Ok, so if the previous model was badly specified then what should we do to fix it? In principle we want the following behavior yt=xt(β+ωzt)+α where zt is a binary state variable indicating whether the current has the first or the second response type. The full model we then might want to consider looks like this.
As is always instructional let’s look at the posteriors of the parameters of our second model. They are depicted below. You can clearly see that the “increase in slope” parameter ω clearly captures the new behavior we wished to model.