# Unit root testing with two structural breaks

I'm investigating two possibly cointegrating time series and need to test for stationarity. Both series have structural breaks, possibly more than one.

My procedure is as follows:

1. Clemente Montanes Reyes (CMR) test for unit root with two breaks

if both breaks are not significant then

2. Perron and Vogelsang (PV) test for unit root with 1 break

3. Augment results with ADF, Philips-Perron, KPSS

4. If all I(1) then Johansen test for cointegration

or if unsure about order of integration then ARDL bounds test of Pesaran and Shin.

5. Test for parameter constancy and usual diagnostics

My question is about the CMR test. One series has unit root with just 1 well defined break with both CMR and PV (low value for rho-1 but high values for the two break points du1 and du2), the other series has unit root with possibly 2 or 1 breaks depending on whether it is additive outlier or innovative outlier.

When I check these same series differenced using CMR and PV I cannot achieve stationarity even after differencing 3 or 4 times.

• Is testing a differenced series a misuse of these tests?
• What interpretative significance, if any, do break points have in a differenced series?
• Should I instead use ADF or KPSS on the differenced series?
• And if so, how do I know that the differenced series do not exhibit structural breaks thus rendering these tests unreliable?

In a nutshell, how do I test to make sure that my series are not I(2) given structural breaks, given that I(2) variables invalidate the ARDL and Johansen techniques with which I am familiar?