We consider nonparametric estimation of a regression function that is identified by requiring a specified quantile of the regression “error” conditional on an instrumental variable to be zero. The resulting estimating equation is a nonlinear integral equation of the first kind, which generates an ill-posed inverse problem. The integral operator and distribution of the instrumental variable are unknown and must be estimated nonparametrically. We show that the estimator is mean-square consistent, derive its rate of convergence in probability, and give conditions under which this rate is optimal in a minimax sense. The results of Monte Carlo experiments show that the estimator behaves well in finite samples.

Tests based on generalized-method-of-moments estimators often have true levels that differ greatly from their nominal levels when asymptotic critical values are used. This paper gives conditions under which the bootstrap provides asymptotic refinements to the critical values of t tests and the test of overidentifying restrictions. Particular attention is given to the case of dependent data. It is shown that, with such data, the bootstrap must sample blocks of data and that the formulae for the bootstrap versions of the test statistics differ from the formulae that apply with the original data. Copyright 1996 by The Econometric Society.

We develop a new test of a parametric model of a conditional mean function against a nonparametric alternative. The test adapts to the unknown smoothness of the alternative model and is uniformly consistent against alternatives whose distance from the parametric model converges to zero at the fastest possible rate. This rate is slower than n[superscript -1/2]. Some existing tests have nontrivial power against restricted classes of alternatives whose distance from the parametric model decreases at the rate n[superscript -1/2]. There are, however, sequences of alternatives against which these tests are inconsistent and ours is consistent. As a consequence, there are alternative models for which the finite-sample power of our test greatly exceeds that of existing tests. This conclusion is illustrated by the results of some Monte Carlo experiments.

Robust estimation aims at developing point estimators that are not highly sensitive to errors in data. However, the population parameters of interest are not identified under the assumptions of robust estimation, so the rationale for point estimation is not apparent. This paper shows that, under error models used in robust estimation, unidentified population parameters can often be bounded. The bounds provide information that is not available in robust estimation. For example, it is possible to bound the population mean under contaminated sampling. It is argued that estimating the bounds is more natural than attempting point estimation of unidentified parameters. Copyright 1995 by The Econometric Society.

We present a method for consistent nonparametric estimation of a demand function with nonseparable unobserved taste heterogeneity subject to the shape restriction implied by the Slutsky inequality. We use the method to estimate gasoline demand in the United States. The results reveal differences in behavior between heavy and moderate gasoline users. They also reveal variation in the responsiveness of demand to plausible changes in prices across the income distribution. We extend our estimation method to permit endogeneity of prices. The empirical results illustrate the improvements in finite-sample performance of a nonparametric estimator from imposing shape restrictions based on economic theory.

Berkson errors are commonplace in empirical microeconomics. In consumer demand, this form of measurement error occurs when the price an individual pays is measured by the (weighted) average price paid by individuals in a group (e.g., a county) rather than the true transaction price. We show the importance of Berkson errors for demand estimation with nonseparable unobserved heterogeneity. We develop a consistent estimator using external information on the true price distribution. Examining gasoline demand in the United States, we document substantial within-market price variability. Accounting for Berkson errors is quantitatively important. Imposing the Slutsky shape constraint reduces sensitivity to Berkson errors.