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 X-2-distribution of the Wald statistic under H-O. As an example, Weld tests for multi-step causality are considered. A variable y(t) is h-step causal for another variable x(t) if the information in y(t) helps improving the j-step forecasts of x(t) for some j = 1,2,...,h. if mon 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 X-2-distributions in general.