Let a linear OLS model: $$Y= X \beta + u$$ Where $u$ is a random disturbance. If we define the residual of the regression as $$e = Y - X \widehat{\beta}$$ where $\widehat{\beta}$ is the OLS vector of estimations of $\beta$, such that $\widehat{\beta} = (X'X)^{-1} X' Y$

Why does $e$ is a good estimator for $u$? In terms of estimator properties, such as efficiency, consistency, sufficiency, unbias, etc.

Thank you in advance


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