3 - Location

Summary

Linear Gaussian unobserved components models, such as the random walk plus noise of (1.1), play a central role in time series modelling. The Kalman filter and associated smoother provide the basis for a comprehensive statistical treatment. The filtered and smoothed estimators of the signal minimize the MSE of the estimation error, the likelihood is given as a byproduct of one-step prediction errors produced by the Kalman filter and the full multistep predictive distribution has a known Gaussian distribution.

We now want to confront a situation in which the location is buried in noise which is non-Gaussian and which may throw up observations that, when judged by the Gaussian yardstick, are outliers. The aim is to develop an observation-driven model that is tractable and retains some of the desirable features of the linear Gaussian model. Letting the dynamics be driven by the conditional score leads to a model which not only is easy to implement, but also facilitates the development of a comprehensive and relatively straightforward theory for the asymptotic distribution of the ML estimator.

Modelling the additive noise with a Student's t distribution is effective and theoretically straightforward. Indeed, the attractions of using the t distribution to guard against outliers in static models are well-documented; see, for example, Lange, Little and Taylor (1989). The general error distribution is less satisfactory for reasons discussed at the end of the chapter. However, it does point to some interesting connections with dynamic quantiles.