Copula Presentation - Department of Agricultural Economics

Report
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

similar documents