Independence between variance and regression coefficients In a regression model with 3 regressors, how can I prove that $\hat{\beta_1}-\hat{\beta_3}$ and $s^2$ are statistically independent? My idea is to start with $Cov((\hat{\beta_1}-\hat{\beta_3}), s^2)=Cov(\hat{\beta_1},s^2)-Cov(\hat{\beta_3},s^2)$ but I can't calculate $Cov(\hat{\beta_1},s^2), Cov(\hat{\beta_3},s^2)$.

$\newcommand{\Cov}{\mathrm{Cov}}$ Calculate the analytical covariance matrix of the estimator for the parameters and see if the corresponding off-diagonal elements are zero. Then you know at least if the parameters are correlated or not, but not necessarily independent. If a Gaussian distribution defines the estimator, then off-diagonal elements equal to zero is equivalent to independence.
For example, consider the linear model:
$$
y = H\theta + \epsilon
$$
with $\epsilon \sim \mathcal{N}(0,Q)$. The least-square estimator is
$$
\hat{\theta} = (H^\top Q^{-1} H)^{-1}H^\top Q^{-1}y
$$
with covariance matrix
$$
\Cov(\hat{\theta}) = (H^\top Q^{-1} H)^{-1}.
$$
The off-diagonal elements of the above matrix then define the correlation between the parameters in $\theta$.
