(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 (x t phi 0 phi 1 x t 1 phi 2 x t 2 w t ), with a constant mean, ( mathrm E left(x ...

The Answer is in the image, click to view ...

(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 [...

Question:

(a) In the AR(1) model \(x_{t}=\phi_{0}+\phi_{1} x_{t-1}+w_{t}\), with \(\phi_{1}<1\) and \(\mathrm{E}\left(x_{t}\right)=\mathrm{E}\left(x_{t-1}\right)\), 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 \(\phi_{1}+\phi_{2} eq 1\).

Fantastic news! We've Found the answer you've been seeking!