I've read many times on this site that high order polynomials (generally more than third) shouldn't be used in linear regression, unless there is a substantial justification to do so.
I understand the issues about extrapolation (and prediction at the boundaries).
Since extrapolation isn't important to me…
- Are high order polynomials also a bad way of approximating the underlying function within the range of the data points? (i.e. interpolation)
- If so, what problems are arising?
I don't mind being redirected to a good book or paper about this.
Thanks.
Best Answer
I cover this in some detail in Chapter 2 of RMS. Briefly, besides extrapolation problems, ordinary polynomials have these problems:
These are reasons that regression splines are so popular, i.e., segmented polynomials tend to work better than unsegmented polynomials. You can also relax a continuity assumption for a spline if you want to have a discontinuous change point in the fit.