Find cointegrating vectors and loadings from a trivariate VAR(1) equation I have the following process:
$\begin{bmatrix}
X_t \\
Y_t \\
Z_t \\
\end{bmatrix}
= \begin{bmatrix}
1 \\
0 \\
1 \\
\end{bmatrix}
+ \begin{bmatrix}
0.5 & 0.5 & 0 \\
0 & 1 & 0 \\
0 & 0 & 0.8 \\
\end{bmatrix}
\begin{bmatrix}
X_{t-1} \\
Y_{t-1} \\
Z_{t-1} \\
\end{bmatrix} 
+ \begin{bmatrix}
\epsilon_{1t} \\
\epsilon_{2t} \\
\epsilon_{3t} \\
\end{bmatrix}
$
When calculating the error correction from I get the following impact matrix:
$\begin{bmatrix}
-0.5 &  0.5  & 0\\
0  &  0 & 0 \\
0  &   0 & -0.2 \\    
\end{bmatrix}
$
Then, I know that the cointegrating rank is 2, and there are two linearly independent cointegrating vectors. 
I should be able to decompose the above matrix into two $3 \times 2$ matrices $\alpha \beta^{T}$ where $\beta$ contains the cointegrating vectors as its columns. However, I cannot figure out how to do this. And I do not know how to find the cointegrating vectors. Any help would be much appreciated.
 A: Checking out equation by equation
$$ \begin{aligned}
X_t &= 1 + 0.5 X_{t-1} + 0.5 &Y_{t-1}    &           &+ \epsilon_{1,t} \\
Y_t &=                       &Y_{t-1}    &           &+ \epsilon_{2,t} \\
Z_t &= 1   &                  &        + 0.8 Z_{t-1} &+ \epsilon_{3,t} \\
\end{aligned} $$
you quickly notice that $Z$ is affected only by its own history and is not affecting $X$ or $Y$. Hence, $Z$ is not cointegrated with the other two. Moreover, it is not even integrated as its autoregressive coefficient is smaller in magnitude than 1. So when it comes to cointegration, we can just remove $Z$ from the system.
$Y$ is driving itself as it depends only on its own history.
$X$ is being driven by $Y$ and own history.

By staring at the equations for $X$ and $Y$ long enough, one can make a guess that $X-Y$ will be integrated of order zero, I(0). Let us see if that works:
$$ (X_t-Y_t) = 1 + 0.5 (X_{t-1} - Y_{t-1}) + (\epsilon_{1,t} - \epsilon_{2,t}). $$
Let us denote $W:=X-Y$ and $u:=\epsilon_1-\epsilon_2$, then
$$ W_t = 1 + 0.5 W_{t-1} + u_t $$
which is I(0) because 


*

*the autoregressive coefficient is below 1 in magnitude;

*the error term is I(0) since it is a sum of two I(0) terms.


So the cointegration vector $\beta=\begin{bmatrix} 1 \\ -1 \\ \end{bmatrix}$ for $(X,Y)$ (or $\tilde\beta=\begin{bmatrix} 1 \\ -1 \\ 0 \\ \end{bmatrix}$ for $(X,Y,Z)$) seems to work.

Now you have to find a vector $\alpha$ to premultiply to $(1,-1)$ to get the matrix $\begin{bmatrix} 0.5 & 0.5 \\ 0 & 1 \\ \end{bmatrix}$. That seems to be $\alpha=\begin{bmatrix} -0.5 \\ 0 \end{bmatrix}$.
