We study linear quantile regression models when regressors and/or dependent variable are not directly observed but estimated in an initial first step and used in the second step quantile regression for estimating the quantile parameters. This general class of generated quantile regression (GQR) covers various statistical applications, for instance, estimation of endogenous quantile regression models and triangular structural equation models, and some new relevant applications are discussed. We study the asymptotic distribution of the two-step estimator, which is challenging because of the presence of generated covariates and/or dependent variable in the non-smooth quantile regression estimator. We employ techniques from empirical process theory to find uniform Bahadur expansion for the two step estimator, which is used to establish the asymptotic results. We illustrate the performance of the GQR estimator through simulations and an empirical application based on auctions.
We propose an econometric environment for structural break detection in nonstationary quantile predictive regressions. We establish the limit distributions for a class of Wald and fluctuation type statistics based on both the ordinary least squares estimator and the endogenous instrumental regression estimator proposed by Phillips and Magdalinos (2009a, Econometric Inference in the Vicinity of Unity. Working paper, Singapore Management University). Although the asymptotic distribution of these test statistics appears to depend on the chosen estimator, the IVX based tests are shown to be asympt…
This paper proposes a new framework to evaluate unconditional quantile effects (UQE) in a data combination model. The UQE measures the effect of a marginal counterfactual change in the unconditional distribution of a covariate on quantiles of the unconditional distribution of a target outcome. Under rank similarity and conditional independence assumptions, we provide a set of identification results for UQEs when the target covariate is continuously distributed and when it is discrete, respectively. Based on these identification results, we propose semiparametric estimators and establish their …
A new Stata command, ldvqreg, is developed to estimate quantile regression models for the cases of censored (with lower and/or upper censoring) and binary dependent variables. The estimators are implemented using a smoothed version of the quantile regression objective function. Simulation exercises show that it correctly estimates the parameters and it should be implemented instead of the available quantile regression methods when censoring is present. An empirical application to women's labor supply in Uruguay is considered.
The paper proposes a quantile-regression inference framework for first-price auctions with symmetric risk-neutral bidders under the independent private-value paradigm. It is first shown that a private-value quantile regression generates a quantile regression for the bids. The private-value quantile regression can be easily estimated from the bid quantile regression and its derivative with respect to the quantile level. This also allows to test for various specification or exogeneity null hypothesis using the observed bids in a simple way. A new local polynomial technique is proposed to estimat…
We propose a dynamic network quantile regression model to investigate the quantile connectedness using a predetermined network information. We extend the existing network quantile autoregression model of Zhu et al. (2019b) by explicitly allowing the contemporaneous network effects and controlling for the common factors across quantiles. To cope with the endogeneity issue due to simultaneous network spillovers, we adopt the instrumental variable quantile regression (IVQR) estimation and derive the consistency and asymptotic normality of the IVQR estimator using the near epoch dependence propert…
The instrumental variable quantile regression (IVQR) model (Chernozhukov and Hansen, 2005) is a popular tool for estimating causal quantile effects with endogenous covariates. However, estimation is complicated by the non-smoothness and non-convexity of the IVQR GMM objective function. This paper shows that the IVQR estimation problem can be decomposed into a set of conventional quantile regression sub-problems which are convex and can be solved efficiently. This reformulation leads to new identification results and to fast, easy to implement, and tuning-free estimators that do not require the…
We introduce the local composite quantile regression (LCQR) to causal inference in regression discontinuity (RD) designs. Kai et al. (2010) study the efficiency property of LCQR, while we show that its nice boundary performance translates to accurate estimation of treatment effects in RD under a variety of data generating processes. Moreover, we propose a bias-corrected and standard error-adjusted t-test for inference, which leads to confidence intervals with good coverage probabilities. A bandwidth selector is also discussed. For illustration, we conduct a simulation study and revisit a class…
Datasets from field experiments with covariate-adaptive randomizations (CARs) usually contain extra covariates in addition to the strata indicators. We propose to incorporate these additional covariates via auxiliary regressions in the estimation and inference of unconditional quantile treatment effects (QTEs) under CARs. We establish the consistency and limit distribution of the regression-adjusted QTE estimator and prove that the use of multiplier bootstrap inference is non-conservative under CARs. The auxiliary regression may be estimated parametrically, nonparametrically, or via regulariza…
Canay (2011)'s two-step estimator of quantile panel data models, due to its simple intuition and low computational cost, has been widely used in empirical studies in recent years. In this paper, we revisit the estimator of Canay (2011) and point out that in his asymptotic analysis the bias of his estimator due to the estimation of the fixed effects is mistakenly omitted, and that such omission will lead to invalid inference on the coefficients. To solve this problem, we propose a similar easy-to-implement estimator based on smoothed quantile regressions. The asymptotic distribution of the new …
This paper studies large $N$ and large $T$ conditional quantile panel data models with interactive fixed effects. We propose a nuclear norm penalized estimator of the coefficients on the covariates and the low-rank matrix formed by the fixed effects. The estimator solves a convex minimization problem, not requiring pre-estimation of the (number of the) fixed effects. It also allows the number of covariates to grow slowly with $N$ and $T$. We derive an error bound on the estimator that holds uniformly in quantile level. The order of the bound implies uniform consistency of the estimator and is …
This paper develops estimation and inference methods for censored quantile regression models with high-dimensional controls. The methods are based on the application of double/debiased machine learning (DML) framework to the censored quantile regression estimator of Buchinsky and Hahn (1998). I provide valid inference for low-dimensional parameters of interest in the presence of high-dimensional nuisance parameters when implementing machine learning estimators. The proposed estimator is shown to be consistent and asymptotically normal. The performance of the estimator with high-dimensional con…
There are many environments in econometrics which require nonseparable modeling of a structural disturbance. In a nonseparable model with endogenous regressors, key conditions are validity of instrumental variables and monotonicity of the model in a scalar unobservable variable. Under these conditions the nonseparable model is equivalent to an instrumental quantile regression model. A failure of the key conditions, however, makes instrumental quantile regression potentially inconsistent. This paper develops a methodology for testing the hypothesis whether the instrumental quantile regression m…
This paper develops bootstrap methods for practical statistical inference in panel data quantile regression models with fixed effects. We consider random-weighted bootstrap resampling and formally establish its validity for asymptotic inference. The bootstrap algorithm is simple to implement in practice by using a weighted quantile regression estimation for fixed effects panel data. We provide results under conditions that allow for temporal dependence of observations within individuals, thus encompassing a large class of possible empirical applications. Monte Carlo simulations provide numeric…
Regression quantiles have asymptotic variances that depend on the conditional densities of the response variable given regressors. This paper develops a new estimate of the asymptotic variance of regression quantiles that leads any resulting Wald-type test or confidence region to behave as well in large samples as its infeasible counterpart in which the true conditional response densities are embedded. We give explicit guidance on implementing the new variance estimator to control adaptively the size of any resulting Wald-type test. Monte Carlo evidence indicates the potential of our approach …
This chapter reviews the instrumental variable quantile regression model of Chernozhukov and Hansen (2005). We discuss the key conditions used for identification of structural quantile effects within this model which include the availability of instruments and a restriction on the ranks of structural disturbances. We outline several approaches to obtaining point estimates and performing statistical inference for model parameters. Finally, we point to possible directions for future research.
In this study, we investigate estimation and inference on a low-dimensional causal parameter in the presence of high-dimensional controls in an instrumental variable quantile regression. Our proposed econometric procedure builds on the Neyman-type orthogonal moment conditions of a previous study Chernozhukov, Hansen and Wuthrich (2018) and is thus relatively insensitive to the estimation of the nuisance parameters. The Monte Carlo experiments show that the estimator copes well with high-dimensional controls. We also apply the procedure to empirically reinvestigate the quantile treatment effect…
The presence of \b{eta}-convergence in European regions is an important issue to be analyzed. In this paper, we adopt a quantile regression approach in analyzing economic convergence. While previous work has performed quantile regression at the national level, we focus on 187 European NUTS2 regions for the period 1981-2009 and use spatial quantile regression to account for spatial dependence.
We propose to smooth the entire objective function, rather than only the check function, in a linear quantile regression context. Not only does the resulting smoothed quantile regression estimator yield a lower mean squared error and a more accurate Bahadur-Kiefer representation than the standard estimator, but it is also asymptotically differentiable. We exploit the latter to propose a quantile density estimator that does not suffer from the curse of dimensionality. This means estimating the conditional density function without worrying about the dimension of the covariate vector. It also all…
In ordinary quantile regression, quantiles of different order are estimated one at a time. An alternative approach, which is referred to as quantile regression coefficients modeling (QRCM), is to model quantile regression coefficients as parametric functions of the order of the quantile. In this paper, we describe how the QRCM paradigm can be applied to longitudinal data. We introduce a two-level quantile function, in which two different quantile regression models are used to describe the (conditional) distribution of the within-subject response and that of the individual effects. We propose a…
The R package quantreg.nonpar implements nonparametric quantile regression methods to estimate and make inference on partially linear quantile models. quantreg.nonpar obtains point estimates of the conditional quantile function and its derivatives based on series approximations to the nonparametric part of the model. It also provides pointwise and uniform confidence intervals over a region of covariate values and/or quantile indices for the same functions using analytical and resampling methods. This paper serves as an introduction to the package and displays basic functionality of the functio…
Extremal quantile regression, i.e. quantile regression applied to the tails of the conditional distribution, counts with an increasing number of economic and financial applications such as value-at-risk, production frontiers, determinants of low infant birth weights, and auction models. This chapter provides an overview of recent developments in the theory and empirics of extremal quantile regression. The advances in the theory have relied on the use of extreme value approximations to the law of the Koenker and Bassett (1978) quantile regression estimator. Extreme value laws not only have been…
Univariate normal regression models are statistical tools widely applied in many areas of economics. Nevertheless, income data have asymmetric behavior and are best modeled by non-normal distributions. The modeling of income plays an important role in determining workers' earnings, as well as being an important research topic in labor economics. Thus, the objective of this work is to propose parametric quantile regression models based on two important asymmetric income distributions, namely, Dagum and Singh-Maddala distributions. The proposed quantile models are based on reparameterizations of…
In this paper, we develop a new censored quantile instrumental variable (CQIV) estimator and describe its properties and computation. The CQIV estimator combines Powell (1986) censored quantile regression (CQR) to deal with censoring, with a control variable approach to incorporate endogenous regressors. The CQIV estimator is obtained in two stages that are non-additive in the unobservables. The first stage estimates a non-additive model with infinite dimensional parameters for the control variable, such as a quantile or distribution regression model. The second stage estimates a non-additive …
Quantile regression is an important tool for estimation of conditional quantiles of a response Y given a vector of covariates X. It can be used to measure the effect of covariates not only in the center of a distribution, but also in the upper and lower tails. This paper develops a theory of quantile regression in the tails. Specifically, it obtains the large sample properties of extremal (extreme order and intermediate order) quantile regression estimators for the linear quantile regression model with the tails restricted to the domain of minimum attraction and closed under tail equivalence a…
The widespread use of quantile regression methods depends crucially on the existence of fast algorithms. Despite numerous algorithmic improvements, the computation time is still non-negligible because researchers often estimate many quantile regressions and use the bootstrap for inference. We suggest two new fast algorithms for the estimation of a sequence of quantile regressions at many quantile indexes. The first algorithm applies the preprocessing idea of Portnoy and Koenker (1997) but exploits a previously estimated quantile regression to guess the sign of the residuals. This step allows f…
Department of Mathematics University of California San Diego（加利福尼亚大学圣地亚哥分校数学系）
Random forests are powerful non-parametric regression method but are severely limited in their usage in the presence of randomly censored observations, and naively applied can exhibit poor predictive performance due to the incurred biases. Based on a local adaptive representation of random forests, we develop its regression adjustment for randomly censored regression quantile models. Regression adjustment is based on new estimating equations that adapt to censoring and lead to quantile score whenever the data do not exhibit censoring. The proposed procedure named censored quantile regression f…
This work proposes new inference methods for a regression coefficient of interest in a (heterogeneous) quantile regression model. We consider a high-dimensional model where the number of regressors potentially exceeds the sample size but a subset of them suffice to construct a reasonable approximation to the conditional quantile function. The proposed methods are (explicitly or implicitly) based on orthogonal score functions that protect against moderate model selection mistakes, which are often inevitable in the approximately sparse model considered in the present paper. We establish the unif…
Nonlinear panel data models with fixed individual effects provide an important set of tools for describing microeconometric data. In a large class of such models (including probit, proportional hazard and quantile regression to name just a few) it is impossible to difference out individual effects, and inference is usually justified in a `large n large T' asymptotic framework. However, there is a considerable gap in the type of assumptions that are currently imposed in models with smooth score functions (such as probit, and proportional hazard) and quantile regression. In the present paper we …
