Correlation in Polynomial Regression

Polynomial regression of degree p in one independent variable χ is considered. Numerically large sample correlations between χα and χβ, α < β, a, β = 1, ···, p, may cause ill-conditioning in the matrix to be inverted in application of the method of least squares. These sample correlations are investigated. It is confirmed that centering of the independent variable to have zero sample mean removes nonessential ill-conditioning. If the sample values of χ are placed symmetrically about their mean, the sample correlation between χα and χβ is reduced to zero by centering when α + β is odd, but may remain large when α + β is even. Some examples and recommendations are given.