Handbook of Economic Forecasting part 31. 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 | 274 V Corradi and . Swanson NV2 i E g yt - 01 1 - 01 -1 E g xt - 0 - 0 -1 V0 01 and ii E g yt - 02 1 - 02 2Xt-1 - 02 3w Z -1 Y inf E g yt - 0- y - t7 y yt-1 - 0J y w Z -1 Y 1 2 1 2 2 1 2 3 for 0 0 y . NV3 T R P and as T x R n with 0 n x . NV4 For any t s V i j k 1 2 and for A rc i e sup g 0 w Zt-1 Y V0g s 0 w Zs-1 Y 4 A 0xyxy e0xTxr 3 where V0k - denotes the kth element of the derivative of its argument with respect to 0 ii e sup y ylegt 0 Jjgs 0 4 a V0 60 7 and iii e sup gi 0 w Zt-1 Y V0 yjgs 0 . 0xyeQxP 7 Assumptions MD1-MD4 are used in Section . MD1 yt Xt with yt scalar and Xt an RZ -valued 0 Z vector is a strictly stationary and absolutely regular -mixing process with size -4 4 y y y 0. MD2 i 0i is uniquely identified . E ln fi yt Zl-1 0 E ln fi yt Zl-1 0i for any 0i 0 ii ln fi is twice continuously differentiable on the interior of i for i 1 . m and for i a compact subset of Re l iii the elements of V0i ln fi and V . ln fi are p-dominated on i with p 2 2 y where y is the same positive constant as defined in Assumption A1 and iv E -Vj. ln fi 0i is positive definite uniformly on i. MD3 T R P and as T P R n withO n . MD4 i Fi ulZt 0i is continuously differentiable on the interior of i and V0i Fi u Z 0 is 2r-dominated on i uniformly in u r 2 i 1 . m 31 and ii let 1 7. . . Vkk u plim Vari V 1 yt 1 u - Fi u Z 01t - u T T t s - 1 yt 1 u - Fk u Z 0kf 2 - p2 u k 2 . m 31 We require that for j 1 . pi E V0 Fi u Zt 0i j Dt u with supt supue E Dt u 2r to. Ch. 5 Predictive Density Evaluation 275 define analogous covariance terms vj k u j k 2 . m and assume that vj k u is positive semi-definite uniformly in u. Appendix B Proofs Proof of Proposition . For brevity we just consider the case of recursive estimation. The case of rolling estimation schemes can be treated in an analogous way. 1 z W rec E 0 F C Z 6t rec r - r t R 1 1 . 1 Ft yt Zt-1 9o F F-1 r Zt-1 9t rec Zt-1 0o Z P t R 1 - r 1 T z - E 1 Ft yt Z-X 9o F F-1 r Z-1 9t Zt-1 9o Z P t 1 - F F-1 r Z-1 .
