# independence of ols coefficient and estimated standard deviation

Is there an error in the reasoning in these lecture notes from a linear models course at EPFL? The author aims to establish that under the standard fixed design linear regression model with gaussian errors, the coefficient estimate $$\hat{\beta}$$ is independent of the residual variance estimate $$S^2$$. He writes,

"If $$e=y-X\hat{\beta}$$ is independent of $$\hat{y}=X\hat{\beta}$$, then $$S^2=e^Te/(n-p)$$ will obviously be independent of $$\hat{\beta}$$. Now notice that: ..."

And he goes on to show that $$e$$ is independent of $$\hat{y}$$ in the usual way, one is a projection onto the column space of $$X$$, the other is the projection onto its orthogonal complement, and the distributions are jointly gaussian so uncorrelatedness implies independence.

But I don't follow what he suggests is "obvious", i.e., why $$e$$ being independent of $$X\hat{\beta}$$ implies $$e$$ is independent of $$\hat{\beta}$$, since $$X$$ need not be invertible. In fact, while it is easy to visualize the independence of $$e$$ and $$\hat{y}$$ in terms of projections, I don't know how to visualize the relationship between $$e$$ and $$\hat{\beta}$$, the projection scaled by the squared norm of $$X$$, in a way that suggests uncorrelatedness.

• $\hat \beta$ is a function of $\hat Y = X \hat \beta$, since $\hat \beta = (X'X)^{-1} X'\hat Y$. Commented Aug 23, 2020 at 18:47
• @BigBendRegion oh right, betahat must be a function of yhat when the columns of X are independent. Commented Aug 23, 2020 at 22:00