Handbook of Economic Forecasting part 30. 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 | 264 V. Corradi and . Swanson where Zp ur 1 k Zp u r 1 k - RPpfiu - p2 u and where p2 u - p2 u is defined as in Equation 53 . Proposition From Proposition 7 in Corradi and Swanson 2006b . Let Assumptions MD1-MD4 hold. Also assume that as T x I x and that 774 0. Then as T P and R x for t 1 2 p a sup I Pp max Vp uuT 1 k v ve k 2 . m - p max Vp uur 1 k v e 0 k 2 . m where Vp j 1 k Vp j 1 k RP J J and where J j2 is defined as in Equation 53 . The above results suggest proceeding in the following manner. For brevity just consider the case of Zp t . For any bootstrap replication compute the bootstrap statistic Z p t . Perform B bootstrap replications B large and compute the quantiles of the empirical distribution of the B bootstrap statistics. Reject H0 if Zp t is greater than the 1 - a th-percentile. Otherwise do not reject. Now for all samples except a set with probability measure approaching zero Zp t has the same limiting distribution as the corresponding bootstrapped statistic when E p2 u - p u 0 Vk ensuring asymptotic size equal to a. On the other hand when one or more competitor models are strictly dominated by the benchmark the rule provides a test with asymptotic size between 0 and a. Under the alternative Zp t diverges to plus infinity while the corresponding bootstrap statistic has a well defined limiting distribution ensuring unit asymptotic power. From the above discussion we see that the bootstrap distribution provides correct asymptotic critical values only for the least favorable case under the null hypothesis that is when all competitor models are as good as the benchmark model. Whenmaxk 2 . m fU Ji u -pl uy p u du 0 but fU p21 u -J2 u u du 0 for some k then the bootstrap critical values lead to conservative inference. An alternative to our bootstrap critical values in this case is the construction of critical values based on subsampling see . Politis Romano and Wolf 1999 Chapter 3 . Heuristically construct T - 2bT statistics using subsamples of
