Exact maximum likelihood estimation of regression models with finite order moving average errors

Abstract

The article presents information on exact maximum likelihood estimation of regression models with finite order moving average errors. A number of procedures for the estimation of models with moving average error specifications already appear in the literature. All these methods are, basically, derived from a consideration of a function which dominates the likelihood function and which is, asymptotically, equivalent to it. Consequently these procedures are applicable to large sample situations. economist M.H. Pesaran, however, has considered the exact likelihood function and his procedure is applicable to small samples. Unfortunately his method is not one that can be easily extended to the case of moving averages of order higher than the first since, for these models, it is not a straightforward matter to set up the orthogonal transformation required to diagonalize the covariance matrix of the disturbance term. Yet, as economist K. Kang has emphasized, it is of some importance to be able to compute the exact maximum likelihood (ML) estimates, since the generalized least squares (GLS) estimates frequently imply a non-invertible process for the disturbance term, and, for identification purposes, it is necessary that the process be invertible. The article is directed at the situation arising when the GLS estimates do not satisfy invertibility whereas the ML estimates do, this lack of invertibility of the OLS estimates being brought about by the omission of the extra terms in the likelihood function