In this chapter we discuss the basic notions about state space models and their use in time series analysis. The dynamic linear model is presented as a special case of a general state space model, being linear and Gaussian. For dynamic linear models, estimation and forecasting can be obtained recursively by the well-known Kalman filter. IntroductionIn recent years there has been an increasing interest in the application of state space models in time series analysis; see, for example, Harvey (1989), West and Harrison (1997), Durbin and Koopman (2001), the recent overviews by Künsch (2001) and Migon et al. (2005), and the references therein. State space models consider a time series as the output of a dynamic system perturbed by random disturbances. They allow a natural interpretation of a time series as the combination of several components, such as trend, seasonal or regressive components. At the same time, they have an elegant and powerful probabilistic structure, offering a flexible framework for a very wide range of applications. Computations can be implemented by recursive algorithms. The problems of estimation and forecasting are solved by recursively computing the conditional distribution of the quantities of interest, given the available information. In this sense, they are quite naturally treated within a Bayesian framework.State space models can be used to model univariate or multivariate time series, also in the presence of non-stationarity, structural changes, and irregular patterns. In order to develop a feeling for the possible applications of state space models in time series analysis, consider for example the data plotted in Figure 2.1. This time series appears fairly predictable, since it repeats quite regularly its behavior over time: we see a trend and a rather regular seasonal component, with a slightly increasing variability. For data of this kind, we would probably be happy with a fairly simple time series model, with a trend

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SUMMARYIn this paper we consider a class of conditionally Gaussian state-space models and discuss how they can provide a #exible and fairly simple tool for modelling "nancial time series, even in the presence of di!erent components in the series, or of stochastic volatility. Estimation can be computed by recursive equations, which provide the optimal solution under rather mild assumptions. In more general models, the "lter equations can still provide approximate solutions. We also discuss how some models traditionally employed for analysing "nancial time series can be regarded in the state-space framework. Finally, we illustrate the models in two examples to real data sets.

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