G ( θ ) = ∂g ∂θ ′ ( θ ) having full rank almost everywhere. Assumption 2.5. The matrix G ( θ ) has full row rank for almost all θ. Before we move to study the asymptotic distribution of W$_{T}$ = W$_{T}$ ( ˆ θ$_{T}$ , ˆ V$_{T}$ ) in general terms we provide examples which show that, indeed, the asymptotic distribution of W$_{T}$ is not regular. In particular, our examples illustrate non- invariance of the asymptotic distribution of the statistic to the form of the restriction and dependence (discontinuous) of the asymptotic distribution on the parameter value, ¯ θ ; we also show that the asymptotic distribution may have either thinner or thicker tails than the standard χ 2 q distribution and can even diverge to infinity under the null. To streamline exposition of the examples we assume that V = I. The following example illustrates lack of invariance of the asymptotic distribution. Example 3.1. Consider two equivalent forms for the null, g ( θ ) = 0 , one is (i) θ = 0 , the other (ii) θ 2 = 0 . Of course, the asymptotic distribution for the Wald test statistic in the case (i) under Assumption 2.3a is χ 2 $_{1}$. By contrast, for (ii) the value of W$_{T}$ = T ˆ θ 4 T 4 ˆ θ 2 T ; the limit distribution then is 1 $_{4}$χ 2 $_{1}$. Below for the multivariate ˆ θ$_{T}$ we suppress dependence of the components, ˆ θ$_{T}$$_{1}$, ..., ˆ θ$_{T p}$, on T. The next example is the one given by Andrews (1987); we develop it to 9