24.5 How to Use a Copula

We can fully describe the joint distribution with:

\[H(x,y) = P(X \leq x \: \& \: Y \leq y)\]

If we know this then we know the entire distⁿ for any pair \((x,y)\)

We can also get the marginal distⁿ:

\[F(x) = \lim \limits_{y \rightarrow \infty} H(x,y)\]

\[G(y) = \lim \limits_{x \rightarrow \infty} H(x,y)\]

and the density:

\[h(x,y) = \dfrac{\partial^2 H(x,y)}{\partial x \partial y}\]

• Describes where the probability lies

• A bit more intuitive to looks at rather than the CDF

It is easier to work with percentiles, so instead of using \(x\) & \(y\), we’ll use the percentiles \(u\) & \(v\) and we’ll compare them to the percentile of the r.v.

Theorem 24.1 (Sklar’s Theorem)

• For any joint distⁿ \(H(x,y)\) \(\exists\) a function \(C(u,v)\) such that

\[H(x,y) = C\left(F(x), G(y)\right)\]

• \(C(u,v)\) is a copula

  • Input is 2 percentiles and output is the joint percentile

Density of a Copula

Graphs of actual copula is difficult to interpret, so focus on the density

\[c(u,v) = \dfrac{\partial^2 C(u,v)}{\partial u \partial v}\]

Relate the density of the copula to the density of the joint distⁿ

\[\begin{equation} h(x,y) = \underbrace{c \left( F(x), G(y) \right)}_{\text{Density of copula}} \cdot \underbrace{\left[f(x) \cdot g(y) \right]}_{\text{Joint dist if }\perp\!\!\!\perp} \tag{24.5} \end{equation}\]

• Think of the \(c(u,v)\) as a multiplier

• Scales up and down the independent distribution

  • \(c(u,v) > 1\) \(\Rightarrow\) Higher density

  • \(c(u,v) = 1\) \(\Rightarrow\) Independence

  • \(c(u,v) < 1\) \(\Rightarrow\) Lower density

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Different copulas have different behavior w.r.t. where the dependencies are located

We’ll use Kendall’s \(\tau\) to measure correlation as it won’t be affected by the underlying distribution