# Copula functions

A neat way of quantifying arbitrary (?) dependence structures between random variables. Useful in, e.g. Quantitative Risk Management.

The trick is simple: Informally, you look at the marginal iCDF of each of $$n$$ variables, and fiddle with the joint distribution of those marginals on $$[0,1]^n$$. (That’s assuming variables are absolutely continuous w.r.t some underlying measure space; distribution with atoms are trickier.)

This is a good trick, although I need to sit down and think through it. I would like to better understand:

• the relationship between the underlying event space and the instrumental one we “sort of” construct in copula modeling.
• Is any information lost with non-monotonic coupling in a copula model?
• conditional copulas and how they work
• the occasionally-mentioned relationship between copula entropy and mutual information.

Chen, Song Xi, and Tzee-Ming Huang. 2007. “Nonparametric Estimation of Copula Functions for Dependence Modelling.” Canadian Journal of Statistics 35 (2): 265–82. https://doi.org/10.1002/cjs.5550350205.

Embrechts, Paul. 2009. “Copulas: A Personal View.” Journal of Risk and Insurance 76 (3): 639–50. https://doi.org/10.1111/j.1539-6975.2009.01310.x.

Embrechts, Paul, Filip Lindskog, and Alexander J McNeil. 2003. “Modelling Dependence with Copulas and Applications to Risk Management.” Handbook of Heavy Tailed Distributions in Finance 8 (329-384): 1. https://people.math.ethz.ch/~embrecht/ftp/copchapter.pdf.

Embrechts, Paul, Alexander J McNeil, and Daniel Straumann. 2002. “Correlation and Dependence in Risk Management: Properties and Pitfalls.” Risk Management: Value at Risk and Beyond, 176–223.

Fan, Yanqin, and Andrew J. Patton. 2014. “Copulas in Econometrics.” Annual Review of Economics 6 (1): 179–200. https://doi.org/10.1146/annurev-economics-080213-041221.

Genest, C, and A C Favre. 2007. “Everything You Always Wanted to Know About Copula Modeling but Were Afraid to Ask.” Journal of Hydrologic Engineering 12: 347. https://doi.org/10.1061/ASCE1084-0699200712:4347.

Härdle, Wolfgang, and Ostap Okhrin. 2010. “De Copulis Non Est Disputandum.” Advances in Statistical Analysis 94 (1): 1–31. https://doi.org/10.1007/s10182-009-0118-1.

Hofert, Marius. 2008. “Sampling Archimedean Copulas.” Computational Statistics & Data Analysis 52 (12): 5163–74. https://doi.org/10.1016/j.csda.2008.05.019.

Hofert, Marius, and Frédéric Vrins. 2013. “Sibuya Copulas.” Journal of Multivariate Analysis 114 (February): 318–37. https://doi.org/10.1016/j.jmva.2012.08.007.

Kraskov, Alexander, Harald Stögbauer, and Peter Grassberger. 2004. “Estimating Mutual Information.” Physical Review E 69: 066138. https://doi.org/10.1103/PhysRevE.69.066138.

Ma, Jian, and Zengqi Sun. 2011. “Mutual Information Is Copula Entropy.” Structural Change and Economic Dynamics 16 (1): 51–54. https://doi.org/10.1016/S1007-0214(11)70008-6.

Mai, Jan-Frederik, and Matthias Scherer. 2012. Simulating Copulas: Stochastic Models, Sampling Algorithms, and Applications. Series in Quantitative Finance, v. 4. London : Hackensack, NJ: Imperial College Press ; World Scientific.

———. 2011. “Reparameterizing Marshall–Olkin Copulas with Applications to Sampling.” Journal of Statistical Computation and Simulation 81 (1): 59–78. https://doi.org/10.1080/00949650903185961.

Malevergne, Yannick, and Didier Sornette. 2003. “Testing the Gaussian Copula Hypothesis for Financial Assets Dependences.” Quantitative Finance 3 (4): 231–50. https://doi.org/10.1088/1469-7688/3/4/301.

McNeil, Alexander J. 2008. “Sampling Nested Archimedean Copulas.” Journal of Statistical Computation and Simulation 78 (6): 567–81. https://doi.org/10.1080/00949650701255834.

Nelsen, Roger B. 1999. An Introduction to Copulas. Lecture Notes in Statistics 139. New York: Springer.

Patton, A J. 2009. “Copula-Based Models for Financial Time Series.” In Handbook of Financial Time Series, 767–85. Berlin, Heidelberg: Springer Berlin Heidelberg.

Patton, Andrew J. 2012. “A Review of Copula Models for Economic Time Series.” Journal of Multivariate Analysis 110: 4–18. https://doi.org/10.1016/j.jmva.2012.02.021.

Rémillard, Bruno, Nicolas Papageorgiou, and Frédéric Soustra. 2012. “Copula-Based Semiparametric Models for Multivariate Time Series.” Journal of Multivariate Analysis 110: 30–42. https://doi.org/10.1016/j.jmva.2012.03.001.

Schmidt, Thorsten. 2006. “Coping with Copulas.” In Copulas from Theory to Applications in Finance.

Shaw, William T. 2006. “Sampling Student’s T Distribution – Use of the Inverse Cumulative Distribution Function.” Journal of Computational Finance, July. http://www.risk.net/journal-of-computational-finance/technical-paper/2160383/sampling-students-t-distribution-inverse-cumulative-distribution-function.

Trivedi, Pravin K, and David M Zimmer. 2006. “Copula Modeling: An Introduction for Practitioners.” Foundations and Trends® in Econometrics 1 (1): 1–111. https://doi.org/10.1561/0800000005.

Whelan, Niall. 2004. “Sampling from Archimedean Copulas.” Quantitative Finance 4: 339–52. https://doi.org/10.1088/1469-7688/4/3/009.

Wu, Florence, Emiliano Valdez, and Michael Sherris. 2007. “Simulating from Exchangeable Archimedean Copulas.” Communications in Statistics - Simulation and Computation 36 (5): 1019–34. https://doi.org/10.1080/03610910701539781.