Consider the ARDL \((2,1)\) model

with auxiliary AR(1) model \(x_{t}=\alpha+\phi x_{t-1}+v_{t}\), where \(I_{t}=\left\{y_{t}, y_{t-1}, \ldots, x_{t}, x_{t-1}, \ldots\right\}, E\left(e_{t} \mid I_{t-1}\right)=0\), \(E\left(v_{t} \mid I_{t-1}\right)=0, \operatorname{var}\left(e_{t} \mid I_{t-1}\right)=\sigma_{e}^{2}, \operatorname{var}\left(v_{t} \mid I_{t-1}\right)=\sigma_{v}^{2}\), and \(v_{t}\) and \(e_{t}\) are independent. Assume that sample observations are available for \(t=1,2, \ldots, T\).

a. Show that the best forecasts for periods \(T+1, T+2\) and \(T+3\) are given by

b. Show that the variances of the forecast errors are given by

Transcribed Image Text:

y=8+01-1+021-2 +81x1 + e,