Lee Bounds with a Continuous Treatment in Sample Selection

Ying-Ying Lee and Chu-An Liu

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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.

Date: 2024-11, Revised 2025-02
New Economics Papers: this item is included in nep-ecm
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