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Handbook of Economic Forecasting part 35

Handbook of Economic Forecasting part 35. 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 | 314 H. Lutkepohl and choosing the cointegrating rank for which the first null hypothesis cannot be rejected in this sequence. For our present purposes it is of interest that Johansen s LR tests can be justified even if a finite-order VAR process is fitted to an infinite order DGP as shown by Lutkepohl and Saikkonen 1999 . It is assumed in this case that the order of the fitted VAR process goes to infinity with the sample size and Lutkepohl and Saikkonen 1999 discuss the choice of the VAR order in this approach. Because the Kronecker indices are usually also unknown choosing the cointegrating rank of a VARMA process by fitting a long VAR process is an attractive approach which avoids knowledge of the VARMA structure at the stage where the cointegrating rank is determined. So far the theory for this procedure seems to be available for processes with nonzero mean term only and not for other deterministic terms such as linear trends. It seems likely however that extensions to more general processes are possible. An alternative way to proceed in determining the cointegrating rank of a VARMA process was proposed by Yap and Reinsel 1995 . They extended the likelihood ratio tests to VARMA processes under the assumption that an identified structure of A L and M L is known. For these tests the Kronecker indices or some other identifying structure has to be specified first. If the Kronecker indices are known already a lower bound for the cointegrating rank is also known see 3.6 . Hence in testing for the cointegrating rank only the sequence of null hypotheses H0 r q H0 r q 1 . H0 r K - 1 is of interest. Again the rank may be chosen as the smallest value for which H0 cannot be rejected. 3.4. Specifying the lag orders and Kronecker indices A number of proposals for choosing the Kronecker indices of ARMAg models were made see for example Hannan and Kavalieris 1984 Poskitt 1992 Nsiri and Roy 1992 and Lutkepohl and Poskitt 1996 for stationary processes and Lutkepohl and Claessen .