Pair-copula constructions of multiple dependence

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Pair-copula constructions of multiple dependence

• Workshop on ''Copulae Theory and Practice''
• Weierstrass Institute for Applied Analysis and
  Stochastics, Berlin 7. December, 2007

Kjersti Aas

Norwegian Computing Center Joint work with


Claudia Czado, Arnoldo Frigessi and
Henrik Bakken
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Dependency modelling

• Appropriate modelling of dependencies is very
  important for quantifying different kinds of
  financial risk.
• The challenge is to design a model that
  represents empirical data well, and at the same
  time is sufficiently simple and robust to be used
  in simulation-based inference for practical risk
  management.

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State-of-the-art

• Parametric multivariate distribution
• Not appropriate when all variables do not have
  the same distribution.
• Marginal distributions copula
• Not appropriate when all pairs of variables do
  not have the same dependency structure.
• In addition, building higher-dimensional copulae
  (especially Archimedean) is generally recognized
  as a difficult problem.

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Introduction

• The pioneering work of Joe (1996) and Bedford and
  Cooke (2001) decomposing a multivariate
  distribution into a cascade of bivariate copulae
  has remained almost completely overlooked.
• We claim that this construction represent a very
  flexible way of constructing higher-dimensional
  copulae.
• Hence, it can be a powerful tool for model
  building.

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Copula

• Definition
  A copula is a
  multivariate distribution C with uniformly
  distributed marginals U(0,1) on 0,1.
• For any joint density f corresponding to an
  absolutely continuous joint distribution F with
  strictly continuous marginal distribution
  functions F1,Fn it holds that
• for some n-variate copula density

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Pair-copula decomposition (I)

• Also conditional distributions might be expressed
  in terms of copulae.
• For two random variables X1 and X2 we have
• And for three random variables X1, X2 and X3
• where the decomposition of f(x1x2) is
  given above.

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Par-copula decomposition (II)
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Pair-copula decomposition (III)
We denote a such decomposition a pair-copula
decomposition
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Example I Three variables

• A three-dimensional pair-copula decomposition is
  given by

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Building blocs

• It is not essential that all the bivariate
  copulae involved belong to the same family. The
  resulting multivariate distribution will be valid
  even if they are of different type.
• One may for instance combine the following types
  of pair-copulae
• Gaussian (no tail dependence)
• Students t (upper and lower tail dependence)
• Clayton (lower tail dependence)
• Gumbel (upper tail dependence)

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Example II Five variables

• A possible pair-copula decomposition of a
  five-dimensional density is
• There are as many as 480 different such
  decompositions in the five-dimensional case..

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Vines

• Hence, for high-dimensional distributions, there
  are a significant number of possible pair-copula
  constructions.
• To help organising them, Bedford and Cooke (2001)
  and (Kurowicka and Cooke, 2004) have introduced
  graphical models denoted
• Canonical vines
• D-vines
• Each of these graphical models gives a specific
  way of decomposing the density.

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General density expressions

• Canonical vine density
• D-vine density

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Five-dimensional canonical vine
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Five-dimensional D-vine
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Conditional distribution functions

• The conditional distribution functions are
  computed using (Joe, 1996)
• For the special case when v is univariate, and x
  and v are uniformly distributed on 0,1, we have
• where ? is the set of copula parameters.

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Simulation
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Uniform variables

• In the rest of this presentation we assume for
  simplicity that the margins of the distributions
  of interest are uniform, i.e. f(xi)1 and
  F(xi)xi for all i.

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Simulation procedure (I)

• For both the canonical and the D-vine, n
  dependent uniform 0,1 variables are sampled as
  follows
• Sample wi i1,,n independent uniform on 0,1
• Set

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Simulation procedure (II)

• The procedures for the canonical and D-vine
  differs in how F(xjx1,x2,,xj-1) is computed.
• For the canonical vine, F(xjx1,x2,,xj-1) is
  computed as
• For the D-vine, F(xjx1,x2,,xj-1) is computed as

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Simulation algorithm for canonical vine
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Parameter estimation
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Three elements

• Full inference for a pair-copula decomposition
  should in principle consider three elements
• The selection of a specific factorisation
• The choice of pair-copula types
• The estimation of the parameters of the chosen
  pair-copulae.

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Which factorisation?

• For small dimensions one may estimate the
  parameters of all possible decompositions and
  comparing the resulting log-likelihood values.
• For higher dimensions, one should instead
  consider the bivariate relationships that have
  the strongest tail dependence, and let this
  determine which decomposition(s) to estimate.
• Note, that in the D-vine we can select more
  freely which pairs to model than in the canonical
  vine.

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Choice of copulae types

• If we choose not to stay in one predefined class,
  we may use the following procedure

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Likelihood evaluation
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Three important expressions

• For each pair-copula in the decomposition, three
  expressions are important
• The bivariate density
• The h-function
• The inverse of the h-function (for simulation).
• For the Gaussian, Students t and Clayton
  copulae, all three are easily derived.
• For other copulae, e.g. Gumbel, the inverse of
  the h-function must be obtained numerically.

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Application Financial returns
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Tail dependence

• Tail dependence properties are often very
  important in financial applications.
• The n-dimensional Students t-copula has been
  much used for modelling financial return data.
• However, it has only one parameter for modelling
  tail dependence, independent of dimension.
• Hence, if the tail dependence of different pairs
  of risk factors in a portfolio are very
  different, we believe the pair-copulae
  decomposition with Students t-copulae for all
  pairs to be better.

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Data set

• Daily data for the period from 04.01.1999 to
  08.07.2003 for
• The Norwegian stock index (TOTX) T
• The MSCI world stock index M
• The Norwegian bond index (BRIX) B
• The SSBWG hedged bond index S
• The empirical data vectors are filtered through a
  GARCH-model, and converted to uniform variables
  using the empirical distribution functions before
  further modeling.
• Degrees of freedom when fitting Students
  t-copulae to each pair of variables

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D-vine structure
Six pair-copulae in the decomposition two
parameters for each copula.
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The six data sets used
cSM cMT
cTB
cSTM cMBT
cSBMT
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Estimated parameters
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Comparison with Students t-copula

• AIC
• 4D Students t-copula -512.33
• 4D Students t pair-copula decomposition -487.42
• Likelihood ratio test statistic
• Likelihood difference is 34.92 with 5 df
• P-value is 1.56e-006 gt 4D Students t-copula is
  rejected in favour of the pair-copula
  decomposition.

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Tail dependence

• Upper and lower tail dependence coefficients for
  the bivariate Students t-copula (Embrechts et
  al., 2001).
• Tail dependence coefficients conditional on the
  two different dependency structures
• For a trader holding a portfolio of
  international stocks and bonds, the practical
  implication of this difference in tail dependence
  is that the probability of observing a large
  portfolio loss is much higher for the
  four-dimensional pair copula decomposition.