Another approach is based on an explicit analysis of the restrictions and represents the limit distribution in a quadratic form; this representation per- mits simple derivation of conservative bounds. In this paper we focus on the second representation. The first subsection gives a few general results about matrices of polyno- mials; the second applies them to matrices related to the Jacobian matrix of the restrictions under test. The third subsection provides the limit distribu- tion for the Wald statistic for polynomial restrictions; this distribution is in general not pivotal and depends on ¯ θ . A polynomial function is either the zero polynomial, when it is identically zero (the coefficient on every monomial term is zero), or it is non-zero a.e.. Consider a q × p matrix G ( y ) of polynomials of variable y ∈ R $^{p}$. When q = p, we will say that the matrix G ( y ) is non-singular if its determinant is a non- zero polynomial. More generally, we will define the rank of the q × p matrix G ( y ) as the largest dimension of a square non-singular submatrix. This section considers q × p matrices G ( y ) , q ≤ p, of full row rank q (Assumption 2.5). We first note that, for any square q × q non-singular matrix S , SG ( y ) is also a matrix of polynomials of rank q : if ˘ G ( y ) is a q × q submatrix of G ( y ) with determinant det ( ˘ G ( y ) ) that is a non-zero polynomial, it is also true for the submatrix S ˘ G y ) of SG ( y ). 14 4.1 Matrices of polynomials