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.

Bielecki, Tomasz R., Jacek Jakubowski, and Mariusz Niewęgłowski. 2020. Fundamentals of the Theory of Structured Dependence Between Stochastic Processes. Encyclopedia of Mathematics and Its Applications. Cambridge ; New York, NY: Cambridge University Press.

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.

MacKenzie, Donald, and Taylor Spears. 2014. “‘The Formula That Killed Wall Street’: The Gaussian Copula and Modelling Practices in Investment Banking.” Social Studies of Science 44 (3): 393–417. https://doi.org/10.1177/0306312713517157.

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.

Watts, Samuel. 2016. “The Gaussian Copula and the Financial Crisis: A Recipe for Disaster or Cooking the Books?” 25.

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.