# Multiplicative unobservable component in state space model

I'm new here and wondering if anyone could give me some hints on how to estimate the time varying coefficient and state variable together. Here is my model:

observation equation: $Y(t)= A(t)X(t)+ w(t)$,
state equation: $X(t)=\phi X(t-1)+v(t)$,

here I have time varying coefficient $A(t)$, it doesn't depend on any predetermined parameter $\theta$, for example. If I treat $A(t)$ as another state variable, then it is nonlinear state space, I have no idea how to estimate multiplicative state variables. Any hints? Thank you.

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@user2510: You may include in the state whatever you please, but you cannot (within the framework of the standard Kalman filter) include non-linear functions of the state in the observation equation; and $A(t)X(t)$ would be non-linear if both $A(t)$ and $X(t)$ are components of the state. –  F. Tusell Dec 24 '10 at 13:59