Modelling Dependence with Copulas - IFOR

3 Dependence
Since the dependence structure among random variables is represented by copulas
it provides a natural way to study and measure dependence and association
between random variables. Many of these properties and measures are invariant
under strictly increasing transforms (as a direct consequence of Theorem 2.6). Linear
correlation (or Pearson’s correlation) is often used in practice as a measure of
dependence. However since linear correlation is not a copula based dependence
measure, it is often quite misleading and should not be taken as the canonical dependence
measure.
We begin by presenting linear correlation, and then we continue with some copula
based measures of dependence.
3.1 Linear Correlation
Definition 8. Let X and Y be two real valued random variables with finite variances.
The linear correlation coefficient between X and Y is
ρ l (X, Y )=
Cov(X, Y )
√
Var(X)
√
Var(Y )
,
where Cov(X, Y )=E(XY ) − E(X)E(Y ) is the covariance between X and Y ,and
Var(X),Var(Y ) denotes the variances of X and Y .
Linear correlation is a measure of linear dependence. In the case of perfect
linear dependence, i.e., Y = aX + b almost surely for a ∈ R\{0},b ∈ R, wehave
ρ l (X, Y )=±1. Otherwise, −1