Autocorrelation of detrended values
In many time series, unusually large values tend to be followed by other similar values, even after taking into account the trend in the series. This will be evident when we examine the detrended values, defined as the residuals after the trend has been subtracted from a time series,
ei = yi − trendi
Even after a smooth trend line has been fitted, it is often found that successive detrended values are similar — if one value is above the trend line, the next value is often above it too. This is called autocorrelation and can be described by the correlation coefficient between pairs of successive residuals.
This similarity of successive residuals has an important impact on forecasting — if the current value is above the trend line, we should forecast that the next value will also be above it.
New company registrations
The time series below shows the rate of new company registrations in New Zealand (per 100,000 population) between 1960 and 1998.
When no attempt is made to model the trend in the series, the residuals are simply the actual values minus their mean. The scatterplot at the bottom plots each residual against the previous residual. Click on any cross to see how this scatterplot relates to the original time series. Observe that the successive residuals are highly correlated — the autocorrelation coefficient is shown on the left.
Use the pop-up menu on the left to model the increasing trend with a linear and then a quadratic model. These remove some of the autocorrelation between the residuals but it remains at 0.7 even after removing a quadratic trend — successive detrended values are still often similar.