Testing for Parallelism Between Trends in Multiple Time Series

TL;DR

This paper introduces a new statistical test for assessing whether multiple nonstationary time series share similar trend patterns, utilizing a parallelism index and simulation-based distribution approximation.

Contribution

It develops a novel parallelism test for trends in multiple time series, including a CLT, simulation-based p-value approximation, and a clustering application.

Findings

The test is consistent and has good finite-sample properties.

Simulation-based approximation outperforms normal approximation.

Application to real data demonstrates practical utility.

Abstract

This paper considers the inference of trends in multiple, nonstationary time series. To test whether trends are parallel to each other, we use a parallelism index based on the L2-distances between nonparametric trend estimators and their average. A central limit theorem is obtained for the test statistic and the test's consistency is established. We propose a simulation-based approximation to the distribution of the test statistic, which significantly improves upon the normal approximation. The test is also applied to devise a clustering algorithm. Finally, the finite-sample properties of the test are assessed through simulations and the test methodology is illustrated with time series from Motorola cell phone activity in the United States.