# Show that estimator is the best linear estimator with smallest MSE

Let's consider we have OLS model $Y=X \beta+\epsilon$ , where $E(\epsilon) = 0$ and $Var(\epsilon)=\sigma^2 I$.

Here $\hat{\beta}$ is the least squares estimator of the parameter $\beta$ . Let $z \neq 0$ be $n$ dimensional vector. The goal is to find the best linear estimator $A$ for $z^T \epsilon$ , where $E(A)=0$ and MSE (mean squared error) is the smallest possible. Actually we have to show that $A=z^T$ ($Y-X\hat{\beta}$) is the best estimator with this properties.

It is clear to me that $E(A)=0$, but since this is not unbiased estimator, I can't use Gauss- Markof theorem and I don't know what I should use to show that this is the best linear estimator (actually that it has the smallest MSE).

Thanks for the help.