Let us consider the following model:

$$ y_{t} = c_{t} + \alpha y_{t-1} + v_{t} \\ c_{t+1} = c_{0} + \beta c_{t} + w_{t} $$ where $v_{t} \in \mathcal{N}(0, \sigma^{2}_{v})$ and $w_{t} \in \mathcal{N}(0, \sigma^{2}_{w})$ are independent.

I am a bit stuck: does this system have a simpler ARMA form or other solvable state-state representation?

If $\alpha = 0$, then the system is AR(1) plus noise, which is equivalent to ARMA(1,1).

The idea is to model a process as a mean reversion taking into account that the mean can change over the time.


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