# Confidence intervals of coefficients in polynomial regression

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 $X'X$.

• If all $x_i$ are distinct both $X$ and $X^\prime X$ are non-singular. In your case it means you can have the same $x$? Have you considered en.wikipedia.org/wiki/Scheff%C3%A9%27s_method? Aug 26, 2011 at 12:26
• The Vandermonde matrix can be terribly ill-conditioned. So, even though $X^T X$ is (theoretically) nonsinguler, it can very easily be numerically singular/unstable. Aug 26, 2011 at 19:16
• I have merged your accounts. Now you can edit all your content directly.
– user88
Aug 27, 2011 at 8:13

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.