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This paper proposes an asymptotically efficient method for estimating models with conditional moment restrictions. Our estimator generalizes the maximum empirical likelihood estimator (MELE) of Qin and Lawless (1994). Using a kernel smoothing method, we efficiently incorporate the information implied by the conditional moment restrictions into our empirical likelihood-based procedure. This yields a one-step estimator which avoids estimating optimal instruments. Our likelihood ratio-type statistic for parametric restrictions does not require the estimation of variance, and achieves asymptotic pivotalness implicitly. The estimation and testing procedures we propose are normalization invariant. Simulation results suggest that our new estimator works remarkably well in finite samples.

This paper develops and implements a nonparametric test of random utility models. The motivating application is to test the null hypothesis that a sample of cross‐sectional demand distributions was generated by a population of rational consumers. We test a necessary and sufficient condition for this that does not restrict unobserved heterogeneity or the number of goods. We also propose and implement a control function approach to account for endogenous expenditure. An econometric result of independent interest is a test for linear inequality constraints when these are represented as the vertices of a polyhedral cone rather than its faces. An empirical application to the U.K. Household Expenditure Survey illustrates computational feasibility of the method in demand problems with five goods.

We show by example that empirical likelihood and other commonly used tests for moment restrictions are unable to control the (exponential) rate at which the probability of a Type I error tends to zero. It follows that the optimality of empirical likelihood asserted in Kitamura (2001) does not hold without additional assumptions. Under stronger assumptions than those in Kitamura (2001), we establish the following optimality result: (i) empirical likelihood controls the rate at which the probability of a Type I error tends to zero and (ii) among all procedures for which the probability of a Type I error tends to zero at least as fast, empirical likelihood maximizes the rate at which probability of a Type II error tends to zero for "most" alternatives.This result further implies that empirical likelihood maximizes the rate at which probability of a Type II error tends to zero for all alternatives among a class of tests that satisfy a weaker criterion for their Type I error probabilities.