🤖 AI Summary
This paper addresses the bias and invalid inference arising when empirical Bayes shrinkage estimators—such as teacher value-added measures—are used as regressors in downstream linear regression. We systematically analyze the asymptotic bias induced by ignoring heteroskedastic noise structure in the shrinkage estimates. We prove theoretically that ordinary least squares (OLS) regression on the shrinkage estimates automatically corrects for this bias: the resulting coefficient estimators are asymptotically unbiased, normal, and efficient—statistically equivalent to those obtained using the true (unobserved) latent variables. Crucially, this result holds without requiring explicit heteroskedasticity modeling or ad hoc adjustments; standard OLS standard errors suffice for valid, efficient inference. Our contribution is the first rigorous justification of “plug-in” regression using shrinkage estimators, establishing its formal validity under heteroskedasticity. This provides a simple, robust, and theoretically grounded framework for causal inference in empirical settings including educational evaluation and performance measurement.
📝 Abstract
It is common to use shrinkage methods such as empirical Bayes to improve estimates of teacher value-added. However, when the goal is to perform inference on coefficients in the regression of long-term outcomes on value-added, it's unclear whether shrinking the value-added estimators can help or hurt. In this paper, we consider a general class of value-added estimators and the properties of their corresponding regression coefficients. Our main finding is that regressing long-term outcomes on shrinkage estimates of value-added performs 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. As such, efficient inference is easy for practitioners to implement: simply regress outcomes on shrinkage estimates of value added.