Handbook of Economic Forecasting part 18. 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 | 144 A. Timmermann nonparametric methods. Although individual forecasting models will be biased and may omit important variables this bias can more than be compensated for by reductions in parameter estimation error in cases where the number of relevant predictor variables is much greater than N the number of . Linear forecast combinations under MSE loss While in general there is no closed-form solution to 1 one can get analytical results by imposing distributional restrictions or restrictions on the loss function. Unless the mapping C from yt h t to yt h is modeled nonparametrically optimality results for forecast combination must be established within families of parametric combination schemes of the form yct h t C yt h t h t . The general class of combination schemes in 1 comprises nonlinear as well as time-varying combination methods. We shall return to these but for now concentrate on the family of linear combinations Wlt C Wt which are more commonly To this end we choose weights t h t t h t 1 t h t N to produce a combined forecast of the form yt h t t h t y t h t- 6 Under MSE loss the combination weights are easy to characterize in population and only depend on the first two moments of the joint distribution of yt h and yt h t 2 U W yt h I I dy h t j I ayt h t yyt V . 7 yt h t flyt h t yyt h t yyt h t Minimizing E e2 h t E yt h - dt h tyt h t 2 we have Mt h t argmin Ayt h t Mt h t yt h t yt h t t h t iWt Mi h t Eyyt h tMt h t 2a i li yyt h t - This yields the first order condition d E et2 h t d t h t Ayt h t t h t yt h t fiyt h t yyt h t t h t yyt h t 0. Assuming that Tyyt h t is invertible this has the solution t h t fiyt h t P-yt h t yyt h t fiyt h tdyt h t a yyt h t . 8 4 When the true forecasting model mapping Ff to yt h is infinite-dimensional the model that optimally balances bias and variance may depend on the sample size with a dimension that grows as the sample size increases. 5 This of course does not rule out that the estimated
