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Suppose that we have the following linear model $\mathbb{E}[Y_{i}]=b_{0}+b_{1}(x_{i}-\bar{x})$ where $i=1,...,n$ and $\bar{x}$ is the mean of $x_{i}$ and $Y_{i}$ are uncorrelated with constant variance $\sigma^{2}$. We derived that the least squares estimators of $b_{0}$ and $b_{1}$ are the following

$\hat{b_{0}} = \bar{Y}$ and $\hat{b_{1}}=\frac{\sum_{i}(x_{i}-\bar{x})Y_{i}}{\sum_{i}(x_{i}-\bar{x})^{2}}$

We also know from the covariance matrix of the estimators that $\mathbb{E}[\hat{b_{0}}\hat{b_{1}}]=0$ are uncorrelated.

Now we are given a different linear function $\tilde{b}=\sum_{i}b_{i}Y_{i}$ where $b_{i}$ are constants. And we are also given the mean square error of $\tilde{b}$ which is

$MSE(\tilde{b}) = var(\tilde{b})+(\mathbb{E}(\tilde{b})-b_{0})^{2}$.

My goal is to prove that $MSE(\tilde{b})$ is independent of $b_{1}$ if and only if $\sum_{i}b_{i}(x_{i}-\bar{x})=0$

Thus we have to prove that

i) $\mathbb{E}[MSE(\tilde{b})b_{1}]=\mathbb{E}[MSE(\tilde{b})]\mathbb{E}[b_{1}]\Rightarrow \sum_{i}b_{i}(x_{i}-\bar{x})=0$

ii) with $\sum_{i}b_{i}(x_{i}-\bar{x})=0 \Rightarrow \mathbb{E}[MSE(\tilde{b})b_{1}]=\mathbb{E}[MSE(\tilde{b})]\mathbb{E}[b_{1}]$

But I am struggling on how to prove it.

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