tailieunhanh - Handbook of Economic Forecasting part 13

Handbook of Economic Forecasting part 13. 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 | 94 . Granger and MJ. Machina xF is set equal to xR so that the optimal action function for the objective function 22 takes the form a x g-1 x . This in turn implies that its canonical form U x xF is given by U x xF U x a xF f x - Jx Ja xFJ f x - L x xF . 23 . Implications of squared-error loss The most frequently used loss function in statistics is unquestionably the squared-error form Lsq xR xF k xr - xf 2 k 0 24 which is seen to satisfy the properties 8 . Theorem 1 thus implies the following result Corollary 1. For arbitrary squared-error function Lsq xR xf k xr - xf 2 with k 0 an objective function U - X x A R1 with strictly monotonic optimal action function a - will generate Lsq - as its loss function if and only if it takes the form U x a f x - k x - g a 2 25 for some function f - X R1 and monotonic function g - A X. Since utility or profit functions of the form 25 are not particularly standard it is worth describing some of their properties. One property which may or may not be realistic for a decision setting is that changes in the level of the choice variable a do not affect the curvature . the second and higher order derivatives of U x a with respect to x but only lead to uniform changes in the level and slope with respect to x -that is to say for any pair of values a1 a2 e A the difference U x a1 - U x a2 is an affine function of x .1 A more direct property of the form 25 is revealed by adopting the forecastequivalent labeling of the choice variable to obtain its canonical form U x xF from 20 which as we have seen specifies the level of utility or profit resulting from an actual realized value of x and the action that would have been optimal for a realized value of xF. Under this labeling the objective function implied by the squared-error loss function LSq xR xF is seen by 23 to take the form U x xf f x - Lsq x xf f x - k x - xf 2. 26 In terms of our earlier example this states that when a firm faces a realized output price of x its shortfall