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Your equation $y_{T+1} = E(y_{T+1}|I_T)$ clearly does not make sense. In any case, $$ E(y_{T+1}|I_T) $$ is the best one step ahead forecast conditional on $Y_1, \cdots, Y_T$, whereas you're asked to find the best linear one step ahead forecast given $Y_T$. They are in general not the same. The best linear forecast for $Y_{T+1}$ given $Y_T$ is $$ \mu + \frac{...

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