I'm using a structural break model (threshold model or regime switching model) to examine the dynamics of a time series. The ADF test shows that the series has a unit root. Right now I'm regressing $y$ on its lags, and the coefficients of lags are allowed to be different in different regimes. My question is, with these types of structural break models, do I still need to first difference the series to make it stationary?
I'm confused because the model is meant to capture shifts in means (or variance), so it seems to make sense to use the original non-stationary series. But it is also true we should use stationary series in a model of AR forms, right? Can anybody help explain whether the original non-stationary series or the differenced stationary series should be used here, and why?