Handbook of Economic Forecasting part 34. 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 | 304 H. Lutkepohl and only if 1 L Lm-1 L1J . . . L 1 - j- -1 Ljm 0 1 QiL j 0 i 0 where Fj 0 for j 0 Lutkepohl 1987 Proposition . In other words the two forecasts are identical and there is no loss in forecast efficiency from using the aggregate directly if the MA operator of yt has the specified multiplicative structure upon multiplication by 1 L Lm-1 . This condition is also satisfied if yt has the purely seasonal structure . However in contrast to what was observed for stock variables the two predictors are generally not identical if the disaggregate process yt is generated by an MA process of order less than m. It is perhaps also interesting to note that if there are both stock and flow variables in one system then even if the underlying disaggregate process yt is the periodic process a forecast based on the disaggregate data may be better than directly forecasting the aggregate Lutkepohl 1987 pp. 177-178 . This result is interesting because for the purely seasonal process using the disaggregate process will not result in superior forecasts if a system consisting either of stock variables only or of flow variables only is considered. So far we have considered temporal aggregation of stationary processes. Most of the results can be generalized to Z 1 processes by considering the stationary process Ayt instead of the original process yt. Recall that forecasts for yt can then be obtained from those of Ayt. Moreover in this context it may be worth taking into account that in deriving some of the conditions for forecast equality the MA operator of the considered disaggregate process may have unit roots resulting from overdifferencing. A result which does not carry over to the I 1 case however is the equality of long horizon forecasts based on aggregate or disaggregate variables. The reason is again that optimal forecasts of I 1 variables do not settle down at zero eventually when h x . Clearly so far we have just discussed forecasting of known
