Lemma 5.1. Consider two random variables T ∼ χ 2 $_{p$_{1}$}$/α$_{1}$ and S ∼ χ 2 $_{p$_{2}$}$/α$_{2}$ , where p$_{2}$ > p$_{1}$, α$_{2}$ > α$_{1}$ > 0 . Then there exists y$_{0}$ such that for y > y$_{0}$ we have This makes it possible to rely only on p and q in indicating when standard critical values provide a conservative test. When there may be a singularity with α ≥ 1 the critical value coming from the standard test will at some level result in a conservative Wald test; the question is whether this holds for conventional test levels. Abstracting from the specific form of restrictions the answer depends on α, p and ˙ q ; the higher the α and the closer together p and q, the easier to obtain conserva- tive asymptotic tests at conventional levels. Since p and q are given by the restrictions, all that is required is to establish α. Comparing the values of p.d.f.$_{χ}$ 2 $_{q}$( y$_{.}$$_{05}$ ) with p.d.f.$_{χ}$ 2 $_{p}$/ (1+ a ) 2 ( y$_{.}$$_{05}$ ) where y$_{.}$$_{05}$ is the critical value for χ 2 q at . 05 level we determine for which max p we get a smaller value for the second p.d.f. ; because of monotonicity in the tail this indicates smaller probability and a conservative test. For one restriction the standard test based on χ 2 1 critical value is conser- vative at . 05 level for p ≤ 6 but may not be not for p = 7 . At . 01 level this test is conservative for p ≤ 10 , but may not be for p = 11 . To show this only a computation of the critical values for χ 2 1 and for overall bound 1 $_{4}$χ 2 p is 32 p.d.f.$_{S}$ ( y ) < p.d.f.$_{T}$ ( y ) .