Forecasting when there is trend
The method described in the previous page works well if there is no trend in the time series. However if there is trend, the forecasting method should specifically model this.
In time series with both trend and autocorrelation, the trend should be removed from the series before an AR(1) model is used for forecasting. The sequence for forecasting a future value is therefore...
(Actually there are more efficient ways to do steps 1 and 2 together, but these are beyond the scope of an introductory course.)
New company registrations
The diagram below illustrates the use of an AR(1) model for forecasting in the presence of trend. The data are new company registrations in New Zealand (per 100,000 population) each year from 1960 to 1998.
The diagram initially ignores the trend and forecasts each residual from the previous residual to be zero, so all forecast values are the same.
Select Linear and then Quadratic from the pop-up menu to fit trend to the data. The scatterplot below the time series shows that the residuals are autocorrelated (but not strongly so).
Click the button Previous to forecast each residual with the previous one. The blue lines on the time series plot show the resulting forecasts (after the trend is added back to the forecast residuals). When any value is above the quadratic trend and therefore has a positive residual, the forecast for the following time is the same distance above the quadratic trend.
Finally click Least squares to find the best forecasts for the residuals. The forecast residuals are a little closer to zero than those from the previous time period, so the blue line is a little closer to the quadratic trend.
(You might like to experiment with the other options in this diagram, but use of a quadratic model and fitting the AR(1) line by least squares are best!)