Consider the (operatorname{AR}(2)) model (y_{t}=delta+theta_{1} y_{t-1}+theta_{2} y_{t-2}+v_{t}). Suppose that [1-theta_{1} z-theta_{2} z^{2}=left(1-c_{1} z ight)left(1-c_{2} z ight)] a.

Consider the \(\operatorname{AR}(2)\) model \(y_{t}=\delta+\theta_{1} y_{t-1}+\theta_{2} y_{t-2}+v_{t}\). Suppose that

\[1-\theta_{1} z-\theta_{2} z^{2}=\left(1-c_{1} z\right)\left(1-c_{2} z\right)\]

a. Show that \(c_{1}+c_{2}=\theta_{1}\) and \(c_{1} c_{2}=-\theta_{2}\).

b. Prove that the \(\operatorname{AR}(2)\) model has a unit root if and only if \(\theta_{1}+\theta_{2}-1=0\). [Hint: The roots of \(1-\theta_{1} z-\theta_{2} z^{2}=0\) are \(1 / c_{1}\) and \(1 / c_{2}\).]

c. Prove that \(\theta_{1}+\theta_{2}-1<0\) if the \(\operatorname{AR}(2)\) process is stationary.

d. Prove that the \(\mathrm{AR}(2)\) model \(y_{t}=\delta+\theta_{1} y_{t-1}+\theta_{2} y_{t-2}+v_{t}\) can also be written as

\[\Delta y_{t}=\delta+\gamma y_{t-1}+a_{1} \Delta y_{t-1}+v_{t}\]

where \(\gamma=\theta_{1}+\theta_{2}-1\) and \(a_{1}=-\theta_{2}\). What are the implications of this result and the results in parts (b) and (c) for unit root tests in an AR(2) model.

e. Show that an \(\operatorname{AR}(p)\) model has a unit root if \(\gamma=\theta_{1}+\theta_{2}+\cdots+\theta_{p}-1=0\).

f. Show that setting \(\gamma=\theta_{1}+\theta_{2}+\cdots+\theta_{p}-1\) in equation (12.23) implies \(a_{j}=-\sum_{r=j}^{p-1} \theta_{r+1}\).