Ordinal Pattern Dependence in the Context of Long-Range Dependence.

Ines Nüßgen,Alexander Schnurr

doi:10.3390/e23060670

Abstract

Ordinal pattern dependence is a multivariate dependence measure based on the co-movement of two time series. In strong connection to ordinal time series analysis, the ordinal information is taken into account to derive robust results on the dependence between the two processes. This article deals with ordinal pattern dependence for a long-range dependent time series including mixed cases of short- and long-range dependence. We investigate the limit distributions for estimators of ordinal pattern dependence. In doing so, we point out the differences that arise for the underlying time series having different dependence structures. Depending on these assumptions, central and non-central limit theorems are proven. The limit distributions for the latter ones can be included in the class of multivariate Rosenblatt processes. Finally, a simulation study is provided to illustrate our theoretical findings.

Highlights

The origin of the concept of ordinal patterns is in the theory of dynamical systems
Ordinal pattern dependence is a multivariate dependence measure that compares the co-movement of two time series based on the ordinal information
Following [11], Section 1, the minimal requirement of the data sets we use for ordinal analysis in the time series context, i.e., for ordinal pattern probabilities as well as for ordinal pattern dependence later on, is ordinal pattern stationarity

Summary

Introduction

The origin of the concept of ordinal patterns is in the theory of dynamical systems. The idea is to consider the order of the values within a data vector instead of the full metrical information. In [5], ordinal patterns were used in order to estimate the Hurst parameter in long-range dependent time series. In [10], ordinal patterns were used for the first time in order to analyze the dependence between two time series. Limit theorems for this new concept were proven in a short-range dependent framework in [11]. Ordinal pattern dependence is a promising tool, which has already been used in financial, biological and hydrological data sets, see in this context, [12] for an analysis of the co-movement of time series focusing on symbols.

Mathematical Framework

Ordinal Pattern Dependence

Limit Theorem for the Estimator of p in Case of Long-Range Dependence is in

Limit Theorem for the Estimator of p in Case of Short-Range Dependence

Simulation Study

Conclusions and Outlook