Testing for Structural Breaks via Ordinal Pattern Dependence

ABSTRACT

We propose new concepts 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, it would indicate that the up-and-down behavior is similar. Hence, our concept can be seen as a way to measure nonlinear “correlation.” We show in the last section how to generalize the concept to capture various other kinds of dependence.

KEYWORDS:

• Limit theorems
• Near epoch dependence
• Nonlinear correlation
• Time series

Funding

The authors gratefully acknowledge financial support of the DFG (German science Foundation) SFB 823: Statistical modeling of nonlinear dynamic processes (projects C3 and C5).