I'm working on a problem,
Consider that a times series $\{y_t\}$ is generated from an $\text{ARIMA}(1,1,1)$ model, so that $$y_t-y_{t-1}=\alpha(y_{t-1}-y_{t-2})+\epsilon_t+\gamma\epsilon_{t-1},$$ where $\alpha$ is the AR parameter, $\gamma$ is the MA parameter and $\{\epsilon_t\}$ is a Gaussian white noise sequence with variance equal to $1$. Define the state space vector $$\beta_t=\begin{bmatrix}y_t\\y_{t-1}\\\epsilon_t\end{bmatrix}.$$ Write down a state space representation for $y_t$, i.e. express $y_t$ as a state space model: $$y_t=x^T\beta_t+\delta_t,\\\beta_t=F\beta_{t-1}+\zeta_t,$$ specifying $x,F,\delta_t$ and $\zeta_t$, and writing down the distributions of $\delta_t$ and $\zeta_t$.
I get \begin{align} y_t & = (1+\alpha)y_{t-1} - \alpha y_{t-2} + \epsilon_t-\gamma\epsilon_{t-1} \\ & = \begin{bmatrix}1&0&0\end{bmatrix}\begin{bmatrix}y_t\\y_{t-1}\\\epsilon_t\end{bmatrix}+0 \\ & = x^T\beta_t+\delta_t \text{ where } x=\begin{bmatrix}1&0&0\end{bmatrix}, \hspace{1em} \delta_t\sim N(0,0); \\[1.5em] \beta_t & = \begin{bmatrix}y_t\\y_{t-1}\\\epsilon_t\end{bmatrix} \\ & = \begin{bmatrix}1+\alpha&-\alpha&\gamma\\1&0&0\\-(1+\alpha)&\alpha&-\gamma\end{bmatrix} \begin{bmatrix}y_{t-1}\\y_{t-2}\\\epsilon_{t-1}\end{bmatrix}+\begin{bmatrix}\epsilon_t\\0\\y_t\end{bmatrix} \\ & = F\beta_{t-1}+\zeta_t \text{ where } F=\begin{bmatrix}1+\alpha&-\alpha&\gamma\\1&0&0\\-(1+\alpha)&\alpha&-\gamma\end{bmatrix},\hspace{1em}\zeta_t\sim N\left(\begin{bmatrix}0\\0\\\hat{y}_t\end{bmatrix},Z\right). \end{align}
Something seems to have gone wrong. I don't know how I can specify the transitional covariance matrix $Z$; I don't think $\hat{y}$ is allowable as a mean at this stage in the calculation; and I'm suspicious about the observation model containing $y_t$ in the parameter vector $\beta$ and the transition model containing it as part of $\zeta_t$.
I notice the model is very similar to the one here, but the latter doesn't have $y_t$ in the $\beta$ equivalent.
Could someone give me a pointer?
