where Outcomei is, alternatively, a series of indicator
variables capturing whether a student took a particular
STEM exam and the score earned; STEMi indicates whether
the student attended a STEM school; and studi is a vector of
student socio-demographic and educational characteristics
that includes 8th grade math score, status with respect
to race or ethnicity, free lunch, Limited English Proficiency
(LEP) and over-age for grade.19 The inclusion of lagged
achievement in middle school provides a value-added
interpretation to our results and serves as a proxy for
unobserved input histories and other unobserved student
characteristics.20 Our coefficient of interest, b1, captures