(a) Show that the denominator in the solutions to the Yule-Walker equations (5.119) and (5.120), \((\gamma(0))^{2}-(\gamma(1))^{2}\), is positive and less than 1.

(b) Suppose that for a certain process, we have \(\gamma(0)=2, \gamma(1)=\) 0.5 , and \(\gamma(2)=0.9\).

If the process is an \(\operatorname{AR}(2)\) model, what are \(\phi_{1}\) and \(\phi_{2}\) ?

(c) Given an \(\mathrm{AR}(2)\) model with \(\phi_{1}\) and \(\phi_{2}\) as in Exercise 5.23b, what is the ACF for lags 1 through 5? You may want to use the \(\mathrm{R}\) function ARMAacf.

\(\phi_1=\frac{\gamma(1) \gamma(2)-\gamma(0) \gamma(1)}{(\gamma(0))^2-(\gamma(1))^2}, \tag{5.119}\)

\(\phi_2=\frac{(\gamma(1))^2-\gamma(0) \gamma(2)}{(\gamma(0))^2-(\gamma(1))^2} \tag{5.120}\)