Consider the following VECM, from a VAR(1) model $$ \Delta y_t=c-AB'y_{t-1}+\epsilon _t $$ and the following VMA representation $$ y_t = \delta t+Cz_t+\eta_t $$ where $z_t=\sum_{s=1}^t \epsilon _t$ and $\eta_t \sim I(0)$.
a. From the VECM specification, find a VMA representation, that is, show that $\delta$ and $C$ can be written in terms of $A$, $B$ and $c$. Furthermore, find the stationary vector $\eta_t$ in terms of $\epsilon_t$ and its history.
Hint: Find a VAR(1) representation for the stationary process $B'y_t$, then find its VMA representationd and replace them in VECM.
b. Prove that $B'C=0$ and $CA=0$
Comments: I don't understand why $B'y_t$ is stationary, wouldn't it be more easier to do the following thing:
\begin{aligned}
y_t-y_{t-1} &= c-AB'y_{t-1}+\epsilon _t \\
y_t &= c+(I_n-AB')+y_{t-1}+\epsilon _t \\
\left [I_n-(I_n-AB')L \right ]y_t &= c+\epsilon _t \\
\Phi (L)y_t &= c+\epsilon _t \\
y_t &= \Phi (L)^{-1}c+\Phi (L)^{-1}\epsilon _t \\
\end{aligned}
And this is the VMA representation of the VECM.
