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This article surveys the development of nonparametric models and methods for estimation of choice models with nonlinear budget sets. The discussion focuses on the budget set regression, that is, the conditional expectation of a choice variable given the budget set. Utility maximization in a nonparametric model with general heterogeneity reduces the curse of dimensionality in this regression. Empirical results using this regression are different from maximum likelihood and give informative inference. The article also considers the information provided by kink probabilities for nonparametric utility with general heterogeneity. Instrumental variable estimation and the evidence it provides of heterogeneity in preferences are also discussed. 

Shape restrictions have played a central role in economics as both testable implications of theory and sufficient conditions for obtaining informative counterfactual predictions. In this paper, we provide a general procedure for inference under shape restrictions in identified and partially identified models defined by conditional moment restrictions. Our test statistics and proposed inference methods are based on the minimum of the generalized method of moments (GMM) objective function with and without shape restrictions. Uniformly valid critical values are obtained through a bootstrap procedure that approximates a subset of the true local parameter space. In an empirical analysis of the effect of childbearing on female labor supply, we show that employing shape restrictions in linear instrumental variables (IV) models can lead to shorter confidence regions for both local and average treatment effects. Other applications we discuss include inference for the variability of quantile IV treatment effects and for bounds on average equivalent variation in a demand model with general heterogeneity.

 

Many causal and structural effects depend on regressions. Examples include policy effects, average derivatives, regression decompositions, average treatment effects, causal mediation, and parameters of economic structural models. The regressions may be high‐dimensional, making machine learning useful. Plugging machine learners into identifying equations can lead to poor inference due to bias from regularization and/or model selection. This paper gives automatic debiasing for linear and nonlinear functions of regressions. The debiasing is automatic in using Lasso and the function of interest without the full form of the bias correction. The debiasing can be applied to any regression learner, including neural nets, random forests, Lasso, boosting, and other high‐dimensional methods. In addition to providing the bias correction, we give standard errors that are robust to misspecification, convergence rates for the bias correction, and primitive conditions for asymptotic inference for estimators of a variety of estimators of structural and causal effects. The automatic debiased machine learning is used to estimate the average treatment effect on the treated for the NSW job training data and to estimate demand elasticities from Nielsen scanner data while allowing preferences to be correlated with prices and income. 

High-dimensional linear models with endogenous variables play an increasingly important role in the recent econometric literature. In this work, we allow for models with many endogenous variables and make use of many instrumental variables to achieve identification. Because of the high-dimensionality in the structural equation, constructing honest confidence regions with asymptotically correct coverage is non-trivial. Our main contribution is to propose estimators and confidence regions that achieve this goal. Our approach relies on moment conditions that satisfy the usual instrument orthogonality condition but also have an additional orthogonality property with respect to specific linear combinations of the endogenous variables which are treated as nuisance parameters. We propose new pivotal procedures for estimating the high-dimensional nuisance parameters which appear in our formulation. We use a multiplier bootstrap procedure to compute critical values and establish its validity for achieving simultaneously valid confidence regions for a potentially high-dimensional set of endogenous variable coefficients. 

We provide adaptive inference methods, based on $\ell _1$ regularization, for regular (semiparametric) and nonregular (nonparametric) linear functionals of the conditional expectation function. Examples of regular functionals include average treatment effects, policy effects, and derivatives. Examples of nonregular functionals include average treatment effects, policy effects, and derivatives conditional on a covariate subvector fixed at a point. We construct a Neyman orthogonal equation for the target parameter that is approximately invariant to small perturbations of the nuisance parameters. To achieve this property, we include the Riesz representer for the functional as an additional nuisance parameter. Our analysis yields weak ‘double sparsity robustness’: either the approximation to the regression or the approximation to the representer can be ‘completely dense’ as long as the other is sufficiently ‘sparse’. Our main results are nonasymptotic and imply asymptotic uniform validity over large classes of models, translating into honest confidence bands for both global and local parameters.

 

There are many economic parameters that depend on nonparametric first steps. Examples include games, dynamic discrete choice, average exact consumer surplus, and treatment effects. Often estimators of these parameters are asymptotically equivalent to a sample average of an object referred to as the influence function. The influence function is useful in local policy analysis, in evaluating local sensitivity of estimators, and constructing debiased machine learning estimators. We show that the influence function is a Gateaux derivative with respect to a smooth deviation evaluated at a point mass. This result generalizes the classic Von Mises (1947) and Hampel (1974) calculation to estimators that depend on smooth nonparametric first steps. We give explicit influence functions for first steps that satisfy exogenous or endogenous orthogonality conditions. We use these results to generalize the omitted variable bias formula for regression to policy analysis for and sensitivity to structural changes. We apply this analysis and find no sensitivity to endogeneity of average equivalent variation estimates in a gasoline demand application. 

Control variables provide an important means of controlling for endogeneity in econometric models with nonseparable and/or multidimensional heterogeneity. We allow for discrete instruments, giving identification results under a variety of restrictions on the way the endogenous variable and the control variables affect the outcome. We consider many structural objects of interest, such as average or quantile treatment effects. We illustrate our results with an empirical application to Engel curve estimation 

The elasticity of taxable income with respect to the net of tax rate is a key parameter for predicting the effect of tax reform or designing an income tax. Bunching at kinks and notches in a single budget set has been used to estimate the taxable income elasticity. We show that when the distribution of preferences is unrestricted the amount of bunching at a kink or a notch is not informative about the size of the taxable income elasticity, and neither is the entire distribution of taxable income for a convex budget set. Kinks do provide information about the size of the elasticity when a priori restrictions are placed on the preference distribution. They can identify the elasticity when the preference distribution is completely and correctly specified across the kink and provide bounds under restrictions on the preference distribution. We find wide estimated bounds in an empirical example using data like Saez (2010) based on upper and lower bounds for the preference density. 

The drift-diffusion model (DDM) is a model of sequential sampling with diffusion signals, where the decision maker accumulates evidence until the process hits either an upper or lower stopping boundary and then stops and chooses the alternative that corresponds to that boundary. In perceptual tasks, the drift of the process is related to which choice is objectively correct, whereas in consumption tasks, the drift is related to the relative appeal of the alternatives. The simplest version of the DDM assumes that the stopping boundaries are constant over time. More recently, a number of papers have used nonconstant boundaries to better fit the data. This paper provides a statistical test for DDMs with general, nonconstant boundaries. As a by-product, we show that the drift and the boundary are uniquely identified. We use our condition to nonparametrically estimate the drift and the boundary and construct a test statistic based on finite samples.

 

Triangular systems with nonadditively separable unobserved heterogeneity provide a theoretically appealing framework for the modeling of complex structural relationships. However, they are not commonly used in practice due to the need for exogenous variables with large support for identification, the curse of dimensionality in estimation, and the lack of inferential tools. This paper introduces two classes of semiparametric nonseparable triangular models that address these limitations. They are based on distribution and quantile regression modeling of the reduced form conditional distributions of the endogenous variables. We show that average, distribution, and quantile structural functions are identified in these systems through a control function approach that does not require a large support condition. We propose a computationally attractive three‐stage procedure to estimate the structural functions where the first two stages consist of quantile or distribution regressions. We provide asymptotic theory and uniform inference methods for each stage. In particular, we derive functional central limit theorems and bootstrap functional central limit theorems for the distribution regression estimators of the structural functions. These results establish the validity of the bootstrap for three‐stage estimators of structural functions, and lead to simple inference algorithms. We illustrate the implementation and applicability of all our methods with numerical simulations and an empirical application to demand analysis.