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TESTING FOR CHANGES IN KENDALL’S TAU

Published online by Cambridge University Press:  04 November 2016

Herold Dehling ,
Daniel Vogel ,
Martin Wendler  and
Dominik Wied
Herold Dehling
Affiliation:
University of Bochum
Daniel Vogel*
Affiliation:
University of Aberdeen
Martin Wendler
Affiliation:
University of Greifswald
Dominik Wied
Affiliation:
University of Cologne
*
*Address correspondence to Daniel Vogel, Institute for Complex Systems and Mathematical Biology, University of Aberdeen, Aberdeen AB24 3UE, UK; e-mail: daniel.vogel@abdn.ac.uk.
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Abstract

For a bivariate time series ((Xi ,Yi))i=1,...,n, we want to detect whether the correlation between Xi and Yi stays constant for all i = 1,...n. We propose a nonparametric change-point test statistic based on Kendall’s tau. The asymptotic distribution under the null hypothesis of no change follows from a new U-statistic invariance principle for dependent processes. Assuming a single change-point, we show that the location of the change-point is consistently estimated. Kendall’s tau possesses a high efficiency at the normal distribution, as compared to the normal maximum likelihood estimator, Pearson’s moment correlation. Contrary to Pearson’s correlation coefficient, it shows no loss in efficiency at heavy-tailed distributions, and is therefore particularly suited for financial data, where heavy tails are common. We assume the data ((Xi ,Yi))i=1,...,n to be stationary and P-near epoch dependent on an absolutely regular process. The P-near epoch dependence condition constitutes a generalization of the usually considered Lp-near epoch dependence allowing for arbitrarily heavy-tailed data. We investigate the test numerically, compare it to previous proposals, and illustrate its application with two real-life data examples.

Type
ARTICLES
Information
Econometric Theory , Volume 33 , Issue 6 , December 2017 , pp. 1352 - 1386
Copyright
Copyright © Cambridge University Press 2016 

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Footnotes

The authors wish to thank their colleague Roland Fried for several very stimulating discussions that motivated this paper. Moreover, we are grateful for helpful comments from the editors and referees, which substantially improved a previous version of the paper. We are also indebted to Alexander Dürre, who did a thorough proofreading of the manuscript. The research was supported in part by the Collaborative Research Grant 823 Statistical modelling of nonlinear dynamic processes of the German Research Foundation.

References

REFERENCES

Aue, A., Hörmann, S., Horváth, L., & Reimherr, M. (2009) Break detection in the covariance structure of multivariate time series models. Annals of Statistics 37, 4046–4087.CrossRefGoogle Scholar
Berkes, I., Hörmann, S., & Schauer, J. (2009) Asymptotic results for the empirical process of stationary sequences. Stochastic Processes and their Applications 119, 1298–1324.CrossRefGoogle Scholar
Bilodeau, M. & Brenner, D. (1999) Theory of Multivariate Statistics. Springer.Google Scholar
Borovkova, S., Burton, R.M., & Dehling, H. (2001) Limit theorems for functionals of mixing processes with applications to U-statistics and dimension estimation. Transactions of the American Mathematical Society 353, 4261–4318.CrossRefGoogle Scholar
Bücher, A., Jäschke, S., & Wied, D. (2015) Nonparametric tests for constant tail dependence with an application to energy and finance. Journal of Econometrics 187, 154–168.CrossRefGoogle Scholar
Bücher, A. & Kojadinovic, I. (2016) Dependent multiplier bootstraps for non-degenerate U-statistics under mixing conditions with applications. Journal of Statistical Planning and Inference 170, 83–105.CrossRefGoogle Scholar
Bücher, A., Kojadinovic, I., Rohmer, T., & Segers, J. (2014) Detecting changes in cross-sectional dependence in multivariate time series. Journal of Multivariate Analysis 132, 111–128.CrossRefGoogle Scholar
Croux, C. & Dehon, C. (2010) Influence functions of the Spearman and Kendall correlation measures. Statistical Methods and Applications 19, 497–515.Google Scholar
Csörgő, M. & Horváth, L. (1997) Limit Theorems in Change-Point Analysis. Wiley.Google Scholar
de Jong, R.M. & Davidson, J. (2000) Consistency of kernel estimators of heteroscedastic and autocorrelated covariance matrices. Econometrica 68, 407–424.CrossRefGoogle Scholar
Dehling, H. & Fried, R. (2012) Asymptotic distribution of two-sample empirical U-quantiles with applications to robust tests for shifts in location. Journal of Multivariate Analysis 105, 124–140.CrossRefGoogle Scholar
Dengler, B. (2010) On the Asymptotic Behaviour of the Estimator of Kendall’s Tau. Ph.D. thesis, Technische Universität Wien, Austria.Google Scholar
Denker, M. & Keller, G. (1986) Rigorous statistical procedures for data from dynamical systems. Journal of Statistical Physics 44, 67–93.CrossRefGoogle Scholar
Fan, J., Liu, H., & Wang, W. (2015) Large covariance estimation through elliptical factor models, submitted.Google Scholar
Gerstenberger, C. & Vogel, D. (2015) On the efficiency of gini’s mean difference. Statistical Methods and Applications 24, 569–596.Google Scholar
Giacomini, E., Härdle, W., & Spokoiny, V. (2009) Inhomogeneous dependence modeling with time-varying copulae. Journal of Business and Economics Statistics 27, 224–234.CrossRefGoogle Scholar
Gombay, E. & Horváth, L. (1995) An application of U-statistics to change-point analysis. Acta Scientiarum Mathematicarum 60, 345–357.Google Scholar
Gombay, E. & Horváth, L. (1999) Change-points and bootstrap. Environmetrics 10, 725–736.3.0.CO;2-K>CrossRefGoogle Scholar
Ibragimov, M., Ibragimov, R., & Kattuman, P. (2013) Emerging markets and heavy tails. Journal of Banking and Finance 37, 2546–2559.CrossRefGoogle Scholar
Inclán, C. & Tiao, G.C. (1994) Use of cumulative sums of squares for retrospective detection of changes of variance. Journal of the American Statistical Association 89, 913–923.Google Scholar
Inoue, A. (2001) Testing for distributional change in time series. Econometric Theory 17, 156–187.CrossRefGoogle Scholar
Kojadinovic, I. (2015) npcp: Some Nonparametric Tests for Change-Point Detection in Possibly Multivariate Observations. R package version 0.1–6. Available at: http://CRAN.R-project.org/package=npcp.Google Scholar
Kojadinovic, I., Quessy, J.-F., & Rohmer, T. (2016) Testing the constancy of Spearman’s rho in multivariate time series. Annals of the Institute of Statistical Mathematics 68, 929–954.CrossRefGoogle Scholar
Loretan, M. & Phillips, P.C.B. (1994) Testing the covariance stationarity of heavy-tailed time series. Journal of Empirical Finance 1, 211–248.CrossRefGoogle Scholar
McNeil, A., Frey, R., & Embrechts, P. (2005) Quantitative Risk Management: Concepts, Techniques, and Tools. Princeton University Press.Google Scholar
Nelsen, R.B. (2006) An Introduction to Copulas, 2nd ed. Springer.Google Scholar
Patton, A. (2006) Modelling asymmetric exchange rate dependence. International Economic Review 47, 527–556.CrossRefGoogle Scholar
Pötscher, B.M. & Prucha, I.R. (1997) Dynamic Nonlinear Econometric Models. Springer-Verlag.CrossRefGoogle Scholar
Quessy, J.-F., Saïd, M., & Favre, A.-C. (2013) Multivariate Kendall’s tau for change-point detection in copulas. Canadian Journal of Statistics 41, 65–82.CrossRefGoogle Scholar
Segers, J. (2012) Asymptotics of empirical copula processes under nonrestrictive smoothness assumptions. Bernoulli 18, 764–782.CrossRefGoogle Scholar
Stock, J. (1994) Unit roots, structural breaks and trends. In Engle, R. & McFadden, D. (eds.), Handbook of Econometrics, vol. IV, pp. 2739–2841. North-Holland.Google Scholar
Stock, J. & Watson, M. (2010) Introduction to Econometrics, 3rd ed. Addison-Wesley.Google Scholar
Stout, W.F. (1974) Almost Sure Convergence. Academic Press.Google Scholar
van der Vaart, A.W. & Wellner, J.A. (1996) Weak Convergence and Empirical Processes. With Applications to Statistics. Springer-Verlag.CrossRefGoogle Scholar
Wied, D., Dehling, H., van Kampen, M., & Vogel, D. (2014) A fluctuation test for constant Spearmans rho with nuisance-free limit distribution. Computational Statistics and Data Analysis 76, 723–736.CrossRefGoogle Scholar
Wied, D., Krämer, W., & Dehling, H. (2012) Testing for a change in correlation at an unknown point in time using an extended delta method. Econometric Theory 28, 570–589.CrossRefGoogle Scholar
Wooldridge, J.M. & White, H. (1988) Some invariance principles and central limit theorems for dependent heterogeneous processes. Econometric Theory 4, 210–230.CrossRefGoogle Scholar
Yoshihara, K.-I. (1976) Limiting behavior of U-statistics for stationary, absolutely regular processes. Zeitschrift für Wahrscheinlichkeistheorie und verwandte Gebiete 35, 237–252.CrossRefGoogle Scholar
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