Vector Autoregressive Model (VAR)

Given $k$ time series $x_1, \ldots, x_t$ with $x_t \in \mathbb{R}^k$, a $p$-th order vector autoregressive model (denoted VAR($p$)) generalizes the univariate AR model. It models the output as linear functions of the input series $x$:

$$x_t = \sum_{i=1}^p A_i x_{t-i} + e_t$$

where $A_i \in \mathbb{R}^{k{\times}k}$ is a $k{\times}k$ matrix. The series $\{e_t\}$ can represent either a controlled external input or noise.