Time Series: Theory and Methods pp 166-197 | Cite as

# Prediction of Stationary Processes

## Abstract

In this chapter we investigate the problem of predicting the values {*X* _{ t }, *t* ≥ *n +* 1} of a stationary process in terms of {*X* _{1},..., *X* _{ n }}. The idea is to utilize observations taken at or before time *n* to forecast the subsequent behaviour of {*X* _{ t }}. Given any closed subspace *ℳ* of *L* ^{2}(Ω, *ℱ*, *P*), the best predictor in *ℳ* of *X* _{ n } _{+h } is defined to be the element of *ℳ* with minimum mean-square distance from *X* _{ n } _{+h. } This of course is not the only possible definition of “best”, but for processes with finite second moments it leads to a theory of prediction which is simple, elegant and useful in practice. (In Chapter 13 we shall introduce alternative criteria which are needed for the prediction of processes with infinite second-order moments.) In Section 2.7, we showed that the projections are respectively the best function of *X* _{1},..., *X* _{ n } and the best linear combination of 1, *X* _{1},..., *X* _{ n } for predicting *X* _{ n+h }. For the reasons given in Section 2.7 we shall concentrate almost exclusively on predictors of the latter type (best linear predictors) instead of attempting to work with conditional expectations.

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