Confidence intervals of coefficients in polynomial regression

Question

Given a model $a_{n}\cdot x^{n}+ a_{n-1}\cdot x^{n-1}+...+ a_0 = y + e$, where $e$ is some unknown (but assumed normally distributed) error and $x$ and $y$ are observations.

Now, how would one to estimate the confidence intervals of the coefficients $a_.$?

AFAIU, a simple procedure based on the covariance $(X'X)^{-1}$ is not sufficient, where $X$ is the $n$'th order (zero mean) Vandermonde matrix of $x$.

First of all $X'X$ is singular. Also calculation $var(a)= (r'r/m)\cdot diag(pinv(X'X))$, where $r$ is the residual and $m$ number of observations, seems to be vague, because high correlation between columns of $X'X$.

Answer 1

I think the practical answer is that in this situation you wouldn't regress using power functions, $x^n$, in the first place, for the reason that you give -- that the Vandermonde matrix is very ill-conditioned. Rather, you use some sort of orthogonal polynomials, which should be better conditioned.

If you specifically need confidence intervals on the coefficients of the power functions, then these will be linear combinations of the coefficients of the orthogonal polynomials, so you can use that to compute the desired confidence intervals.