Quadratic models

A linear trend is not appropriate for all time series. Many trends have curvature which must be described with a more complex model. We now briefly describe fitting a quadratic trend of the form...

trend   =   b0  +  b1 time  +  b2 time2

A quadratic curve of this form has three parameters that can be adjusted to improve the fit of the model. We again define residuals to be the differences between the actual time series values, y, and the model's predictions,

ei  =  yi − trendi

The least squares estimates of the three parameters are again the values that minimise the residual sum of squares,

Σ ei2

Credit card debt

The time series below shows the total debt owed to credit cards in New Zealand at the end of each December from 1990 to 2001. The trend is clearly nonlinear, so a quadratic model might be appropriate. (Note that we have again subtracted 1990 from the years to make the coefficients more manageable.)

Drag the three red arrows to position the quadratic curve close to the data.

Click Least squares to position the line to minimise the residual sum of squares.

The equation of the trend line is shown under the graph. It can be used to predict credit card debt in future years.