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arXiv2020

Locally Robust Semiparametric Estimation

Many economic and causal parameters depend on nonparametric or high dimensional first steps. We give a general construction of locally robust/orthogonal moment functions for GMM, where moment conditions have zero derivative with respect to first steps. We show that orthogonal moment functions can be constructed by adding to identifying moments the nonparametric influence function for the effect of the first step on identifying moments. Orthogonal moments reduce model selection and regularization bias, as is very important in many applications, especially for machine learning first steps. We give debiased machine learning estimators of functionals of high dimensional conditional quantiles and of dynamic discrete choice parameters with high dimensional state variables. We show that adding to identifying moments the nonparametric influence function provides a general construction of orthogonal moments, including regularity conditions, and show that the nonparametric influence function is robust to additional unknown functions on which it depends. We give a general approach to estimating the unknown functions in the nonparametric influence function and use it to automatically debias estimators of functionals of high dimensional conditional location learners. We give a variety of new doubly robust moment equations and characterize double robustness. We give general and simple regularity conditions and apply these for asymptotic inference on functionals of high dimensional regression quantiles and dynamic discrete choice parameters with high dimensional state variables.

math.ST↗

arXiv2024

Double/Debiased Machine Learning for Treatment and Causal Parameters

Most modern supervised statistical/machine learning (ML) methods are explicitly designed to solve prediction problems very well. Achieving this goal does not imply that these methods automatically deliver good estimators of causal parameters. Examples of such parameters include individual regression coefficients, average treatment effects, average lifts, and demand or supply elasticities. In fact, estimates of such causal parameters obtained via naively plugging ML estimators into estimating equations for such parameters can behave very poorly due to the regularization bias. Fortunately, this regularization bias can be removed by solving auxiliary prediction problems via ML tools. Specifically, we can form an orthogonal score for the target low-dimensional parameter by combining auxiliary and main ML predictions. The score is then used to build a de-biased estimator of the target parameter which typically will converge at the fastest possible 1/root(n) rate and be approximately unbiased and normal, and from which valid confidence intervals for these parameters of interest may be constructed. The resulting method thus could be called a "double ML" method because it relies on estimating primary and auxiliary predictive models. In order to avoid overfitting, our construction also makes use of the K-fold sample splitting, which we call cross-fitting. This allows us to use a very broad set of ML predictive methods in solving the auxiliary and main prediction problems, such as random forest, lasso, ridge, deep neural nets, boosted trees, as well as various hybrids and aggregators of these methods.

stat.ML↗

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.

stat.ME↗

arXiv2019

Fixed-Effect Regressions on Network Data

This paper considers inference on fixed effects in a linear regression model estimated from network data. An important special case of our setup is the two-way regression model. This is a workhorse technique in the analysis of matched data sets, such as employer-employee or student-teacher panel data. We formalize how the structure of the network affects the accuracy with which the fixed effects can be estimated. This allows us to derive sufficient conditions on the network for consistent estimation and asymptotically-valid inference to be possible. Estimation of moments is also considered. We allow for general networks and our setup covers both the dense and sparse case. We provide numerical results for the estimation of teacher value-added models and regressions with occupational dummies.

stat.ME↗

arXiv2018

Generic Inference on Quantile and Quantile Effect Functions for Discrete Outcomes

Quantile and quantile effect functions are important tools for descriptive and causal analyses due to their natural and intuitive interpretation. Existing inference methods for these functions do not apply to discrete random variables. This paper offers a simple, practical construction of simultaneous confidence bands for quantile and quantile effect functions of possibly discrete random variables. It is based on a natural transformation of simultaneous confidence bands for distribution functions, which are readily available for many problems. The construction is generic and does not depend on the nature of the underlying problem. It works in conjunction with parametric, semiparametric, and nonparametric modeling methods for observed and counterfactual distributions, and does not depend on the sampling scheme. We apply our method to characterize the distributional impact of insurance coverage on health care utilization and obtain the distributional decomposition of the racial test score gap. We find that universal insurance coverage increases the number of doctor visits across the entire distribution, and that the racial test score gap is small at early ages but grows with age due to socio economic factors affecting child development especially at the top of the distribution. These are new, interesting empirical findings that complement previous analyses that focused on mean effects only. In both applications, the outcomes of interest are discrete rendering existing inference methods invalid for obtaining uniform confidence bands for observed and counterfactual quantile functions and for their difference -- the quantile effects functions.

stat.ME↗

arXiv2016

The structure of the climate debate

First-best climate policy is a uniform carbon tax which gradually rises over time. Civil servants have complicated climate policy to expand bureaucracies, politicians to create rents. Environmentalists have exaggerated climate change to gain influence, other activists have joined the climate bandwagon. Opponents to climate policy have attacked the weaknesses in climate research. The climate debate is convoluted and polarized as a result, and climate policy complex. Climate policy should become easier and more rational as the Paris Agreement has shifted climate policy back towards national governments. Changing political priorities, austerity, and a maturing bureaucracy should lead to a more constructive climate debate.

econ.GN↗

arXiv2020

Model Selection for Treatment Choice: Penalized Welfare Maximization

This paper studies a penalized statistical decision rule for the treatment assignment problem. Consider the setting of a utilitarian policy maker who must use sample data to allocate a binary treatment to members of a population, based on their observable characteristics. We model this problem as a statistical decision problem where the policy maker must choose a subset of the covariate space to assign to treatment, out of a class of potential subsets. We focus on settings in which the policy maker may want to select amongst a collection of constrained subset classes: examples include choosing the number of covariates over which to perform best-subset selection, and model selection when approximating a complicated class via a sieve. We adapt and extend results from statistical learning to develop the Penalized Welfare Maximization (PWM) rule. We establish an oracle inequality for the regret of the PWM rule which shows that it is able to perform model selection over the collection of available classes. We then use this oracle inequality to derive relevant bounds on maximum regret for PWM. An important consequence of our results is that we are able to formalize model-selection using a "hold-out" procedure, where the policy maker would first estimate various policies using half of the data, and then select the policy which performs the best when evaluated on the other half of the data.

math.ST↗

arXiv2018

Generalized Random Forests

We propose generalized random forests, a method for non-parametric statistical estimation based on random forests (Breiman, 2001) that can be used to fit any quantity of interest identified as the solution to a set of local moment equations. Following the literature on local maximum likelihood estimation, our method considers a weighted set of nearby training examples; however, instead of using classical kernel weighting functions that are prone to a strong curse of dimensionality, we use an adaptive weighting function derived from a forest designed to express heterogeneity in the specified quantity of interest. We propose a flexible, computationally efficient algorithm for growing generalized random forests, develop a large sample theory for our method showing that our estimates are consistent and asymptotically Gaussian, and provide an estimator for their asymptotic variance that enables valid confidence intervals. We use our approach to develop new methods for three statistical tasks: non-parametric quantile regression, conditional average partial effect estimation, and heterogeneous treatment effect estimation via instrumental variables. A software implementation, grf for R and C++, is available from CRAN.

stat.ME↗

arXiv2018

Best Subset Binary Prediction

We consider a variable selection problem for the prediction of binary outcomes. We study the best subset selection procedure by which the covariates are chosen by maximizing Manski (1975, 1985)'s maximum score objective function subject to a constraint on the maximal number of selected variables. We show that this procedure can be equivalently reformulated as solving a mixed integer optimization problem, which enables computation of the exact or an approximate solution with a definite approximation error bound. In terms of theoretical results, we obtain non-asymptotic upper and lower risk bounds when the dimension of potential covariates is possibly much larger than the sample size. Our upper and lower risk bounds are minimax rate-optimal when the maximal number of selected variables is fixed and does not increase with the sample size. We illustrate usefulness of the best subset binary prediction approach via Monte Carlo simulations and an empirical application of the work-trip transportation mode choice.

stat.ME↗

arXiv2017

probitfe and logitfe: Bias corrections for probit and logit models with two-way fixed effects

We present the Stata commands probitfe and logitfe, which estimate probit and logit panel data models with individual and/or time unobserved effects. Fixed effect panel data methods that estimate the unobserved effects can be severely biased because of the incidental parameter problem (Neyman and Scott, 1948). We tackle this problem by using the analytical and jackknife bias corrections derived in Fernandez-Val and Weidner (2016) for panels where the two dimensions ($N$ and $T$) are moderately large. We illustrate the commands with an empirical application to international trade and a Monte Carlo simulation calibrated to this application.

stat.ME↗

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.

stat.CO↗

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.

stat.CO↗

arXiv2019

Specification Tests for the Propensity Score

This paper proposes new nonparametric diagnostic tools to assess the asymptotic validity of different treatment effects estimators that rely on the correct specification of the propensity score. We derive a particular restriction relating the propensity score distribution of treated and control groups, and develop specification tests based upon it. The resulting tests do not suffer from the "curse of dimensionality" when the vector of covariates is high-dimensional, are fully data-driven, do not require tuning parameters such as bandwidths, and are able to detect a broad class of local alternatives converging to the null at the parametric rate $n^{-1/2}$, with $n$ the sample size. We show that the use of an orthogonal projection on the tangent space of nuisance parameters facilitates the simulation of critical values by means of a multiplier bootstrap procedure, and can lead to power gains. The finite sample performance of the tests is examined by means of a Monte Carlo experiment and an empirical application. Open-source software is available for implementing the proposed tests.

stat.ME↗

arXiv2016

The Factor-Lasso and K-Step Bootstrap Approach for Inference in High-Dimensional Economic Applications

We consider inference about coefficients on a small number of variables of interest in a linear panel data model with additive unobserved individual and time specific effects and a large number of additional time-varying confounding variables. We allow the number of these additional confounding variables to be larger than the sample size, and suppose that, in addition to unrestricted time and individual specific effects, these confounding variables are generated by a small number of common factors and high-dimensional weakly-dependent disturbances. We allow that both the factors and the disturbances are related to the outcome variable and other variables of interest. To make informative inference feasible, we impose that the contribution of the part of the confounding variables not captured by time specific effects, individual specific effects, or the common factors can be captured by a relatively small number of terms whose identities are unknown. Within this framework, we provide a convenient computational algorithm based on factor extraction followed by lasso regression for inference about parameters of interest and show that the resulting procedure has good asymptotic properties. We also provide a simple k-step bootstrap procedure that may be used to construct inferential statements about parameters of interest and prove its asymptotic validity. The proposed bootstrap may be of substantive independent interest outside of the present context as the proposed bootstrap may readily be adapted to other contexts involving inference after lasso variable selection and the proof of its validity requires some new technical arguments. We also provide simulation evidence about performance of our procedure and illustrate its use in two empirical applications.

stat.ME↗

arXiv2020

Nonparametric Tests for Treatment Effect Heterogeneity with Duration Outcomes

This article proposes different tests for treatment effect heterogeneity when the outcome of interest, typically a duration variable, may be right-censored. The proposed tests study whether a policy 1) has zero distributional (average) effect for all subpopulations defined by covariate values, and 2) has homogeneous average effect across different subpopulations. The proposed tests are based on two-step Kaplan-Meier integrals and do not rely on parametric distributional assumptions, shape restrictions, or on restricting the potential treatment effect heterogeneity across different subpopulations. Our framework is suitable not only to exogenous treatment allocation but can also account for treatment noncompliance - an important feature in many applications. The proposed tests are consistent against fixed alternatives, and can detect nonparametric alternatives converging to the null at the parametric $n^{-1/2}$-rate, $n$ being the sample size. Critical values are computed with the assistance of a multiplier bootstrap. The finite sample properties of the proposed tests are examined by means of a Monte Carlo study and an application about the effect of labor market programs on unemployment duration. Open-source software is available for implementing all proposed tests.

stat.ME↗

arXiv2021

Maximum Likelihood Estimation in Markov Regime-Switching Models with Covariate-Dependent Transition Probabilities

This paper considers maximum likelihood (ML) estimation in a large class of models with hidden Markov regimes. We investigate consistency of the ML estimator and local asymptotic normality for the models under general conditions which allow for autoregressive dynamics in the observable process, Markov regime sequences with covariate-dependent transition matrices, and possible model misspecification. A Monte Carlo study examines the finite-sample properties of the ML estimator in correctly specified and misspecified models. An empirical application is also discussed.

math.ST↗

arXiv2016

Population and trends in the global mean temperature

The Fisher Ideal index, developed to measure price inflation, is applied to define a population-weighted temperature trend. This method has the advantages that the trend is representative for the population distribution throughout the sample but without conflating the trend in the population distribution and the trend in the temperature. I show that the trend in the global area-weighted average surface air temperature is different in key details from the population-weighted trend. I extend the index to include urbanization and the urban heat island effect. This substantially changes the trend again. I further extend the index to include international migration, but this has a minor impact on the trend.

econ.GN↗

arXiv2017

Estimating Average Treatment Effects: Supplementary Analyses and Remaining Challenges

There is a large literature on semiparametric estimation of average treatment effects under unconfounded treatment assignment in settings with a fixed number of covariates. More recently attention has focused on settings with a large number of covariates. In this paper we extend lessons from the earlier literature to this new setting. We propose that in addition to reporting point estimates and standard errors, researchers report results from a number of supplementary analyses to assist in assessing the credibility of their estimates.

stat.ME↗