(a) i. Generate a random walk of length 10 and perform an augmented Dickey-Fuller test on the series for a unit root.

ii. Generate a random walk of length 100 and perform an augmented Dickey-Fuller test on the series for a unit root.

iii. Generate a random walk of length 1,000 and perform an augmented Dickey-Fuller test on the series for a unit root.

(b) Determine the coefficients for an \(\mathrm{AR}(2)\) process with roots 1.00008 and 3.9968, and simulate a series of length 500 with this model. Perform an augmented Dickey-Fuller test for unit root.