Autocopulas: Investigating the Interdependence Structure of Stationary Time Series

Here we present a novel approach to the description of the lagged interdependence structure of stationary time series. The idea is to extend the use of copulas to the lagged (one-dimensional) series, to the analogy of the autocorrelation function. The use of such autocopulas can reveal the specifics of the lagged interdependence in a much finer way. However, the lagged interdependence is resulted from the dynamics, governing the series, therefore the known and popular copula models have little to do with that type of interdependence. True though, it seems rather cumbersome to calculate the exact form of the autocopula even for the simplest nonlinear time series models, so we confine ourselves here to an empirical and simulation based approach. The advantage of using autocopulas lays in the fact that they represent nonlinear dependencies as well, and make it possible e.g. to study the interdependence of high (or low) values of the series separately. The presented methods are capable to check whether autocopulas of an observed process can be distinguished significantly from the autocopulas a of given time series model. The proposed approach is based on the Kendall's transform which reduces the multivariate problem to one dimension. After illustrating the use of our approach in detecting conditional heteroscedasticity in the AR-ARCH vs. AR case, we apply the proposed methods to investigate the lagged interdependence of river flow time series with particular focus on model choice based on the synchronized appearance of high values.