Granger Causality

A sequence $X$ is said to Granger-cause a sequence $Y$ if

• $\mathbb{P}[Y(t+1)\in A\mid \mathcal{I}(t)]\ne \mathbb{P}[Y(t+1)\in A\mid {\mathcal{I}}_{-X}(t)]$

where:

• $\mathbb{P}$ denotes the probability of an event
• $X$, $Y$ are sequences
• $t$ denotes time
• $A$ is an arbitrary non-empty set
• $\mathcal{I}(t)$ and ${\mathcal{I}}_{-\mathcal{X}}(t)$ denote Filtrations