The framework considered and the test statistics are defined in Section 2. A number of examples are presented in Section 3; they illustrate the prop- erties of the Wald test in singular cases. In Section 4 we discuss some gen- eral algebraic and analytic features of matrices of polynomials and quadratic forms and derive the asymptotic distribution of the Wald statistic. Bounds are derived in Section 5. An adaptive strategy for determining the asymptotic distribution and the bounds is developed in Section 6. Proofs are presented in the Appendix. We consider testing q restrictions in a situation where an asymptotically non-singular estimator ˆ θ$_{T}$ is available for a p × 1 parameter of interest ¯ θ that satisfies the restrictions; q ≤ p. Assumption 2.1. The function g ( θ ) = [ g$_{1}$ ( θ ) , . . . , g$_{q}$ ( θ )] ′ is a continu- ously differentiable function from Θ to R $^{q}$, where Θ is an open subset of R p and q ≤ p. Assumption 2.1a. The function g ( θ ) = [ g$_{1}$ ( θ ) , . . . , g$_{q}$ ( θ )] ′ is such that each g$_{i}$ ( θ ) is a polynomial of order m in the components of θ, i.e. g$_{i}$ ( θ ) = m ∑ k =0 g$_{ik}$ ( θ ) , (1) g$_{ik}$ ( θ ) = i$_{1}$ + ··· + i$_{p}$ = k A$_{ik}$ ( i$_{1}$, . . . , i$_{p}$ ) θ i$_{1}$ 1 · · · θ i$_{p}$ p , k = 0 , 1 , . . . , m, i = 1 , . . . , q, (2) 6 2 Framework