Granger causality exists when past values of a variable $X$ contains information about another variable $Y$ beyond the information in past values of $Y$.

The Granger causality test seeks to establish the directional relation between two time-series $\lbrace X_t\rbrace^{T}_{t=1}$ and $\lbrace Y_t\rbrace^{T}_{t=1}$ by testing whether one of the two is useful in predicting the other. Consider the two regressions: $$\begin{align} Y_t &= \sum^m_{k=1}\alpha_k Y_{t-k} + \sum^m_{k=1}\beta_k X_{t-k} + \varepsilon_{1,t} \\[5pt] X_t &= \sum^m_{k=1}\gamma_k X_{t-k} + \sum^m_{k=1}\delta_k Y_{t-k} + \varepsilon_{2,t} \end{align}$$ where the errors $\varepsilon_{1,t}$ and $\varepsilon_{2,t}$ are uncorrelated. If $\beta_k$ are jointly significant in the first regression but $\delta_k$ are jointly insignificant in the second, we say that $X$ Granger-causes $Y$. This is referred to as unidirectional causality. Equally $Y$ may Granger-cause $X$ if $\beta_k = 0$ jointly and $\delta_k \neq 0$ jointly.

Bilateral causality refers to the case when both $\beta_k \neq 0$ and $\delta_k \neq 0$, whilst $\beta_k = 0$ and $\delta_k = 0$ is called independence.

Note that Granger causality does not necessarily prove a true causal relationship between the two variables. It just tests whether one event frequently occurs before the other and thus is a reliable predictor of it. The post hoc ergo propter hoc ("after this, therefore because of this") principle is generally not regarded as "true" causality.