Title: Paircopula constructions of multiple dependence
1Pair-copula constructions of multiple dependence
- Vine Copula Workshop
- Delft Institute of Applied Mathematics,
19. November, 2007
Kjersti Aas
Norwegian Computing Center Joint work with
Claudia Czado, Daniel Berg, Ingrid
Hobæk Haff, Arnoldo Frigessi and Henrik Bakken
2Dependency 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.
3State-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 recognised
as a difficult problem.
4Introduction
- 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 overseen. - 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.
5Copula
- 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
6Pair-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.
7Par-copula decomposition (II)
8Pair-copula decomposition (III)
We denote a such decomposition a pair-copula
decomposition
9Example I Three variables
- A three-dimensional pair-copula decomposition is
given by
10Building 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)
11Example II Five variables
- A possible pair-copula decomposition of a
five-dimensional density is - There are as many as 240 different such
decompositions in the five-dimensional case..
12Vines
- 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.
13General density expressions
- Canonical vine density
- D-vine density
14Five-dimensional canonical vine
15Five-dimensional D-vine
16Conditional 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.
17Simulation
18Uniform 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.
19Simulation 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
20Simulation 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
21Simulation algorithm for canonical vine
22Parameter estimation
23Three 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.
24Which 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 are
most important to model correctly, 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.
25Choice of copulae types
- If we choose not to stay in one predefined class,
we may use the following procedure
26Likelihood evaluation
27Three 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.
28Application Financial returns
29Tail 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.
30Data 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
31D-vine structure
Six pair-copulae in the decomposition two
parameters for each copula.
32The six data sets used
cSM cMT
cTB
cSTM cMBT
cSBMT
33Estimated parameters
34Comparison 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. - May be used since the 4D Students t-copula is
a special case of the 4-dimensional Students t
pair-copula decomposition
35Tail 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.
36Some robustness studies
37Robustness studies
- Different factorisations
- Different copula families
38Other factorisations (I)
- We also estimated the parameters for the 11 other
D-vine factorisations for the 4-dimensional data
set.
p-value for likelihood ratio test is 1.56e-006.
Maximum difference is 4.5
p-value for likelihood ratio test is 0.06.
39Other factorisations (II)
- We also did the following experiment
- Do 100 times
- Simulate 1094 observations from the D-vine
corresponding to the highest likelihood. - For combination 1 to 12
- Estimate parameters and compute likelihood
- Compute difference between highest and lowest
likelihood. - End do
40Other factorisations (III)
- Histogram of differences between highest and
lowest likelihood
Min 1.78 Max 19.34 Mean
8.51 Observed 12.18
41Other factorisations (IV)
- We simulated 50,000 realisations of the
dependency structure corresponding to the
portfolio
P 0.25S - 0.25M -
0.25B 0.25T
- one day ahead using the combinations that
gave the highest and lowest likelihood, and
compared some quantiles with the corresponding
ones obtained for the data and the Student copula.
42Other factorisations (V)
- Not very large differences between different
combinations. - However, the worst combination is not
significantly better than the Student copula. - We still believe in modelling the pairs with the
strongest tail dependence at the base level.
43Copulae from different families (I)
Clayton copula?
44Copulae from different families (II)
- To investigate whether the Clayton copula is a
better choice than the Students t-copula for the
pair F(xSxM), F(xBxT), we examine the degree of
closeness of the parametric and non-parametric
versions of the distribution function K(z)
defined by - For the Clayton copula K(z) is given by an
explicit expression, while for the Students
t-copula it has to be numerically derived. - We plot ?(z) z K(z)
45Copulae from different families (III)
Clayton
Students t
Students t-copula remains the best choice!
46Copulae from different families (IV)
- Assume that C12 is a Student copula
- Assume that C23 is a Student copula
- Let F(u1u2)? C12(u1,u2)/? u2
- Let F(u3u2)? C23(u2,u3)/? u2
- Do we then have any prior knowledge of the copula
C(F(u1u2), F(u3u2))? - Will it be best modelled by a Student copula?
- Will it have both upper and lower tail dependence?
47Comparison with other constructions.
48Studies
- Berg and Aas (2007)
- Compare pair-copula constructions (PCCs) with
nested Archimedean constructions (NACs) for
rainfall data and equity returns (both data sets
are 4-dimensional). - PCC superior for both data sets.
- Fischer, Köck, Schlüter, Weigert (2007)
- Compare PCCs with 5 other types of multivariate
copulas (including NACs) for stock returns,
exchange rate returns and metal returns (all data
sets are 4-dimensional). - PCC superior for all three data sets.