Autocorrelation of detrended values

In many time series, there is a tendency for unusually large values to be followed by other similar values, even after detrending the series,

ei  =  yi − trendi

Any tendency for these residuals to be followed by others of similar size is called autocorrelation and can be described by the correlation coefficient between pairs of adjacent residuals.

In this example, the autocorrelation is 0.71, so there is some tendency for values above the trend line to be followed by others on the same side of it.

Positive autocorrelation has an important impact on forecasting — if the most recent value is above the trend line, we should forecast the next value will also be above it.