We define the Wald test statistic: when λ 2 T = T, this is If the distribution Q ( ¯ θ ) has a finite variance, we can assume without loss of generality that its variance is the identity matrix. However, when the rate of convergence λ$_{T}$ is not the standard T 1 / $^{2}$, a factor λ 2 T shows up instead of T . The statistic W$_{T}$ is not well defined when the estimator ˆ θ$_{T}$ falls into the set of singularity points at which ∂g ∂θ ′ ( ˆ θ$_{T}$ ) ˆ V$_{T}$ ∂g ′ ∂θ ( ˆ θ$_{T}$ ) is non-invertible (of rank less than q ). Andrews (1987) studied the case where [ ∂g ∂θ ′ ( ˆ θ$_{T}$ ) ˆ V$_{T}$ ∂g ′ ∂θ ( ˆ θ$_{T}$ ) ] − 1 is replaced by a generalized inverse ( e.g. , the Moore-Penrose inverse) and gave conditions under which the asymptotic distribution is chi-square. The main result there is that the asymptotic distribution of W$_{T}$ under H$_{0}$ is chi-square χ $^{2}$( r$_{0}$ ) with r$_{0}$ = rank [ ∂g ∂θ ′ ( θ )] when rank [ ∂g ∂θ ′ ( ˆ θ$_{T}$ )] converges to r$_{0}$ under H$_{0}$. This will be the case in particular when ∂g ∂θ ′ ( θ ) has rank r$_{0}$ in some open neighborhood of ¯ θ. Here we study situations where the matrix ∂g ∂θ ′ ( ˆ θ$_{T}$ ) ˆ V$_{T}$ ∂g ′ ∂θ ( ˆ θ$_{T}$ ) is non-singular in finite samples (with probability 1) but may converge to a singular ma- trix. Under Assumption 2.4 this non-singularity is equivalent to the matrix 8 W$_{T}$ = λ 2 $_{T}$g $^{′}$( ˆ θ$_{T}$ ) [ ∂g ∂θ $_{′}$( ˆ θ$_{T}$ ) ˆ V$_{T}$ ∂g ′ ∂θ ( ˆ θ$_{T}$ ) ] − 1 g ( ˆ θ$_{T}$ ) , (6) W$_{T}$ = T g $^{′}$( ˆ θ$_{T}$ ) [ ∂g ∂θ $_{′}$( ˆ θ$_{T}$ ) ˆ V$_{T}$ ∂g ′ ∂θ ( ˆ θ$_{T}$ ) ] − 1 g ( ˆ θ$_{T}$ ) . (7)