Common Copulas: Gaussian, t and Archimedean Copulas Explained

Understand Common Copulas: Gaussian, t and Archimedean

Key rules to remember

Sklar's theorem

F(x, y) = C(F_X(x), F_Y(y))

C is unique when the margins are continuous.

Archimedean copula from generator

C(u, v) = φ⁻¹(φ(u) + φ(v))

φ is continuous, strictly decreasing and convex, with φ(1) = 0.

Clayton copula

C(u, v) = (u^(−α) + v^(−α) − 1)^(−1/α), α > 0

Generator φ(t) = t^(−α) − 1. Lower tail dependence λ_L = 2^(−1/α). Upper tail dependence is 0.

Gumbel copula

C(u, v) = exp{ −[(−ln u)^α + (−ln v)^α]^(1/α) }, α ≥ 1

Generator φ(t) = (−ln t)^α. Upper tail dependence λ_U = 2 − 2^(1/α). Lower tail dependence is 0. α = 1 gives independence.

Frank copula

C(u, v) = −(1/α) ln[1 + (e^(−αu) − 1)(e^(−αv) − 1) ÷ (e^(−α) − 1)], α ≠ 0

Generator φ(t) = −ln[(e^(−αt) − 1) ÷ (e^(−α) − 1)]. No tail dependence. α > 0 positive dependence, α < 0 negative.

Kendall's tau for Clayton and Gumbel

Clayton: τ = α ÷ (α + 2). Gumbel: τ = 1 − 1/α

Use to get α from a sample τ.

Tail dependence of the t copula

λ_L = λ_U = 2 × t_(ν+1)( −√((ν + 1)(1 − ρ) ÷ (1 + ρ)) )

Positive for every ρ > −1 and finite ν. The Gaussian copula has λ = 0 for ρ < 1.

Gaussian and t copula link to Kendall's tau

ρ = sin(πτ ÷ 2)

Holds for elliptical copulas, including Gaussian and t.

How to solve Common Copulas: Gaussian, t and Archimedean questions

1. 1Read what the data or scenario shows: do extreme values cluster at the high end, the low end, both ends, or neither?
2. 2Match the tail pattern to a family: lower tail Clayton, upper tail Gumbel, both tails t, no tail dependence Gaussian or Frank.
3. 3Write the copula formula with its parameter range, and state any generator if asked.
4. 4If a sample Kendall's tau or Spearman's rho is given, convert it to the parameter using the family's relationship.
5. 5Compute the required quantity: C(u, v), a joint probability, or the tail dependence coefficient.
6. 6For joint probabilities, convert margins to uniforms first: u = F_X(x), v = F_Y(y).
7. 7State the result with units and one line of interpretation, such as how much more likely joint extremes are than under independence.

Quickest way: Tail-pattern shortcut

Common mistakes in Common Copulas: Gaussian, t and Archimedean

• Saying the Gaussian copula has tail dependence when correlation is high.

  High ρ gives strong dependence in the centre, which feels like tail dependence.

  Fix: Remember that for ρ < 1 the Gaussian copula has λ = 0. Only the t copula (and Clayton, Gumbel) give tail dependence.

• Mixing up which tail Clayton and Gumbel affect.

  Both names sound similar and students memorise them without the shape.

  Fix: Clayton is lower tail, Gumbel is upper tail. Link Clayton with joint falls and losses at the bad end.

• Using a wrong parameter range, such as α = 0.5 for Gumbel.

  Students forget each family has its own allowed range.

  Fix: Clayton α > 0, Gumbel α ≥ 1, Frank α ≠ 0. Check the range before substituting.

• Putting x and y values straight into the copula.

  Students forget the copula takes uniform inputs.

  Fix: Compute u = F_X(x) and v = F_Y(y) first, then apply C(u, v).

• Applying ρ = sin(πτ/2) to Clayton or Gumbel.

  The formula is learnt as a general link between tau and the parameter.

  Fix: That relationship is for elliptical copulas. Use τ = α ÷ (α + 2) for Clayton and τ = 1 − 1/α for Gumbel.

• Thinking the t copula with large ν still has strong tail dependence.

  Students recall that t has tail dependence but forget the effect of ν.

  Fix: Tail dependence falls as ν rises. The t copula tends to the Gaussian copula as ν → ∞.

Worked examples