The autocorrelation function of a time series with a deterministic component

The standard calculation of the autocorrelation function (acf) assumes stationarity. This implies that the series does not contain trends or cycles. Nevertheless, computer programs are used to calculate acf’s for any series, irrespective of whether the stationarity condition is satisfied or not. It has long been observed that the acf of a series that contains a trend shows a slowly declining pattern and that the acf of a series that contains cycles displays periodic behaviour. This is explained here. Equations are given for the autocorrelation function of a series that contains a deterministic component. The case of trends and cycles is examined in detail.