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

Berg11
Bergsma, W. (2011) Nonparametric testing of conditional independence by means of the partial copula. arXiv:1101.4607.
CaVi09
Calsaverini, R. S., & Vicente, R. (2009) An information-theoretic approach to statistical dependence: Copula information. EPL (Europhysics Letters), 88(6), 68003. DOI.
Char03
Charpentier, A. (2003) Tail distribution and dependence measures. (pp. 24–27). Presented at the XXXIV International Astin Colloquium, Berlin, August
ChHu07
Chen, S. X., & Huang, T.-M. (2007) Nonparametric estimation of copula functions for dependence modelling. Canadian Journal of Statistics, 35(2), 265–282. DOI.
ChFa06
Chen, X., & Fan, Y. (2006) Estimation of copula-based semiparametric time series models. Journal of Econometrics, 130(2), 307–335. DOI.
CGHH14
Chernozhukov, V., Galichon, A., Hallin, M., & Henry, M. (2014) Monge-Kantorovich Depth, Quantiles, Ranks, and Signs. arXiv:1412.8434 [math, Stat].
Embr09
Embrechts, P. (2009) Copulas: A Personal View. Journal of Risk and Insurance, 76(3), 639–650. DOI.
EmLM03
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.
EmMS02
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.
FaPa14
Fan, Y., & Patton, A. J.(2014) Copulas in Econometrics. Annual Review of Economics, 6(1), 179–200. DOI.
FeWe12
Fermanian, J.-D., & Wegkamp, M. H.(2012) Time-dependent copulas. Journal of Multivariate Analysis, 110, 19–29. DOI.
GeFa07
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.
Hofe10
Hofert, J. M.(2010) Sampling Nested Archimedean Copulas: With Applications to CDO Pricing.
Hofe08
Hofert, M. (2008) Sampling Archimedean copulas. Computational Statistics & Data Analysis, 52(12), 5163–5174. DOI.
HoVr13
Hofert, M., & Vrins, F. (2013) Sibuya copulas. Journal of Multivariate Analysis, 114, 318–337. DOI.
KoYa10
Kojadinovic, I., & Yan, J. (2010) Modeling multivariate distributions with continuous margins using the copula R package. Journal of Statistical Software, 34(9), 1–20.
KrSG04
Kraskov, A., Stögbauer, H., & Grassberger, P. (2004) Estimating mutual information. Physical Review E, 69, 066138. DOI.
LaVa03
Landsman, Z. M., & Valdez, E. A.(2003) Tail conditional expectations for elliptical distributions. North American Actuarial Journal, 7(4), 55–71. DOI.
MaSc12
Mai, J.-F., & Scherer, M. (2012) Simulating copulas: stochastic models, sampling algorithms, and applications. . London : Hackensack, NJ: Imperial College Press ; World Scientific
MaSu11
Ma, J., & Sun, Z. (2011) Mutual Information Is Copula Entropy. Structural Change and Economic Dynamics, 16(1), 51–54. DOI.
MaSo03
Malevergne, Y., & Sornette, D. (2003) Testing the Gaussian copula hypothesis for financial assets dependences. Quantitative Finance, 3(4), 231–250. DOI.
Mcne08
McNeil, A. J.(2008) Sampling nested Archimedean copulas. Journal of Statistical Computation and Simulation, 78(6), 567–581. DOI.
MiLi02
Miller, D. J., & Liu, W. (2002) On the recovery of joint distributions from limited information. Journal of Econometrics, 107(1-2), 259–274. DOI.
Nels99
Nelsen, R. B.(1999) An introduction to copulas. . New York: Springer
OwRa83
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.
PaHo05
Palaro, H., & Hotta, L. (2005) Using conditional copula to estimate value at risk.
Patt01
Patton, A. J.(2001) Modelling time-varying exchange rate dependence using the conditional copula. . University of California, San Diego
Patt06
Patton, A. J.(2006) Modelling asymmetric exchange rate dependence. International Economic Review, 47(2), 527–556.
Patt09
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
Patt12
Patton, A. J.(2012) A review of copula models for economic time series. Journal of Multivariate Analysis, 110, 4–18. DOI.
RéPS12
Rémillard, B., Papageorgiou, N., & Soustra, F. (2012) Copula-based semiparametric models for multivariate time series. Journal of Multivariate Analysis, 110, 30–42. DOI.
ScFe02
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
Schm06
Schmidt, T. (2006) Coping with copulas. In Copulas from theory to applications in finance.
Shaw06
Shaw, W. T.(2006) Sampling Student’s T distribution – use of the inverse cumulative distribution function. Journal of Computational Finance.
Shaw00
Shaw, W. T.(n.d.) New Methods for Simulating the Student T-Distribution- Direct Use of the Inverse Cumulative Distribution.
SoDi11
Sokolinskiy, O., & Dijk, D. J. C.(2011) Forecasting Volatility with Copula-Based Time Series Models.
TrZi06
Trivedi, P. K., & Zimmer, D. M.(2006) Copula Modeling: An Introduction for Practitioners. Foundations and Trends® in Econometrics, 1(1), 1–111. DOI.
WuVS07
Wu, F., Valdez, E., & Sherris, M. (2007) Simulating from Exchangeable Archimedean Copulas. Communications in Statistics - Simulation and Computation, 36(5), 1019–1034. DOI.
Yan07
Yan, J. (2007) Enjoy the joy of copulas: With a package copula. Journal of Statistical Software, 21(4), 1–21.
ZeDu11
Zeng, X., & Durrani, T. S.(2011) Estimation of mutual information using copula density function. Electronics Letters, 47(8), 493. DOI.