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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
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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
Estimation Inference for margins (IFM) Maximum likelihood Simulation
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Summary Copulas can give us a multivariate cdf for nonnormal distributions Agricultural economists should use copulas
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