# Linearisation of Kalman filter

Assume we have the following state-space model: $$z_{k} = \theta_{k} z_{k-1} + v_{k}\\ \theta_{k} = \phi \theta_{k-1} + w_{k},$$ where $$v_{k}$$ and $$w_{k}$$ are independent and normal for all $$k$$. The space equation is the first one, i.e. the one with $$z_{k}$$. This state-space system is not linear, nevertheless, $$cov(z_{k-1},v_{k}) = 0$$.

Therefore, I rewrite the first equation as $$z_{k} = H_{k} \theta_{k} + v_{k},$$ where $$H_{k} = z_{k-1}$$ and I can use a standard linear Kalman filter. Basically, we end up with Kalman regression model, where the slope is the state variable. Is this correct?