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Assume $e_i=(\epsilon_i+\epsilon_{i+1})/2, i=1,...,n$ where $\epsilon_1,...,\epsilon_{n+1}$ are iid with mean zero and variance $\sigma^2$. Then $e_i$ are the moving average errors. Now, consider the model $Y_i=\mu+e_i, i=1,...,n$. I want to show two statements:

1. $$E(Y_{i+1}|Y_1,...,Y_i)=\frac{1}{2}(\mu+Y_i)$$ I figured that this means the the optimal MSPE predictor for future $Y_{i+1}$ given the past $Y_1,...,Y_i$ is $\frac{1}{2}(\mu+Y_i)$ but I don't know how to show it.

  1. That $\bar{Y}$ is a multivariate method of moments estimate of $\mu$.

Thanks for your help in advance!

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