Handbook of Economic Forecasting part 25. 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 | 214 V. Corradi and . Swanson Theorem From Theorem 1 in Corradi and Swanson 2006a . Let CS1 CS2 i - ii and CS3 hold. Then i Under Ho Vit suprS o 1 I V1 r where V is a zero mean Gaussian process with covariance kernel K1 r r given by E V1 r V1 r K1 r r e 1 F j1 Zo 0o r - r s - X x 1 F y. Zs-1 0o r - r E eF x r Z-1 0o A 0o x E q1 0o qs 0o A 0o E VeF x r Z-1 0o S - X - 2E VeF x r Zt-1 0o A 0o x E 1 F y1 Z 0o r - r qs 0o s - with qs 0o V0 ln fs ys Zs-1 0o x r F-1 r Zt-1 0o and A 0o E 0qs 0o 0qs 0o -1. ii Under Ha there exists an o such that . Pr TT 2 V1t 1 Notice that the limiting distribution is a zero mean Gaussian process with a covariance kernel that reflects both dynamic misspecification as well as the contribution of parameter estimation error. Thus the limiting distribution is not nuisance parameter free and so critical values cannot be tabulated. Corradi and Swanson 2oo6a also suggest another Kolmogorov test which is no longer based on the probability integral transformation but can be seen as an extension of the conditional Kolmogorov CK test of Andrews 1997 to the case of time series data and possible dynamic misspecification. In a related important paper Li and Tkacz 2oo6 discuss an interesting approach to testing for correct specification of the conditional density which involves comparing a nonparametric kernel estimate of the conditional density with the density implied under the null hypothesis. As in Hong and Li 2oo3 and Hong 2oo1 the Tkacz and Li test is characterized by a nonparametric rate. Of further note is that Whang 2ooo 2oo1 also proposes a version of Andrews CK test for the correct specification although his focus is on conditional mean and not conditional distribution. Ch. 5 Predictive Density Evaluation 215 A conditional distribution version of the CK test is constructed by comparing the empirical joint distribution of yt and Z -1 with the product of the distribution of yt Z and the empirical CDF of Z -1. In practice the empirical joint
