Shakeeb Khan

Irregular Identification, Support Conditions, and Inverse Weight Estimation

In weighted moment condition models, we show a subtle link between identification and estimability that limits the practical usefulness of estimators based on these models. In particular, if it is necessary for (point) identification that the weights take arbitrarily large values, then the parameter of interest, though point identified, cannot be estimated at the regular (parametric) rate and is said to be irregularly identified. This rate depends on relative tail conditions and can be as slow in some examples as n−1/4. This nonstandard rate of convergence can lead to numerical instability and/or large standard errors. We examine two weighted model examples: (i) the binary response model under mean restriction introduced by Lewbel (1997) and further generalized to cover endogeneity and selection, where the estimator in this class of models is weighted by the density of a special regressor, and (ii) the treatment effect model under exogenous selection (Rosenbaum and Rubin (1983)), where the resulting estimator of the average treatment effect is one that is weighted by a variant of the propensity score. Without strong relative support conditions, these models, similar to well known “identified at infinity” models, lead to estimators that converge at slower than parametric rate, since essentially, to ensure point identification, one requires some variables to take values on sets with arbitrarily small probabilities, or thin sets. For the two models above, we derive some rates of convergence and propose that one conducts inference using rate adaptive procedures that are analogous to Andrews and Schafgans (1998) for the sample selection model.

Quantile regression under random censoring

Censored regression models have received a great deal of attention in both the theoretical and applied econometric literature. Most of the existing estimation procedures for either cross-sectional or panel data models are designed only for models with fixed censoring. In this paper, a new procedure for adapting these estimators designed for fixed censoring to models with random censoring is proposed. This procedure is then applied to the CLAD and quantile estimators of Powell (J. Econom. 25 (1984) 303, 32 (1986a) 143) to obtain an estimator of the coefficients under a mild conditional quantile restriction on the error term that is applicable to samples exhibiting fixed or random censoring. The resulting estimator is shown to have desirable asymptotic properties, and performs well in a small-scale simulation study.

Two-step estimation of semiparametric censored regression models

Root-n-consistent estimators of the regression coecients in the linear censored regression model under conditional quantile restrictions on the error terms were pro- posed by Powell (Journal of Econometrics 25 (1984) 303-325, 32 (1986a) 143-155). While those estimators have desirable asymptotic properties under weak regularity conditions, simulation studies have shown these estimators to exhibit a small sam- ple bias in the opposite direction of the least squares bias for censored data. This paper introduces two-step estimators for these models which minimize convex objec- tive functions, and are designed to overcome thisnite-sample bias. The paper gives regularity conditions under which the proposed two-step estimators are consistent and asymptotically normal; a Monte Carlo study compares thenite sample behavior of the proposed methods with their one-step counterparts. ? 2001 Elsevier Science S.A. All rights reserved.

Two-stage rank estimation of quantile index models ☆

Abstract This paper estimates a class of models which satisfy a monotonicity condition on the conditional quantile function of the response variable. This class includes as a special case the monotonic transformation model with the error term satisfying a conditional quantile restriction, thus allowing for very general forms of conditional heteroscedasticity. A two-stage approach is adopted to estimate the relevant parameters. In the first stage the conditional quantile function is estimated nonparametrically by the local polynomial estimator discussed in Chaudhuri (Journal of Multivariate Analysis 39 (1991a) 246–269; Annals of Statistics 19 (1991b) 760–777) and Cavanagh (1996, Preprint). In the second stage, the monotonicity of the quantile function is exploited to estimate the parameters of interest by maximizing a rank-based objective function. The proposed estimator is shown to have desirable asymptotic properties and can then also be used for dimensionality reduction or to estimate the unknown structural function in the context of a transformation model.

SEMIPARAMETRIC ESTIMATION OF A PARTIALLY LINEAR CENSORED REGRESSION MODEL

In this paper we propose an estimation procedure for a censored regression model where the latent regression function has a partially linear form. Based on a conditional quantile restriction, we estimate the model by a two stage procedure. The first stage nonparametrically estimates the conditional quantile function at in-sample and appropriate out-of-sample points, and the second stage involves a simple weighted least squares procedure. The proposed procedure is shown to have desirable asymptotic properties under regularity conditions that are standard in the literature. A small scale simulation study indicates that the estimator performs well in moderately sized samples.

Semiparametric estimation of a heteroskedastic sample selection model

This paper considers estimation of a sample selection model subject to conditional heteroskedasticity in both the selection and outcome equations. The form of heteroskedasticity allowed for in each equation is multiplicative, and each of the two scale functions is left unspecified. A three-step estimator for the parameters of interest in the outcome equation is proposed. The first two stages involve nonparametric estimation of the “propensity score†and the conditional interquartile range of the outcome equation, respectively. The third stage reweights the data so that the conditional expectation of the reweighted dependent variable is of a partially linear form, and the parameters of interest are estimated by an approach analogous to that adopted in Ahn and Powell (1993, Journal of Econometrics 58, 3–29). Under standard regularity conditions the proposed estimator is shown to be -consistent and asymptotically normal, and the form of its limiting covariance matrix is derived.We are grateful to B. HonorA©, R. Klein, E. Kyriazidou, L.-F. Lee, J. Powell, two anonymous referees, and the co-editor D. Andrews and also to seminar participants at Princeton, Queens, UCLA, and the University of Toronto for helpful comments. Chens research was supported by RGC grant HKUST 6070/01H from the Research Grants Council of Hong Kong.

Weighted and Two Stage Least Squares Estimation of Semiparametric Truncated Regression Models

This paper provides a root-n consistent, asymptotically normal weighted least squares estimator of the coefficients in a truncated regression model. The distribution of the errors is unknown and permits general forms of unknown heteroskedasticity. Also provided is an instrumental variables based two stage least squares estimator for this model, which can be used when some regressors are endogenous, mismeasured, or otherwise correlated with the errors. A simulation study indicates the new estimators perform well in finite samples. Our limiting distribution theory includes a new asymptotic trimming result addressing the boundary bias in first stage density estimation without knowledge of the support boundary.

Nonparametric Identification and Estimation of a Censored Location-Scale Regression Model

In this article we consider identification and estimation of a censored nonparametric location scale-model. We first show that in the case where the location function is strictly less than the (fixed) censoring point for all values in the support of the explanatory variables, the location function is not identified anywhere. In contrast, when the location function is greater or equal to the censoring point with positive probability, the location function is identified on the entire support, including the region where the location function is below the censoring point. In the latter case we propose a simple estimation procedure based on combining conditional quantile estimators for various higher quantiles. The new estimator is shown to converge at the optimal nonparametric rate with a limiting normal distribution. A small-scale simulation study indicates that the proposed estimation procedure performs well in finite samples. We also present an empirical illustration on unemployment insurance duration using administrative-level data from New Jersey.

Estimating censored regression models in the presence of nonparametric multiplicative heteroskedasticity

Abstract Powells (1984, Journal of Econometrics 25, 303–325) censored least absolute deviations (CLAD) estimator for the censored linear regression model has been regarded as a desirable alternative to maximum likelihood estimation methods due to its robustness to conditional heteroskedasticity and distributional misspecification of the error term. However, the CLAD estimation procedure has failed in certain empirical applications due to the restrictive nature of the ‘full rank’ condition it requires. This condition can be especially problematic when the data are heavily censored. In this paper we introduce estimation procedures for heteroskedastic censored linear regression models with a much weaker identification restriction than that required for the LCAD, and which are flexible enough to allow for various degrees of censoring. The new estimators are shown to have desirable asymptotic properties and perform well in small-scale simulation studies, and can thus be considered as viable alternatives for estimating censored regression models, especially for applications in which the CLAD fails.

Rates of convergence for estimating regression coefficients in heteroskedastic discrete response models

Abstract In this paper, we consider estimation of discrete response models exhibiting conditional heteroskedasticity of a multiplicative form, where the latent error term is assumed to be the product of an unknown scale function and a homoskedastic error term. It is first shown that for estimation of the slope coefficients in a binary choice model under this type of restriction, the semiparametric information bound is zero, even when the homoskedastic error term is parametrically specified. Hence, it is impossible to attain the parametric convergence rate for the parameters of interest. However, for ordered response models where the response variable can take at least three different values, the parameters of interest can be estimated at the parametric rate under the multiplicative heteroskedasticity assumption. Two estimation procedures are proposed. The first estimator, based on a parametric restriction on the homoskedastic component of the error term, is a two-step maximum likelihood estimators, where the unknown scale function is estimated nonparametrically in the first stage. The second procedure, which does not require the parametric restriction, estimates the parameters by a kernel weighted least-squares procedure. Under regularity conditions which are standard in the literature, both estimators are shown to be n -consistent and asymptotically normal.