# Rss and sample variance indipendence in simple linear regression

Suppose that $(X_1 ,Y_1...X_n,Y_n)$ is an i.i.d. random sample from a simple homoschedastic linear model $Y=\alpha +\beta X+e$ , with $e|X \sim N(0,\sigma_e^2)$.

I want to understand if $\frac{rss}{\sigma_Y^2}=\frac{1}{\sigma_Y^2}\sum_i (Y_i - \hat{Y_i} )^2$ is independent from $\frac{nS^2}{\sigma_X^2} =\sum_i (X_i - \bar X_i )^2$.

I can easily show that $rss$ $\sim \sigma_Y^2 \chi^2_{n-2}$ and $nS^2 \sim \sigma_X^2 \chi^2_{n-1}$ via Cochran theorem.

I think that a feasible option should be Basu Theorem: both sample statistics should be ancillary, but how can I demonstrate completeness and sufficiency of one of the two?

Are any other options possible?