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In this note, we offer an approach to estimating structural parameters in the presence of many instruments and controls based on methods for estimating sparse high-dimensional models. We use these high-dimensional methods to select both which instruments and which control variables to use. The approach we take extends Belloni et al. (2012), which covers selection of instruments for IV models with a small number of controls, and extends Belloni, Chernozhukov and Hansen (2014), which covers selection of controls in models where the variable of interest is exogenous conditional on observables, to accommodate both a large number of controls and a large number of instruments. We illustrate the approach with a simulation and an empirical example. Technical supporting material is available in a supplementary appendix.

We derive general, yet simple, sharp bounds on the size of the omitted variable bias for a broad class of causal parameters that can be identified as linear functionals of the conditional expectation function of the outcome. Such functionals encompass many of the traditional targets of investigation in causal inference studies, such as, for example, (weighted) average of potential outcomes, average treatment effects (including subgroup effects, such as the effect on the treated), (weighted) average derivatives, and policy effects from shifts in covariate distribution -- all for general, nonparametric causal models. Our construction relies on the Riesz-Frechet representation of the target functional. Specifically, we show how the bound on the bias depends only on the additional variation that the latent variables create both in the outcome and in the Riesz representer for the parameter of interest. Moreover, in many important cases (e.g, average treatment effects and avearage derivatives) the bound is shown to depend on easily interpretable quantities that measure the explanatory power of the omitted variables. Therefore, simple plausibility judgments on the maximum explanatory power of omitted variables (in explaining treatment and outcome variation) are sufficient to place overall bounds on the size of the bias. Furthermore, we use debiased machine learning to provide flexible and efficient statistical inference on learnable components of the bounds. Finally, empirical examples demonstrate the usefulness of the approach.

We revisit the classic semiparametric problem of inference on a low dimensional parameter [theta]_0 in the presence of high-dimensional nuisance parameters [eta]_0. We depart from the classical setting by allowing for [eta]_0 to be so high-dimensional that the traditional assumptions, such as Donsker properties, that limit complexity of the parameter space for this object break down. To estimate [eta]_0, 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 [eta]_0 cause a heavy bias in estimators of [theta]_0 that are obtained by naively plugging ML estimators of [eta]_0 into estimating equations for [theta]_0. This bias results in the naive estimator failing to be N^(-1/2) consistent, where N is the sample size. We show that the impact of regularization bias and overfitting on estimation of the parameter of interest [theta]_0 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 [theta]_0, and (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 a N^(-1/2)-neighborhood 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 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, and 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.

A wide variety of important distributional hypotheses can be assessed using the empirical quantile regression processes. In this paper, a very simple and practical resampling test is offered as an alternative to inference based on Khmaladzation, as developed in Koenker and Xiao (2002). This alternative has better or competitive power, accurate size, and does not require estimation of non-parametric sparsity and score functions. It applies not only to iid but also time series data. Computational experiments and an empirical example that re-examines the effect of re-employment bonus on the unemployment duration strongly support this approach. Keywords: bootstrap, subsampling, quantile regression, quantile regression process, Kolmogorov-Smirnov test, unemployment duration. JEL Classification: C13, C14, C30, C51, D4, J24, J31.

The paper develops estimation and inference methods for econometric models with partial identification, focusing on models defined by moment inequalities and equalities. Main applications of this framework include analysis of game-theoretic models, regression with missing and mismeasured data, bounds in structural quantile models, and bounds in asset pricing, among others. Keywords: Set estimator, contour sets, moment inequalities, moment equalities. JEL Classifications: C13, C14, C21, C41, C51, C53.

We propose methods for inference on the average effect of a treatment on a scalar outcome in the presence of very many controls. Our setting is a partially linear regression model containing the treatment/policy variable and a large number p of controls or series terms, with p that is possibly much larger than the sample size n, but where only s “n unknown controls or series terms are needed to approximate the regression function accurately. The latter sparsity condition makes it possible to estimate the entire regression function as well as the average treatment effect by selecting an approximately the right set of controls using Lasso and related methods. We develop estimation and inference methods for the average treatment effect in this setting, proposing a novel "post double selection" method that provides attractive inferential and estimation properties. In our analysis, in order to cover realistic applications, we expressly allow for imperfect selection of the controls and account for the impact of selection errors on estimation and inference. In order to cover typical applications in economics, we employ the selection methods designed to deal with non-Gaussian and heteroscedastic disturbances. We illustrate the use of new methods with numerical simulations and an application to the effect of abortion on crime rates. -- treatment effects ; high-dimensional regression ; inference under imperfect model selection

Quantile regression is an increasingly important empirical tool in economics and other sciences for analyzing the impact of a set of regressors on the conditional distribution of an outcome. Extremal quantile regression, or quantile regression applied to the tails, is of interest in many economic and financial applications, such as conditional value-at-risk, production efficiency, and adjustment bands in (S, s) models. In this paper we provide feasible inference tools for extremal conditional quantile models that rely upon extreme value approximations to the distribution of self-normalized quantile regression statistics. The methods are simple to implement and can be of independent interest even in the non-regression case. We illustrate the results with two empirical examples analyzing extreme fluctuations of a stock return and extremely low percentiles of live infants' birth weights in the range between 250 and 1500 grams.