arXiv subjects

Victor Chernozhukov

Publications and source records attributed to Victor Chernozhukov.

At least 109 records · Page 6Linked to original sources

arXiv2017

Confidence Bands for Coefficients in High Dimensional Linear Models with Error-in-variables

We study high-dimensional linear models with error-in-variables. Such models are motivated by various applications in econometrics, finance and genetics. These models are challenging because of the need to account for measurement errors to avoid non-vanishing biases in addition to handle the high dimensionality of the parameters. A recent growing literature has proposed various estimators that achieve good rates of convergence. Our main contribution complements this literature with the construction of simultaneous confidence regions for the parameters of interest in such high-dimensional linear models with error-in-variables. These confidence regions are based on the construction of moment conditions that have an additional orthogonal property with respect to nuisance parameters. We provide a construction that requires us to estimate an additional high-dimensional linear model with error-in-variables for each component of interest. We use a multiplier bootstrap to compute critical values for simultaneous confidence intervals for a subset $S$ of the components. We show its validity despite of possible model selection mistakes, and allowing for the cardinality of $S$ to be larger than the sample size. We apply and discuss the implications of our results to two examples and conduct Monte Carlo simulations to illustrate the performance of the proposed procedure.

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arXiv2017

Extremal Quantile Regression: An Overview

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 shown to provide more accurate approximations than Gaussian laws at the tails, but also have served as the basis to develop bias corrected estimators and inference methods using simulation and suitable variations of bootstrap and subsampling. The applicability of these methods is illustrated with two empirical examples on conditional value-at-risk and financial contagion.

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arXiv2017

Double/Debiased/Neyman Machine Learning of Treatment Effects

Chernozhukov, Chetverikov, Demirer, Duflo, Hansen, and Newey (2016) provide a generic double/de-biased machine learning (DML) approach for obtaining valid inferential statements about focal parameters, using Neyman-orthogonal scores and cross-fitting, in settings where nuisance parameters are estimated using a new generation of nonparametric fitting methods for high-dimensional data, called machine learning methods. In this note, we illustrate the application of this method in the context of estimating average treatment effects (ATE) and average treatment effects on the treated (ATTE) using observational data. A more general discussion and references to the existing literature are available in Chernozhukov, Chetverikov, Demirer, Duflo, Hansen, and Newey (2016).

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arXiv2016

quantreg.nonpar: An R Package for Performing Nonparametric Series Quantile Regression

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 functions contained within.

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arXiv2016

Counterfactual: An R Package for Counterfactual Analysis

The Counterfactual package implements the estimation and inference methods of Chernozhukov, Fernández-Val and Melly (2013) for counterfactual analysis. The counterfactual distributions considered are the result of changing either the marginal distribution of covariates related to the outcome variable of interest, or the conditional distribution of the outcome given the covariates. They can be applied to estimate quantile treatment effects and wage decompositions. This paper serves as an introduction to the package and displays basic functionality of the commands contained within.

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arXiv2016

Vector quantile regression beyond correct specification

This paper studies vector quantile regression (VQR), which is a way to model the dependence of a random vector of interest with respect to a vector of explanatory variables so to capture the whole conditional distribution, and not only the conditional mean. The problem of vector quantile regression is formulated as an optimal transport problem subject to an additional mean-independence condition. This paper provides a new set of results on VQR beyond the case with correct specification which had been the focus of previous work. First, we show that even under misspecification, the VQR problem still has a solution which provides a general representation of the conditional dependence between random vectors. Second, we provide a detailed comparison with the classical approach of Koenker and Bassett in the case when the dependent variable is univariate and we show that in that case, VQR is equivalent to classical quantile regression with an additional monotonicity constraint.

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arXiv2016

High-Dimensional Metrics in R

The package High-dimensional Metrics (\Rpackage{hdm}) is an evolving collection of statistical methods for estimation and quantification of uncertainty in high-dimensional approximately sparse models. It focuses on providing confidence intervals and significance testing for (possibly many) low-dimensional subcomponents of the high-dimensional parameter vector. Efficient estimators and uniformly valid confidence intervals for regression coefficients on target variables (e.g., treatment or policy variable) in a high-dimensional approximately sparse regression model, for average treatment effect (ATE) and average treatment effect for the treated (ATET), as well for extensions of these parameters to the endogenous setting are provided. Theory grounded, data-driven methods for selecting the penalization parameter in Lasso regressions under heteroscedastic and non-Gaussian errors are implemented. Moreover, joint/ simultaneous confidence intervals for regression coefficients of a high-dimensional sparse regression are implemented, including a joint significance test for Lasso regression. Data sets which have been used in the literature and might be useful for classroom demonstration and for testing new estimators are included. \R and the package \Rpackage{hdm} are open-source software projects and can be freely downloaded from CRAN: \texttt{http://cran.r-project.org}.

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arXiv2016

hdm: High-Dimensional Metrics

In this article the package High-dimensional Metrics (\texttt{hdm}) is introduced. It is a collection of statistical methods for estimation and quantification of uncertainty in high-dimensional approximately sparse models. It focuses on providing confidence intervals and significance testing for (possibly many) low-dimensional subcomponents of the high-dimensional parameter vector. Efficient estimators and uniformly valid confidence intervals for regression coefficients on target variables (e.g., treatment or policy variable) in a high-dimensional approximately sparse regression model, for average treatment effect (ATE) and average treatment effect for the treated (ATET), as well for extensions of these parameters to the endogenous setting are provided. Theory grounded, data-driven methods for selecting the penalization parameter in Lasso regressions under heteroscedastic and non-Gaussian errors are implemented. Moreover, joint/ simultaneous confidence intervals for regression coefficients of a high-dimensional sparse regression are implemented. Data sets which have been used in the literature and might be useful for classroom demonstration and for testing new estimators are included.

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arXiv2016

Valid Post-Selection Inference in High-Dimensional Approximately Sparse Quantile Regression Models

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 uniform validity of the proposed confidence regions for the quantile regression coefficient. Importantly, these methods directly apply to more than one variable and a continuum of quantile indices. In addition, the performance of the proposed methods is illustrated through Monte-Carlo experiments and an empirical example, dealing with risk factors in childhood malnutrition.

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arXiv2016

Post-Selection Inference for Generalized Linear Models with Many Controls

This paper considers generalized linear models in the presence of many controls. We lay out a general methodology to estimate an effect of interest based on the construction of an instrument that immunize against model selection mistakes and apply it to the case of logistic binary choice model. More specifically we propose new methods for estimating and constructing confidence regions for a regression parameter of primary interest $α_0$, a parameter in front of the regressor of interest, such as the treatment variable or a policy variable. These methods allow to estimate $α_0$ at the root-$n$ rate when the total number $p$ of other regressors, called controls, potentially exceed the sample size $n$ using sparsity assumptions. The sparsity assumption means that there is a subset of $s<n$ controls which suffices to accurately approximate the nuisance part of the regression function. Importantly, the estimators and these resulting confidence regions are valid uniformly over $s$-sparse models satisfying $s^2\log^2 p = o(n)$ and other technical conditions. These procedures do not rely on traditional consistent model selection arguments for their validity. In fact, they are robust with respect to moderate model selection mistakes in variable selection. Under suitable conditions, the estimators are semi-parametrically efficient in the sense of attaining the semi-parametric efficiency bounds for the class of models in this paper.

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arXiv2016

Central Limit Theorems and Bootstrap in High Dimensions

This paper derives central limit and bootstrap theorems for probabilities that sums of centered high-dimensional random vectors hit hyperrectangles and sparsely convex sets. Specifically, we derive Gaussian and bootstrap approximations for probabilities $\Pr(n^{-1/2}\sum_{i=1}^n X_i\in A)$ where $X_1,\dots,X_n$ are independent random vectors in $\mathbb{R}^p$ and $A$ is a hyperrectangle, or, more generally, a sparsely convex set, and show that the approximation error converges to zero even if $p=p_n\to \infty$ as $n \to \infty$ and $p \gg n$; in particular, $p$ can be as large as $O(e^{Cn^c})$ for some constants $c,C>0$. The result holds uniformly over all hyperrectangles, or more generally, sparsely convex sets, and does not require any restriction on the correlation structure among coordinates of $X_i$. Sparsely convex sets are sets that can be represented as intersections of many convex sets whose indicator functions depend only on a small subset of their arguments, with hyperrectangles being a special case.

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arXiv2015

Vector Quantile Regression: An Optimal Transport Approach

We propose a notion of conditional vector quantile function and a vector quantile regression. A \emph{conditional vector quantile function} (CVQF) of a random vector $Y$, taking values in $\mathbb{R}^d$ given covariates $Z=z$, taking values in $\mathbb{R}% ^k$, is a map $u \longmapsto Q_{Y\mid Z}(u,z)$, which is monotone, in the sense of being a gradient of a convex function, and such that given that vector $U$ follows a reference non-atomic distribution $F_U$, for instance uniform distribution on a unit cube in $\mathbb{R}^d$, the random vector $Q_{Y\mid Z}(U,z)$ has the distribution of $Y$ conditional on $Z=z$. Moreover, we have a strong representation, $Y = Q_{Y\mid Z}(U,Z)$ almost surely, for some version of $U$. The \emph{vector quantile regression} (VQR) is a linear model for CVQF of $Y$ given $Z$. Under correct specification, the notion produces strong representation, $Y=β\left(U\right) ^\top f(Z)$, for $f(Z)$ denoting a known set of transformations of $Z$, where $u \longmapsto β(u)^\top f(Z)$ is a monotone map, the gradient of a convex function, and the quantile regression coefficients $u \longmapsto β(u)$ have the interpretations analogous to that of the standard scalar quantile regression. As $f(Z)$ becomes a richer class of transformations of $Z$, the model becomes nonparametric, as in series modelling. A key property of VQR is the embedding of the classical Monge-Kantorovich's optimal transportation problem at its core as a special case. In the classical case, where $Y$ is scalar, VQR reduces to a version of the classical QR, and CVQF reduces to the scalar conditional quantile function. An application to multiple Engel curve estimation is considered.

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arXiv2015

Monge-Kantorovich Depth, Quantiles, Ranks, and Signs

We propose new concepts of statistical depth, multivariate quantiles, ranks and signs, based on canonical transportation maps between a distribution of interest on $R^d$ and a reference distribution on the $d$-dimensional unit ball. The new depth concept, called Monge-Kantorovich depth, specializes to halfspace depth in the case of spherical distributions, but, for more general distributions, differs from the latter in the ability for its contours to account for non convex features of the distribution of interest. We propose empirical counterparts to the population versions of those Monge-Kantorovich depth contours, quantiles, ranks and signs, and show their consistency by establishing a uniform convergence property for empirical transport maps, which is of independent interest.

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arXiv2015

Empirical and multiplier bootstraps for suprema of empirical processes of increasing complexity, and related Gaussian couplings

We derive strong approximations to the supremum of the non-centered empirical process indexed by a possibly unbounded VC-type class of functions by the suprema of the Gaussian and bootstrap processes. The bounds of these approximations are non-asymptotic, which allows us to work with classes of functions whose complexity increases with the sample size. The construction of couplings is not of the Hungarian type and is instead based on the Slepian-Stein methods and Gaussian comparison inequalities. The increasing complexity of classes of functions and non-centrality of the processes make the results useful for applications in modern nonparametric statistics (Giné and Nickl, 2015), in particular allowing us to study the power properties of nonparametric tests using Gaussian and bootstrap approximations.

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arXiv2015

Valid Post-Selection and Post-Regularization Inference: An Elementary, General Approach

Here we present an expository, general analysis of valid post-selection or post-regularization inference about a low-dimensional target parameter, $α$, in the presence of a very high-dimensional nuisance parameter, $η$, which is estimated using modern selection or regularization methods. Our analysis relies on high-level, easy-to-interpret conditions that allow one to clearly see the structures needed for achieving valid post-regularization inference. Simple, readily verifiable sufficient conditions are provided for a class of affine-quadratic models. We focus our discussion on estimation and inference procedures based on using the empirical analog of theoretical equations $$M(α, η)=0$$ which identify $α$. Within this structure, we show that setting up such equations in a manner such that the orthogonality/immunization condition $$\partial_ηM(α, η) = 0$$ at the true parameter values is satisfied, coupled with plausible conditions on the smoothness of $M$ and the quality of the estimator $\hat η$, guarantees that inference on for the main parameter $α$ based on testing or point estimation methods discussed below will be regular despite selection or regularization biases occurring in estimation of $η$. In particular, the estimator of $α$ will often be uniformly consistent at the root-$n$ rate and uniformly asymptotically normal even though estimators $\hat η$ will generally not be asymptotically linear and regular. The uniformity holds over large classes of models that do not impose highly implausible "beta-min" conditions. We also show that inference can be carried out by inverting tests formed from Neyman's $C(α)$ (orthogonal score) statistics.

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arXiv2015

Some New Asymptotic Theory for Least Squares Series: Pointwise and Uniform Results

In applications it is common that the exact form of a conditional expectation is unknown and having flexible functional forms can lead to improvements. Series method offers that by approximating the unknown function based on $k$ basis functions, where $k$ is allowed to grow with the sample size $n$. We consider series estimators for the conditional mean in light of: (i) sharp LLNs for matrices derived from the noncommutative Khinchin inequalities, (ii) bounds on the Lebesgue factor that controls the ratio between the $L^\infty$ and $L_2$-norms of approximation errors, (iii) maximal inequalities for processes whose entropy integrals diverge, and (iv) strong approximations to series-type processes. These technical tools allow us to contribute to the series literature, specifically the seminal work of Newey (1997), as follows. First, we weaken the condition on the number $k$ of approximating functions used in series estimation from the typical $k^2/n \to 0$ to $k/n \to 0$, up to log factors, which was available only for spline series before. Second, we derive $L_2$ rates and pointwise central limit theorems results when the approximation error vanishes. Under an incorrectly specified model, i.e. when the approximation error does not vanish, analogous results are also shown. Third, under stronger conditions we derive uniform rates and functional central limit theorems that hold if the approximation error vanishes or not. That is, we derive the strong approximation for the entire estimate of the nonparametric function. We derive uniform rates, Gaussian approximations, and uniform confidence bands for a wide collection of linear functionals of the conditional expectation function.

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arXiv2015

Sparse Models and Methods for Optimal Instruments with an Application to Eminent Domain

We develop results for the use of Lasso and Post-Lasso methods to form first-stage predictions and estimate optimal instruments in linear instrumental variables (IV) models with many instruments, $p$. Our results apply even when $p$ is much larger than the sample size, $n$. We show that the IV estimator based on using Lasso or Post-Lasso in the first stage is root-n consistent and asymptotically normal when the first-stage is approximately sparse; i.e. when the conditional expectation of the endogenous variables given the instruments can be well-approximated by a relatively small set of variables whose identities may be unknown. We also show the estimator is semi-parametrically efficient when the structural error is homoscedastic. Notably our results allow for imperfect model selection, and do not rely upon the unrealistic "beta-min" conditions that are widely used to establish validity of inference following model selection. In simulation experiments, the Lasso-based IV estimator with a data-driven penalty performs well compared to recently advocated many-instrument-robust procedures. In an empirical example dealing with the effect of judicial eminent domain decisions on economic outcomes, the Lasso-based IV estimator outperforms an intuitive benchmark. In developing the IV results, we establish a series of new results for Lasso and Post-Lasso estimators of nonparametric conditional expectation functions which are of independent theoretical and practical interest. We construct a modification of Lasso designed to deal with non-Gaussian, heteroscedastic disturbances which uses a data-weighted $\ell_1$-penalty function. Using moderate deviation theory for self-normalized sums, we provide convergence rates for the resulting Lasso and Post-Lasso estimators that are as sharp as the corresponding rates in the homoscedastic Gaussian case under the condition that $\log p = o(n^{1/3})$.

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arXiv2015

A lava attack on the recovery of sums of dense and sparse signals

Common high-dimensional methods for prediction rely on having either a sparse signal model, a model in which most parameters are zero and there are a small number of non-zero parameters that are large in magnitude, or a dense signal model, a model with no large parameters and very many small non-zero parameters. We consider a generalization of these two basic models, termed here a "sparse+dense" model, in which the signal is given by the sum of a sparse signal and a dense signal. Such a structure poses problems for traditional sparse estimators, such as the lasso, and for traditional dense estimation methods, such as ridge estimation. We propose a new penalization-based method, called lava, which is computationally efficient. With suitable choices of penalty parameters, the proposed method strictly dominates both lasso and ridge. We derive analytic expressions for the finite-sample risk function of the lava estimator in the Gaussian sequence model. We also provide an deviation bound for the prediction risk in the Gaussian regression model with fixed design. In both cases, we provide Stein's unbiased estimator for lava's prediction risk. A simulation example compares the performance of lava to lasso, ridge, and elastic net in a regression example using feasible, data-dependent penalty parameters and illustrates lava's improved performance relative to these benchmarks.

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