# Proof of variance of ridge estimate with only one predictor!

Let's consider Ridge with only one predictor (extreme and simple case). I would like to proof that $$V(B_r)=\sigma^2/(1+\lambda)$$, so its variance it less than OLS variance, that is $$V(B_{OLS})=\sigma^2$$. I don't want the proof in the matrix form, but in it that mathematical way.

Thank you in advance to everyone!!!

• Could you elaborate on what "not ... matrix but ... mathematical" might mean? BTW, a matrix proof appears at stats.stackexchange.com/questions/11009. It's a straightforward matter of (slight) changes in notation to make it "non-matrix," although the benefits of such a translation are dubious. – whuber Dec 1 '18 at 23:38