MAXIMUM LIKELIHOOD ESTIMATORS IN THE MULTIVARIATE AUTOREGRESSIVE MOVING‐AVERAGE MODEL FROM A GENERALIZED LEAST SQUARES VIEWPOINT

Abstract

Abstract. Explicit expressions are derived for the gradient vector and (approximate) Hessian matrix of the log likelihood function for the multivariate autoregressive moving‐average (ARMA) model. Based on these expressions an explicit description of the Gauss‐Newton iterative procedure to obtain maximum likelihood (ML) estimates of the parameters in the multivariate ARMA model is presented. The resulting computational procedure has the form of a generalized least squares (GLS) estimation involving lagged values of the observed vector series and of the residual series as independent variables. This direct form of the estimator is found to be appealing and useful in understanding and interpreting the ML estimation procedure from a regression point of view, and in comparing the ML procedure with other ‘linear’ estimation procedures that have recently been presented. Simulation results are also presented for a univariate and a multivariate ARMA model to illustrate the ML‐GLS estimation procedure and to compare it with other linear estimation procedures.