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

Jiafeng Chen, Jiaying Gu, Soonwoo Kwon

Abstract

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.

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

Abstract

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.

Paper Structure

This paper contains 15 sections, 9 theorems, 61 equations, 3 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Linear regression coefficients
  3. When is $\tilde{\beta}$ consistent?
  4. Efficiency and uniqueness of $\hat{\beta}$
  5. Implementation
  6. Regression coefficients involving nonlinear functions of $\mu_i$
  7. Simulation
  8. Empirical applications
  9. Impact of examiner on patent outcome
  10. Teacher value-added in a low-income country
  11. Conclusion
  12. Proof of \ref{['lemma:implication']}
  13. Proof of \ref{['thm:eff']}
  14. Proof of \ref{['thm:minimax']}
  15. Details for the empirical application to bau2020teacher

Key Result

lemma 1

Under basicassp, as:strong_low_level(1) implies as:strong_high_level; as:strong_low_level(2) implies as:strong_high_level(1), and as:strong_low_level(3) implies as:strong_low_level(2).

Figures (3)

  • Figure 1: Simulation results for linear-in-$\mu$ regressions.
  • Figure 2: Simulation results for regressions on nonlinear transformations of $\mu$.
  • Figure 3: Robust estimators for the predictive relationship between teacher value-added and log salary (Table 8, bau2020teacher)

Theorems & Definitions (20)

  • proof
  • proof
  • proof
  • lemma 1
  • theorem 1
  • proof
  • theorem 2
  • proof : Notes
  • lemma 1
  • proof
  • ...and 10 more