Solved – Estimating joint distributions using copula package in R

copuladistributionsestimationjoint distributionr

I am trying to estimate the joint distribution of stock returns using the copula package. I have read a couple of papers on copulae, but alas my lack of math knowledge prevents me from understanding beyond the basics if that! I am thinking of estimating the joint distribution using the following steps and have a couple of questions about the process in general:

Download stock prices for two companies
Convert prices to continuous returns
Transform returns to marginal uniform distribution.
1. How do I do this??
Create the copula object
1. When creating the Copula object using ellipCopula() or archmCopula(), I need to specifiy the type of copula and the parameters. How do I know which copula to use? And how do I obtain the parameters… I thought the point of fitting a copula to the data was to obtain the estimated parameters, so why do I need to specify here? Am I just providing initial estimates of the parameters which are then corrected in the actual fitting?
Specifiy the bivariate distribution
1. When specifying the bivariate distribution, I need to specify the margins in mvdc(), since, according to a number of sources, I should convert the data (returns) to uniform distributions, do I specify the margins as 'unif' in the function?
Define the data to be fitted
1. I am assuming the will be a matrix with 2 columns in this case, each column being the uniforms
Fit the data

As you can see I am very new at this and any help would be greatly appreciated!

Best Answer

Check out A Short, Comprehensive, Practical Guide to Copulas by Atillio Meucci. The paper provides further references in case you'd like to learn more. Steps #3 and #5 are addressed by this paper.

Step #4 is specific to this particular function which I am not too familiar. Since you have so many questions you'd probably have better success breaking down the problem into related parts. You will be in a better place to frame question #4 after some further background reading. As such you really can't answer the question without going into the background of copulas.

There are parametric and non-parametric methods to estimating copulas. In my view, parametric copulas impose tight restrictions that are not respected by the data (non-stationarity, fat tails, etc.). More recent research on time-varying copulas and Meucci's non-parametric copulas I believe can better cope with these issues.

Related Solutions

Solved – Plot of copula (based on data set) – R

Sklar's Theorem is very powerful and reads quite easy, but in practice one will have to know the (analytical) distributions before any "inverse" can be applied. Without knowing the exact joint and marginal distributions, one can still plot the empirical copula of a data set. At times, these plots reveal e.g. asymmetric patterns that can not be captured with correlation coefficients.

Typically, the scaled ranks of a data set (scaled to the interval (0,1)) are a good estimate of a non-parametric inverse of the marginal distributions. An empirical copula's density can then be represented as a scatter plot of the data's scaled ranks. As the spread of (many) points is at times hard to grasp, a smoothed scatter plot provides a visual proxy of the empirical copula's density (see e.g. the function "smoothScatter" from the (base) graphics package or "dependencePlot" in the package spcopula available from r-forge). In case one does know the marginal distributions, one can of'course replace the scaled ranks by the marginally "inverted" data.

In case one does know the copula's family and corresponding parameter(s) (sometimes a 1-1 relationship with Kendall's tau), 3D-plots of the copula can be obtained using the function "persp" with a copula and PDF/CDF function:

library(copula)
persp(claytonCopula(2), dCopula) # plotting the PDF
persp(claytonCopula(2), pCopula) # plotting the CDF

The package copula provides as well the function "fitCopula" that helps to estimate a copula's parameter(s) using different estimators.

Solved – Copula for non-standard distributions in R

About the perspective and contour plots looking strange: It's true that the maximum of the density is not at $(0,0)$, but also you are plotting only a tiny part of the distribution. Try different xlim and ylim arguments, e.g.,

persp(my_dist, dMvdc, xlim = c(0, 300), ylim = c(0, 300), n.grid = 51)

About custom marginal distributions for the copula model: That's possible simply defining a density function dsomething, a distribution function psomething, and (if you need sampling) a quantile function qsomething in the global environment of R. Here is an example with a Gaussian mixture distribution; I made up some parameter values:

dgmix <- function(x, p1, mean1, sd1, mean2, sd2) {
  p1 * dnorm(x, mean1, sd1) + (1 - p1) * dnorm(x, mean2, sd2)
}

pgmix <- function(x, p1, mean1, sd1, mean2, sd2) {
  p1 * pnorm(x, mean1, sd1) + (1 - p1) * pnorm(x, mean2, sd2)
}

curve(dgmix(x, p1 = 0.3, mean1 = 0, sd1 = 20, mean2 = 200, sd2 = 100),
      from = -50, to = 300, n = 1001)

paramMargins <- list(list(shape = 0.652, rate = 0.014),
                     list(p1 = 0.3, mean1 = 0, sd1 = 20,
                          mean2 = 200, sd2 = 100))

my_dist <- mvdc(copula = frankCopula(param = -0.228),
                margins = c("gamma", "gmix"),
                paramMargins = paramMargins)

persp(my_dist, dMvdc, xlim = c(0, 300), ylim = c(-50, 300), n.grid = 51)

Best Answer

Related Solutions

Solved – Plot of copula (based on data set) – R

Solved – Copula for non-standard distributions in R

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