What Gaussian Copulas Can Represent in Joint Distributions

Summary

The document discusses the range of bivariate joint distributions that can be modeled with a Gaussian copula, also called the Nataf transformation. Its answer distinguishes the marginal behavior of each variable from how the variables are joined: the marginals can have different locations, scales, skewness, and kurtosis, while the copula describes their dependence structure.

The practical guidance is to choose marginal distributions that fit the data before using the copula for simulation. The response does not develop a formal characterization of the dependence structures a Gaussian copula can capture, nor does it present a derivation, comparison, or empirical evidence. Consequently, it offers only a partial answer to the original question about approximation limits. Researchers should not take the discussion of flexible marginals as evidence that a Gaussian copula can reproduce every possible joint density; the dependence model itself imposes constraints.

Key ideas

• A Gaussian copula separates marginal distributions from the dependence structure joining variables.
• Marginals may differ in location, scale, skewness, and kurtosis.
• Choose marginal distributions that fit the data when using a copula for simulation.
• The response does not specify the dependence patterns that Gaussian copulas cannot represent.

Tags

# What are the general limitations of Gaussian copulas with regards to the range of joint pdf's it can approximate?


# What are the general limitations of Gaussian copulas with regards to the range of joint pdf's it can approximate?












I'm working with the nataf transformation - AkA Gaussian copula - and trying to establish the range of joint bivariate pdf's it can approximate, and what limitations it puts on those joint pdf's. I've scoured the net and library’s but am unable to find such information. Does anyone know? Or know of a research that looks specifically at this?

## Answer by user6430 (score 1)

https://quant.stackexchange.com/a/9808

The joint pdfs can exhibit whatever the characteristics are of the two random variables. This includes location (mean), spread(sigma), skewness, kurtosis, other moments, etc. As was pointed out above however, you need to ensure that normal is the best fitting distribution for your data. Copulas are used for simulation, which requires knowing the appropriate distribution of the data you are trying to simulate.

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This summary was written by Stratmill's research agent from the original; it is not a copy of the source.