Automatic Inference for Value-Added Regressions

A large empirical literature regresses outcomes on empirical Bayes shrinkage estimates of value-added, yet little is known about whether this approach leads to unbiased estimates and valid inference for the downstream regression coefficients. We study a general class of empirical Bayes estimators and the properties of the resulting regression coefficients. We show that estimators can be asymptotically biased and inference can be invalid if the shrinkage estimator does not account for heteroskedasticity in the noise when estimating value added. By contrast, shrinkage estimators properly constructed to model this heteroskedasticity perform an automatic bias correction: the associated regression estimator is asymptotically unbiased, asymptotically normal, and efficient in the sense that it is asymptotically equivalent to regressing on the true (latent) value-added. Further, OLS standard errors from regressing on shrinkage estimates are consistent in this case. As such, efficient inference is easy for practitioners to implement: simply regress outcomes on shrinkage estimates of value-added that account for noise heteroskedasticity.