An extension of the asymmetric causality tests for dealing with deterministic trend components

This article extends the asymmetric causality tests, as developed by Hatemi-J (2012), for dealing with deterministic trend parts. It is shown how integrated variables up to three degrees with deterministic trend parts can be transformed into positive and negative cumulative partial components. These cumulative components can be used for implementing the asymmetric causality tests based on a Wald test statistic that is shown to follow a chi-square distribution asymptotically. Each solution is expressed as a proposition and a mathematic proof is provided for each underlying proposition. This issue is important because most economic or financial variables seem to be characterized by both stochastic as well as deterministic trend parts. An empirical application is provided in order to show how the oil prices and the exchange rates as integrated variables with drift and trend can be transformed into cumulative partial sums of positive and negative components. The conducted causality tests reveal that allowing for asymmetry has important repercussions for the underlying causal inference between these two variables.