Clustering of financial time series in risky scenarios. (English) Zbl 1414.62241

Summary: A methodology is presented for clustering financial time series according to the association in the tail of their distribution. The procedure is based on the calculation of suitable pairwise conditional Spearman’s correlation coefficients extracted from the series. The performance of the method has been tested via a simulation study. As an illustration, an analysis of the components of the Italian FTSE-MIB is presented. The results could be applied to construct financial portfolios that can manage to reduce the risk in case of simultaneous large losses in several markets.


62H30 Classification and discrimination; cluster analysis (statistical aspects)
62H20 Measures of association (correlation, canonical correlation, etc.)
62M10 Time series, auto-correlation, regression, etc. in statistics (GARCH)


QRM; clusfind
Full Text: DOI


[1] Bastos J, Caiado J (2013) Clustering financial time series with variance ratio statistics. Quant Financ (in press) · Zbl 1402.62246
[2] Bernard, C.; Brechmann, E.; Czado, C.; Fouque, JP (ed.); Langsam, J. (ed.), Statistical assessments of systemic risk measures, 165-179, (2013), Cambridge
[3] Billio, M.; Caporin, M., A generalized dynamic conditional correlation model for portfolio risk evaluation, Math Comput Simul, 79, 2566-2578, (2009) · Zbl 1162.91364
[4] Billio, M.; Caporin, M.; Gobbo, M., Flexible dynamic conditional correlation multivariate GARCH models for asset allocation, Appl Financ Econ Lett, 2, 123-130, (2006)
[5] Bock, HH, Special issue on “time series clustering”, Adv Data Anal Classif, 5, 247-249, (2011)
[6] Bonanno, G.; Caldarelli, G.; Lillo, F.; Miccichè, S.; Vandewalle, N.; Mantegna, R., Networks of equities in financial markets, Eur Phys J B, 38, 363-371, (2004)
[7] Bradley, B.; Taqqu, M., Framework for analyzing spatial contagion between financial markets, Financ Lett, 2, 8-16, (2004)
[8] Brechmann E (2013) Hierarchical Kendall copulas: properties and inference. Can J Stat (to appear) · Zbl 1316.62015
[9] Brida, J.; Adrián-Risso, W., Hierarchical structure of the German stock market, Expert Syst Appl, 37, 3846-3852, (2010)
[10] Caiado J, Crato N (2010) Identifying common dynamic features in stock returns. Quant Financ 10(7): 797-807
[11] Chen, X.; Fan, Y., Estimation and model selection of semiparametric copula-based multivariate dynamic models under copula misspecification, J Econom, 135, 125-154, (2006) · Zbl 1418.62425
[12] Cherubini U, Mulinacci S, Gobbi F, Romagnoli S (2012) Dynamic Copula methods in finance. Wiley finance series, Wiley, Chichester
[13] Corduas M, Piccolo D (2008) Time series clustering and classification by the autoregressive metric. Comput Statist Data Anal 52(4):1860-1872 · Zbl 1452.62624
[14] Croux, C.; Dehon, C., Influence functions of the Spearman and Kendall correlation measures, Stat Methods Appl, 19, 497-515, (2010) · Zbl 1332.62186
[15] Czado C (2010) Pair-copula constructions of multivariate copulas. In: Jaworski P, Durante F, Härdle W, Rychlik T (eds) Copula theory and its applications, vol 198, Lecture notes in statistics—proceedings. Springer, Berlin, pp 93-109
[16] Angelis, L., Latent class models for financial data analysis: some statistical developments, Stat Methods Appl, 22, 227-242, (2013) · Zbl 1332.91113
[17] Luca, G.; Zuccolotto, P., A tail dependence-based dissimilarity measure for financial time series clustering, Adv Data Anal Classif, 5, 323-340, (2011)
[18] De Luca G, Rivieccio G, Zuccolotto P (2010) Combining random forest and copula functions: a heuristic approach for selecting assets from a financial crisis perspective. Intell Sys Acc Financ Manage 17(2): 91-109
[19] Dobrić J, Frahm G, Schmid F (2007) Dependence of stock returns in bull and bear markets. Discussion Papers in Statistics and Econometrics 9/07, University of Cologne, Department for Economic and Social Statistics. http://ideas.repec.org/p/zbw/ucdpse/907.html
[20] Durante, F.; Foscolo, E., An analysis of the dependence among financial markets by spatial contagion, Int J Intell Syst, 28, 319-331, (2013)
[21] Durante, F.; Jaworski, P., Spatial contagion between financial markets: a copula-based approach, Appl Stoch Models Bus Ind, 26, 551-564, (2010) · Zbl 1226.91085
[22] Durante F, Sempi C (2010) Copula theory: an introduction. In: Jaworki P, Durante F, Härdle W, Rychlik T (eds) Copula theory and its applications, vol 198, Lecture notes in statistics—proceedings, Springer, Berlin, pp 3-31
[23] Durante F, Foscolo E, Sabo M (2013) A spatial contagion test for financial markets. In: Kruse R, Berthold M, Moewes C, Gil M, Grzegorzewski P, Hryniewicz O (eds) Synergies of soft computing and statistics for intelligent data analysis, vol 190, Advances in intelligent systems and computing, Springer, Berlin, pp 313-320
[24] Embrechts P, McNeil AJ, Straumann D (2002) Correlation and dependence in risk management: properties and pitfalls. In: Dempster M (ed) Risk management: value at risk and beyond. Cambridge University Press, Cambridge, pp 176-223
[25] Engle, R., Dynamic conditional correlation: a simple class of multivariate generalized autoregressive conditional heteroskedasticity models, J Bus Econ Statist, 20, 339-350, (2002)
[26] Everitt, BS, Unresolved problems in cluster analysis, Biometrics, 35, 169-181, (1979) · Zbl 0406.62042
[27] Forbes, KJ; Rigobon, R., No contagion, only interdependence: measuring stock market comovements, J Financ, 57, 2223-2261, (2002)
[28] Genest, C.; Favre, AC, Everything you always wanted to know about copula modeling but were afraid to ask, J Hydrol Eng, 12, 347-368, (2007)
[29] Glosten, L.; Jagannathan, R.; Runkle, D., On the relation between the expected value and the volatility of the nominal excess return on stocks, J Financ, 48, 1779-1801, (1993)
[30] Gordon AD (1999) Classification, 2nd edn. CRC, Boca Raton · Zbl 0929.62068
[31] Härdle W, Simar L (2012) Applied multivariate statistical analysis, 3rd edn. Springer, Berlin · Zbl 1266.62032
[32] Hubert, L.; Arabie, P., Comparing partitions, J Classif, 2, 193-218, (1985)
[33] Jaworski P, Pitera M (2013) On spatial contagion and multivariate GARCH models. Appl Stoch Models Bus Ind (in press)
[34] Jaworski P, Durante F, Härdle WK, Rychlik T (eds) (2010) Copula theory and its applications. Lecture notes in statistics proceedings, vol 198. Springer, Berlin
[35] Jaworski P, Durante F, Härdle WK (eds) (2013) Copulae in mathematical and quantitative finance. Lecture notes in statistics, proceedings, vol 213. Springer, Berlin
[36] Joe H (1997) Multivariate models and dependence concepts, vol 73, Monographs on statistics and applied probability. Chapman & Hall, London · Zbl 0990.62517
[37] Jondeau, E.; Rockinger, M., The copula-GARCH model of conditional dependencies: an international stock market application, J Int Money Financ, 25, 827-853, (2006)
[38] Kaufman L, Rousseeuw P (1990) Finding groups in data. An introduction to cluster analysis. Wiley series in probability and mathematical statistics: applied probability and statisticsWiley, New York
[39] Liao, T., Clustering of time series data—a survey, Pattern Recogn, 38, 1857-1874, (2005) · Zbl 1077.68803
[40] Longin, F.; Solnik, B., Extreme correlation of international equity markets, J Financ, 56, 649-676, (2001)
[41] Malevergne Y, Sornette D (2006) Extreme financial risks. Springer, Berlin · Zbl 1093.62098
[42] Mantegna, R., Hierarchical structure in financial markets, Euro Phys J B, 11, 193-197, (1999)
[43] McNeil AJ, Frey R, Embrechts P (2005) Quantitative risk management. Concepts, techniques and tools. Princeton series in finance, Princeton University Press, Princeton · Zbl 1089.91037
[44] Milligan, GW; Cooper, MC, An examination of procedures for determining the number of clusters in a data set, Psychometrica, 50, 159-179, (1985)
[45] Nelsen RB (2006) An introduction to copulas, 2nd edn. Springer series in statistics. Springer, New York
[46] Otranto, E., Clustering heteroskedastic time series by model-based procedures, Comput Statist Data Anal, 52, 4685-4698, (2008) · Zbl 1452.62784
[47] Pattarin, F.; Paterlini, S.; Minerva, T., Clustering financial time series: an application to mutual funds style analysis, Comput Statist Data Anal, 47, 353-372, (2004) · Zbl 1429.62476
[48] Patton, A., A review of copula models for economic time series, J Multivariate Anal, 110, 4-18, (2012) · Zbl 1244.62085
[49] Patton A (2013) Copula methods for forecasting multivariate time series. In: Handbook of economic forecasting II, Elsevier, Amsterdam (to appear)
[50] Piccolo, D., A distance measure for classifying ARIMA models, J Time Ser Anal, 11, 153-164, (1990) · Zbl 0691.62083
[51] Rand, WM, Objective criteria for the evaluation of clustering methods, J Am Statist Assoc, 66, 846-850, (1971)
[52] Remillard B (2010) Goodness-of-fit tests for copulas of multivariate time series. SSRN eLibrary URL: http://ssrn.com/abstract=1729982
[53] Schmid, F.; Schmidt, R., Multivariate conditional versions of Spearman’s rho and related measures of tail dependence, J Multivariate Anal, 98, 1123-1140, (2007) · Zbl 1116.62061
[54] Schmid F, Schmidt R, Blumentritt T, Gaisser S, Ruppert M (2010) Copula-based measures of multivariate association. In: Jaworski P, Durante F, Härdle W, Rychlik T (eds) Copula theory and its applications, vol 198. Lecture notes in statistics, Proceedings, Springer, Berlin, pp 209-236
[55] Sneath PHA, Sokal RR (1973) Numerical taxonomy. Freeman, San Francisco · Zbl 0285.92001
[56] Tola, V.; Lillo, F.; Gallegati, M.; Mantegna, R., Cluster analysis for portfolio optimization, J Econom Dyn Control, 32, 235-258, (2008) · Zbl 1181.91303
[57] Wilks, SS, Order statistics, Bull Am Math Soc, 54, 6-50, (1948) · Zbl 0034.07305
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.