On causality I: Sampling and noise

We discuss, apparently for the first time, the impact of sampling and noise on Granger causality. We show that strong Granger causality is preserved under sampling. We also show that strong Granger causality is preserved when the two processes of interest are observed through independent but coloured additive noises. Weak Granger causality is not preserved in these situations. These results have important implications suggesting that Granger causal relations can still be unravelled even when the system time constants are faster than sampling rates and even in the presence of noise. The results also cast serious doubts on the validity of current approaches based on vector autoregressive modeling and suggest that state space or vector ARMA models are needed instead.