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

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.

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.

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.

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 F 1,…F n it holds that for some n-variate copula density

Pair-copula decomposition (I) ► Also conditional distributions might be expressed in terms of copulae. ► For two random variables X 1 and X 2 we have ► And for three random variables X 1, X 2 and X 3 where the decomposition of f(x 1 |x 2 ) is given above.

Par-copula decomposition (II)

Pair-copula decomposition (III) We denote a such decomposition a pair-copula decomposition

Example I:Three variables ► A three-dimensional pair-copula decomposition is given by

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) ▪Student’s t (upper and lower tail dependence) ▪Clayton (lower tail dependence) ▪Gumbel (upper tail dependence)

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…..

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.

General density expressions ► Canonical vine density ► D-vine density

Five-dimensional canonical vine

Five-dimensional D-vine

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.

Simulation

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(x i )=1 and F(x i )=x i for all i.

Simulation procedure (I) ► For both the canonical and the D-vine, n dependent uniform [0,1] variables are sampled as follows: ► Sample w i ; i=1,…,n independent uniform on [0,1] ► Set

Simulation procedure (II) ► The procedures for the canonical and D-vine differs in how F(x j |x 1,x 2,…,x j-1 ) is computed. ► For the canonical vine, F(x j |x 1,x 2,…,x j-1 ) is computed as ► For the D-vine, F(x j |x 1,x 2,…,x j-1 ) is computed as

Simulation algorithm for canonical vine

Parameter estimation

Three elements ► Full inference for a pair-copula decomposition should in principle consider three elements: 1. The selection of a specific factorisation 2. The choice of pair-copula types 3. The estimation of the parameters of the chosen pair-copulae.

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.

Choice of copulae types ► If we choose not to stay in one predefined class, we may use the following procedure:

Likelihood evaluation

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, Student’s 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.

Application: Financial returns

Tail dependence ► Tail dependence properties are often very important in financial applications. ► The n-dimensional Student’s 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 Student’s t-copulae for all pairs to be better.

Data set ► Daily data for the period from to for ▪The Norwegian stock index (TOTX)T ▪The MSCI world stock indexM ▪The Norwegian bond index (BRIX)B ▪The SSBWG hedged bond indexS ► 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 Student’s t-copulae to each pair of variables

D-vine structure Six pair-copulae in the decomposition – two parameters for each copula.

The six data sets used c SM c MT c TB c ST|M c MB|T c SB|MT

Estimated parameters

Comparison with Student’s t-copula ► AIC ▪4D Student’s t-copula ▪4D Student’s t pair-copula decomposition ► Likelihood ratio test statistic ▪Likelihood difference is with 5 df ▪P-value is 1.56e-006 => 4D Student’s t-copula is rejected in favour of the pair-copula decomposition.

Tail dependence ► Upper and lower tail dependence coefficients for the bivariate Student’s 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.