I appreciated Marco's elegant answer explaining why the variance of a local polynomial regression increases monotonically in the degree. However, in the end of the proof, I find difficult to calculate each term in order to conclude the desired inequality.
If I pursue the proof from the last equation there, we can see that $$ \sigma^2 z^\top \left(X^\top X − X^\top q(q^\top q)^{−1}q^\top X\right)^{−1} z \geq \sigma^2 z^\top \left(X^\top X\right)^{−1}z = \mbox{Var}\left[\hat{y}_t\right],$$ by the arguments he mentioned. Then the following inequality holds for all $t\in \mathbb{R}$ $$\mbox{Var}[\hat{y}^∗_t] ≥ \mbox{Var}[\hat{y}_t]+ 2\sigma^2 t^k z^\top B_∗+\sigma^2 t^{2k} D_∗.$$ The conclusion is immediate if $2\sigma^2 t^k z^\top B_∗+\sigma^2 t^{2k} D_∗\geq 0$. Obviously $\sigma^2 t^{2k} D_∗≥0$ by construction of the Schur complement but I don't see how I can show that $2\sigma^2 t^k z^\top B_∗$ is positive for every $t$.
I don't see what I am missing.
Can anyone explain how to finish the proof?
