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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

$$Y|\hat{\beta}_0,\hat{\beta}_1$$

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

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

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  • $\begingroup$ 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