# Should simulation from a student-t copula distribution yield the input correlation matrix

I am using mathematica to simulate random variates from a student-t copula distribution. Assuming that I input in the correlation matrix R, after generating a certain number of random variates, should the correlation matrix formed by my simulations fit to the initial correlation matrix R.

My results have indicated lack of fit, but I am unsure if this is true or if I am simply not generating enough results.

Is there any theory behind this?.

• Are you using Gaussian copula? Are we talking about Pearson correlation? – Aksakal Dec 16 '14 at 22:24
• I am not using the Gaussian copula, but the student T copula. I could easily switch between the 2. And yes, we are talking about the standard Pearson correlation – Jim Dec 16 '14 at 22:36
• Copulas convey only and entirely the dependency structure. Correlation matrices convey information about a mix of linear dependence ($E(XY)\lesseqgtr E(X)E(Y)$) and marginal distributions. So probably you have some poor specification of the marginal distributions. – Horst Grünbusch Dec 16 '14 at 23:23
• Great insight Horst. What if my specified marginal distributions are known to be correct (say, based on empirical information). What other factors could cause the discrepency between the input correlation matrix and the simulated correlation matrix. – Jim Dec 16 '14 at 23:43
• How are you judging that 'lack of fit'? – Glen_b Dec 17 '14 at 1:57

I'm not familiar with Mathematica's copula functions. If this were Matlab, then you would have to provide the parameters to t-copula. These parameters wouldn't have been Spearman correlation coefficients. That's the reason I asked if you were using Gaussian copula and Spearman, which does use Spearman correlation matrix as input.

Matlab's t-copula uses correlation parameters which it gets from rank correlation coefficients, not Spearman. The other way to fit copulas is to fit them strauight to data with copulafit function. This way if you additionally calculate Spearman correlation matrix, the simulated numbers will be close to it. They will not converge though.

• Is there a reason why they would not converge. I have tried using Gaussian copula as well and that has yielded similar results to the t-copula. – Jim Dec 17 '14 at 3:34
• @aksakal "which it gets from rank correlation coefficients, not Spearman" --- Spearman correlation is a rank correlation coefficient. Which other rank correlation coefficient do you mean? Kendall? – Glen_b Aug 29 '16 at 7:24