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Published byElliott Goldthwaite Modified about 1 year ago

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Presented by : Mohsen Ben Hassine

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1. Definitions and Basic Properties 2. Dependence 3. Important copulas 4. Methods of Constructing Copulas 5. Choice of Models 2

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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 3

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

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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 F x (x) and F Y (y) respectively, then U = F X (x) ~ U(0, 1) and V = F Y (y) ~ U(0, 1) and the distribution function of (U, V ) is a copula. 5

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Properties Example: Independent Copula 6

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The graph of Independent Copula 7

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The Frechet-Hoeffding Bounds for Joint Distribution Fréchet Upper bound Copula ( Counter monotonic) Fréchet Lower bound Copula ( Comonotonic) 8

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Any copula will be bounded by Fréchet lower and upper bound copulas U1=U2 U2=1-U1 9

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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 10

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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 11

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S pearman’s Rho Kendall’s Tau 12

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A. Gaussian (Normal) copula For ρ=0.3 13

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

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C. 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, ∞] 15

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C. Archimedean copulas 16

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

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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 = F 1 −1 (u1), and y2 = F 2 −1 (u2), 18

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The Inversion Method (example) hence y1 = −log(−log(u1)) and y2 = −log (−log(u2)). After substituting these expressions for y 1 and y 2 into the distribution function 19

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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 C L and C U, and the product copula Cp, a new copula can be constructed using a convex sum 20

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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 21

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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,

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