Models for construction of multivariate dependence Workshop on Copulae and Multivariate Probability distributions in Finance – Theory, Applications, Opportunities and Problems, Warwick, 14. September Kjersti Aas, Norwegian Computing Center Joint work with Daniel Berg

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 (as far as we know, both of them were originally proposed by Harry Joe): ▪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:

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 All bivariate copulae that have not been directly specified will have copula C 21.

HNAC ► The 12-dimensional case shown in the figure:

Parameter estimation ► 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 is often 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. ► Most of the algorithms proposed include higher- order derivatives of the generator, inverse generator or copula functions. These are usually extremely complex for high dimensions. ► There are some exceptions for special cases: ▪McNeil (2007) uses the Laplace-transform method for the FNAC (only Gumbel and Clayton). ▪McNeil (2007) also uses the Laplace-transform method for the 4-dimensional PNAC, but does not extend this algorithm to higher-dimensional PNACs.

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 two main types of PCCs. The density above corresponds to a D-vine. There is also a type denoted canonical vines.

Parameter estimation ► 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 this restriction.

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.

Precipitation ► Kendall’s tau for pairs of variables

Precipitation data We use Gumbel-Hougaard copulae for all pairs. The copulae at level one in both constructions are those corresponding to the largest tail dependence coefficients. HNAC PCC We compare: We use Gumbel-Hougaard copulae for all pairs.

Precipitation data The goodness-of-fit test suggested by Genest and Rémilliard (2005) and Genest et. al (2007) strongly rejects the HNAC (P-value is 0.000), while the PCC is not rejected (P-value is ).

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.

Equity returns ► Kendall’s tau for pairs of variables

Equity returns HNAC PCC We compare: We use Gumbel- Hougaard for all pairs. We use the Student copula for all pairs. The copulae at level one in both constructions are those corresponding to the largest tail dependence coefficients.

Equity returns HNAC PCC The goodness-of-fit test strongly rejects the HNAC (P-value is 0.000), while the PCC is not rejected (P-value is ). The P-value for a PCC with Gumbel copulae is

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 two important restrictions ▪The level of dependence must decrease with the level of nesting. ▪The involved copulae have to be Archimedean. ► 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.