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## 6.2 Adaptively estimated asymptotic Wald statistic 

The case of one restriction is given by the following Lemma. 

With q = 1 represent each component { G ( θ ) }$_{i}$ of G ( θ ) as a polynomial of form (29) and consider the corresponding k$_{G}$$_{i}$ defined in (31) and the corre- sponding estimator, ˆ k$_{G}$$_{i}$. Then define ˆ k = min { ˆ k$_{G}$$_{i}$ } and the corresponding vector ¯ G$_{ˆ}$ $_{k}$( y ) with components given by the homogeneous polynomials of order ˆ k (some could be zero). 

Lemma 6.2. Under the Assumptions of Theorem 4.1 if (a) ˆ k = 0 , then with probability approaching 1 as T → ∞ there is no singularity, the distribution of the asymptotic statistic is standard and under the normality assumption is χ 2 $_{1}$; if (b) ˆ k > 0 the estimated asymptotic statistic is 

<img src='content_image/295762.jpg'>

and its distribution converges to the non-standard asymptotic distribution for the Wald statistic at ¯ θ as T → ∞ . 

The next theorem considers the general case of the Wald test for several restrictions. Denote by ˆ k$_{det}$ the estimator of k$_{P}$ applied to ¯ P $_{θ}$( θ ) in (27) that represents the polynomial det[ G ( θ ) G ( θ ) $^{′}$]; as T → ∞ Pr ( ˆ k$_{det}$ = k$_{det}$ ) → 1 . Set A = I then for every q × q submatrix ˆ G$_{l}$ ( y ) of ˆ G ( y ) the estimator of k$_{det}$$_{,l}$ for the corresponding determinant polynomial as T → ∞ equals the true k$_{det}$$_{,l}$ with probability approaching 1, and so does then the estimated value of ¯ a , as well as the estimated α$_{i}$ defined in proof of Lemma 4.3. It 

