- We observe some realization of the (discrete-time) \(p\)-dimensional stochastic process
\[
\{Y(t): t \in T\}
\]
- with state space \(Y(t) \in \mathbb{R}^p\)
- Vector autoregressive (VAR) model assumes
\[
\mathbf{y}_t = \mathbf{\nu} + \mathbf{A}_1 \mathbf{y}_{t-1} + \ldots + \mathbf{A}_l \mathbf{y}_{t-1} + \mathbf{\epsilon}_t
\]
- \(\nu \in \mathbb{R}^p\) is the intercept
- \(A_l \in \mathbb{R}^{p \times p}\) describes the coefficients at lag \(l\)
- \(\mathbf{\epsilon}_t\) are the stochastic innovations at time \(t\), for which
\[
\begin{align}
\mathbf{\epsilon}_t \sim \mathcal{N}(0, \Sigma) \\[0.5em]
\mathbb{E}[\mathbf{\epsilon}_t\mathbf{\epsilon}_{t-1}] = 0
\end{align}
\]
- VAR process is covariance-stationary, i.e. its first and second moment are time-invariant