Ch5 Modelling Dependence: Correlations and Copulas

**Gaussian Copula: http://www.youtube.com/watch?v=z43_pf5Y6A8

1. Correlation as a Measure of Dependence
The most common way to measure the dependence is to use standard (linear) correlation.
The linear correlation is a good measure of dependence when random variables are distributed as multivariate elliptical.
*elliptical distributions: normal & t-distributions

1-1. Limitation even in an elliptical
- If risks are independent, the correlation is zero. But, the reverse does not necessarily hold except in the special case where we have a multivariate normal distribution.
=> zero correlation does not imply that risks are independent unless we have multivariate normality.
- The correlation is not invariant to transformations of underlying variables. For instance, the correlation between X and Y will not in general be the same as the correlation between ln(X) and ln(Y). Hence, transformations of the data can affect correlation estimates.

1-2. Non-elliptical distribution
- correlation is not defined unless variances are finite (infinite variance); heavy-tailed distribution with an infinite variance or trended return series that are not co-integrated
- we cannot count on correlations [-1, 1] out side an elliptical even where defined
- marginal distributions & correlations no longer suffice to determine the joint multivariate distribution -> correlation does not tell us about dependence; spurious correlations (a correlation between two variables that does not result from any direct relation between them but from their relation to other variables) -> correlation does not imply causation
- outliers can affect correlations significantly

2. Copula Theory
2-1. Basics of Copula Theory
A copula is a function that joins a multivariate distribution function to a collection of univariate marginal distribution functions. Copulas enable us to extract the dependence structure from the joint distribution function and separate out the dependence structure from the marginal distribution functions.

**Modeling Joint Distribution Function
- Specify marginal distributions
- Choose a copula to represent the dependence structure
- Estimate parameters involved
- Apply the copula function to the maginals

2-2. Common Copulas
2-2-1. Simplest Copulas
- Independence Copula: X & Y are independent
- Minimum Copula: positively dependent or comonotonic
- Maximum Copula: negatively dependent or countermonotonic

2-2-2. Other Copulas
- Gaussian(Normal) Copula: The copula depends only on the correlation coefficient which confirms that the correlation coefficient is sufficient to determine the whole dependence structure; no closed-form solution
- t-Copula: generalization of the normal copula
- Gumbel or Logistic Copula: beta determines the amount of dependence between variables.
beta = 1 -> variables are independent
beta > 0 -> limited dependence
beta = 0 -> perfect dependence

- Elliptical & Archimedean Copulas
Archimedean: distribution function is strictly decreasing & convex; easy to use; it fits a wide range of dependence behavior

- EV(extreme-value) Copula: minimum & Gumbel copula, Gumbel II, Galambos copulas

2-3. Tail Dependence
Copulas can be used to investigate tail dependence, which is an asymptotic measure of the dependence of extreme values.