# How to prove that conditional distribution for Y given OLS for simple linear regression do not depends on original parameters? [duplicate]

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