Pair-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

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 recognised 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 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.

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 240 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 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.

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. *May be used since the 4D Student’s t-copula is a special case of the 4-dimensional Student’s t 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.

Some robustness studies

Robustness studies ► Different factorisations ► Different copula families

Other factorisations (I) ► We also estimated the parameters for the 11 other D-vine factorisations for the 4-dimensional data set. Maximum difference is 4.5% p-value for likelihood ratio test is p-value for likelihood ratio test is 1.56e-006.

Other 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

Other factorisations (III) ► Histogram of differences between highest and lowest likelihood Min 1.78 Max Mean 8.51 Observed 12.18

Other factorisations (IV) ► We simulated 50,000 realisations of the dependency structure corresponding to the portfolio P = 0.25*S *M *B *T 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.

Other 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.

Copulae from different families (I) Clayton copula?

Copulae from different families (II) ► To investigate whether the Clayton copula is a better choice than the Student’s t-copula for the pair F(x S |x M ), F(x B |x T ), 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 Student’s t-copula it has to be numerically derived. ► We plot (z) = z – K(z)

Copulae from different families (III) Student’s t Clayton Student’s t-copula remains the best choice!

Copulae from different families (IV) ► Assume that C 12 is a Student copula ► Assume that C 23 is a Student copula ► Let F(u 1 |u 2 )=  C 12 (u 1,u 2 )/  u 2 ► Let F(u 3 |u 2 )=  C 23 (u 2,u 3 )/  u 2 ► Do we then have any prior knowledge of the copula C(F(u 1 |u 2 ), F(u 3 |u 2 ))? ▪Will it be best modelled by a Student copula? ▪Will it have both upper and lower tail dependence?

Comparison with other constructions.

Studies ► 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.