Handbook of Economic Forecasting part 33. Research on forecasting methods has made important progress over recent years and these developments are brought together in the Handbook of Economic Forecasting. The handbook covers developments in how forecasts are constructed based on multivariate time-series models, dynamic factor models, nonlinear models and combination methods. The handbook also includes chapters on forecast evaluation, including evaluation of point forecasts and probability forecasts and contains chapters on survey forecasts and volatility forecasts. Areas of applications of forecasts covered in the handbook include economics, finance and marketing | 294 H. Lutkepohl process is the state space representation which will not be used in this review however. The relation between state space models and VARMA processes is considered for example by Aoki 1987 Hannan and Deistler 1988 Wei 1990 and Harvey 2006 in this Handbook Chapter 7. . Cointegrated I 1 processes If the DGP is not stationary but contains some I 1 variables the levels VARMA form is not the most convenient one for inference purposes. In that case det A z 0 for z 1. Therefore we write the model in EC form by subtracting A0yt - 1 on both sides and re-arranging terms as follows AcAyt nyt-1 T1Ayt-1 r _1Ayt -p 1 MgUt M1Ut-1 MqUt-q t e N where n - Aq - A1------------- Ap - A 1 and r - - A 1 Ap i 1 . p - 1 Lutkepohl and Claessen 1997 . Here I yt-1 is the EC term and r rk n is the cointegrating rank of the system which specifies the number of linearly independent cointegration relations. The process is assumed to be started at time t 1 from some initial values y0 . y-p 1 u0 . u-q 1 to avoid infinite moments. Thus the initial values are now of some importance. Assuming that they are zero is convenient because in that case the process is easily seen to have a pure EC-VAR or VECM representation of the form t-1 Ayt n yt-1 j yt-j A-lMgut t e N j 1 where n and j j 1 2 . are such that IkA - n L - jALj a-1mqm L -1 aqa - nL - r1 al--------------rp-1ALp-1 . A similar representation can also be obtained if nonzero initial values are permitted see Saikkonen and Lutkepohl 1996 . Bauer and Wagner 2003 present a state space representation which is especially suitable for cointegrated processes. . Linear transformations of VARMA processes As mentioned in the introduction a major advantage of the class of VARMA processes is that it is closed with respect to linear transformations. In other words linear transformations of VARMA processes have again a finite order VARMA representation. These transformations are very common and are useful to study problems of .
