The examples show that even for the simplest restrictions the limit dis- tribution of the Wald statistic may be quite complex and far from standard. A number of applications require the Wald test of polynomial restriction functions where singularity could not be excluded and thus the non-standard features illustrated by the simple examples above may be present. 

Several applications involve test of one restriction, such as tests of deter- minants and other polynomial functions in coefficients in Gourieroux, Mon- fort, Renault (1988, 1993), Galbraith and Zinde-Walsh (1992), Gourieroux and Jasiak (2013). In tests of Granger noncausality in VARMA models by Boudjellaba, Dufour and Roy (1992,1994) several polynomial restrictions need to hold under the null, similarly in testing noncausality at various hori- zons in Dufour, Renault (1998) and Dufour, Pelletier and Renault (2005). 

## 4 Limit distribution of the Wald statistic 

The asymptotic behavior of the Wald statistic has not been generally exam- ined in the literature for full functional classes of nonlinear restrictions. Here we provide a characterization of the asymptotic distribution for restrictions given by polynomial functions. We shall work under the Assumptions 2.1a, 2.2,2.3, 2.4 and 2.5. 

Two approaches are possible. The Wald statistic can be represented as a ratio of two polynomial functions in random variables; such a representa- tion implicitely incorporates the information in the polynomial restrictions. 

