e cient goodness-of- t tests in multi-dimensional vine ... filee cient goodness-of- t tests in...

Efficient goodness-of-fit tests in multi-dimensional vinecopula models

Ulf Schepsmeier

Technische Universitat MunchenLehrstuhl fur Mathematische Statistik

January 5, 2014International Workshop on High-Dimensional Dependence and Copulas:

Theory, Modeling, and Applications, Beijing, China

Schepsmeier (TUM) Gof-tests for R-vines January 5, 2014, Beijing 1 / 23

Motivation

Vines: very flexible class of multivariate copulas

We can do estimation and selection for vine copulas

But: Little validation tools for vine copulas are available

So far: likelihood, AIC, BIC, Vuong-test, Clarke-test

Goodness-of-fit tests for bivariate copulas

New: Goodness-of-fit test for vine copulas


1 Motivation

2 Short introduction to R-vines

3 Goodness-of-fit tests for R-vines

4 Power study

5 Application


Copulas

Consider d random variables X = (X1, ...,Xd) with

density function distribution function

marginal fi (xi ), i = 1, ..., d Fi (xi ), i = 1, ..., d

joint f (x1, ..., xd) F (x1, ..., xd)

conditional fi |j(xi |xj), i 6= j Fi |j(xi |xj), i 6= j

Copula

A d-dimensional copula C is a multivariate distribution on [0, 1]d withuniformly distributed marginals.

Copula density function: c(u1, ..., ud) := ∂d

∂u1...∂udC (u1, ..., ud)


Sklar’s theorem

Theorem (Sklar 1959)

F (x1, ..., xd) = C (F1(x1), ...,Fd(xd))

f (x1, ..., xd) = c(F1(x1), ...,Fd(xd))f1(x1)...fd(xd)

for some d-dimensional copula C .

d = 2 :f (x1, x2) = c12(F1(x1),F2(x2))f1(x1)f2(x2)

f2|1(x2|x1) = c12(F1(x1),F2(x2))f2(x2)

cij ;D is bivariate copula pdf associated with (Xi ,Xj) given XD .


Pair-copula construction in 3 dimensions

One possible decomposition of f (x1, x2, x3) is:

f (x1, x2, x3) = f3|12(x3|x1, x2)f2|1(x2|x1)f1(x1)

We can represent the density f asa product of copula densities and marginal densities!

f2|1(x2|x1) = c12(F1(x1),F2(x2))f2(x2)

f3|12(x3|x1, x2) = c13;2(F1|2(x1|x2),F3|2(x3|x2))f3|2(x3|x2)

f3|2(x3|x2) = c23(F2(x2),F3(x3))f3(x3)

⇒ f3|12(x3|x1, x2) = c13;2(F1|2(x1|x2),F3|2(x3|x2))c23(F2(x2),F3(x3))f3(x3)

f (x1, x2, x3) = f3(x3)f2(x2)f1(x1) (marginals)

× c12(F1(x1),F2(x2)) · c23(F2(x2),F3(x3)) (unconditional pairs)

× c13;2(F1|2(x1|x2),F3|2(x3|x2))(conditional pair)


R-vine structure (d = 5)

1 3 4

2 5

1,3 3,4

1,2

1,5

T1

1,2 1,3 3,4

1,5

2,3;1 1,4;3

3,5;1 T2

2,3;1 1,4;3 3,5;12,4;1,3 4,5;1,3

T3

2,4;1,3 4,5;1,32,5;1,3,4

T4

Pair-copulas:

1 c12, c13, c34, c15

2 proximity condition If two

nodes are joined by an edge in tree

i + 1, the corresponding edges in tree

i share a node.

3 c23;1, c14;3, c35;1

4 c24;13, c45;13

5 c25;134


Goodness-of-fit tests for R-vines

H0 : −H = C versus H1 : −H 6= C H0 : C ∈ C0 versus H1 : C /∈ C0

multivariate PITut ∼ C 7→ yt ∼ C⊥

t = 1, . . . , T

test statistic based onempirical copula process using u

(ECP test)

original aggregationy1, . . . ,yn 7→ s1, . . . , sn

st =∑di=1 Γ(yti), t = 1, . . . , n

ordered aggregationyT(1), . . . ,y

T(d) 7→ s1, . . . , sn

st =∑di=1 Γ(yt(i)), t = 1, . . . , n

test statisticbased on H + C(White test)

test statisticbased on C−1H

(IR test)

Γ(·) = Φ−1(·)2(Breymann tests)

Γ(·) = | · −0.5|(Berg tests)

Γ(·) = (· − 0.5)α

α = 2, 4, . . .(Berg2 tests)

test statistic based onempirical copula process using y

(ECP2 test)mCvM mKS

mCvM mKS

AD CvM KS AD CvM KS AD CvM KS

Legend:

AD univariate Anderson-Darling test CvM univariate Cramer-von Mises test KS univariate Kolmogoroov-Smirnov test

mCvM multivariate Cramer-von Mises test mKS multivariate Kolmogorov-Smirnov test


The misspecification test (White 1982)

Let U = (U1, . . . ,Ud)T ∈ [0, 1]d be a d-dimensional random vector withcopula distribution Cθ(u1, . . . , ud). Then

H(θ) = E[∂2θ `(θ|U)

],

C(θ) = E[∂θ`(θ|U)

(∂θ`(θ|U)

)T ]

are the expected Hessian matrix and the expected outer product ofgradient, respectively, and `(θ|u) := ln(cθ(u1, . . . , ud))Under correct model specification (θ = θ0)

−H(θ0) = C(θ0)

⇒ Test problem:

H0 : H(θ0) + C(θ0) = 0 against H1 : H(θ0) + C(θ0) 6= 0,


Test statistic

Proposition (Huang and Prokhorov 2013; Schepsmeier 2013b)

Under correct vine copula specification, given margins and suitableregularity conditions (A1-A10 in White 1982) the information matrix teststatistic is

T = n(d (θ)

)TV−1

θnd (θ), (1)

where V−1

θnis a consistent estimate for the inverse asymptotic covariance

matrix. Further, T is asymptotically χ2p(p+1)/2 distributed.

d (θ) := vech(H(θ) + C(θ)), ∇Dθ := E [∂θkd `(θ|ut)]`=1,...,p(p+1)

2,k=1,...,p

,

Vθ0 := E

[(d(θ0|ut)−∇Dθ0H(θ0)

−1∂θ0`(θ0|ut))(d(θ0|ut)−∇Dθ0H(θ0)

−1∂θ0`(θ0|ut))T]

,

where Vθ0 is the asymptotic covariance matrix of√nd (θn)

vech() vectorizes the lower triangular of a matrix (including the diagonal).


Information matrix ratio test (IR test)

Zhou et al. (2012) followed a different approach: Information matrix ratio:

Ψ(θ) := −H(θ)−1C(θ)

⇒ Test problem:

H0 : Ψ(θ0) = Ip against H1 : Ψ(θ0) 6= Ip,

where Ip is the p-dimensional identity matrix.Ψ(θ0) corresponds to the information ratio (IR) statistic

IR := tr(Ψ(θ0))/p.

⇒ Test problem:

H0 : IR = 1 against H1 : IR 6= 1,


Information matrix ratio test (IR test)

Further letW := (W1, . . . ,Wp(p+1))T = (vech(C(θ)), vech(H(θ)))T ∈ Rp(p+1), thenPresnell and Boos (2004) showed that

Σ−1/2W

√n(W − µW )

d→ Np(p+1)(0p(p+1), Ip(p+1)),

where µW is the mean vector and ΣW is the asymptotic covariance matrixof W .Furthermore let

D(θ) :=

(∂IR

∂Wi

)

i=1,...,p(p+1)

∈ Rp(p+1).

the partial derivatives of IR with respect to the components of W .


Test statistic

Proposition (Presnell and Boos 2004; Schepsmeier 2013a)

Let U ∼ RV (V,B(V), θ(B(V))) satisfy some regularity conditionsSchepsmeier (2013a). Then the IR test statistic

Zn :=IRn − 1

σIR

d→ N(0, 1) as n→∞,

where σIR is the standard error of the IR test statistic, defined as

σ2IR :=

1

nDTΣWD.

Here ΣW is the asymptotic covariance matrix of W , andD := D(θn)|

θnP→θ0

.


Rosenbatt’s transform tests

u11 . . . u1d

......

un1 . . . und

Rosenblatt−−−−−−→

(PIT )

y11 . . . y1d

......

yn1 . . . ynd

Aggregation−−−−−−−−→

st=∑d

j=1 Γ(ytj )

s1

...sn

univariate−−−−−−→

GOF tests

Breymann et al. (2003): Γ(ytj) = Φ−1(ytj)2

Berg and Bakken (2007): Γ(ytj) = |ytj − 0.5| orΓ(ytj) = (ytj − 0.5)α, α = (2, 4, . . .)

Use univariate Cramer-von Mises (CvM), Kolmogorov-Smirnov (KS) orAnderson-Darling (AD) GOF tests in the last step.New:

Probability integral transform (PIT) for Vines (Schepsmeier 2013a)

Extension to vine copulas


Empirical copula process testsLet Cθn(u) be the copula distribution function with estimated parameter θ

and the empirical copula Cn(v) = 1n+1

∑nt=1 1{ut1≤v1,...,utd≤vd}

ECP-mCvM: nω2ECP := n

∫

[0,1]d(

empirical copula process︷︸︸︷Cn(u)− Cθn(u) )2dCn(u) and

ECP-mKS: Dn,ECP := supu∈[0,1]d

|Cn(u)− Cθn(u)|.

Problem: Need double bootstrap procedure to estimate p-valuesVariant: use transformed data y = (y1, . . . , yd) of the PIT approach:

ECP2-mCvM: nω2ECP2 := n

∫

[0,1]d(Cn(y)− C⊥(y))2dCn(y) and

ECP2-mKS: Dn,ECP2 := supy∈[0,1]d

|Cn(y)− C⊥(y)|,

Advantage: Calculation of the independence copula C⊥(y) is easy.


Simulation study - Test models

T1

2

1

3

4 5

1,2 1,3

1,4

4,5

T2

1,2 1,3

1,4

4,5

2, 4; 1 3,4;1

1,5;4

T3

2, 4; 1 1, 5; 4

3, 4; 12, 3; 1, 4 3,

5;1,4

T4

2, 3; 1, 4

3, 5; 1, 4

2,5;1,3,4

Mixed Kendall’s τ in a range from 0.1 to 0.75

R-package: VineCopula

Null hypothesis: M1 = R-Vine

Alternatives: M2 = C-vine, D-vine or multivariate Gauss copula

Most interesting properties are size and power.


Power study - ResultsAll results are based on bootstrapped p-values!

● WhiteIR

Breymann−ADBreymann−CvMBreymann−KS

Berg−ADBerg−CvMBerg−KS

Berg2−ADBerg2−CvMBerg2−KS

ECP−mCvMECP−mKS

ECP2−mCvMECP2−mKS

●

●

●

●

model

pow

er

R−vine D−vine C−vine Gauss

0.05

0.15

0.30

0.45

0.60

0.75

0.90

(a) d = 5, n = 500

●

●

●

●

model

pow

er


0.05

0.15

0.30

0.45

0.60

0.75

0.90

(b) d = 5, n = 750


Power study - ResultsAll results are based on bootstrapped p-values!

● WhiteIR

Breymann−ADBreymann−CvMBreymann−KS

Berg−ADBerg−CvMBerg−KS

Berg2−ADBerg2−CvMBerg2−KS

ECP−mCvMECP−mKS

ECP2−mCvMECP2−mKS

●

●

●

●

model

pow

er


0.05

0.15

0.30

0.45

0.60

0.75

0.90

(c) d = 5, n = 1000

●

● ● ●

model

pow

er


0.05

0.15

0.30

0.45

0.60

0.75

0.90

(d) d = 5, n = 2000


Power study - Results

All proposed GOF tests maintain their given size.

Power > size for White, IR, ECP and ECP2 ⇒ good performancein mean; Breymann undecided

Power < size for Berg and Berg2 ⇒ poor performance

↑ number of observations ⇒ ↑ power

Given sufficient data points the White, IR and ECP2 tests areconsistent

Power seems to depend on the alternative

For ECP, ECP2 or Breymann tests CvM performs better than KS.

Not in the plots

↑ Kendall’s τ ⇒ ↑ power

↑ dimension d ⇒ no influence on the top performing GOF tests(White, IR, ECP, ECP2)


Application: Indices and volatility indices

Daily log returns of 4 major stock indices and their correspondingvolatility indices.• German DAX and VDAX-NEW

• European EuroSTOXX50 and VSTOXX

• US S&P500 and VIX

• Swiss SMI and VSMI

Observed from August, 9th, 2007 until April 30th, 2013 (currentfinancial crisis; 1405 observations).

Time series are filtered using AR(1)-GARCH(1,1) with Student’s tinnovations.

Data set of standardized residuals transformed to [0,1].


Application

DAX

0.0 0.4 0.8

xy

x

y

0.0 0.4 0.8

x

y

x

y

0.0 0.4 0.8

x

y

x

y

0.0 0.4 0.8

0.0

0.4

0.8

x

y

0.0

0.4

0.8 0.01

0.0

1775

51

STOXX50

xy

x

y

x

y

x

y

x

y

x

y

0.01 0.01

0.0

1775

51

SMI

xy

x

y

x

y

x

y

0.0

0.4

0.8

x

y

0.0

0.4

0.8 0.01 0.01 0.01

S&P500

xy

x

y

x

y

x

y

0.01

0.01387755

0.01

0.01387755

0.01

0.01387755

0.01 0.01387755 VDAX−NEW

xy

x

y

0.0

0.4

0.8

x

y

0.0

0.4

0.8

0.01

0.01387755

0.01

0.01387755

0.01 0.01 0.01387755

0.01

0.0

1775

51

VSTOXX

xy

x

y

0.01

0.01387755

0.01

0.01387755

0.01

0.01387755

0.01 0.01387755

0.01

0.01387755

0.01

0.01387755

VSMI

0.0

0.4

0.8

xy

0.0 0.4 0.8

0.0

0.4

0.8

0.01

0.01387755

0.01

0.01387755

0.0 0.4 0.8

0.01

0.0177551

0.01

0.01387755

0.0 0.4 0.8

0.01

0.01387755

0.01

0.01387755

0.0 0.4 0.8

0.01

0.01387755

VIX


Application - Results

log-lik #par AIC BIC

R-vine 7652 33 -15238 -15065C-vine 7585 42 -15086 -14865D-vine 7654 41 -15226 -15011Gauss 7320 28 -14584 -14445

White ECP ECP2 IR

CvM KS CvM KS

R-vine 0.002 0.18 0.98 0.30 0.67 0.75C-vine 0.14 0.51 0.36 0.01 < 0.01 0.74D-vine 0.41 0.82 0.24 0.55 0.67 0.52Gauss < 0.01 0.60 0.28 < 0.01 < 0.01 < 0.01

Tabelle: Bootstrapped p-values of the White, ECP, ECP2 and IR goodness-of-fittest for the 4 considered (vine) copula models


Summary/Outlook

Summary

New GOF tests for vine copulas are introduced

For comparison further GOF tests were extended from the bivariatecase to the vine copula case

Information matrix based GOF tests perform very well

Empirical copula process based GOF tests work well and are fast

Outlook

Extension of further copula GOF tests to the vine copula case, e.g.tests based on Kendall’s process, likelihood ratio based tests,...

Hybrid approach of Zhang et al. (2013)

GOF tests for classes of vines (here the vine structure was fixed)

Thank you for your attention!


Berg, D. and H. Bakken (2007).A copula goodness-of-fit approach based on the conditional probability integral

transformation.http://www.danielberg.no/publications/Btest.pdf.

Breymann, W., A. Dias, and P. Embrechts (2003).Dependence structures for multivariate high-frequency data in finance.Quantitative Finance 3(1), 1–14.

Huang, W. and A. Prokhorov (2013).A goodness-of-fit test for copulas.Economic Reviews to appear.

Presnell, B. and D. D. Boos (2004).The ios test for model misspecification.Journal of the American Statistical Association 99(465), 216–227.

Schepsmeier, U. (2013a).Efficient goodness-of-fit tests in multi-dimensional vine copula models.preprint, available at http://arxiv.org/abs/1309.5808.

Schepsmeier, U. (2013b).A goodness-of-fit test for regular vine copula models.preprint, available at: http://arxiv.org/abs/1306.0818.

Sklar, M. (1959).Fonctions de repartition a n dimensions et leurs marges.Publ. Inst. Statist. Univ. Paris 8, 229–231.


White, H. (1982).Maximum likelihood estimation of misspecified models.Econometrica 50, 1–26.

Zhang, S., O. Okhrin, Q. M. Zhou, and P. X.-K. Song (2013).Goodness-of-fit Test For Specification of Semiparametric Copula Dependence Models.personal communication

http://sfb649.wiwi.hu-berlin.de/papers/pdf/SFB649DP2013-041.pdf.

Zhou, Q. M., P. X.-K. Song, and M. E. Thompson (2012).Information ratio test for model misspecification in quasi-likelihood inference.Journal of the American Statistical Association 107(497), 205–213.


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