We introduce two empirical strategies harnessing the randomness in school assignment mechanisms to measure school value-added. The first estimator controls for the probability of school assignment, treating take-up as ignorable. We test this assumption using randomness in assignments. The second approach uses assignments as instrumental variables (IVs) for low-dimensional models of value-added and forms empirical Bayes posteriors from these IV estimates. Both strategies solve the underidentification challenge arising from school undersubscription. Models controlling for assignment risk and lagged achievement in Denver and New York City yield reliable value-added estimates. Estimates from models with lower-quality achievement controls are improved by IV.

The problem of when to control for continuous or high-dimensional discrete covariate vectors arises in both experimental and observational studies. Large-cell asymptotic arguments suggest that full control for covariates or stratification variables is always efficient, even if treatment is assigned independently of covariates or strata. Here, we approximate the behavior of different estimators using a panel-data-type asymptotic sequence with fixed cell sizes and the number of cells increasing to infinity. Exact calculations in simple examples and Monte Carlo evidence suggest this generates a substantially improved approximation to actual finite-sample distributions. Under this sequence, full control for covariates is dominated by propensity-score matching when cell sizes are small, the explanatory power of the covariates conditional on the propensity score is low, and/or the probability of treatment is close to 0 or 1. Our panel-asymptotic framework also provides an explanation for why propensity-score matching can dominate covariate matching even when there are no empty cells. Finally, we introduce a random-effects estimator that provides finite-sample efficiency gains over both covariate matching and propensity-score matching.