hold in any part of the parameter space and using the corresponding critical values may lead to a severely oversized test. Example 3.3. Suppose that g ( θ ) = θ 2 $_{1}$+ ... + θ 2 $_{p}$; then G ( θ ) = [2 θ$_{1}$, ...., 2 θ$_{p}$ ] and Then the limit distribution is that of 1 4 ‖ Z ‖ 1 2 ; under normality this is 1 $_{4}$χ 2 $_{p}$; it is a pivotal distribution even though non-standard. If p is large enough, the χ 2 1 will not provide an upper bound. In the case of more than one restriction in addition to all the non-standard features that can arise for a single restriction it is also possible that the test statistic diverges even under H$_{0}$. Example 3.4. Suppose that q = p = 2 and g ( θ ) = [ θ 2 1 : θ$_{1}$θ 2 2 ] ′ . Then it follows that Then if (i) ¯ θ$_{1}$ = ¯ θ$_{2}$ = 0 the asymptotic distribution is 1 $_{4}$Z 2 1 + 1 $_{16}$Z 2 and thus under normality is a linear combination of two independent χ 2 1 and is bounded by 1 $_{4}$χ 2 $_{2}$. However, if (ii) ¯ θ$_{1}$ = 0 , but ¯ θ$_{2}$ = 0 the null still holds, but as T → ∞ the Wald statistic diverges to + ∞ . 12 G ( θ ) =    2 θ$_{1}$ 0 θ 2 2 2 θ$_{1}$θ$_{2}$    ; W$_{T}$ = T 4 ˆ θ 2 1 + ˆ θ 2 2 16 . W$_{T}$ = T ( Σ p i $_{=1}$ˆ θ 2 i ) 2 4Σ p i $_{=1}$ˆ θ 2 i .