The Perils of Polynomials

It is extremely dangerous to extrapolate a polynomial trend. Although there are other sources of danger, this study concentrates on the statistical error, which is very large for end values and increases alarmingly with extrapolations. Relative standard errors of polynomial trend values are summarized, and generalizations are made concerning the relative magnitude of the standard errors with changes in: (1) the degree of equation; (2) the number of observations; (3) the position of the trend value in time. The most interesting generalization is that the relative variance of a polynomial of degree m, as the position of the trend value departs farther and farther from the central position in time, may be described by a polynomial equation of degree 2m.