We present evidence of a positive relationship between school starting age and children's cognitive development from ages 6 to 18 using a fuzzy regression discontinuity design and large-scale population-level birth and school data from the state of Florida. We estimate effects of being old for grade (being born in September vs. August) that are remarkably stable--always around 0.2 SD difference in test scores--across a wide range of heterogeneous groups, based on maternal education, poverty at birth, race/ethnicity, birth weight, gestational age, and school quality. While the September-August difference in kindergarten readiness is dramatically different by subgroup, by the time students take their first exams, the heterogeneity in estimated effects on test scores effectively disappears. We do, however, find significant heterogeneity in other outcome measures such as disability status and middle and high school course selections. We also document substantial variation in compensatory behaviors targeted towards young-for-grade children. While the more affluent families tend to redshirt their children, young-for-grade children from less affluent families are more likely to be retained in grades prior to testing. School district practices regarding retention and redshirting are correlated with improved outcomes for the groups less likely to use those remediation approaches (i.e., retention in the case of more affluent families and redshirting in the case of less affluent families.) Finally, we find that very few school policies or practices mitigate the test score advantage of September-born children.