# MSPE predictor of MA model

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 $$\mu$$.