Let us consider two scalar stochastic processes $x_t$ and $y_t$, $t \in \mathbb{Z}$ , each observed for $T$ realizations. We assume that $x_t$ and $y_t$ are covariance stationary or that $Δx_t$ and $Δy_t$ are covariance stationary.

If we exclude the possibility that the future can cause the past, but allow contemporaneous feedback loops due for example to temporal aggregation, there are several possibilities as regards the causal structure between $x_t$ and $y_t$

• (i) The series $x_t$ has a contemporaneous or lagged causal effect on $y_t$ , i.e. $x_i → y_{i + s}$ for some $i , s$ such that $i ≥ 0, s ≥ 0$.
• (ii) The series $y_t$ has a contemporaneous or lagged causal effect on $x_t$ , i.e. $y_i → x_{i + s}$ for some $i , s$ such that $i ≥ 0, s ≥ 0$.
• (iii) A not-measured series $z_t$ has a contemporaneous or lagged causal effect on both $x_t$ and $y_t$ .
• (iv) The causal structure between $x_t$ and $y_t$ can be described by any combination of (i)–(iii).
• (v) There is no causal link or path (of any type) linking $x_t$ and $y_{t + s}$ , for any $s \in \mathbb{N}$.