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Introduction to Copulas B. Wade Brorsen Oklahoma State University

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

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

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

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Agricultural Economics Taylor (1990) Richardson/Simetar Heuristic

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

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

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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.

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

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Example

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Estimation Inference for margins (IFM) Maximum likelihood Simulation

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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;

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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;

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

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