Transcript Copula Presentation - Department of Agricultural Economics

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
Most Copulas are Bivariate
Two Main Multivariate Copulas
Gaussian Copula
Multivariate-t Copula
A copula C(u, v) is C:[0, 1]2 →[0, 1]
Other properties
Sklar’s Theorem
Any cdf H(X1, X2) with margins
F(X1) and G(X2) can be represented as
H(X1, X2) = C[F(X1), G(X2)]
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
X 1 ~ N ( 1 ,  )
2
1
X 2 ~ N (2 , )
2
2
Corr ( X 1 , X 2 )  
ˆ1  2, ˆ1  2
ˆ 2  5, ˆ 2  5
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