a. Show that the mean-squared forecast error (Eleft[left(hat{y}_{T+1}-y_{T+1} ight)^{2} mid I_{T} ight]) for a forecast (hat{y}_{T+1}), that

a. Show that the mean-squared forecast error \(E\left[\left(\hat{y}_{T+1}-y_{T+1}\right)^{2} \mid I_{T}\right]\) for a forecast \(\hat{y}_{T+1}\), that depends only on past information \(I_{T}\), can be written as

b. Show that \(E\left[\left(\hat{y}_{T+1}-y_{T+1}\right)^{2} \mid I_{T}\right]\) is minimized by choosing \(\hat{y}_{T+1}=E\left(y_{T+1} \mid I_{T}\right)\).

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