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?
Solved – Analysis after Gregory-Hansen cointegration test
cointegrationecm
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From your question it seems like you want to estimate the effect of a treatment variable on some outcome variable. If that is indeed the case, a cointegration analysis won't do you much good. Here's why: You say that your treatment variable is binary variable, so I take it it takes the value 1 if an individual was treated and 0 otherwise. You are correct in your hesitation regarding the unit root testing of that variable; it's not meaningful. Think especially in terms of cointegration and how it is defined.
We say that two processes, $X_t$ and $Y_t$, say, are cointegrated when both are integrated of some order larger than zero but there exist a linear combination of them that is integrated of a lower order. But if one of them is constant over time, cointegration not possible since a constant is trivially stationary. Each treatment dummy in your data (one for each cross-section dimension / observed individual) is indeed constant over time.
With that in mind, RDD seems like the better choice. However, RDD is not always implementable (you need some sort of discontinuity to exploit, for one) so in general you cannot have a rule saying "either I do cointegration testing or exploit an RDD design". It all depends on the data you have or can get hold of. This is also what I mean in my comment: if you want advice on which approach to use, you have to give some details about what data you have and which question are you trying to answer.
Edit:
In response to your comment: the party membership of the governor cannot be cointegrated with the level of environmental expenditure because it cannot be integrated of any order $p>0$. Even if the value changes over time AND between individuals, the variable only takes two values and, thus, cannot include a stochastic trend. In order to be cointegrated with the dependent variable, they must share the same stochastic trend, but they cannot possibly do that, then.
Best Answer
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.