22th February, 2019
There is no simple mapping between the two (Molenaar, 2004)
\[ \{Y(t): t \in T\} \]
\[ \mathbf{y}_t = \mathbf{\nu} + \mathbf{A}_1 \mathbf{y}_{t-1} + \ldots + \mathbf{A}_l \mathbf{y}_{t-1} + \mathbf{\epsilon}_t \]
\[ \begin{align} \mathbf{\epsilon}_t \sim \mathcal{N}(0, \Sigma) \\[0.5em] \mathbb{E}[\mathbf{\epsilon}_t\mathbf{\epsilon}_{t-1}] = 0 \end{align} \]
\[ A_{\text{VAR}} = \begin{pmatrix} \alpha_{11} & \alpha_{12} & \alpha_{13} & \alpha_{14} & \alpha_{15} & \alpha_{16} \\ \alpha_{21} & \alpha_{22} & \alpha_{23} & \alpha_{24} & \alpha_{25} & \alpha_{26} \\ \alpha_{31} & \alpha_{32} & \alpha_{33} & \alpha_{34} & \alpha_{35} & \alpha_{36} \\ \alpha_{41} & \alpha_{42} & \alpha_{43} & \alpha_{44} & \alpha_{45} & \alpha_{46} \\ \alpha_{51} & \alpha_{52} & \alpha_{53} & \alpha_{54} & \alpha_{55} & \alpha_{56} \\ \alpha_{61} & \alpha_{62} & \alpha_{63} & \alpha_{64} & \alpha_{65} & \alpha_{66} \end{pmatrix} \]
\[ A_{\text{AR}} = \begin{pmatrix} \alpha_{11} & 0 & 0 & 0 & 0 & 0 \\ 0 & \alpha_{22} & 0 & 0 & 0 & 0 \\ 0 & 0 & \alpha_{33} & 0 & 0 & 0 \\ 0 & 0 & 0 & \alpha_{44} & 0 & 0 \\ 0 & 0 & 0 & 0 & \alpha_{55} & 0 \\ 0 & 0 & 0 & 0 & 0 & \alpha_{66} \\ \end{pmatrix} \]
\[ \begin{align} y_{1t} &= \mathbf{\nu}_1 + \mathbf{a}_1^T \cdot \mathbf{y}_{t-1} + \mathbf{\epsilon}_{1t} \\[.5em] y_{2t} &= \mathbf{\nu}_2 + \mathbf{a}_2^T \cdot \mathbf{y}_{t-1} + \mathbf{\epsilon}_{2t} \\[.5em] \vdots &= \hspace{3em}\vdots \\[.5em] y_{pt} &= \mathbf{\nu}_p + \mathbf{a}_p^T \cdot \mathbf{y}_{t-1} + \mathbf{\epsilon}_{pt} \\ \end{align} \]
\[ \begin{align*} \alpha_{ij} \mid \lambda_{ij}, \tau &\sim \text{Normal}(0, \tau^2 \lambda_{ij}^2) \\ \lambda_{ij} &\sim \text{Cauchy}^+(0, 1) \end{align*} \]
\[
\begin{align*}
\alpha_{ij} \mid \lambda_{ij}, \tau &\sim \text{Normal}(0, \tau^2 \lambda_{ij}^2) \\
\lambda_{ij} &\sim \text{Bernoulli}(\theta)
\end{align*}
\]
\[ \begin{align*} \alpha_{ij} &\mid \lambda_{ij}, c_{ij} \sim \lambda_{ij} \, \text{Normal}(0, \tau_{ij}^2) + (1 - \lambda_{ij}) \, \delta_0 \\[0.5em] \lambda_{ij} &\sim \text{Bern}(\theta) \\[0.5em] \theta &\sim \text{Beta}(1, 1) \\[0.5em] \tau_{ij}^2 &\sim \text{Inverse-Gamma}(1/2, s/2) \\[0.5em] \Sigma &\sim \text{Inverse-Wishart}(S, \nu) \end{align*} \]
\[ \text{RMSE}_{\text{est}} := \sqrt{\frac{1}{p^2} \sum_{i=1}^p \sum_{j=1}^p \left(A_{i, j} - \hat{A}_{i, j}\right)^2} \]
\[ \text{RMSE}_{\text{pred}} := \sqrt{\frac{1}{200} \sum_{i=1}^{200} \frac{1}{p} \sum_{j=1}^p \left(Y_{i, j} - \hat{Y}_{i, j}\right)^2} \enspace \]
