# 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 more tricky.)

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.

## Cite me baby

Berg11
Bergsma, W. (2011) Nonparametric testing of conditional independence by means of the partial copula. arXiv:1101.4607.
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Calsaverini, R. S., & Vicente, R. (2009) An information-theoretic approach to statistical dependence: Copula information. EPL (Europhysics Letters), 88(6), 68003. DOI.
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Charpentier, A. (2003) Tail distribution and dependence measures. (pp. 24–27). Presented at the XXXIV International Astin Colloquium, Berlin, August
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Chen, S. X., & Huang, T.-M. (2007) Nonparametric estimation of copula functions for dependence modelling. Canadian Journal of Statistics, 35(2), 265–282. DOI.
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Chen, X., & Fan, Y. (2006) Estimation of copula-based semiparametric time series models. Journal of Econometrics, 130(2), 307–335. DOI.
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Chernozhukov, V., Galichon, A., Hallin, M., & Henry, M. (2014) Monge-Kantorovich Depth, Quantiles, Ranks, and Signs. arXiv:1412.8434 [math, Stat].
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Embrechts, P. (2009) Copulas: A Personal View. Journal of Risk and Insurance, 76(3), 639–650. DOI.
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Embrechts, P., Lindskog, F., & McNeil, A. J.(2003) Modelling dependence with copulas and applications to risk management. Handbook of Heavy Tailed Distributions in Finance, 8(329-384), 1.
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Embrechts, P., McNeil, A. J., & Straumann, D. (2002) Correlation and dependence in risk management: properties and pitfalls. Risk Management: Value at Risk and beyond, 176–223.
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Fan, Y., & Patton, A. J.(2014) Copulas in Econometrics. Annual Review of Economics, 6(1), 179–200. DOI.
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Fermanian, J.-D., & Wegkamp, M. H.(2012) Time-dependent copulas. Journal of Multivariate Analysis, 110, 19–29. DOI.
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Genest, C., & Favre, A. C.(2007) Everything you always wanted to know about copula modeling but were afraid to ask. Journal of Hydrologic Engineering, 12, 347. DOI.
HäOk10
Härdle, W., & Okhrin, O. (2010) De copulis non est disputandum. AStA Advances in Statistical Analysis, 94(1), 1–31. DOI.
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Hofert, J. M.(2010) Sampling Nested Archimedean Copulas: With Applications to CDO Pricing.
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Hofert, M. (2008) Sampling Archimedean copulas. Computational Statistics & Data Analysis, 52(12), 5163–5174. DOI.
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Hofert, M., & Vrins, F. (2013) Sibuya copulas. Journal of Multivariate Analysis, 114, 318–337. DOI.
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Kojadinovic, I., & Yan, J. (2010) Modeling multivariate distributions with continuous margins using the copula R package. Journal of Statistical Software, 34(9), 1–20.
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Kraskov, A., Stögbauer, H., & Grassberger, P. (2004) Estimating mutual information. Physical Review E, 69, 066138. DOI.
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Landsman, Z. M., & Valdez, E. A.(2003) Tail conditional expectations for elliptical distributions. North American Actuarial Journal, 7(4), 55–71. DOI.
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Mai, J.-F., & Scherer, M. (2012) Simulating copulas: stochastic models, sampling algorithms, and applications. . London : Hackensack, NJ: Imperial College Press ; World Scientific
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Ma, J., & Sun, Z. (2011) Mutual Information Is Copula Entropy. Structural Change and Economic Dynamics, 16(1), 51–54. DOI.
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Malevergne, Y., & Sornette, D. (2003) Testing the Gaussian copula hypothesis for financial assets dependences. Quantitative Finance, 3(4), 231–250. DOI.
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McNeil, A. J.(2008) Sampling nested Archimedean copulas. Journal of Statistical Computation and Simulation, 78(6), 567–581. DOI.
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Miller, D. J., & Liu, W. (2002) On the recovery of joint distributions from limited information. Journal of Econometrics, 107(1-2), 259–274. DOI.
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Nelsen, R. B.(1999) An introduction to copulas. . New York: Springer
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Owen, J., & Rabinovitch, R. (1983) On the Class of Elliptical Distributions and their Applications to the Theory of Portfolio Choice. The Journal of Finance, 38(3), 745–752. DOI.
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Palaro, H., & Hotta, L. (2005) Using conditional copula to estimate value at risk.
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Patton, A. J.(2001) Modelling time-varying exchange rate dependence using the conditional copula. . University of California, San Diego
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Patton, A. J.(2006) Modelling asymmetric exchange rate dependence. International Economic Review, 47(2), 527–556.
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Patton, A. J.(2009) Copula-based models for financial time series. In Handbook of financial time series (pp. 767–785). Berlin, Heidelberg: Springer Berlin Heidelberg
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Patton, A. J.(2012) A review of copula models for economic time series. Journal of Multivariate Analysis, 110, 4–18. DOI.
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Rémillard, B., Papageorgiou, N., & Soustra, F. (2012) Copula-based semiparametric models for multivariate time series. Journal of Multivariate Analysis, 110, 30–42. DOI.
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Scaillet, O., & Fermanian, J.-D. (2002) Nonparametric Estimation of Copulas for Time Series (SSRN Scholarly Paper No. ID 372142). . Rochester, NY: Social Science Research Network
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Schmidt, T. (2006) Coping with copulas. In Copulas from theory to applications in finance.
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Shaw, W. T.(2006) Sampling Student’s T distribution – use of the inverse cumulative distribution function. Journal of Computational Finance.
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Sokolinskiy, O., & Dijk, D. J. C.(2011) Forecasting Volatility with Copula-Based Time Series Models.
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Trivedi, P. K., & Zimmer, D. M.(2006) Copula Modeling: An Introduction for Practitioners. Foundations and Trends® in Econometrics, 1(1), 1–111. DOI.
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Wu, F., Valdez, E., & Sherris, M. (2007) Simulating from Exchangeable Archimedean Copulas. Communications in Statistics - Simulation and Computation, 36(5), 1019–1034. DOI.
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Yan, J. (2007) Enjoy the joy of copulas: With a package copula. Journal of Statistical Software, 21(4), 1–21.
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Zeng, X., & Durrani, T. S.(2011) Estimation of mutual information using copula density function. Electronics Letters, 47(8), 493. DOI.