Discussion
Let \(\{X_{t}\}\) be a stationary time series model given by :
\(X_{t} = \mu + \phi X_{t-1} + Z_{t}\), \(Z_{t} \sim\) iid normal \((0, \sigma^{2})\). Then, which of the following statements is not true if \(P_{X_{t}}(X_{t+k})\) is the forecast of \(X_{t+k}\). Given \(X_{1}, X_{2} \ldots X_{t}\) ?
Answer: A
The statement that is not true is the forecast mean square error approaches \(\mu^{2} + \sigma^{2}/(1 - \phi^{2})\).
For a stationary AR(1) with constant, the \(k\)-step forecast is
\[ \begin{aligned} P(X_{t+k}) &= m + \phi^{k}(X_{t} - m), \\ m &= \frac{\mu}{1 - \phi} \end{aligned} \]
a weighted average of the mean \(E(X_{t}) = m\) and the latest value \(X_{t}\) (D), which tends to \(m\) as \(k \to \infty\) (C). Its error variance is \(\sigma^{2}(1 - \phi^{2k})/(1 - \phi^{2})\), which tends to the process variance \(\sigma^{2}/(1 - \phi^{2})\) (B).
A forecast error has mean 0, so the constant \(\mu\) never adds \(\mu^{2}\) to the mean square error; (A) is the false statement.
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