Handbook of Economic Forecasting part 40. 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 | 364 A. Harvey . Innovations The joint density function for the T sets of observations y1 . yT is -L p Y t 0 P yt I Yt-1 68 where p yt Yt -1 denotes the distribution of yt conditional on the information set at time t - 1 that is Yt-1 yt-1 yt-2 . y1 . In the Gaussian state space model the conditional distribution of yt is normal with mean yt t -1 and covariance matrix Ft. Hence the N x 1 vector of prediction errors or innovations vt yt -yt t-1 t 1 . T 69 is serially independent with mean zero and covariance matrix Ft that is vt NID 0 Ft . Re-arranging 69 57 and 60 gives the innovations form representation yt Zt at t-1 dt v t 70 at 1 t Tt at t-1 ct Kt vt. This mirrors the original SSF with the transition equation as in 55 except that at t-1 appears in the place of the state and the disturbances in the measurement and transition equations are perfectly correlated. Since the model contains only one disturbance vector it may be regarded as a reduced form with Kt subject to restrictions coming from the original structural form. The SSOE models discussed in Section are effectively in innovations form but if this is the starting point of model formulation some way of putting constraints on Kt has to be found. . Time-invariant models In many applications the state space model is time-invariant. In other words the system matrices Zt dt Ht Tt Ct Rt and Qt are all independent of time and so can be written without a subscript. However most of the properties in which we are interested apply to a system in which ct and dt are allowed to change over time and so the class of models under discussion is effectively yt Zat dt et Var et H 71 and at Tat-1 Ct Rnt Var nt Q 72 with E et n st 0 for all s t and P1 0 H and Q . Ch. 7 Forecasting with Unobserved Components Time Series Models 365 The principal STMS are time invariant and easily put in SSF with a measurement equation that for univariate models will be written yt Zat st t 1 . T 73 with Var st H of Thus state space .
