Time Series Analysis: Forecasting and Control, Fourth Edition

CHAPTER FIVE

Forecasting

Having considered in Chapter 4 some of the properties of ARIMA models, we now show how they may be used to forecast future values of an observed time series. In Part Two we consider the problem of fitting the model to actual data. For the present, however, we proceed as if the model were known exactly, bearing in mind that estimation errors in the parameters will not seriously affect the forecasts unless the number of data points, used to fit the model, is small.

In this chapter we consider nonseasonal time series. The forecasting, as well as model fitting, of seasonal time series is described in Chapter 9. We show how minimum mean square error forecasts may be generated directly from the difference equation form of the model. A further recursive calculation yields probability limits for the forecasts. It is to be emphasized that for practical computation of the forecasts, this approach via the difference equation is the simplest and most elegant. However, to provide insight into the nature of the forecasts, we also consider them from other viewpoints.

5.1 MINIMUM MEAN SQUARE ERROR FORECASTS AND THEIR PROPERTIES

In Section 4.2 we discussed three explicit forms for the general ARIMA model:

where φ(B) = ϕ(B)∇^d. We begin by recalling these three forms since each one sheds light on a different aspect of the forecasting problem.

We shall be concerned with forecasting ...