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 and derive its asymptotic distribution under the null hypothesis of no change by means a new U-statistic invariance principle for dependent processes. The asymptotic distribution depends on the long run variance of Kendall's tau, for which we propose an estimator and show its consistency. Furthermore, 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 coefficient. Contrary to Pearson's correlation coefficient, it has excellent robustness properties and shows no loss in efficiency at heavy-tailed distributions. 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, $p \ge 1$, that does not require the existence of any moments. It is therefore very well suited for our objective to efficiently detect changes in correlation for arbitrarily heavy-tailed data.