Some Remarks on the Causality Definitions in the Non-linear Theory of Stochastic Process

The Granger causality concept is extensively used in econometrics and several works have applied the Granger causality definition to test hypotheses about economic structures. The applications have shown controversial results depending on the test structures, the time series length, the lag orders and the filtering. In particular different test structures were originated by different definitions on causality derived from Sims and Pierce-Haugh. The knowledge of theoretical equivalences between the definitions avoids the attribution of the discrepancies of the results, using the same data but different tests, to the different definitions. The aim of this paper is to propose an extension of Pierce-Haugh causality definition in the non-linear case, showing that this extended definition is implied by the non-linear definition of Granger causality.