I came across an article with the following question:
Consider a linear regression with one single covariate, y=β0+β1x1+ε and the least-square estimates. The variance of the slope is Var[β1]. Do we decrease this variance if we add one variable, and consider y=β0+β1x1+β2x2+ε ?
They showed, using simulation, that the answer is no as it depends on how correlated
x2 are. Is there a way to prove this without simulation?