Cholesky decomposition-Monte Carlo simulation

[Simon Mathieu]

Hi there,
I went on the wikipedia page looking for a method to correlate two random variables in a Monte-Carlo simulation.
Here is the method they show (simplified example)
You have two random variables z1 and z2 distributed normally. Once you found the good correlation coefficient (rhô) for the relation, you program it like :
x1=z1 and x2=rhô*z1+(sqrt(1-(rhô)^2))*z2
So that they now move in a correlated way.
My question is this: what happens when one of them (z1 and z2) is not distributed normally? For example if one is normal and the other is uniform? Or if they are both following a student-t law.

Would it remain the same equation, or would it change completely? If it does change completely, how do I know which formula to use for every situation? I know my question is quite vague, if you have a book/article to propose me that would be great.

Thanks in advance,

Simon

[TomDoan]

There are very few distributions which behave like the Normal when you move from univariate to multivariate. A multivariate t, which is basically a multivariate Normal divided by a common gamma, is the only other significant distribution which can be handled in roughly the same way. (Dividing by a common scalar doesn't change the correlation). The RATS functions for drawing a multivariate Normal and a multivariate t are

%RANMVNORMAL and
%RANMVT

Both take a Cholesky factor of the covariance matrix (for the t, it's the Cholesky factor of the covariance matrix of the underlying Normal, since the t itself might not have a variance).

[Simon Mathieu]

Thanks Tom!

To be honest with you I have difficulties processing what you have said. I'm quite a beginner in those kind of things.
Tell me if I'm wrong, but if I understand correctly the computation I posted earlier would work if the relation between the two variables (the multivariate distribution) would be a multivariate t because it is very close to the normal multivariate relation and dividing by a scalar (common gamma) wouldn't change the correlation.

So what you're saying is that I have to measure which multivariate distribution rules the two variables. And what if it is not the multivariate t or the normal multivariate? What should I do then?

Thanks,

Simon

[TomDoan]

The types of distributions which generalize the multivariate Normal are called
elliptical distributions. With all of those, the multivariate distribution can be simulated by picking a direction in n-space (based upon the correlation matrix) then picking a distance along that. As I said above, the two main examples of that are the multivariate Normal and the multivariate t (multivariate Cauchy is a special case of the t).

In general, there's no simple way to generate correlated multivariate draws from other types of distributions by drawing marginals and combining. The multivariate Normal has the very convenient (and highly unusual) property that uncorrelated implies independent so you can transform to uncorrelated marginals, draw those independently, and then transform back. You can't do that for other distributions. You can't even do that with the multivariate t---note that you do the t by drawing the Normal then applying a common scale. If you draw the marginals independently, you don't get the common scale. And you really can't do it when you have marginals with very different distributions.

[Simon Mathieu]

Thanks again Tom,

Wow I am definitely not ready to process all this, which is why I am gonna read a few articles and continue my research (learning) specifically on those matter. I will probably ask you and the people of the forum some questions later.

But before I go, just teasing myself I guess, you are talking about elliptical distributions and stuff. It reminds me of concepts I have seen on articles about Copulas. Are the Copulae good tools to monitor the dependence between two variables when they are both not distributed normally? Or is it completely different and there is no link between those concepts?

Best regards,

Simon

[TomDoan]

Modeling correlated series and simulating correlated series are too completely different methodologies. In a practical sense, almost all of econometrics is for modeling correlated series, so there are a huge number of techniques available for that. By contrast, there are very few (correlated) multivariate distributions which can be simulated easily---that's why Markov Chain Monte Carlo techniques exist.