Identification and Estimation of Time-Varying Treatment Effects: How The Timing of Grade Retention Affects Outcomes∗

This paper builds a framework for analyzing models with multiple time-varying treatments when selection into treatment is sequential and varies across treatment statuses. The key challenge is to separate the time-varying effect of treatment from dynamic selection into treatment based on unobservables. To address this problem, we develop a method that is a hybrid between a control function and a generalized fixed effects approach. Using a factor structure, we recover the distribution of the unobservables that jointly determine selection into treatment and the effect of treatment. A useful feature of our method is that it can recover the distribution of heterogeneous treatment effects across unobservable types. We apply our strategy to study grade retention using the Early Childhood Longitudinal Study of Kindergartners. We find evidence of dynamic selection into retention and that the treatment effect of retention varies considerably across grades and unobservable abilities of students.