= Three-coordinate state realization of an ARMA(1,1) process
{title2=$S_t=(X_{t-1},\varepsilon_t,\varepsilon_{t-1})^\top$}
For an <autoregressive moving-average model> of order $(1,1)$, this state has observation row $F=(\phi,1,\theta)$ and transition <matrix>
$$
G=\begin{pmatrix}\phi&1&\theta\\0&0&0\\0&1&0\end{pmatrix}.
$$
The state noise is $(0,\varepsilon_t,0)^\top$. Its first transition row reconstructs $X_{t-1}$ and its third row shifts the previous <white noise> value.
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