1. INTRODUCTION TO COPULAS
Presented by : Mohsen Ben Hassine
2011

2. Outline Definitions and Basic Properties Dependence Important copulas
Methods of Constructing Copulas
Choice of Models

3. Definitions and Basic Properties
The Word Copula is a Latin noun that means ''A link, tie, bond'‘
Appeared for the first time (Sklar 1959)
Non-linear dependence
Be able to measure dependence for heavy tail distributions

4. Definitions and Basic Properties
Marginal distribution function
Joint distribution function
For each pair (x, y), we can associate three numbers: F(x), G(y) and H(x, y)

5. Definitions and Basic Properties
An n-dimensional copula is a distribution function on [0, 1]n, with standard uniform marginal distributions.
If (X, Y ) is a pair of continuous random variables with distribution function H(x, y) and marginal distributions Fx(x) and FY (y) respectively, then U = FX(x) ~ U(0, 1) and V = FY(y) ~ U(0, 1) and the distribution function of (U, V ) is a copula.

6. Definitions and Basic Properties
Example: Independent Copula

7. Definitions and Basic Properties
The graph of Independent Copula

8. Definitions and Basic Properties
The Frechet-Hoeffding Bounds for Joint Distribution
Fréchet Upper bound Copula ( Countermonotonic)
Fréchet Lower bound Copula (Comonotonic)

9. Definitions and Basic Properties
Any copula will be bounded by Fréchet lower and upper bound copulas
U1=U U2=1-U1

10. Definitions and Basic Properties
Sklar's Theorem
Let H be a joint df with marginal dfs F and G, Then there exists a copula C such that
If F and G are continuous, then the copula is unique

11. Dependence Rank correlation Linear correlation
The correlation coefficient between a pair of variables
(X,Y ), defined as
Rank correlation
Spearman’s rho : Pearson applied to ranks
Kendall’s Tau
Tail dependence

12. Spearman’s Rho Kendall’s Tau
Dependence
Spearman’s Rho
Kendall’s Tau

13. Gaussian (Normal) copula For ρ=0.3
Important copulas
Gaussian (Normal) copula
For ρ=0.3

14. Student’s t-copula For v=2, ρ=0.3
Important copulas
Student’s t-copula
For v=2, ρ=0.3

15. General form : Important copulas
Archimedean copulas
The family of Archimedean copulas is a useful tool to generate copulas
General form :
where ф was a decreasing function mapping [0, 1] into [0, ∞]

16. Important copulas
Archimedean copulas

17. Important copulas Archimedean copulas
Densities of the Gumbel (upper left), Clayton (upper right), Frank (lower left) and generalized Clayton (lower right) copulas.

18. Methods of Constructing Copulas
The Inversion Method
By Sklar’s theorem, given continuous margins F1 and F2 and the joint continuous distribution function F(y1,y2) = C(F1(y1),F2(y2)), the corresponding copula is generated using the unique inverse transformations y1 = F1−1 (u1), and y2 = F2−1 (u2),

19. Methods of Constructing Copulas
The Inversion Method (example)
hence y1 = −log(−log(u1)) and y2 = −log (−log(u2)). After substituting these expressions for y1 and y2 into the distribution function

20. Methods of Constructing Copulas
Other Methods
Algebraic Methods
Some derivations of copulas begin with a relationship between marginals based on independence. Then this relationship is modified by introducing a dependence parameter and the corresponding copula is obtained.
Mixtures and Convex Sums
Given a copula C, its lower and upper bounds CL and CU, and the product copula Cp, a new copula can be constructed using a convex sum

21. If a researcher wants to explore the structure of dependence, he may estimate several copulas and choose one on the basis of best fit to the data
Model selection approach
Genest and Rivest (1993) : minimizing the distance function (Archimedean copulas)
Ané and Kharoubi (2003) (all copulas) :
Find a copula minimizing
Choice of Models

22. Everything you always wanted to know about copula, christian genest and anne-catherine favre, journal of hydrologic engineering © asce / july/august 2007
Copula modeling: an introduction for practitioners ,pravin k. trivedi and david m. zimmer, foundations and trends in econometrics 2007
Coping with copulas, thorsten schmidt, forthcoming in risk books "copulas - from theory to applications in finance“, 2006
An introduction to copulas, Roger B. Nelsen, springer series in statistics 2006
Understanding relationships using copulas, edward w. frees and emiliano a. valdez, north american actuarial journal, volume 2, number 1, 1998
References