Posterior Effects in Linear Random Coefficients Models

Accurately measuring heterogeneous effects is key to improving public policy design. We focus on predicting individual heterogeneity in linear random coefficients models, conditional on the sample. One application is estimating how sensitive each teacher’s value-added is to their knowledge of the program, conditional on the latter and their students’ test scores. We establish two new characterizations of these posterior effects, depending on whether the covariates are continuous or discrete. The first expresses these effects directly as a function of the data. Our associated series estimator is minimax adaptive, with various alternatives providing some robustness to the baseline assumptions. The second formulation characterizes the distribution of the random coefficients in terms of minimum distance, which is then used to compute the posterior. The associated estimator is consistent and implementable using optimal transport. Our methods reveal highly heterogeneous effects, identify the teachers most likely to benefit from training, thus providing tools to make personal development more cost-effective.