Estimation, prediction and smoothing for univariate structural time series models
Andrew C. Harvey · Cambridge University Press eBooks · 1990
The general properties of state space models were set out in chapter 3. The opening section of this chapter shows how the structural models introduced in section 2.3 can be put in state space form. The state space form provides the key to the statistical treatment of structural models. It enables ML estimators of the unknown parameters in a Gaussian model to be computed via the Kaiman filter and the prediction error decomposition. Once estimates of these parameters have been obtained, it provides algorithms for prediction of future observations and estimation of the unobserved components. Section 4.2 describes various ways in which the unknown parameters in structural models can be estimated in the time domain. Estimation can also be carried out in the frequency domain. This latter approach has a number of attractions and is described in detail in section 4.3. (Note that even if frequency-domain methods are used for ML estimation, the state space form is still needed for prediction and estimation of the unobserved components.) Frequency-domain methods are also important in determining the asymptotic properties of ML estimators. Both asymptotic and small sample properties of estimators for structural models are considered in section 4.5. The preceding material, in section 4.4, is primarily to assure the reader that the models under consideration are identifiable. Finally, sections 4.6 and 4.7 discuss various aspects of prediction and the estimation of unobserved components.