constructing a copula given information for the underlying random variables

gupta · August 10th, 2005, 3:38 pm

Dear All,I could not find this question answered in the forum anywhere, although it may seem a simple one.Given random vector u = [u_1; u_2] which is normally distributed with means [0;0] and covariance Q (a 2x2 matrix), can I find a copula describing u? If this is not enough information, what more would I need? And how do I find the copula and is it unique?I'm a bit of a newbie to copulas so I would appreciate a higher level of detail.And many thanks in advance.. I hope the question makes sense!-Gupta

erstwhile · August 10th, 2005, 5:01 pm

i don't understand precisely what you are asking, but if you are asking this question:given two random variables x and y, with probability density functions p(x) and q(y), as well as their correlation, is there a unique copula function that will allow me to construct their joint probability density P(x,y)?the answer is "no".people often use the gaussian copula as it is simple.the qualitative difference between copulas i think is mainly in the way that tail events end up being linked.if you already know the joint distribution then you can deduce the copula uniquely.

madmax · August 10th, 2005, 7:43 pm

I don't see any problem, since you already have the joint distribution as you said it is a normal with mean [0,0] and covariance matrix Q.So you have a gaussian copula. you just need to extract your correlation parameter from Q.

zarnywhoop · August 11th, 2005, 6:25 am

I don't think this query really belongs in the Technical forum

erstwhile · August 11th, 2005, 6:29 am

madmax - my understanding is that he has the two marginal distributions and the cov matrix. this is is less information than the joint distribution, right?

madmax · August 11th, 2005, 6:57 am

QuoteGiven random vector u = [u_1; u_2] which is normally distributed with means [0;0] and covariance Q (a 2x2 matrix)a normally distributed random vector. so the vector[\b] is normally distributed, so you have your joint distribution.

mrmelchi · August 13th, 2005, 10:57 pm

I wrote the paper "Which Archimedean Copula is the right one?". It was published in October 2003 by .YieldCurve.com. Currently, paper and spreadsheets are freely available on the following links:PaperExcel SheetExcel SheetPerhaps you could find out a reply to your question.I hope it helps.