[Paper Review] Testing for Structural Breaks via Ordinal Pattern Dependence

Alexander Schnurr, Herold Dehling|arXiv (Cornell University)|Jan 1, 2015

Complex Systems and Time Series Analysis16 references2 citations

TL;DR

This paper introduces a robust, nonparametric method for detecting structural breaks in the dependence between two time series using ordinal pattern dependence. By analyzing the order structure of consecutive data points (via permutations of ranks), the method captures nonlinear, monotonic dependence without assuming second moments. The key contribution is a formal test for structural breaks in this dependence, proven to converge asymptotically to a functional of Brownian motion, with applications in finance and biomedical data.

ABSTRACT

We propose new concepts in order to analyze and model the dependence structure between two time series. Our methods rely exclusively on the order structure of the data points. Hence, the methods are stable under monotone transformations of the time series and robust against small perturbations or measurement errors. Ordinal pattern dependence can be characterized by four parameters. We propose estimators for these parameters, and we calculate their asymptotic distributions. Furthermore, we derive a test for structural breaks within the dependence structure. All results are supplemented by simulation studies and empirical examples. For three consecutive data points attaining different values, there are six possibilities how their values can be ordered. These possibilities are called ordinal patterns. Our first idea is simply to count the number of coincidences of patterns in both time series, and to compare this with the expected number in the case of independence. If we detect a lot of coincident patterns, this means that the up-and-down behavior is similar. Hence, our concept can be seen as a way to measure non-linear `correlation'. We show in the last section, how to generalize the concept in order to capture various other kinds of dependence.

Motivation & Objective

• To develop a nonparametric, robust method for detecting changes in the dependence structure between two time series.
• To extend ordinal pattern dependence to model nonlinear, monotonic dependence without requiring second-order moments.
• To provide consistent estimators and asymptotic distributions for key dependence parameters (p, q, r, s).
• To construct a formal statistical test for structural breaks in the ordinal pattern dependence structure.
• To demonstrate the method's applicability through simulation studies and real-world financial data examples.

Proposed method

• Define ordinal patterns as rank permutations of h+1 consecutive data points in each time series.
• Measure positive dependence via the probability p that both series exhibit the same ordinal pattern.
• Measure negative dependence via the probability r that one series exhibits the inverse (reflected) pattern of the other.
• Estimate p and r using empirical relative frequencies of coincident or reflected patterns.
• Establish asymptotic normality of the estimators using the delta method under near-epoch dependence and 1-approximating functional conditions.
• Construct a test statistic Tn based on partial sums of pattern coincidence indicators, proving weak convergence to a functional of Brownian motion under the null of no structural break.

Experimental results

Research questions

• RQ1 Can ordinal pattern dependence effectively detect nonlinear, monotonic dependence between time series without assuming finite second moments?
• RQ2 How can structural breaks in the dependence structure be formally tested using ordinal patterns?
• RQ3 What are the asymptotic distributions of the estimators for the key parameters (p, q, r, s) under weak dependence assumptions?
• RQ4 How robust is the method to measurement errors and monotone transformations of the data?
• RQ5 Can the test detect meaningful changes in dependence, such as those observed in financial market dynamics?

Key findings

• The estimator for the probability of coincident ordinal patterns (p) is asymptotically normal with a derived variance-covariance structure.
• The test statistic Tn for structural breaks converges in distribution to σ sup₀≤λ≤1 |W(λ) − λW(1)|, where W is a standard Brownian motion.
• The method is robust under monotone transformations and stable under small perturbations or measurement errors.
• The test maintains validity even when the underlying time series lack second moments, unlike classical correlation-based tests.
• Empirical applications on S&P 500 and VIX data reveal detectable structural breaks in dependence, confirming the method's practical relevance.
• The approach captures dependence structures not detected by Pearson correlation, Kendall’s tau, or Spearman’s rho, demonstrating its unique sensitivity to non-linear monotonic relationships.