Cubic and higher-degree polynomial models

If a quadratic model does not adequately describe the shape of the trend in a time series, it is tempting to try to further increase the order of the polynomial,

trend   =   b0  +  b1 time  +  b2 time2  +  b3 time3  +  ...

This kind of polynomial model can also be fitted by least squares.

A polynomial of degree 3 or 4 often provides a fairly smooth description of trend but polynomial models usually behave badly (with sudden increases or decreases) beyond the data points, so

Polynomial models of degree greater than 2 should not be used for forecasting.

Fatal traffic crashes in New Zealand

The next data set gives the number of fatal vehicle crashes in New Zealand per 100,000 population between 1970 and 2005.

Use the pop-up menu to increase the degree of the polynomial model. As the order increases, the curve describes more subtleties in the form of the trend.

However observe how the shape of the curve varies after 2005 when the number of polynomial terms changes. For example, polynomials of degree 5 and 6 both fit the data similarly but the behaviours of these polynomials after 2005 are totally different.

Several of the polynomials seem reasonable descriptions of the changes in fatal vehicle crashes between 1970 and 2005. However none of the polynomials would provide reasonable predictions after 2005.