Handbook of Economic Forecasting part 26. 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 | 224 V. Corradi and . Swanson where 0t rec and Qt roi are defined as in 19 and 20 respectively. Also define W reo r VP rec r - s C-1 s g t Wp ds and W rol r Vp roi r - g s C-1 s g s g t dVp roi r ds. Let BAI1 BAI2 and BAI4 be as given in Appendix A and modify BAI3 as follows BAI3 fit rec - 90 OP P-1 2 uniformly in BAI3 fit rol - 6o Op P-1 2 uniformly in Given this setup the following proposition holds. Proposition . Let BAI1 BAI2 BAI4 hold and assume that as T œ P R n with n œ. Then i If BAI3 hold under the null hypothesis in 1 suprS o 1 WP rec r -P suprS 0 1 W r . ii If BAI3 hold under the null hypothesis in 1 suprs o 1 -4-suprS 0 1 W r . Proof. See Appendix B. Turning now to an out-of-sample version of the Hong and Li test note that these tests can be defined as in Equations 8 - 11 above by replacing Ut in 8 with Ut rec 14 Note that BAI3 is satisfied under mild conditions provided P R 5 n with n to. In particular 1 - 1 P1 2 t P1 2pt - efi -E v2 E Veqfido . j 1 j 1 Now by uniform law of large numbers 7 Jj 1 v2qj 9t -1 7 I2j 1 E v2qj eo -1 -5 0. Let t Tr with 1 i 1 g r g 1. Then P1 2 Tr P1 1 Tr TFTEveqj eo Tveqj eo . Tr V T r fiT j 1 j 1 For any r 1 -p V Tr Veqi 9oo satisfies a CLT and so is Op T 1 2 and so O P-1 2 . As r is bounded r T J 1 away from zero and because of stochastic equicontinuity in r sup Jr1 - iveqj eoo OP P 1 2 . re 1 n - 1 1p 1 r S1 j 1 15 BAI3 is also satisfied under mild assumptions by the same arguments used in the footnote above. Ch. 5 Predictive Density Evaluation 225 and Ut rol respectively where Ut 1 rec Ft 1 yt 11Zt t rec and t 1 rol Ft 1 yt 11Z fy rol 23 with 0t rec and 9t rol defined as in 19 and 20 . Thus for the recursive estimation case it follows that T-1 t rec U1 U2 P - j -1 Kh u1 Ur reo K U2 Ur-j rec T R j 1 where n T R P. For the rolling estimation case it follows that T-1 t rol U1 U2 P - j - 2 Kh u1 UT rol Kh u2 UT-j rol T R j 1 Also define Mrec j j y Crec U1 U2 - 1 2 du1 du2 Mrol j ol 1 u2 - 1 2 du1 du2 JO
