Kalman filter: correct estimation of time-varying regression

[Yohan]

Hello altogether,

i want to estimate a time-varying parameter regression model via the kalman filter.
Furthermore, i intend to drop the gaussian assumption about the error terms in the corresponding state space model.
So, i derive the kalman filter via linear projection.

This means, that the explanatory (exogenous) variables act as the measurement system matrix in the state space model, whereas the corresponding parameters are included in the state vector.
O.k., when the explanatory variables contain lagged observed variables, then the state variable is non-linear in the observations and the estimate of the state variable cannot be interpreted as linear projection. With Gaussian assumption on the error terms, the state space model is still conditionally Gaussian.

But what is about other exogenous variables?
I thought they were handled as "pre-determined" and therefore do not create any problem.
But Hamilton 1994 wrote, that with stochastic explanatory variables linear projection is not feasible. The reason is, that he conditioned the estimate of the state vector on both, the observed variables and the explanatory variables. But he didn't exclude lagged observed variables from the vector of explanatory variables. Perhaps, this is the reason for his comment on linear projection.

My Question: Is linear projection in general correct with explanatory variables which do not include lagged observed variables?
Which problems can arise?

Thank you for your comments!!!
Many thanks and best regards from Germany!
Yohan.