Chapter VII - ESTIMATION OF UNOBSERVED-COMPONENTS AND CANONICAL MODELS

Abstract

This chapter describes the maximum-likelihood estimation of univariate autoregressive moving-average (ARMA) and unobserved-components (UC) models in both the time and frequency domain and with multivariate time-series models in the frequency domain. The method for the estimation of UC models in the time domain involves reducing an UC model to its canonical form, given suitable restrictions on the parameters of the canonical ARMA model. The frequency domain methods turn out to be more efficient computationally, even for ARMA models, and lead to nearly the same results as do time domain estimates. The chapter illustrates the case of the standard case of a mixed ARMA model to illustrate the estimation of ARMA models by maximum-likelihood methods. It discusses how the method can be adapted to handle ARMA models. The method for estimating an ordinary ARMA model may be adapted for the estimation of an UC model by expressing the parameters of the canonical form of the latter in terms of the original parameters of the unobserved-components form.