This paper provides conditions for identification of functionals in nonparametric simultaneous equations models with nonadditive unobservable random terms. The conditions are derived from a characterization of observational equivalence between models. We show that, in the models considered, observational equivalence can be characterized by a restriction on the rank of a matrix. The use of the new results is exemplified by deriving previously known results about identification in parametric and nonparametric models as well as new results. A stylized method for analyzing identification, which is useful in some situations, is also presented.

This paper is concerned with inference about a function g that is identified by a conditional moment restriction involving instrumental variables. The paper presents a test of the hypothesis that g belongs to a finite-dimensional parametric family against a nonparametric alternative. The test does not require nonparametric estimation of g and is not subject to the ill-posed inverse problem of nonparametric instrumental variables estimation. Under mild conditions, the test is consistent against any alternative model. In large samples, its power is arbitrarily close to 1 uniformly over a class of alternatives whose distance from the null hypothesis is O(n−1/2), where n is the sample size. In Monte Carlo simulations, the finite-sample power of the new test exceeds that of existing tests.

We present estimators for nonparametric functions that are nonadditive in unobservable random terms. The distributions of the unobservable random terms are assumed to be unknown. We show that when a nonadditive, nonparametric function is strictly monotone in an unobservable random term, and it satisfies some other properties that may be implied by economic theory, such as homogeneity of degree one or separability, the function and the distribution of the unobservable random term are identified. We also present convenient normalizations, to use when the properties of the function, other than strict monotonicity in the unobservable random term, are unknown. The estimators for the nonparametric function and for the distribution of the unobservable random term are shown to be consistent and asymptotically normal. We extend the results to functions that depend on a multivariate random term. The results of a limited simulation study are presented.

The block bootstrap is the best known bootstrap method for time-series data when the analyst does not have a parametric model that reduces the data generation process to simple random sampling. However, the errors made by the block bootstrap converge to zero only slightly faster than those made by first-order asymptotic approximations. This paper describes a bootstrap procedure for data that are generated by a Markov process or a process that can be approximated by a Markov process with sufficient accuracy. The procedure is based on estimating the Markov transition density nonparametrically. Bootstrap samples are obtained by sampling the process implied by the estimated transition density. Conditions are given under which the errors made by the Markov bootstrap converge to zero more rapidly than those made by the block bootstrap.

The proportional hazard model with unobserved heterogeneity gives the hazard function of a random variable conditional on covariates and a second random variable representing unobserved heterogeneity. This paper shows how to estimate the baseline hazard function and the distribution of the unobserved heterogeneity nonparametrically. The baseline hazard function and heterogeneity distribution are assumed to satisfy smoothness conditions but are not assumed to belong to known, finite-dimensional, parametric families. Existing estimators assume that the baseline hazard function or heterogeneity distribution belongs to a known parametric family. Thus, the estimators presented here are more general than existing ones.

This paper is about the economic theory of biodiversity preservation. A cost-effectiveness methodology is constructed, which results in a ranking criterion sufficiently operational to be useful in suggesting what to look at when determining actual conservation priorities. The formula is firmly rooted in a mathematically rigorous optimization framework, so that its theoretical underpinnings are clear. The underlying model, called the 'Noah's Ark Problem, ' is intended to be a kind of canonical form that hones down to its analytical essence the problem of best preserving diversity under a limited budget constraint.

The least-absolute-deviations (LAD) estimator for a median-regression model does not satisfy the standard conditions for obtaining asymptotic refinements through use of the bootstrap because the LAD objective function is not smooth. This paper overcomes this problem by smoothing the objective function. The smoothed estimator is asymptotically equivalent to the standard LAD estimator. With bootstrap critical values, the rejection probabilities of symmetrical t and X 2 tests based on the smoothed estimator are correct through O(n -γ ) under the null hypothesis, where γ<1 but can be arbitrarily close to 1. In contrast, first-order asymptotic approximations make errors of size O(n -γ ). These results also hold for symmetrical t and X 2 tests for censored median regression models.

This paper shows how to estimate a model in which an unknown transformation of the dependent variable is a linear function of explanatory variables plus an unobserved random variable, U, whose distribution is unknown. The model nests many familiar parametric and semiparametric models, including models with Box-Cox transformed dependent variables and proportional hazards models with and without unobserved heterogeneity. The paper develops root-n consistent, asymptotically normal estimators of the transformation function, coefficients of the explanatory variables, and distribution of U. The results of Monte Carlo experiments indicate that the estimators work well in samples of size one hundred. Copyright 1996 by The Econometric Society.

Robin L. Lumsdaine, Consistency and Asymptotic Normality of the Quasi-Maximum Likelihood Estimator in IGARCH(1,1) and Covariance Stationary GARCH(1,1) Models, Econometrica, Vol. 64, No. 3 (May, 1996), pp. 575-596