Models for construction of multivariate dependence 2nd Vine Copula Workshop, Delft, 16. December 2008 Kjersti Aas, Norwegian Computing Center Joint work with Daniel Berg Paper is accepted for publication in European Journal of Finance.

Introduction (I) ► Apart from the Gaussian and Student copulae, the set of higher-dimensional copulae proposed in the literature is rather limited. ► When it comes to Archimedean copulae, the most common multivariate extension, the exchangeable one, is extremely restrictive, allowing only one parameter regardless of dimension.

Introduction (II) ► There have been some attempts at constructing more flexible multivariate Archimedean copula extensions. ► In this talk we examine two such hierarchical constructions ▪The nested Archimedean constructions (NACs) ▪The pair-copula constructions (PCCs) ► In both constructions, the multivariate data set is modelled using a cascade of lower-dimensional copulae. ► They differ however in their modelling of the dependency structure.

Content ► The nested Archimedean constructions (NACs) ► The pair-copula constructions (PCCs) ► Comparison ► Applications ▪Precipitation data ▪Equity returns

The nested Archimedean constructions (NACs)

Content ► The fully nested construction (FNAC) ► The partially nested construction (PNAC) ► The hierarchically nested construction (HNAC) ► Parameter estimation ► Simulation

The FNAC ► The FNAC was originally proposed by Joe (1997) and is also discussed in Embrechts et al. (2003), Whelan (2004), Savu and Trede (2006) and McNeil (2007). ► Allows for the specification of at most d-1 copulae, while the remaining unspecified copulae are implicitly given through the construction. ► All bivariate margins are Archimedean copulae.

The FNAC The pairs (u 1,u 3 ) and (u 2,u 3 ) both have copula C 21. The pairs (u 1,u 4 ), (u 2,u 4 ) and (u 3,u 4 ) all have copula C 31. Decreasing dependence

The FNAC ► The 4-dimensional case shown in the figure: ► The d-dimensional case:

Restrictions

The PNAC ► The PNAC was originally proposed by Joe (1997) and is also discussed in Whelan (2004), McNeil et. al. (2006) and McNeil (2007). ► Allows for the specification of at most d-1 copulae, while the remaining unspecified copulae are implicitly given through the construction. ► Can be understood as a composite between the exchangeable copula and the FNAC, since it is partly exchangeable.

The PNAC Decreasing dependence All pairs (u 1,u 3 ), (u 1,u 4 ), (u 2,u 3 ) and (u 2,u 4 ) have copula C 2,1. Exchangeable between u 1 and u 2 Exchangeable between u 3 and u 4

The PNAC ► The 4-dimensional case shown in the figure:

The HNAC ► The HNAC was originally proposed by Joe (1997) and is also mentioned in Whelan (2004). However, Savu and Trede (2006) were the first to work out the idea in full generality. ► This structure is an extension of the PNAC in that the copulae involved do not need to be bivariate. ► Both the FNAC and the PNAC are special cases of the HNAC.

The HNAC Decreasing dependence

HNAC ► The 9-dimensional case shown in the figure:

Parameter estimation (I) ► Full estimation of a NAC should in principle consider the following three steps: ▪The selection of a specific factorisation ▪The choice of copula families ▪The estimation of the copula parameters.

Parameter estimation (II) ► For all NACs parameters may be estimated by maximum likelihood. ► However, it is in general not straightforward to derive the density. One usually has to resort to a computer algebra system, such as Mathematica. ► Moreover, the density must often be obtained by a recursive approach. This means that the number of computational steps needed to evaluate the density increases rapidly with the complexity of the copula.

Simulation ► Simulation from higher-dimensional NACs is not straightforward in general. ► The Laplace-transform method may be used for some specific NACs, e.g. when all copulae are Gumbel. ► Otherwise the conditional distribution method must be used. ▪This method involves the d-1’th derivative of the copula function (which usually is extremely complex) and in most cases numerical inversion. ▪Hence, simulation is challenging even for small dimensions.

The pair-copula constructions (PCCs)

PCCs ► The PCC was originally proposed by Joe (1996) and it has later been discussed in detail by Bedford and Cooke (2001, 2002), Kurowicka and Cooke (2006) (simulation) and Aas et. al. (2007) (inference). ► Allows for the specification of d(d-1)/2 bivariate copulae, of which the first d-1 are unconditional and the rest are conditional. ► The bivariate copulae involved do not have to belong to the same class.

PCC No restrictions on dependence C 2,1 is the copula of F(u 1 |u 2 ) and F(u 3 |u 2 ). C 2,2 is the copula of F(u 2 |u 3 ) and F(u 4 |u 3 ). C 3,1 is the copula of F(u 1 |u 2,u 3 ) and F(u 4 |u 2,u 3 ).

PCC ► The density corresponding to the figure is ► where

PCC ► The d-dimensional density is given by ► where ► Note that there are other types of PCCs. The density above corresponds to a D-vine. D-vines belong to a broader class denoted regular vines.

Parameter estimation (I) ► Full estimation of a PCC should in principle consider the same three steps as for the NACs: ▪The selection of a specific factorisation ▪The choice of copula families ▪The estimation of the copula parameters.

Parameter estimation (II) ► The parameters of the PCC may be estimated by maximum likelihood. ► Since the density is explicitly given, the procedure is simpler than the one for the NACs. ► However, the likelihood must be numerically maximised, and parameter estimation becomes time consuming in higher dimensions.

Simulation ► The simulation algorithm for the D-vine is straightforward and simple to implement. ► Like for the NACs, the conditional inversion method is used. ► However, to determine each of the conditional distribution functions involved, only the first partial derivative of a bivariate copula needs to be computed. ► Hence, the simulation procedure for the PCC is in general much simpler and faster than for the NACs.

Comparison

Flexibility When looking for appropriate data sets for the comparison of these structures, it turned out to be quite difficult to find real- world data sets satisfying the constraints for the NACs.

Computational efficiency Computational times (seconds) in R. Estimation and likelihood: 4-dimensional data set with 2065 observations. Simulation: 1000 observations

Structure ► The multivariate distribution defined through a NAC will always by definition be an Archimedean copula and all bivariate margins will belong to a known parametric family. ► For the PCCs, neither the multivariate distribution nor the unspecified bivariate margins will belong to a known parametric family in general.

Applications

Applications ► Precipitation data ▪Parameter estimation ▪Goodness-of-fit ► Equity returns ▪Parameter estimation ▪Goodness-of-fit ▪Out-of-sample validation

Precipitation data Four Norwegian weather stations Daily data from to observ. Convert precipitation vectors to uniform pseudo-observations before further modelling.

Pseudo-observations

Precipitation ► Kendall’s tau for pairs of variables Ski and Vestby are closely located Hurdal and Nannestad are closely located

Precipitation data We use either Gumbel or Frank copulae for all pairs. The copulae at the bottom level in both constructions are those corresponding to the largest Kendall’s tau values. NAC PCC We compare: We use either Gumbel, Frank or Student copulae for all pairs.

Estimated parameters S n is the statistic suggested by Genest and Rémilliard (2005) T n is the statistic suggested by Genest et. al. (2006).

Equity returns Four stocks; two from oil sector and two from telecom. Daily data from to observ. Log-returns are processed through a GARCH-NIG-filter and converted to uniform pseudo-observations before further modelling.

Pseudo-observations

Equity returns ► Kendall’s tau for pairs of variables Oil sector : British Petroleum (BP) and Exxon Mobile Corp (XOM). Telecom sector : Deutsche Telekom AG (DT) and France Telecom (FTE).

Equity returns HNAC PCC We compare: We use Frank copula for all pairs. We use either Student or Frank copula for all pairs. The copulae at the bottom level in both constructions are those corresponding to the largest tail dependence coefficients.

Parameter estimates

Equity returns ► With increasing complexity of models, there is always the risk of overfitting the data. ► The examine whether this is the case for our equity example, we validate the GARCH-NIG-PCC model out-of-sample. ► We put together an equally-weighted portfolio of the four stocks. ► The estimated model is used to forecast 1-day VaR for each day in the period from to

Equity returns 5% VaR 1% VaR 0.5% VaR We use the likelihood ratio statistic by Kupiec (1995) to compute the P-values PCC works well out of sample!

Summary

Summary ► The NACs have three important restrictions ▪There are strong limitations on the parameters. ▪The involved copulae have to be Archimedean. ▪The Archimedean copulae can not be freely mixed. ► The PCCs are in general more computationally efficient than the NACs both for simulation and parameter estimation. ► The NAC is strongly rejected for two different four- dimensional data sets (rain data and equity returns) while the PCC provides an appropriate fit. ► The PCC does not seem to overfit data.