Short Run and Long Run Causality in Time Series: Theory

Causality in Granger's sense is defined in terms of predictibility one period ahead. The notion of causality is generalized by considering causality at any given horizon 1 <= h <= inifinity, providing a rigorous formalization of indirect causal effects and causality chains in (possibly) nonstationary time series. The authors derive necessary and sufficient conditions for noncausality between vectors up to any given horizon. They observe that coefficients of lagged variables in forecasts at various horizons can be interpreted as generalized impulse response coefficients yielding a complete picture of linear causality, in contrast with usual impulse coefficients which can be misleading.