# AR(1) model with autoregressive intercept

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.

• The autoregressive intercept is simply a second AR process, and $y_t$ can be considered the sum of two AR processes. As discussed in Hamilton, Time Series Analysis p.107/108, this system can be represented as an ARMA(2,1). As for the changing mean over time, if both processes are stationary then the unconditional mean will not be time dependent. – Henry Sep 12 at 20:20