ABSTRACT
As we argued in Chapter 1, most of the dynamic economic models based on optimizing behavior are de…ned in terms of a set of conditional moment restrictions of the form E[h(y; x; 0)jz] = 0:However, the law of iterated expectations is often applied to the population conditional moment restrictions to obtain an unconditional moment restriction of the form E[ (z)h(y; x; 0)] = 0, where (z) is usually chosen in an arbitrary manner or is restricted by data availability, and this unconditional moment restriction is subsequently used for GMM estimation. If (z) is not selected using optimality considerations, such as attaining the lower asymptotic variance bound, this approach will result in e¢ ciency losses. As a consequence, a substantial research e¤ort has been directed towards …nding and operationalizing the matrix of optimal instruments (z) so that the asymptotic variance of the method of moments estimator attains the semiparametric e¢ ciency bound. The …rst part of this chapter is devoted to derivation of optimal instruments and their operationalization for both linear and nonlinear models. In addition to e¢ ciency considerations, exploiting fully the information in
conditional moment restrictions may also have important implications for the consistency of the estimator. This observation has spurred a new interest in developing parameter estimators in models de…ned by conditional moment restrictions that are consistent and asymptotically e¢ cient. Some of the estimators are based on feasible procedures for estimating explicitly the matrix of optimal instruments for a …nite number or a continuum of moment conditions. Other methods operate directly on the conditional moment restriction and achieve e¢ ciency by implicit estimation of the optimal instruments. While most of these approaches are asymptotically equivalent, they vary substantially in terms of their initial motivation, …nite sample properties and the degree of numerical complexity and computation. These recent approaches are discussed in the second part of this chapter.
