a. Consider the stationary \(\operatorname{AR}(1)\) model \(y_{t}=ho y_{t-1}+v_{t},|ho|)=ho^{s}\). Given \(ho=0.9\), find the autocorrelations for observations 1 period apart, 2 periods apart, etc., up to 10 periods apart.

b. Consider the nonstationary random walk model \(y_{t}=y_{t-1}+v_{t}\). Assuming a fixed \(y_{0}=0\), rewrite \(y_{t}\) as a function of all past errors \(v_{t-1}, v_{t-2}, \ldots, v_{1}\).

c. Use the result in part (b) to find (i) the mean of \(y_{t}\), (ii) the variance of \(y_{t}\), and (iii) the covariance between \(y_{t}\) and \(y_{t+s}\).

d. Use the results from part (c) to show that \(\operatorname{corr}\left(y_{t}, y_{t+s}\right)=\sqrt{t /(t+s)}\).

e. Assume \(t=100\) (the random walk has been operating for 100 periods). Find the correlations between \(y_{100}\) and \(y\) in each of the next 10 periods (up to \(y_{110}\) ). Compare these correlations with those obtained in part (a).

f. Find \(\operatorname{corr}\left(y_{100}, y_{200}\right)\) for each of the two models and comment on their magnitudes.

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W X 16 30 14- 12- 10- 25 20 00 8 6 15 10- 4 5 2 () 0 25 50 75 100 125 150 175 200 25 50 75 100 125 150 175 200 12. 8 4 4 Y 40 30 20 10 2 25 50 75 100 125 150 175 200 25 50 75 100 125 150 175 200 FIGURE 12.8 Time series for Exercise 12.3.