Time varying long memory parameter.
Self-similarity and related fields, Le Touquet, 2010.
June 7, 2011
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
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.
Increment stationary processes
Generalized processes indexed by sequences
Given a processX ={Xk}k∈Z, 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.
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)
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).
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).
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| → ∞.
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
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)(θ1),B(H)(θ2)
= Z
R
|ξ|−2dθ1∗(ξ)θ∗2(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|θ∗(ξ)|2dξ <∞.
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|θ∗(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)(δt−δ0) is a H-self-similar Gaussian process, with stationary increments; thus
n
B(H)t ,t ∈Ro d
∝n
B(H)(δt−δ0),t ∈Ro .
It follows that cov
B(H)t ,B(H)u
∝ Z
R
|ξ|−2d(1−e−iξt) (1−eiξu)dξ .
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.
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.
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.
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
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).
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).
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
X¯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).
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.
Fourier and wavelet analysis
Periodogram of a long memory process
FARIMA(0, 0.25 ,0)
Time
0 100 200 300 400 500 600
−1012
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1e−051e−031e−01
I(λλ)
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|ψ∗(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.
Fourier and wavelet analysis
Strang Fix conditions
Under (W-1), (W-3) is equivalent to
|ψ∗(ξ)|=O(|ξ|M) as ξ →0. (4) Under (W-1), (W-4) is equivalent to
sup
k6=0
|φ∗(ξ+ 2kπ)|=O(|ξ|M) as ξ→0. (5)
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.
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)
Fourier and wavelet analysis
Daubechies Filters at different scales j
We plot 2−j h∗j(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
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
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
Fourier and wavelet analysis
DWT of Teletraffic data
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
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)
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.
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)
Fourier and wavelet semiparametric estimation
Semiparametric estimation
The basic tool is to study the joint convergence of (conveniently normalized)
σb2j −σ222dj =
bσ2j −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 .
Fourier and wavelet semiparametric estimation
Error bound
In this case one finds, asj and nj → ∞,
bσ2j −σ222dj = 22dj n
OP
n−1/2j
+O
2−βj
o .
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−1/2j bσ2j+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 .
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)
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).
The model
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
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).
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)
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)
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
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∗(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 .
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 .
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:
(1−z)−d=X
k≥0
a(d)k zk, |z|<1.
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
bσ2j,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.
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).
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 → ∞.
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).
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.
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.
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.
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.