Key words :nonlinear restriction; deficient rank; singular covariance ma- trix; Wald test ; nonstandard asymptotic theory; bound. Journal of Eco- nomic Literature classification: C3. December 3, 2013 Victoria Zinde-Walsh § Jean-Marie Dufour † Eric Renault ‡ arXiv:1312.0569v1 [math.ST] 2 Dec 2013 Wald tests when restrictions are locally singular ∗ 1 $^{∗}$This work was supported by the Willam Dow Chair in Political Economy (McGill University), the Bank of Canada Research Fellowship, The Toulouse School of Economics Pierre-de-Fermat Chair of Excellence, A Guggenheim Fellowship, Conrad-Adenauer Fel- lowship from Alexander-von-Humboldt Foundation, the Canadian Network of Centres of Excellence program on Mathematics of Information Technology and Complex Systems, the Natural Sciences and Engineering Research Council of Canada, the Social Sciences and Humanities Research Council of Canada and the Fonds de recherche sur la soci´et´e et la culture (Qu´ebec). The authors also thank the research centres CIREQ and CIRANO for providing support and meeting space for the joint work. We thank Purevdorj Tuvaandorj for very useful comments. $^{†}$William Dow Professor of Economics, McGill University, Centre interuniversitaire de recherche en analyse des organisations (CIRANO) and Centre interuniversitaire de recherche en ´economie quatative (CIREQ). $^{‡}$Brown University $^{§}$McGill University and CIREQ