Data from a Multivariate State Space Model (p = 1)
Format
A matrix with 100 rows (time points) and 5 columns
(eta1, and eta2 for latent states,
y1, and y2 for observed data,
and
time for discrete time from 1 to 100)
generated from the state space model given by
$$
\left(\begin{array}{c} y_{1_{t}} \\ y_{2_{t}} \end{array} \right)
=
\left(\begin{array}{c} \eta_{1_{t}} \\ \eta_{2_{t}} \end{array} \right)
+
\left(
\begin{array}{c} \varepsilon{1_{t}} \\ \varepsilon{2_{t}} \end{array}
\right)
\quad
\text{with}
\quad
\left(
\begin{array}{c} \varepsilon_{1_{t}} \\ \varepsilon_{2_{t}} \end{array}
\right)
\sim
\mathcal{N}
\left(
\left(\begin{array}{c} 0 \\ 0 \end{array} \right) ,
\left(\begin{array}{cc} 1 & 0.0 \\ 0.0 & 1 \end{array} \right)
\right)
$$
$$
\left(\begin{array}{c} \eta_{1_{t}} \\ \eta_{2_{t}} \end{array} \right)
=
\left(\begin{array}{cc} 0.8 & 0.0 \\ 0.0 & 0.8 \end{array} \right)
\left(
\begin{array}{c} \eta_{1_{t - 1}} \\ \eta_{2_{t - 1}} \end{array}
\right)
+
\left(\begin{array}{c} \zeta_{1_{t}} \\ \zeta_{2_{t}} \end{array} \right)
\quad
\text{with}
\quad
\left(\begin{array}{c} \zeta{1_{t}} \\ \zeta{2_{t}} \end{array} \right)
\sim
\mathcal{N}
\left(
\left(\begin{array}{c} 0 \\ 0 \end{array} \right) ,
\left(\begin{array}{cc} 1 & 0.0 \\ 0.0 & 1 \end{array} \right)
\right)
$$