fourier and wavelet semiparametric estimation

Semiparametric estimation

The basic tool is to study the joint convergence of (conveniently normalized)

σb2j −σ222dj =

2j −σ2j

(fluctuation term) +

h

σ2j −σ222dj i

(bias term), forJ0 ≤j ≤J as J0 andn → ∞.

TheBias termis given by (9);

Thefluctuation term can be studied using additional assumptions, e.g. if X is Gaussian, or alinear process,

Xt =X

s

αst−s .

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Fourier and wavelet semiparametric estimation

Error bound

In this case one finds, asj and nj → ∞,

2j −σ222dj = 22dj n

OP

n−1/2j

+O

2−βj

o .

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Fourier and wavelet semiparametric estimation

Asymptotic normality as n, j → ∞ [Roueff and Taqqu, 2009, Theorem 2]

Suppose thatX is a linear process,Xt =P

sαst−s, with long memory parameterd. Suppose that (s) is i.i.d. and E[40]<∞.

Then, asj,n→ ∞, n

σ−2j n−1/2j2j+m−σ2j+m

, m∈Z

ofidi

→ {Zm,m∈Z}

whereZ is a centered Gaussian process and, for all m≤p cov(Zm,Zp) = 4π24dp

Z π

−π

|D(d)∞,u|2dλ .

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Fourier and wavelet semiparametric estimation

Asymptotic behavior: non-linear cases

I Stochastic volatility processes: a Fourier estimator has been proposed and studied in Hurvich et al. [2005]. Results are similar to the linear case.

I Infinite Source Poisson Process: wavelet estimation of the long memory parameter has been studied in Fa¨y et al. [2007].

Rates areslower than in the linear case.

I Subordinated Gaussian process: Rates areslower than in the linear case. (work in progress)

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The model

Long memory in volatility

The following Figure is borrowed from Granger and Ding (1996). It is a plot of the evolution of the estimated long memory parameter for absolute values of S&P stock market returns between 1928 and 1991 (10 estimates over non-overlapping 6 years periods).

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The model

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The model

Daily realized log volatility YEN/US$ 1986–2004.

−4

−3

−2

−1 0 1 2 3 4

Realized Log volatility YEN/USD exchange rate from June 1986 to Sept. 2004

(40)

The model

Question

Is long memory a good indicator of a structural change such as the 1997 asian crisis ?

To answer this, we need

I A model allowing atime varying long memory parameterin a non-parametric fashion.

I We use the concept of local stationarity introduced by Dahlhaus: d d(u),u[0,1].

I Long memory estimators easily adapted to vary along time.

I We localize the scalogram: bσ2j σb2j(u).

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The model

The change point model

A first approach for introducing a time varyingd is to introduce one (or more) change point(s) as follows.

TakeT observations X1, . . . ,XT and suppose that Change point model

I X1, . . . ,Xk is a k-length sample with long memory parameter d(1)

I Xk+1, . . . ,XT is a T −k-length sample with long memory parameter d(2)

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The model

The change point model: rescaled

To allow a consistent estimation ofk andd1 asT → ∞, one needs anarray asymptotic or in-fill parameterization:

TakeT observations X1,T, . . . ,XT,T and suppose that Change point model

Letu ∈(0,1) be fixed and setk = [uT]. Then

I X1,T, . . . ,Xk,T is a k-length sample with long memory parameter d(1)

I Xk+1,T, . . . ,XT,T is a T −k-length sample with long memory parameter d(2)

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The model

The change point model: drawbacks

However

1. the process start afresh in a stationary state after a change point;

2. the parameter d jumps instantaneously ;

Dahlhaus’s approach circumvents these two points

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Outline Long memory processes Semi-parametric estimation Time varying long memory Bibliography References

The model

Local long memory parameter

Local stationarity is based on thespectral representation of linear processes

Xt

,T

= Z π

−π

A(λ) eiλtdZ(λ) , t = 1, . . .T, with

local

spectral density defined on [−π, π]

×[0,1]

by f(λ

,u

) =|A(λ

,u

)|2 =|1−e−iλ|−2d

(u)

f

,u

).

,

(10)

where, for someD <1/2,

A0t,T(λ)−A(λ,t/T)

≤c T−1 |λ|−D, 1≤t ≤T,−π ≤λ≤π , and

|A(λ,u)−A(λ,v)| ≤c |v−u| |λ|−D, 0≤u,v ≤1,−π≤λ≤π .

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Outline Long memory processes Semi-parametric estimation Time varying long memory Bibliography References

The model

Local long memory parameter

Local stationarity is based on thespectral representation of linear processes

Xt,T = Z π

−π

A0t,T(λ) eiλtdZ(λ) , t = 1, . . .T, withlocalspectral density defined on [−π, π]×[0,1] by

f(λ,u) =|A(λ,u)|2 =|1−e−iλ|−2d(u)f(λ,u)

.

, (10)

where, for someD <1/2, A0t,T(λ)−A(λ,t/T)

≤c T−1|λ|−D, 1≤t ≤T,−π ≤λ≤π , and

|A(λ,u)−A(λ,v)| ≤c |v−u| |λ|−D, 0≤u,v ≤1,−π≤λ≤π .

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The model

Examples

I d ≡0 (locally stationary short memory processes).

I tvMA(q) process

Xt,T =σ(t/T){t+θ1(t/T)t−1+· · ·+θq(t/T)t−q}, whereσ,θare defined on [0,1].

I tvAR(p) process

Xt,T =σ(t/T)t+θ1(t/T)Xt−1,T+· · ·+θq(t/T)Xt−q,T}, whereσ,θare defined on [0,1].

I d : [0,1]→(−∞,D) Lipschitz.

I tvFARIMA(0,d,0) processXt,T=σ(t/T)P

k≥0a(d(t/T))k t−k where (a(d)k )k≥0are thed fractional integration coefficients:

(1z)−d=X

k≥0

a(d)k zk, |z|<1.

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Local long memory estimation

Local scalogram

1. Computestandard wavelet coefficients X1,T, . . . ,XT,T

DFT−→Wj,k;T, j = 1, . . . ,J, k = 1, . . . ,Tj , 2. Compute a local scalogram

2j,T(u) =

Tj

X

k=1

γj,T(k)W2j,k;T ,

where{γj,T(k)} are some non-negative weights localized at indicesk ≈uTj and normalized in such a way that

PTj

k=1γj,T(k) = 1.

3. Apply the usual steps to obtain bd(u) from σb2j,T(u),;j = 1,2, . . . ,J.

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Local long memory estimation

The tangent process

Why should this work ?

Letu ∈(0,1) and let Xt(u) be thestationary approximationof Xt,T defined by

Xt(u) = Z π

−π

A(λ,u) eiλt dZ(λ) , t= 1, . . .T,

ThenX(u) ={Xt(u),t ∈Z} has spectral density given by (10) and thus long memory parameterd(u).

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Local long memory estimation

The tangent wavelet coefficients and the local wavelet spectrum

LetWj,k(u) denotes the wavelet coefficients ofX(u).

SinceXt(u)'Xt,T as T → ∞ with t/T 'u we have Wj,k;T 'Wj,k(u)

and thusσb2j,T(u) is an estimator of the local wavelet spectrum σ2j(u) =E[W2j,0(u)]'C22d(u)j

asj → ∞.

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Local long memory estimation

Asymptotic behavior : scalogram

Asymptotic results rely on the convergence ofσb2j,T(u) toσ2j(u) as Tj,j → ∞ andb →0, whereb denotes the bandwidth of the weightsγj,T(k).

In Roueff and von Sachs [2011],consistencyandasymptotic normalityunder certain conditions: bias must be negligible (b small enough,j large enough).

Confidence intervals are obtained forσ2j(u) and then ford(u).

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Local long memory estimation

Local memory parameter for the realized log volatility YEN/US$.

1988 1990 1992 1994 1996 1998 2000 2002 2004

0 0.2 0.4 0.6 0.8 1

Kernel Estimator of d using scales 1 2 3 to 3 4 5 resp.

scales j=1 to j=3 scales j=2 to j=4 scales j=3 to j=5

1988 1990 1992 1994 1996 1998 2000 2002 2004

10−2 10−1 100 101

Kernel Estimator of the wavelet spectrum for 5 scales.

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Local long memory estimation

Asymptotic behavior: estimation of d (u)

The following bound is obtained bdT(b,j) =d(u) +OP

(Tb)−1+ 2−βj +b2j(3/2−d(u))

. An additional error term implies that

b cannot be tuned independently ofj (they cannot be too large at the same time)

It illustrates the competition between thelarge scaleanalysis for catchinglong memory and thesmall scale analysis for catching local stationarity.

Open questions

1. Lower bound for the rate.

2. Data-driven selection ofb,j.

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Further reading I

G. Fa¨y, F. Roueff, and P. Soulier. Estimation of the memory parameter of the infinite-source Poisson process. Bernoulli, 13 (2):473–491, 2007. ISSN 1350-7265.

G. Fa¨y, E. Moulines, F. Roueff, and M.S. Taqqu. Estimators of long-memory: Fourier versus wavelets. J. of Econometrics, 151 (2):159–177, 2009. doi: 10.1016/j.jeconom.2009.03.005. URL http://dx.doi.org/10.1016/j.jeconom.2009.03.005.

C. M. Hurvich, E. Moulines, and P. Soulier. Estimating long memory in volatility. Econometrica, 73(4):1283–1328, 2005.

ISSN 0012-9682.

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Further reading II

F. Roueff and M. S. Taqqu. Central limit theorems for arrays of decimated linear processes. Stoch. Proc. App., 119(9):

3006–3041, 2009.

F. Roueff and R. von Sachs. Locally stationary long memory estimation. Stochastic Processes and their Applications, 121(4):

813 – 844, 2011. ISSN 0304-4149. doi:

DOI:10.1016/j.spa.2010.12.004. URL

http://www.sciencedirect.com/science/article/

B6V1B-51R4SX9-1/2/74b83473ada9e80d606edcd7cf917c82.

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