copula based spectral analysis - sfb649.wiwi.hu-berlin.de

Serial Dependence Spectral Analysis Asymptotic Properties Data Example Conclusions Extensions

Copula based spectral analysis

Holger Dette, Ruhr-Universitat BochumMarc Hallin, Universite Libre de Bruxelles

Tobias Kley, Ruhr-Universitat BochumStefan Skowronek, Ruhr-Universitat Bochum

Stanislav Volgushev, Ruhr-Universitat Bochum

July , 2015


Outline

1 Serial DependenceThe Traditional ApproachA Quantile-based Approach

2 Spectral Analysis of Time SeriesA Least Squares Interpretation of the PeriodogramA Quantile-based Approach

3 Asymptotic Properties

4 Data Example

5 Conclusions

6 ExtensionsDistributional properties of smoothed periodogramsLocal stationarity


The Traditional Approach

Traditional Time Series Models

Most traditional time series models are of the conditionallocation/scale type:

Xt = ψ(Xt−1,Xt−2, . . .) + σ(Xt−1,Xt−2, . . .)εt

where

the innovations (εt)t∈Z are white noise,

εt is independent of Xt−1,Xt−2, . . .,

Holger Dette Copula based spectral analysis 1 / 58



Implications

The distribution of Xt conditional on Xt−1,Xt−2, . . . is thedistribution of εt rescaled and shifted (by conditionalparameters). Therefore

Xt is a linear function of εt

All standardized conditional distributions of Xt |Xt−1,Xt−2, . . .coincide with the distribution of εt ⇒all conditional quantiles (hence values at risk) follow fromthose of ε by a linear transformation

Interpretation of ψ and σ depends on the identificationconstraints on ε.

Interpretation as conditional mean and variance correspondsto (Gaussian) L2-legacy, which is widely used in time seriesanalysis.




Example: spectral density for 3 time series

Yt = Xt/(Var(Xt))1/2,where

Xt is i.i.d.Xt is ARCH(1)Xt = 0.1Φ−1(Ut) + 1.9(Ut − 0.5)Xt−1 is QAR (1)(Ut i.i.d. uniform, Φ cdf of the standard normal distribution

0.0 0.1 0.2 0.3 0.4 0.5

0.10

0.14

0.18

0.22

i.i.d.

0.0 0.1 0.2 0.3 0.4 0.5

0.10

0.14

0.18

0.22

QAR(1)

0.0 0.1 0.2 0.3 0.4 0.5

0.10

0.14

0.18

0.22

ARCH(1)




Preceding analysis based on L2 methods and auto-covariances.

works well for Gaussian time series, not so for heavy tails

is good at capturing linear dynamics

is concerned with mean/variance (specific location-scale)effects

In this talk: robustness, richer view of dynamics, tails?

replace covariances joint distributions/copulas.

replace L2-loss by L1-based loss functions, quantile regression.

ranks.




The final result:

A family of copula spectral densities : ARCH(1)

0.0 0.1 0.2 0.3 0.4 0.5

0.01

00.

020

0.03

0

0.0 0.1 0.2 0.3 0.4 0.5

0.00

40.

008

0.01

2

0.0 0.1 0.2 0.3 0.4 0.5

−0.0

15−0

.005

0.00

5

0.0 0.1 0.2 0.3 0.4 0.5

−0.0

03−0

.001

0.00

10.

003

0.0 0.1 0.2 0.3 0.4 0.5

0.03

00.

040

0.05

0

0.0 0.1 0.2 0.3 0.4 0.5

0.00

40.

008

0.01

2

0.0 0.1 0.2 0.3 0.4 0.5

−0.0

020.

000

0.00

2

0.0 0.1 0.2 0.3 0.4 0.5

−0.0

030.

000

0.00

20.

004

0.0 0.1 0.2 0.3 0.4 0.50.

010

0.02

00.

030

τ 1=0

.1τ 1

=0.5

τ 1=0

.9

τ2=0.1 τ2=0.5 τ2=0.9

ω 2π




The final result:

A family of copula spectral densities : QAR

0.0 0.1 0.2 0.3 0.4 0.5

0.00

80.

012

0.01

6

0.0 0.1 0.2 0.3 0.4 0.5

−0.0

050.

005

0.01

5

0.0 0.1 0.2 0.3 0.4 0.5

−0.0

020.

000

0.00

20.

004

0.0 0.1 0.2 0.3 0.4 0.5

−0.0

08−0

.004

0.00

0

0.0 0.1 0.2 0.3 0.4 0.5

0.03

00.

040

0.05

0

0.0 0.1 0.2 0.3 0.4 0.5

0.00

00.

010

0.02

0

0.0 0.1 0.2 0.3 0.4 0.5

−0.0

06−0

.003

0.00

0

0.0 0.1 0.2 0.3 0.4 0.5

−0.0

10−0

.006

−0.0

020.0 0.1 0.2 0.3 0.4 0.5

0.00

50.

015

0.02

5

τ 1=0

.1τ 1

=0.5

τ 1=0

.9

τ2=0.1 τ2=0.5 τ2=0.9

ω 2π




Being less restrictive

Modeling the entire distribution of the strictly stationary process(Xt)t∈Z?

Allow the conditional distribution of Xt |Xt−1,Xt−2, . . . to bearbitrary, but

make structural assumptions.

Example: (Xt)t∈Z a stationary, Markovian process of order one.

Then the distribution of (Xt)t∈Z is fully characterized by either

the joint distribution F1 of (Xt−1,Xt), or

the marginal distribution F of Xt andthe pair copula C1 of lag one, i. e. the joint distribution of(F (Xt−1),F (Xt)).



A Quantile-based Approach

Pair Copulae as a Measure for serial dependence

Notation:F denotes the distribution function of Xt

qτ = F−1(τ) is the τ -quantile of FFk denotes the joint distribution function of (Xt−k ,Xt)Ck denotes the copula of Fk , i.e the distribution of

(F (Xt−k),F (Xt))

(pair copula of lag k)

Note:In the previous example all the pair copulae Ck of lag k aredetermined by C1 (by Markov assumption).For other processes this is not necessarily the case; the paircopulae (Ck) vary freely.The copulae (Ck) are well suited to quantify serial dependence.Offer much richer information than the autocorrelations only.




Clipped Processes(1Xt ≤ 0.5)t∈N

t

Xt

-1

1

2




Clipped Processes(1Xt ≤ 0.5)t∈N and (1Xt ≤ −1)t∈N

t

Xt

-1

1

2

t

Xt

-1

1

2




Copula and Laplace Cross-covariance Kernels

Let (Xt)t∈Z be a real-valued, strictly stationary process (where Xt

has a positive density).

Definition

Laplace cross-covariance kernel of lag k ∈ Z

γk(q1, q2) = Cov(1Xt ≤ q1,1Xt+k ≤ q2)= Fk(q1, q2)− F (q1)F (q2)

= Ck(F (q1),F (q2))− F (q1)F (q2), q1, q2 ∈ R

Copula cross-covariance kernel of lag k ∈ Z

γk(τ1, τ2) = Cov(1Xt ≤ F−1(τ1),1Xt+k ≤ F−1(τ2))= γk(qτ1 , qτ2)

= Ck(τ1, τ2)− τ1τ2, τ1, τ2 ∈ (0, 1)




Covariance and Cross-covariance

We use terminology of classical time series

But only covariances of indicators are considered

γk and γk always exist (no assumptions about moments)

Note:γk(q1, q2)| q1, q2 ∈ R

entirely characterizes the joint distribution of (Xt ,Xt+k)

γk(τ1, τ2)| τ1, τ2 ∈ (0, 1)

and F entirely characterize the joint distributionof (Xt ,Xt+k)




Example: AR(1) Process with independent Innovations

AR(1) process

Xt = ϑXt−1 + εt , t ∈ Z.

ϑ = −0.3

independent innovations with

(i) t1-distribution

(ii) standard normal distribution




Sample path: AR(1) Process, ϑ = −0.3; t1-distributed(left) and normal distributed innovations (right)

0 50 100 150 200

−80

−60

−40

−20

020

4060

t

Xt=

−0.

3Xt−

1+

ε t

0 50 100 150 200−

2−

10

12

34

t

Xt=

−0.

3Xt−

1+

ε t




γ1(τ1, τ2) = C1(τ1, τ2)− τ1τ2 = F1(qτ1, qτ2

)− τ1τ2


γ1(τ

1,τ

2)

τ1

τ2

τ1

τ2



γ2(τ1, τ2) = C2(τ1, τ2)− τ1τ2 = F2(qτ1, qτ2

)− τ1τ2


γ2(τ

1,τ

2)

τ1

τ2

τ1

τ2



Properties

Laplace and Copula cross-covariance kernel

If EX 2t <∞, then (γk) and F determine the autocovariance

function of (Xt)t∈Z.

“Symmetry”

γk(q1, q2) = γ−k(q2, q1)

γk(τ1, τ2) = γ−k(τ2, τ1)

Invariance of γk under continuous monotone transformation




The Laplace Spectral Density Kernel

Assume that the Laplace cross-covariance kernels γk , k ∈ Z satisfy

∞∑k=−∞

|γk(q1, q2)| <∞ for all q1, q2 ∈ R

Laplace spectral density kernel

fq1,q2(ω) :=1

2π

∞∑k=−∞

γk(q1, q2)e−ikω




Properties

fq1,q2(−ω) = fq2,q1(ω) = fq1,q2(ω)

If =fq1,q2 ≡ 0, then γk(q1, q2) = γ−k(q1, q2), ∀k

“Time reversibility”:

If =fq1,q2 ≡ 0, ∀q1, q2, then (Xt ,Xt+k)d= (Xt ,Xt−k).

Copula spectral density kernel

fqτ1 ,qτ2(ω) :=

1

2π

∞∑k=−∞

γk(τ1, τ2)e−ikω

:=1

2π

∞∑k=−∞

Ck(τ1, τ2)− τ1τ2e−ikω



A Least Squares Interpretation of the Periodogram

Traditional Spectral Analysis

For the analysis of serial dependencies the autocovariance functionand its spectral representation are often considered.

Autocovariances measure linear serial dependencies.

The same is true for the corresponding spectral density.

Estimation of the spectral density is (often) based on theperiodogram

In(ωj) :=1

n

∣∣∣∣∣n∑

t=1

Xteitωj

∣∣∣∣∣2

where ωj := 2πjn ∈ (−π, π], j ∈ Z are the Fourier

frequencies.





Note (representation as a quadratic form, i =√−1):

In(ωj) =n

4bn(ωj)

′(

1 −ii 1

)bn(ωj)

where bn(ωj) = (b1n(ωj), b2n(ωj))′,

(an(ωj), b1n(ωj), b2n(ωj)) = arg minb∈R3

n∑t=1

(Xt − ct(ωj)

′b)2

and ct(ωj) = (1, cos(tωj), sin(tωj))′.




The Laplace Periodogram Kernel

Use weighted L1 instead of L2 projections:

(aτn(ω), bτ1n(ω), bτ2n(ω)) = arg minb∈R3

n∑t=1

ρτ(Xt − ct(ω)′b

)where ρτ (u) := u(τ − 1(−∞,0](u)) is the check function:

Use an inner product for two quantiles τ1, τ2




The Laplace Periodogram Kernel

Main idea: Use weighted L1 projections for two differentvalues τ1, τ2

This gives two vectors:

bτn(ω) = (bτ1n(ω), bτ2n(ω))′ , τ = τ1, τ2.

Definition

The Laplace periodogram kernel is defined as

Lτ1,τ2n (ω) =

n

4bτ1n (ω)′

(1 −ii 1

)bτ2n (ω), τ1, τ2 ∈ (0, 1)




Remarks and questions:

Computation of Lτ1,τ2n (ω): Simplex algorithm

A special case (τ1 = τ2 = 12 ) of this L1 approach to spectral

analysis was suggested previously by Li (JASA 2008).

What object is this statistic “estimating”?

Consistency?



Asymptotics of the Laplace Periodogram Kernels

Technical assumptions

Assume that (Xt)t=1,...,n, n ∈ N is

strictly stationary

mn-decomposable (which includes linear processes) orβ-mixing or dependence concept of Wu and Shao (2004)

Let F be the absolutely continuous cdf of Xt ,

admitting the non-vanishing density f and

qτ := F−1(τ) is the quantile of F .

Assume that the Laplace cross-covariance kernels (γk) areabsolutely summable, such that the Laplace spectral densitykernel fq1,q2(ω) : q1, q2 ∈ R exists.




Theorem 1 (part 1/2)

For any ω1, . . . , ων ∈ (0, π) and any τ1, τ2 ∈ (0, 1) we have

(Lτ1,τ2n (ω1), . . . , Lτ1,τ2

n (ων)) (Lτ1,τ2(ω1), . . . , Lτ1,τ2(ων))

where Lτ1,τ2(ωj) are independent random variables such that

Lτ1,τ2(ωj) ∼1

2f τ1,τ2(ωj)χ

22 if τ1 = τ2,

and

f τ1,τ2(ω) := 2πfqτ1 ,qτ2

(ω)

f (qτ1)f (qτ2),




Theorem 1 (part 2/2)

and

Lτ1,τ2(ωj)d=

1

4

(Z11

Z12

)′(1 −ii 1

)(Z21

Z22

)if τ1 6= τ2

where (Z11,Z12,Z21,Z22) ∼ N4(0,Σ4(ωj)), with

Σ4(ωj) =1

2

f τ1,τ1(ωj) 0 <f τ1,τ2(ωj) −=f τ1,τ2(ωj)

0 f τ1,τ1(ωj) =f τ1,τ2(ωj) <f τ1,τ2(ωj)<f τ1,τ2(ωj) =f τ1,τ2(ωj) f τ2,τ2(ωj) 0−=f τ1,τ2(ωj) <f τ1,τ2(ωj) 0 f τ2,τ2(ωj)

..




The bias?

Corollary

For all τ1, τ2 ∈ (0, 1) and ω ∈ (0, π) we have

limn→∞

ELτ1,τ2n (ω) = f τ1,τ2(ω) = 2π

fqτ1 ,qτ2(ω)

f (qτ1)f (qτ2)

Note: The Laplace periodogram Lτ1,τ2(ωj) is not an asymptot-ically unbiased estimator of the copula spectral density fqτ1 ,qτ2

(ω),but of the quantity

2πfqτ1 ,qτ2

(ω)

f (qτ1)f (qτ2)




Finite Sample Properties

data from, AR(1) time series (ϑ = −0.3, n = 500)

t1- and N (0, 1) distributed innovations

Get an idea on the remaining bias of the estimate

ELτ1,τ2n (ω)

Corollary toTheorem 1−−−−−−−→

n→∞ELτ1,τ2(ω) = 2π

fqτ1 ,qτ2(ω)

f (qτ1)f (qτ2)


Laplace Periodogram for AR(1) data (n = 500, εt ∼ t1)

0.0 0.1 0.2 0.3 0.4 0.5

010

0030

0050

00

τ 1=

0.05

0.0 0.1 0.2 0.3 0.4 0.5

−50

050

100

150

200

τ 1=

0.5

0.0 0.1 0.2 0.3 0.4 0.5

−20

000

2000

4000

τ2=0.05

τ 1=

0.95

0.0 0.1 0.2 0.3 0.4 0.5

−10

0−

500

5010

0

0.0 0.1 0.2 0.3 0.4 0.5

05

1015

2025

0.0 0.1 0.2 0.3 0.4 0.5

−50

050

100

150

200

τ2=0.5

0.0 0.1 0.2 0.3 0.4 0.5

−15

00−

500

050

015

00

0.0 0.1 0.2 0.3 0.4 0.5

−10

0−

500

5010

0

0.0 0.1 0.2 0.3 0.4 0.5

010

0030

0050

00τ2=0.95

ω

L n, Nτ 1, τ

2 (ω)

Laplace Periodogram for AR(1) data (n = 500,εt ∼ N (0, 1))

0.0 0.1 0.2 0.3 0.4 0.5

0.02

0.06

0.10

τ 1=

0.05

0.0 0.1 0.2 0.3 0.4 0.5

−0.

050.

050.

15

τ 1=

0.5

0.0 0.1 0.2 0.3 0.4 0.5

−0.

040.

000.

04

τ2=0.05

τ 1=

0.95

0.0 0.1 0.2 0.3 0.4 0.5

−0.

100.

000.

050.

10

0.0 0.1 0.2 0.3 0.4 0.5

0.0

0.2

0.4

0.6

0.8

0.0 0.1 0.2 0.3 0.4 0.5

−0.

050.

050.

15

τ2=0.5

0.0 0.1 0.2 0.3 0.4 0.5

−0.

040.

000.

020.

04

0.0 0.1 0.2 0.3 0.4 0.5

−0.

100.

000.

050.

10

0.0 0.1 0.2 0.3 0.4 0.5

0.02

0.06

0.10

τ2=0.95

ω

L n, Nτ 1, τ

2 (ω)



Estimating the marginal distributions

Recall:

The Laplace periodogram Lτ1,τ2(ω) is not an asymptoticallyunbiased estimator for fqτ1 ,qτ2

(ω), but for

2πfqτ1 ,qτ2

(ω)

f (qτ1)f (qτ2)

If F ∼ U([0, 1]), then the Laplace periodogram Lτ1,τ2(ω) is anasymptotically unbiased estimator for 2πfqτ1 ,qτ2

(ω)

Idea: Use probability integral transformation Xt → F (Xt)!

Replace F (Xt) by Fn(Xt), where Fn is the empiricaldistribution function of X1, . . .Xn.

Note: nFn(Xt) gives the rank Rt(n) of Xt among X1, . . . ,Xn




Rank based Periodogram

The rank-based periodogram kernel

Lτ1,τ2

n,R (ω)

is the Laplace periodogram kernel calculated from

1

n + 1R1(n), . . . ,

1

n + 1Rn(n)

(instead of from X1, . . . ,Xn), where Rt(n) denotes the rank of Xt

among X1, . . . ,Xn.

(aτn(ω), bτ1n(ω), bτ2n(ω)) = arg minb∈R3

n∑t=1

ρτ

(1

n+1 Rt(n) − ct(ω)′b)




Theorem 2

For any ω1, . . . , ων ∈ (0, π) and any τ1, τ2 ∈ (0, 1) we have

(Lτ1,τ2

n,R (ω1), . . . , Lτ1,τ2

n,R (ων)) (Lτ1,τ2

R (ω1), . . . , Lτ1,τ2

R (ων))

where Lτ1,τ2

R (ωj) denote independent random variables distributedas Lτ1,τ2(ωj) , where

f τ1,τ2(ω) = 2πfqτ1 ,qτ2

(ω)

f (qτ1)f (qτ2)

has to be replaced by 2πfqτ1 ,qτ2(ω)

In particular the rank-based periodogram Lτ1,τ2

n,R (ω) is anasymptotically unbiased estimator for 2πfqτ1 ,qτ2

(ω).




Investigation of Finite Sample Properties

Consider the AR(1) time series with t1-distributed innovations(ϑ = −0.3) from the example

Get an idea on the remaining bias of the estimate

ELτ1,τ2n (ω) Theorem 2−−−−−−→

n→∞ELτ1,τ2(ω) = 2πfqτ1 ,qτ2

(ω)


AR(1) Example: Rank-based Periodograms, n = 500,t1-distributed innovations

0.0 0.1 0.2 0.3 0.4 0.5

0.02

0.06

0.10

0.14

τ 1=

0.05

0.0 0.1 0.2 0.3 0.4 0.5

−0.

050.

050.

150.

25

τ 1=

0.5

0.0 0.1 0.2 0.3 0.4 0.5

−0.

050.

000.

050.

10

τ2=0.05

τ 1=

0.95

0.0 0.1 0.2 0.3 0.4 0.5

−0.

100.

000.

10

0.0 0.1 0.2 0.3 0.4 0.5

0.0

0.2

0.4

0.6

0.8

1.0

0.0 0.1 0.2 0.3 0.4 0.5

−0.

050.

050.

150.

25

τ2=0.5

0.0 0.1 0.2 0.3 0.4 0.5

−0.

04−

0.02

0.00

0.02

0.04

0.0 0.1 0.2 0.3 0.4 0.5

−0.

15−

0.05

0.05

0.0 0.1 0.2 0.3 0.4 0.5

0.02

0.06

0.10

0.14

τ2=0.95

ω

L n, Nτ 1, τ

2 (ω)



Smoothed Periodogram

Sequence of positive weights Wn = Wn(j) : |j | ≤ NnWn(k) = Wn(−k) for all k∑|k|≤Nn

Wn(k) = 1

Definition

smoothed rank-based periodogram kernel:

f τ1,τ2

n,R (ωj) :=∑|k|≤Nn

Wn(k)Lτ1,τ2

n,R (ωj+k)




Theorem 3

For any ω ∈ (0, π) and any τ1, τ2 ∈ (0, 1) we have

f τ1,τ2

n,R (gn(ω)) = 2πfqτ1 ,qτ2(ω) + oP(1),

where gn(ω) denotes the Fourier frequency closest to ω.


AR(1) Example: Means of the Smoothed Rank-basedPeriodograms, n = 500, t1− distributed innovations

0.0 0.1 0.2 0.3 0.4 0.5

0.00

0.05

0.10

0.15

τ 1=

0.05

0.0 0.1 0.2 0.3 0.4 0.5

0.0

0.1

0.2

0.3

τ 1=

0.5

0.0 0.1 0.2 0.3 0.4 0.5

−0.

050.

000.

050.

10

τ2=0.05

τ 1=

0.95

0.0 0.1 0.2 0.3 0.4 0.5

−0.

100.

000.

10

0.0 0.1 0.2 0.3 0.4 0.5

0.0

0.2

0.4

0.6

0.8

1.0

1.2

0.0 0.1 0.2 0.3 0.4 0.5

0.0

0.1

0.2

0.3

τ2=0.5

0.0 0.1 0.2 0.3 0.4 0.5

−0.

04−

0.02

0.00

0.02

0.04

0.0 0.1 0.2 0.3 0.4 0.5

−0.

15−

0.05

0.05

0.0 0.1 0.2 0.3 0.4 0.5

0.00

0.05

0.10

0.15

τ2=0.95

ω

f nτ 1, τ

2 (ω)

AR(1) Example: Means of the Smoothed Rank-basedPeriodograms, n = 500, N (0, 1)-distributed innovations

0.0 0.1 0.2 0.3 0.4 0.5

0.00

0.04

0.08

0.12

τ 1=

0.05

0.0 0.1 0.2 0.3 0.4 0.5

−0.

050.

050.

15

τ 1=

0.5

0.0 0.1 0.2 0.3 0.4 0.5

−0.

040.

000.

040.

08

τ2=0.05

τ 1=

0.95

0.0 0.1 0.2 0.3 0.4 0.5

−0.

100.

000.

050.

10

0.0 0.1 0.2 0.3 0.4 0.5

0.0

0.2

0.4

0.6

0.8

1.0

0.0 0.1 0.2 0.3 0.4 0.5

−0.

050.

050.

150.

25

τ2=0.5

0.0 0.1 0.2 0.3 0.4 0.5

−0.

040.

000.

020.

04

0.0 0.1 0.2 0.3 0.4 0.5

−0.

100.

000.

050.

10

0.0 0.1 0.2 0.3 0.4 0.5

0.00

0.04

0.08

0.12

τ2=0.95


Standard & Poor’s 500: Daily (log-)Returns, 1963–2009

Year

S&

P 5

00 R

etur

n

1970 1980 1990 2000 2010

−0.

050.

000.

05



Traditional Spectral Analysis (Returns)

0.0 0.2 0.4 0.6 0.8 1.0

0.0

0.2

0.4

0.6

0.8

1.0

Lag

AC

F

0.0 0.1 0.2 0.3 0.4 0.5

0.00

007

0.00

008

0.00

009

0.00

011

frequency

spec

trum

bandwidth = 0.0142



Traditional Spectral Analysis (Quadratic (log-)Returns)

0.0 0.2 0.4 0.6 0.8 1.0

0.0

0.2

0.4

0.6

0.8

1.0

Lag

AC

F

0.0 0.1 0.2 0.3 0.4 0.5

1e−

072e

−07

3e−

075e

−07

frequency

spec

trum

bandwidth = 0.0142


Smoothed Rank-based Periodograms

0.0 0.1 0.2 0.3 0.4 0.5

−0.

10.

00.

10.

20.

3

τ 1=

0.05

0.0 0.1 0.2 0.3 0.4 0.5

−0.

10.

00.

10.

20.

3

τ 1=

0.5

0.0 0.1 0.2 0.3 0.4 0.5

−0.

10.

00.

10.

20.

3

τ2=0.05

τ 1=

0.95

0.0 0.1 0.2 0.3 0.4 0.5

−0.

10.

00.

10.

20.

3

0.0 0.1 0.2 0.3 0.4 0.5

−0.

10.

00.

10.

20.

3

0.0 0.1 0.2 0.3 0.4 0.5

−0.

10.

00.

10.

20.

3

τ2=0.5

0.0 0.1 0.2 0.3 0.4 0.5

−0.

10.

00.

10.

20.

3

0.0 0.1 0.2 0.3 0.4 0.5

−0.

10.

00.

10.

20.

30.0 0.1 0.2 0.3 0.4 0.5

−0.

10.

00.

10.

20.

3

τ2=0.95


Comments

The classical approach does not detect any serial structure

... the Laplace periodogram does ...

For extreme quantiles the smoothed periodograms peak at lowfrequencies

This indicates long-range dependence in the tails (ornon-stationarity)

Imaginary parts have smaller (absolute) values than the realparts, which might indicate “time reversibility”



Summary: “model-free” and “nonlinear” Spectral Analysis

The rank-based periodogram kernel seems to inherit much ofthe properties of the ordinary periodogram

Robustness can be expected due to the L1-nature of the toolsinvolved

Analysis of conditional distributions, not simply conditionalmeans and variances

No linearity, distribution, nor even moment assumptions arerequired

Separation of serial dependencies and marginal features

Invariance under monotone transformations of theobservations



Much work remains on the research agenda

Distributional properties of smoothed periodograms

Locally stationary processes

Prediction

Testing time reversibility

Extreme quantiles - tail dependence - tail copulas

Integrated spectra, higher order spectra

Graphical models




Distributional properties of smoothed periodograms:

In(ωj) :=1

n

∣∣∣ n∑t=1

Xte−itωj

∣∣∣2So far we have used generalizations of L2 projections

In(ωj) =n

4bn(ωj)

′(

1 −ii 1

)bn(ωj)

where bn(ωj) = (b1n(ωj), b2n(ωj))′ with

(an(ωj), b1n(ωj), b2n(ωj)) = arg minb∈R3

n∑t=1

(Xt − ct(ωj)

′b)2

and ct(ωj) = (1, cos(tωj), sin(tωj))′.




Alternative representation of periodogram (DFT of ACF):

Fourier transform of empirical auto-covariance

In(ωj) =1

n

∣∣∣ n∑t=1

Xte−itωj

∣∣∣2 =∑|k|<n

e−ikωjn − k

nγk

for Fourier frequencies ωj = 2πjn ∈ (0, π)

γk is empirical auto-covariance at lag k, i.e.

γk :=1

n − |k |

n−|k|∑t=1

(Xt − X )(Xt+|k| − X ).




Alternative estimator:

Discrete Fourier transform: dτn,R(ω) :=∑n

t=1 I 1n+1 Rt(n) ≤ τe−iωt

Copula periodogram

I τ1,τ2

n,R (ω) :=1

ndτ1

n,R(ω)dτ2

n,R(−ω), ω ∈ (0, π), (τ1, τ2) ∈ [0, 1]2,

For ω = 2πj/n with j = 1, ..., (n − 1)

I τ1,τ2

n,R (ω) =∑|k|<n

n − k

ne iωk Γτ1,τ2

k

where (k > 0)

Γτ1,τ2

k :=1

n − k

n−k∑t=1

I 1n+1 Rt(n) ≤ τ1I 1

n+1 Rt+k(n) ≤ τ2

is an estimator of the pair copula at lag k (but not the empiricalcopula!).




Alternative smoothed estimator:

Definition

The smoothed copula periodogram kernel:

fτ1,τ2

n,R (ω) :=2π

n

n−1∑s=1

Wn

(ω − 2πs/n

)I τ1,τ2

n,R (2πs/n)

where

Wn(u) :=∞∑

j=−∞b−1n W (b−1

n [u + 2πj ])

for a kernel W and bandwidth bn.




Theorem 4

Under suitable assumptions, for any fixed ω ∈ (0, π)√nbn

(fτ1,τ2

n,R (ω)− fqτ1,qτ2

(ω)− B(k)n (τ1, τ2;ω)

)τ1,τ2∈[0,1]

H(·, ·;ω)

in `∞([0, 1]2) where

B(k)n (τ1, τ2;ω) :=

k∑j=1

bjn

j!

∫v jW (v)dv

dj

dωjfqτ1

,qτ2(ω),

and H(·, ·;ω) is a centered Gaussian process with

Cov(H(x1, y1;ω

),H(x2, y2, ω)) = fqx1

,qy1(ω)fqx2

,qy2 (ω)

∫W 2(u)du.

Weak convergence above holds jointly for any finite collections offrequencies (asymptotic independence).

Conjecture: similar result holds for rank based periodogram





Estimates of copula spectra and point wise confidence intervals forsimulated ARCH-data

0.0 0.1 0.2 0.3 0.4 0.5

0.01

00.

020

0.03

0

0.0 0.1 0.2 0.3 0.4 0.5

0.00

40.

008

0.01

2

0.0 0.1 0.2 0.3 0.4 0.5

−0.0

100.

000

0.00

5

0.0 0.1 0.2 0.3 0.4 0.5

−0.0

03−0

.001

0.00

10.

003

0.0 0.1 0.2 0.3 0.4 0.5

0.03

00.

035

0.04

00.

045

0.05

0

0.0 0.1 0.2 0.3 0.4 0.5

0.00

40.

008

0.01

2

0.0 0.1 0.2 0.3 0.4 0.5

−0.0

020.

000

0.00

2

0.0 0.1 0.2 0.3 0.4 0.5

−0.0

020.

000

0.00

20.

004

0.0 0.1 0.2 0.3 0.4 0.5

0.01

00.

020

0.03

0

τ 1=0

.1τ 1

=0.5

τ 1=0

.9

τ2=0.1 τ2=0.5 τ2=0.9

ω 2πHolger Dette Copula based spectral analysis 52 / 58




Estimates of copula spectra and point wise confidence intervals forsimulated e QAR-data

0.0 0.1 0.2 0.3 0.4 0.5

0.00

80.

012

0.01

6

0.0 0.1 0.2 0.3 0.4 0.5

0.00

00.

005

0.01

00.

015

0.0 0.1 0.2 0.3 0.4 0.5

−0.0

020.

000

0.00

20.

004

0.0 0.1 0.2 0.3 0.4 0.5

−0.0

08−0

.004

0.00

0

0.0 0.1 0.2 0.3 0.4 0.5

0.03

50.

040

0.04

50.

050

0.0 0.1 0.2 0.3 0.4 0.5

0.00

00.

010

0.02

0

0.0 0.1 0.2 0.3 0.4 0.5

−0.0

05−0

.003

−0.0

010.

001

0.0 0.1 0.2 0.3 0.4 0.5

−0.0

10−0

.006

−0.0

02

0.0 0.1 0.2 0.3 0.4 0.5

0.01

00.

015

0.02

00.

025

τ 1=0

.1τ 1

=0.5

τ 1=0

.9

τ2=0.1 τ2=0.5 τ2=0.9

ω 2πHolger Dette Copula based spectral analysis 53 / 58


Local stationarity

Local stationarity:

Definition

A triangular array (Xt,T )t∈ZT∈N of processes is called locally strictlystationary (of order two) if there exists a constant L > 0 and, forevery ϑ ∈ (0, 1), a strictly stationary process Xϑ

t , t ∈ Z such that, forevery 1 ≤ r , s ≤ T ,∥∥Fr ,s;T (·, ·)− Gϑ

r−s(·, ·)∥∥∞ ≤ L

(max(|r/T − ϑ|, |s/T − ϑ|) + 1/T

),

where

‖ · ‖∞ is the supremum norm

Fr ,s;T (·, ·) is the joint distribution functions of (Xr ,T ,Xs,T )

Gϑk (·, ·) is the joint distribution functions of (Xϑ

0 ,Xϑ−k)



Local stationarity

Local smoothed copula periodograms:

Data: dailylog-returns of S&P500 since the sixties,13000 observations.

Compute spectrafrom local windowsand plot as heatplots.

Colours: indicatedeviations from whitenoise



Local stationarity




Local stationarity

Some references

- Davis, R. A., Mikosch, T. and Zhao, Y. (2013). Measures of serial extremal dependence and theirestimation. Stochastic Processes and their Applications 123 2575-2602.

- Dette, H., Hallin, M., Kley, T., Volgushev, S. (2011,2014). Of copulas, quantiles, ranks and spectra: AnL1-approach to spectral analysis. Available on Arxiv. To appear in: Bernoulli

- Hagemann, A. (2011). Robust Spectral Analysis (arXiv:1111.1965v1).

- Li, T. H. (2008). Laplace periodogram for time series analysis. Journal of the American StatisticalAssociation 103, 757- 768.

- Li, T.-H. (2012). Quantile periodograms. Journal of the American Statistical Association 107, 765-776.

- Kley, T. Volgushev, S. , Dette, H., Hallin, M., (2013). Quantile spectral processes - asymptotic analysisand inference. Available on Arxiv. To appear in: Bernoulli

- Skowronek, S., Volgushev, S., Kley, T., Dette, H., Hallin, M., (2013). Quantile Spectral Analysis forLocally Stationary Time Series,. Available on Arxiv.

- Kley, T. (2014). quantspec: Quantile-based Spectral Analysis Functions. R package version 0.1.Available on http://cran.r-project.org/web/packages/quantspec/index.html.


http://cran.r-project.org/web/packages/quantspec/index.html


Copula based spectral analysis

Holger Dette, Ruhr-Universitat BochumMarc Hallin, Universite Libre de Bruxelles

Tobias Kley, Ruhr-Universitat BochumStefan Skowronek, Ruhr-Universitat Bochum

Stanislav Volgushev, Ruhr-Universitat Bochum

July , 2015

Proof of Theorem 1 Proof of Theorem 3

Backup Slides

7 Proof of Theorem 1

8 Proof of Theorem 3


Proof of Theorem 1 (main idea)

We show for Ω := ω1, . . . , ων ⊂ Fn ⊂ (0, π) and allT := τ1, . . . , τp ⊂ (0, 1)

√n(

bτn(ω))τ∈T , ω∈Ω

L−−−→n→∞

(Nτ (ω)

)τ∈T , ω∈Ω

Here (Nτ (ω))τ∈T , ω∈Ω is a vector of centered (bivariate) normaldistributed random variables with Cov(Nτ1(ω1),Nτ2(ω2)) = 0 ifω1 6= ω2 and

Cov(Nτ1(ω1),Nτ2(ω1)) = 2

(<f τ1,τ2(ω1) =f τ1,τ2(ω1)−=f τ1,τ2(ω1) <f τ1,τ2(ω1)

)otherwise

Holger Dette Copula based spectral analysis BU 32 / 36


Most important step: uniform linearization

Define

Zn,τ,ω(δ) :=∑n

t=1

(ρτ (Xt − qτ − n−1/2c′t(ω)δ)− ρτ (Xt − qτ )

)ZX ′n,τ,ω(δ) = −δ′ζX ′

n,τ,ω + 12δ′QX ′

n,τ,ωδ

where

ζX ′

n,τ,ω = n−1/2∑n

t=1 ct(ω)(τ − IX ′t,n ≤ qn,τ)QX ′

n,τ,ω = fn,X ′(qn,τ ) n−1∑n

t=1 ct(ω)c′t(ω)fn,X ′ is the is the density of X ′t,n

then

supω∈Fn

sup‖δ−δX ′

n,τ,ω‖≤ε|Zn,τ,ω(δ)−ZX ′

n,τ,ω(δ)| = OP

((n−1/4∨(n−1/2mn))(log n)2

)where δX

′n,τ,ω = (QX ′

n,τ,ω)−1ζX′

n,τ,ω



Consequences:

If

δn,τ,ω = arg minδ

Zn,τ,ω(δ)

δX′

n,τ,ω = arg minδ

ZX ′n,τ,ω(δ) = (QX ′

n,τ,ω)−1ζX′

n,τ,ω

then

supω∈Fn

‖δn,τ,ω − δX′

n,τ,ω‖ = OP

((n−1/8 ∨ (n−1/4m

1/2n )) log n

).

Therefore the asymptotic properties of√

nbn,τ (ωj) can be obtainedfrom those of the random variables δX

′n,τ,ω




Let Fn denote the empirical distribution function of X1, . . . ,Xn,We have to minimize

n∑t=1

(ρτ (Fn(Xt)− τ − n−1/2c′t(ω)δ)− ρτ (Fn(Xt)− τ)

)Replace Fn(Xt) by Fn(X ′t,n), where Xt = X ′t,n + X ′,′t,n refers to mn

decomposibility

n∑t=1

(ρτ (Fn(X ′t,n)− τ − n−1/2c′t(ω)δ)− ρτ (Fn(X ′t,n)− τ))

Replace Fn(Xt,n) by Ut,n = Fn,X (X ′t,n), where Fn,X denotes theempirical distribution function of X ′1,n, . . . ,X

′n,n




n∑t=1

(ρτ (Ut,n−τ−n−1/2c′t(ω)δ)−ρτ (Ut,n−τ)

)−δ1

√n(Fn,X (F−1

n (τ))−τ)

where δ = (δ1, δ2, δ3)′.

Linearization (e1 = (1, 0, 0)′)

−δ′(ζUn,τ,ω + e1

√n(Fn,X (F−1

n (τ))− τ))

+1

2δ′QU

n,ωδ

where

QUn,ω =

1

n

n∑t=1

ct(ω)c′t(ω)

ζUn,τ,ω = n−1/2n∑

t=1

ct(ω)(τ − IUt,n ≤ τ

)Holger Dette Copula based spectral analysis BU 36 / 36

copula based spectral analysis - sfb649.wiwi.hu-berlin.de

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