Consider a simple bivariate VECM [ begin{aligned} & y_{1 t}-y_{1 t-1}=delta_{1}+alpha_{1}left(y_{2 t-1}-beta y_{1 t-1}-mu ight) & y_{2 t}-y_{2 t-1}=delta_{2}+alpha_{2}left(y_{2 t-1}-beta y_{1 t-1}-mu ight) end{aligned}

Consider a simple bivariate VECM

\[ \begin{aligned} & y_{1 t}-y_{1 t-1}=\delta_{1}+\alpha_{1}\left(y_{2 t-1}-\beta y_{1 t-1}-\mu\right) \\ & y_{2 t}-y_{2 t-1}=\delta_{2}+\alpha_{2}\left(y_{2 t-1}-\beta y_{1 t-1}-\mu\right) \end{aligned} \]

(a) Using the initial conditions for the endogenous variables \(y_{1}=100\) and \(y_{2}=110\) simulate the model for 30 periods using the parameters

\[ \delta_{1}=\delta_{2}=0 ; \alpha_{1}=-0.5 ; \alpha_{2}=0.1 ; \beta=1 ; \mu=0 \]

Compare the two series. Also check to see that the long-run value of \(y_{2}\) is given by \(\beta y_{1}+\mu\).

(b) Simulate the model using the following parameters:

\[ \delta_{1}=\delta_{2}=0 ; \alpha_{1}=-1.0 ; \alpha_{2}=0.1 ; \beta=1 ; \mu=0 \]

Compare the resultant series with the those in (a) and hence comment on the role of the error correction parameter \(\alpha_{1}\).

(c) Simulate the model using the following parameters:

\[ \delta_{1}=\delta_{2}=0 ; \alpha_{1}=1.0 ; \alpha_{2}=-0.1 ; \beta=1 ; \mu=0 \]

Compare the resultant series with the previous ones and hence comment on the relationship between stability and cointegration.

(d) Simulate the model using the following parameters:

\[ \delta_{1}=\delta_{2}=0 ; \alpha_{1}=-1.0 ; \alpha_{2}=0.1 ; \beta=1 ; \mu=10 \]

Comment on the role of the parameter \(\mu\). Also check to see that the long-run value of \(y_{2}\) is given by \(\beta y_{1}+\mu\).

(e) Simulate the model using the following parameters:

\[ \delta_{1}=\delta_{2}=1 ; \alpha_{1}=-1.0 ; \alpha_{2}=0.1 ; \beta=1 ; \mu=0 \]

Comment on the role of the parameters \(\delta_{1}\) and \(\delta_{2}\).

(f) Explore a richer class of models which also includes short-run dynamics. For example, consider the model \[ \begin{aligned} y_{1 t}-y_{1 t-1}= & \delta_{1}+\alpha_{1}\left(y_{2 t-1}-\beta y_{1 t-1}-\mu\right)+\phi_{11}\left(y_{1 t-1}-y_{1 t-2}\right) \\ & +\phi_{12}\left(y_{2 t-1}-y_{2, t-2}\right) \\ y_{2 t}-y_{2 t-1}= & \delta_{2}+\alpha_{2}\left(y_{2 t-1}-\beta y_{1 t-1}-\mu\right)+\phi_{21}\left(y_{1 t-1}-y_{1, t-2}\right) \\ & +\phi_{22}\left(y_{2 t-1}-y_{2 t-2}\right) \end{aligned} \]