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I am interested in an ARMA$(2,2)$ model with an additional input variable, which I want to put in state-space form. If $w_t$ is white noise, and $x_t$ is a known input, the model is given by:

$$y_t = \beta_0 + \beta_1 \cdot x_{t-1} + \alpha_1 \cdot y_{t-1} + \alpha_2 \cdot y_{t-2} + w_t + \theta_1 \cdot w_{t-1} + \theta_2 \cdot w_{t-2}.$$

Can someone please show how to write this in state-space form?

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One way to do it is to define the state vector as $$ \xi_t = \begin{pmatrix} y_t \\ y_{t-1} \\ w_{t} \\ w_{t-1} \\ 1 \\ \end{pmatrix} $$ The measurement equation is just $$ y_t = \begin{pmatrix} 1 & 0 & 0 & 0 & 0 \end{pmatrix} \, \xi_t $$ i.e. there is no noise term. The state transition equation is then $$ \underbrace{\begin{pmatrix} y_t \\ y_{t-1} \\ w_{t} \\ w_{t-1} \\ 1 \\ \end{pmatrix}}_{\xi_t} = \begin{pmatrix} \alpha_1 & \alpha_2 & \theta_1 & \theta_2 & \beta_0+\beta_1 x_{t-1} \\ 1 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 0 & 1 \\ \end{pmatrix} \underbrace{\begin{pmatrix} y_{t-1} \\ y_{t-2} \\ w_{t-1} \\ w_{t-2} \\ 1 \\ \end{pmatrix}}_{\xi_{t-1}} + \begin{pmatrix} 1 \\ 0 \\ 1 \\ 0 \\ 0 \\ \end{pmatrix} w_t $$

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