Three-coordinate state realization of an ARMA(1,1) process ($S_t=(X_{t-1},{\epsilon }_t,{\epsilon }_{t-1}{)}^{\top }$)

For an autoregressive moving-average model of order $(1,1)$, this state has observation row $F=(\varphi ,1,\theta )$ and transition matrixThe state noise is $(0,{\epsilon }_t,0{)}^{\top }$. Its first transition row reconstructs $X_{t-1}$ and its third row shifts the previous white noise value.

For $|\varphi |<1$ and centered independent white noise values, the state above has expected value zero and this stationary covariance matrix, where ${\gamma }_0={\sigma }^2(1+{\theta }^2+2\varphi \theta )/(1-{\varphi }^2)$. The cross-covariance with the future white noise value is zero and that with the contemporaneous white noise value is ${\sigma }^2$. Direct multiplication verifies $P=GP{G}^{\top }+Q$.