# 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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## 1 Answer

I think this is related to a previously asked question. One of the answers suggested to use the AD Model Builder software. Although I haven't used it myself, looking at the manual it looks like an alternative.

I wonder though if your problem is sufficiently specified. How does the coefficient At change? You need to put some structure on it, perhaps a smoothness constraint, it it is to be estimated at all.

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Hi, thank you very much for the reply. A(t) is time varying coefficient associated with state variable X(t). For some textbooks, when they deal with time varying coefficient, they assume the coefficients is a function of constant variable lamda, however, in this particular question, I don't know the function. Should I treat A(t) as another state variable which follows AR(1) process? Thank you. –  user2510 Dec 24 '10 at 13:05
@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