Short run and long run causality in time series: Theory

Causality in the sense of Granger is typically defined in terms of predictibility of a vector of variables one period ahead. Recently, Lutkepohl (1993) proposed to define noncausality between two variables in terms of nonpredictibility at any number of periods ahead. When more than two vectors are considered (i.e., when the information set contains auxiliary variables), these two notions are not equivalent. In this paper, we first generalize the notion of causality by considering causality at a given (arbitrary) horizon h. Then we derive necessary and sufficient conditions for noncausality between vectors of variables (inside a larger vector) up to any given horizon h, where h can be infinite. In particular, for general possibly nonstationary processes with finite second moments, relatively simple exhaustivity and separation conditions, which are sufficient for noncausality at all horizons, are provided. To deal with cases where such conditions do not apply, we consider a more specific, although still very wide, class of vector autoregressive processes (possibly of infinite order, stationary or nonstationary), which include multivariate ARIMA processes, and we derive general parametric characterizations of noncausality at various horizons for this class (including a causality chain characterization). We also observe that the coefficients of lagged variables in forecasts at various horizons h greater than or equal to 1 can be interpreted as generalized impulse response coefficients which yield a complete picture of linear causality properties, in contrast with usual response coefficients which can be quite misleading in this respect.