Testing for Changes in Kendall’s Tau

Abstract

For a bivariate time series ((X i , Y i )) i=1,...,n we want to detect whether the correlation between X i and Y i 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 ((X i , Y i )) 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 L p -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.Keywords: Change-point analysis, Kendall's tau, U -statistic, functional limit theorem, near epoch dependence in probability the association of financial price processes and, within reasonable time frames, re-estimate the correlation parameters. Particularly, in times of global financial crises, the price processes of most financial assets tend to be highly dependent, united in their common downward trend, causing the hedging powers of investment diversification to cease -an effect for which the term diversification meltdown has been coined. The problem of detecting changes in the distribution of sequential observations has a long history in statistics, see e.g. Csörgő and Horváth (1997). However, particularly detecting changes in the dependence structure of multivariate time series has attracted the focus of statistical research only recently. Examples for such detection procedures are Loretan and Phillips (1994), who test for covariance stationarity of a possibly heavy-tailed time series, Giacomini, Härdle, and Spokoiny (2009), who consider tests for homogeneity of time-varying copulae, Aue, Hörmann, Horváth, and Reimherr (2009), who propose a test for a constant covariance matrix, and Wied, Krämer, and Dehling (2012), who suggest a change-point test for correlations between two random variables based on Pearson's correlation coefficient. With this paper, we want to contribute to the literature by proposing a new test for constant Kendall's tau that can be applied to dependent series. We recommend to use the rank correlation measure Kendall's tau instead of Pearson's correlation coefficient because it is almost as efficient as the moment correlation at normality, but is significantly more efficient at heavytailed distributions. For details see Section 5. This issue is very im...

Testing Kendall's τ for a large class of dependent sequences

“…is not too slow, this provides an easy and reasonable choice for a n . To conclude this remark, note that an estimator of V similar to V n is also proposed in [6].…”

“…The advantage of our approach is that we are able to prove that our condition on the coefficient β 2 is optimal in some sense (see Proposition 2 below). Moreover, If we consider only the class of β-mixing sequences, then our results are valid under the condition k>0 β(k) < ∞, while the condition in [6] cannot be better than k>0 kβ(k) < ∞.…”

“…To conclude this section, let us quote that the asymptotic normality of Kendall's U -statistic for dependent sequences has been recently established by Dehling et al [6] (note that these authors are able to deal with a large class of U -statistics, and that they also prove a functional central limit theorem for U -processes). Their result is valid for a large class of dependent sequences (including non-mixing sequences in the sense of Rosenblatt [18]), with a large intersection with our class of β 2 -dependent sequences.…”

The factors driving the synergy value in mergers of mechanical engineering companies in the Czech Republic

“…According to these authors, the Kendall's tau correlation coefficient is more robust and also more efficient than the Spearman's rank correlation coefficient. Another reason for using this test are the findings of Dehling et al (2017), who stated that the Kendall's tau test is particularly suitable for small research data sets, and they considered it appropriate for the analysis of financial ratios. Statistica software was used for testing.…”

Quantifying the data-dredging bias in structural break tests

“…For the leading case k n = n, it holds (e.g.) for the mean (Davidson 1994), the variance , correlation ) and Kendall's tau (Dehling et al 2017). For extreme value quantities, whose estimators typically depend only on a vanishing fraction of the sample, a scaling different from √ n is required in Assumption 1.…”