## Selecting and estimating regular vine copulae and application to financial returns.(English)Zbl 1400.62114

Summary: Regular vine distributions which constitute a flexible class of multivariate dependence models are discussed. Since multivariate copulae constructed through pair-copula decompositions were introduced to the statistical community, interest in these models has been growing steadily and they are finding successful applications in various fields. Research so far has however been concentrating on so-called canonical and D-vine copulae, which are more restrictive cases of regular vine copulae. It is shown how to evaluate the density of arbitrary regular vine specifications. This opens the vine copula methodology to the flexible modeling of complex dependencies even in larger dimensions. In this regard, a new automated model selection and estimation technique based on graph theoretical considerations is presented. This comprehensive search strategy is evaluated in a large simulation study and applied to a 16-dimensional financial data set of international equity, fixed income and commodity indices which were observed over the last decade, in particular during the recent financial crisis. The analysis provides economically well interpretable results and interesting insights into the dependence structure among these indices.

### MSC:

 62H12 Estimation in multivariate analysis 62H05 Characterization and structure theory for multivariate probability distributions; copulas 62G32 Statistics of extreme values; tail inference 62P05 Applications of statistics to actuarial sciences and financial mathematics

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 [1] Aas, K.; Czado, C.; Frigessi, A.; Bakken, H., Pair-copula constructions of multiple dependence, Insurance: Mathematics and Economics, 44, 2, 182-198, (2009) · Zbl 1165.60009 [2] Akaike, H., Information theory and an extension of the likelihood ratio principle, (Petrov, B. N., Proceedings of the Second International Symposium of Information Theory, (1973), Akademiai Kiado Budapest), 257-281 · Zbl 0283.62006 [3] Anderson, T. W., An introduction to multivariate statistical analysis, (2003), Wiley Chichester · Zbl 1039.62044 [4] Ang, A.; Bekaert, G., International asset allocation with regime shifts, Review of Financial Studies, 15, 4, 1137-1187, (2002) [5] Beaudoin, D.; Lakhal-Chaieb, L., Archimedean copula model selection under dependent truncation, Statistics in Medicine, 27, 22, 4440-4454, (2008) [6] Bedford, T.; Cooke, R. M., Probability density decomposition for conditionally dependent random variables modeled by vines, Annals of Mathematics and Artificial Intelligence, 32, 245-268, (2001) · Zbl 1314.62040 [7] Bedford, T.; Cooke, R. M., Vines—a new graphical model for dependent random variables, Annals of Statistics, 30, 4, 1031-1068, (2002) · Zbl 1101.62339 [8] Berg, D.; Aas, K., Models for construction of higher-dimensional dependence: a comparison study, European Journal of Finance, 15, 639-659, (2009) [9] Brechmann, E.C., 2010. Truncated and simplified regular vines and their applications. Diploma Thesis, Technische Universität München. [10] Brechmann, E.C., Czado, C., 2011. Risk management with high-dimensional vine copulas: an analysis of the Euro Stoxx 50 (submitted for publication). · Zbl 1429.62462 [11] Brechmann, E. C.; Czado, C.; Aas, K., Truncated regular vines in high dimensions with applications to financial data, Canadian Journal of Statistics, 40, 1, 68-85, (2012) · Zbl 1274.62381 [12] Cherubini, U.; Luciano, E.; Vecchiato, W., Copula methods in finance, (2004), Wiley Chichester · Zbl 1163.62081 [13] Chollete, L.; Heinen, A.; Valdesogo, A., Modeling international financial returns with a multivariate regime switching copula, Journal of Financial Econometrics, 7, 437-480, (2009) [14] Cormen, T. H.; Leiserson, C. E.; Rivest, R. L.; Stein, C., Introduction to algorithms, (2001), The MIT Press Cambridge · Zbl 1047.68161 [15] Czado, C., Pair-copula constructions of multivariate copulas, (Jaworski, P.; Durante, F.; Härdle, W.; Rychlik, T., Copula Theory and its Applications, (2010), Springer Berlin) [16] Czado, C.; Schepsmeier, U.; Min, A., Maximum likelihood estimation of mixed C-vines with application to exchange rates, Statistical Modelling, 12, 3, 229-255, (2010) [17] Demarta, S.; McNeil, A. J., The $$t$$ copula and related copulas, International Statistical Review, 73, 1, 111-129, (2005) · Zbl 1104.62060 [18] Devroye, L., Non-uniform random variate generation, (1986), Springer New York · Zbl 0593.65005 [19] Diestel, R., Graph theory, (2006), Springer Berlin · Zbl 1086.05001 [20] Dißmann, J., 2010. Statistical inference for regular vines and application. Diploma Thesis, Technische Universität München. [21] Erdorf, S., Hartmann-Wendels, T., Heinrichs, N., 2011. Diversification in firm valuation: a multivariate copula approach. Cologne Graduate School Working Paper Series 02-01, Cologne Graduate School in Management, Economics and Social Sciences. URL: http://econpapers.repec.org/RePEc:cgr:cgsser:02-01. [22] Fang, H. B.; Fang, K. T.; Kotz, S., The meta-elliptical distributions with given marginals, Journal of Multivariate Analysis, 82, 1-16, (2002) · Zbl 1002.62016 [23] Fischer, M.; Köck, C.; Schlüter, S.; Weigert, F., An empirical analysis of multivariate copula models, Quantitative Finance, 9, 7, 839-854, (2009) · Zbl 1180.91314 [24] Frahm, G.; Junker, M.; Szimayer, A., Elliptical copulas: applicability and limitations, Statistics & Probability Letters, 63, 3, 275-286, (2003) · Zbl 1116.62352 [25] Garcia, R., Tsafack, G., 2009. Dependence structure and extreme comovements in international equity and bond markets. CIRANO Working Papers 2009s-21, CIRANO. URL: http://ideas.repec.org/p/cir/cirwor/2009s-21.html. [26] Genest, C.; Favre, A.-C., Everything you always wanted to know about copula modeling but were afraid to ask, Journal of Hydrologic Engineering, 12, 4, 347-368, (2007) [27] Heinen, A., Valdesogo, A., 2009. Asymmetric CAPM dependence for large dimensions: the canonical vine autoregressive model. CORE Discussion Papers 2009069, Université catholique de Louvain, Center for Operations Research and Econometrics, CORE. [28] Hobæk Haff, I., 2011. Parameter estimation for pair-copula constructions. Bernoulli (forthcoming). · Zbl 1456.62033 [29] Hobæk Haff, I.; Aas, K.; Frigessi, A., On the simplified pair-copula construction—simply useful or too simplistic?, Journal of Multivariate Analysis, 101, 5, 1296-1310, (2010) · Zbl 1184.62079 [30] Hofert, M., Efficiently sampling nested Archimedean copulas, Computational Statistics & Data Analysis, 55, 1, 57-70, (2011) · Zbl 1247.62132 [31] Hofmann, M., Czado, C., 2010. Assessing the VaR of a portfolio using D-vine copula based multivariate GARCH models (submitted for publication). [32] Joe, H., Families of $$m$$-variate distributions with given margins and $$m(m - 1) / 2$$ bivariate dependence parameters, (Rüschendorf, L.; Schweizer, B.; Taylor, M. D., Distributions with Fixed Marginals and Related Topics, (1996), Institute of Mathematical Statistics Hayward), 120-141 [33] Joe, H., Multivariate models and dependence concepts, (1997), Chapman & Hall London · Zbl 0990.62517 [34] Joe, H.; Li, H.; Nikoloulopoulos, A. K., Tail dependence functions and vine copulas, Journal of Multivariate Analysis, 101, 1, 252-270, (2010) · Zbl 1177.62072 [35] Kazianka, H.; Pilz, J., Bayesian spatial modeling and interpolation using copulas, Computational Geosciences, 37, 310-319, (2011), URL: http://dx.doi.org/10.1016/j.cageo.2010.06.005 [36] Kurowicka, D., Optimal truncation of vines, (Kurowicka, D.; Joe, H., Dependence Modeling: Handbook on Vine Copulae, (2011), World Scientific Publishing Co. Singapore) [37] Kurowicka, D.; Cooke, R., Uncertainty analysis with high dimensional dependence modelling, (2006), Wiley Chichester · Zbl 1096.62073 [38] (Kurowicka, D.; Joe, H., Dependence Modeling: Handbook on Vine Copulae, (2011), World Scientific Publishing Co. Singapore) [39] Li, D. X., On default correlation: a copula function approach, Journal of Fixed Income, 9, 4, 43-54, (2000) [40] Longin, F.; Solnik, B., Is the correlation in international equity returns constant: 1960–1990?, Journal of International Money and Finance, 14, 3-26, (1995) [41] Longin, F.; Solnik, B., Extreme correlation of international equity markets, Journal of Finance, 56, 2, 649-676, (2001) [42] Manner, H., 2007. Estimation and model selection of copulas with an application to exchange rates. METEOR Research Memorandum 07/056, Maastricht University. [43] McNeil, A. J.; Frey, R.; Embrechts, P., Quantitative risk management: concepts techniques and tools, (2005), Princeton University Press Princeton · Zbl 1089.91037 [44] Mendes, B. V.d. M.; Semeraro, M. M.; Leal, R. P.C., Pair-copulas modeling in finance, Financial Markets and Portfolio Management, 24, 2, 193-213, (2010) [45] Mercier, G., Frison, P.-L., 2009. Statistical characterization of the Sinclair matrix: application to polarimetric image segmentation. In: IGARSS (3)’09, pp. 717–720. [46] Min, A.; Czado, C., Bayesian inference for multivariate copulas using pair-copula constructions, Journal of Financial Econometrics, 8, 4, (2010) [47] Min, A.; Czado, C., Bayesian model selection for multivariate copulas using pair-copula constructions, Canadian Journal of Statistics, 39, 2, 239-258, (2011) · Zbl 1219.62048 [48] Morales-Nápoles, O., 2008. Bayesian belief nets and vines in aviation safety and other applications. Ph.D. Thesis, Technische Universiteit Delft. [49] Morales-Nápoles, O., Cooke, R., Kurowicka, D., 2010. About the number of vines and regular vines on n nodes (submitted for publication). [50] Nelsen, R. B., Dependence modeling with Archimedean copulas, (Kolev, N.; Morettin, P., Proceedings of the Second Brazilian Conference on Statistical Modelling in Insurance and Finance, (2005), Institute of Mathematics and Statistics University of São Paulo), 45-54 [51] Nelsen, R. B., An introduction to copulas, (2006), Springer New York · Zbl 1152.62030 [52] Nikoloulopoulos, A. K.; Joe, H.; Li, H., Vine copulas with asymmetric tail dependence and applications to financial return data, Computational Statistics & Data Analysis, 56, 11, 3659-3673, (2012) · Zbl 1254.91613 [53] Salinas-Gutiérrez, R.; Hernández-Aguirre, A.; Villa-Diharce, E. R., D-vine EDA: a new estimation of distribution algorithm based on regular vines, (Proceedings of the 12th Annual Conference on Genetic and Evolutionary Computation, GECCO’10, (2010), ACM New York, NY, USA), 359-366, URL: http://doi.acm.org/10.1145/1830483.1830550 [54] Salmon, F., Recipe for disaster: the formula that killed wall street, Wired Magazine, 17, 3, (2009), URL: http://www.wired.com/techbiz/it/magazine/17-03/wp_quant [55] Salvadori, G.; De Michele, C.; Kottegoda, N. T.; Rosso, R., Extremes in nature: an approach using copulas, (2007), Springer Dordrecht [56] Sato, M.; Ichiki, K.; Takeuchi, T. T., Precise estimation of cosmological parameters using a more accurate likelihood function, Physical Review Letters, 105, 25, (2010) [57] Savu, C.; Trede, M., Hierarchical Archimedean copulas, Quantitative Finance, 10, 295-304, (2010) · Zbl 1270.91086 [58] Schölzel, C.; Friederichs, P., Multivariate non-normally distributed random variables in climate research—introduction to the copula approach, Nonlinear Processes in Geophysics, 15, 761-772, (2008) [59] Sklar, A., Fonctions de répartition à n dimensions et leurs marges, Publications de l’Institut de Statistique de l’Université de Paris, 8, 229-231, (1959) · Zbl 0100.14202 [60] Smith, M.; Min, A.; Czado, C.; Almeida, C., Modeling longitudinal data using a pair-copula decomposition of serial dependence, Journal of the American Statistical Association, 105, 492, 1467-1479, (2010) · Zbl 1388.62171 [61] Vuong, Q. H., Likelihood ratio tests for model selection and non-nested hypotheses, Econometrica, 57, 2, 307-333, (1989) · Zbl 0701.62106
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