content_image/326752.jpg content_image/326751.jpg content_image/326749.jpg
The Wald test statistic in (6) for θ = ˆ θ$_{T}$, λ = λ$_{T}$, Ω = ˆ V$_{T}$ and with y$_{T}$ = A ( ˆ θ$_{T}$ − ¯ θ ) can be written as 

<img src='content_image/326752.jpg'>

Consider g ¯ θ, $_{0}$( y ) for A$_{0}$ = V − 1 2 , and g ¯ θ, $_{1}$( y ) for some nonsingular A ; then the distribution of g ¯ θ, $_{1}$( V 1 2 AZ ) is the same as g ¯ θ, $_{0}$( Z ) . The distribution of G ¯ θ, $_{1}$( V 1 2 AZ ) is the same as G ¯ θ, $_{0}$( Z ) V − 1 2 A − $^{1}$, since we consider non-singular reparametrizations of continuous functions. Thus the limit distribution does not depend on A. Below we write g ¯ $_{θ}$, G ¯ θ for g ¯ θ, 0 and G ¯ θ, $_{0}$. 

Theorem 4.1. Under the Assumptions 2.1a, 2.2, 2.3 and 2.4 if (a) at ¯ θ the CLDR property holds for G ¯ $_{θ}$( y ) (for any non-singular A ) then the limit distribution of W$_{T}$ as T → ∞ is given by the distribution of 

<img src='content_image/326751.jpg'>

if (b) at ¯ θ the deficient rank property holds, then W$_{T}$ diverges to infinity as T → ∞ . 

Corollary 4.1. When the CLDR property holds the limit distribution can be represented as the distribution of 

This follows from the Euler formula (19) . 

<img src='content_image/326749.jpg'>

