statistical inference of vine copulas using the r-package … · statistical inference of vine...

66
Statistical inference of vine copulas using the R-package VineCopula Eike Christian Brechmann [email protected] Technische Universit¨ at M¨ unchen May 23, 2013 Eike Brechmann (TUM) The R-package VineCopula May 23, 2013 1 / 26

Upload: others

Post on 21-Jun-2020

7 views

Category:

Documents


0 download

TRANSCRIPT

Page 1: Statistical inference of vine copulas using the R-package … · Statistical inference of vine copulas using the R-package VineCopula Eike Christian Brechmann brechmann@ma.tum.de

Statistical inference of vine copulas using theR-package VineCopula

Eike Christian [email protected]

Technische Universitat Munchen

May 23, 2013

Eike Brechmann (TUM) The R-package VineCopula May 23, 2013 1 / 26

Page 2: Statistical inference of vine copulas using the R-package … · Statistical inference of vine copulas using the R-package VineCopula Eike Christian Brechmann brechmann@ma.tum.de

Available software for vines

In general...

I “Uncertainty analysis with Correlations” (UNICORN, TU Delft)includes some functionality for vines.

In R...

I Packages for bivariate and multivariate copulas (copula, fCopulae,QRMlib,...).

I Daniel Berg (U Oslo/NR): copulaGOF/CopulaLib.

The R-packages VineCopula and CDVine fill this gap.

Eike Brechmann (TUM) The R-package VineCopula May 23, 2013 2 / 26

Page 3: Statistical inference of vine copulas using the R-package … · Statistical inference of vine copulas using the R-package VineCopula Eike Christian Brechmann brechmann@ma.tum.de

Available software for vines

In general...

I “Uncertainty analysis with Correlations” (UNICORN, TU Delft)includes some functionality for vines.

In R...

I Packages for bivariate and multivariate copulas (copula, fCopulae,QRMlib,...).

I Daniel Berg (U Oslo/NR): copulaGOF/CopulaLib.

The R-packages VineCopula and CDVine fill this gap.

Eike Brechmann (TUM) The R-package VineCopula May 23, 2013 2 / 26

Page 4: Statistical inference of vine copulas using the R-package … · Statistical inference of vine copulas using the R-package VineCopula Eike Christian Brechmann brechmann@ma.tum.de

Scope of VineCopula

VineCopula provides functions for bivariate analysis...

graphical tools

analytical tools

selection and estimation of bivariate copulas

simulation of bivariate copulas

BiCop...

...and for multivariate analysis using (simplified) regular vine copulas.

sequential and joint maximum likelihood estimation

simulation of vine copulas

model selection

illustration of vine trees

p

RVine...

p

General assumption: Data lies in the unit hypercube [0, 1]d .

Eike Brechmann (TUM) The R-package VineCopula May 23, 2013 3 / 26

Page 5: Statistical inference of vine copulas using the R-package … · Statistical inference of vine copulas using the R-package VineCopula Eike Christian Brechmann brechmann@ma.tum.de

Scope of VineCopula

VineCopula provides functions for bivariate analysis...

graphical tools

analytical tools

selection and estimation of bivariate copulas

simulation of bivariate copulas

BiCop...

...and for multivariate analysis using (simplified) regular vine copulas.

sequential and joint maximum likelihood estimation

simulation of vine copulas

model selection

illustration of vine trees

p

RVine...

p

General assumption: Data lies in the unit hypercube [0, 1]d .

Eike Brechmann (TUM) The R-package VineCopula May 23, 2013 3 / 26

Page 6: Statistical inference of vine copulas using the R-package … · Statistical inference of vine copulas using the R-package VineCopula Eike Christian Brechmann brechmann@ma.tum.de

Scope of VineCopula

VineCopula provides functions for bivariate analysis...

graphical tools

analytical tools

selection and estimation of bivariate copulas

simulation of bivariate copulas

BiCop...

...and for multivariate analysis using (simplified) regular vine copulas.

sequential and joint maximum likelihood estimation

simulation of vine copulas

model selection

illustration of vine trees

p

RVine...

p

General assumption: Data lies in the unit hypercube [0, 1]d .

Eike Brechmann (TUM) The R-package VineCopula May 23, 2013 3 / 26

Page 7: Statistical inference of vine copulas using the R-package … · Statistical inference of vine copulas using the R-package VineCopula Eike Christian Brechmann brechmann@ma.tum.de

The package CDVine

Functionality for the sub-classes of C- and D-vines.

Links to the package VineCopula: C2RVine and D2RVine.

Vignette:

Brechmann & Schepsmeier (2013).Modeling dependence with C- and D-vine copulas:The R-package CDVine.Journal of Statistical Software 52(3), 1–27.

Eike Brechmann (TUM) The R-package VineCopula May 23, 2013 4 / 26

Page 8: Statistical inference of vine copulas using the R-package … · Statistical inference of vine copulas using the R-package VineCopula Eike Christian Brechmann brechmann@ma.tum.de

The building blocks: Bivariate copula families

Each family is denoted by a number to shorten notation (0 = indep.).

Elliptical copulas:

(parameters: par, par2 (degrees of freedom))

family =

0

1 Gaussian copulafamily =

0

2 Student’s t copula

One parameter Archimedean copulas:

(parameter: par)

Two parameter Archimedean copulas:

(parameters: par, par2)

Density, distribution & h-functions: BiCopPDF, BiCopCDF & BiCopHfunc.

Eike Brechmann (TUM) The R-package VineCopula May 23, 2013 5 / 26

Page 9: Statistical inference of vine copulas using the R-package … · Statistical inference of vine copulas using the R-package VineCopula Eike Christian Brechmann brechmann@ma.tum.de

The building blocks: Bivariate copula families

Each family is denoted by a number to shorten notation (0 = indep.).

Elliptical copulas:

(parameters: par, par2 (degrees of freedom))

family =

0

1 Gaussian copulafamily =

0

2 Student’s t copula

One parameter Archimedean copulas:

(parameter: par)

family =

0

3 Clayton copulafamily =

0

4 Gumbel copulafamily =

0

5 Frank copulafamily =

0

6 Joe copula

Two parameter Archimedean copulas:

(parameters: par, par2)

Density, distribution & h-functions: BiCopPDF, BiCopCDF & BiCopHfunc.

Eike Brechmann (TUM) The R-package VineCopula May 23, 2013 5 / 26

Page 10: Statistical inference of vine copulas using the R-package … · Statistical inference of vine copulas using the R-package VineCopula Eike Christian Brechmann brechmann@ma.tum.de

The building blocks: Bivariate copula families

Each family is denoted by a number to shorten notation (0 = indep.).

Elliptical copulas:

(parameters: par, par2 (degrees of freedom))

family =

0

1 Gaussian copulafamily =

0

2 Student’s t copula

One parameter Archimedean copulas:

(parameter: par)

family =

0

3 Clayton copulafamily =

0

4 Gumbel copulafamily =

0

5 Frank copulafamily =

0

6 Joe copula

Two parameter Archimedean copulas:

(parameters: par, par2)

family =

0

7 Clayton-Gumbel (BB1) copulafamily =

0

8 Joe-Gumbel (BB6) copulafamily =

0

9 Joe-Clayton (BB7) copulafamily = 10 Joe-Frank (BB8) copula

Density, distribution & h-functions: BiCopPDF, BiCopCDF & BiCopHfunc.

Eike Brechmann (TUM) The R-package VineCopula May 23, 2013 5 / 26

Page 11: Statistical inference of vine copulas using the R-package … · Statistical inference of vine copulas using the R-package VineCopula Eike Christian Brechmann brechmann@ma.tum.de

The building blocks: Bivariate copula families

Each family is denoted by a number to shorten notation (0 = indep.).

Elliptical copulas:

(parameters: par, par2 (degrees of freedom))

family =

0

1 Gaussian copulafamily =

0

2 Student’s t copula

One parameter Archimedean copulas:

(parameter: par)

family = 13 survival Clayton copulafamily = 14 survival Gumbel copulafamily =

0

5 Frank copulafamily = 16 survival Joe copula

Two parameter Archimedean copulas:

(parameters: par, par2)

family = 17 survival BB1 copulafamily = 18 survival BB6 copulafamily = 19 survival BB7 copulafamily = 20 survival BB8 copula

Density, distribution & h-functions: BiCopPDF, BiCopCDF & BiCopHfunc.

Eike Brechmann (TUM) The R-package VineCopula May 23, 2013 5 / 26

Page 12: Statistical inference of vine copulas using the R-package … · Statistical inference of vine copulas using the R-package VineCopula Eike Christian Brechmann brechmann@ma.tum.de

The building blocks: Bivariate copula families

Each family is denoted by a number to shorten notation (0 = indep.).

Elliptical copulas:

(parameters: par, par2 (degrees of freedom))

family =

0

1 Gaussian copulafamily =

0

2 Student’s t copula

One parameter Archimedean copulas:

(parameter: par)

family = 23 rotated Clayton copula (90 degrees)family = 24 rotated Gumbel copula (90 degrees)family =

0

5 Frank copulafamily = 26 rotated Joe copula (90 degrees)

Two parameter Archimedean copulas:

(parameters: par, par2)

family = 27 rotated BB1 copula (90 degrees)family = 28 rotated BB6 copula (90 degrees)family = 29 rotated BB7 copula (90 degrees)family = 30 rotated BB8 copula (90 degrees)

Density, distribution & h-functions: BiCopPDF, BiCopCDF & BiCopHfunc.

Eike Brechmann (TUM) The R-package VineCopula May 23, 2013 5 / 26

Page 13: Statistical inference of vine copulas using the R-package … · Statistical inference of vine copulas using the R-package VineCopula Eike Christian Brechmann brechmann@ma.tum.de

The building blocks: Bivariate copula families

Each family is denoted by a number to shorten notation (0 = indep.).

Elliptical copulas:

(parameters: par, par2 (degrees of freedom))

family =

0

1 Gaussian copulafamily =

0

2 Student’s t copula

One parameter Archimedean copulas:

(parameter: par)

family = 33 rotated Clayton copula (270 degrees)family = 34 rotated Gumbel copula (270 degrees)family =

0

5 Frank copulafamily = 36 rotated Joe copula (270 degrees)

Two parameter Archimedean copulas:

(parameters: par, par2)

family = 37 rotated BB1 copula (270 degrees)family = 38 rotated BB6 copula (270 degrees)family = 39 rotated BB7 copula (270 degrees)family = 40 rotated BB8 copula (270 degrees)

Density, distribution & h-functions: BiCopPDF, BiCopCDF & BiCopHfunc.

Eike Brechmann (TUM) The R-package VineCopula May 23, 2013 5 / 26

Page 14: Statistical inference of vine copulas using the R-package … · Statistical inference of vine copulas using the R-package VineCopula Eike Christian Brechmann brechmann@ma.tum.de

The building blocks: Bivariate copula families

Each family is denoted by a number to shorten notation (0 = indep.).

Elliptical copulas: (parameters: par, par2 (degrees of freedom))

family =

0

1 Gaussian copulafamily =

0

2 Student’s t copula

One parameter Archimedean copulas: (parameter: par)

family = 33 rotated Clayton copula (270 degrees)family = 34 rotated Gumbel copula (270 degrees)family =

0

5 Frank copulafamily = 36 rotated Joe copula (270 degrees)

Two parameter Archimedean copulas: (parameters: par, par2)

family = 37 rotated BB1 copula (270 degrees)family = 38 rotated BB6 copula (270 degrees)family = 39 rotated BB7 copula (270 degrees)family = 40 rotated BB8 copula (270 degrees)

Density, distribution & h-functions: BiCopPDF, BiCopCDF & BiCopHfunc.

Eike Brechmann (TUM) The R-package VineCopula May 23, 2013 5 / 26

Page 15: Statistical inference of vine copulas using the R-package … · Statistical inference of vine copulas using the R-package VineCopula Eike Christian Brechmann brechmann@ma.tum.de

The building blocks: Bivariate copula families

Each family is denoted by a number to shorten notation (0 = indep.).

Elliptical copulas: (parameters: par, par2 (degrees of freedom))

family =

0

1 Gaussian copulafamily =

0

2 Student’s t copula

One parameter Archimedean copulas: (parameter: par)

family = 33 rotated Clayton copula (270 degrees)family = 34 rotated Gumbel copula (270 degrees)family =

0

5 Frank copulafamily = 36 rotated Joe copula (270 degrees)

Two parameter Archimedean copulas: (parameters: par, par2)

family = 37 rotated BB1 copula (270 degrees)family = 38 rotated BB6 copula (270 degrees)family = 39 rotated BB7 copula (270 degrees)family = 40 rotated BB8 copula (270 degrees)

Density, distribution & h-functions: BiCopPDF, BiCopCDF & BiCopHfunc.

Eike Brechmann (TUM) The R-package VineCopula May 23, 2013 5 / 26

Page 16: Statistical inference of vine copulas using the R-package … · Statistical inference of vine copulas using the R-package VineCopula Eike Christian Brechmann brechmann@ma.tum.de

Rotation of copulas

I Rotate Archimedean copulas to capture negative dependence:if (U1,U2) ∼ C90◦ , then (1− U1,U2) ∼ C0◦ .

I Survival copulas correspond to rotation by 180 degrees.

Clayton copulas rotated by 0, 90, 180 and 270 degrees

> dat0 = BiCopSim(N=500, family=3, par=2)

> dat90 = BiCopSim(N=500, family=23, par=-2)

> dat180 = BiCopSim(N=500, family=13, par=2)

> dat270 = BiCopSim(N=500, family=33, par=-2)

Eike Brechmann (TUM) The R-package VineCopula May 23, 2013 6 / 26

Page 17: Statistical inference of vine copulas using the R-package … · Statistical inference of vine copulas using the R-package VineCopula Eike Christian Brechmann brechmann@ma.tum.de

Rotation of copulas

I Rotate Archimedean copulas to capture negative dependence:if (U1,U2) ∼ C90◦ , then (1− U1,U2) ∼ C0◦ .

I Survival copulas correspond to rotation by 180 degrees.

Clayton copulas rotated by 0, 90, 180 and 270 degrees

> dat0 = BiCopSim(N=500, family=3, par=2)

> dat90 = BiCopSim(N=500, family=23, par=-2)

> dat180 = BiCopSim(N=500, family=13, par=2)

> dat270 = BiCopSim(N=500, family=33, par=-2)

●●

●●

● ●●

●●

●●

●●

● ●

●●

●●

●●

●●

●●

●●

● ●

●●

●●

●●

●●

●●

● ●●

●●

●●

●●

0.0 0.2 0.4 0.6 0.8 1.0

0.0

0.2

0.4

0.6

0.8

1.0

0 degrees

u1

u2

●●

●●

●●

●●

●● ●

●●

●●

●●

● ●

●●

●●

●●

●●

●●

●●

●●

●●

●●

●●

●●

0.0 0.2 0.4 0.6 0.8 1.0

0.0

0.2

0.4

0.6

0.8

1.0

90 degrees

u1

u2

●●

●● ●

● ●

● ●

●●

●●

●●

●●

●●

●●

● ●

●●

● ●

●●

● ●

●●

● ●

●●

●●

● ●

●●

● ●

●●

●●

● ●

●●

● ●

0.0 0.2 0.4 0.6 0.8 1.0

0.0

0.2

0.4

0.6

0.8

1.0

180 degrees

u1

u2

●●

●●

●●

●●

● ●

●●

●●

● ● ●●

●●

●●

●●

●●

●●

●●

● ●●

●●

● ●

●●

●●

●●

0.0 0.2 0.4 0.6 0.8 1.0

0.0

0.2

0.4

0.6

0.8

1.0

270 degrees

u1

u2

Eike Brechmann (TUM) The R-package VineCopula May 23, 2013 6 / 26

Page 18: Statistical inference of vine copulas using the R-package … · Statistical inference of vine copulas using the R-package VineCopula Eike Christian Brechmann brechmann@ma.tum.de

Two parameter Archimedean copulasI The BB1 and the BB7 copula can model tail-asymmetric dependence

with different non-zero lower and upper tail dependence.I Density expressions and derivatives are however numerically involved.

Clayton-Gumbel (BB1) copula

> BiCopPar2Tau(family=7,

+ par=0.25, par2=2.5)

[1] 0.64

> BiCopPar2TailDep(family=7,

+ par=0.25, par2=2.5)

$lower

[1] 0.33

$upper

[1] 0.68

Eike Brechmann (TUM) The R-package VineCopula May 23, 2013 7 / 26

Page 19: Statistical inference of vine copulas using the R-package … · Statistical inference of vine copulas using the R-package VineCopula Eike Christian Brechmann brechmann@ma.tum.de

Two parameter Archimedean copulasI The BB1 and the BB7 copula can model tail-asymmetric dependence

with different non-zero lower and upper tail dependence.I Density expressions and derivatives are however numerically involved.

Clayton-Gumbel (BB1) copula

> BiCopPar2Tau(family=7,

+ par=0.25, par2=2.5)

[1] 0.64

> BiCopPar2TailDep(family=7,

+ par=0.25, par2=2.5)

$lower

[1] 0.33

$upper

[1] 0.68

Eike Brechmann (TUM) The R-package VineCopula May 23, 2013 7 / 26

Page 20: Statistical inference of vine copulas using the R-package … · Statistical inference of vine copulas using the R-package VineCopula Eike Christian Brechmann brechmann@ma.tum.de

Two parameter Archimedean copulasI The BB1 and the BB7 copula can model tail-asymmetric dependence

with different non-zero lower and upper tail dependence.I Density expressions and derivatives are however numerically involved.

Clayton-Gumbel (BB1) copula

> BiCopPar2Tau(family=7,

+ par=0.25, par2=2.5)

[1] 0.64

> BiCopPar2TailDep(family=7,

+ par=0.25, par2=2.5)

$lower

[1] 0.33

$upper

[1] 0.68

> BiCopMetaContour(family=7,

+ par=0.25, par2=2.5)

0.01

0.05

0.1 0.15

0.2

−3 −2 −1 0 1 2 3

−3

−2

−1

01

23

Eike Brechmann (TUM) The R-package VineCopula May 23, 2013 7 / 26

Page 21: Statistical inference of vine copulas using the R-package … · Statistical inference of vine copulas using the R-package VineCopula Eike Christian Brechmann brechmann@ma.tum.de

Reminder: R-vine copulasThree components: vine trees, pair copulas, copula parameters

2

5 1 4

3 T1

Density

c = c14 · c15 · c24 · c34

· c12;4 · c13;4 · c45;1

· c23;14 · c35;14

· c25;134

Eike Brechmann (TUM) The R-package VineCopula May 23, 2013 8 / 26

Page 22: Statistical inference of vine copulas using the R-package … · Statistical inference of vine copulas using the R-package VineCopula Eike Christian Brechmann brechmann@ma.tum.de

Reminder: R-vine copulasThree components: vine trees, pair copulas, copula parameters

2

5 1 4

3

1,5 1,4

2,4

3,4 T1

Density

c = c14 · c15 · c24 · c34

· c12;4 · c13;4 · c45;1

· c23;14 · c35;14

· c25;134

Eike Brechmann (TUM) The R-package VineCopula May 23, 2013 8 / 26

Page 23: Statistical inference of vine copulas using the R-package … · Statistical inference of vine copulas using the R-package VineCopula Eike Christian Brechmann brechmann@ma.tum.de

Reminder: R-vine copulasThree components: vine trees, pair copulas, copula parameters

2

5 1 4

3

1,5 1,4

2,4

3,4 T1

2,4

1,5 1,4

3,4 T2

Density

c = c14 · c15 · c24 · c34

· c12;4 · c13;4 · c45;1

· c23;14 · c35;14

· c25;134

Eike Brechmann (TUM) The R-package VineCopula May 23, 2013 8 / 26

Page 24: Statistical inference of vine copulas using the R-package … · Statistical inference of vine copulas using the R-package VineCopula Eike Christian Brechmann brechmann@ma.tum.de

Reminder: R-vine copulasThree components: vine trees, pair copulas, copula parameters

2

5 1 4

3

1,5 1,4

2,4

3,4 T1

2,4

1,5 1,4

3,4 T2

Density

c = c14 · c15 · c24 · c34

· c12;4 · c13;4 · c45;1

· c23;14 · c35;14

· c25;134

Eike Brechmann (TUM) The R-package VineCopula May 23, 2013 8 / 26

Page 25: Statistical inference of vine copulas using the R-package … · Statistical inference of vine copulas using the R-package VineCopula Eike Christian Brechmann brechmann@ma.tum.de

Reminder: R-vine copulasThree components: vine trees, pair copulas, copula parameters

2

5 1 4

3

1,5 1,4

2,4

3,4 T1

2,4

1,5 1,4

3,4

4,5|1

1,2|4

1,3|4 T2

Density

c = c14 · c15 · c24 · c34

· c12;4 · c13;4 · c45;1

· c23;14 · c35;14

· c25;134

Eike Brechmann (TUM) The R-package VineCopula May 23, 2013 8 / 26

Page 26: Statistical inference of vine copulas using the R-package … · Statistical inference of vine copulas using the R-package VineCopula Eike Christian Brechmann brechmann@ma.tum.de

Reminder: R-vine copulasThree components: vine trees, pair copulas, copula parameters

2

5 1 4

3

1,5 1,4

2,4

3,4 T1

2,4

1,5 1,4

3,4

4,5|1

1,2|4

1,3|4 T2

4,5|1 1,3|4 1,2|4 T3

Density

c = c14 · c15 · c24 · c34

· c12;4 · c13;4 · c45;1

· c23;14 · c35;14

· c25;134

Eike Brechmann (TUM) The R-package VineCopula May 23, 2013 8 / 26

Page 27: Statistical inference of vine copulas using the R-package … · Statistical inference of vine copulas using the R-package VineCopula Eike Christian Brechmann brechmann@ma.tum.de

Reminder: R-vine copulasThree components: vine trees, pair copulas, copula parameters

2

5 1 4

3

1,5 1,4

2,4

3,4 T1

2,4

1,5 1,4

3,4

4,5|1

1,2|4

1,3|4 T2

4,5|1 1,3|4 1,2|43,5|14 2,3|14

T3

3,5|14 2,3|142,5|134

T4

Density

c = c14 · c15 · c24 · c34

· c12;4 · c13;4 · c45;1

· c23;14 · c35;14

· c25;134

Eike Brechmann (TUM) The R-package VineCopula May 23, 2013 8 / 26

Page 28: Statistical inference of vine copulas using the R-package … · Statistical inference of vine copulas using the R-package VineCopula Eike Christian Brechmann brechmann@ma.tum.de

Storing R-vine copulas in matrix notation

Efficient enconding of R-vine models needed for statistical inference.

I Matrix notation by Morales-Napoles et al. (2010) and Dißmann et al.(2013).

25 33 5 41 1 5 54 4 1 1 1

1 {2, 4}, {3, 4}, {4, 1}, {5, 1}2 {2, 1|4}, {3, 1|4}, {4, 5|1}3 {2, 3|14}, {3, 5|14}4 {2, 5|314}

2

5 1 4

3

1,5 1,4

2,4

3,4 T1

2,4

1,5 1,4

3,4

4,5|1

1,2|4

1,3|4 T2

4,5|1 1,3|4 1,2|43,5|14 2,3|14

T3

3,5|14 2,3|142,5|134

T4

Eike Brechmann (TUM) The R-package VineCopula May 23, 2013 9 / 26

Page 29: Statistical inference of vine copulas using the R-package … · Statistical inference of vine copulas using the R-package VineCopula Eike Christian Brechmann brechmann@ma.tum.de

Storing R-vine copulas in matrix notation

Efficient enconding of R-vine models needed for statistical inference.

I Matrix notation by Morales-Napoles et al. (2010) and Dißmann et al.(2013).

25 33 5 41 1 5 54 4 1 1 1

1 {2, 4}, {3, 4}, {4, 1}, {5, 1}2 {2, 1|4}, {3, 1|4}, {4, 5|1}3 {2, 3|14}, {3, 5|14}4 {2, 5|314}

2

5 1 4

3

1,5 1,4

2,4

3,4 T1

2,4

1,5 1,4

3,4

4,5|1

1,2|4

1,3|4 T2

4,5|1 1,3|4 1,2|43,5|14 2,3|14

T3

3,5|14 2,3|142,5|134

T4

Eike Brechmann (TUM) The R-package VineCopula May 23, 2013 9 / 26

Page 30: Statistical inference of vine copulas using the R-package … · Statistical inference of vine copulas using the R-package VineCopula Eike Christian Brechmann brechmann@ma.tum.de

Storing R-vine copulas in matrix notation

Efficient enconding of R-vine models needed for statistical inference.

I Matrix notation by Morales-Napoles et al. (2010) and Dißmann et al.(2013).

25 33 5 41 1 5 54 4 1 1 1

1 {2, 4}, {3, 4}, {4, 1}, {5, 1}2 {2, 1|4}, {3, 1|4}, {4, 5|1}3 {2, 3|14}, {3, 5|14}4 {2, 5|314}

2

5 1 4

3

1,5 1,4

2,4

3,4 T1

2,4

1,5 1,4

3,4

4,5|1

1,2|4

1,3|4 T2

4,5|1 1,3|4 1,2|43,5|14 2,3|14

T3

3,5|14 2,3|142,5|134

T4

Eike Brechmann (TUM) The R-package VineCopula May 23, 2013 9 / 26

Page 31: Statistical inference of vine copulas using the R-package … · Statistical inference of vine copulas using the R-package VineCopula Eike Christian Brechmann brechmann@ma.tum.de

Storing R-vine copulas in matrix notation

Efficient enconding of R-vine models needed for statistical inference.

I Matrix notation by Morales-Napoles et al. (2010) and Dißmann et al.(2013).

25 33 5 41 1 5 54 4 1 1 1

1 {2, 4}, {3, 4}, {4, 1}, {5, 1}2 {2, 1|4}, {3, 1|4}, {4, 5|1}3 {2, 3|14}, {3, 5|14}4 {2, 5|314}

2

5 1 4

3

1,5 1,4

2,4

3,4 T1

2,4

1,5 1,4

3,4

4,5|1

1,2|4

1,3|4 T2

4,5|1 1,3|4 1,2|43,5|14 2,3|14

T3

3,5|14 2,3|142,5|134

T4

Eike Brechmann (TUM) The R-package VineCopula May 23, 2013 9 / 26

Page 32: Statistical inference of vine copulas using the R-package … · Statistical inference of vine copulas using the R-package VineCopula Eike Christian Brechmann brechmann@ma.tum.de

Storing R-vine copulas in matrix notation

Efficient enconding of R-vine models needed for statistical inference.

I Matrix notation by Morales-Napoles et al. (2010) and Dißmann et al.(2013).

25 33 5 41 1 5 54 4 1 1 1

1 {2, 4}, {3, 4}, {4, 1}, {5, 1}2 {2, 1|4}, {3, 1|4}, {4, 5|1}3 {2, 3|14}, {3, 5|14}4 {2, 5|314}

2

5 1 4

3

1,5 1,4

2,4

3,4 T1

2,4

1,5 1,4

3,4

4,5|1

1,2|4

1,3|4 T2

4,5|1 1,3|4 1,2|43,5|14 2,3|14

T3

3,5|14 2,3|142,5|134

T4

Eike Brechmann (TUM) The R-package VineCopula May 23, 2013 9 / 26

Page 33: Statistical inference of vine copulas using the R-package … · Statistical inference of vine copulas using the R-package VineCopula Eike Christian Brechmann brechmann@ma.tum.de

R-vine copula and parameter matrices

Copula families and parameters can be stored in associated matrices.25 33 5 41 1 5 54 4 1 1 1

−→

C25;314

C23;14 C35;14

C21;4 C31;4 C45;1

C24 C34 C41 C51

Eike Brechmann (TUM) The R-package VineCopula May 23, 2013 10 / 26

Page 34: Statistical inference of vine copulas using the R-package … · Statistical inference of vine copulas using the R-package VineCopula Eike Christian Brechmann brechmann@ma.tum.de

R-vine matrix objectsAn RVineMatrix object contains all required matrices:

> Matrix = c(2,5,3,1,4,0,3,5,1,4,0,0,4,5,1,

+ 0,0,0,5,1,0,0,0,0,1)

> Matrix = matrix(Matrix,5,5)

>

> family = c(0,1,3,4,4,0,0,3,4,1,0,0,0,4,1,

+ 0,0,0,0,3,0,0,0,0,0)

> family = matrix(family,5,5)

>

> par = c(0,0.2,0.9,1.5,3.9,0,0,1.1,1.6,0.9,0,0,0,1.9,0.5,

+ 0,0,0,0,4.8,0,0,0,0,0)

> par = matrix(par,5,5)

>

> par2 = matrix(0,5,5)

>

> RVM = RVineMatrix(Matrix=Matrix, family=family, par=par,

+ par2=par2, names=c("V1","V2","V3","V4","V5"))

Eike Brechmann (TUM) The R-package VineCopula May 23, 2013 11 / 26

Page 35: Statistical inference of vine copulas using the R-package … · Statistical inference of vine copulas using the R-package VineCopula Eike Christian Brechmann brechmann@ma.tum.de

Simulation

> simdat = RVineSim(500, RVM)

> head(simdat)

V1 V2 V3 V4 V5

[1,] 0.51 0.24 0.42 0.33 0.45

[2,] 0.23 0.14 0.16 0.12 0.20

[3,] 0.65 0.38 0.46 0.29 0.70

[4,] 0.43 0.18 0.08 0.08 0.26

[5,] 0.86 0.86 0.85 0.86 0.87

[6,] 0.71 0.71 0.80 0.68 0.88

Eike Brechmann (TUM) The R-package VineCopula May 23, 2013 12 / 26

Page 36: Statistical inference of vine copulas using the R-package … · Statistical inference of vine copulas using the R-package VineCopula Eike Christian Brechmann brechmann@ma.tum.de

Simulation

> simdat = RVineSim(500, RVM)

> head(simdat)

V1 V2 V3 V4 V5

[1,] 0.51 0.24 0.42 0.33 0.45

[2,] 0.23 0.14 0.16 0.12 0.20

[3,] 0.65 0.38 0.46 0.29 0.70

[4,] 0.43 0.18 0.08 0.08 0.26

[5,] 0.86 0.86 0.85 0.86 0.87

[6,] 0.71 0.71 0.80 0.68 0.88

V1

0.0 0.4 0.8

●●

●●

●●

● ●

● ●

● ●●

●● ●

●●

●●

●●

●●

●●●

●●

● ●

●●

●●

●●

●●

●●

●●

●●

●●

●●

●●

●●

● ●

●●

●●

●●

●●

● ●

●●

●●

●●

●●

● ●

●●

● ●●

●●

●●

●●

● ●

●●

●●●

● ●

●●

● ●

● ●

●●

●●

●●

● ●

●●

●●

●● ●

●●

●●

●●

● ●

●●

●●

●●

●●

●●

●●

●●

●●

● ●

● ●

● ●●

●● ●

●●

●●

●●

●●

● ●●

●●

●●

●●

●●

●●

●●

●●

●●

●●

●●

●●

●●

●●

● ●

●●

●●

●●

●●

● ●

●●

●●

●●

●●

● ●

●●

● ●●

●●●

●●

●●

● ●

●●

●●●

● ●

●●

● ●

● ●

●●

●●

●●

● ●

●●

●●

●● ●

●●

●●

●●

● ●

●●

●●

●●

●●

●●

0.0 0.4 0.8

●●

●●

●●

● ●

● ●

● ●●

●● ●

●●

●●

●●

●●

● ●●

●●

● ●

●●

●●

●●

●●

●●

●●

●●

●●

●●

●●

●●

● ●

●●

●●

●●

●●

● ●

●●

●●

●●

●●

●●

●●

● ●●

●●

●●

●●

● ●

●●

●●●

● ●

●●

● ●

●●

●●

●●

●●

● ●

●●

●●

●● ●

●●

●●

●●

● ●

●●

● ●

●●

●●

●●

0.0

0.4

0.8

●●

●●

●●

●●

● ●

● ●●

●● ●

●●

●●

●●

●●

● ●●

●●

●●

●●

●●

●●

●●

●●

●●

●●

●●

●●

●●

●●

●●

●●

●●

●●

●●

● ●

●●

●●

●●

●●

●●

●●

●●●

●●

●●

●●

● ●

●●

●●●

● ●

●●

● ●

● ●

●●

●●

●●

● ●

●●

●●

●●●

●●

●●

●●

● ●

●●

●●

●●

●●

●●

0.0

0.4

0.8

0.43 V2●

●●

●●

●●

●●

●●

●●

●●

●●

●●

●●

●●

●●

●●

●●

●●

●●

●●

●●

● ●●

●●

●●●●

●●

●●

●●

●●

●●

●●

● ●●

●●

●●

●●

● ●

●●

●●

●●

●●

●●

●●

●●

● ●

●● ●

●●

● ●

●●

●●

● ●

●●

● ●

●● ●●

●●

●●

●●

●●

●●

●●

●●

● ●

●●

●●

●●

●●

●●

● ●●

●●●

●●

●●

●●

●●

●●●

●●

●●

●●

●●

●●

●●

●●

●●

●●

●●

●●

●●

●●

●●●

●●

●●

●●

●●

● ●

●●

●●

● ●●

●●

●●

●●

●●

●●

●●

●●

●●

●●

●●●

●●

●●

● ●

●●

●●

●●

●●

● ●

●●

●●

●●

●●

●●●

●●

●●

●●

● ●

●●

●●

●●

●● ● ●

●●

●●

●●

●●

●●

●●

●●

●●

● ●

●●

●●

●●

●●

●●●

●●●

●●

●●

●●

●●

●●●

●●

●●

●●

●●

●●

●●

●●

●●

●●

●●

● ●

●●

●●

●●

●●

●●

●●

●●

●●

●●

●●

● ●●

●●

●●

●●

●●

●●

●●

●●

●●

●●

● ●●

●●

●●

● ●

● ●

●●

●●

●●

● ●

●●

●●

●●

●●

●● ●

●●

● ●

●●

●●

●●

● ●

● ●

●● ●●

●●

●●

●●

●●

●●

●●

●●

● ●

●●

●●

●●

●●

●●

● ●●

● ●●

●●

●●

●●

●●

●●●

●●

●●

●●

0.44 0.78 V3 ●

●●

●●

●●

●●

● ●

●●

●●

●●

●●

●●

●●

●●

●●

●●

●●

●●

●●

● ●

●●

● ●●● ●

●●

●●●

●●

● ●

●●

●●

●●

●●

●●

●●

●●

●●

●●

●●

●●

●●

●●

● ●

●●

●●

●●●●

●●

● ●

●●

●●

●●

●●

●●

●●

●●

● ●●

●●

●●

●●

●●

●●

●●

●●●

●●

● ●

●●

●●

●●

●●

● ●

●●

●●

●●

●●

●●

●● ●●

● ●

●●

0.0

0.4

0.8

●●

●●

●●

●●

● ●

●●

●●

●●

●●

●●

●●

●●

●●

●●

●●

●●

●●

● ●

●●

●● ●● ●

●●●

● ●●

●●

●●

● ●

●●

●●

●●

●●

●●

●●

●●

●●

●●

●●

●●

●●

●●

● ●

●●

● ●●●

●●

● ●

● ●

●●

●●

●●

●●

● ●

●●

●●●

●●

● ●

●●

●●

●●

●●

●●●

●●

● ●

●●

●●

●●

●●

●●

● ●

●●

●●

●●

●●

●●●●

● ●

●●

0.0

0.4

0.8

0.31 0.74 0.70 V4●

●●

●●

● ●

●●

●●

●●

● ●

●●

● ●

●●

● ●●

● ●

●●

●●

●●

●●

●●

●●

●●

●●

●●

●●

●●

● ●●

●●

●●

●●

●●

● ●● ●

●●

●●

● ●

● ●

●●

●● ●

● ●

●●

● ●

●●

●●

●●●

●●

●●

●●

●●●

●●

●● ●

●●

●●

●●

●●

●●

●●

●●●

●●

●●

●●

● ●

●●

●●

●●

● ●

●●

●●

●●

●●

●●

●●

●●

●●

0.0 0.4 0.8

0.72 0.60

0.0 0.4 0.8

0.62 0.47

0.0 0.4 0.8

0.0

0.4

0.8

V5

Eike Brechmann (TUM) The R-package VineCopula May 23, 2013 12 / 26

Page 37: Statistical inference of vine copulas using the R-package … · Statistical inference of vine copulas using the R-package VineCopula Eike Christian Brechmann brechmann@ma.tum.de

Example

Daily log returns of 15 major German stocks.

Observed from January 2005 to August 2009 (1158 observations).

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

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

Load into workspace:

> data(daxreturns)

Now: selection of trees, pair copulas and parameters in inverse order.

Eike Brechmann (TUM) The R-package VineCopula May 23, 2013 13 / 26

Page 38: Statistical inference of vine copulas using the R-package … · Statistical inference of vine copulas using the R-package VineCopula Eike Christian Brechmann brechmann@ma.tum.de

A first look at the data

ALV.DE

0.0 0.4 0.8

●●

●●

●●

●●

●●

●●

● ●

● ●

● ●

●●

●●

●●

●●

● ●

● ●●

●●

●●

● ●●

●●

●●●

●●

● ●●

●●

●●

●●

●●

● ●

●●

●● ●

●●

●●●

● ●

●●●

●●

●●

●●

●●

●●

●●

●●

●●

●●

●●

●●

●●

●●

●●

●● ●

●●

●●

●●●

●●

● ●

●●

● ●

●●

●●

●●●

●●

●●●

●●

●●

●●

●●●

●●

●●●●

●●

●●

●●

● ●

●●

●●

●●

●●

●●●

●●●

● ●

●●

●●

●●

● ●●

●●

●● ●

● ●

●●●

●●

●●

● ●

●●

●●

●●●

●●

●●

●●

●●

●●

●●

●●

●●

●●

● ●

●●

●●

●●

●●

● ●

● ●

● ●

●●

● ● ●

●●

●●

● ●

●●

●●

●●

●● ●●

●●

● ●

● ●●

●●

●●

● ●

●●

●●

●●

●●

●●

● ●

●●

●●

●●●

●●

●●

●●

●●

●●

●●

● ●

●●

●●

●●

●●

●●

●●

● ●

●●

●●

●●

●●

●●

●●

●●●

●●

●●

●●

●●

● ●

●●●●

●●

●●

●●

●●●

●●

●●

● ●

●●

●●

●●

●●

●●●

●●

●●●

● ●

●●

●●

●●

●●

●●

●●

●●

●●

●●

●●

●●●●

●●

●●

●●

●●

●●

●●

● ●

●●

●●

●●

●●

●●

●●

●●

●●

●●

●●

● ●●

●●

●●

● ●●

●●

●● ●

●●

● ●●

●●

●●

●●

●●

● ●

●●

●● ●

●●

●●●

● ●

●●●

●●

●●

● ●

●●

●●

● ●

●●

●●

●●

●●

●●

●●

●●

●●

●● ●

●●

●●

●●●

●●

● ●

●●

● ●

● ●

●●

●●●

●●

● ●●

●●

●●

●●

●● ●

●●

●●●●

●●

●●

●●

● ●

●●

●●

●●

●●

●●●

●●●

●●

●●

●●

●●

● ●●

●●

● ●●

● ●

● ●●

●●

●●

● ●

● ●

●●

●●●

● ●

●●

●●

●●

●●

●●

●●

●●

●●

● ●

●●

●●

●●

●●

●●

● ●

●●

●●

●● ●

●●

● ●

● ●

●●

●●

●●

●●●●

● ●

● ●

● ●●

●●

●●

●●

●●

●●

●●

●●

● ●

●●

●●

●●

●● ●

●●

●●

●●

●●

●●

●●

● ●

●●

●●

●●

●●

●●

●●

●●

●●

●●

●●

●●

● ●

● ●

●● ●

●●

●●

●●

●●

● ●

●●●

●●

●●

●●

● ●●

●●

●●

● ●

●●

●●

●●

●●

●●●

●●

●●●

● ●

●●

●●

●●

●●

●●

●●

●●

●●

●●

●●

●●

● ●

●●

●●

●●

0.0 0.4 0.8

●●

●●

●●

●●

●●

●●

● ●

● ●

● ●

●●

● ●

●●

●●

●●

●●●

● ●

●●

● ●●

● ●

●●●

●●

●●●

●●

●●

●●

●●

● ●

●●

●●●

●●

●●●

● ●

●●●

●●

●●

●●

●●

●●

●●

●●

●●

●●

●●

●●

●●

●●

●●

●●●

●●

●●

●●●

● ●

● ●

●●

●●

●●

●●

●●●

●●

●●●

●●

●●

●●

●●●

●●

●●●●

●●

●●

●●

● ●

●●

●●

●●

● ●

●●●

●● ●

● ●

●●

●●

●●

●●●

●●

●●●

●●

●●●

●●

●●

●●

● ●

●●

●● ●

●●

●●

●●

●●

●●

●●

●●

●●

●●

●●

●●

●●

●●

●●

● ●

● ●

● ●

● ●

● ● ●

●●

●●

●●

●●

●●

●●

●●●●

●●

● ●

● ●●

●●

●●

●●

●●

●●

●●

●●●●

●●

● ●

●●

●●●

●●

●●

●●

●●

●●

●●

●●

●●

●●

●●

●●

●●

●●

●●

●●

●●

●●

●●

● ●

●●

●●●

●●

●●

●●

●●

● ●

● ●●

●●

●●

● ●

●●●

●●

●●

●●

●●

●●

●●

●●

●●

●●

●● ●

●●

●●

●●

●●

●●

●●

●●

●●

●●

●●

●●

●●

●●

●●

●●

●●

0.0

0.4

0.8

●●

●●

●●

● ●

●●

●●

● ●

●●

● ●

●●

●●

●●

●●

●●

●●●

●●

●●

●● ●

● ●

●●●

●●

●●●

●●

●●

●●

●●

●●

●●

●●●

●●

●●●

● ●

●●●

●●

●●

● ●

●●

●●

●●

●●

●●

●●

●●

●●

●●

●●

●●

●●●

●●

●●

●●●

●●

● ●

●●

●●

●●

●●

●●●

●●

● ●●

●●

●●

●●

●●●

●●

●●●●

●●

●●

●●

● ●

●●

●●

●●

●●

●●●

●●●

● ●

●●

●●

●●

● ●●

●●

● ●●

● ●

●●●

●●

●●

●●

●●

●●

●●●

●●

●●

●●

●●

●●

●●

●●

●●

●●

● ●

●●

●●

●●

●●

●●

●●

● ●

●●

● ● ●

●●

● ●

●●

●●

●●

●●

● ●●●

● ●

●●

●●●

●●●

●●

●●

●●

●●

●●

●●

●●

● ●

● ●

●●

●●●

●●

●●

●●

●●

●●

●●

● ●

●●

●●

●●

●●

●●

●●

●●

●●

●●

●●

●●

● ●

●●

●●●

●●

●●

●●

●●

● ●

● ●●

●●

●●

●●

●●●

●●

●●

● ●

●●

●●

●●

●●

●●

●●

●●●

●●

●●

●●

●●

●●

●●

●●

●●

●●

●●

●●

●●●●

●●

●●

●●

0.0

0.4

0.8

0.40 BAS.DE ●

●●

● ●●●

●●

● ●●

● ●

●● ●

●●

●●

●●

●●

●●

●●

●●

●●

●● ●

●●●

● ●

●●

● ●

●●

●●●

●●

●●

●●

● ●

●●

●●

●●

● ●

●●

●●

● ●

●●

● ●

●●

●●

● ●●

●●

●●

●●

●●

●●

●●

● ●

●●

●●

●●

●●●

●●

●●

●●

●●

●●

●●●

●●

●●

●●

●●

●●

● ●

●●

●●

●●

● ●

●●

● ●

●● ●

● ●

●●

● ●

●●

●●

●●

●●

●●

●● ●

●●

●●

●● ●●

●●

●●

● ● ●

● ●

●●

●●

●●

●●

●●

●●

●●

●●

●●

●●

●●

●●

●●

●●

● ●●●

● ●

●●

●●

●●

●●

●●

●●

●●

● ●

●●

●●

●●

● ●

●●

●●

● ●

●●

●●

●●

●●

● ●

● ●

● ●●

●●

●●

●●

●●

●●

● ●

●●

● ●

● ● ●

●●

●●

● ●

●●

●●

●● ●

●●

●●

● ●

● ●

●●

●●●●

●●

●●

●●

●●●

●●

●● ●

●●

●●

●●

●●

● ●

●●

●● ●

●●

●●

●●

● ●

●●

●●

●●

●●

● ●

●●

●●

●●

●●

●●

●●● ●

●●

●●

● ●

●●

● ●●●

●●

● ●●

● ●

●●●

●●

●●

●●

●●

●●

●●

●●

●●

●● ●

●●●

●●

●●

●●

●●

●●●

●●

●●

●●

●●

●●

●●

●●

●●

●●

●●

● ●

●●

●●

●●

● ●

●●●

●●

●●

● ●

●●

●●

●●

●●

●●

●●

●●

●●●

●●

●●

●●

●●

●●

●●●

●●

●●

●●

●●

●●

●●

●●

●●

●●

●●

●●

●●

●●●

● ●

●●

● ●

●●

●●

●●

●●

●●

●● ●

●●

●●

● ●● ●

●●

● ●

●● ●

● ●

● ●

●●

●●

●●

●●

●●

●●

●●

●●

●●

●●

●●

●●

●●

● ●●●

● ●

● ●

● ●

●●

●●

●●

●●

●●

●●

●●

●●

● ●

●●

●●

●●

●●

●●

●●

●●

●●

●●

● ●

●●●

●●

● ●

●●

●●

●●

●●

●●

●●

● ●●

● ●

●●

● ●

●●

●●

●● ●

●●

●●

● ●

● ●

●●

● ●●●

● ●

●●

● ●

●● ●

●●

●● ●

●●

●●

●●

●●

●●

●●

●● ●

● ●

●●

●●

● ●

●●

●●

●●

●●

●●

●●

●●

●●

●●

●●

●● ●●

●●

●●

●●

●●

● ●● ●

●●

● ●●

●●

●●●

●●

●●

●●

●●

●●

●●

●●

● ●

●● ●

●●●

● ●

● ●

●●

●●

●●●

●●

●●

●●

●●

●●

●●

●●

●●

●●

●●

● ●

●●

●●

●●

●●

●●●

●●

●●

● ●

●●

●●

●●

● ●

●●

●●

●●

●●●

●●

●●

●●

●●

●●

●●●

●●

●●

●●

●●

●●

● ●

● ●

●●

●●

● ●

●●

●●

●●●

● ●

●●

● ●

●●

●●

●●

●●

●●

●●●

●●

●●

●●●●

●●

●●

●●●

●●

● ●

●●

●●

●●

●●

●●

● ●

●●

●●

●●

●●

●●

●●

●●

●●●●

●●

● ●

● ●

●●

●●

●●

●●

●●

● ●

●●

●●

● ●

●●

● ●

●●

●●

●●

●●

●●

●●

●●

● ●

●●●

●●

● ●

●●

●●

●●

● ●

●●

●●

● ●●

● ●

●●

● ●

●●

●●

●● ●

●●

●●

● ●

● ●

●●

● ●●●

●●

●●

●●

●●●

●●

●● ●

●●

●●

●●

●●

●●

●●

●● ●

● ●

●●

●●

● ●

●●

●●

●●

●●

● ●

●●

●●

●●

●●

●●

●●●●

●●

●●

● ●

0.32 0.42 BAYN.DE●

●●

● ●

●●

●●

●●

●●

● ●

● ●

●●

●●

●●

●●

●●

●●

●●

●●

● ●

●●●

●●

●●

●●

●●

● ●

●●

●●●

●●

●●

●●

●●

●●●

● ●

●●

●●

●●

●●●

●●

●●

●●

●●●

●●

●●

●●●

●●

●●

●●

●●

●●

●●

●●●

● ●●

●●

●●

●●

●●

●●

●●

● ●

●●●●●

●●

● ●

●●

●●

●● ●

●●

●●

●●

●●

●●

●●

●●●

●●

●●

●●

● ●

●●

●●

●●

● ●

●●

●●

●●

● ●

●●

●●

●●

●●

●●●

●●

●●

● ●

●●

●●

●●

● ●

● ●●

●●

●●

●●

● ●

● ●

● ●

●●

●●

●●

●●

●●

●●

● ●

●●

●●

●●

●●

●●

●●

● ●

●●

●●●

● ●

●●

● ●●

●●

●●

●●

●●

●●

●●

●●

●●

●●

●●

●●

●●

● ●●

● ●

●●

● ●

●●

●●

● ●

●●

● ●

●●

●●

●●

●●

●●

●●

●●

●●●

●●●

● ●

●● ●●

●●●

●●

●●

● ●

● ●●

● ● ●

●●

●●

●●

●●

●●

●●

●● ●

● ●

●●

● ●

●●

●●

●●

●●

●●

●●

●●

●●

●●

●●

●●

●●

●●

0.0

0.4

0.8

●●

●●

●●

●●

●●

●●

●●

● ●

●●

●●

●●

●●

●●

●●

●●

●●

●●

●●●

●●

●●

● ●

●●

● ●

●●

●●●

●●

●●

●●

●●

●●●

● ●

●●

● ●

●●

●●

●●

●●

●●

●● ●

●●

●●

●●●

●●

●●

●●

●●

●●

●●

●●●

● ●●

●●

●●

●●

●●

●●

●●

●●

●●●●●

●●

● ●

●●

●●

●● ●

●●

●●

●●

●●

●●

●●

●●●

●●

●●

●●

● ●

●●

●●

●●

●●

●●

●●

●●

●●

●●

●●

●●

●●

●●●

●●

●●

● ●

●●

●●

●●

● ●

● ●●

●●

●●

●●● ●

●●

● ●

●●

● ●

●●

●●

●●

●●

● ●

●●

●●

●●

●●

●●

●●

● ●

●●

●●●

● ●

●●

●●●

●●

●●

●●

●●

●●

●●●

●●

●●

●●

●●

●●

●●

●●●

● ●

●●

● ●

●●

●●

● ●

●●

● ●

●●

●●

●●

●●

●●

●●

●●

●●

●●●

●●

●●● ●

●●●

●●

●●

● ●

●●●

● ●●

●●

●●

●●

●●

●●

●●

●● ●

●●

●●

●●

●●

●●

●●●

●●

●●

●●

●●

●●

●●

●●

●●

●●

●●

0.0

0.4

0.8

0.40 0.35 0.29 BMW.DE

● ●

●●

●●

●●●

●●

● ●

●●

●●

● ●●

●●

●●

●●

●●

●●

● ●●

●●

●●●

●●

●●

●●

●●

●●

●●

●●

●●

●●

●●

●●

●●

●●

●●●●

● ●●

●●

●●

●●

●●

● ●

● ●

● ●●

● ●●

● ●

●●

●●

●●

●●

●●●

●● ●

●●

●●

●●

●●

●●

●●

●●●

●●

●●

● ●● ● ●

●●

●●

● ●●

● ●●

●●

●●

●●

● ●

●●

●●

●●

●●

●●

●●

●●

● ●

●●

●●●

● ●

● ●

●●

●●

●●

●●●

● ●

●●

●●

● ●

●●

●●

●●

●●

●●

●●

●●

● ●

●●

●●

●●

●●

●●

●●

●●

●●

● ●

●●

●●

●●

● ●

●●

●●

●●

● ●

●●

●●

●●

● ●●

● ●

●●

●●

●●

●●

●●

●●

●●

●●●●

●●● ●●

●●

●●

●●

● ●

●●

●●

●●

●●

●●●

●●

●●

●●

●●

●●

●●

●●

●●

●●

●●

●●

●●

●●

● ●

●●

●●

●●

●●

●●

●●

●●

●●

●●

●●

●●

●●

● ●

●●

●●

●●

●●

● ●

●●

●●

●●

●●

●●

●●

●●

●●

● ●

●●

● ●

● ●

● ●

●●

●●

● ●

●●

●●

●●●

●●

●●

●●

●●

● ●

●●

●●

●●

●●

●●

0.0 0.4 0.8

0.43 0.38

0.0 0.4 0.8

0.30 0.49

0.0 0.4 0.8

0.0

0.4

0.8

DAI.DE

Eike Brechmann (TUM) The R-package VineCopula May 23, 2013 14 / 26

Page 39: Statistical inference of vine copulas using the R-package … · Statistical inference of vine copulas using the R-package VineCopula Eike Christian Brechmann brechmann@ma.tum.de

Parameter estimation I

Sequential estimation (based on BiCopEst)

either using bivariate inversion of Kendall’s τ :

> RVineSeqEst(data, RVM, method="itau")

or bivariate maximum likelihood estimation:

> RVineSeqEst(data, RVM, method="mle")

I Very fast, since only bivariate estimation.

I Provides good starting values for joint maximum likelihood estimation.

Eike Brechmann (TUM) The R-package VineCopula May 23, 2013 15 / 26

Page 40: Statistical inference of vine copulas using the R-package … · Statistical inference of vine copulas using the R-package VineCopula Eike Christian Brechmann brechmann@ma.tum.de

Parameter estimation II

Maximum likelihood estimation of all parameters jointly(log-likelihood computation: RVineLogLik).

> RVineMLE(data, RVM, start, start2, maxit,

+ grad, hessian, se)

I Starting values can be calculated using sequential estimation.

I Analytical gradient can be used for numerical optimization(see RVineGrad).

I Standard errors can be computed based on the analytical Hessian(see RVineStdError and RVineHessian).

Eike Brechmann (TUM) The R-package VineCopula May 23, 2013 16 / 26

Page 41: Statistical inference of vine copulas using the R-package … · Statistical inference of vine copulas using the R-package VineCopula Eike Christian Brechmann brechmann@ma.tum.de

Parameter estimation II

Maximum likelihood estimation of all parameters jointly(log-likelihood computation: RVineLogLik).

> RVineMLE(data, RVM, start=0, start2=0, maxit,

+ grad, hessian, se)

I Starting values can be calculated using sequential estimation.

I Analytical gradient can be used for numerical optimization(see RVineGrad).

I Standard errors can be computed based on the analytical Hessian(see RVineStdError and RVineHessian).

Eike Brechmann (TUM) The R-package VineCopula May 23, 2013 16 / 26

Page 42: Statistical inference of vine copulas using the R-package … · Statistical inference of vine copulas using the R-package VineCopula Eike Christian Brechmann brechmann@ma.tum.de

Parameter estimation II

Maximum likelihood estimation of all parameters jointly(log-likelihood computation: RVineLogLik).

> RVineMLE(data, RVM, start, start2, maxit,

+ grad=TRUE, hessian, se)

I Starting values can be calculated using sequential estimation.

I Analytical gradient can be used for numerical optimization(see RVineGrad).

I Standard errors can be computed based on the analytical Hessian(see RVineStdError and RVineHessian).

Eike Brechmann (TUM) The R-package VineCopula May 23, 2013 16 / 26

Page 43: Statistical inference of vine copulas using the R-package … · Statistical inference of vine copulas using the R-package VineCopula Eike Christian Brechmann brechmann@ma.tum.de

Parameter estimation II

Maximum likelihood estimation of all parameters jointly(log-likelihood computation: RVineLogLik).

> RVineMLE(data, RVM, start, start2, maxit,

+ grad, hessian=TRUE, se=TRUE)

I Starting values can be calculated using sequential estimation.

I Analytical gradient can be used for numerical optimization(see RVineGrad).

I Standard errors can be computed based on the analytical Hessian(see RVineStdError and RVineHessian).

Eike Brechmann (TUM) The R-package VineCopula May 23, 2013 16 / 26

Page 44: Statistical inference of vine copulas using the R-package … · Statistical inference of vine copulas using the R-package VineCopula Eike Christian Brechmann brechmann@ma.tum.de

Parameter estimation III

> mle = RVineMLE(data=daxreturns[,1:5], RVM, start=0, start2=0,

+ grad=TRUE, hessian=TRUE, se=TRUE)

> mle$RVM$par

[,1] [,2] [,3] [,4] [,5]

[1,] 0.00 0.000 0.00 0.00 0

[2,] 0.14 0.000 0.00 0.00 0

[3,] 0.70 0.079 0.00 0.00 0

[4,] 1.30 1.206 1.42 0.00 0

[5,] 1.42 0.366 0.52 0.99 0

> mle$se

[,1] [,2] [,3] [,4] [,5]

[1,] 0.000 0.000 0.000 0.000 0

[2,] 0.029 0.000 0.000 0.000 0

[3,] 0.059 0.037 0.000 0.000 0

[4,] 0.031 0.028 0.036 0.000 0

[5,] 0.032 0.024 0.020 0.059 0

Eike Brechmann (TUM) The R-package VineCopula May 23, 2013 17 / 26

Page 45: Statistical inference of vine copulas using the R-package … · Statistical inference of vine copulas using the R-package VineCopula Eike Christian Brechmann brechmann@ma.tum.de

Parameter estimation III

> mle = RVineMLE(data=daxreturns[,1:5], RVM, start=0, start2=0,

+ grad=TRUE, hessian=TRUE, se=TRUE)

> mle$RVM$par

[,1] [,2] [,3] [,4] [,5]

[1,] 0.00 0.000 0.00 0.00 0

[2,] 0.14 0.000 0.00 0.00 0

[3,] 0.70 0.079 0.00 0.00 0

[4,] 1.30 1.206 1.42 0.00 0

[5,] 1.42 0.366 0.52 0.99 0

> mle$se

[,1] [,2] [,3] [,4] [,5]

[1,] 0.000 0.000 0.000 0.000 0

[2,] 0.029 0.000 0.000 0.000 0

[3,] 0.059 0.037 0.000 0.000 0

[4,] 0.031 0.028 0.036 0.000 0

[5,] 0.032 0.024 0.020 0.059 0

Eike Brechmann (TUM) The R-package VineCopula May 23, 2013 17 / 26

Page 46: Statistical inference of vine copulas using the R-package … · Statistical inference of vine copulas using the R-package VineCopula Eike Christian Brechmann brechmann@ma.tum.de

Parameter estimation III

> mle = RVineMLE(data=daxreturns[,1:5], RVM, start=0, start2=0,

+ grad=TRUE, hessian=TRUE, se=TRUE)

> mle$RVM$par

[,1] [,2] [,3] [,4] [,5]

[1,] 0.00 0.000 0.00 0.00 0

[2,] 0.14 0.000 0.00 0.00 0

[3,] 0.70 0.079 0.00 0.00 0

[4,] 1.30 1.206 1.42 0.00 0

[5,] 1.42 0.366 0.52 0.99 0

> mle$se

[,1] [,2] [,3] [,4] [,5]

[1,] 0.000 0.000 0.000 0.000 0

[2,] 0.029 0.000 0.000 0.000 0

[3,] 0.059 0.037 0.000 0.000 0

[4,] 0.031 0.028 0.036 0.000 0

[5,] 0.032 0.024 0.020 0.059 0

Eike Brechmann (TUM) The R-package VineCopula May 23, 2013 17 / 26

Page 47: Statistical inference of vine copulas using the R-package … · Statistical inference of vine copulas using the R-package VineCopula Eike Christian Brechmann brechmann@ma.tum.de

Pair copula selection

Manually using tools for bivariate analysis (e.g., contour plots orgoodness-of-fit tests: BiCopMetaContour, BiCopGofTest,...).

Automatically using AIC or BIC from a set of copula families:BiCopSelect (bivariate) or RVineCopSelect (multivariate).

> cops = RVineCopSelect(data=daxreturns[,1:5], familyset=NA,

+ Matrix=Matrix, selectioncrit="AIC",

+ indeptest=FALSE, level=0.05)

> cops$family

[,1] [,2] [,3] [,4] [,5]

[1,] 0 0 0 0 0

[2,] 5 0 0 0 0

[3,] 2 5 0 0 0

[4,] 2 20 2 0 0

[5,] 17 2 14 20 0

Eike Brechmann (TUM) The R-package VineCopula May 23, 2013 18 / 26

Page 48: Statistical inference of vine copulas using the R-package … · Statistical inference of vine copulas using the R-package VineCopula Eike Christian Brechmann brechmann@ma.tum.de

Pair copula selection

Manually using tools for bivariate analysis (e.g., contour plots orgoodness-of-fit tests: BiCopMetaContour, BiCopGofTest,...).

Automatically using AIC or BIC from a set of copula families:BiCopSelect (bivariate) or RVineCopSelect (multivariate).

> cops = RVineCopSelect(data=daxreturns[,1:5], familyset=NA,

+ Matrix=Matrix, selectioncrit="AIC",

+ indeptest=FALSE, level=0.05)

> cops$family

[,1] [,2] [,3] [,4] [,5]

[1,] 0 0 0 0 0

[2,] 5 0 0 0 0

[3,] 2 5 0 0 0

[4,] 2 20 2 0 0

[5,] 17 2 14 20 0

Eike Brechmann (TUM) The R-package VineCopula May 23, 2013 18 / 26

Page 49: Statistical inference of vine copulas using the R-package … · Statistical inference of vine copulas using the R-package VineCopula Eike Christian Brechmann brechmann@ma.tum.de

Pair copula selection

Manually using tools for bivariate analysis (e.g., contour plots orgoodness-of-fit tests: BiCopMetaContour, BiCopGofTest,...).

Automatically using AIC or BIC from a set of copula families:BiCopSelect (bivariate) or RVineCopSelect (multivariate).

> cops = RVineCopSelect(data=daxreturns[,1:5], familyset=NA,

+ Matrix=Matrix, selectioncrit="AIC",

+ indeptest=FALSE, level=0.05)

> cops$family

[,1] [,2] [,3] [,4] [,5]

[1,] 0 0 0 0 0

[2,] 5 0 0 0 0

[3,] 2 5 0 0 0

[4,] 2 20 2 0 0

[5,] 17 2 14 20 0

Eike Brechmann (TUM) The R-package VineCopula May 23, 2013 18 / 26

Page 50: Statistical inference of vine copulas using the R-package … · Statistical inference of vine copulas using the R-package VineCopula Eike Christian Brechmann brechmann@ma.tum.de

Pair copula selection

Manually using tools for bivariate analysis (e.g., contour plots orgoodness-of-fit tests: BiCopMetaContour, BiCopGofTest,...).

Automatically using AIC or BIC from a set of copula families:BiCopSelect (bivariate) or RVineCopSelect (multivariate).

> cops = RVineCopSelect(data=daxreturns[,1:5], familyset=NA,

+ Matrix=Matrix, selectioncrit="AIC",

+ indeptest=FALSE, level=0.05)

> cops$family

[,1] [,2] [,3] [,4] [,5]

[1,] 0 0 0 0 0

[2,] 5 0 0 0 0

[3,] 2 5 0 0 0

[4,] 2 20 2 0 0

[5,] 17 2 14 20 0

Eike Brechmann (TUM) The R-package VineCopula May 23, 2013 18 / 26

Page 51: Statistical inference of vine copulas using the R-package … · Statistical inference of vine copulas using the R-package VineCopula Eike Christian Brechmann brechmann@ma.tum.de

Vine tree selection I

Sequential tree-by-tree selection according to Dißmann et al. (2013):

1 Select maximum spanning tree (respecting the proximity condition) interms of the absolute empirical pairwise Kendall’s τ values.

2 Select and estimate pair copulas of the tree.3 Compute transformed observations using h-functions and go back to

Step 1.

I R- and C-vine copulas can be selected.

I The vine copula can be truncated to reduce the model complexity.

Eike Brechmann (TUM) The R-package VineCopula May 23, 2013 19 / 26

Page 52: Statistical inference of vine copulas using the R-package … · Statistical inference of vine copulas using the R-package VineCopula Eike Christian Brechmann brechmann@ma.tum.de

Vine tree selection I

Sequential tree-by-tree selection according to Dißmann et al. (2013):

1 Select maximum spanning tree (respecting the proximity condition) interms of the absolute empirical pairwise Kendall’s τ values.

2 Select and estimate pair copulas of the tree.3 Compute transformed observations using h-functions and go back to

Step 1.

> RVineStructureSelect(data, familyset, type,

+ selectioncrit, indeptest,

+ level, trunclevel)

I R- and C-vine copulas can be selected.

I The vine copula can be truncated to reduce the model complexity.

Eike Brechmann (TUM) The R-package VineCopula May 23, 2013 19 / 26

Page 53: Statistical inference of vine copulas using the R-package … · Statistical inference of vine copulas using the R-package VineCopula Eike Christian Brechmann brechmann@ma.tum.de

Vine tree selection I

Sequential tree-by-tree selection according to Dißmann et al. (2013):

1 Select maximum spanning tree (respecting the proximity condition) interms of the absolute empirical pairwise Kendall’s τ values.

2 Select and estimate pair copulas of the tree.3 Compute transformed observations using h-functions and go back to

Step 1.

> RVineStructureSelect(data, familyset, type="RVine",

+ selectioncrit, indeptest,

+ level, trunclevel)

I R- and C-vine copulas can be selected.

I The vine copula can be truncated to reduce the model complexity.

Eike Brechmann (TUM) The R-package VineCopula May 23, 2013 19 / 26

Page 54: Statistical inference of vine copulas using the R-package … · Statistical inference of vine copulas using the R-package VineCopula Eike Christian Brechmann brechmann@ma.tum.de

Vine tree selection I

Sequential tree-by-tree selection according to Dißmann et al. (2013):

1 Select maximum spanning tree (respecting the proximity condition) interms of the absolute empirical pairwise Kendall’s τ values.

2 Select and estimate pair copulas of the tree.3 Compute transformed observations using h-functions and go back to

Step 1.

> RVineStructureSelect(data, familyset, type,

+ selectioncrit, indeptest,

+ level, trunclevel=2)

I R- and C-vine copulas can be selected.

I The vine copula can be truncated to reduce the model complexity.

Eike Brechmann (TUM) The R-package VineCopula May 23, 2013 19 / 26

Page 55: Statistical inference of vine copulas using the R-package … · Statistical inference of vine copulas using the R-package VineCopula Eike Christian Brechmann brechmann@ma.tum.de

Vine tree selection II

> rvm = RVineStructureSelect(data=daxreturns)

> rvm$Matrix

[,1] [,2] [,3] [,4] [,5] [,6] [,7] [,8] [,9] [,10] [,11] [,12] [,13] [,14] [,15]

[1,] 15 0 0 0 0 0 0 0 0 0 0 0 0 0 0

[2,] 11 4 0 0 0 0 0 0 0 0 0 0 0 0 0

[3,] 9 11 3 0 0 0 0 0 0 0 0 0 0 0 0

[4,] 7 9 11 11 0 0 0 0 0 0 0 0 0 0 0

[5,] 13 7 9 12 9 0 0 0 0 0 0 0 0 0 0

[6,] 3 13 7 8 12 7 0 0 0 0 0 0 0 0 0

[7,] 12 3 13 10 8 12 13 0 0 0 0 0 0 0 0

[8,] 8 12 5 5 10 8 12 5 0 0 0 0 0 0 0

[9,] 10 8 1 2 5 10 8 12 1 0 0 0 0 0 0

[10,] 2 10 12 13 2 5 10 8 12 12 0 0 0 0 0

[11,] 1 2 8 9 13 2 5 10 8 10 8 0 0 0 0

[12,] 14 1 6 7 1 13 2 2 10 6 10 10 0 0 0

[13,] 6 14 14 14 14 1 14 1 2 14 6 6 2 0 0

[14,] 4 6 10 6 6 14 6 14 14 2 14 14 6 14 0

[15,] 5 5 2 1 7 6 1 6 6 8 2 2 14 6 6

Eike Brechmann (TUM) The R-package VineCopula May 23, 2013 20 / 26

Page 56: Statistical inference of vine copulas using the R-package … · Statistical inference of vine copulas using the R-package VineCopula Eike Christian Brechmann brechmann@ma.tum.de

Vine tree selection II

> rvm = RVineStructureSelect(data=daxreturns)

> rvm$Matrix

[,1] [,2] [,3] [,4] [,5] [,6] [,7] [,8] [,9] [,10] [,11] [,12] [,13] [,14] [,15]

[1,] 15 0 0 0 0 0 0 0 0 0 0 0 0 0 0

[2,] 11 4 0 0 0 0 0 0 0 0 0 0 0 0 0

[3,] 9 11 3 0 0 0 0 0 0 0 0 0 0 0 0

[4,] 7 9 11 11 0 0 0 0 0 0 0 0 0 0 0

[5,] 13 7 9 12 9 0 0 0 0 0 0 0 0 0 0

[6,] 3 13 7 8 12 7 0 0 0 0 0 0 0 0 0

[7,] 12 3 13 10 8 12 13 0 0 0 0 0 0 0 0

[8,] 8 12 5 5 10 8 12 5 0 0 0 0 0 0 0

[9,] 10 8 1 2 5 10 8 12 1 0 0 0 0 0 0

[10,] 2 10 12 13 2 5 10 8 12 12 0 0 0 0 0

[11,] 1 2 8 9 13 2 5 10 8 10 8 0 0 0 0

[12,] 14 1 6 7 1 13 2 2 10 6 10 10 0 0 0

[13,] 6 14 14 14 14 1 14 1 2 14 6 6 2 0 0

[14,] 4 6 10 6 6 14 6 14 14 2 14 14 6 14 0

[15,] 5 5 2 1 7 6 1 6 6 8 2 2 14 6 6

Eike Brechmann (TUM) The R-package VineCopula May 23, 2013 20 / 26

Page 57: Statistical inference of vine copulas using the R-package … · Statistical inference of vine copulas using the R-package VineCopula Eike Christian Brechmann brechmann@ma.tum.de

Illustrating R-vine copula modelsSelected R-vine trees:> RVineTreePlot(data=NULL, RVM=rvm, tree=1,

+ edge.labels=c("family","theotau"))

Eike Brechmann (TUM) The R-package VineCopula May 23, 2013 21 / 26

Page 58: Statistical inference of vine copulas using the R-package … · Statistical inference of vine copulas using the R-package VineCopula Eike Christian Brechmann brechmann@ma.tum.de

Illustrating R-vine copula modelsSelected R-vine trees:> RVineTreePlot(data=NULL, RVM=rvm, tree=1,

+ edge.labels=c("family","theotau"))

Tree 1

t,0.45

t,0.43

t,0.4

t,0.37

t,0.55

t,0.52

t,0.44

t,0.37

SBB8,0.34

SBB1,0.22

t,0.52

t,0.42

t,0.48

t,0.35

DBK.DE

SIE.DE

BAS.DE

LIN.DE

EOAN.DE

RWE.DE

ALV.DE

DAI.DE

SAP.DE

DTE.DE

FME.DE

MUV2.DE

BAYN.DE

BMW.DE

VOW3.DE

Eike Brechmann (TUM) The R-package VineCopula May 23, 2013 21 / 26

Page 59: Statistical inference of vine copulas using the R-package … · Statistical inference of vine copulas using the R-package VineCopula Eike Christian Brechmann brechmann@ma.tum.de

Illustrating R-vine copula modelsSelected R-vine trees:> RVineTreePlot(data=NULL, RVM=rvm, tree=2,

+ edge.labels=FALSE)

Tree 2

BAS.DE,SIE.DE

DBK.DE,SIE.DE

BAS.DE,LIN.DE

BAS.DE,EOAN.DE

EOAN.DE,RWE.DE

ALV.DE,DBK.DE

DAI.DE,DBK.DE

ALV.DE,SAP.DE

DBK.DE,DTE.DE

DTE.DE,FME.DE

ALV.DE,MUV2.DE

BAS.DE,BAYN.DE

BMW.DE,DAI.DE

DAI.DE,VOW3.DE

Eike Brechmann (TUM) The R-package VineCopula May 23, 2013 21 / 26

Page 60: Statistical inference of vine copulas using the R-package … · Statistical inference of vine copulas using the R-package VineCopula Eike Christian Brechmann brechmann@ma.tum.de

Discrimination among vine copula models I

AIC and BIC: RVineAIC and RVineBIC.

Tests by Vuong (1989) and by Clarke (2007) for non-nestedcomparisons of two R-vine models RVM1 and RVM2:

> RVineVuongTest(data, RVM1, RVM2)

> RVineClarkeTest(data, RVM1, RVM2)

Eike Brechmann (TUM) The R-package VineCopula May 23, 2013 22 / 26

Page 61: Statistical inference of vine copulas using the R-package … · Statistical inference of vine copulas using the R-package VineCopula Eike Christian Brechmann brechmann@ma.tum.de

Discrimination among vine copula models IISelect a C-vine copula for comparison:> cvm = RVineStructureSelect(daxreturns, type="CVine")

Compare the models in terms of AIC values and the Vuong test:> c(RVineAIC(daxreturns,rvm)$AIC, RVineAIC(daxreturns,cvm)$AIC)

[1] -9808.44 -9804.42

> RVineVuongTest(daxreturns, rvm, cvm)

$statistic

[1] 0.27

$statistic.Akaike

[1] 0.068

$statistic.Schwarz

[1] -0.44

...

The models are essentially indistinguishable.

Eike Brechmann (TUM) The R-package VineCopula May 23, 2013 23 / 26

Page 62: Statistical inference of vine copulas using the R-package … · Statistical inference of vine copulas using the R-package VineCopula Eike Christian Brechmann brechmann@ma.tum.de

Discrimination among vine copula models IISelect a C-vine copula for comparison:> cvm = RVineStructureSelect(daxreturns, type="CVine")

Compare the models in terms of AIC values and the Vuong test:> c(RVineAIC(daxreturns,rvm)$AIC, RVineAIC(daxreturns,cvm)$AIC)

[1] -9808.44 -9804.42

> RVineVuongTest(daxreturns, rvm, cvm)

$statistic

[1] 0.27

$statistic.Akaike

[1] 0.068

$statistic.Schwarz

[1] -0.44

...

The models are essentially indistinguishable.

Eike Brechmann (TUM) The R-package VineCopula May 23, 2013 23 / 26

Page 63: Statistical inference of vine copulas using the R-package … · Statistical inference of vine copulas using the R-package VineCopula Eike Christian Brechmann brechmann@ma.tum.de

Outlook

Current projects and plans:

Move code from C to C++.

Parallelize numerical maximum likelihood estimation.

Implement asymmetric copulas (Tawn).

Bayesian estimation and model selection.

Thank you to my package coauthors: Ulf Schepsmeier & Jakob Stober

Eike Brechmann (TUM) The R-package VineCopula May 23, 2013 24 / 26

Page 64: Statistical inference of vine copulas using the R-package … · Statistical inference of vine copulas using the R-package VineCopula Eike Christian Brechmann brechmann@ma.tum.de

Outlook

Current projects and plans:

Move code from C to C++.

Parallelize numerical maximum likelihood estimation.

Implement asymmetric copulas (Tawn).

Bayesian estimation and model selection.

Thank you to my package coauthors: Ulf Schepsmeier & Jakob Stober

Eike Brechmann (TUM) The R-package VineCopula May 23, 2013 24 / 26

Page 65: Statistical inference of vine copulas using the R-package … · Statistical inference of vine copulas using the R-package VineCopula Eike Christian Brechmann brechmann@ma.tum.de

Bibliography

Brechmann, E. C. and C. Czado (2013).Risk management with high-dimensional vine copulas: An analysis of the Euro Stoxx 50.Statistics & Risk Modeling , forthcoming.

Brechmann, E. C. and U. Schepsmeier (2013).Modeling dependence with C- and D-vine copulas: The R-package CDVine.Journal of Statistical Software 52(3), 1–27.

Clarke, K. A. (2007).A simple distribution-free test for nonnested model selection.Political Analysis 15(3), 347–363.

Dißmann, J., E. C. Brechmann, C. Czado, and D. Kurowicka (2013).Selecting and estimating regular vine copulae and application to financial returns.Computational Statistics & Data Analysis 59(1), 52–69.

Morales-Napoles, O., R. M. Cooke, and D. Kurowicka (2010).About the number of vines and regular vines on n nodes.Working paper .

Vuong, Q. H. (1989).Ratio tests for model selection and non-nested hypotheses.Econometrica 57(2), 307–333.

Eike Brechmann (TUM) The R-package VineCopula May 23, 2013 25 / 26

Page 66: Statistical inference of vine copulas using the R-package … · Statistical inference of vine copulas using the R-package VineCopula Eike Christian Brechmann brechmann@ma.tum.de

Thank you very much for your attention!

Visit: http://cran.r-project.org/web/packages/VineCopula/

Eike Brechmann (TUM) The R-package VineCopula May 23, 2013 26 / 26