If you run a Engle and Granger test with the regression: $$ y_t=b_0+b_1x_t+\varepsilon_t $$ and you know that $y_t$ is integrated of order 2, what can you say about the order of integration of $x_t$, knowing that, from the Augmented DF, $\varepsilon_t$ is non-stationary (integrated of order 1)?
And what can you say about the cointegration between the two time series?
I cannot get the logic behind it. I mean, I would say that if you fail to reject the Augmented DF then you fail to reject the null hypothesis of non-cointegration between the time series and you cannot say anything about the order of integration of $x_t$. Is that correct or am I missing something?