Handbook of Economic Forecasting part 28. 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 | 244 V. Corradi and . Swanson Proposition Parts i and iii are from Proposition in White 2000 . Let W1-W2 and WH hold. Then under Ho max Sp 1 k -RPE g ui t i - g uk t i -1 max S 1 k 37 k 2 . m k 2 . m where S S 1 2 . S 1 n is a zero mean Gaussian process with covariance kernel given by V with V a m x m matrix and i If parameter estimation error vanishes . if either P R goes to zero and or the same loss function is used for estimation and model evaluation g q where q is again the objective function then for i 1 . m 1 V v Sgigi and ii If parameter estimation error does not vanish . if P R 1 0 and g q then for i j 1 . m 1 V v Sgigi 2n 1At1C11 a 1 IU CuA i 4ni1AjC1iAj ii 2nSgiq1 Al 2nSgiqiA ii where . . . . Sgigi E E g u1 1 g ui 1 g u1 1 t g T TO C E E qi y1 s Zs 0 yetqt y1 s r Zs T 0 T TO Sgiq E E g u1 1 g ut 1 yeiqi y1 s T Zs T 0 t T TO Bi E 0iqi yt Zt 1 0 t 1 i E e g ui t 1y and n 1 n 1 ln 1 n . iii Under Ha Pr Sp s 1 1 as P 1 x . Proof. For the proof of part ii see Appendix B. Note that under the null the least favorable case arises when E g u1 t 1 g ukt 1 0 Vk. In this case the distribution of SP coincides with that of maxk 2 . m SP 1 k RPE g u1 t 1 g uk t 1 so that SP has the above limiting distribution which is a functional of a Gaussian process with a covariance kernel that reflects uncertainty due to dynamic misspecification and possibly to parameter estimation error. Additionally when all competitor models are worse than the benchmark the statistic diverges to minus infinity at rate VP. Finally when only some competitor models are worse than the benchmark the limiting distribution provides a conservative Ch. 5 Predictive Density Evaluation 245 test as SP will always be smaller than max Sp 1 k - d PE g ui t 1 - g ik t 1 k 2 . m asymptotically. Of course when HA holds the statistic diverges to plus infinity at rate vP. We now outline how to obtain valid asymptotic critical values for the limiting distribution on the right-hand side of 37 .
