Figure 6.

(a) Illustration of Granger noncausality in an example with $p=2$ and $m_1=m_2=3$. Since the tensor represents conditional probabilities, the columns of the front face of the tensor, the vertical $x_{1t}$ axis, must sum to one. Here, $x_2$ is not Granger causal for $x_1$ since each slice of the conditional probability tensor along the $x_2$ mode is equal. (b) Schematic of the mixture transition distribution (MTD) factorization of the conditional probability tensor $p\left(x_{1t}\mid x_{(t-1)1},x_{(t-1)2}\right)$. Figure adapted with permission from Tank et al. (2021b).