ABSTRACT
There are several approaches to introducing the models presented in this chapter, and many of the underlying ideas have a long history. We introduced and motivated the VAR(p) model by considering the prediction of xt based upon past series values up to a finite lag. This is an example of finite memory prediction. One of the main ideas of this extension is to base the prediction upon values extending to much greater lags but still using a limited number of coefficients. One of the earliest forms of predictor based on this idea was the exponentially weighted moving average (EWMA) of Brown (1962), or simply the exponential smoother . Past values are given positive weights which sum to one and decay exponentially with discount factor (1− λ). There have been various generalizations of this idea, some of which are based on ideas closely related to the models we propose; see, for example, Cogger (1974), Lu¨tkepohl (1982) and Burman and Shumway (2006). Some of these generalizations use several selected discount factors, but in our models a single one is specified. A prediction may be constructed as a linear combination of variables obtained by applying the exponential smoother repeatedly to past values-the so-called single and double smoothing, etc. However, this leads to multicollinearity in determining the coefficients of the required linear combination.
