Conditional moment restrictions can be combined through GMM estimation to construct more efficient semiparametric estimators. This paper is about attainable efficiency for such estimators. We define and use a moment tangent set, the directions of departure from the truth allowed by the moments, to characterize when the semiparametric efficiency bound can be attained. The efficiency condition is that the moment tangent set equals the model tangent set. We apply these results to transformed, censored, and truncated regression models, e.g., finding that the conditional moment restrictions from Powell's (1986) censored regression quantile estimators can be combined to approximate efficiency when the disturbance is independent of regressors. Copyright The Econometric Society 2004.

Knowledge of the asymptotic variance of an estimator is important for large sample inference, efficiency, and as a guide to the specification of regularity conditions.The purpose of this paper is the presentation of a general formula for the asymptotic variance of a semiparametric estimator.A particularly important feature of this formula is a way of accounting for the presence of nonparametric estimates of nuisance functions.The general form of an adjustment factor for nonparametric estimates is derived and analyzed.The usefulness of the formula is illustrated by deriving propositions on asymptotic equivalence for different nonparametric estimators of the same function, conditions for estimation of the nuisance functions to have no effect on the asymptotic variance, and the form of a correction term for the presence of linear function of a conditional expectation estimator, or other projection estimator (e.g.partially linear and/or additive nonparametric projections), and for a function of a density.Specific results cover a semiparametric random effects model for binary panel data, nonparametric consumer surplus, nonparametric prediction, and average derivatives.Regularity conditions are given for many of the propositions.These include primitive conditions for v'n-consistency, asymptotic normality, and consistency of an asymptotic variance estimator with series estimators of conditional expectations (or projections), in each of the examples.

This paper considers asymptotically efficient instrumental variables estimation of nonlinear models in an i.i.d. environment. The class of models includes nonlinear simultaneous equations models and other models of interest. A problem in constructing efficient instrumental variables estimators for such models is that the optimal instruments involve a conditional expectation, calculation of which can require functional form assumptions for the conditional distribution of endogenous variables, as well as integration. Nonparametric methods provide a way of avoiding this difficulty. Here it is shown that nonparametric estimates of the optimal instruments can give asymptotically efficient instrumental variables estimators. Also, ways of choosing the nonparametric estimate in applications are discussed. Two types of nonparametric estimates of the optimal instruments are considered. Each involves nonparametric regression, one by nearest neighbor and the other by series approximation. The finite sample properties of the estimators are considered in a small sampling experiment involving an endogenous dummy variable model.

[This paper examines the detection of misspecification in the context of maximum likelihood models. The power properties of specification tests based on moment conditions are explicitly considered. Tests of conditional moment restrictions are also discussed and are shown to be particularly useful when exogenous variables are present. The form of optimal conditional moment tests is presented. The general results are then applied to specification tests for probit.]

Nonlinear regression with measurement error is important for estimation from microeconomic data. One approach to identification and estimation is a causal model, in which the unobserved true variable is predicted by observable variables. This paper details the estimation of such a model using simulated moments and a flexible disturbance distribution. An estimator of the asymptotic variance is given for parametric models. Also, a semiparametric consistency result is given. The value of the estimator is demonstrated in a Monte Carlo study and an application to estimating Engel Curves.

In many economic models, objects of interest are functions which satisfy conditional moment restrictions. Economics does not restrict the functional form of these models, motivating nonparametric methods. In this paper we review identification results and describe a simple nonparametric instrumental variables (NPIV) estimator. We also consider a simple method of inference. In addition we show how the ability to uncover nonlinearities with conditional moment restrictions is related to the strength of the instruments. We point to applications where important nonlinearities can be found with NPIV and applications where they cannot.

Individual heterogeneity is an important source of variation in demand. Allow-ing for general heterogeneity is needed for correct welfare comparisons. We consider general heterogenous demand where preferences and linear budget sets are statis-tically independent. Only the marginal distribution of demand for each price and income is identified from cross-section data where only one price and income is observed for each individual. Thus, objects that depend on varying price and/or income for an indiviual are not generally identified, including average exact con-sumer surplus. We use bounds on income effects to derive relatively simple bounds on the average surplus, including for discrete/continous choice. We also sketch an approach to bounding surplus that does not use income effect bounds. We apply the results to gasoline demand. We find tight bounds for average surplus in this application but wider bounds for average deadweight loss.

Properties of instrumental variable estimators are sensitive to the choice of valid instruments, even in large cross-section applications. In this paper we address this problem by deriving simple mean-square error criteria that can be minimized to choose the instrument set. We develop these criteria for two-stage least squares (2SLS), limited information maximum likelihood (LIML), and a bias adjusted version of 2SLS (B2SLS). We give a theoretical derivation of the mean-square error and show optimality. In Monte Carlo experiments we find that the instrument choice generally yields an improvement in performance. Also, in the Angrist and Krueger (1991) returns to education application, when the instrument set is chosen in the way we consider, it turns out that both 2SLS and LIML give similar (large) returns to education.