Models for construction of multivariate dependence

1
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.
2
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.

3
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.

4
Content

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

5
The nested Archimedean constructions (NACs)
6
Content

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

7
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.

8
The FNAC
The pairs (u1,u3) and (u2,u3) both have copula
C21.
The pairs (u1,u4), (u2,u4) and (u3,u4) all have
copula C31.
Decreasing dependence
9
The FNAC

• The 4-dimensional case shown in the figure
• The d-dimensional case

10
Restrictions
11
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.

12
The PNAC
All pairs (u1,u3), (u1,u4), (u2,u3) and (u2,u4)
have copula C2,1.
Decreasing dependence
Exchangeable between u1 and u2
Exchangeable between u3 and u4
13
The PNAC

• The 4-dimensional case shown in the figure

14
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.

15
The HNAC
Decreasing dependence
16
HNAC

• The 9-dimensional case shown in the figure

17
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.

18
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.

19
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-1th derivative of the
  copula function (which usually is extremely
  complex) and in most cases numerical inversion.
• Hence, simulation is challenging even for small
  dimensions.

20
The pair-copula constructions (PCCs)
21
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.

22
PCC
C2,1 is the copula of F(u1u2) and F(u3u2).
C2,2 is the copula of F(u2u3) and F(u4u3).
No restrictions on dependence
C3,1 is the copula of F(u1u2,u3) and F(u4u2,u3).
23
PCC

• The density corresponding to the figure is
• where

24
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.

25
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.

26
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.

27
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.

28
Comparison
29
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.
30
Computational efficiency
Computational times (seconds) in R.
Estimation and likelihood 4-dimensional data
set with 2065 observations.
Simulation 1000 observations
31
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.

32
Applications
33
Applications

• Precipitation data
• Parameter estimation
• Goodness-of-fit
• Equity returns
• Parameter estimation
• Goodness-of-fit
• Out-of-sample validation

34
Precipitation data
Four Norwegian weather stations
Daily data from 01.01.90 to 31.12.06 2065 observ.
Convert precipitation vectors to uniform
pseudo-observations before further modelling.
35
Pseudo-observations
36
Precipitation

• Kendalls tau for pairs of variables

Ski and Vestby are closely located Hurdal and
Nannestad are closely located
37
Precipitation data
We compare
NAC
PCC
We use either Gumbel or Frank copulae for all
pairs.
We use either Gumbel, Frank or Student copulae
for all pairs.
The copulae at the bottom level in both
constructions are those corresponding to the
largest Kendalls tau values.
38
Estimated parameters
Sn is the statistic suggested by Genest and
Rémilliard (2005) Tn is the statistic suggested
by Genest et. al. (2006).
39
Equity returns
Four stocks two from oil sector and two from
telecom.
Daily data from 14.08.03 to 29.12.06 852 observ.
Log-returns are processed through a
GARCH-NIG-filter and converted to uniform
pseudo-observations before further modelling.
40
Pseudo-observations
41
Equity returns

• Kendalls 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).
42
Equity returns
We compare
HNAC
PCC
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.
43
Parameter estimates
44
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 30.12.06 to
  11.06.07.

45
Equity returns
PCC works well out of sample!
We use the likelihood ratio statistic by Kupiec
(1995) to compute the P-values
46
Summary
47
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.