Consider the error correction form for the vector time series $Y_t$:
$\Delta Y_t = \Pi Y_{t-1} + \Phi_1 \Delta Y_{t-1} + ... + \Phi_p \Delta Y_{t-p} + \epsilon_t$.
Here, $Y_t$ are non-stationary and $I(1)$. Clearly, all terms except the $\Pi Y_{t-1}$ are $I(0)$. I have two questions now:
How does $\Pi$ being rank deficient ensure that $\Pi Y_{t-1}$ becomes $I(0)$? I can see that using SVD one can write this as $AB^T$, where $A$ and $B$ are reduced rank compared to the size of $\Pi$. But then what?
What if $Y_{t}$ is $I(2)$? Can co-integrating relationships exist now? What are the conditions on $\Pi$ then?
Thank you for your response.