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Asymptotically, you should be able to interchange the role of $Y_t$ and $X_t$ and estimate lets say $$X_t =\alpha^* + \beta^* Y_t+u^*_t$$ where $\alpha^* = -\alpha/\beta$ and $\beta^*= 1/\beta$ and still get cointegrated relationship. However, crucial caveat here is that this holds only asymptotically and only for $R^2$ close to unity. In finite samples you ...

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