Large-sample estimation of parameters for movingaverage models

Abstract

Let {xJ}, t = O, + 1, + 2, ... , be a stationary normal moving-average process, defined by Xt = # +'61 + fli 6f-1 + * e + lhe 6f:--h) (1) where (et) is a set of independent random variables, each distributed normally with mean zero and variance C-2. The problem of making inferences about the parameters fl%, ... *) *A, given a sample of consecutive observations (xl, x2, ..., x.) from the process, is a well-known one in time-series analysis. Little progress with this seems to have been made except under the assumption that n is large, but in the large-sample case the work of Vhittle (1951, Chapter 7, 1953, 1954, pp. 211-18) enables one to obtain, at least in principle, a solution which for most purposes can be regarded as complete. From this work it follows that, provided that the roots of the equation zh +jl.Zh-l + ... + /h = 0 al have moduli less than unity, the logarithm of the likelihood of the sample is given asymptotically by