Now, we learn about trivariate copula. This is an extension of the bivariate copula. Here we learn about trivariate copulas through the exchangeable approach, or also called symmetric copulas. For more details, please read [1].
Let’s start from step by step.
Load Data
load stockreturns
X1 = stocks(:,1);
X2 = stocks(:,2);
X3 = stocks(:,3);
Marginal Distribution and Probability Transform
Just like bivariate copula, the fitting step begins by determining the marginal distribution, then transforming the data.
F1 = fitter(X1); U1 = cdf(F1,X1);
F2 = fitter(X2); U2 = cdf(F2,X2);
F3 = fitter(X3); U3 = cdf(F3,X3);
% ===== OUTPUT ======
Domain = Real
Sort by = Anderson-Darling Stastistics
fittest distribution = Logistic
Parameters: mu = -0.19364, sigma = 0.58158
Decision: fails to reject h0 (AD pval=0.9990)
Domain = Real
Sort by = Anderson-Darling Stastistics
fittest distribution = Generalized Extreme Value
Parameters: k = -0.21362, sigma = 1.1975, mu = -0.66838
Decision: fails to reject h0 (AD pval=0.9949)
Domain = Real
Sort by = Anderson-Darling Stastistics
fittest distribution = Generalized Extreme Value
Parameters: k = -0.36064, sigma = 1.6488, mu = -0.92152
Decision: fails to reject h0 (AD pval=0.9536)
Fitting Copula
Then, fit the copula function using the transformed variables.
C = copfitter([U1,U2,U3],'verbosity',3);
% ===== OUTPUT ======
Case = Symmetric Trivariate
Sort by = Akaike Information Criterion
Fittest copula = Gaussian
Parameter: param1 =
1.0000 0.7223 0.3642
0.7223 1.0000 0.3652
0.3642 0.3652 1.0000
Decision: fails to reject h0 (CvM pval=0.7248)
Summary =
copulaName param1 param2 CvM pValue RMSE AIC
____________ ____________ ______________ ________ _______ ________ _______
{'Gaussian'} {3×3 double} {0×0 double } 0.023096 0.72483 0.015197 -88.511
{'t' } {3×3 double} {[1.2699e+08]} 0.031971 0.68688 0.017881 -86.512
{'Gumbel' } {[ 1.4082]} {0×0 double } 0.06333 0.568 0.025165 -58.982
{'Frank' } {[ 2.8937]} {0×0 double } 0.044194 0.63784 0.021022 -54.603
{'Joe' } {[ 1.5664]} {0×0 double } 0.12503 0.39082 0.035359 -48.461
{'Clayton' } {[ 0.5799]} {0×0 double } 0.11185 0.4233 0.033444 -43.63
Or, we provide a more concise way with just one-line code.
Download: this example is available on demo4.m. Visit Github
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