# Kalman filter-ish model, is this identifiable?

Time series of observations $$y_t$$. The proposed model is that there's unobservable scalar series $$x_t$$: $$x_t=\phi x_{t-1}+B_tu_t+w_t$$ where $$u_t$$ - vector predictirs and $$w_t$$ - noise.

Then there's observable scalar: $$y_t=Z_t\times(x_t\beta_1+(1-x_t)\beta_2)+v_t$$ where $$Z_t$$ - another vector of regressors, potentially, could be the same as $$u_t$$ if it helps; $$v_t$$ - noise, and $$\beta_1,\beta_2$$ - parameter vectors I'm trying to estimate.

Is this model identifiable? Can estimate betas and the unobservable $$x_t$$?

It could be depending on what your state regressors are ($$u_t$$). Consider the flipped around latent process $$\tilde{x}_t = 1-x_t$$ whose dynamics can be found using your first equation: $$\tilde{x}_t= (1-\phi) + \phi \tilde{x}_{t-1} + \tilde{B}u_t + \tilde{w}_t.$$ where $$\tilde{B}_t = -B_t$$ and $$\tilde{w}_t = -w_t$$.
The associated observation equation is $$y_t=Z_t\times((1-\tilde{x}_t)\beta_1+ \tilde{x}_t\beta_2 )+v_t.$$