A Gini-based time series analysis and test for reversibility

Time reversibility is a fundamental hypothesis in time series. In this paper, Gini-based equivalents for time series concepts that enable to construct a Gini-based test for time reversibility under merely first-order moment assumptions are developed. The key idea is that the relationship between two variables using Gini (as measured by Gini autocorrelations and partial autocorrelations) can be measured in two directions, which are not necessarily equal. This implies a built-in capability to discriminate between looking at forward and backward directions in time series. The Gini creates two bi-directional Gini autocorrelations (and partial autocorrelations), looking forward and backward in time, which are not necessarily equal. The difference between them may assist in identifying models with underlying heavy-tailed and non-normal innovations. Gini-based test and Gini-based correlograms, which serve as visual tools to examine departures from the symmetry assumption, are constructed. Simulations are used to illustrate the suggested Gini-based framework and to validate the statistical test. An application to a real data set is presented.