Denis Chetverikov

Conditional quantile processes based on series or many regressors

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 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 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.

IV QUANTILE REGRESSION FOR GROUP-LEVEL TREATMENTS, WITH AN APPLICATION TO THE DISTRIBUTIONAL EFFECTS OF TRADE

We present a methodology for estimating the distributional effects of an endogenous treatment that varies at the group level when there are group‐level unobservables, a quantile extension of Hausman and Taylor, 1981. Because of the presence of group‐level unobservables, standard quantile regression techniques are inconsistent in our setting even if the treatment is independent of unobservables. In contrast, our estimation technique is consistent as well as computationally simple, consisting of group‐by‐group quantile regression followed by two‐stage least squares. Using the Bahadur representation of quantile estimators, we derive weak conditions on the growth of the number of observations per group that are sufficient for consistency and asymptotic zero‐mean normality of our estimator. As in Hausman and Taylor, 1981, micro‐level covariates can be used as internal instruments for the endogenous group‐level treatment if they satisfy relevance and exogeneity conditions. Our approach applies to a broad range of settings including labor, public finance, industrial organization, urban economics, and development; we illustrate its usefulness with several such examples. Finally, an empirical application of our estimator finds that low‐wage earners in the United States from 1990 to 2007 were significantly more affected by increased Chinese import competition than high‐wage earners.

Uniformly valid post-regularization confidence regions for many functional parameters in z-estimation framework

In this paper, we develop procedures to construct simultaneous confidence bands for p ˜ potentially infinite-dimensional parameters after model selection for general moment condition models where p ˜ is potentially much larger than the sample size of available data, n. This allows us to cover settings with functional response data where each of the p ˜ parameters is a function. The procedure is based on the construction of score functions that satisfy Neyman orthogonality condition approximately. The proposed simultaneous confidence bands rely on uniform central limit theorems for high-dimensional vectors (and not on Donsker arguments as we allow for p ˜ ≫ n ). To construct the bands, we employ a multiplier bootstrap procedure which is computationally efficient as it only involves resampling the estimated score functions (and does not require resolving the high-dimensional optimization problems). We formally apply the general theory to inference on regression coefficient process in the distribution regression model with a logistic link, where two implementations are analyzed in detail. Simulations and an application to real data are provided to help illustrate the applicability of the results.

IV Quantile Regression for Group-Level Treatments, with an Application to the Effects of Trade on the Distribution of Wages

We present a methodology for estimating the distributional effects of an endogenous treatment that varies at the group level when there are group-level unobservables, a quantile extension of Hausman and Taylor (1981). Standard quantile regression techniques are inconsistent in this setting, even if the treatment is exogenous. Using the Bahadur representation of quantile estimators, we derive weak conditions on the growth of the number of observations per group that are sufficient for consistency and asymptotic normality. Simulations confirm the superiority of this grouped instrumental variables quantile regression estimator to standard quantile regression. An empirical application finds that low-wage earners in the U.S. from 1990-2007 were significantly more affected by increased Chinese import competition than high-wage earners. We also illustrate the usefulness of the estimation approach with additional empirical examples from urban economics, labor, regulation, and empirical auctions.

Double/debiased machine learning for treatment and structural parameters: Double/debiased machine learning

We revisit the classic semi‐parametric problem of inference on a low‐dimensional parameter θ in the presence of high‐dimensional nuisance parameters η. We depart from the classical setting by allowing for η to be so high‐dimensional that the traditional assumptions (e.g. Donsker properties) that limit complexity of the parameter space for this object break down. To estimate η, we consider the use of statistical or machine learning (ML) methods, which are particularly well suited to estimation in modern, very high‐dimensional cases. ML methods perform well by employing regularization to reduce variance and trading off regularization bias with overfitting in practice. However, both regularization bias and overfitting in estimating η cause a heavy bias in estimators of θ that are obtained by naively plugging ML estimators of η into estimating equations for θ. This bias results in the naive estimator failing to be consistent, where N is the sample size. We show that the impact of regularization bias and overfitting on estimation of the parameter of interest θ can be removed by using two simple, yet critical, ingredients: (1) using Neyman‐orthogonal moments/scores that have reduced sensitivity with respect to nuisance parameters to estimate θ; (2) making use of cross‐fitting, which provides an efficient form of data‐splitting. We call the resulting set of methods double or debiased ML (DML). We verify that DML delivers point estimators that concentrate in an ‐neighbourhood of the true parameter values and are approximately unbiased and normally distributed, which allows construction of valid confidence statements. The generic statistical theory of DML is elementary and simultaneously relies on only weak theoretical requirements, which will admit the use of a broad array of modern ML methods for estimating the nuisance parameters, such as random forests, lasso, ridge, deep neural nets, boosted trees, and various hybrids and ensembles of these methods. We illustrate the general theory by applying it to provide theoretical properties of the following: DML applied to learn the main regression parameter in a partially linear regression model; DML applied to learn the coefficient on an endogenous variable in a partially linear instrumental variables model; DML applied to learn the average treatment effect and the average treatment effect on the treated under unconfoundedness; DML applied to learn the local average treatment effect in an instrumental variables setting. In addition to these theoretical applications, we also illustrate the use of DML in three empirical examples.

An adaptive test of stochastic monotonicity

We propose a new nonparametric test of stochastic monotonicity which adapts to the unknown smoothness of the conditional distribution of interest, possesses desirable asymptotic properties, is conceptually easy to implement, and computationally attractive. In particular, we show that the test asymptotically controls size at a polynomial rate, is non-conservative, and detects certain smooth local alternatives that converge to the null with the fastest possible rate. Our test is based on a data-driven bandwidth value and the critical value for the test takes this randomness into account. Monte Carlo simulations indicate that the test performs well in finite samples. In particular, the simulations show that the test controls size and, under some alternatives, is significantly more powerful than existing procedures. ∗This version: September 30, 2019. We thank Ivan Canay and Whitney Newey for useful comments. †Department of Economics, University of California at Los Angeles, 315 Portola Plaza, Bunche Hall 8369, Los Angeles, CA 90024, USA; E-Mail address: chetverikov@econ.ucla.edu. ‡Department of Economics, University College London, Gower Street, London WC1E 6BT, United Kingdom; E-Mail address: d.wilhelm@ucl.ac.uk. The author gratefully acknowledges financial support from the ESRC Centre for Microdata Methods and Practice at IFS (RES-589-28-0001) and the European Research Council (ERC-2014-CoG-646917-ROMIA and ERC-2015-CoG-682349). §Department of Economics, Simon Fraser University, 8888 University Drive, Burnaby, BC, V5A 16S, Canada; E-Mail address: dongwook@sfu.ca.