Discussion
Let \(\{X_{t}\}\) be a stationary AR(1) model with \(X_{t} = \mu + \phi X_{t-1} + Z_{t}\), where \(Z_{t} \sim \mathrm{WN}(0, \sigma^{2})\). Then, which of the following statements is true ?
(i) \(E(X_{t}) = \mu\)
(ii) \(V(X_{t}) = \sigma^{2}/1 - \phi^{2}\)
(iii) \(\mathrm{Corr}(X_{t}, X_{t+s}) = \phi^{s}\)
(iv) \(V(X_{t}) = 1 - \phi^{2}\)
Answer: A
The true statements are (ii) and (iii).
Taking expectations of \(X_{t} = \mu + \phi X_{t-1} + Z_{t}\) under stationarity gives \(E(X_{t}) = \mu/(1 - \phi)\), not \(\mu\), so (i) is false. Taking variances gives \(\gamma(0) = \phi^{2}\gamma(0) + \sigma^{2}\), so \(V(X_{t}) = \sigma^{2}/(1 - \phi^{2})\), statement (ii) as printed without brackets. The autocorrelation of an AR(1) at lag \(s\) is \(\phi^{s}\) (iii).
(iv) leaves out \(\sigma^{2}\). So the options including (i) or (iv) are wrong.
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