On the Cramér-Wold factorization

Abstract

SUMMARY Several authors have considered the problem of deriving the q moving average parameters of an autoregressive-moving average process of order ( p, q ), given estimates of the p auto-regressive parameters and the first q members of the parent autocorrelation function; the problem was first raised by H. Cramér and H. Wold. The problem is discussed in connexion with the parameter estimation procedures of Walker (1961, 1962) and of Box & Jenkins (1970). An approach to the problem is described which is most suitable in the common cases where q is small; otherwise a factorization routine due to Wold (1954) is found to be preferable for most practical purposes when q is moderately large, whilst an iterative algorithm due to Tunnicliffe Wilson (1969) has more desirable properties in all other cases.