For x_t = 0.6*x_{t-1} + e_t with e_t white noise of variance 1, give the autocorrelation at lag 3, the unconditional variance of x_t, and the condition on the coefficient that makes the process stationary.

For x_t = 0.6*x_{t-1} + e_t with e_t white noise of variance 1, give the autocorrelation at lag 3, the unconditional variance of x_t, and the condition on the coefficient that makes the process stationary.

Approach: Solve the Yule-Walker recursion for the autocorrelation, then take variances of both sides of the recursion to pin the unconditional variance.

0.216. For a stationary AR(1) with coefficient phi, multiplying the recursion by x_{t-k} and taking expectations gives the autocorrelation recursion rho(k) = phi*rho(k-1), so rho(k) = phi^k and rho(3) = 0.6^3 = 0.216. Taking variances of both sides, and using that e_t is uncorrelated with x_{t-1}, gives Var(x) = phi^2*Var(x) + 1, so Var(x) = 1/(1 - 0.36) = 1.5625. Stationarity needs |phi| < 1, which makes the moving-average representation x_t = sum phi^j * e_{t-j} converge in mean square. At |phi| = 1 the process is a random walk whose variance grows linearly in t and which has no fixed mean to revert to, so no unconditional variance exists.

Follow-up: For an AR(2) with coefficients 1.2 and -0.35, what are the stationarity conditions and what shape does the autocorrelation function take?

Key concepts: autocorrelation, stationarity, AR(1), random walk.