James L. Powell

LEAST ABSOLUTE DEVIATIONS ESTIMATION FOR THE CENSORED REGRESSION MODEL

This paper proposes an alternative to maximum likelihood estimation of the parameters of the censored regression (or censored ‘Tobit’) model. The proposed estimator is a generalization of least absolute deviations estimation for the standard linear model, and, unlike estimation methods based on the assumption of normally distributed error terms, the estimator is consistent and asymptotically normal for a wide class of error distributions, and is also robust to heteroscedasticity. The paper gives the regularity conditions and proofs of these large-sample results, and proposes classes of consistent estimators of the asymptotic covariance matrix for both homoscedastic and heteroscedastic disturbances.

Censored regression quantiles

Abstract The object of this paper is to demonstrate how the LAD estimation method for the censored regression model can be extended to more general quantiles. In this paper, the form of the conditional quantiles for the censored regression models is heuristically derived and discussed. The resulting estimators of the regression coefficients, which include the censored LAD estimator as a special case, are shown to be consistent and asymptotically normally distributed under appropriately ‘translated’ versions of the corresponding assumptions for the former approach; consistent estimation of the asymptotic covariance matrix when the error terms are i.i.d. is also treated. The paper discussed how several quantile estimators might be combined to improve efficiency when the error terms are identically distributed, and how tests of homoskedasticity and symmetry of the error distribution can be constructed using particular differences of the estimated coefficients.

Semiparametric estimation of index coefficients

This paper gives a solution to the problem of estimating coefficients of index models, through the estimation of the density-weighted average derivative of a general regression function. A normalized version of the density-weighted average derivative can be estimated by certain linear instrumental variables coefficients. The estimators, based on sample analogies of the product moment representation of the average derivative, are constructed using nonparametric kernel estimators of the density of the regressors. Consistent estimators of the asymptotic variance-covariance matrices of the estimators are given, and a limited Monte Carlo simulation is used to study the practical performance of the procedures. Copyright 1989 by The Econometric Society.

Instrumental Variable Estimation of Nonparametric Models

In econometrics there are many occasions where knowledge of the structural relationship among dependent variables is required to answer questions of interest. This paper gives identification and estimation results for nonparametric conditional moment restrictions. We characterize identification of structural functions as completeness of certain conditional distributions, and give sufficient identification conditions for exponential families and discrete variables. We also give a consistent, nonparametric estimator of the structural function. The estimator is nonparametric two-stage least squares based on series approximation, which overcomes an ill-posed inverse problem by placing bounds on integrals of higher-order derivatives. Copyright The Econometric Society 2003.

Asymmetric Least Squares Estimation and Testing

This paper considers estimation and testing using location measures for regression m odels that are based on an asymmetric least-squares criterion functio n. These estimators have properties that are analogous to regression quantiles, but are easier to calculate, as are the corresponding test statistics. Asymmetric least-squares tests of homoskedasticity and s ymmetry compare quite favorably with other tests of these hypotheses in terms of asymptotic relative efficiency. Consequently, asymmetric least-squares estimation provides a convenient and relatively efficie nt method of characterizing the conditional distributi on of a dependent variable given some regressors. Copyright 1987 by The Econometric Society.

Semiparametric estimation of censored selection models with a nonparametric selection mechanism

Abstract In this paper, estimation of the coefficients in a ‘single-index selectivity bias’ model is considered under the assumption that the selection correction function depends on the conditional mean of some observable ‘selection’ variable. The estimation method follows a familiar ‘two-step’ strategy: the first step uses a nonparametric regression estimator for the selection variable, while the second step uses a weighted instrumental variables estimator for the coefficients in the equation of interest. The paper gives conditions under which the proposed estimator is root- n -consistent and asymptotically normal. The proposed method is applied to data on labor supply.

Nonparametric Estimation of Triangular Simultaneous Equations Models

This paper presents a simple two-step nonparametric estimator for a triangular simultaneous equation model. Our approach employs series approximations that exploit the additive structure of the model. The first step comprises the nonparametric estimation of the reduced form and the corresponding residuals. The second step is the estimation of the primary equation via nonparametric regression with the reduced form residuals included as a regressor. We derive consistency and asymptotic normality results for our estimator, including optimal convergence rates. Finally we present an empirical example, based on the relationship between the hourly wage rate and annual hours worked, which illustrates the utility of our approach.

Estimation of semiparametric models

A semiparametric model for observational data combines a parametric form for some component of the data generating process (usually the behavioral relation between the dependent and explanatory variables) with weak nonparametric restrictions on the remainder of the model (usually the distribution of the unobservable errors). This chapter surveys some of the recent literature on semiparametric methods, emphasizing microeconometric applications using limited dependent variable models. An introductory section defines semiparametric models more precisely and reviews the techniques used to derive the large-sample properties of the corresponding estimation methods. The next section describes a number of weak restrictions on error distributions -- conditional mean, conditional quantile, conditional symmetry, independence, and index restrictions -- and show how they can be used to derive identifying restrictions on the distributions of observables. This general discussion is followed by a survey of a number of specific estimators proposed for particular econometric models, and the chapter concludes with a brief account of applications of these methods in practice.

Nonlinear errors in variables : estimation of some Engel curves

Abstract The most common solution to the errors in variables problem for the linear regression model is the use of instrumental variable estimation. However, this methodology cannot be applied in the nonlinear regression framework. In this paper we develop consistent estimators for nonlinear regression specifications when errors in variables are present. We apply our methodology to estimation of Engel curves on household data. First, we find that the ‘Lesser-Working’ specification of budget shares regressed on the log of income or expenditure should be generalized to higher-order terms in log income. Also, we find that errors in variables in either reported income or expenditure should be accounted for. Lastly and perhaps most interesting, we find rather strong support for the Gorman rank restriction on the matrix of coefficients for the polynomial terms in income.

Nonparametric and Semiparametric Methods in Econometrics and Statistics

This collection of papers delivered at the Fifth International Symposium in Economic Theory and Econometrics in 1988 is devoted to the estimation and testing of models that impose relatively weak restrictions on the stochastic behaviour of data. Particularly in highly non-linear models, empirical results are very sensitive to the choice of the parametric form of the distribution of the observable variables, and often nonparametric and semiparametric models are a preferable alternative. Methods and applications that do not require string parametric assumptions for their validity, that are based on kernels and on series expansions, and methods for independent and dependent observations are investigated and developed in these essays by renowned econometricians.