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I am slightly confused about the requirements necessary to conduct a cointegration test with structural breaks such as the Gregory-Hansen test.

Suppose I have two I(1) variables. Variable 1 follows a trend, then the trend slopes up and drops back down. Variable 2 follows a similar trend, the trend similarly slopes back up and then similarly drops back down.

In other words, the variables behave very similarly even though their trend isn't constant over time. Does the behavior described above imply that one has to test for cointegration in the presence of a structural break or does that mean that you want to test whether the variables cointegrate if there is a structural break in only one of the variables?

Also given the above description of the variables is it possible to model cointegration between these variables given their behavior? If it is does a standard Granger 2 step cointegration test or Johansen Test still apply or is a Gregory-Hansen test necessary to be performed?

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