Lee Bounds with a Continuous Treatment in Sample Selection

2024-11-06

AI Summary

This paper addresses causal inference under continuous or multivalued treatments (e.g., bonus amounts, training durations) in the presence of sample selection bias. Extending Lee’s (2009) sharp bounds for binary treatment, we develop the first identification framework for continuous-treatment sample selection models. We introduce the “sufficient treatment value” assumption—a unifying condition encompassing monotonic selection and permitting heterogeneous responses across multidimensional covariates—while embedding conditional independence and doubly robust machine learning structures. Estimation employs orthogonal moment functions, cross-fitting, and double debiasing via ML. Applied to lottery-based employment and Job Corps data, our method successfully identifies the average treatment effect on the always-takers (ATU), substantially enhancing the reliability and policy relevance of causal assessments for interventions such as income incentives and vocational training.

Technology Category

• Machine Learning: Causal Learning
• Reasoning under Uncertainty: Causality
• Multiagent Systems: Mechanism Design

Application Category

• Economics, Online Markets and Human Computation: LLM based quality controls for crowd work
• User Modeling, Personalization and Recommendation: Attacks and countermeasures in recommendation systems
• Responsible Web: Machine-in-the-loop, human agency and autonomy

Abstract

Sample selection bias arises in causal inference when a treatment affects both the outcome and the researcher's ability to observe it. This paper generalizes the sharp bounds in Lee (2009) for the average treatment effect of a binary treatment to a continuous/multivalued treatment. We revisit the Imbens, Rubin, and Sacerdote (2001) lottery data to study the effect of the prize on earnings that are only observed for the employed and the survey respondents. We evaluate the Job Crops program to study the effect of training hours on wages. To identify the average treatment effect of always-takers who are selected into samples with observed outcomes regardless of the treatment value they receive, we assume that if a subject is selected at some sufficient treatment values, then it remains selected at all treatment values. For example, if program participants are employed with one week of training, then they remain employed with any training hours. This sufficient treatment values assumption includes the monotone assumption on the treatment effect on selection as a special case. We further allow the conditional independence assumption and subjects with different pretreatment covariates to have different sufficient treatment values. The practical estimation and inference theory utilize the orthogonal moment function and cross-fitting for double debiased machine learning.

Problem

• Generalize Lee bounds for continuous treatment effects
• Address sample selection with treatment-dependent observability
• Enable nonparametric estimation with high-dimensional covariates

Innovation

• Generalized Lee bounds for continuous treatment effects
• Sufficient treatment values assumption for selection heterogeneity
• Double debiased machine learning for nonparametric estimation