# Prove variance of locally weighted regression increases with degree

I am interested in proving the following fact for locally weighted polynomial regression from The Elements of Statistical Learning by Hastie et. al.

It can be shown that $$||l(x_0)||$$ increases with $$d$$...

The same fact is also mentioned in Local Regression and Likelihood by Loader:

... the influence function infl(x) increases as the degree of the local polynomial increases.

Where $$d$$ is the degree of local polynomial and

$$||l(x_0)||^2 = b(x_0)^T(B^TWB)^{-1}B^TWW^TB(B^TWB)^{-1}b(x_0)$$ where $$b(x)^T = (1, x, x^2, ..., x^{d-1})$$, $$B$$ is the $$N \times d$$ regression matrix with ith row $$b(x_i)^T$$, and $$W(x_0)$$ is an $$N \times N$$ diagonal weight matrix.

This question's accepted solution solves a related variance problem in the special case when $$W = I_{N \times N}$$, however this doesn't cover the more general case as it uses the fact that $$I_{N \times N} \times I_{N \times N} = I_{N \times N}$$ which is not the case for $$W$$.