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Randomized and natural experiments are commonly used in economics and other social science fields to estimate the effect of programs and interventions. Even when employing experimental data, assessing the impact of a treatment is often complicated by the presence of sample selection (outcomes are only observed for a selected group) and noncompliance (some treatment group individuals do not receive the treatment while some control individuals do). We address both of these identification problems simultaneously and derive nonparametric bounds for average treatment effects within a principal stratification framework. We employ these bounds to empirically assess the wage effects of Job Corps (JC), the most comprehensive and largest federally funded job training program for disadvantaged youth in the United States. Our results strongly suggest positive average effects of JC on wages for individuals who comply with their treatment assignment and would be employed whether or not they enrolled in JC (the \"always-employed compliers\"). Under relatively weak monotonicity and mean dominance assumptions, we find that this average effect is between 5.7% and 13.9% 4 years after randomization, and between 7.7% and 17.5% for non-Hispanics. Our results are consistent with larger effects of JC on wages than those found without adjusting for noncompliance.

What should researchers do when their baseline model is falsified? We recommend reporting the set of parameters that are consistent with minimally nonfalsified models. We call this the falsification adaptive set (FAS). This set generalizes the standard baseline estimand to account for possible falsification. Importantly, it does not require the researcher to select or calibrate sensitivity parameters. In the classical linear IV model with multiple instruments, we show that the FAS has a simple closed-form expression that only depends on a few 2SLS coefficients. We apply our results to an empirical study of roads and trade. We show how the FAS complements traditional overidentification tests by summarizing the variation in estimates obtained from alternative nonfalsified models.

We illustrate how partial identification methods can be used to provide credible inferences on the causal impacts of food assistance programs, focusing on the impact that the Supplemental Nutrition Assistance Program (SNAP, formerly known as the Food Stamp Program) has on food insecurity among households with children. Recent research applies these methods to address two key issues confounding identification: missing counterfactuals and nonrandomly misclassified treatment status. In this paper, we illustrate and extend the recent literature by using data from the Survey of Income and Program Participation (SIPP) to study the robustness of prior conclusions. The SIPP confers important advantages: the detailed information about income and eligibility allows us to apply a modified discontinuity design to sharpen inferences, and the panel nature allows us to reduce uncertainty about true participation status. We find that SNAP reduces the prevalence of food insecurity in households with children by at least six percentage points.

Given a set of baseline assumptions, a breakdown frontier is the boundary between the set of assumptions which lead to a specific conclusion and those which do not. In a potential outcomes model with a binary treatment, we consider two conclusions: First, that ATE is at least a specific value (e.g., nonnegative) and second that the proportion of units who benefit from treatment is at least a specific value (e.g., at least 50%). For these conclusions, we derive the breakdown frontier for two kinds of assumptions: one which indexes relaxations of the baseline random assignment of treatment assumption, and one which indexes relaxations of the baseline rank invariance assumption. These classes of assumptions nest both the point identifying assumptions of random assignment and rank invariance and the opposite end of no constraints on treatment selection or the dependence structure between potential outcomes. This frontier provides a quantitative measure of the robustness of conclusions to relaxations of the baseline point identifying assumptions. We derive ÍN consistent sample analog estimators for these frontiers. We then provide two asymptotically valid bootstrap procedures for constructing lower uniform confidence bands for the breakdown frontier. As a measure of robustness, estimated breakdown frontiers and their corresponding confidence bands can be presented alongside traditional point estimates and confidence intervals obtained under point identifying assumptions. We illustrate this approach in an empirical application to the effect of child soldiering on wages. We find that sufficiently weak conclusions are robust to simultaneous failures of rank invariance and random assignment, while some stronger conclusions are fairly robust to failures of rank invariance but not necessarily to relaxations of random assignment.

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 non-parametric 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 is about measuring state dependence in dynamic discrete outcomes. I develop a nonparametric dynamic potential outcomes (DPO) model and propose an array of parameters and identifying assumptions that can be considered in this model. I show how to construct sharp identified sets under combinations of identifying assumptions by using a flexible linear programming procedure. I apply the analysis to study state dependence in unemployment for working age high school educated men using an extract from the 2008 Survey of Income and Program Participation (SIPP). Using only non-parametric assumptions, I estimate that state dependence accounts for at least 30-40% of the four-month persistence in unemployment among high school educated men.

We propose an alternative dynamic price experimentation policy that extends multiarmed bandit (MAB) algorithms from statistical machine learning to include microeconomic choice theory. Pricing managers at online retailers face a unique challenge. They must decide on real-time prices for a large number of products with incomplete demand information. The manager runs price experiments to learn about each product’s demand curve and the profit-maximizing price. In practice, balanced field price experiments can create high opportunity costs, because a large number of customers are presented with suboptimal prices. In this paper, we propose an alternative dynamic price experimentation policy. The proposed approach extends multiarmed bandit (MAB) algorithms from statistical machine learning to include microeconomic choice theory. Our automated pricing policy solves this MAB problem using a scalable distribution-free algorithm. We prove analytically that our method is asymptotically optimal for any weakly downward sloping demand curve. In a series of Monte Carlo simulations, we show that the proposed approach performs favorably compared with balanced field experiments and standard methods in dynamic pricing from computer science. In a calibrated simulation based on an existing pricing field experiment, we find that our algorithm can increase profits by 43% during the month of testing and 4% annually. Data files and the online appendix are available at https://doi.org/10.1287/mksc.2018.1129 .

Conditional independence of treatment assignment from potential outcomes is a commonly used but nonrefutable assumption. We derive identified sets for various treatment effect parameters under nonparametric deviations from this conditional independence assumption. These deviations are defined via a conditional treatment assignment probability, which makes it straightforward to interpret. Our results can be used to assess the robustness of empirical conclusions obtained under the baseline conditional independence assumption.

It is well understood that classical sample selection models are not semiparametrically identified without exclusion restrictions. Lee (2009) developed bounds for the parameters in a model that nests the semiparametric sample selection model. These bounds can be wide. In this paper, we investigate bounds that impose the full structure of a sample selection model with errors that are independent of the explanatory variables but have unknown distribution. The additional structure can significantly reduce the identified set for the parameters of interest. Specifically, we construct the identified set for the parameter vector of interest. It is a one-dimensional line segment in the parameter space, and we demonstrate that this line segment can be short in practice. We show that the identified set is sharp when the model is correct and empty when there exist no parameter values that make the sample selection model consistent with the data. We also provide non-sharp bounds under the assumption that the model is correct. These are easier to compute and associated with lower statistical uncertainty than the sharp bounds. Throughout the paper, we illustrate our approach by estimating a standard sample selection model for wages.

The literature assessing the efficacy of the Supplemental Nutrition Assistance Program (SNAP), formerly known as the Food Stamp Program, has long puzzled over positive associations between SNAP receipt and various undesirable health outcomes such as food insecurity. Assessing the causal impacts of SNAP, however, is hampered by two key identification problems: endogenous selection into participation and extensive systematic underreporting of participation status. Using data from the National Health and Nutrition Examination Survey (NHANES), we extend partial identification bounding methods to account for these two identification problems in a single unifying framework. Specifically, we derive informative bounds on the average treatment effect (ATE) of SNAP on child food insecurity, poor general health, obesity, and anemia across a range of different assumptions used to address the selection and classification error problems. In particular, to address the selection problem, we apply relatively weak nonparametric assumptions on the latent outcomes, selected treatments, and observed covariates. To address the classification error problem, we formalize a new approach that uses auxiliary administrative data on the size of the SNAP caseload to restrict the magnitudes and patterns of SNAP reporting errors. Layering successively stronger assumptions, an objective of our analysis is to make transparent how the strength of the conclusions varies with the strength of the identifying assumptions. Under the weakest restrictions, there is substantial ambiguity; we cannot rule out the possibility that SNAP increases or decreases poor health. Under stronger but plausible assumptions used to address the selection and classification error problems, we find that commonly cited relationships between SNAP and poor health outcomes provide a misleading picture about the true impacts of the program. Our tightest bounds identify favorable impacts of SNAP on child health.