On the Hsiao definition of non-causality

Granger (Granger, C.W.J., 1969. Investigating causal relations by econometric models and cross-spectral methods. Econometrica 37, 424–438.) defined causality between two variables X and Y in terms of predictability. A difficulty with this definition is that it is restricted to one-step ahead prediction. In the presence of a third environment variable Z the non-causality properties depend on the horizon of the involved prediction. Hsiao (Hsaio, C., 1982. Autoregressive modelling and causal ordering of economic variables. Journal of Economic Dynamic and Control 4, 243–259.) proposed a generalization of the Granger notion of causality. The main purpose of this paper is to show that the Hsiao non-causality properties do not depend on the horizon of involved prediction.  2000 Elsevier Science S.A. All rights reserved.