# Autoregressive model for time series with structural breaks

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?

• I was going through my old answers and noticed this one was not accepted. Do you perhaps need further clarification? Feb 15 '17 at 11:41

## 1 Answer

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.

If you indeed have structural breaks, the standard ADF test is not appropriate. It is known that the ADF test may not reject the null of a unit root when there actually are structural breaks (such as level shifts) but no unit root.

My question is, with these types of structural break models, do I still need to first difference the series to make it stationary?

Differencing addresses the problem of a unit root. If your variable has shifts in mean or variance but no unit root, differencing will not be the appropriate approach.

But it is also true we should use stationary series in a model of AR forms, right?

Right, but you said you have a threshold model to capture different regimes. If the process is stationary within each particular regime (and the nonstationarity is only due to switching between the regimes), you may use AR models within each regime.

• This makes sense. But how about I use a Breakpoint unit root test like the ones incorporated in Eviews, instead of a standard ADF test? The result is unstable though: if we set that there is only breaking in intercept, then the unit root hypothesis cannot be rejected; if we set that breakings happen to both intercept and trend, then the unit root hypothesis is rejected. In this case, should we be concerned that there actually is unit root and the series is non-stationary even within each regime? (unit root hypothesis is always rejected for the first difference of the series) May 6 '16 at 17:52
• You would have to decide which specification of the unit root test is correct and go on from there. May 6 '16 at 22:58
• Just to clarify: if the breakpoint unit root test shows there is a unit root, then first difference is necessary even with a threshold model, right? May 6 '16 at 23:37
• Yes, I would think so. May 7 '16 at 9:11