Conditional Quantile Estimators: A Small Sample Theory (original) (raw)

Bias correction for quantile regression estimators

arXiv (Cornell University), 2020

We study the bias of classical quantile regression and instrumental variable quantile regression estimators. While being asymptotically first-order unbiased, these estimators can have non-negligible second-order biases. We derive a higher-order stochastic expansion of these estimators using empirical process theory. Based on this expansion, we derive an explicit formula for the second-order bias and propose a feasible bias correction procedure that uses finite-difference estimators of the bias components. The proposed bias correction method performs well in simulations. We provide an empirical illustration using Engel's classical data on household expenditure.

Inference Approaches for Instrumental Variable Quantile Regression

We consider asymptotic and finite sample confidence bounds in instrumental variables quantile regressions of wages on schooling with relatively weak instruments. We find practically important differences between the asymptotic and finite sample interval estimates.

Finite Sample Inference for Quantile Regression Models

Under minimal assumptions finite sample confidence bands for quantile regression models can be constructed. These confidence bands are based on the "conditional pivotal property" of estimating equations that quantile regression methods aim to solve and will provide valid finite sample inference for both linear and nonlinear quantile models regardless of whether the covariates are endogenous or exogenous. The confidence regions can be computed using MCMC, and confidence bounds for single parameters of interest can be computed through a simple combination of optimization and search algorithms. We illustrate the finite sample procedure through a brief simulation study and two empirical examples: estimating a heterogeneous demand elasticity and estimating heterogeneous returns to schooling. In all cases, we find pronounced differences between confidence regions formed using the usual asymptotics and confidence regions formed using the finite sample procedure in cases where the usual asymptotics are suspect, such as inference about tail quantiles or inference when identification is partial or weak. The evidence strongly suggests that the finite sample methods may usefully complement existing inference methods for quantile regression when the standard assumptions fail or are suspect.

Bias Transmission and Variance Reduction in Two-Stage Quantile Regression

2012

In this paper, we propose a variance reduction method for quantile regressions with endogeneity problems. First, we derive the asymptotic distribution of two-stage quantile estimators based on the fitted-value approach under very general conditions on both error terms and exogenous variables. Second, we exhibit a bias transmission property derived from the asymptotic representation of our estimator. Third, using a reformulation of the dependent variable, we improve the efficiency of the two-stage quantile estimators by exploiting a trade-off between an asymptotic bias confined to the intercept estimator and a reduction of the variance of the slope estimator. Monte Carlo simulation results show the excellent performance of our approach. In particular, by combining quantile regressions with first-stage trimmed least-squares estimators, we obtain more accurate slope estimates than 2SLS, 2SLAD and other estimators for a broad range of distributions.

Efficient Semiparametric Seemingly Unrelated Quantile Regression Estimation

Econometric Theory, 2009

We propose an efficient semiparametric estimator for the coefficients of a multivariate linear regression model -with a conditional quantile restriction for each equation -in which the conditional distributions of errors given regressors are unknown. The procedure can be used to estimate multiple conditional quantiles of the same regression relationship. The proposed estimator is asymptotically as efficient as if the true optimal instruments were known. Simulation results suggest that the estimation procedure works well in practice and dominates an equation-by-equation efficiency correction if the errors are dependent conditional on the regressors.

Nonparametric Instrumental Variables Estimation of a Quantile Regression Model

Econometrica, 2007

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. JEL Codes: C13, C31

Quantile Regression Estimates for a Class of Linear and Partially Linear Errors-in-Variables Models

1997

We consider the problem of estimating quantile regression coefficients in errors-in-variables models. When the error variables for both the response and the manifest variables have a joint distribution that is spherically symmetric but otherwise unknown, the regression quantile estimates based on orthogonal residuals are shown to be consistent and asymptotically normal. We also extend the work to partially linear models when the response is related to some additional covariate.

A simple approach to quantile regression for panel data

This paper provides a set of sufficient conditions that point identify a quantile regression model with fixed effects. It also proposes a simple transformation of the data that gets rid of the fixed effects under the assumption that these effects are location shifters. The new estimator is consistent and asymptotically normal as both n and T grow. Journal The Econometrics A simple approach to quantile regression for panel data 369 2. THE MODEL Consider the following model Y it = X it θ (U it ) + α i , t = 1, . . . , T , i = 1, . . . , n,

Semiparametric Efficient Estimation of Partially Linear Quantile Regression Models

2005

Lee (2003) develops a n-consistent estimator of the parametric component of a partially linear quantile regression model, which is used to obtain his one-step semiparametric efficient estimator. As a result, how well the efficient estimator performs depends on the quality of the initial n-consistent estimator. In this paper, we aim to improve the small sample performance of the one-step efficient