Forecasting with structural time series models

Structural time series models are formulated directly in terms of unobserved components, such as trends, cycles and seasonals, that have a natural interpretation and represent the salient features of the series under investigation. An explicit link with other approaches such as the ARIMA approach and the regression methodology can usually be made. As far as the former is concerned, linear univariate structural models have a reduced form ARIMA representation, but the latter is subject to restrictions on the parameter space, which play a relevant role for forecasting and signal extraction, providing a sensible way of weighting the available information. Moreover, structural models can easily be extended to handle any frequency of observation (weekly, daily, hourly) and specific features of the series that are difficult to deal with in the ARIMA framework (heteroscedasticity, nonlinearity, non-Gaussianity). From the second standpoint, structural models are set up as regression models in which the explanatory variables are functions of time and the coefficients are allowed to vary over time, thereby encompassing the traditional decomposition of a time series into deterministic components. A thorough presentation of the main ideas and methodological aspects underlying structural time series models is contained in Harvey (1989); other important references are West and Harrison (1997) and Kitagawa and Gersch (1996). The material presented in this chapter is organised as follows: the next three sections deal with the specifications of time series models respectively for the trend, the cycle, and the seasonal component, and how they can be combined into the main univariate structural models; multivariate extensions are discussed in section 5. The disturbances driving the different components are assumed independent and this can be viewed as an identification restriction. However, models with correlated disturbances can be

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