FORECASTING:BOX-JENKINS METHOD

BOX-JENKINS METHOD

In practice, a forecasting method often is chosen without adequately checking whether the underlying model is an appropriate one for the application. The beauty of the Box-Jenkins method is that it carefully coordinates the model and the procedure. (Practitioners often use this name for the method because it was developed by G.E.P. Box and G.M. Jenkins. An alternative name is the ARIMA method, which is an acronym for autoregressive integrated moving average.) This method employs a systematic approach to identifying an appropriate model, chosen from a rich class of models. The historical data are used to test the validity of the model. The model also generates an appropriate forecasting procedure.

To accomplish all this, the Box-Jenkins method requires a great amount of past data (a minimum of 50 time periods), so it is used only for major applications. It also is a sophisticated and complex technique, so we will provide only a conceptual overview of the method. (See Selected References 2 and 8 for further details.)

The Box-Jenkins method is iterative in nature. First, a model is chosen. To choose this model, we must compute autocorrelations and partial autocorrelations and examine their pat- terns. An autocorrelation measures the correlation between time series values separated by a fixed number of periods. This fixed number of periods is called the lag. Therefore, the autocorrelation for a lag of two periods measures the correlation between every other observation; i.e., it is the correlation between the original time series and the same series moved forward two periods. The partial autocorrelation is a conditional autocorrelation between the original time series and the same series moved forward a fixed number of periods, holding the effect of the other lagged times fixed. Good estimates of both the autocorrelations and the partial autocorrelations for all lags can be obtained by using a computer to calculate the sample autocorrelations and the sample partial autocorrelations. (These are “good” estimates because we are assuming large amounts of data.)

From the autocorrelations and the partial autocorrelations, we can identify the functional form of one or more possible models because a rich class of models is characterized by these quantities. Next we must estimate the parameters associated with the model by using the historical data. Then we can compute the residuals (the forecasting errors when the forecasting is done retrospectively with the historical data) and examine their behavior. Similarly, we can examine the behavior of the estimated parameters. If both the residuals and the estimated parameters behave as expected under the presumed model, the model appears to be validated. If they do not, then the model should be modified and the procedure repeated until a model is validated. At this point, we can obtain an actual forecast for the next period.

For example, suppose that the sample autocorrelations and the sample partial auto- correlations have the patterns shown in Fig. 27.6. The sample autocorrelations appear to decrease exponentially as a function of the time lags, while the same partial autocorrela- tions have spikes at the first and second time lags followed by values that seem to be of negligible magnitude. This behavior is characteristic of the functional form

The Box-Jenkins procedure appears to be a complex one, and it is. Fortunately, computer software is available. The programs calculate the sample autocorrelations and the sample partial autocorrelations necessary for identifying the form of the model. They also estimate the parameters of the model and do the diagnostic checking. These programs, how- ever, cannot accurately identify one or more models that are compatible with the autocorrelations and the partial autocorrelations. Expert human judgment is required. This expertise can be acquired, but it is beyond the scope of this text. Although the Box-Jenkins method is complicated, the resulting forecasts are extremely accurate and, when the time horizon is short, better than most other forecasting methods. Furthermore, the procedure produces a measure of the forecasting error.