Question: (a) In the AR(1) model (x_{t}=phi_{0}+phi_{1} x_{t-1}+w_{t}), with (phi_{1} <1) and (mathrm{E}left(x_{t} ight)=mathrm{E}left(x_{t-1} ight)), show that [ mathrm{E}left(x_{t} ight)=frac{phi_{0}}{1-phi_{1}} ] (b) In the (mathrm{AR}(2)) model

(a) In the AR(1) model $x_{t}=\phi_{0}+\phi_{1} x_{t-1}+w_{t}$, with $\phi_{1}<1$ and $\mathrm{E}\left(x_{t}\right)=\mathrm{E}\left(x_{t-1}\right)$, show that

\[ \mathrm{E}\left(x_{t}\right)=\frac{\phi_{0}}{1-\phi_{1}} \]

(b) In the $\mathrm{AR}(2)$ model $x_{t}=\phi_{0}+\phi_{1} x_{t-1}+\phi_{2} x_{t-2}+w_{t}$, with a constant mean, $\mathrm{E}\left(x_{t}\right)=\mathrm{E}\left(x_{t-1}\right)=\mathrm{E}\left(x_{t-2}\right)$, show that

\[ \mathrm{E}\left(x_{t}\right)=\frac{\phi_{0}}{1-\phi_{1}-\phi_{2}} \]

so long as $\phi_{1}+\phi_{2} eq 1$.

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