This question already has an answer here:

How to prove that for a simple linear regression model: $$y_i=\beta_0+\beta_1 x_i+\varepsilon_i,$$ the conditional distribution


do not depends on $\beta_0$ and $\beta_1$, the original parameters for $Y$ and $\hat{\beta}_0$, $\hat{\beta}_1$ are the OLS.


marked as duplicate by kjetil b halvorsen, Michael Chernick, usεr11852, mdewey, Alexis Mar 24 at 16:41

This question has been asked before and already has an answer. If those answers do not fully address your question, please ask a new question.

  • $\begingroup$ Thanks, it's really about sufficiency statistics for $\beta$. I think The factorization theorem Will help with the proof. $\endgroup$ – Anselmo Alves de Sousa Mar 23 at 21:50