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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 $x_t$?

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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 $$ y_t=Z_t\times((1-\tilde{x}_t)\beta_1+ \tilde{x}_t\beta_2 )+v_t. $$

If you forced the state process to be zero mean, and not have an intercept, then this flipped process wouldn't yield a state space model of the same form, so you would be okay.

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