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Bootstrap methods for copulas, with applications tostock index data
László VargaAndrás Zempléni
László Sajtos
Eötvös Loránd University (ELTE)Faculty of Science
Institute of MathematicsDepartment of Probability Theory and Statistics
Budapest, Hungary
Applied Stochastic Models and Data Analysis (ASMDA)7 - 10 June 2011
László Varga (ELTE) Bootstrap methods for copulas, with applications to stock index dataASMDA 2011 1 / 24
![Page 2: Bootstrap methods for copulas, with applications to stock ...vargal4.elte.hu/Rome.pdf · László Varga András Zempléni László Sajtos Eötvös Loránd University (ELTE) Faculty](https://reader033.vdocument.in/reader033/viewer/2022050300/5f6943e0abfd0b231270a744/html5/thumbnails/2.jpg)
Outline
1 Copulas
2 Bootstrap methods for Goodness-of-fit testsGoF tests in generalEmpirical copula process, limit distributionCvM tests based on Kendall’s process
3 Analysis of stock index data
4 Final remarks
László Varga (ELTE) Bootstrap methods for copulas, with applications to stock index dataASMDA 2011 2 / 24
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Copulas
C is a copula, if it is a d-dimensional random vector with marginals∼ Unif[0,1]Existence (Sklar′s Theorem): to any d-dimensional randomvariable X with c.d.f. F and marginals Fi (i=1,...,d) there exists acopula C : F (x1, ..., xd ) = C(F1(x1), ...,Fd (xd ))
Uniqueness: if Fi are continuous (i=1,...,d)Separation of the marginal model and the dependence
Archimedean copulasCopula generator function: φ : [0,1]→ [0,∞]
where φ is continuous, strictly decreasing and φ(1)=0
d-variate Archimedean copula: Cφ(u) = φ−1(
d∑i=1
φ(ui)
)Gumbel: φ(u) = (−log(u))θ where θ ∈ [1,∞]
Clayton: φ(u) = u−θ − 1 where θ > 0László Varga (ELTE) Bootstrap methods for copulas, with applications to stock index dataASMDA 2011 3 / 24
![Page 4: Bootstrap methods for copulas, with applications to stock ...vargal4.elte.hu/Rome.pdf · László Varga András Zempléni László Sajtos Eötvös Loránd University (ELTE) Faculty](https://reader033.vdocument.in/reader033/viewer/2022050300/5f6943e0abfd0b231270a744/html5/thumbnails/4.jpg)
Copulas
C is a copula, if it is a d-dimensional random vector with marginals∼ Unif[0,1]Existence (Sklar′s Theorem): to any d-dimensional randomvariable X with c.d.f. F and marginals Fi (i=1,...,d) there exists acopula C : F (x1, ..., xd ) = C(F1(x1), ...,Fd (xd ))
Uniqueness: if Fi are continuous (i=1,...,d)Separation of the marginal model and the dependence
Archimedean copulasCopula generator function: φ : [0,1]→ [0,∞]
where φ is continuous, strictly decreasing and φ(1)=0
d-variate Archimedean copula: Cφ(u) = φ−1(
d∑i=1
φ(ui)
)Gumbel: φ(u) = (−log(u))θ where θ ∈ [1,∞]
Clayton: φ(u) = u−θ − 1 where θ > 0László Varga (ELTE) Bootstrap methods for copulas, with applications to stock index dataASMDA 2011 3 / 24
![Page 5: Bootstrap methods for copulas, with applications to stock ...vargal4.elte.hu/Rome.pdf · László Varga András Zempléni László Sajtos Eötvös Loránd University (ELTE) Faculty](https://reader033.vdocument.in/reader033/viewer/2022050300/5f6943e0abfd0b231270a744/html5/thumbnails/5.jpg)
Copulas
C is a copula, if it is a d-dimensional random vector with marginals∼ Unif[0,1]Existence (Sklar′s Theorem): to any d-dimensional randomvariable X with c.d.f. F and marginals Fi (i=1,...,d) there exists acopula C : F (x1, ..., xd ) = C(F1(x1), ...,Fd (xd ))
Uniqueness: if Fi are continuous (i=1,...,d)Separation of the marginal model and the dependence
Archimedean copulasCopula generator function: φ : [0,1]→ [0,∞]
where φ is continuous, strictly decreasing and φ(1)=0
d-variate Archimedean copula: Cφ(u) = φ−1(
d∑i=1
φ(ui)
)Gumbel: φ(u) = (−log(u))θ where θ ∈ [1,∞]
Clayton: φ(u) = u−θ − 1 where θ > 0László Varga (ELTE) Bootstrap methods for copulas, with applications to stock index dataASMDA 2011 3 / 24
![Page 6: Bootstrap methods for copulas, with applications to stock ...vargal4.elte.hu/Rome.pdf · László Varga András Zempléni László Sajtos Eötvös Loránd University (ELTE) Faculty](https://reader033.vdocument.in/reader033/viewer/2022050300/5f6943e0abfd0b231270a744/html5/thumbnails/6.jpg)
Copulas
C is a copula, if it is a d-dimensional random vector with marginals∼ Unif[0,1]Existence (Sklar′s Theorem): to any d-dimensional randomvariable X with c.d.f. F and marginals Fi (i=1,...,d) there exists acopula C : F (x1, ..., xd ) = C(F1(x1), ...,Fd (xd ))
Uniqueness: if Fi are continuous (i=1,...,d)Separation of the marginal model and the dependence
Archimedean copulasCopula generator function: φ : [0,1]→ [0,∞]
where φ is continuous, strictly decreasing and φ(1)=0
d-variate Archimedean copula: Cφ(u) = φ−1(
d∑i=1
φ(ui)
)Gumbel: φ(u) = (−log(u))θ where θ ∈ [1,∞]
Clayton: φ(u) = u−θ − 1 where θ > 0László Varga (ELTE) Bootstrap methods for copulas, with applications to stock index dataASMDA 2011 3 / 24
![Page 7: Bootstrap methods for copulas, with applications to stock ...vargal4.elte.hu/Rome.pdf · László Varga András Zempléni László Sajtos Eötvös Loránd University (ELTE) Faculty](https://reader033.vdocument.in/reader033/viewer/2022050300/5f6943e0abfd0b231270a744/html5/thumbnails/7.jpg)
Copulas - examples
László Varga (ELTE) Bootstrap methods for copulas, with applications to stock index dataASMDA 2011 4 / 24
![Page 8: Bootstrap methods for copulas, with applications to stock ...vargal4.elte.hu/Rome.pdf · László Varga András Zempléni László Sajtos Eötvös Loránd University (ELTE) Faculty](https://reader033.vdocument.in/reader033/viewer/2022050300/5f6943e0abfd0b231270a744/html5/thumbnails/8.jpg)
Outline
1 Copulas
2 Bootstrap methods for Goodness-of-fit testsGoF tests in generalEmpirical copula process, limit distributionCvM tests based on Kendall’s process
3 Analysis of stock index data
4 Final remarks
László Varga (ELTE) Bootstrap methods for copulas, with applications to stock index dataASMDA 2011 5 / 24
![Page 9: Bootstrap methods for copulas, with applications to stock ...vargal4.elte.hu/Rome.pdf · László Varga András Zempléni László Sajtos Eötvös Loránd University (ELTE) Faculty](https://reader033.vdocument.in/reader033/viewer/2022050300/5f6943e0abfd0b231270a744/html5/thumbnails/9.jpg)
Goodness-of-fit (GoF) tests in general
Let X1, . . . ,Xn be random vectors, with c.d.f. F : Rd → RWe want to test the following hypothesis:
H0 : F ∈ F = {Fθ : θ ∈ Θ} where Θ ⊂ Rp is an open set
Estimate of the parameter: θn = Tn(X1, . . . ,Xn)
Cramér-von Mises statistics based on the empirical process or itsfunctional
GFn =√
n(Fn − Fθn )Sn = φ(GF
n )
Problem: the asymptotic null distribution of Sn is rarely known, butif known, usually it is rather complicated
László Varga (ELTE) Bootstrap methods for copulas, with applications to stock index dataASMDA 2011 6 / 24
![Page 10: Bootstrap methods for copulas, with applications to stock ...vargal4.elte.hu/Rome.pdf · László Varga András Zempléni László Sajtos Eötvös Loránd University (ELTE) Faculty](https://reader033.vdocument.in/reader033/viewer/2022050300/5f6943e0abfd0b231270a744/html5/thumbnails/10.jpg)
Goodness-of-fit (GoF) tests in general
Let X1, . . . ,Xn be random vectors, with c.d.f. F : Rd → RWe want to test the following hypothesis:
H0 : F ∈ F = {Fθ : θ ∈ Θ} where Θ ⊂ Rp is an open set
Estimate of the parameter: θn = Tn(X1, . . . ,Xn)
Cramér-von Mises statistics based on the empirical process or itsfunctional
GFn =√
n(Fn − Fθn )Sn = φ(GF
n )
Problem: the asymptotic null distribution of Sn is rarely known, butif known, usually it is rather complicated
László Varga (ELTE) Bootstrap methods for copulas, with applications to stock index dataASMDA 2011 6 / 24
![Page 11: Bootstrap methods for copulas, with applications to stock ...vargal4.elte.hu/Rome.pdf · László Varga András Zempléni László Sajtos Eötvös Loránd University (ELTE) Faculty](https://reader033.vdocument.in/reader033/viewer/2022050300/5f6943e0abfd0b231270a744/html5/thumbnails/11.jpg)
Goodness-of-fit (GoF) tests in general
Solution: parametric bootstrap - generation of N bootstraprepetitions: (k ∈ 1, . . . ,N)
1 Estimate of the parameter: θn2 Generate n independent observations from distribution Fθn :
X ∗1,k , . . . ,X∗n,k
3 Estimate the parameter and the empirical c.d.f. from the sample:θ∗n,k = Tn(X ∗1,k , . . . ,X
∗n,k )
F ∗n,k (x) = 1n
n∑i=1
I(X ∗i,k ≤ x)
4 GF∗
n,k =√
n(F ∗n,k − Fθ∗n,k )
S∗n,k = φ(GF∗
n,k )5 Calculate the test statistics and the critical values
Instead of Fθ an abstract quantity Aθ depending on Pθ
Let’s suppose that Φn =√
n(θn − θ) Φ andAn =
√n(An − A) A
For the convergence of the bootstrap alteregos we need furtherregularity assumptions.László Varga (ELTE) Bootstrap methods for copulas, with applications to stock index dataASMDA 2011 7 / 24
![Page 12: Bootstrap methods for copulas, with applications to stock ...vargal4.elte.hu/Rome.pdf · László Varga András Zempléni László Sajtos Eötvös Loránd University (ELTE) Faculty](https://reader033.vdocument.in/reader033/viewer/2022050300/5f6943e0abfd0b231270a744/html5/thumbnails/12.jpg)
Goodness-of-fit (GoF) tests in general
Solution: parametric bootstrap - generation of N bootstraprepetitions: (k ∈ 1, . . . ,N)
1 Estimate of the parameter: θn2 Generate n independent observations from distribution Fθn :
X ∗1,k , . . . ,X∗n,k
3 Estimate the parameter and the empirical c.d.f. from the sample:θ∗n,k = Tn(X ∗1,k , . . . ,X
∗n,k )
F ∗n,k (x) = 1n
n∑i=1
I(X ∗i,k ≤ x)
4 GF∗
n,k =√
n(F ∗n,k − Fθ∗n,k )
S∗n,k = φ(GF∗
n,k )5 Calculate the test statistics and the critical values
Instead of Fθ an abstract quantity Aθ depending on Pθ
Let’s suppose that Φn =√
n(θn − θ) Φ andAn =
√n(An − A) A
For the convergence of the bootstrap alteregos we need furtherregularity assumptions.László Varga (ELTE) Bootstrap methods for copulas, with applications to stock index dataASMDA 2011 7 / 24
![Page 13: Bootstrap methods for copulas, with applications to stock ...vargal4.elte.hu/Rome.pdf · László Varga András Zempléni László Sajtos Eötvös Loránd University (ELTE) Faculty](https://reader033.vdocument.in/reader033/viewer/2022050300/5f6943e0abfd0b231270a744/html5/thumbnails/13.jpg)
GoF tests - regularity assumptions
Def.: A P = {θ ∈ Θ} is said to belong to the class S(λ) (given λreference measure), if
1 The measure Pθ is abs. continuous with respect to λ for all θ ∈ Θ.2 The density pθ = dPθ/dλ admits first and second order derivatives
with respect to θ ∈ Θ. Gradient vector: pθ, Hessian matrix: pθ3 For arbitrary u ∈ Rd and every θ ∈ Θ, the mappingsθ → pθ(u)/pθ(u) and θ → pθ(u)/pθ(u) are continuous at θ0, Pθ0 -a.s.
4 For every θ0 ∈ Θ, there exist a neighborhood N of θ0 and aλ-integrable function h : Rd → R such that for all u ∈ Rd ,supθ∈N ||pθ(u)|| = h(u).
5 For every θ0 ∈ Θ, there exist a neighborhood N of θ0 andPθ0 -integrable functions h1,h2 : Rd → R such that for every u ∈ Rd ,supθ∈N|| pθ(u)pθ(u)
||2 ≤ h1(u) and supθ∈N|| pθ(u)pθ(u)
|| ≤ h2(u).
Coroll.: Let P ∈ S(λ), if P ∈ P and n→∞ then
WP,n := n−1/2n∑
i=1
pT (Xi )p(Xi )
WP ∼ N(0, IP)
where IP : Fisher information matrixLászló Varga (ELTE) Bootstrap methods for copulas, with applications to stock index dataASMDA 2011 8 / 24
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GoF tests - regularity assumptions
Def.: A P = {θ ∈ Θ} is said to belong to the class S(λ) (given λreference measure), if
1 The measure Pθ is abs. continuous with respect to λ for all θ ∈ Θ.2 The density pθ = dPθ/dλ admits first and second order derivatives
with respect to θ ∈ Θ. Gradient vector: pθ, Hessian matrix: pθ3 For arbitrary u ∈ Rd and every θ ∈ Θ, the mappingsθ → pθ(u)/pθ(u) and θ → pθ(u)/pθ(u) are continuous at θ0, Pθ0 -a.s.
4 For every θ0 ∈ Θ, there exist a neighborhood N of θ0 and aλ-integrable function h : Rd → R such that for all u ∈ Rd ,supθ∈N ||pθ(u)|| = h(u).
5 For every θ0 ∈ Θ, there exist a neighborhood N of θ0 andPθ0 -integrable functions h1,h2 : Rd → R such that for every u ∈ Rd ,supθ∈N|| pθ(u)pθ(u)
||2 ≤ h1(u) and supθ∈N|| pθ(u)pθ(u)
|| ≤ h2(u).
Coroll.: Let P ∈ S(λ), if P ∈ P and n→∞ then
WP,n := n−1/2n∑
i=1
pT (Xi )p(Xi )
WP ∼ N(0, IP)
where IP : Fisher information matrixLászló Varga (ELTE) Bootstrap methods for copulas, with applications to stock index dataASMDA 2011 8 / 24
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Convergence of bootstrap alteregos
Def.: Let U1, . . . ,Un be a random sample from P = Pθ0 .A sequence An is said to be Pθ0-regular for A = Aθ0 , if the process(An,WP,n) (A,WP) and A(t) = E(A(t)WT
P) for every t .Def.: The An sequence is said to be P-regular for A(A = {Aθ0 : θ0 ∈ Θ}) if it is Pθ0-regular at all A = Aθ0 .Theorem: Assume that
P ∈ S(λ)if P ∈ P and n→∞, (An,Φn) (A,Φ), where the limit is acentered Gaussian process(An, θn) is P-regular for A×Θ
Then (GAn ,GA∗
n ) (GA,GA∗)
where GA∗is an independent copy of GA.
Prop.: The assumptions hold for the empirical copula process(Genest&Rémillard (2005))
László Varga (ELTE) Bootstrap methods for copulas, with applications to stock index dataASMDA 2011 9 / 24
![Page 16: Bootstrap methods for copulas, with applications to stock ...vargal4.elte.hu/Rome.pdf · László Varga András Zempléni László Sajtos Eötvös Loránd University (ELTE) Faculty](https://reader033.vdocument.in/reader033/viewer/2022050300/5f6943e0abfd0b231270a744/html5/thumbnails/16.jpg)
Convergence of bootstrap alteregos
Def.: Let U1, . . . ,Un be a random sample from P = Pθ0 .A sequence An is said to be Pθ0-regular for A = Aθ0 , if the process(An,WP,n) (A,WP) and A(t) = E(A(t)WT
P) for every t .Def.: The An sequence is said to be P-regular for A(A = {Aθ0 : θ0 ∈ Θ}) if it is Pθ0-regular at all A = Aθ0 .Theorem: Assume that
P ∈ S(λ)if P ∈ P and n→∞, (An,Φn) (A,Φ), where the limit is acentered Gaussian process(An, θn) is P-regular for A×Θ
Then (GAn ,GA∗
n ) (GA,GA∗)
where GA∗is an independent copy of GA.
Prop.: The assumptions hold for the empirical copula process(Genest&Rémillard (2005))
László Varga (ELTE) Bootstrap methods for copulas, with applications to stock index dataASMDA 2011 9 / 24
![Page 17: Bootstrap methods for copulas, with applications to stock ...vargal4.elte.hu/Rome.pdf · László Varga András Zempléni László Sajtos Eötvös Loránd University (ELTE) Faculty](https://reader033.vdocument.in/reader033/viewer/2022050300/5f6943e0abfd0b231270a744/html5/thumbnails/17.jpg)
Convergence of bootstrap alteregos
Def.: Let U1, . . . ,Un be a random sample from P = Pθ0 .A sequence An is said to be Pθ0-regular for A = Aθ0 , if the process(An,WP,n) (A,WP) and A(t) = E(A(t)WT
P) for every t .Def.: The An sequence is said to be P-regular for A(A = {Aθ0 : θ0 ∈ Θ}) if it is Pθ0-regular at all A = Aθ0 .Theorem: Assume that
P ∈ S(λ)if P ∈ P and n→∞, (An,Φn) (A,Φ), where the limit is acentered Gaussian process(An, θn) is P-regular for A×Θ
Then (GAn ,GA∗
n ) (GA,GA∗)
where GA∗is an independent copy of GA.
Prop.: The assumptions hold for the empirical copula process(Genest&Rémillard (2005))
László Varga (ELTE) Bootstrap methods for copulas, with applications to stock index dataASMDA 2011 9 / 24
![Page 18: Bootstrap methods for copulas, with applications to stock ...vargal4.elte.hu/Rome.pdf · László Varga András Zempléni László Sajtos Eötvös Loránd University (ELTE) Faculty](https://reader033.vdocument.in/reader033/viewer/2022050300/5f6943e0abfd0b231270a744/html5/thumbnails/18.jpg)
Convergence of bootstrap alteregos
Def.: Let U1, . . . ,Un be a random sample from P = Pθ0 .A sequence An is said to be Pθ0-regular for A = Aθ0 , if the process(An,WP,n) (A,WP) and A(t) = E(A(t)WT
P) for every t .Def.: The An sequence is said to be P-regular for A(A = {Aθ0 : θ0 ∈ Θ}) if it is Pθ0-regular at all A = Aθ0 .Theorem: Assume that
P ∈ S(λ)if P ∈ P and n→∞, (An,Φn) (A,Φ), where the limit is acentered Gaussian process(An, θn) is P-regular for A×Θ
Then (GAn ,GA∗
n ) (GA,GA∗)
where GA∗is an independent copy of GA.
Prop.: The assumptions hold for the empirical copula process(Genest&Rémillard (2005))
László Varga (ELTE) Bootstrap methods for copulas, with applications to stock index dataASMDA 2011 9 / 24
![Page 19: Bootstrap methods for copulas, with applications to stock ...vargal4.elte.hu/Rome.pdf · László Varga András Zempléni László Sajtos Eötvös Loránd University (ELTE) Faculty](https://reader033.vdocument.in/reader033/viewer/2022050300/5f6943e0abfd0b231270a744/html5/thumbnails/19.jpg)
Outline
1 Copulas
2 Bootstrap methods for Goodness-of-fit testsGoF tests in generalEmpirical copula process, limit distributionCvM tests based on Kendall’s process
3 Analysis of stock index data
4 Final remarks
László Varga (ELTE) Bootstrap methods for copulas, with applications to stock index dataASMDA 2011 10 / 24
![Page 20: Bootstrap methods for copulas, with applications to stock ...vargal4.elte.hu/Rome.pdf · László Varga András Zempléni László Sajtos Eötvös Loránd University (ELTE) Faculty](https://reader033.vdocument.in/reader033/viewer/2022050300/5f6943e0abfd0b231270a744/html5/thumbnails/20.jpg)
Limit distribution of the empirical copula process
Let X1, . . . ,Xn be i.i.d. bivariate random vectors with continuousc.d.f. F, marginal distribution functions F1,F2 and copula C:
C(u1,u2) = F (F−1 (u1),F−2 (u2)) (Sklar’s theorem)Empirical copula:Cn(u1,u2) = Fn(F−n1(u1),F−n2(u2)),
where Fn(x) = Fn(x1, x2) = 1n
n∑i=1
I{Xi1 ≤ x1,Xi2 ≤ x2},
Fnp(xp) = 1n
n∑i=1
I{Xip ≤ xp}, p = 1,2.
denote weak convergence in space l∞([0,1]2) of all uniformlybounded functions on [0,1]2
László Varga (ELTE) Bootstrap methods for copulas, with applications to stock index dataASMDA 2011 11 / 24
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The limit distribution of the empirical copula processand the pdm method
Theorem: If the copula C possesses continuous partial derivatives∂1C, ∂2C on [0,1]2 then the empirical copula process convergesweakly:
αn :=√
n(Cn − C) GC
where GC(u1,u2) =BC(u1,u2)− ∂1C(u1,u2)BC(u1,1)− ∂2C(u1,u2)BC(1,u2)where BC is a centered Gaussian field with the following covariancestructure:Cov(BC(u1,u2),BC(v1, v2)) = C(u1∧v1,u2∧v2)−C(u1,u2)C(v1, v2).
The pdm method (Bücher&Dette (2010)):Let Z1, . . . ,Zn be centered i.i.d. random variables; D2Zi = 1;
independent of X1, . . . ,Xn;∞∫0
√P(|Z1| > x)dx <∞.
C∗n (u) = 1n
n∑i=1
ZiI{Fn1(Xi1) ≤ u1,Fn2(Xi2) ≤ u2}
László Varga (ELTE) Bootstrap methods for copulas, with applications to stock index dataASMDA 2011 12 / 24
![Page 22: Bootstrap methods for copulas, with applications to stock ...vargal4.elte.hu/Rome.pdf · László Varga András Zempléni László Sajtos Eötvös Loránd University (ELTE) Faculty](https://reader033.vdocument.in/reader033/viewer/2022050300/5f6943e0abfd0b231270a744/html5/thumbnails/22.jpg)
The limit distribution of the empirical copula processand the pdm method
Theorem: If the copula C possesses continuous partial derivatives∂1C, ∂2C on [0,1]2 then the empirical copula process convergesweakly:
αn :=√
n(Cn − C) GC
where GC(u1,u2) =BC(u1,u2)− ∂1C(u1,u2)BC(u1,1)− ∂2C(u1,u2)BC(1,u2)where BC is a centered Gaussian field with the following covariancestructure:Cov(BC(u1,u2),BC(v1, v2)) = C(u1∧v1,u2∧v2)−C(u1,u2)C(v1, v2).
The pdm method (Bücher&Dette (2010)):Let Z1, . . . ,Zn be centered i.i.d. random variables; D2Zi = 1;
independent of X1, . . . ,Xn;∞∫0
√P(|Z1| > x)dx <∞.
C∗n (u) = 1n
n∑i=1
ZiI{Fn1(Xi1) ≤ u1,Fn2(Xi2) ≤ u2}
László Varga (ELTE) Bootstrap methods for copulas, with applications to stock index dataASMDA 2011 12 / 24
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The pdm method
Approximate the partial derivatives:∂1C(u, v) := Cn(u+h,v)−Cn(u−h,v)
2h
∂2C(u, v) := Cn(u,v+h)−Cn(u,v−h)2h
where h = n−1/2 → 0Estimate of BC : βn :=
√n(C∗n − Z nCn)
Estimate of GC : αpdmn (u1,u2) :=
βn(u1,u2)− ∂1C(u1,u2)βn(u1,1)− ∂2C(u1,u2)βn(1,u2)
Theorem: Using the foregoing notations(αn, α
pdmn ) (GC ,G′C) in l∞([0,1]2)2
where G′C is an independent copy of GC .
Cramér-von Mises teststatistics:Ln =
∫[0,1]2 α
2n(x)dx
Lpdmn =
∫[0,1]2(αpdm
n )2(x)dx
László Varga (ELTE) Bootstrap methods for copulas, with applications to stock index dataASMDA 2011 13 / 24
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The pdm method
Approximate the partial derivatives:∂1C(u, v) := Cn(u+h,v)−Cn(u−h,v)
2h
∂2C(u, v) := Cn(u,v+h)−Cn(u,v−h)2h
where h = n−1/2 → 0Estimate of BC : βn :=
√n(C∗n − Z nCn)
Estimate of GC : αpdmn (u1,u2) :=
βn(u1,u2)− ∂1C(u1,u2)βn(u1,1)− ∂2C(u1,u2)βn(1,u2)
Theorem: Using the foregoing notations(αn, α
pdmn ) (GC ,G′C) in l∞([0,1]2)2
where G′C is an independent copy of GC .
Cramér-von Mises teststatistics:Ln =
∫[0,1]2 α
2n(x)dx
Lpdmn =
∫[0,1]2(αpdm
n )2(x)dx
László Varga (ELTE) Bootstrap methods for copulas, with applications to stock index dataASMDA 2011 13 / 24
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The pdm method
Approximate the partial derivatives:∂1C(u, v) := Cn(u+h,v)−Cn(u−h,v)
2h
∂2C(u, v) := Cn(u,v+h)−Cn(u,v−h)2h
where h = n−1/2 → 0Estimate of BC : βn :=
√n(C∗n − Z nCn)
Estimate of GC : αpdmn (u1,u2) :=
βn(u1,u2)− ∂1C(u1,u2)βn(u1,1)− ∂2C(u1,u2)βn(1,u2)
Theorem: Using the foregoing notations(αn, α
pdmn ) (GC ,G′C) in l∞([0,1]2)2
where G′C is an independent copy of GC .
Cramér-von Mises teststatistics:Ln =
∫[0,1]2 α
2n(x)dx
Lpdmn =
∫[0,1]2(αpdm
n )2(x)dx
László Varga (ELTE) Bootstrap methods for copulas, with applications to stock index dataASMDA 2011 13 / 24
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Outline
1 Copulas
2 Bootstrap methods for Goodness-of-fit testsGoF tests in generalEmpirical copula process, limit distributionCvM tests based on Kendall’s process
3 Analysis of stock index data
4 Final remarks
László Varga (ELTE) Bootstrap methods for copulas, with applications to stock index dataASMDA 2011 14 / 24
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Kendall’s process
Kendall’s transform (K function):K (θ, t) = P(Cθ(F1(X1), ...,F1(X1)) ≤ t)
advantage: one-dimensional
For Archim. copulas K (θ, t) = t +d−1∑i=1
(−1)i
i!
[φθ(t)i]fi(θ, t)
where fi(θ, t) = d i
dx i φ−1θ (x)
∣∣∣x=φθ(t)
Kendall’s process: κn(t) =√
n(K (θn, t)− Kn(t))where Kn(t): empirical version of K (θ, t)
Cramér von Mises statistics:Sn =
∫ 10 (κn(t))2Φ(t)dt
Focused Regions Φ(t)Global 1Upper Tail 1− K (θn, t)Lower Tail K (θn, t)Lower and Upper Tail K (θn, t)(1− K (θn, t))
László Varga (ELTE) Bootstrap methods for copulas, with applications to stock index dataASMDA 2011 15 / 24
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Kendall’s process
Kendall’s transform (K function):K (θ, t) = P(Cθ(F1(X1), ...,F1(X1)) ≤ t)
advantage: one-dimensional
For Archim. copulas K (θ, t) = t +d−1∑i=1
(−1)i
i!
[φθ(t)i]fi(θ, t)
where fi(θ, t) = d i
dx i φ−1θ (x)
∣∣∣x=φθ(t)
Kendall’s process: κn(t) =√
n(K (θn, t)− Kn(t))where Kn(t): empirical version of K (θ, t)
Cramér von Mises statistics:Sn =
∫ 10 (κn(t))2Φ(t)dt
Focused Regions Φ(t)Global 1Upper Tail 1− K (θn, t)Lower Tail K (θn, t)Lower and Upper Tail K (θn, t)(1− K (θn, t))
László Varga (ELTE) Bootstrap methods for copulas, with applications to stock index dataASMDA 2011 15 / 24
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Kendall’s process
Kendall’s transform (K function):K (θ, t) = P(Cθ(F1(X1), ...,F1(X1)) ≤ t)
advantage: one-dimensional
For Archim. copulas K (θ, t) = t +d−1∑i=1
(−1)i
i!
[φθ(t)i]fi(θ, t)
where fi(θ, t) = d i
dx i φ−1θ (x)
∣∣∣x=φθ(t)
Kendall’s process: κn(t) =√
n(K (θn, t)− Kn(t))where Kn(t): empirical version of K (θ, t)
Cramér von Mises statistics:Sn =
∫ 10 (κn(t))2Φ(t)dt
Focused Regions Φ(t)Global 1Upper Tail 1− K (θn, t)Lower Tail K (θn, t)Lower and Upper Tail K (θn, t)(1− K (θn, t))
László Varga (ELTE) Bootstrap methods for copulas, with applications to stock index dataASMDA 2011 15 / 24
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Analysis of stock index data
2 stock index daily maxima for 9 years (2002-2011):NASDAQ Composite & Dow Jones Industrial Average
Method: copula fitting and goodness-of-fit testingAnalysed copulas: Gumbel, ClaytonTests:
1 Cramér-von Mises teststatistics based on Kendall’s process2 Cramér-von Mises teststatistics based on empirical copula process
estimated with parametric bootstrap3 Cramér-von Mises teststatistics based on empirical copula process
estimated using αn4 Testing correlation structure with multivariate Bartlett test
László Varga (ELTE) Bootstrap methods for copulas, with applications to stock index dataASMDA 2011 16 / 24
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Analysis of stock index data
2 stock index daily maxima for 9 years (2002-2011):NASDAQ Composite & Dow Jones Industrial Average
Method: copula fitting and goodness-of-fit testingAnalysed copulas: Gumbel, ClaytonTests:
1 Cramér-von Mises teststatistics based on Kendall’s process2 Cramér-von Mises teststatistics based on empirical copula process
estimated with parametric bootstrap3 Cramér-von Mises teststatistics based on empirical copula process
estimated using αn4 Testing correlation structure with multivariate Bartlett test
László Varga (ELTE) Bootstrap methods for copulas, with applications to stock index dataASMDA 2011 16 / 24
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Stock index data and fitted copulas
László Varga (ELTE) Bootstrap methods for copulas, with applications to stock index dataASMDA 2011 17 / 24
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CvM test based on Kendall’s process
Gumbel is betterFor some periods Gumbel
can’t be rejected
László Varga (ELTE) Bootstrap methods for copulas, with applications to stock index dataASMDA 2011 18 / 24
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CvM test based on empirical copula processestimated with parametric bootstrap and with αn
mostly rejected always rejected
László Varga (ELTE) Bootstrap methods for copulas, with applications to stock index dataASMDA 2011 19 / 24
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Testing correlation structure – multivariate Bartlett test
Multivariate Bartlett test: H0 : Σαn = Σ
Critical values and teststatistics with simulationAnalysed pairs: (0.1,0.11); (0.1,0.9); (0.89,0.9)Results (Gumbel copula):
Years (0.1,0.11) (0.1,0.9) (0.89,0.9)2010-2011 0.1263 0.0421 0.10032009-2011 0.0021 0.0000 0.00402008-2011 0.0001 0.0000 0.00262007-2011 0.0017 0.0009 0.00112006-2011 0.0091 0.0057 0.01532005-2011 0.0007 0.0002 0.00092004-2011 0.0001 0.0000 0.00002003-2011 0.0000 0.0000 0.00002002-2011 0.0000 0.0000 0.0000
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Testing correlation structure – multivariate Bartlett test
Multivariate Bartlett test: H0 : Σαn = Σ
Critical values and teststatistics with simulationAnalysed pairs: (0.1,0.11); (0.1,0.9); (0.89,0.9)Results (Gumbel copula):
Years (0.1,0.11) (0.1,0.9) (0.89,0.9)2010-2011 0.1263 0.0421 0.10032009-2011 0.0021 0.0000 0.00402008-2011 0.0001 0.0000 0.00262007-2011 0.0017 0.0009 0.00112006-2011 0.0091 0.0057 0.01532005-2011 0.0007 0.0002 0.00092004-2011 0.0001 0.0000 0.00002003-2011 0.0000 0.0000 0.00002002-2011 0.0000 0.0000 0.0000
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Final remarks
ConclusionsK process based tests are the fastest but not the bestThe null hypotesis was not rejected between 2010 and 2011
Future workCvM test based on empirical copula process estimated with pdmmethodWorking with Gaussian and t-copulas (problem: estimating thepartial derivatives)
AcknowledgementAndrás ZempléniLászló SajtosDoctoral School of Mathematics of ELTE for supporting myparticipation at ASMDA2011 ConferenceAndrás Zempléni’s work was supported by the European UnionSocial Fund (Grant Agreement No.TÁMOP4.2.1/B-09/1/KMR-2010-0003)
László Varga (ELTE) Bootstrap methods for copulas, with applications to stock index dataASMDA 2011 21 / 24
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Final remarks
ConclusionsK process based tests are the fastest but not the bestThe null hypotesis was not rejected between 2010 and 2011
Future workCvM test based on empirical copula process estimated with pdmmethodWorking with Gaussian and t-copulas (problem: estimating thepartial derivatives)
AcknowledgementAndrás ZempléniLászló SajtosDoctoral School of Mathematics of ELTE for supporting myparticipation at ASMDA2011 ConferenceAndrás Zempléni’s work was supported by the European UnionSocial Fund (Grant Agreement No.TÁMOP4.2.1/B-09/1/KMR-2010-0003)
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References I
S.N. Lahiri:Resampling Methods for Dependent Data.Springer, 2003.
P. Rakonczai, L. Varga, A. Zempléni:Copula Fitting to Autocorrelated Data, with Applications to WindSpeed Modelling.Working paper, november 11., 2010.
P. Rakonczai, A. Zempléni:Copulas and goodness of fit tests. Recent advances in stochasticmodeling and data analysis.World Scientific, pp. 198-206, 2007.
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References II
A. Bücher, H. Dette:A note on bootstrap approximations for the empirical copulaprocess.German Research Foundation July 29., 2010.
C. Genest, B. Rémillard:Validity of the parametric bootstrap for goodness-of-fit testing insemiparametric models.Annales de lInstitut Henri Poincaré - Probabilités et Statistiques,Vol. 44, No. 6, 10961127, 2008.
C. Genest, J-F. Quessy, B. Rémillard:Goodness-of-fit Procedures for Copula Models Based on theProbability Integral Transformation.Scandinavian Journal of Statistics, Vol. 33, 337-366, 2006.
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References III
C. Genest, B. Rémillard:Validity of the parametric bootstrap for goodness-of-fit testing insemiparametric models.Les Cahiers du GERAD, July 2005.
R.B.V. Silva, D.F. Ferreira, D.A. Nogueira:Robustness of asymptotic and bootstrap tests for multivariatehomogeneity of covariance matrices.Cienc. agrotec., Lavras, Vol. 32, No. 1, 157-166, jan./febr. 2008.
László Varga (ELTE) Bootstrap methods for copulas, with applications to stock index dataASMDA 2011 24 / 24