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 $\Delta x_t$ and $\Delta y_t$ 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 regarding 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 rightarrow y_{i + s} for some i, s such that $i \geq 0, s \geq 0$.
• (ii) The series $y_t$ has a contemporaneous or lagged causal effect on $x_t$, i.e., y_i rightarrow x_{i + s} for some i, s such that $i \geq 0, s \geq 0$.
• (iii) An unmeasured 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 $s \in \mathbb{N}$.