What differences a day can make : quantile regression estimates of the distribution of daily learning gains

Recent research exploits a variety of natural experiments that create exogenous variation in annual school days to estimate the average effect of formal schooling on students' academic achievement. However, the extant literature's focus on average effects masks potentially important variation in the effect of formal schooling across the achievement distribution. We address this gap in the literature by estimating quantile regressions that exploit quasi-random variation in the number of school days between kindergarten students' fall and spring tests in the nationally representative Early Childhood Longitudinal Study – Kindergarten Cohort (ECLS-K). The marginal effect of a typical 250-day school-year on kindergarten students' math and reading gains varies significantly, and monotonically, across the achievement distribution. For example, the marginal effect on the 10th percentile of the reading achievement distribution is 0.9 test score standard deviation (SD), while the marginal effect on the 90th percentile is 2.1 test score SD. We find analogous results for math achievement.