Let \(e_{t}\) denote the error term in a time series regression. We wish to compare the autocorrelations from an AR(1) error model \(e_{t}=ho e_{t-1}+v_{t}\) with those from an MA(1) error model \(e_{t}=\phi v_{t-1}+v_{t}\). In both cases, we assume that \(E\left(v_{t} v_{t-s}\right)=0\) for \(s eq 0\) and \(E\left(v_{t}^{2}\right)=\sigma_{v}^{2}\). Let \(ho_{s}=E\left(e_{t} e_{t-s}\right) / \operatorname{var}\left(e_{t}\right)\) be the \(s\)-th order autocorrelation for \(e_{t}\). Show that,

a. for an \(\operatorname{AR}(1)\) error model, \(ho_{1}=ho, ho_{2}=ho^{2}, ho_{3}=ho^{3}, \ldots\)

b. for an MA(1) error model, \(ho_{1}=\phi /\left(1+\phi^{2}\right), ho_{2}=0, ho_{3}=0, \ldots\)

Describe in words the difference between the two autocorrelation structures.