GMM Estimation with Optimal Weights

Abstract

Having established the lower bound Λ_u of the GMM variance-covariance matrix for given unconditional moment functions in Section 5.2 which is attained by an optimal choice of the weight matrix ${\rm{\hat W}}$ such that W = V ^-1₀ , a consistent estimator ${{\rm{\hat V}}^{{\rm{ - 1}}}}$ of V ^-1₀ remains to be derived in order to obtain a feasible GMM estimator. A simple estimator for V₀ has been already introduced at the end of Section 3.2. By continuity of matrix inversion a consistent estimator of V ^-1₀ results from

$$ \hat V^{{\text{ - 1}}} = \left[ {\tfrac{1} {n}\sum\limits_{i = 1}^n \psi \left( {Z_i ,\hat \theta _1 } \right)\psi \left( {Z_i ,\hat \theta _1 } \right)^\prime } \right]^{ - 1} $$

((7.1.1))

with ${{\rm{\hat \theta }}_{\rm{1}}}$ being some consistent first step estimator. The usual procedure in applied work consists of computing ${{\rm{\hat \theta }}_{\rm{1}}}$ in a first step by minimizing the GMM objective function (2.1.6) for a weight matrix which is independent of $\hat \theta$ , e.g. the identity matrix, and obtaining the final GMM estimator ${{\rm{\hat \theta }}_{\rm{2}}}$ which reaches the lower bound of the asymptotic variance-covariance matrix in a second step using the weight matrix ${\rm{\hat W}}$ = ${{\rm{\hat V}}^{{\rm{ - 1}}}}$. A consistent estimator ${\hat \Lambda _{\rm{u}}}$ of the asymptotic variancecovariance matrix of the stabilizing transformation of ${{\rm{\hat \theta }}_{\rm{2}}}$ is obtained afterwards by substituting the elements of Λ _u = (G₀′V ^-1₀ G₀)^-1 with consistent plug-in estimators. The matrix V ^-1₀ can be estimated using either (7.1.1) or a corresponding expression evaluated at the final estimator ${{\rm{\hat \theta }}_{\rm{2}}}$. Newey and McFadden (1994, p. 2161) point out that there seems to be no evidence if any of these two methods creates efficiency advantages in small samples. A consistent estimator of G₀ was introduced in Section 3.2 and replaces the population moment by a sample moment.