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

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Introduction to Copulas B. Wade Brorsen Oklahoma State University

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

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

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

Agricultural Economics Taylor (1990) Richardson/Simetar Heuristic

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

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

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.

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

Example

Estimation Inference for margins (IFM) Maximum likelihood Simulation

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;

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;

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

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