Exogenous Regime Switching

#1
I currently have a model that measures inflation expectations as follows:

y(t) = a*x(t) + (1-a)*y(t-1)

I'm looking to create a regime switching models that change the values of "a" based upon a shock to an exogenous variable (let's say "z(t)"). Are there any methods which estimate the threshold of z(t) that would merit an switch in the regime of this particular model??

As far as I'm aware, threshold models are purely endogenous, rather than exogenous...for Markov Switching Models though, the switches are based upon random probabilities. Is there a way to switch a model based upon an exogenous variable passing a specific, tested value??

Thanks!!

Abraham