Residuals
The residuals for a time series model are the differences between the actual and fitted values from the model. They describe how the model would have over- or under-estimated the data.
ei = yi − trendi
The residuals form the detrended series, and the process of removing the trend is called detrending. Detrending will often reveal interesting features that were obscured by the trend, and which may be important in explaining the past or forecasting the future.
It is therefore useful to look for patterns in a time series plot of the residuals. If the model under consideration fits well, there should be no pattern in the residuals — each should have the same chance of being positive or negative.
If there are systematic patterns in the residuals, it may be possible to use a different model for the trend (e.g. a quadratic rather than a linear model), but time series often exhibit patterns that cannot be explained with simple models for the trend.
Fatal traffic crashes in New Zealand
The diagram below shows the number of fatal vehicle crashes in New Zealand per 100,000 population between 1970 and 2005.
The residuals are the vertical distances between the points and the line. Move the slider to the right to display a time series of these residuals.
Repeat with a quadratic model and observe how the residuals effectively straighten out the curvature.
After detrending in this way, the residual for 1973 stands out as being unusually high, especially in relation to the numbers in the adjacent years. We might further investigate whether there were any particular changes in legislation that may have contributed to this.
In a similar way, there is a run of positive residuals between 1985 and 1990. Was there something unusual about these years?