Vector exponential smoothing

In earlier chapters we have considered only univariate models; we now proceed to examine multi-series extensions and to compare the multi-series innovations models with other multi-series schemes. We shall refer to our approach as the vector exponential smoothing (VES) framework. The innovations framework is similar to the structural time series models advocated by Harvey (1989) in that both rely upon unobserved components, but there is a fundamental difference: in keeping with the earlier developments in this book, each time series has only one source of error. The VES models are introduced in Sect. 17.1; special cases of the general model are then discussed in Sect. 17.2. An inferential framework is then developed in Sect. 17.3 for the VES models, building upon our earlier results for the univariate schemes. The most commonly used multivariate time series models are those defined within the ARIMA framework. Interestingly, this approach also has only one source of randomness for each time series. Thus, the vector versions of the ARIMA framework (VARIMA), and special cases such as vector autoregression (VAR) and vector moving average (VMA), may be classified as innovations approaches to time series analysis (L?tkepohl 2005).We compare the VES frameworkwith existing approaches in Sect. 17.4. As in Chap. 11, when we consider equivalences between vector innovations models and the VARIMA forms, we will make the infinite start-up assumption. Finally we compare the performance of VES models to VAR and other existing state space alternatives, first in an empirical study of exchange rates (Sect. 17.5), and then across a range of different time series taken froma large macroeconomic database, in Sect. 17.6.