Empirical Bayes shrinkage (mostly) does not correct the measurement error in regression

In the value-added literature, it is often claimed that regressing on empirical Bayes shrinkage estimates corrects for the measurement error problem in linear regression. We clarify the conditions needed; we argue that these conditions are stronger than the those needed for classical measurement error correction, which we advocate for instead. Moreover, we show that the classical estimator cannot be improved without stronger assumptions. We extend these results to regressions on nonlinear transformations of the latent attribute and find generically slow minimax estimation rates.

💡 Research Summary

The paper critically examines the widespread belief in the education, health, and patent‑examination literatures that using Empirical Bayes (EB) shrinkage estimates as regressors automatically corrects for measurement error. The authors set up a canonical model where the researcher observes an outcome (Y_i), a noisy proxy (X_i) for an unobserved latent attribute (\mu_i), and the standard error (\sigma_i) of that proxy. Under the standard assumption that (X_i\mid\mu_i,\sigma_i\sim N(\mu_i,\sigma_i^2)) and that the data are i.i.d., the target parameter is the linear projection coefficient (\beta_0 = \operatorname{Cov}(Y_i,\mu_i)/\operatorname{Var}(\mu_i)).

The classical errors‑in‑variables correction estimates (\beta_0) by
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