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.
CITATION STYLE
Brockwell, P. J., & Davis, R. A. (1987). Prediction of Stationary Processes (pp. 159–190). https://doi.org/10.1007/978-1-4899-0004-3_5
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