A zero correlation across time also doesn't prove that there isn't causality. Suppose that there's (e.g.) a positive relationship due to an unobserved third fac

A zero correlation across time also doesn't prove that there isn't causality. Suppose that there's (e.g.) a positive relationship due to an unobserved third factor, and a negative relationship which is genuinely causal. These two could cancel out in the observed data.