In Chapter 9, we found that, given a time series of observations, (I_{T}=left{left(y_{1}, x_{1} ight),left(y_{2}, x_{2} ight),

In Chapter 9, we found that, given a time series of observations, \(I_{T}=\left\{\left(y_{1}, x_{1}\right),\left(y_{2}, x_{2}\right), \ldots,\left(y_{T}, x_{T}\right)\right\}\), the best one-period and two-period ahead forecasts for \(y_{T+1}\) and \(y_{T+2}\) were given by \(E\left(y_{T+1} \mid I_{T}\right)\) and \(E\left(y_{T+2} \mid I_{T}\right)\), respectively. Given that \(T=29, y_{T}=10, y_{T-1}=12, x_{T+2}=x_{T+1}=x_{T}=5\), and \(x_{T-1}=6\), find forecasts for \(y_{T+1}\) and \(y_{T+2}\) from each of the following models. In each case, assume that \(v_{t}\) are independent random errors distributed as \(N\left(0, \sigma_{v}^{2}=4\right)\).

a. The random walk \(y_{t}=y_{t-1}+v_{t}\).

b. The random walk with drift \(y_{t}=5+y_{t-1}+v_{t}\).

c. The random walk \(\ln \left(y_{t}\right)=\ln \left(y_{t-1}\right)+v_{t}\).

d. The deterministic trend model \(y_{t}=10+0.1 t+v_{t}\).

e. The ARDL model \(y_{t}=6+0.6 y_{t-1}+0.3 x_{t}+0.1 x_{t-1}+v_{t}\).

f. The error correction model \(\Delta y_{t}=-0.4\left(y_{t-1}-15-x_{t-1}\right)+0.3 \Delta x_{t}+v_{t}\). In addition, find the long-run equilibrium value for \(y\) when \(x=5\).

g. The first difference model \(\Delta y_{t}=0.6 \Delta y_{t-1}+0.3 \Delta x_{t}+0.1 \Delta x_{t-1}+v_{t}\).