Joint distributions and copulas

More than one variable at once

A joint distribution assigns probabilities to pairs $(X,Y)$. Summing it over one variable gives the marginal distribution of the other. $X$ and $Y$ are independent exactly when the joint probability is always the product of the marginals, $P(X=x,Y=y)=P(X=x)P(Y=y)$. Otherwise they are dependent, and the covariance and correlation of the previous topics summarise the linear part of that dependence in one number.

Correlation is a poor summary when the dependence is not linear, or is strongest in the tails. A copula $C$ separates the two ingredients of a joint distribution: the marginals $F_X,F_Y$ say how each variable behaves alone, and the copula, a joint distribution of uniform variables, says how they move together (Sklar's theorem). The Gaussian copula has no tail dependence, however high the correlation: extreme moves in one variable do not make extreme moves in the other more likely in the limit. A Student-$t$ copula does, which is why it is preferred for assets that crash together.

• Marginals come from summing the joint; independence means the joint is their product.
• Copula: the dependence structure left after removing the marginals (Sklar).
• Tail dependence: Gaussian copula has none; Student-$t$ copula has it.

Lab · what's graded

• A marginal probability10 pts
• The covariance15 pts
• The correlation15 pts
• Are they independent10 pts
• Facts about copulas15 pts
• A Gaussian copula simulation15 pts
• Low correlation, joint crashes20 pts