Estimating and predicting treatment-effect heterogeneity across sites, in multi-site randomized experiments with few randomization units per site

We seek to estimate and predict treatment-effect heterogeneity across sites, in multi-site randomized controlled trials, with a large number of sites but few randomization units per site. As is well-known, an Empirical-Bayes (EB) estimator can be used to estimate the variance of the treatment effect across sites. We propose consistent estimators of the coefficients from ridge and OLS regressions of site-level effects on site-level characteristics that are unobserved but can be unbiasedly estimated, such as sites' average outcome without treatment, or site-specific treatment effects on mediator variables. In experiments with imperfect compliance, we also propose a non-parametric and partly testable assumption under which the variance of local average treatment effects (LATEs) across sites can be estimated. We revisit Behaghel et al (2014), who study the effect of counseling programs on job seekers job-finding rate, in 200 job placement agencies in France. We find considerable treatment-effect heterogeneity, both for intention to treat and LATE effects, and the treatment effect is negatively correlated with sites' job-finding rate without treatment.