Exact Maximum Likelihood Estimation of a Regression Equation with a First-Order Moving-Average Error

Abstract

Durbin [5] or its modification by Walker [14] in spite of its approximate nature. Other procedures for estimating regression models with moving-average errors have also been suggested by Phillips [12], Box and Jenkins [3] and Hannan [7], all of which are ML estimators only in large samples and they are based, in one way or another, on the truncation of the autoregressive representation of the moving-average process (when it exists) which was originally proposed by Whittle [16]. The small sample properties of the Phillips estimating procedure have recently been extensively studied by Hendry and Trivedi [9]. It is the purpose of this paper to suggest a procedure by which one can obtain, in small samples, the exact ML estimators of the parameters of the regression models with first order moving-average disturbances by means of an orthogonal transformation which reduces the computational burden to a great extent and which requires neither any estimation of the initial value of the disturbances as suggested by Phillips [12] nor any use of the method of backward forecasting of the initial disturbances proposed by Box and Jenkins [3].3 In the last section we shall demonstrate the proposed estimation procedure by estimating earnings equations for UK manufacturing industries where a casual comparison of movingaverage and autoregressive error specifications will also be made.