Module 18: Copulas

Question 1

Copula

Answer 1

A copula is a joint cumulative distribution function expressed in terms of the marginal cumulative distribution functions.

A copula in n-dimensions would be expressed as:
C(u) = C(u₁, u₂, …, uₙ) = P(U₁ ≤ u₁, U₂ ≤ u₂, …, Uₙ ≤ uₙ)

Where uᵢ = Fₓᵢ(𝑥ᵢ), ie the values of the individual cumulative distribution functions (CDFs), each of which lie in the range [0,1].

A copula is determined by the relative order of the observations rather than by the exact shape of the marginal distribution (the invariance property).

Question 2

A copula must have 3 properties

Answer 2

• It must be an increasing function of its inputs.
• If the values of all but one of the marginal CDFs are equal to 1, then the copula is equal to the value of the remaining marginal CDF.
• The copula must always return a non-negative probability.

Question 3

Sklar’s Theorem

Answer 3

Say’s that if F is a joint CDF and F₁, …, Fₙ are marginal CDFs, then there exists a copula such that for all 𝑥₁, …, 𝑥ₙ ϵ [-∞, ∞]

F_{X₁, …, Xₙ} ( 𝑥₁, …, 𝑥ₙ) = C_{X₁, …, Xₙ} [ F_𝑥₁ (𝑥₁), …, F_𝑥ₙ (𝑥ₙ) ]

Furthermore if the marginal distributions are continuous, then the copula is unique.

Question 4

Survival Copula

Answer 4

For each copula, there is a corresponding survival copula expressing the joint survival probability in terms of the marginal survival probabilities.

Question 5

Concordance (or association)

Answer 5

Concordance between two variables, X and Y, does not necessarily imply dependence (eg the two variables may instead be linked by a third variable).

Question 6

3 Measures of concordance

Answer 6

• Pearson’s rho
• Spearman’s rho
• Kendall’s tau

Question 7

Explicit copulas

Answer 7

have simple closed-form expressions.

An important subclass is that of Archimedian copulas

Question 8

4 Examples of Archimedian copulas

Answer 8

1. Gumbel
2. Frank
3. Clayton
4. Generalised Clayton

Question 9

Archimedian copulas

Answer 9

Define closed-form probability distributions, but are limited by the small number of parameters available to describe multivariate relationships.

Question 10

Implicit Copulas

Answer 10

Based on well-known multivariate distributions, but no simple closed-form expression exists for them. Examples are the Normal (or Gaussian) copula and the (Student’s) t-copula.

The t-copula allows for more flexibility than the normal copula in the level of tail dependency. The smaller the value of γ, the greater the level of tail dependency. As γ →∞, the t-copula tends to the normal copula.

Question 11

Choosing copulas

Answer 11

Possible copula candidates might be selected based on features of the sample data:

• upper, lower (and other) ‘tail’ dependencies
• patterns of dependence, eg general positive co-dependence.

Question 12

Fitting copulas

Answer 12

Copulas might be fitted using::

• maximum likelihood
• parameterisation based on rank correlations in the sample