The VEC model is a special form of the VAR for I(1) variables that are cointegrated. Consider the following VEC model:

\[\begin{aligned}& \Delta y_{t}=\alpha_{10}+\alpha_{11}\left(y_{t-1}-\beta_{0}-\beta_{1} x_{t-1}\right)+v_{t}^{y} \\& \Delta x_{t}=\alpha_{20}+\alpha_{21}\left(y_{t-1}-\beta_{0}-\beta_{1} x_{t-1}\right)+v_{t}^{x}\end{aligned}\]

The VEC model may also be rewritten as a VAR, but the two equations will contain common parameters:

\[\begin{aligned}& y_{t}=\alpha_{10}+\left(\alpha_{11}+1\right) y_{t-1}-\alpha_{11} \beta_{0}-\alpha_{11} \beta_{1} x_{t-1}+v_{t}^{y} \\& x_{t}=\alpha_{20}+\alpha_{21} y_{t-1}-\alpha_{21} \beta_{0}-\left(\alpha_{21} \beta_{1}-1\right) x_{t-1}+v_{t}^{x}\end{aligned}\]

a. Suppose you were given the following results from an estimated VEC model:

\[\begin{aligned}& \widehat{\Delta y_{t}}=2-0.5\left(y_{t-1}-1-0.7 x_{t-1}\right) \\& \widehat{\Delta x_{t}}=3+0.3\left(y_{t-1}-1-0.7 x_{t-1}\right)\end{aligned}\]

Rewrite the model in the VAR form.

b. Now suppose you were given the following results of an estimated VAR model, but you were also told that \(y\) and \(x\) are cointegrated.

\[\begin{aligned}& \hat{y}_{t}=0.7 y_{t-1}+0.3+0.24 x_{t-1} \\& \hat{x}_{t}=0.6 y_{t-}-0.6+0.52 x_{t-1}\end{aligned}\]

Rewrite the model in the VEC form.