I am trying to construct a state-space model and ran into an issue regarding how to deal with a constant term. I have three equations as follows:
- $ \pi_t =\tilde\rho_{t} \pi_{t-1}+\nu_t^{\pi}$ - AR(1) process
- $\tilde\rho_t =\rho +q_{t} $ -- $\tilde\rho_t $ is sum of a constant $\rho$ and time-varying $ q_{t} $.
- $ q_t =q_{t-1} +\nu_t^{q}$ - random walk process
In my setting, $\pi_t,\tilde\rho_t,q_t $ are unobservables so they can be tracked by signal extractions(The Kalman filter). The observation equation is $s_t =\pi_t +\epsilon_t $. \begin{pmatrix} \begin{bmatrix} 1 &0 &0& 0\\ 0 & 1 & 0& 0 \\ 0 & -1 &1 & -1 \\ 0 & 0 &0 & 1 \end{bmatrix} \begin{bmatrix} \pi_{t} \\ q_{t} \\ \tilde\rho_t\\ \rho \end{bmatrix}= \begin{bmatrix} \tilde\rho_t &0 &0& 0\\ 0 & 1 & 0& 0 \\ 0 & 0 &0 & 0 \\ 0 & 0 &0 & 1 \end{bmatrix} \begin{bmatrix} \pi_{t-1} \\ q_{t-1} \\ \tilde\rho_{t-1}\\ \rho \end{bmatrix} + \begin{bmatrix} \nu_t^{\pi} \\ \nu_t^{q} \\ 0\\ 0 \end{bmatrix} \end{pmatrix}
Agents aim to learn the values of these parameters over time. My question centers on how to construct the state vector in this context. I have attempted to do so as described above, but I am uncertain about whether to include the true value of $\rho$ in the state vector. This is because agents are not able to infer the true $\rho$; even after processing the signal, they can only learn $\tilde\rho_t = \rho+q_t$
