Calculating a trivariate copula involves modeling the joint distribution of three random variables using a copula function. Copulas are used to describe the dependency structure between variables while separating it from their marginal distributions. Here's a step-by-step guide on how to calculate a trivariate copula:

Step 1: Understand the Basics of Copulas:

Before diving into trivariate copulas, make sure you understand the basics of copula functions, including Archimedean copulas (e.g., Clayton, Gumbel, Frank) or elliptical copulas (e.g., Gaussian).

Step 2: Gather Data:

Collect data for your three variables (X, Y, and Z). These variables should be continuous and have non-degenerate marginal distributions.

Step 3: Rank the Data:

Rank the data for each variable individually. Convert each observation to its corresponding rank, ranging from 1 (smallest) to n (largest), where n is the number of observations.

Step 4: Select a Copula Family:

Choose an appropriate copula family based on the dependency structure you want to model (e.g., Clayton for positive dependence, Gumbel for extreme value dependence, Gaussian for symmetric dependence, etc.).

Step 5: Estimate Parameters:

Estimate the parameters of the selected copula using the ranked data. Various methods can be used for parameter estimation, such as the method of moments, maximum likelihood estimation, or goodness-of-fit techniques.

Step 6: Generate Copula Values:

Generate trivariate copula values using the estimated copula parameters. You can do this by applying the copula function to the pairs of uniform variables corresponding to each pair of original data variables.

For example, if you have three variables X, Y, and Z, and you've estimated a Clayton copula, you would generate trivariate copula values (u1, u2, u3) using the Clayton copula function with the estimated parameter. Here, u1, u2, and u3 are uniform random variables in [0, 1] corresponding to the ranked data of X, Y, and Z.

Step 7: Transform Back to Original Scale:

Transform the copula values back to the original scale of your variables using the inverse cumulative distribution functions (CDFs) of the marginal distributions. This will give you trivariate data that respects the dependency structure modeled by the copula.

Step 8: Analyze and Interpret:

Analyze the generated trivariate data to understand the dependency structure between the three variables. You can compute various statistics, such as correlation coefficients, tail dependencies, and conditional probabilities, to gain insights into the joint behavior of the variables.

Keep in mind that working with copulas requires a good understanding of statistical theory and multivariate analysis. Depending on the complexity of your analysis, you may also need to consider issues such as model selection, goodness-of-fit tests, and sensitivity analysis.

Additionally, various software packages (e.g., R, Python with libraries like copulas) provide tools for estimating and simulating copulas, which can simplify the implementation of these steps.

Dr Ronnie

What I know that C2 is the bivariate copula between u1 and u2. let say the bivariate copula u1 and u2 is clayton. But I not sure how to determine C2 value and how to combine u3 with the clayton copula that I have.