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Autocopulas: investigating the interdependence structure of stationary time series. (English) Zbl 1241.62126
Summary: We present a novel approach to the description of the lagged interdependence structure of stationary time series. The idea is to extend the use of copulas to the lagged (one-dimensional) series, to an analogy of the autocorrelation function. The use of such autocopulas can reveal the specifics of the lagged interdependence in a much finer way. However, the lagged interdependence is resulted from the dynamics, governing the series, therefore the known and popular copula models have little to do with that type of interdependence. True though, it seems rather cumbersome to calculate the exact form of the autocopula even for the simplest nonlinear time series models, so we confine ourselves here to an empirical and simulation based approach. The advantage of using autocopulas lays in the fact that they represent nonlinear dependencies as well, and make it possible, e.g., to study the interdependence of high (or low) values of the series separately. The presented methods are capable to check whether autocopulas of an observed process can be distinguished significantly from the autocopulas a of given time series model. The proposed approach is based on the Kendall transform which reduces the multivariate problem to one dimension. After illustrating the use of our approach in detecting conditional heteroscedasticity in the AR-ARCH vs. the AR case, we apply the proposed methods to investigate the lagged interdependence of river flow time series with particular focus on model choice based on the synchronized appearance of high values.

MSC:
62M10 Time series, auto-correlation, regression, etc. in statistics (GARCH)
62H05 Characterization and structure theory for multivariate probability distributions; copulas
62G10 Nonparametric hypothesis testing
65C60 Computational problems in statistics (MSC2010)
62P12 Applications of statistics to environmental and related topics
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[1] Beirlant J, Goegebeur Y, Segers J, Teugels J (2004) Statistics of extremes: theory and applications. Wiley series in probability and statistcs pp 342–348 · Zbl 1070.62036
[2] Borkovec M, Klüppelberg K (2001) The tail of the stationary distribution of an autoregressive process with ARCH(1) errors. Ann Appl Probab 11:1220–1241 · Zbl 1010.62083
[3] Chen XH, Fan YQ (2006) Estimation of copula based semiparametric time series models. J Econom 130:307–335 · Zbl 1337.62201
[4] Cherubini U, Luciano E, Vecchiato W (2004) Copula methods in finance. WileyFinance, West Sussex · Zbl 1163.62081
[5] Diks C, Panchenko V (2008) Rank-based entropy tests for serial independence. Stud Nonlinear Dyn Econom 12(1):1–19. Article 2, Berkeley Electronic Press · Zbl 1194.62065
[6] Elek P, Márkus L (2008) A light-tailed conditionally heteroscedastic time series model with an application to river flows. J Time Ser Anal 29:14–36 · Zbl 1164.62044
[7] Francq C, Zakoïan JM (1998) Estimating linear representations of nonlinear processes. J Stat Plan Inference 68:145–165 · Zbl 0942.62100
[8] Francq C, Roy R, Zakoïan JM (2005) Diagnostic checking in ARMA models with uncorrelated errors. J Am Stat Assoc 100:532–544 · Zbl 1117.62336
[9] Genest C, Rémillard B (2004) Tests of independence and randomness based on the empirical copula process. Test 13(2):335–369 · Zbl 1069.62039
[10] Genest C, Rémillard B (2006) Discussion of ”Copulas: tales and facts” by Thomas Mikosch. Extremes 9:27–36
[11] Genest C, Quessy JF, Rémillard B (2006) Goodnes-of-fit procedures for copula models based on the integral probability transformation. Scand J Stat 33:337–366 · Zbl 1124.62028
[12] Genest C, Rémillard B, Beaudoin D (2009) Goodness-of-fit tests for copulas: a review and a power study. Insur, Math Econ 4:199–213 · Zbl 1161.91416
[13] Ghoudi K, Rémillard B (2004) Empirical processes based on pseudo-observations II: the multivariate case. In: Horváth L, Szyszkowicz B (eds) Asymptotic methods in stochastics: Festschrift for Miklós Csörgö, vol 44, pp 381–406 · Zbl 1079.60024
[14] Hoeffding W (1940) Masstabinvariante korrelationstheorie. Schriften des Mathematischen Instituts und des Instituts für Angewandte Mathematik der Universitat Berlin 5(3):179–233
[15] Joe H (1997) Multivariate models and dependence concepts. Chapman & Hall, London · Zbl 0990.62517
[16] Kotz S, Nadarajah S (2000) Extreme value distributions. Imperial College Press, London · Zbl 0960.62051
[17] McNeil AJ, Frey R, Embrechts P (2005) Quantitative risk management: concepts, techniques, tools. Princeton University Press, Princeton · Zbl 1089.91037
[18] Meitz M, Saikkonen P (2008) Stability of AR-GARCH models. J Time Ser Anal 29:453–475 · Zbl 1199.62014
[19] Mikosch T (2006) Copulas: tales and facts. Extremes 9:21–53 (Discussion), 55–62 (Rejoinder)
[20] Nelsen RB (2006) An introduction to copulas, 2nd edn. Wiley, New York · Zbl 1152.62030
[21] Patton AJ (2008) Copula-based models for financial time series. In: Andersen TG, Davis RA, Kreiss JP, Mikosch T (eds) Handbook of financial time series. Springer, Berlin
[22] Sklar A (1959) Fonctions de répartition à n dimensions et leurs marges. Publ Inst Stat Univ Paris 8:229–231 · Zbl 0100.14202
[23] Van den Goorbergh RWJ (2004) A copula-based autoregressive conditional dependence model of international stock markets. Working Paper No. 022/2004, De Nederlandsche Bank NV, Amsterdam, The Netherlands
[24] Vasas K, Elek P, Márkus L (2007) A two-state regime switching autoregressive model with an application to river flow analysis. J Stat Plan Inference 137(10):3113–3126 · Zbl 1114.62093
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