(a cointegrated VAR(1) by hand)

Let $y_t = Ay_{t-1}+\varepsilon_t$ with $A = \begin{pmatrix}0.8 & 0.2\\ 0.1 & 0.9\end{pmatrix}$. (a) Write $A - I = \alpha\beta'$ with $\beta = (1,-1)'$. Find $\alpha$. (b) Find the AR(1) coefficient of the spread $\beta'y_t$ and its half-life in periods. (c) Find $\alpha_\perp$, the common trend, and the long-run impact matrix $C$ of (13A.13). Verify that $C = \lim_{h\to\infty}A^h$. Give the long-run effect on both prices of a unit shock to $\varepsilon_1$, and of a unit shock to $\varepsilon_2$. (d) Which price does more of the adjusting? Interpret for a trader choosing which leg to trade aggressively.