Vine copulas, or pair-copula constructions, have become an important tool in high-dimensional dependence modeling. Commonly, it is assumed that the data generating copula can be represented by a simplified vine copula (SVC). In this paper, we study the simplifying assumption and investigate the approximation of multivariate copulas by SVCs. We introduce the partial vine copula (PVC) which is a particular SVC where to any edge a $j$-th order partial copula is assigned. The PVC generalizes the partial correlation matrix and plays a major role in the approximation of copulas by SVCs. We investigate to what extent the PVC describes the dependence structure of the underlying copula. We show that, in general, the PVC does not minimize the Kullback-Leibler divergence from the true copula if the simplifying assumption does not hold. However, under regularity conditions, stepwise estimators of pair-copula constructions converge to the PVC irrespective of whether the simplifying assumption holds or not. Moreover, we elucidate why the PVC is often the best feasible SVC approximation in practice.
References
[1] Aas, K. and Berg, D. (2009). Models for construction of multivariate dependence – a comparison study., The European Journal of Finance 15 639–659.[1] Aas, K. and Berg, D. (2009). Models for construction of multivariate dependence – a comparison study., The European Journal of Finance 15 639–659.
[2] Aas, K., Czado, C., Frigessi, A. and Bakken, H. (2009). Pair-Copula Constructions of Multiple Dependence., Insurance: Mathematics and Economics 44 182–198. 1165.60009 10.1016/j.insmatheco.2007.02.001[2] Aas, K., Czado, C., Frigessi, A. and Bakken, H. (2009). Pair-Copula Constructions of Multiple Dependence., Insurance: Mathematics and Economics 44 182–198. 1165.60009 10.1016/j.insmatheco.2007.02.001
[3] Acar, E. F., Genest, C. and Nešlehová, J. (2012). Beyond simplified pair-copula constructions., Journal of Multivariate Analysis 110 74–90. 1243.62067 10.1016/j.jmva.2012.02.001[3] Acar, E. F., Genest, C. and Nešlehová, J. (2012). Beyond simplified pair-copula constructions., Journal of Multivariate Analysis 110 74–90. 1243.62067 10.1016/j.jmva.2012.02.001
[4] Bedford, T. and Cooke, R. M. (2002). Vines – A New Graphical Model for Dependent Random Variables., The Annals of Statistics 30 1031–1068. 1101.62339 10.1214/aos/1031689016 euclid.aos/1031689016[4] Bedford, T. and Cooke, R. M. (2002). Vines – A New Graphical Model for Dependent Random Variables., The Annals of Statistics 30 1031–1068. 1101.62339 10.1214/aos/1031689016 euclid.aos/1031689016
[5] Bergsma, W. P. (2004). Testing conditional independence for continuous random variables. URL:, https://www.eurandom.tue.nl/reports/2004/048-report.pdf.[5] Bergsma, W. P. (2004). Testing conditional independence for continuous random variables. URL:, https://www.eurandom.tue.nl/reports/2004/048-report.pdf.
[6] Dißmann, J., Brechmann, E. C., Czado, C. and Kurowicka, D. (2013). Selecting and estimating regular vine copulae and application to financial returns., Computational Statistics & Data Analysis 59 52–69. MR3000041 1400.62114 10.1016/j.csda.2012.08.010[6] Dißmann, J., Brechmann, E. C., Czado, C. and Kurowicka, D. (2013). Selecting and estimating regular vine copulae and application to financial returns., Computational Statistics & Data Analysis 59 52–69. MR3000041 1400.62114 10.1016/j.csda.2012.08.010
[7] Fermanian, J.-D. and Wegkamp, M. H. (2012). Time-dependent copulas., Journal of Multivariate Analysis 110 19–29. 1352.62073 10.1016/j.jmva.2012.02.018[7] Fermanian, J.-D. and Wegkamp, M. H. (2012). Time-dependent copulas., Journal of Multivariate Analysis 110 19–29. 1352.62073 10.1016/j.jmva.2012.02.018
[9] Gijbels, I., Omelka, M. and Veraverbeke, N. (2015a). Estimation of a Copula when a Covariate Affects only Marginal Distributions., Scandinavian Journal of Statistics 42 1109–1126. 06527648 10.1111/sjos.12154[9] Gijbels, I., Omelka, M. and Veraverbeke, N. (2015a). Estimation of a Copula when a Covariate Affects only Marginal Distributions., Scandinavian Journal of Statistics 42 1109–1126. 06527648 10.1111/sjos.12154
[10] Gijbels, I., Omelka, M. and Veraverbeke, N. (2015b). Partial and average copulas and association measures., Electronic Journal of Statistics 9 2420–2474. 1327.62208 10.1214/15-EJS1077[10] Gijbels, I., Omelka, M. and Veraverbeke, N. (2015b). Partial and average copulas and association measures., Electronic Journal of Statistics 9 2420–2474. 1327.62208 10.1214/15-EJS1077
[11] Grothe, O. and Nicklas, S. (2013). Vine constructions of Lévy copulas., Journal of Multivariate Analysis 119 1–15. 1283.60075 10.1016/j.jmva.2013.04.002[11] Grothe, O. and Nicklas, S. (2013). Vine constructions of Lévy copulas., Journal of Multivariate Analysis 119 1–15. 1283.60075 10.1016/j.jmva.2013.04.002
[12] Hobæk Haff, I. (2012). Comparison of estimators for pair-copula constructions., Journal of Multivariate Analysis 110 91–105. 1244.62072 10.1016/j.jmva.2011.08.013[12] Hobæk Haff, I. (2012). Comparison of estimators for pair-copula constructions., Journal of Multivariate Analysis 110 91–105. 1244.62072 10.1016/j.jmva.2011.08.013
[13] Hobæk Haff, I. (2013). Parameter estimation for pair-copula constructions., Bernoulli 19 462–491. 06168760 10.3150/12-BEJ413 euclid.bj/1363192035[13] Hobæk Haff, I. (2013). Parameter estimation for pair-copula constructions., Bernoulli 19 462–491. 06168760 10.3150/12-BEJ413 euclid.bj/1363192035
[14] Hobæk Haff, I., Aas, K. and Frigessi, A. (2010). On the simplified pair-copula construction – Simply useful or too simplistic?, Journal of Multivariate Analysis 101 1296–1310. 1184.62079 10.1016/j.jmva.2009.12.001[14] Hobæk Haff, I., Aas, K. and Frigessi, A. (2010). On the simplified pair-copula construction – Simply useful or too simplistic?, Journal of Multivariate Analysis 101 1296–1310. 1184.62079 10.1016/j.jmva.2009.12.001
[15] Hobæk Haff, I. and Segers, J. (2015). Nonparametric estimation of pair-copula constructions with the empirical pair-copula., Computational Statistics & Data Analysis 84 1–13. 06984141 10.1016/j.csda.2014.10.020[15] Hobæk Haff, I. and Segers, J. (2015). Nonparametric estimation of pair-copula constructions with the empirical pair-copula., Computational Statistics & Data Analysis 84 1–13. 06984141 10.1016/j.csda.2014.10.020
[16] Joe, H. (1996). Families of m-Variate Distributions with Given Margins and m(m-1)/2 Bivariate Dependence Parameters., Lecture Notes-Monograph Series 28 120–141.[16] Joe, H. (1996). Families of m-Variate Distributions with Given Margins and m(m-1)/2 Bivariate Dependence Parameters., Lecture Notes-Monograph Series 28 120–141.
[18] Joe, H., Li, H. and Nikoloulopoulos, A. K. (2010). Tail dependence functions and vine copulas., Journal of Multivariate Analysis 101 252–270. 1177.62072 10.1016/j.jmva.2009.08.002[18] Joe, H., Li, H. and Nikoloulopoulos, A. K. (2010). Tail dependence functions and vine copulas., Journal of Multivariate Analysis 101 252–270. 1177.62072 10.1016/j.jmva.2009.08.002
[19] Kauermann, G., Schellhase, C. and Ruppert, D. (2013). Flexible Copula Density Estimation with Penalized Hierarchical B-splines., Scandinavian Journal of Statistics 40 685–705. 1364.62084 10.1111/sjos.12018[19] Kauermann, G., Schellhase, C. and Ruppert, D. (2013). Flexible Copula Density Estimation with Penalized Hierarchical B-splines., Scandinavian Journal of Statistics 40 685–705. 1364.62084 10.1111/sjos.12018
[20] Kauermann, G. and Schellhase, C. (2014). Flexible pair-copula estimation in D-vines using bivariate penalized splines., Statistics and Computing 24 1081–1100. 1332.62117 10.1007/s11222-013-9421-5[20] Kauermann, G. and Schellhase, C. (2014). Flexible pair-copula estimation in D-vines using bivariate penalized splines., Statistics and Computing 24 1081–1100. 1332.62117 10.1007/s11222-013-9421-5
[21] Krupskii, P. and Joe, H. (2013). Factor copula models for multivariate data., Journal of Multivariate Analysis 120 85–101. 1280.62070 10.1016/j.jmva.2013.05.001[21] Krupskii, P. and Joe, H. (2013). Factor copula models for multivariate data., Journal of Multivariate Analysis 120 85–101. 1280.62070 10.1016/j.jmva.2013.05.001
[22] Kurowicka, D. and Cooke, R. (2006)., Uncertainty Analysis with High Dimensional Dependence Modelling. Wiley, Chichester. 1096.62073[22] Kurowicka, D. and Cooke, R. (2006)., Uncertainty Analysis with High Dimensional Dependence Modelling. Wiley, Chichester. 1096.62073
[24] Kurz, M. S. and Spanhel, F. (2017). Testing the simplifying assumption in high-dimensional vine copulas. ArXiv e-prints, arXiv:1706.02338.[24] Kurz, M. S. and Spanhel, F. (2017). Testing the simplifying assumption in high-dimensional vine copulas. ArXiv e-prints, arXiv:1706.02338.
[25] McNeil, A. J., Frey, R. and Embrechts, P. (2005)., Quantitative Risk Management. Princeton series in finance. Princeton Univ. Press, Princeton NJ. 1089.91037[25] McNeil, A. J., Frey, R. and Embrechts, P. (2005)., Quantitative Risk Management. Princeton series in finance. Princeton Univ. Press, Princeton NJ. 1089.91037
[26] Mesfioui, M. and Quessy, J.-F. (2008). Dependence structure of conditional Archimedean copulas., Journal of Multivariate Analysis 99 372–385. MR2396969 1133.60010 10.1016/j.jmva.2006.10.007[26] Mesfioui, M. and Quessy, J.-F. (2008). Dependence structure of conditional Archimedean copulas., Journal of Multivariate Analysis 99 372–385. MR2396969 1133.60010 10.1016/j.jmva.2006.10.007
[27] Nagler, T. and Czado, C. (2016). Evading the curse of dimensionality in nonparametric density estimation with simplified vine copulas., Journal of Multivariate Analysis 151 69–89. 1346.62071 10.1016/j.jmva.2016.07.003[27] Nagler, T. and Czado, C. (2016). Evading the curse of dimensionality in nonparametric density estimation with simplified vine copulas., Journal of Multivariate Analysis 151 69–89. 1346.62071 10.1016/j.jmva.2016.07.003
[28] Nelsen, R. B. (2006)., An Introduction to Copulas. Springer Series in Statistics. Springer, New York. 1152.62030[28] Nelsen, R. B. (2006)., An Introduction to Copulas. Springer Series in Statistics. Springer, New York. 1152.62030
[29] Nikoloulopoulos, A. K., Joe, H. and Li, H. (2012). Vine copulas with asymmetric tail dependence and applications to financial return data., Computational Statistics & Data Analysis 56 3659–3673. 1254.91613 10.1016/j.csda.2010.07.016[29] Nikoloulopoulos, A. K., Joe, H. and Li, H. (2012). Vine copulas with asymmetric tail dependence and applications to financial return data., Computational Statistics & Data Analysis 56 3659–3673. 1254.91613 10.1016/j.csda.2010.07.016
[31] Portier, F. and Segers, J. (2018). On the weak convergence of the empirical conditional copula under a simplifying assumption., Journal of Multivariate Analysis 166 160–181. 1401.62082 10.1016/j.jmva.2018.03.002[31] Portier, F. and Segers, J. (2018). On the weak convergence of the empirical conditional copula under a simplifying assumption., Journal of Multivariate Analysis 166 160–181. 1401.62082 10.1016/j.jmva.2018.03.002
[32] Remillard, B. (2013)., Statistical Methods for Financial Engineering. CRC Press, Boca Raton, FL. 1273.91010[32] Remillard, B. (2013)., Statistical Methods for Financial Engineering. CRC Press, Boca Raton, FL. 1273.91010
[33] Schellhase, C. and Spanhel, F. (2018). Estimating non-simplified vine copulas using penalized splines., Statistics and Computing 28 387–409. 1384.62170 10.1007/s11222-017-9737-7[33] Schellhase, C. and Spanhel, F. (2018). Estimating non-simplified vine copulas using penalized splines., Statistics and Computing 28 387–409. 1384.62170 10.1007/s11222-017-9737-7
[34] Schepsmeier, U., Stoeber, J., Brechmann, E. C., Graeler, B., Nagler, T. and Erhardt, T. (2016). VineCopula: Statistical Inference of Vine Copulas URL:, https://CRAN.R-project.org/package=VineCopula, R package version 2.0.5.[34] Schepsmeier, U., Stoeber, J., Brechmann, E. C., Graeler, B., Nagler, T. and Erhardt, T. (2016). VineCopula: Statistical Inference of Vine Copulas URL:, https://CRAN.R-project.org/package=VineCopula, R package version 2.0.5.
[35] Spanhel, F. and Kurz, M. S. (2016a). The partial copula: Properties and associated dependence measures., Statistics & Probability Letters 119 76–83. 1398.62150 10.1016/j.spl.2016.07.014[35] Spanhel, F. and Kurz, M. S. (2016a). The partial copula: Properties and associated dependence measures., Statistics & Probability Letters 119 76–83. 1398.62150 10.1016/j.spl.2016.07.014
[37] Stöber, J., Joe, H. and Czado, C. (2013). Simplified pair copula constructions—Limitations and extensions., Journal of Multivariate Analysis 119 101–118. 1277.62139 10.1016/j.jmva.2013.04.014[37] Stöber, J., Joe, H. and Czado, C. (2013). Simplified pair copula constructions—Limitations and extensions., Journal of Multivariate Analysis 119 101–118. 1277.62139 10.1016/j.jmva.2013.04.014
[38] Vatter, T. and Nagler, T. (2018). Generalized Additive Models for Pair-Copula Constructions., Journal of Computational and Graphical Statistics 27 715–727.[38] Vatter, T. and Nagler, T. (2018). Generalized Additive Models for Pair-Copula Constructions., Journal of Computational and Graphical Statistics 27 715–727.
[39] White, H. (1982). Maximum Likelihood Estimation of Misspecified Models., Econometrica 50 1–25. 0478.62088 10.2307/1912526[39] White, H. (1982). Maximum Likelihood Estimation of Misspecified Models., Econometrica 50 1–25. 0478.62088 10.2307/1912526
