GMM Estimation with Optimal Instruments

It has been shown in Section 6.3 that a GMM estimator attains the semiparametric efficiency bound (5.3.11) for given conditional moment functions if the instruments are chosen optimally. For the case of strict exogeneity, these optimal instruments were given in (5.3.12) as B(X)=D₀′Ω₀-1 , ignoring the transformation matrix F, with D₀ = E[∂ρ (Z,θ₀)/∂θ′|X] and Ω₀ = E[ρ (Z,θ₀)ρ (Z,θ₀)′|X].32 For the derivation of the lower efficiency bound it has been assumed that the conditional probability density function of Y depends on the parameters of interest, θ , and possibly on additional parameters, η . In the current section this dependence is explicitly taken into account by writing D(X,τ) and Ω(X,τ) with τ = (θ′,η′)′ , hence D₀ = D(X,τ₀) and Ω₀ = Ω(X,τ₀). Note that the conditional expectations are usually functions of X which justifies these expressions. Obviously, the optimal instruments are not available and have to be estimated in order to obtain a feasible GMM estimator. Two estimation strategies can be distinguished and will be discussed throughout this chapter. The first strategy, presented in this section, rests on substituting the unknown τ₀ with some consistent first step estimator $$ \hat \tau $$. Assuming that the functional form of D₀ and Ω₀ is known, estimators of the unknown conditional expectations follow from $$ D\left( X \right) = D\left( {X,\hat \tau } \right) $$ and $$ \hat \Omega \left( X \right) = \Omega \left( {X,\hat \tau } \right) $$. The second estimation strategy, which will be discussed throughout the Sections 8.2 − 8.5, rests on an application of nonparametric estimation techniques to obtain the estimators $$ \hat D\left( X \right) $$ and $$ \hat \Omega \left( X \right) $$ of D₀ and Ω₀.