Modified wald tests under nonregular conditions

Under nonregular conditions Wald tests are known to have incorrect size even asymptotically in part of the parameter space. Modifications are discussed which ensure an asymptotic η2-distribution of the Wald statistic under H0. As an example, Wald tests for multi-step causality are considered. A variable yt is h-step causal for another variable xt if the information in yt helps improving the j-step forecasts of xt for some j=1,2, …, h. If more than two variables are involved and are generated by a finite order vector autoregressive (VAR) process, this type of multi-step noncausality implies a set of highly nonlinear restrictions on the VAR coefficient matrices. For this type of nonlinear restrictions standard Wald tests fail to have limiting η2-distributions in general.