Experimental Design
Applications for enrollment in adult ESOL courses regularly exceed the program’s capacity. Offers of admission to the adult ESOL program were determined by a random lottery conducted twice yearly. We will compare outcomes for applicants whose lottery number were drawn—and who were offered seats in oversubscribed ESOL classes as a result—to those whose lottery numbers were not drawn. Specifically, we will estimate the effect of winning a lottery to attend the adult ESOL program on outcomes of interest using an ordinary least squares regression of the form:
Y_(it) = α_(0) + α_(1)*WonLottery_(i) + α_(2)*X_(i) + ν_(clt) + e_(iclt)
where Y_(it) represents a health outcome of interest and α_(1) represents the estimated impact of winning the lottery to attend the adult ESOL program on that outcome, X_(i) is a vector of individual demographic controls, and ν_(clt) is a set of lottery fixed effects that interact the class difficulty level, time of day, and term at which an individual first applied to the lottery. If we replace Y_(it) with a measure of program enrollment or attendance, we can re-estimate the prior equation to measure the “first-stage” impact of winning one’s first lottery on ESOL participation:
Enrolled_(i) = β_(0) + β_(1)*WonLottery_(i) + β_(2)*X_(i) + ψ_(clt) + υ_(iclt)
where β_(1) represents the impact of an individual’s first lottery outcome on the ESOL enrollment or attendance outcome on the left-hand side of the equation. The predicted value of ESOL enrollment (or attendance) from this first-stage regression can be used as an instrument for ESOL enrollment (or attendance) in a second-stage of an instrumental variables regression of the form:
Y_(it) = δ_(0) + δ_(1)*Enrolled_(i) + δ_(2)*X_(i) + γ_(clt) + ϵ_(iclt)
where δ_(1) represents the estimated impact of ESOL enrollment on outcomes of interest for those who are induced to enroll by the outcome of their admissions lottery (i.e., the local average treatment effect for this group of applicants who “comply” with their lottery outcome).