The Living Thing / Notebooks : 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:

Cite me baby

Bergsma, W. (2011) Nonparametric testing of conditional independence by means of the partial copula. arXiv:1101.4607.
Calsaverini, R. S., & Vicente, R. (2009) An information-theoretic approach to statistical dependence: Copula information. EPL (Europhysics Letters), 88(6), 68003. DOI.
Charpentier, A. (2003) Tail distribution and dependence measures. (pp. 24–27). Presented at the XXXIV International Astin Colloquium, Berlin, August
Chen, S. X., & Huang, T.-M. (2007) Nonparametric estimation of copula functions for dependence modelling. Canadian Journal of Statistics, 35(2), 265–282. DOI.
Chen, X., & Fan, Y. (2006) Estimation of copula-based semiparametric time series models. Journal of Econometrics, 130(2), 307–335. DOI.
Chernozhukov, V., Galichon, A., Hallin, M., & Henry, M. (2014) Monge-Kantorovich Depth, Quantiles, Ranks, and Signs. arXiv:1412.8434 [math, Stat].
Embrechts, P. (2009) Copulas: A Personal View. Journal of Risk and Insurance, 76(3), 639–650. DOI.
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.
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.
Fan, Y., & Patton, A. J.(2014) Copulas in Econometrics. Annual Review of Economics, 6(1), 179–200. DOI.
Fermanian, J.-D., & Wegkamp, M. H.(2012) Time-dependent copulas. Journal of Multivariate Analysis, 110, 19–29. DOI.
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.
Härdle, W., & Okhrin, O. (2010) De copulis non est disputandum. AStA Advances in Statistical Analysis, 94(1), 1–31. DOI.
Hofert, J. M.(2010) Sampling Nested Archimedean Copulas: With Applications to CDO Pricing.
Hofert, M. (2008) Sampling Archimedean copulas. Computational Statistics & Data Analysis, 52(12), 5163–5174. DOI.
Hofert, M., & Vrins, F. (2013) Sibuya copulas. Journal of Multivariate Analysis, 114, 318–337. DOI.
Kojadinovic, I., & Yan, J. (2010) Modeling multivariate distributions with continuous margins using the copula R package. Journal of Statistical Software, 34(9), 1–20.
Kraskov, A., Stögbauer, H., & Grassberger, P. (2004) Estimating mutual information. Physical Review E, 69, 066138. DOI.
Landsman, Z. M., & Valdez, E. A.(2003) Tail conditional expectations for elliptical distributions. North American Actuarial Journal, 7(4), 55–71. DOI.
Mai, J.-F., & Scherer, M. (2012) Simulating copulas: stochastic models, sampling algorithms, and applications. . London : Hackensack, NJ: Imperial College Press ; World Scientific
Ma, J., & Sun, Z. (2011) Mutual Information Is Copula Entropy. Structural Change and Economic Dynamics, 16(1), 51–54. DOI.
Malevergne, Y., & Sornette, D. (2003) Testing the Gaussian copula hypothesis for financial assets dependences. Quantitative Finance, 3(4), 231–250. DOI.
McNeil, A. J.(2008) Sampling nested Archimedean copulas. Journal of Statistical Computation and Simulation, 78(6), 567–581. DOI.
Miller, D. J., & Liu, W. (2002) On the recovery of joint distributions from limited information. Journal of Econometrics, 107(1-2), 259–274. DOI.
Nelsen, R. B.(1999) An introduction to copulas. . New York: Springer
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.
Palaro, H., & Hotta, L. (2005) Using conditional copula to estimate value at risk.
Patton, A. J.(2001) Modelling time-varying exchange rate dependence using the conditional copula. . University of California, San Diego
Patton, A. J.(2006) Modelling asymmetric exchange rate dependence. International Economic Review, 47(2), 527–556.
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
Patton, A. J.(2012) A review of copula models for economic time series. Journal of Multivariate Analysis, 110, 4–18. DOI.
Rémillard, B., Papageorgiou, N., & Soustra, F. (2012) Copula-based semiparametric models for multivariate time series. Journal of Multivariate Analysis, 110, 30–42. DOI.
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
Schmidt, T. (2006) Coping with copulas. In Copulas from theory to applications in finance.
Shaw, W. T.(2006) Sampling Student’s T distribution – use of the inverse cumulative distribution function. Journal of Computational Finance.
Shaw, W. T.(n.d.) New Methods for Simulating the Student T-Distribution- Direct Use of the Inverse Cumulative Distribution.
Sokolinskiy, O., & Dijk, D. J. C.(2011) Forecasting Volatility with Copula-Based Time Series Models.
Trivedi, P. K., & Zimmer, D. M.(2006) Copula Modeling: An Introduction for Practitioners. Foundations and Trends® in Econometrics, 1(1), 1–111. DOI.
Wu, F., Valdez, E., & Sherris, M. (2007) Simulating from Exchangeable Archimedean Copulas. Communications in Statistics - Simulation and Computation, 36(5), 1019–1034. DOI.
Yan, J. (2007) Enjoy the joy of copulas: With a package copula. Journal of Statistical Software, 21(4), 1–21.
Zeng, X., & Durrani, T. S.(2011) Estimation of mutual information using copula density function. Electronics Letters, 47(8), 493. DOI.