Selection of the Relevant Information Set for Predictive Relationships Analysis between Time Series

In time series analysis, a vector Y is often called causal for another vector X if the former helps to improve the k-step-ahead forecast of the latter. If this holds for k D 1, vector Y is commonly called Granger-causal for X. It has been shown in several studies that the finding of causality between two (vectors of) variables is not robust to changes of the information set. In this paper, using the concept of Hilbert spaces, we derive a condition under which the predictive relationships between two vectors are invariant to the selection of a bivariate or trivariate framework. In more detail, we provide a condition under which the finding of causality (improved predictability at forecast horizon 1) respectively non-causality of Y for X is unaffected if the information set is either enlarged or reduced by the information in a third vector Z. This result has a practical usefulness since it provides a guidance to validate the choice of the bivariate system fX,Yg in place of fX,Y, Zg. In fact, to test the ‘goodness’ of fX,Yg we should test whether Z Granger cause X not requiring the joint analysis of all variables in fX,Y,Zg.