How do I write a state space model and how do you find the unknown parameters of phi, mu, and matrix A$_t,$ along with covariance matrices Q and R?

Question

Consider a system process given by $x_t=-0.9x_{t-2}+z_t$,$t=1,2,…,n$ with observation $y_t=x_t+v_t$ where ${z_t}$ and ${v_t}$ are independent white noise with variances $σ^2$ and $σ_v^2$. Assume that $x_0\sim \mathcal N(0,σ_0^2)$ and $x_{-1}\sim \mathcal N(0,σ_1^2)$, and that $x_0$ and $x_{-1}$ are independent. Write the system and observation equations as state-space model with clearly defined parameters $\Phi$, $A_t$, $Q$, $R$, $μ_0$, and $Σ_0$.

Answer 1

Normally, the state value at $t$ should depend on its value at $t-1$. That's why we must consider the state at $t$ as being $X_t := \begin{bmatrix} x_t \\ x_{t-1}\end{bmatrix}$ so that we obtain the measurement equation: $$y_t = \begin{bmatrix}1 & 0\\ 0 & 0\end{bmatrix}X_t+v_t$$ and the state equation: $$X_t = \begin{bmatrix}0 & -0.9\\ 1 & 0\end{bmatrix}X_{t-1}+\begin{bmatrix}z_t \\ 0\end{bmatrix}.$$ Especially, $X_0\sim \mathcal N\left(\begin{bmatrix} 0 \\ 0\end{bmatrix},\begin{bmatrix} \sigma_0^2 & 0 \\ 0 & \sigma_1^2 \end{bmatrix}\right)$ since $x_0\sim \mathcal N(0,\sigma_0^2)$ and $x_{-1}\sim \mathcal N(0,\sigma_1^2)$ are independent.