Handbook of Economic Forecasting part 89. 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 | 854 T. G. Andersen et al. the forecaster him herself as a diagnostic tool on the forecasting model. A more general discussion and review of the forecast evaluation literature can be found in Diebold and Lopez 1996 and Chapter 3 by West in this Handbook. Below we will first introduce a general loss function framework and then highlight the particular issues involved when forecasting volatility itself is the direct object of interest. We then discuss several other important forecasting situations where volatility dynamics are crucial including Value-at-Risk probability and density forecasting. . Point forecast evaluation from general loss functions Consider the general forecast loss function L yt 1 yt 1 t discussed in Section 2 in which the arguments are the univariate discrete-time real-valued stochastic variable yt 1 as well as its forecast yt 1 t. From the optimization problem solved by the optimal forecast yt 1 t must satisfy the generic first order condition Et dL yt i j f i t 9y 0. The partial derivative of the loss function - the term inside the conditional expectation - is sometimes referred to as the generalized forecast error. Realizations of this partial derivative should fluctuate unpredictably around zero directly in line with the standard optimality condition that regular forecasts display uncorrelated prediction errors. Specifically consider the situation in which we observe a sequence of out-of-sample forecasts and subsequent realizations yt 1 yt 1 t t2 1. A natural diagnostic on is then given by the simple regression version of the conditional expectation that is dL yt 1 yt 1 t -------- -------- a b xt et 1 dy where xt denotes a vector of candidate explanatory variables in the time t information set observed by the forecaster Ft and b is a vector of regression coefficients. An appropriately calibrated forecast should then have a b 0 which can be tested using standard t - and F -tests properly robustified to allow for heteroskedasticity
