Lecture Notes: Copulas Beyond Two Dimensions

Overview

• Discussed fitting data to parametric families of copulas
• Focus on moving beyond two dimensions

Importance of Copulas

• Useful for modeling dependencies between random variables
• Sklar's theorem is foundational and applies to multiple dimensions

Sklar’s Theorem

• Multi-dimensional forms exist for joint CDFs
• Involves copulas defining dependence structure and marginal distributions
• Inverse form also applicable

Technicalities to Note

• Frechet-Hoeffding bounds for multi-dimensional copulas
• More relevant to theoretical researchers

Challenges in Multi-Dimensional Copulas

• Fewer parametric forms available in higher dimensions
• Fitting empirical copulas is slower (curve fitting operation)
• Need for exponentially more data as dimensions increase
• Constraints of parametric forms can hinder model fitting

Approaches to Address Challenges

• Two major thrusts:
  1. Copulobasian networks
  2. Vine copulas

Copulobasian Networks

• Introduced by Gal Elidan
• Combines Bayesian networks with copula framework for complex dependencies

Vine Copulas

• Introduced by Bedford and Cook in 2002
• More widely adopted than copulobasian networks
• Focus of this lecture

Joint Density Functions

• Joint density can be represented in different conditional forms
• For 3 variables:
  • Six equivalent representations for joint density of x1, x2, x3

Vine Copulas Explained

• Basic Idea: Model high-dimensional distributions using conditional densities
• Relationship between conditional densities and copulas helps establish vine copulas
• A vine copula factorizes multidimensional distributions into bivariate copulas

Types of Vine Copulas

• R-vine: Regular vine copulas with various properties
• C-vine: Canonical vine, hub-and-spoke architecture
• D-vine: Directed vine copulas

C-Vine Structure

• Central node connecting other variables
• Example with three variables: dependence modeled through pairs and conditioned on central node
• For four variables: will have n-1 trees (n is number of variables)

Example Application: C-Vine Copula with Crypto Prices

• Modeling dependencies between Bitcoin, Ethereum, and Filecoin
• Assumption: Bitcoin as primary driver for both Ethereum and Filecoin
• Data sourced from Yahoo Finance, focusing on log returns

Steps Taken

1. Fit a time series model (using Facebook's Prophet)
2. Generate residuals from predictions
3. Scatterplots to evaluate relationships between residuals
4. Build C-vine copula model to predict residuals
5. Fit data to pseudo observations and evaluate copula fit

Results Evaluation

• Various copulas were fitted to pairwise interactions
• Generated samples from the fitted copula model
• Observationally matched residuals with model predictions

Conclusion

• C-vine copulas effectively model dependencies in large datasets
• Practical application shown with crypto asset prices
• Recommendation to explore more literature on copulas for further understanding

Additional Notes

• Jupyter Notebook with example will be uploaded to GitHub
• Open for comments and requests for further topics covered

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