Handbook of Economic Forecasting part 38. 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 | 344 A. Harvey and indeterministic with zero mean variance o o2 1 - P2 and autocorrelation function ACF P t pT coskcr t 0 1 2 . 25 For 0 kc n the spectrum of displays a peak centered around c which becomes sharper as p moves closer to one see Harvey 1989 p. 60 . The period corresponding to kc is 2n c. Higher order cycles have been suggested by Harvey and Trimbur 2003 . The nth order stochastic cycle n t for positive integer n is KK cos kc sin kc K1 t-1 Kt .Kt. p - sin kc cos kc L Kt-u K. Kt cos kc sin kc K-1 Ki- 1 t-1 i 2 . Kit. p - sin cc cos Cc Kh-1. Ki- 1 t-1. . n. 26 The variance of the cycle for n 2 is a 1 p2 1 - p2 3ic 2. while the ACF is P t pT cos kcr 1 - P2 1 p2 T t 0 1 2 . 27 1 The derivation and expressions for higher values of n are in Trimbur 2006 . For very short term forecasting transitory fluctuations may be captured by a local linear trend. However it is usually better to separate out such movements by including a stochastic cycle. Combining the components in an additive way that is yt Pt K 8t t 1 . T 28 provides the usual basis for trend-cycle decompositions. The cycle may be regarded as measuring the output gap. Extracted higher order cycles tend to be smoother with more noise consigned to the irregular. The cyclical trend model incorporates the cycle into the slope by moving it from 28 to the equation for the level pt pt-1 K-1 Pt-1 nt. 29 The damped trend is a special case corresponding to kc 0. . Forecasting components A UC model not only yields forecasts of the series itself it also provides forecasts for the components and their MSEs. US GDP. A trend plus cycle model 28 was fitted to the logarithm of quarterly seasonally adjusted real per capita US GDP using STAMP. Figure 4 shows the forecasts for the series itself with one RMSE on either side while Figures 5 and 6 show the forecasts for the logarithms of the cycle and the trend together with their smoothed 345 Ch. 7 Forecasting with Unobserved Components Time Series Models Figure 4. US GDP
