New Prospects on Vines P.A. Maugis D. Guégan CES-MSE, Université Paris 1 Panthéon-Sorbonne

December 17, 2008

Introduction

Introduction

I

The Vine Model

Introduction

I

The Vine Model I

The Vine Decompositions

Introduction

I

The Vine Model I I

The Vine Decompositions The Vine Estimators

Introduction

I

The Vine Model I I

I

The Vine Decompositions The Vine Estimators

Prospects

Introduction

I

The Vine Model I I

I

The Vine Decompositions The Vine Estimators

Prospects I

Issues

Introduction

I

The Vine Model I I

I

The Vine Decompositions The Vine Estimators

Prospects I I

Issues Proposed Solutions

Introduction

I

The Vine Model I I

I

Prospects I I

I

The Vine Decompositions The Vine Estimators Issues Proposed Solutions

Example

The Vine Decompositions A Vine is a decomposition of a n-variate density in a product of bivariate copulas 1 .

1

Aas et al. (2007), Berg and Aas (2007), Joe (1997) and Bedford and Cooke (2002, 2001).

The Vine Decompositions A Vine is a decomposition of a n-variate density in a product of bivariate copulas 1 . f (x1 , . . . , xn )

1

Aas et al. (2007), Berg and Aas (2007), Joe (1997) and Bedford and Cooke (2002, 2001).

The Vine Decompositions A Vine is a decomposition of a n-variate density in a product of bivariate copulas 1 . f (x1 , . . . , xn ) = f (x1 , . . . , xn−1 ) · f (xn |x1 , . . . , xn−1 )

1

Aas et al. (2007), Berg and Aas (2007), Joe (1997) and Bedford and Cooke (2002, 2001).

The Vine Decompositions A Vine is a decomposition of a n-variate density in a product of bivariate copulas 1 . f (x1 , . . . , xn ) = f (x1 , . . . , xn−1 ) · f (xn |x1 , . . . , xn−1 ) = f (x1 , . . . , xn−1 ) · f ((xn , xn−1 )|(xn−1 |x1 , . . . , xn−2 ))

1

Aas et al. (2007), Berg and Aas (2007), Joe (1997) and Bedford and Cooke (2002, 2001).

The Vine Decompositions A Vine is a decomposition of a n-variate density in a product of bivariate copulas 1 . f (x1 , . . . , xn ) = f (x1 , . . . , xn−1 ) · f (xn |x1 , . . . , xn−1 ) = f (x1 , . . . , xn−1 ) · f ((xn , xn−1 )|(xn−1 |x1 , . . . , xn−2 )) = f (x1 , . . . , xn−1 ) · f (x1 , . . . , xn−2 , xn )/f (x1 , . . . , xn−2 ) · cn,n−1|1...n−2 (F (xn |x1 . . . xn−2 ), F (xn−1 |x1 . . . xn−2 ))

1

Aas et al. (2007), Berg and Aas (2007), Joe (1997) and Bedford and Cooke (2002, 2001).

The Vine Decompositions A Vine is a decomposition of a n-variate density in a product of bivariate copulas 1 . f (x1 , . . . , xn ) = f (x1 , . . . , xn−1 ) · f (xn |x1 , . . . , xn−1 ) = f (x1 , . . . , xn−1 ) · f ((xn , xn−1 )|(xn−1 |x1 , . . . , xn−2 )) = f (x1 , . . . , xn−1 ) · f (x1 , . . . , xn−2 , xn )/f (x1 , . . . , xn−2 ) · cn,n−1|1...n−2 (F (xn |x1 . . . xn−2 ), F (xn−1 |x1 . . . xn−2 ))

1

Aas et al. (2007), Berg and Aas (2007), Joe (1997) and Bedford and Cooke (2002, 2001).

The Vine Decompositions A Vine is a decomposition of a n-variate density in a product of bivariate copulas 1 . f (x1 , . . . , xn ) = f (x1 , . . . , xn−1 ) · f (xn |x1 , . . . , xn−1 ) = f (x1 , . . . , xn−1 ) · f ((xn , xn−1 )|(xn−1 |x1 , . . . , xn−2 )) = f (x1 , . . . , xn−1 ) · f (x1 , . . . , xn−2 , xn )/f (x1 , . . . , xn−2 ) · cn,n−1|1...n−2 (F (xn |x1 . . . xn−2 ), F (xn−1 |x1 . . . xn−2 )) Dimension 3:

1

Aas et al. (2007), Berg and Aas (2007), Joe (1997) and Bedford and Cooke (2002, 2001).

The Vine Decompositions A Vine is a decomposition of a n-variate density in a product of bivariate copulas 1 . f (x1 , . . . , xn ) = f (x1 , . . . , xn−1 ) · f (xn |x1 , . . . , xn−1 ) = f (x1 , . . . , xn−1 ) · f ((xn , xn−1 )|(xn−1 |x1 , . . . , xn−2 )) = f (x1 , . . . , xn−1 ) · f (x1 , . . . , xn−2 , xn )/f (x1 , . . . , xn−2 ) · cn,n−1|1...n−2 (F (xn |x1 . . . xn−2 ), F (xn−1 |x1 . . . xn−2 )) Dimension 3: f (x1 , x2 , x3 ) =f (x1 ).f (x2 ).f (x3 ) .c1,2 (F (x1 ), F (x2 )).c1,3 (F (x1 ), F (x3 )) .c1,3|2 (F (x1 |x2 ), F (x3 |x2 )). 1

Aas et al. (2007), Berg and Aas (2007), Joe (1997) and Bedford and Cooke (2002, 2001).

The Vine Estimators To compute the conditional distributions we use this formula the necessary number of times 2 : F (x|ν) =

2 3

∂Cx,νj |ν−j (F (x|ν−j ), F (νj |ν−j )) ∂F (νj |ν−j )

Joe (1997) Bedford and Cooke (2002) and Chen and Fan (2006)

The Vine Estimators To compute the conditional distributions we use this formula the necessary number of times 2 : F (x|ν) =

∂Cx,νj |ν−j (F (x|ν−j ), F (νj |ν−j )) ∂F (νj |ν−j )

Then the estimation is done with a n-stage ML 3 :

2 3

Joe (1997) Bedford and Cooke (2002) and Chen and Fan (2006)

The Vine Estimators To compute the conditional distributions we use this formula the necessary number of times 2 : F (x|ν) =

∂Cx,νj |ν−j (F (x|ν−j ), F (νj |ν−j )) ∂F (νj |ν−j )

Then the estimation is done with a n-stage ML 3 : I

2 3

First the bivariate copulas without conditions.

Joe (1997) Bedford and Cooke (2002) and Chen and Fan (2006)

The Vine Estimators To compute the conditional distributions we use this formula the necessary number of times 2 : F (x|ν) =

∂Cx,νj |ν−j (F (x|ν−j ), F (νj |ν−j )) ∂F (νj |ν−j )

Then the estimation is done with a n-stage ML 3 : I

First the bivariate copulas without conditions.

I

Then the bivariate copulas with one conditions.

2 3

Joe (1997) Bedford and Cooke (2002) and Chen and Fan (2006)

The Vine Estimators To compute the conditional distributions we use this formula the necessary number of times 2 : F (x|ν) =

∂Cx,νj |ν−j (F (x|ν−j ), F (νj |ν−j )) ∂F (νj |ν−j )

Then the estimation is done with a n-stage ML 3 : I

First the bivariate copulas without conditions.

I

Then the bivariate copulas with one conditions.

I

And so on ...

2 3

Joe (1997) Bedford and Cooke (2002) and Chen and Fan (2006)

Facts

Facts

I

There are about

Qn

i=1 i!

Vines in dimension n.

Facts

I I

There are about The are

n2

to

n3

Qn

i=1 i!

Vines in dimension n.

copula parameters to estimate.

Facts

I

There are about n2

to

n3

Qn

i=1 i!

Vines in dimension n.

I

The are

copula parameters to estimate.

I

A Vine copula function takes < 1/2s to estimate.

Facts

I

There are about n2

to

n3

Qn

i=1 i!

Vines in dimension n.

I

The are

copula parameters to estimate.

I

A Vine copula function takes < 1/2s to estimate.

I

We compute a pdf , so the VaR takes long to compute.

Issues Spike Let x1 , x2 , x3 ∼ U(0, 1) c1,2;θ (F (x1 ), F (x2 )) Gumbel Copula with θ = 50.

Issues Spike Let x1 , x2 , x3 ∼ U(0, 1) c1,2;θ (F (x1 ), F (x2 )) Gumbel Copula with θ = 50.

f (x1 , x2 , x3 ) =f (x1 ).f (x2 ).f (x3 ) .c1,2 (F (x1 ), F (x2 )).c1,3 (F (x1 ), F (x3 )) .c1,3|2 (F (x1 |x2 ), F (x3 |x2 )).

Issues Spike Let x1 , x2 , x3 ∼ U(0, 1) c1,2;θ (F (x1 ), F (x2 )) Gumbel Copula with θ = 50.

f (x1 , x2 , x3 ) =f (x1 ).f (x2 ).f (x3 ) .c1,2 (F (x1 ), F (x2 )).c1,3 (F (x1 ), F (x3 )) .c1,3|2 (F (x1 |x2 ), F (x3 |x2 )).

Figure: f (x1 |x2 = 0.5)

Issues Spike Let x1 , x2 , x3 ∼ U(0, 1) c1,2;θ (F (x1 ), F (x2 )) Gumbel Copula with θ = 50.

f (x1 , x2 , x3 ) =f (x1 ).f (x2 ).f (x3 ) .c1,2 (F (x1 ), F (x2 )).c1,3 (F (x1 ), F (x3 )) .c1,3|2 (F (x1 |x2 ), F (x3 |x2 )).

Figure: f (x1 |x2 = 0.5) c1,3|2;θ0 (F (x1 |x2 ), F (x3 |x2 )) will be evaluated almost only on extreme events, making the estimation erroneous, and in increasing θ0 , make the whole estimation biased.

Issues Datastream Morgan Stanley evaluation of the French, German and British price indexes from 01/01/06 to 01/12/07.

Issues Datastream Morgan Stanley evaluation of the French, German and British price indexes from 01/01/06 to 01/12/07.

Copula I

Issues Datastream Morgan Stanley evaluation of the French, German and British price indexes from 01/01/06 to 01/12/07.

Copula I

Copula II

Solutions More: I

More Vine formulas.

Solutions More: I

More Vine formulas.

I

More pair copulas.

Solutions More: I

More Vine formulas.

I

More pair copulas.

Solutions More: I

More Vine formulas.

I

More pair copulas.

Solutions More: I

More Vine formulas.

I

More pair copulas.

Solutions More: I

More Vine formulas.

I

More pair copulas.

Solutions More: I

More Vine formulas.

I

More pair copulas.

Solutions Model selection approach:

Lattice structure on the Vine set4 and a test5 .

4 5

Garbiel,Edwards Chen et al. (2004) and Chen and Fan (2006)

Solutions Model selection approach:

Lattice structure on the Vine set4 and a test5 .

Example:

4 5

Garbiel,Edwards Chen et al. (2004) and Chen and Fan (2006)

Solutions Model selection approach:

Lattice structure on the Vine set4 and a test5 .

Example: Z = αX

4 5

Garbiel,Edwards Chen et al. (2004) and Chen and Fan (2006)

Solutions Model selection approach:

Lattice structure on the Vine set4 and a test5 .

Example: Z = αX

4 5

Z = α0 X + β 0 Y

Garbiel,Edwards Chen et al. (2004) and Chen and Fan (2006)

Solutions Model selection approach:

Lattice structure on the Vine set4 and a test5 .

Example: Z = αX I

4 5

Z = α0 X + β 0 Y

We assume that specifications of a false model are false, and that generalizations of valid models are valid.

Garbiel,Edwards Chen et al. (2004) and Chen and Fan (2006)

Solutions Model selection approach:

Lattice structure on the Vine set4 and a test5 .

Example: Z = αX

Z = α0 X + β 0 Y

I

We assume that specifications of a false model are false, and that generalizations of valid models are valid.

I

This allow for an efficient research algorithm and does not require all the models to be tested.

4 5

Garbiel,Edwards Chen et al. (2004) and Chen and Fan (2006)

Example

6

Gaussian proved unnecessary and Student too unstable

Example

I

6

Data: Datastream spot prices of Total, BNP-Paribas, Sanofi-Aventis, GDF-Suez and France-Télécom.

Gaussian proved unnecessary and Student too unstable

Example

I

Data: Datastream spot prices of Total, BNP-Paribas, Sanofi-Aventis, GDF-Suez and France-Télécom.

I

In Sample: from 02/08 to 09/08.

6

Gaussian proved unnecessary and Student too unstable

Example

I

Data: Datastream spot prices of Total, BNP-Paribas, Sanofi-Aventis, GDF-Suez and France-Télécom.

I

In Sample: from 02/08 to 09/08.

I

Out of Sample: from 09/08 to 11/09.

6

Gaussian proved unnecessary and Student too unstable

Example

I

Data: Datastream spot prices of Total, BNP-Paribas, Sanofi-Aventis, GDF-Suez and France-Télécom.

I

In Sample: from 02/08 to 09/08.

I

Out of Sample: from 09/08 to 11/09.

I

Univariate analysis GARCH(1, 4); Bivariate analysis Clayton and Gumbel copulas6 .

6

Gaussian proved unnecessary and Student too unstable

Example

I

Data: Datastream spot prices of Total, BNP-Paribas, Sanofi-Aventis, GDF-Suez and France-Télécom.

I

In Sample: from 02/08 to 09/08.

I

Out of Sample: from 09/08 to 11/09.

I

Univariate analysis GARCH(1, 4); Bivariate analysis Clayton and Gumbel copulas6 .

I

We present the VaR of BNP and Total knowing the others during the out of sample. We see robustness before and after the crisis, though we fail to protect during the crisis.

6

Gaussian proved unnecessary and Student too unstable

Figure: Total

Figure: BNP-Paribas

CONCLUSION & QUESTIONS

Thank you for your attention.

Bibliography I K. Aas, C. Czadob, Frigessic, and H. A., Bakkend. Pair-copula constructions of multiple dependence. Insurance: Mathematics and Economics, 2007. T. Bedford and R.M. Cooke. Probability density decomposition for conditionally dependent random variables modeled by vines. Annals of Mathematics and Artificial Intelligence, 32:245–268, 2001. T. Bedford and R.M. Cooke. Vines: A New Graphical Model for Dependent Random Variables. The Annals of Statistics, 30(4): 1031–1068, 2002. D. Berg and K. Aas. Models for construction of multivariate dependence. Working Paper, 2007. X. Chen and Y. Fan. Estimation of copula-based semiparametric time series models. Journal of Econometrics, 130:307–335, 2006.

Bibliography II

X. Chen, Y. Fan, and A.J. Patton. Simple Tests for Models of Dependence Between Multiple Financial Time Series, with Applications to U.S. Equity Returns and Exchange Rates. Working Paper, 2004. H. Joe. Multivariate Models and Dependence Concepts. Chapman & Hall, 1997.

Test I

H0 : Pr (C (U1 , . . . , Un ) = C0 (U1 , . . . , Un )) = 1 H1 : Pr (C (U1 , . . . , Un ) = C0 (U1 , . . . , Un )) < 1 We define then {Zi }i
n X

[Φ−1 (Zj )]2

1

W follows a χ2d distribution; our test will be based on this result. We defin gW as a kernel estimation of its density function: T

1 X gW (ω) = Kh (ω, Fχ2 (Wt )) d n∗h t

Test II Where Kh (x, y ) is an univariate boundary kernel built on k:  R1 x−y   k( h )/ − xh k(u)du Kh (x, y ) = k( x−y h )   k( x−y )/ R 1 x k(u)du h −

if x ∈ [0, h) if x ∈ [h, 1 − h] if x ∈ (1 − h, 1]

h

R1 The alternative test is based on: Jn = 0 [gW (ω) − 1]2 d ω Then under classical hypothesis we have that: Statn =

(nh1/2 Jn − rn ) → N(0, 1) σ

Where: h i R1 R1Rz rn = h1/2 (h−1 − 2) −1 k 2 (ω)d ω + 2 0 −1 kz2 (y )dy dz i2 R 1 hR 1 2 σ = 2 −1 −1 k(u + v )k(v )dv du Rz kz (y ) = k(y )/ −1 k(u)du

Lattice I

I

We call C-index (for copula index) any I such that I ∈ [0, n]2 × P([0, n]) and I = (A, B) ⇒ A ∩ B = ∅. It represents the index of the copulas. The first set contains the number of the variables it is the copula of, the second one which indeces are known. For example the C-index of c1,2|3,4 is {{1, 2} , {3, 4}} We call Model any set M of C-index such that there exists a tree where each element of M is the index of one of the pair copulas in the resulting vine formula. The n-variate copula associated with M is : Y cm m∈M

I

We call M n the set of Models in dimension n.

Afterward the lattice structure we choose on the vine set is that induced by (M n , ⊂). In this case, if (Φ, Θ) ∈M 2n and Φ ⊂ Θ, then we say that Θ is a specification or submodel of Φ because more pair copulas are specified.