I've always had only a vague idea how copulas work, so it was really refreshing and enlightening to read "Everything you always wanted to know about copula modeling but were afraid to ask" in the Journal of Hydrologic Engineering. Here is a non-gated version. Here is how I understand the concept:
Say you have a data set consisting of 500 observations of the returns of two assets (500x2). A scatterplot with marginal densities looks as follows:
Asset 1 has a mean of 0 and a standard deviation of 1% and Asset 2 has a mean of 0 and a standard deviation of 2%. The marginal distributions appear normal but the data are clearly not bivariate normal. There seems to be tail-dependence, i.e. we see co-movements in extreme situations. Now since we know that the t copula is capable of modeling exactly such behaviour, we tell our statistical software to find for our individually normally distributed data the parameters (ρ, ν) for the t copula which produces a joint cumulative distribution function closest to our empirical one:
.
With our parameter estimates for the t copula (ρe=0.5, νe=2.5) we can know simulate as much data (with hopefully the same statistical properties as our small sample) as we wish. Here is the plot of 10,000 random variates:
As you can imagine, in most cases it's easier to run a simulation then to come up with the analytic solution.
Unfortunatley, the t copula produces ugly "wings", i.e. dependence in the upper-left and lower-right quadrant. This is something you actually don't observe in financial data and gives you reason to believe that my small sample of returns was actually generated using a t copula. A counter plot will help you see how ugly those wings really are. Looks like a pad:
So if you can recommend certain copulas for modelling tail-dependence, please let me know!
R: A Language and Environment for Statistical Computing, R Development Core Team, R Foundation for Statistical Computing, Vienna, Austria
related items:
Enjoy the Joy with Copulas: With a Package Copula, J. of Statistical Software
The t Copula and Related Copulas, ETH
Say you have a data set consisting of 500 observations of the returns of two assets (500x2). A scatterplot with marginal densities looks as follows:
Asset 1 has a mean of 0 and a standard deviation of 1% and Asset 2 has a mean of 0 and a standard deviation of 2%. The marginal distributions appear normal but the data are clearly not bivariate normal. There seems to be tail-dependence, i.e. we see co-movements in extreme situations. Now since we know that the t copula is capable of modeling exactly such behaviour, we tell our statistical software to find for our individually normally distributed data the parameters (ρ, ν) for the t copula which produces a joint cumulative distribution function closest to our empirical one:.
With our parameter estimates for the t copula (ρe=0.5, νe=2.5) we can know simulate as much data (with hopefully the same statistical properties as our small sample) as we wish. Here is the plot of 10,000 random variates:
As you can imagine, in most cases it's easier to run a simulation then to come up with the analytic solution.Unfortunatley, the t copula produces ugly "wings", i.e. dependence in the upper-left and lower-right quadrant. This is something you actually don't observe in financial data and gives you reason to believe that my small sample of returns was actually generated using a t copula. A counter plot will help you see how ugly those wings really are. Looks like a pad:
So if you can recommend certain copulas for modelling tail-dependence, please let me know!
R: A Language and Environment for Statistical Computing, R Development Core Team, R Foundation for Statistical Computing, Vienna, Austria
related items:
Enjoy the Joy with Copulas: With a Package Copula, J. of Statistical Software
The t Copula and Related Copulas, ETH
Mahalanobis - am 2009-01-07 08:14 - Rubrik: finance
