Classify these processes as stationary or not

Question

Classify each of the following as weakly stationary or not, with a one-line reason: (a) white noise; (b) AR(1) with \phi = 0.5; (c) AR(1) with \phi = 1.2; (d) a deterministic seasonal term st = A\sin(2\pi t/12) + \varepsilont; (e) an MA(2) process Xt = \varepsilont + 	heta1\varepsilon{t-1} + 	heta2\varepsilon{t-2}.

Accepted Answer

(a) White noise: stationary. Constant mean 0, constant variance \sigma^2, and zero autocovariance at every nonzero lag. This is the benchmark stationary process. (b) AR(1) with \phi = 0.5: stationary. Because \phi < 1, the process has a stable stationary distribution with mean 0, variance \sigma^2/(1-\phi^2), and geometric ACF \rhok = 0.5^k. Shocks decay. (c) AR(1) with \phi = 1.2: not stationary (explosive). With \phi 1, past shocks are amplified: the variance grows geometrically like \phi^{2t}, so no finite stationary variance exists. (d) Deterministic seasonal A\sin(2\pi t/12) + \varepsilo…

Classify these processes as stationary or not

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