1. Introduction to Copulas B. Wade Brorsen Oklahoma State University

2. Problem Multivariate pdf or cdf when marginal distributions are not normally distributed and not independent.

3. Where Used? Risk and Simulation Value at Risk (VaR) Valuing Derivatives Insurance

4. Extreme Value Theory Tail Dependence –Housing bubble –Collateralized Debt Obligations (CDO) –Hurricane –Crop disease –Bank failures –Long Term Capital Management

5. Agricultural Economics Taylor (1990) Richardson/Simetar Heuristic

6. Gaussian Copula Multivariate-t Copula Most Copulas are Bivariate Two Main Multivariate Copulas

7. A copula C(u, v) is C:[0, 1] 2 →[0, 1] Other properties

8. Sklar’s Theorem Any cdf H(X 1, X 2 ) with margins F(X 1 ) and G(X 2 ) can be represented as H(X 1, X 2 ) = C[F(X 1 ), G(X 2 )] Where C[ ] is a unique copula function.

9. Gaussian Copula H(Ψ -1 (u), Ψ -1 (v)) H is bivariate normal cdf Ψ -1 is inverse of a univariate normal cdf

10. Example

11. Estimation Inference for margins (IFM) Maximum likelihood Simulation

12. SAS Program u = cdf (‘normal’, x1, 2, 2); v = cdf (‘normal’, x2, 5, 5); z1 = probit (u); z2 = probit (v); PROC CORR; /* IFM Method */ Var z1, z2;

13. SAS Program u = cdf (‘gamma’, x1, r1, lambda1); v = cdf (‘gamma’, x2, r2, lambda 2); z1 = probit (u); z2 = probit (v); PROC CORR; Var z1, z2;

15. Summary Copulas can give us a multivariate cdf for nonnormal distributions Agricultural economists should use copulas