Models for construction of multivariate dependence

1
Models for construction of multivariate
dependence

• Workshop on Copulae and Multivariate Probability
  distributions in Finance Theory, Applications,
  Opportunities and Problems, Warwick, 14.
  September 2007.
  

Kjersti Aas, Norwegian Computing Center
Joint work with Daniel Berg
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 (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.

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
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.

11
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
12
The PNAC

• The 4-dimensional case shown in the figure

13
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.

14
The HNAC
All bivariate copulae that have not been directly
specified will have copula C21.
Decreasing dependence
15
HNAC

• The 12-dimensional case shown in the figure

16
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.

17
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.

18
The pair-copula constructions (PCCs)
19
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.

20
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).
21
PCC

• The density corresponding to the figure is
• where

22
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.

23
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.

24
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.

25
Comparison
26
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.
27
Computational efficiency
Computational times (seconds) in R.
Estimation and likelihood 4-dimensional data
set with 2065 observations.
Simulation 1000 observations
28
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.

29
Applications
30
Applications

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

31
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.
32
Precipitation

• Kendalls tau for pairs of variables

33
Precipitation data
We compare
HNAC
PCC
We use Gumbel-Hougaard copulae for all pairs.
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.
34
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 0.1635).
35
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.
36
Equity returns

• Kendalls tau for pairs of variables

37
Equity returns
We compare
HNAC
PCC
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.
38
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 0.3142). The P-value for a
PCC with Gumbel copulae is 0.0885.
39
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.

40
Equity returns
PCC works well out of sample!
We use the likelihood ratio statistic by Kupiec
(1995) to compute the P-values
41
Summary
42
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.