Search

Quantile regression (QR) is a principal regression method for analyzing the impact of covariates on outcomes. The impact is described by the conditional quantile function and its functionals. In this paper we develop the nonparametric QR-series framework, covering many regressors as a special case, for performing inference on the entire conditional quantile function and its linear functionals. In this framework, we approximate the entire conditional quantile function by a linear combination of series terms with quantile-specific coefficients and estimate the function-valued coefficients from the data. We develop large sample theory for the QR-series coefficient process, namely we obtain uniform strong approximations to the QR-series coefficient process by conditionally pivotal and Gaussian processes. Based on these two strong approximations, or couplings, we develop four resampling methods (pivotal, gradient bootstrap, Gaussian, and weighted bootstrap) that can be used for inference on the entire QR-series coefficient function. We apply these results to obtain estimation and inference methods for linear functionals of the conditional quantile function, such as the conditional quantile function itself, its partial derivatives, average partial derivatives, and conditional average partial derivatives. Specifically, we obtain uniform rates of convergence and show how to use the four resampling methods mentioned above for inference on the functionals. All of the above results are for function-valued parameters, holding uniformly in both the quantile index and the covariate value, and covering the pointwise case as a by-product. We demonstrate the practical utility of these results with an empirical example, where we estimate the price elasticity function and test the Slutsky condition of the individual demand for gasoline, as indexed by the individual unobserved propensity for gasoline consumption.

In this paper, we consider estimation of general modern moment-condition problems in econometrics in a data-rich environment where there may be many more control variables available than there are observations. The framework we consider allows for a continuum of target parameters and for Lasso-type or Post-Lasso type methods to be used as estimators of a continuum of highdimensional nuisance functions. As an important leading example of this environment, we first provide detailed results on estimation and inference for relevant treatment effects, such as local average and quantile treatment effects. The setting we work in is designed expressly to handle many control variables, endogenous receipt of treatment, heterogeneous treatment effects, and possibly function-valued outcomes. To make informative inference possible, we assume that key reduced form predictive relationships are approximately sparse. That is, we require that the relationship between the control variables and the outcome, treatment status, and instrument status can be captured up to a small approximation error by a small number of the control variables whose identities are unknown to the researcher. This condition permits estimation and inference to proceed after datadriven selection of control variables. We provide conditions under which post-selection inference is uniformly valid across a wide-range of models and show that a key condition underlying the uniform validity of post-selection inference allowing for imperfect model selection is the use of orthogonal moment conditions. We illustrate the use of the proposed methods with an application to estimating the effect of 401(k) participation on accumulated assets. We generalize the results from the treatment effects setting to accommodate more general moment condition models in a second part of the paper. In particular, we establish a functional central limit theorem for robust estimators of a continuum of target parameters that holds uniformly in a wide range of data generating processes (dgp's), i.e. over dgp's P ϵ P where P include dgp's where perfect model selection is theoretically impossible. We prove that the use of orthogonal moment conditions is key to achieving uniform validity. We also establish a functional central limit theorem for the multiplier bootstrap that resamples first-order approximations to the proposed estimators that holds uniformly over P ϵ P. We propose a notion of differentiability, together with a functional delta method, that allows us to derive approximate distributions for smooth functionals of a continuum of target parameters that hold uniformly over P ϵ P and to establish the validity of the multiplier bootstrap for approximating these distributions uniformly over P ϵ P. Finally, we establish rate and consistency results for continua of Lasso or Post-Lasso type estimators for continua of (nuisance) regression functions and provide practical, theoretically justified choices for the penalty parameters used in these methods. Each of these results is new and of independent interest.

This paper investigates the impact of income on an individual's subjective self-assessment of own health. We employ recently developed methods in the non linear panel data literature to account for the endogeneity of income and the presence of individual heterogeneity. We examine a panel data set of individuals living in Australia and find no statistically significant relationship between income and health responses. Moreover, the evidence suggests that the variation in the individual specific effects, comprising both observed and unobserved time invariant factors, is primarily responsible for the variation across individuals' responses.

This paper gives identification and estimation results for marginal effects in nonlinear panel models. We find that linear fixed effects estimators are not consistent, due in part to marginal effects not being identified. We derive bounds for marginal effects and show that they can tighten rapidly as the number of time series observations grows. We also show in numerical calculations that the bounds may be very tight for small numbers of observations, suggesting they may be useful in practice. We propose two novel inference methods for parameters defined as solutions to linear and nonlinear programs such as marginal effects in multinomial choice models. We show that these methods produce uniformly valid confidence regions in large samples. We give an empirical illustration.

This paper introduces bias-corrected estimators for nonlinear panel data models with both time invariant and time varying heterogeneity. These include limited dependent variable models with both unobserved individual effects and endogenous explanatory variables, and sample selection models with unobserved individual effects. Our two-step approach first estimates the reduced form by fixed effects procedures to obtain estimates of the time variant heterogeneity underlying the endogeneity/selection bias. We then estimate the primary equation by fixed effects including an appropriately constructed control function from the reduced form estimates as an additional explanatory variable. The fixed effects approach in this second step captures the time invariant heterogeneity while the control function accounts for the time varying heterogeneity. Since either or both steps might employ nonlinear fixed effects procedures it is necessary to bias adjust the estimates due to the incidental parameters problem. This problem is exacerbated by the two step nature of the procedure. As these two step approaches are not covered in the existing literature we derive the appropriate correction thereby extending the use of large-T bias adjustments to an important class of models. Simulation evidence indicates our approach works well in finite samples and an empirical example illustrates the applicability of our estimator.