How to combine Gaussian marginals with Gaussian copula to obtain multivariate normals?

Question

in the book "Numerical Methods and Optimization in Finance" I red the following: "Combining the Gaussian copula with Gaussian marginal gives a fancy way of expressing multivariate normals. However, the Gaussian copula can also be combined with other marginals, and Gaussian marginals can be linked via any copula”.

I would like to combine the Gaussian copula with Gaussian marginals, to obtain multivariate normals for my 7 asset classes. In addition, I would like to combine t-marginals with t-copula, to obtain a multivariate t-distribution. Does anyone know how to do this in MatLab?? I kinda struggle with this for quite some time!

This is how I approached the problem for the t marginals & t copula:

---

%% Define univariate process by t-distribution

for i = 1:nAssets

marginal{i} = fitdist(returns(:,i),'tlocationscale');

end

%% Copula calibration

for i = 1:nAssets

U(:,i) = marginal{i}.cdf(returns(:,i)); % transform margin to uniform

end

[rhoT, DoF] = copulafit('t', U, 'Method', 'ApproximateML');

%% Reverse transformation on each index

U = copularnd('t', rhoT, DoF, NumObs * NumSim);

for j = 1:nAssets

ExpReturns(:,:,j) = reshape(marginal{j}.icdf(U(:,j), DoF), NumObs, NumSim);

end

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Does my approach make sense?? Any help is very much appreciated, especially on the MatLab code!!!

Best regards

Answer 1

You can express the Normal distribution by Sklar's Theorem in terms of Gaussian Marginals and Gaussian Copula as follows:

$$F(x_1,...,x_n)=C(F(x_1),...,F(x_n))=C^{Gau}(N(x_1),...,N(x_n))$$

So the distribution equals the copula function with the respective inverse marginals as arguments.

You can aswell combine any types of Copula and (continuous) different Marginals to form new distributions by this formula:

$$F(x_1,...,x_n)=C(F(x_1),...,F(x_n))$$

So for Student-t-Copula:

$$F(x_1,...,x_n)=C_t(F_t(x_1),...,F_t(x_n))$$

Remark: You can also combine different types of marginals and copula, e.g. Gauss Copula with t-Marginals.

The MATLAB-function to generate Copula values can be found here:

Y = copulacdf('Gaussian',U,rho)

Y = copulacdf('t',U,rho,NU)

Answer 2

Just keep in mind that Gaussian marginals with Gaussian copula is nothing more than the multivariate Gaussian distribution (details e.g. here). For t-marginals with t-copula (with the same degree of freedom) you get the multivariate t-distribution.

Both multivariate distributions are characterized by their covariance matrix. The t-distribution has the additional parameter degrees of freedom and will thus produce tail dependence.

All in all you don't need the copula concept in these cases. The only benefit would be to use different degrees of freedom for the marginals and the copula in the t-distribution case.