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Instrumental variable (IV) strategies are widely used to estimate causal effects in economics, political science, epidemiology, sociology, psychology, and other fields. When there is unobserved heterogeneity in causal effects, standard linear IV estimators only represent effects for complier subpopulations (Imbens and Angrist, 1994). Marginal treatment effect (MTE) methods (Heckman and Vytlacil, 1999, 2005) allow researchers to use additional assumptions to extrapolate beyond complier subpopulations. We discuss a flexible framework for MTE methods based on linear regression and the generalized method of moments. We show how to implement the framework using the ivmte package for R. 

We develop a new nonparametric approach for discrete choice and use it to analyze the demand for health insurance in the California Affordable Care Act marketplace. The model allows for endogenous prices and instrumental variables, while avoiding parametric functional form assumptions about the unobserved components of utility. We use the approach to estimate bounds on the effects of changing premiums or subsidies on coverage choices, consumer surplus, and government spending on subsidies. We find that a $10 decrease in monthly premium subsidies would cause a decline of between 1.8% and 6.7% in the proportion of subsidized adults with coverage. The reduction in total annual consumer surplus would be between $62 and $74 million, while the savings in yearly subsidy outlays would be between $207 and $602 million. We estimate the demand impacts of linking subsidies to age, finding that shifting subsidies from older to younger buyers would increase average consumer surplus, with potentially large impacts on enrollment. We also estimate the consumer surplus impact of removing the highly‐subsidized plans in the Silver metal tier, where we find that a nonparametric model is consistent with a wide range of possibilities. We find that comparable mixed logit models tend to yield price sensitivity estimates toward the lower end of the nonparametric bounds, while producing consumer surplus impacts that can be both higher and lower than the nonparametric bounds depending on the specification of random coefficients. 

This paper considers the problem of testing whether there exists a non‐negative solution to a possibly under‐determined system of linear equations with known coefficients. This hypothesis testing problem arises naturally in a number of settings, including random coefficient, treatment effect, and discrete choice models, as well as a class of linear programming problems. As a first contribution, we obtain a novel geometric characterization of the null hypothesis in terms of identified parameters satisfying an infinite set of inequality restrictions. Using this characterization, we devise a test that requires solving only linear programs for its implementation, and thus remains computationally feasible in the high‐dimensional applications that motivate our analysis. The asymptotic size of the proposed test is shown to equal at most the nominal level uniformly over a large class of distributions that permits the number of linear equations to grow with the sample size. 

We discuss the ivcrc command, which implements an instrumental-variables (IV) estimator for the linear correlated random-coefficients model. The correlated random-coefficients model is a natural generalization of the standard linear IV model that allows for endogenous, multivalued treatments and unobserved heterogeneity in treatment effects. The estimator implemented by ivcrc uses recent semiparametric identification results that allow for flexible functional forms and permit instruments that may be binary, discrete, or continuous. The ivcrc command also allows for the estimation of varying-coefficient regressions, which are closely related in structure to the proposed IV estimator. We illustrate the use of ivcrc by estimating the returns to education in the National Longitudinal Survey of Young Men.