Is this moving-average series stationary?

Question

A series is built as Xt = \varepsilont + 0.5\,\varepsilon{t-1}, where \varepsilont is white noise with mean 0 and variance \sigma^2. Is \{Xt\} weakly stationary? Answer yes or no, and justify.

Accepted Answer

Check the three requirements: constant mean, constant finite variance, and covariance that depends only on the lag. Mean. E[Xt] = E[\varepsilont] + 0.5\,E[\varepsilon{t-1}] = 0 for every t. Constant. Variance. Since the noise terms are uncorrelated, \operatorname{Var}(Xt) = \sigma^2 + 0.5^2\sigma^2 = 1.25\,\sigma^2, finite and the same at every t. Covariances. \operatorname{Cov}(Xt, X{t+1}) = 0.5\,\sigma^2 (the two windows share the term \varepsilont), and for any lag k \ge 2 the windows share nothing, so the covariance is 0. These values depend only on the gap k, not on t. All three conditio…

Is this moving-average series stationary?

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