Discussion
In an AR(1) model \(X_{t} = 0.5X_{t-1} + Z_{t}\), \(Z_{t} \sim\) iid normal (0, 1), the best linear predictor \(aX_{1} + bX_{3}\) of \(X_{2}\) using \((X_{1}, X_{3})\) will have :
Answer: A
The coefficients satisfy \(a = b\).
For AR(1) with \(\phi = 0.5\), the autocorrelations are \(\rho(1) = 0.5\) and \(\rho(2) = 0.25\). The best linear predictor of \(X_{2}\) from \(X_{1}, X_{3}\) solves
\[ \begin{aligned} a + \rho(2)b &= \rho(1) \\ \rho(2)a + b &= \rho(1) \end{aligned} \]
The system is symmetric in \(a\) and \(b\), so \(a = b = \rho(1)/(1 + \rho(2)) = 0.5/1.25 = 0.4\), which is the general \(\phi/(1 + \phi^{2})\). The reason is time reversibility: a stationary Gaussian AR(1) looks the same run backwards, so the past and future neighbours of \(X_{2}\) carry equal weight. It can be determined, so D is wrong.
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