Relating Granger causality to long-term causal effects

In estimation of causal couplings between observed processes, it is important to characterize coupling roles at various time scales. The widely used Granger causality reflects short-term effects: it shows how strongly perturbations of a current state of one process affect near future states of another process, and it quantifies that via prediction improvement (PI) in autoregressive models. However, it is often more important to evaluate the effects of coupling on long-term statistics, e.g., to find out how strongly the presence of coupling changes the variance of a driven process as compared to an uncoupled case. No general relationships between Granger causality and such long-term effects are known. Here, we pose the problem of relating these two types of coupling characteristics, and we solve it for a class of stochastic systems. Namely, for overdamped linear oscillators, we rigorously derive that the above long-term effect is proportional to the short-term effects, with the proportionality coefficient depending on the prediction interval and relaxation times. We reveal that this coefficient is typically considerably greater than unity so that small normalized PI values may well correspond to quite large long-term effects of coupling. The applicability of the derived relationship to wider classes of systems, its limitations, and its value for further research are discussed. To give a real-world example, we analyze couplings between large-scale climatic processes related to sea surface temperature variations in equatorial Pacific and North Atlantic regions.