Analysis after Gregory-Hansen cointegration test I have two time series which fail E-G cointegration test over the whole sample, but G-H cointegration test indicates the presence of cointegration when regime shift is allowed. Can I set up a dummy taking on value 0 before and 1 after the regime shift (where the date of regime shift is identified by G-H) and include it in the ECM to estimate the parameters or is there  other superior approach?
 A: Yes, this method does make sense, as it is similar in spirit to the standard Engle and Granger VECM estimation where you first obtain the $\beta$ parameters and then include the residuals in a "ECT augmented VAR" that you estimate by OLS. So you can use instead the residuals from your regression that takes into account the structural break. Technically, this is justified since the cointegration as well as the structural break estimators are super-convergent, so that you do not need to perform adjustments at the second stage. 
Now what you might want to do is also ask whether the structural break affected the $\alpha$ parameters, in which case you would just add two separate ECT (assuming here that the date for the $\beta$ and $\alpha$ is the same). 
I don't think overall there is a superior approach, there has been a very large literature on structural break and cointegration, especially with the work of Perron (among others Perron and Kejriwal 2010, see also the survey paper by Perron (2008), but I am not sure if they feature the same setup than GH. 


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*Perron, Kejriwal (2010) "Testing for Multiple Structural Changes in Cointegrated Regression Models," , Journal of Business and Economic Statistics 28 (2010), 503-522

*"Structural Change" in The New Palgrave Dictionary of Economics, 2nd ed, S. Durlauf and L. Blume (eds.), 2008, Palgrave Macmillan.

