# Assume Y=X beta_0+epsilon where ϵ is zero mean and X is fixed. I know that under certain conditions on the design matrix X in OLS, the sample mean of the residuals e is 0. Can we say the same for the true population mean of residuals as well?

Assume $Y=X{\beta }_{0}+ϵ$ where $ϵ$ is zero mean and $X$ is fixed. I know that under certain conditions on the design matrix $X$ in OLS, the sample mean of the residuals $\overline{e}$ is $0.$ Can we say the same for the true population mean of residuals as well?
$\stackrel{^}{Y}=X\stackrel{^}{\beta }$ and
$E\left[e\right]=E\left(Y-\stackrel{^}{Y}\right)=E\left[X{\beta }_{0}-X\stackrel{^}{\beta }\right]=X\left({\beta }_{0}-E\left(\stackrel{^}{\beta }\right)\right)=0$
since $\stackrel{^}{\beta }$ is an unbiased estimator of ${\beta }_{0}$.
Am I making a mistake?
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If we are treating $X$ as fixed, which I take to mean non-random, then by assumption
$E\left[ϵ\right]=0$
You are sort of going in a circle because you say $E\left[Y\right]=X{\beta }_{0}$, but this is because of our prior assumption. It is a result not an assumption. We have
$\begin{array}{rl}Y& =X{\beta }_{0}+ϵ\\ E\left[Y\right]& =E\left[X{\beta }_{0}+ϵ\right]\\ E\left[Y\right]& =E\left[X{\beta }_{0}\right]+E\left[ϵ\right]\\ E\left[Y\right]& =X{\beta }_{0}+0\\ E\left[Y\right]& =X{\beta }_{0}\end{array}$
where the fourth line is a result of our assumption and $X$ being non-random (as well as ${\beta }_{0}$ being a constant).