The ill-posedness of the inverse problem of recovering a regression function in a nonparametric instrumental variable (NPIV) model leads to estimators that may suffer from poor statistical performance.In this paper, we explore the possibility of imposing shape restrictions to improve the performance of the NPIV estimators.We assume that the regression function is monotone and consider sieve estimators that enforce the monotonicity constraint.We define a restricted measure of ill-posedness that is relevant for the constrained estimators and show that under the monotone IV assumption and certain other conditions, our measure of ill-posedness is bounded uniformly over the dimension of the sieve space, in stark contrast with a well-known result that the unrestricted sieve measure of ill-posedness that is relevant for the unconstrained estimators grows to infinity with the dimension of the sieve space.Based on this result, we derive a novel non-asymptotic error bound for the constrained estimators.The bound gives a set of data-generating processes where the monotonicity constraint has a particularly strong regularization effect and considerably improves the performance of the estimators.The bound shows that the regularization effect can be strong even in large samples and for steep regression functions if the NPIV model is severely ill-posed -a finding that is confirmed by our simulation study.We apply the constrained estimator to the problem of estimating gasoline demand from U.S. data.

We develop a new method for selecting the penalty parameter for \ell_1-penalized M-estimators in high dimensions, which we refer to as bootstrapping after cross-validation. We derive rates of convergence for the corresponding \ell_1-penalized M-estimator and also for the post-\ell_1-penalized M-estimator, which refits the non-zero parameters of the former estimator without penalty in the criterion function. We demonstrate via simulations that our method is not dominated by cross-validation in terms of estimation errors and outperforms cross-validation in terms of inference. As an illustration, we revisit Fryer Jr (2019), who investigated racial differences in police use of force, and confirm his findings.

This article considers the problem of testing many moment inequalities where the number of moment inequalities, denoted by p, is possibly much larger than the sample size n. There is a variety of economic applications where solving this problem allows to carry out inference on causal and structural parameters; a notable example is the market structure model of Ciliberto and Tamer (2009) where p=2^m+1 with m being the number of firms that could possibly enter the market. We consider the test statistic given by the maximum of p Studentized (or t-type) inequality-specific statistics, and analyse various ways to compute critical values for the test statistic. Specifically, we consider critical values based upon (1) the union bound combined with a moderate deviation inequality for self-normalized sums, (2) the multiplier and empirical bootstraps, and (3) two-step and three-step variants of (1) and (2) by incorporating the selection of uninformative inequalities that are far from being binding and a novel selection of weakly informative inequalities that are potentially binding but do not provide first-order information. We prove validity of these methods, showing that under mild conditions, they lead to tests with the error in size decreasing polynomially in n while allowing for p being much larger than n; indeed p can be of order $\exp (n^c)$ for some $c > 0$. Importantly, all these results hold without any restriction on the correlation structure between p Studentized statistics, and also hold uniformly with respect to suitably large classes of underlying distributions. Moreover, in the online supplement, we show validity of a test based on the block multiplier bootstrap in the case of dependent data under some general mixing conditions.

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.

Chernozhukov et al. (2016) provide a generic double/de-biased machine learning (ML) approach for obtaining valid inferential statements about focal parameters, using Neyman-orthogonal scores and cross-fitting, in settings where nuisance parameters are estimated using ML methods. In this note, we illustrate the application of this method in the context of estimating average treatment effects and average treatment effects on the treated using observational data.