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Empirical Bayes shrinkage (mostly) does not correct the measurement error in regression
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Heterogeneity of individuals is a vital element in many important areas of inquiry within economics. Empirical Bayes methods robbins56 are applicable in many such settings to denoise individual fixed effects from noisy data. These methods are increasingly widely applied: Researchers use them to estimate individual effects of teachers (kane2008does, chetty2014measuringa, gilraine2020new), mobility of geographies (chetty2018impacts), value-added of hospitals (chandra2016health), skill of patent examiners (feng2020crafting), quality of managers (fenizia2022managers), and individual income dynamics (gu2017unobserved), among others.
Researchers also often hope to quantify how unobserved individual attributes, like a teacher value-added, predicts downstream economic outcomes, like long-term student outcomes. In such settings, there is a common intuition in the empirical literature that empirical Bayes shrinkage provides a correction for attenuation caused by statistical noise in a linear regression estimator jacob2005principle,kane2008estimating,chetty2014measuringa,angrist2023methods. For instance, angrist2023methods write “shrinkage corrects measurement error in models that treat school value-added as a regressor,” and this intuition appears widespread. On the other hand, the classical literature in errors-in-variable regression offers simple corrections for measurement error that apply in these settings fuller2009measurement, and it is unclear how including shrinkage estimates on the right-hand side of a regression compares to the classical approach.
This paper clarifies the conditions under which the regress-on-shrinkage estimator corrects for measurement error. Suppose the researcher would like to compute an infeasible regression of $Y_i$ on $\mu_i$, but only has access to noisy measurements $ (X_i, \sigma_i)$, where $X_i \sim \Norm(\mu_i, \sigma_i^2).$ Commonly, researchers fit the regression of $Y_i$ on $\hat\mu_i(X_i, \sigma_i)$, where $\hat\mu_i(X_i, \sigma_i)$ is a linear shrinkage estimate. Importantly, an often-ignored condition for the consistency of this regress-on-shrinkage estimator is that the degree of shrinkage---implicitly a function of the noise level $\sigma_i$---does not correlate with the outcome $Y_i$ once $\mu_i$ is controlled. Under such a condition, adding flexible functions in $\sigma_i$ to the infeasible regression would not change the coefficient. This is a condition much like precision independence walters2024empirical,chen2023empirical and can be economically unrealistic. On the other hand, in this setting, because the $\sigma_i^2$ are observed, the attentuation bias in the regression of $Y_i$ on $X_i$ is estimable---and thus can be directly corrected. Such classical measurement error corrections does not impose any precision independence assumption. This makes the classical corrections more robust and---in our view---preferable.
Second, we show that without stronger assumptions, this classical estimator is effectively the only reasonable estimator for the regression coefficient in the infeasible regression, up to asymptotic equivalence; it is thus (vacuously) semiparametrically efficient. This result shows that the deficiencies of the regress-on-shrinkage estimator are not due to insufficiently flexible empirical Bayes modeling. No amount of flexible modeling---despite tools proposed by, e.g., chen2023empirical,gilraine2020new,kwon2023optimalshrinkageestimationfixed---can improve on the classical measurement error correction.
Third, we extend our results to more complex settings, where the infeasible regression involves known nonlinear transforms $f(\mu_i)$ (e.g., indicators like $\one(\mu_i > \mu_0)$ may represent “high value-added” teachers). Under strong assumptions---that both the infeasible regression and parametric empirical Bayes models are correctly specified--–a regression of $Y_i$ on $\E[f(\mu_i) \mid X_i, \sigma_i]$ recovers the infeasible regression coefficients. However, we caution that these strong assumptions are difficult to relax without relinquishing appealing features of the resulting estimator, due to the fundamental statistical difficulty of deconvolution cai2011testing. To that end, we show that if the infeasible regression coefficient is estimable at polynomial $(n^{-\alpha})$ rates of convergence uniformly across data-generating processes that impose few assumptions, then $f (\cdot)$ must necessarily be an analytic function---an extreme smoothness requirement. Put differently, if $f(\cdot)$ is not smooth, then there is some data-generating process under which one would need exponentially-in-$k$ larger sample sizes to reduce uncertainty by a factor of $k$.
We compare these methods in simulation and across two empirical applications. The simulation results confirm that the classical estimator performs well, and substantially better than the regress-on-shrinkage estimator, across a wide range of data generating processes. Regress-on-shrinkage estimates can be severely biased even under settings that are calibrated to real data. Moreover, we confirm that regressions involving nonlinear transformations $f(\mu_{i})$ are indeed difficult, with widely dispersed estimates even with a reasonably large sample size.
We then compare the methods on two empirical applications. The first one revisits feng2020crafting which concerns about patent examination practise on patent outcomes. The second application revisits bau2020teacher, who analyze teacher value added in a low income country. We find potentially substantive economic differences when using the classical estimator rather than the regress-on-shrinkage estimator in one of the applications. We also find supportive evidence that the conditions needed for the regress-on-shrinkage estimator to provide reliable estimates are violated.
This paper is related to the classical errors-in-variable regression literature fuller2009measurement,bickel1987efficient as well as a recent literature on generated regressors. We highlight and compare a few. rose2022effects study multidimensional teacher value-added (e.g., value-added on criminal justice event, on math performance, etc.); they advocate for estimating the variance-covariance matrix of teacher value-added by correcting for measurement error, rather than by taking the variance-covariance matrix of empirical Bayes posterior means. We show that the same extends to regressions of downstream variables on value-added. deeb2021framework studies the regress-on-shrinkage estimator and proposes corrections to its standard error that accounts for the uncertainty in estimating empirical Bayes hyperparameters. In a similar setting, xie2025automatic establishes conditions under which regression-on-shrinkage estimators automatically yield valid inference without requiring additional adjustments. battaglia2024inference study the generated regressor problem with machine learning predictions for $\mu_i$. The setting is related but distinct---they do not impose the Gaussian structure commonly imposed in the empirical Bayes literature. To our knowledge, our efficiency and minimax rate results have not appeared in the literature.
This paper proceeds as follows. (ref) studies regression on latent $\mu_i$. (ref) presents our results for regression on $f(\mu_i)$ with nonlinear $f (\cdot)$. (ref) illustrate our results using a simulation and two empirical applications.
Suppose we observe $(Y_i, X_i, \sigma_i)_{i=1}^n$ for a sample of individuals. Throughout, we will refer to these individuals as teachers, but the applications extend beyond teacher value-added. Here, $Y_i$ is some outcome variable, $X_i$ is a noisy measure of an unobserved teacher attribute $\mu_i$, and $\sigma_i$ is the observed standard error for $X_i$. For instance, $Y_i$ would be some attribute of a teacher $i$ (e.g. teacher-level mean of student outcomes), $X_i$ would be the estimated teacher value-added of $i$, $\mu_i$ is the true teacher value-added for teacher $i$, and $\sigma_i$ is the estimated standard error for $X_i$. For expositional simplicity, our main results shall restrict to setups of the data that aggregate to the teacher level. (ref) discusses implementations of analogous approaches with disaggregated data.
Following the empirical Bayes literature, we assume $X_i \mid Y_i, \mu_i, \sigma_i \sim \Norm(\mu_i, \sigma_i^2)$ is Normally distributed and unrelated to $Y_i$, collected in the following assumption:
(ref)(2) and (3) are standard regularity assumptions. (ref)(1) is common in the empirical Bayes literature. The Normality of $X_i$ is motivated by the fact that $X_i$ is typically an estimate of $\mu_i$ with micro-data within a teacher: For instance, $X_i$ may be a teacher-level mean of student scores, and the central limit theorem provides a Gaussian approximation, where $\sigma_i^2$ is the estimated standard error walters2024empirical.
(ref)(1) also implies that the outcome $Y_i$ does not predict the noise component of $X_i$, i.e., $X_i \indep Y_i \mid \mu_i, \sigma_i^2$. This assumption may be violated if, for instance, $X_i$ is the sample mean of student test scores at the teacher level, but $Y_i$ is the average downstream outcome of that same set of students. Meanwhile, if $X_i$ and $Y_i$ come from different sets of students (and we assume student outcomes are independent) or if $Y_i$ is an underlying characteristic of teacher $i$,\footnote{For instance, chandra2016health consider a regression of hospital $i$ market size ($Y_i$) on hospital quality ($\mu_i$), which are then estimated from clinical outcome of patients ($X_i$). Here, it is reasonable to assume that hospital size is independent from the noise component of patient outcomes $X_i - \mu_i$. } then this assumption is reasonable. (ref) considers general cases where $Y_i$ needs not be independent of the noise in $X_i$.
Empirical researchers are often interested in the population infeasible regression of $Y_i$ on $\mu_i$: \[ Y_i = \alpha_0 + \beta_0 \mu_i + \eta_i, \addtocounter{equation}{1}\tag{\theequation} \label{model} \] where the population regression coefficient $\beta_0 = \frac{\cov(Y_i, \mu_i)}{\var (\mu_i)}$ by definition. Here, we do not treat (ref) as a linear restriction on $\E[Y_i \mid \mu_i]$, but simply as a definition for $\beta_0$. That is, $\beta_0$ is the OLS regression coefficient one would have obtained had one access to the true unobserved attribute $\mu_i$ and infinitely many observations. Regressions of this form appear in, among others, chandra2016health, chetty2014measuringa,jacob2005parents,jackson2018test,warnick2024instructor,mulhern2023beyond.\footnote{We take as given that the infeasible regression coefficient (ref) is the target parameter. It is possible, however, that empirical researchers may prefer other estimands. For instance, since decisions are only functions of $(X_i, \sigma_i)$, we might be interested instead in the statistical relationship between $Y_i$ and functions of $(X_i, \sigma_i)$. For instance, we might form a prediction $\hat\mu_i (X_i, \sigma_i)$ and we might want to assess how predictive the predicted teacher value-added is for $Y_i$, instead of how predictive the true $\mu_i$ is for $Y_i$. The former is more relevant, say, if we are more interested in assessing the quality of feasible predictions for teacher effects, rather than the inherent relationship between true teacher effects and outcomes.}
For instance, when $Y_i$ is the mean student long-term outcome for those taught by teacher $i$ and $\mu_i$ is a teacher value-added for a teacher experienced by student $i$, then $\beta_0$ is the coefficient of the best linear prediction the long-term outcome from true teacher value-added. Under appropriate identifying assumptions such that $Y_i$ is unbiased for the mean potential outcome of students assigned to teacher $i$, $\beta_0$ admits a causal interpretation as the best linear approximation to the conditional mean of teacher causal effects on true teacher value added.
Immediately, (ref) implies that $\beta_0$ is identified by the following formula.
(ref) suggests a simple analogue estimator for $\beta_0$ that replaces population covariances, variances, and expectations with their sample counterparts:\footnote{We shorthand $\E_n[W_i] = \frac{1}{n} \sum_{i=1}^n W_i$, $\var_n (W_i) = \E_n[W_i^2] - (\E_n[W_i])^2$, and $\cov_n(W_i, Z_i) = \E_n[W_i Z_i] - \E_n[W_i] \E_n[Z_i]$.} \[ \hat\beta = \frac{{\cov}_n(Y_i, X_i)}{{\var}_n(X_i) - {\E}_n [\sigma_i^2]}. \addtocounter{equation}{1}\tag{\theequation} \label{eq:inf_est} \] Under standard conditions, $\hat\beta$ is consistent and asymptotically Normal, whose asymptotic distribution can be consistently estimated by a nonparametric bootstrap given $(Y_i, X_i, \sigma_i)$. This estimator is classical from the errors-in-variable regression literature (e.g., Section 3.1 of fuller2009measurement, (23) in deaton1985panel) and more recently advocated by de2024estimating. Moreover, the consistency and Normality of $\hat\beta$ does not require $X_i$ to be Normally distributed.
Our first main point is to advocate for this classical estimator $\hat\beta$ as opposed to common estimation approaches for (ref). A popular---and standard---alternative estimator of $\beta_0$ is the regression coefficient of $Y_i$ on estimated empirical Bayes posterior means $\hat\mu_i(X_i,\sigma_i)$, following a parametric empirical Bayes procedure:
This approach is followed in, e.g., jacob2005parents, kane2008estimating, warnick2024instructor, jackson2018test, bau2020teacher,angelova2023algorithmic. The estimator $\tilde \beta$ is widely thought to be consistent for $\beta_0$, which we argue rests strong assumptions, often implicitly imposed.\footnote{ jacob2005parents (Appendix C) states that “one can easily show that using the EB estimates as an explanatory variable in a regression context will yield point estimates that are unaffected by the attenuation bias that would exist if one used simple OLS estimates.”
angrist2023methods write that “shrinkage corrects measurement error in models that treat school value-added as a regressor. Putting the unbiased but noisy estimate [in our notation, $X_i$] on the right-hand side of a regression results in attenuation bias toward zero due to classical measurement error; the posterior mean introduces non-classical measurement error that corrects this so that a regression with [$\mu_i (X_i, \sigma_i)$] on the right yields the same coefficient as using the true [$\mu_i$].” } We also note that when $\sigma_i^2 = \sigma^2$ are constant for all $i$, then $\tilde \beta = \hat\beta$, but they are no longer equal in heteroskedastic settings.
To this end, we show that, first, the estimator $\tilde \beta$ is consistent for $\beta_0$ only under much stronger conditions than (ref), rendering it less robust and less preferable to $\hat\beta$. Second, we show that it is impossible to improve upon $\hat\beta$ with any other procedure, including more flexible empirical Bayes procedures kwon2023optimalshrinkageestimationfixed,gilraine2020new,chen2023empirical, at least without imposing stronger assumptions. Under (ref), any consistent and asymptotically Normal estimator of $\beta_0$ is in fact asymptotically equivalent to $\hat\beta$. Thus, any empirical Bayes procedure can at most match the performance of $\hat\beta$.
Since $\mu \mathrel{\ensurestackMath{\stackon[1.5pt]{=}{\scriptscriptstyle\Delta}}} \E[\mu], \sigma_\mu^2\mathrel{\ensurestackMath{\stackon[1.5pt]{=}{\scriptscriptstyle\Delta}}} \var(\mu)$ are consistently estimable, let us restrict attention to an oracle counterpart of $\tilde \beta$ where $\mu,\sigma_\mu^2$ are known, namely \[ \tilde\beta^* \mathrel{\ensurestackMath{\stackon[1.5pt]{=}{\scriptscriptstyle\Delta}}} \frac{\cov_n(Y_i, \mu_i^*)}{\var_n(\mu_i^*)} \text{, with population limit } \tilde\beta_0\mathrel{\ensurestackMath{\stackon[1.5pt]{=}{\scriptscriptstyle\Delta}}} \frac{\cov_{P_0}(Y_i, \mu_i^*)}{\var_{P_0}(\mu_i^*)}. \]
We first formalize the folklore in the empirical literature on the consistency of $\tilde \beta$.
(ref)(1) is often referred to as forecast unbiasedness chetty2014measuringa,chetty2014measuringb,stigler19901988. It assumes that the empirical Bayes posterior means are reasonable predictors of the true unobserved attribute, in the sense that a hypothetical regression of the unobserved attribute $\mu_i$ on its predicted value $\mu_i^*$ returns a regression coefficient of $1$.\footnote{When the empirical Bayes prior is well-specified, i.e. $\mu_i^* = \E[\mu_i \mid X_i, \sigma_i]$, we have that $\E[\mu_i \mid \mu_i^*] = \mu_i^*$ by the law of iterated expectations, and (ref)(1) is satisfied. For Gaussian-prior empirical Bayes, even if the empirical Bayes model is misspecified, (ref)(1) would be true if we only assume (ref), meaning that $\sigma_i$ does not predict the first two moments of $\mu_i$.}
(ref)(2) is a key assumption that seems to be implicitly taken for granted in the literature. Importantly, the condition $\E[\eta_i X_i] = 0$---which we impose by definition of $\beta_0$ as the population projection coefficient---does not on its own justify (ref)(2).\footnote{Under homoskedasticity, i.e. $\sigma_i = \sigma$ for all $i$, $\mu_i^*$ is simply a linear function of $X_i$, and $\E[\mu_i^* \eta_i] = 0$ holds by construction; however, the same is not true when $\sigma_i$ are heterogeneous and can be correlated with $Y_i$.} The reason is that $\mu_i^*$ is a function of both $X_i$ and $\sigma_i$, and we have made no assumptions on how $\sigma_i$ interacts with $\eta_i$.
Under (ref), we review and formalize two arguments in the literature for (ref), and point out where each of the assumptions in (ref) is used. In particular, we have not found any work in the literature that explicitly mentions (ref)(2) or sufficient conditions for it, even though---as we shall see---(ref)(2) is crucial for the consistency of $\tilde\beta$.
We formalize the two main arguments used to justify the consistency of $\tilde{\beta}$. First, many papers justify $\tilde \beta_0 = \beta_0$ using an argument from jacob2005parents.
A second popular intuition justifies $\tilde\beta_0 = \beta_0$ by appealing to instrumental variables chetty2014measuringb.
(ref) is a set of high-level assumptions imposed to justify $\tilde \beta_0 = \beta_0$. The following assumptions are natural sufficient conditions for (ref). Though they are stronger than (ref), scenarios that satisfy (ref) but violate (ref) are economically knife-edge. Moreover, all assumptions in (ref) are testable.
A strong sufficient condition for (ref) is directly that $\sigma_i$ is completely independent of the joint distribution of $(Y_i, \mu_i)$.
(ref)(2)--(3) are in some respects weaker. (ref)(2) imposes that $\sigma_i$ does not predict $\mu_i$ at least in its first two conditional moments. This is a moment version of the precision independence condition discussed in chen2023empirical, who shows examples in which (ref)(2) fails to hold. This precision independence assumption alone does not involve $Y_i$, and is thus insufficient to justify $\tilde\beta_0 = \beta_0$.
(ref)(3) imposes two strong assumptions on $P_0$. The first is that the conditional expectation function of $Y_i$ given $\mu_i$ is in fact linear in $\mu_i$, viewing $\beta_0$ instead as the slope of that linear relationship rather than the population best linear approximation of $\E[Y_i \mid \mu_i]$. Second, (ref) imposes that $\sigma_i$ does not have additional predictive power over $Y_i$ given $\mu_i$. This is the analogue of a precision independence assumption for the outcome variable $Y_i$, and can fail due to similar reasons outlined in chen2023empirical. This assumption is rejected in our empirical application.
Since $\hat\beta$ is consistent for $\beta_0$ without imposing (ref) or (ref) at all, it should be preferred over the estimator $\tilde\beta$ on robustness grounds. Nevertheless, motivated by (ref)(2), we might wonder whether the defects of the regress-on-shrinkage estimator $\tilde \beta$ is due to insufficiently flexible empirical Bayes modeling. Perhaps the Normality assumption in (ref) can be exploited to yield efficiency benefits. The next subsection answers this question in the negative. Specifically, we show that under (ref), not only is $\hat\beta$ a semiparametrically efficient estimator for $\beta_0$, but all well-behaved consistent estimators for $\beta_0$ must be asymptotically equivalent to $\hat\beta$. This provides strong justification for $\hat\beta$ as the only estimator for $\beta_0$ asymptotically.
Let $\mathcal P_0$ be the set of distributions $P_0$ that satisfy (ref) as well as some technical conditions, stated in (ref). Let $\mathcal P$ be the set of distributions on the observed data $(Y_i, X_i, \sigma_i)$ that is induced by members of $\mathcal P_0$.
Following standard semiparametric theory van2000asymptotic,tsiatis2006semiparametric, we restrict our attention to regular and asymptotically linear (RAL) estimators, defined formally in (ref). RAL estimators are asymptotically Normal along all local perturbations of $P_0$ within $\mathcal P_0$. Indeed, most $\sqrt{n}$-consistent and asymptotically Normal estimators are RAL, and semiparametrically efficient estimators are exactly the optimal estimators among the class of RAL estimators.
We also recall Definition 2.2 in chen2018overidentification on local just identification, again formally stated in (ref). Local just identification is the semiparametric analogue to just identification in parametric GMM models. In a just identified GMM model, there are no additional moment conditions to exploit, and all GMM weightings yield the same estimator. Exactly analogously, in just identified semiparametric models, there are no (local) testable restrictions to exploit. Just like all weighted GMM estimators are exactly the same under just identification, under local just identification, all RAL estimators are asymptotically equivalent. They have the same asymptotic distribution, and they are all (vacuously) semiparametrically efficient.
The next theorem shows that many members in $\mathcal P$ are locally just identified by $\mathcal P$. The results of chen2018overidentification then imply that all RAL estimators for the regression coefficient $\beta_0$ are asymptotically equivalent to each other and to $\hat\beta$, which we also verify in the following theorem. This also implies that $\hat\beta$ is semiparametrically efficient, though in a vacuous sense.
(ref), an application of Example 25.35 in van2000asymptotic, rules out efficiency gains from alternative estimators under the minimal assumptions (ref). As a result, $\hat\beta$ is strongly justified for estimating $\beta$, since it is in fact the unique estimator for $\beta$ in an asymptotic sense. The implications of (ref) are not limited to the regression coefficient $\beta_0$; indeed, any regular parameter of $P$ admits unique RAL estimators in the sense of (ref)(1).
Imposing stronger assumptions than (ref) amounts to shrinking the model $\mathcal P$. Doing so would generally result in testable overidentification restrictions and weakly decrease the efficiency bound. Under (ref)(1) or similar assumptions, for instance, the estimator $\tilde \beta$ is indeed more efficient than $\hat\beta$ sullivan2001note, though yet more efficient estimators exist bickel1987efficient. As discussed, (ref) is much stronger than (ref).
One assumption that is arguably reasonable is the conditional Normality structure of $Y_i$, given that $i$ is a teacher and sometimes $Y_i$ is a sample mean of student outcomes at the teacher level. In particular, we might consider imposing $Y_i \mid \theta_i, X_i, \mu_i, \sigma_i, \nu_i \sim \Norm(\theta_i, \nu_i^2)$ for some unknown $\theta_i$ and known $\nu_i^2$, and assume that $(Y_i, \theta_i, \nu_i, X_i, \mu_i, \sigma_i) \iid P_0^*$ under the additional conditional Normality assumption. Simple modification to the proof of (ref) shows that such a restriction does not alter the conclusion of (ref).
We close this section with a discussion of implementing $\hat\beta$ in richer environments. First, given teacher-level data $(Y_i, X_i, Z_i, \sigma_i)$, we may want to include covariates $Z_i$ in the infeasible regression: \[ Y_i = \alpha_0 + \beta_0 \mu_i + \gamma_0' Z_i + \epsilon_i. \] Suppose $Z_i$ does not predict the noise component of $X_i$, then by Frisch--Waugh--Lovell, \[ Y_i = \beta_0 (\mu_i - \operatorname{proj}(\mu_i \mid Z_i)) + \epsilon_i \] and $X_i - \operatorname{proj}(X_i \mid Z_i) \sim \Norm(\mu_i - \operatorname{proj}(\mu_i \mid Z_i), \sigma_i^2)$, where $\operatorname{proj}(\cdot \mid Z_i)$ is the population linear projection of a random variable onto $Z_i$. Thus, our analysis above applies to the partialled out teacher effects $\mu_i- \operatorname{proj}(\mu_i \mid Z_i)$ and its measurement $X_i - \operatorname{proj}(X_i \mid Z_i)$.
More generally, it is common practice to consider a student-level regression. For $j$ a student associated with teacher $i$, we consider the infeasible regression \[ Y_{ij} = \alpha_0 + \beta_0 \mu_j + \gamma_0'Z_{ij} + \epsilon_{ij} \addtocounter{equation}{1}\tag{\theequation} \label{eq:infeasible_disagg} \] where $X_{ij}$ is unbiased for $\mu_j$.\footnote{One plausible and precise sampling process is as follows. Suppose \[ (N_i, \mu_i, (Y_{ij}, Z_{ij}, X_{ij})_{j=1}^{N_i}) \iid P_0 \] and $P_0$ is such that $\E_{P_0}[X_{ij} \mid \mu_i, N_i] = \mu_i$. We also assume that conditional on $\mu_i, N_i$, the student-level variables $(Y_{ij}, Z_{ij}, X_{ij})_{j=1}^ {N_i}$ are uncorrelated across $j$.
To connect the aggregate and disaggregated setups, suppose $Z_{ij} = Z_i$ is not a function of the student. Then the population OLS regression (ref) is equivalent to the teacher-level weighted least squares regression \[ Y_i = \alpha_0 + \beta_0 \mu_j + \gamma_0' Z_i + \epsilon_{ij} \] where $Y_i = \frac{1}{N_i} \sum_{j} Y_{ij}$ and observations are weighted by $N_i$. } Here, we do not assume $X_ {ij}, Y_ {ij}, Z_ {ij}$ are uncorrelated within a teacher $i$, and so $Y,Z$ may predict the noise component of $X$. This infeasible regression coefficient $\theta_0 = (\alpha_0, \beta_0, \gamma_0')'$ can be written as \[ \theta_0 = \E_{P_0}\bk{ \sum_{j=1}^{N_i} W_{ij} W_{ij}' }^{-1} \E_{P_0}\bk{\sum_{j=1}^{N_i} W_{ij} Y_{ij}}. \] The quantities $\E_{P_0}\bk{ \sum_{j=1}^{N_i} W_{ij} W_{ij}' }$, $\E_{P_0}\bk{\sum_{j=1}^{N_i} W_{ij} Y_{ij}}$ involves infeasible terms such as $\E \sum_{j} \mu_i^2$, $\E \sum_j \mu_i Z_{ij}$, $\E\sum_j \mu_iY_{ij}$. Fortunately, they can be similarly estimated from $X_{ij}$: For instance, for $ X_i = \frac{1} {N_i} \sum_{j=1}^{N_i} X_{ij}$, \[ \E\bk{\sum_{j=1}^{N_i} \mu_i Z_{ij}} = \E\bk{ \sum_{j=1}^{N_i} X_i Z_{ij}} - \E\bk{\underbrace{\frac{1}{N_{i}-1} \sum_{j=1}^{N_i} Z_ {ij} (X_ {ij} - X_i)}_{\text{Empirical covariance between $X_{ij}$ and $Z_{ij}$}} }. \] Thus, we may use $\sum_{j=1}^{N_i} X_i Z_{ij} - \frac{1}{N_{i}-1} \sum_{j=1}^{N_i} Z_ {ij} (X_ {ij} - X_i)$ to substitute for $\sum_j \mu_i Z_{ij}$, and similarly for other terms in $\theta_0$. This construction is very similar to $\hat\beta$, since it essentially debiases an empirical moment with $X_i$ by adjusting for the impact of second moments. In a slightly different context, recent work by de2024estimating proposes this estimator as well.
Even more conveniently, it turns out that such an analogue can be simply implemented by instrumenting for $X_ {ij}$ in following regression with a leave-one-out instrument $X_{i,-j} = \frac{1} {N_i-1} \sum_{k\neq j} X_{ik}$ \[ Y_{ij} = \alpha_0 + \delta_0 X_{ij} + \gamma_0'Z_{ij} + \epsilon_{ij}. \] devereux2007improved shows\footnote{We thank Patrick Kline for this reference.} that these two estimates are in fact numerically equivalent. This equivalence means that researchers can conveniently implement the measurement error correction by constructing the leave-one-out instrument and use off-the-shelf routines, without other multistep procedures.
Sometimes empirical researchers are interested in the projection coefficient on some known nonlinear function $f(\cdot)$ of the latent quantities $\mu_i$: \[ Y_i = \rho_0 + \tau_0 f(\mu_i) + \eta_i. \] For instance, we might be interested in how being a “good teacher” predicts outcomes, and being a good teacher is defined as $\one(\mu_i > \mu_0)$ for some threshold $\mu_0$. For an example of setting where $\beta_{0}$ is of interest under such specification, see Section 3.2.2 of bruhn2022regulatory.
Unfortunately, estimating $\tau_0$ involves some unpleasanat tradeoffs for the analyst, as we no longer have access to a simple estimator like $\hat\beta$. On the one hand, imposing a strong assumption does allow us to recover $\tau_0$ by regressing $Y_i$ on the correctly specified parametric empirical Bayes posterior means for $f (\mu_i)$. The downside of this approach is that the assumptions involved can be quite strong, and are unlikely to be justified. On the other hand, without these assumptions, estimating $\tau_0$ is fundamentally difficult. This difficulty is in the sense that the the error of the best possible estimator contracts at subpolynomial rates in the sample size, meaning that reducing uncertainty requires exponentially large sample sizes.
We begin with the first approach and document a simple estimator under strong assumptions on model specification.
Operationizing (ref) requires us to estimate $\E[f(\mu_i) \mid X_i, \sigma_i]$ (importantly, distinct from $f(\E[\mu_i \mid X_i, \sigma_i])$). Traditional empirical Bayes methods often specify a parametric model, e.g., $\mu_i \mid \sigma_i \sim \Norm (\mu, \sigma_\mu^2)$. Regressing $Y_i$ on the estimated empirical Bayes posterior means under such models can be interpreted as estimating (ref). Nevertheless, the price of such an interpretation is the strong assumptions imposed: (ref) and well-specified parametric prior.
Without these strong assumptions, $\tau_0$ remains identified, as it is a function of the joint distribution of $(Y_i, \mu_i)$. This joint distribution is identified from the joint distribution of $(Y_i, X_i, \sigma_i)$ via deconvolution. However, as is typical with deconvolution, the problem of estimating $\tau_0$ becomes prohibitively difficult without making parametric restrictions fan1993nonparametric. We illustrate this point with a minimax lower bound for $\tau_0$.
Let us first define the class of distributions the lower bound is over. Fix some function $f: [-1,1] \to \R$ such that it is bounded ($\norm{f}_\infty < \infty$) and nonconstant in the sense that $V(f) \mathrel{\ensurestackMath{\stackon[1.5pt]{=}{\scriptscriptstyle\Delta}}} \var_{U \sim \Unif[-1,1]}(f(U)) > 0$. Let $\mathcal Q_0$ collect all distributions $Q_0$ for $(Y_i ,\mu_i, X_i, \sigma_i)$ satisfying (ref) where (a) $\mu_i$ and $Y_i$ are supported within the interval $ [-1,1]$, (b) $\var_{Q_0}(f(\mu_i)) > \frac{1}{2} V(f)$ to avoid degeneracy, and (c) $\sigma_i \le 1$.\footnote{The restrictions made on $\mathcal Q_0$ is for convenience. Note that the minimax rate over a larger set of distributions is necessarily bounded below by the minimax rate over $\mathcal Q_0$. } Let $\mathcal Q = \br{P^\mathrm{obs} (P_0) : P_0 \in \mathcal Q_0}$. Relative to $f$, let $\tau_0(Q_0) = \frac{\cov_{Q_0}(Y, f(\mu))}{\var_{Q_0}(f(\mu))}$. The minimax risk of estimating $\tau_0 (Q_0)$ is the worst-case squared error of a given estimator $T$ over $\mathcal Q_0$, \[ R_n(\mathcal Q_0, f) = \inf_T \sup_{Q_0 \in \mathcal Q_0} \E_{Q_0}\bk{ \br{T(Y_{1:n}, X_{1:n}, \sigma_{1:n}) - \tau_0(Q_0)}^2 }, \] optimized over choices of all estimators.
The minimax rate measures the difficulty of estimating $\tau_0(Q_0)$ over $Q_0$. One way to interpret $R_n$ is as the value of a zero-sum game where an analyst moves first and chooses and estimator $T$, and an adversary moves second and chooses a distribution $Q_0 \in \mathcal Q_0$. If different distributions $Q_0, Q_1 \in \mathcal Q_0$ produce very different $\tau_0$ but very similar data, then the analyst would suffer large losses as they would be unable to distinguish these scenarios.
For well-behaved estimands (e.g. when $f(\mu) = \mu$), $R_n = O(1/\sqrt{n})$ contracts at the familiar parametric rate.\footnote{To wit, the restrictions on $\mathcal Q_0$ implies that $\tau_0(Q_0)$ is bounded by some $M > 0$. Thus, we can consider the estimator $T = \max(\min(\hat\beta, M), -M)$. The truncation at $M$ is so that expectations always exist. } In sharp contrast, when $f (\cdot)$ is not an analytic function, the minimax rate vanishes slower than any polynomial in $n$. To reduce the uncertainty in our estimates proportionally by $t \in (0,1)$, we therefore would require sample sizes exponential in $1/t$.
Analytic functions are infinitely differentiable and admit Taylor expansions everywhere. However, many---if not most---transformations of interest are not analytic as they are typically non-smooth. For these functions, (ref) is then a negative result, in the sense that the regression coefficient $\tau_0$ associated with them is fundamentally difficult to estimate without assuming further structure.
The proof to (ref) uses Le Cam's two-point method by constructing $Q_1, Q_0 \in \mathcal Q_0$ that generate similar data but very different $\tau_0(Q)$. A key step of the proof constructs $Q_1, Q_0$ so as to reduce the problem of estimating the regression coefficient $\tau_0$ to the problem of estimating a mean $\E_Q[f(\mu)]$. Minimax rates for the latter problem then follow from the techniques developed by cai2011testing and presented in wu2020polynomial.\footnote{cai2011testing specifically study estimating $\E|\mu|$. Their proof technique extends to $\E[f(\mu)]$ by applying some results in approximation theory known as Bernstein's and Jackson's theorems. We have not seen these results stated in the statistical literature. Our proof of (ref) makes it precise. } The proof to (ref) similarly applies to regression problems where $f(\mu)$ appears on the left-hand side. That is, the minimax rate for the population regression coefficient $\tau_0$ in the infeasible specification\[ f(\mu_i) = \rho_0 + \tau_0 W_i + \eta_i \] suffers from similar subpolynomial rates if $f$ is non-analytic.
We follow chen2023empirical and calibrate our data generating process (DGP) to the Opportunity Atlas (chetty2020opportunity), which provides (unshrunk) economic mobility estimates $X_{i}$ and their standard errors $\sigma_{i}$. One measure of economic mobility $\mu_{i}$ of tract $i$ they consider is the probability that a Black individual becomes relatively high-income (i.e., has family income in the top 20 percentiles nationally) after growing up relatively poor in tract $i$. (i.e., with parents at the 25th percentile nationally). We have 10,058 tracts in the data.
Taking this mobility measure as our measure of interest, we estimate the conditional mean function $ \E [\mu_{i} \mid \sigma_{i} ] = \E[X_{i} \mid \sigma_{i} ]$ and the conditional variance function $\var(\mu_{i} \mid\sigma_{i}) = \var(X_{i}\mid \sigma_{i}) -\sigma_{i}^{2}$ via local linear regression implemented by calonico2019nprobust; denote the estimates of the conditional mean and conditional variance functions as $\hat{m}(\cdot)$ and $\hat{s}^{2}(\cdot)$, respectively. We use these estimates to draw $\mu_{i}$'s from the distribution $\mu_{i} \mid \sigma_{i} \sim \Norm(\hat{m}(\sigma_{i}), \hat{s}^{2}(\sigma_{i}))$. The outcome variable $Y_{i}$ for the regression of interest is generated by \[ Y_{i} = \beta_{\mu} \mu_{i} + \beta_{\sigma} \log_{10} \sigma_{i} + u_{i}, \] where $u_{i} \sim \Norm(0, 1)$. Under this DGP, the linear projection coefficient---in a hypothetical regression of $Y_i$ on $\mu_i$---we wish to estimate is given by \[\beta_{0} = \beta_{\mu} + \frac{\cov(\mu_{i}, \log_{10} \sigma_{i})}{\var(\mu_{i})} \beta_{\sigma}.\] We vary $(\beta_{\mu},\beta_{\sigma})$\footnote{Specifically, we vary $(\beta_{\mu},\beta_{\sigma})$ so that $\var(\mu)^{1/2}\beta_{\mu}$ ranges from $-.3$ to $.3$ and $\var(\log_{10}(\sigma))^{1/2}\beta_{\sigma}$ from $0$ to $.3$. Combined with the fact that we take $\var(u_{i}) = 1$, this results in a DGP where the regressors have realistic explanatory power.} and compare the performance of our proposed estimator $\hat{\beta}$, given in (ref), with the regression-on-shrinkage estimator $\tilde{\beta}$, given in (ref).
Figure (ref) summarizes the results of our simulation exercise in this setting by comparing the MSE and bias of the two estimators across the different DGPs. As expected, the MSE of $\hat{\beta}$ is smaller than the $\tilde{\beta}$ in almost all specifications, and this improvement is substantial in a wide range of simulations. For example, when $\var(\mu)^{1/2}\beta_{\mu} = .2$ and $\var(\log_{10}(\sigma))^{1/2}\beta_{\sigma} = .05$, which is a setting where, roughly speaking, $\log_{10}(\sigma)$ has a low explanatory power compared to $\mu$, the log MSE ratio is around $4.36$, which indicates that the MSE of $\tilde{\beta}$ is about $78$ times greater than that of $\hat{\beta}$. Similarly, the bias plot shows that $\hat{\beta}$ having a smaller bias than $\tilde{\beta}$ across all DGPs, which is expected given that $\hat{\beta}$ is unbiased across all DGPs we consider.
For the simulations regarding nonlinear transformations of $\mu_{i}$, we use a simpler DGP for $\mu_{i}$. Specifically, we take an estimate of the unconditional mean and variance implied by the previous DGP, $\hat{m} = n^{-1}\sum_{i=1}^{n}\hat{m}(\sigma_{i})$ and $\hat{s}^{2} = n^{-1}\sum_{i=1}^{n}\hat{s}^{2}(\sigma_{i}) + n^{-1}\sum_{i=1}^{n}(\hat{m}(\sigma_{i}) - \hat{m})^{2}$ and draw $\mu_{i} \sim G$, where $G$ is the distribution function of $\Norm(\hat{m}, \hat{s}^{2})$. Then, we generate the outcome variable as \[ Y_{i} = \tau_{0}\one(\mu_i > \mu_0) + u_{i}, \] where $u_{i} \sim N(0, 1)$ is independent of $(\mu_{i}, \sigma_{i})$, and $\mu_{0}$ is some fixed threshold assumed to be known. We vary $\tau_{0}$ and $\mu_{0}$ to learn about the performance of the different estimators we describe below.
We consider three distinct estimators. First is the oracle estimator $\hat{\tau}_{0}^{\mathrm{oracle}}$ obtained by regressing $Y_{i}$ on the true $\E_{G}[\one(\mu_i > \mu_0) \mid X_{i}, \sigma_{i}]$ using knowledge of the true $G$. By Proposition (ref), $\hat{\tau}_{0}^{\mathrm{oracle}}$ is consistent at the usual $n^{-1/2}$-rate. The second estimator we consider is the nonparametric empirical Bayes (NPEB) estimator $\hat{\tau}^{\mathrm{NPEB}}_{0}$ obtained by regressing $Y_{i}$ on $\E_{\hat{G}}[\one(\mu_i > \mu_0) \mid X_{i}, \sigma_{i}]$, where $\hat{G}$ is obtained by using nonparametric maximum likelihood as in gilraine2020new. This estimator is consistent, but potentially at a slower rate, as implied by (ref). Finally, we also consider a plug-in estimator $\hat{\tau}_{0}^{\text{plug-in}}$ which is obtained by regressing $Y_{i}$ on $\one(\E_{G}[\mu_{i} \mid X_{i}, \sigma_{i}] > \mu_{0})$.\footnote{We use an “infeasible” plug-in estimator in the sense that we take the true $G$ to calculate $\E_{G}[\mu_{i} \mid X_{i}, \sigma_{i}]$. Hence, the results should be considered an upper bound on how well a plug-in rule can do.} This estimator mimics the rather common empirical practice of plugging in the shrunk estimates $\hat{\mu}_{i}$ for $\mu_{i}$ in various downstream analyses.
Figure (ref) shows the simulation results for two DGPs where we set $\tau_{0}$ so that $\var(\one(\mu_i > \mu_0))^{1/2}\tau_{0} = 1$. We consider two threshold levels for $\mu_{0}$, the .75-quantile and .90-quantile of $G$. The plug-in estimator $\hat{\tau}_{0}^{\text{plug-in}}$ is substantially biased, as expected since $\hat{\tau}_{0}^{\text{plug-in}}$ plugging the posterior of $\mu_{i}$ into nonlinear transformation does not in general correct the measurement error. On the other hand, we see that $\hat{\tau}_{0}^{\mathrm{oracle}}$ is unbiased, as expected by Proposition (ref). Note that $\hat{\tau}_{0}^{\mathrm{oracle}}$ shows higher accuracy when $\mu_{0}$ is set at the .75-quantile of $G$, because this results in regressors with higher variance. Finally, while the estimates generated by $\hat{\tau}_{0}^{\mathrm{NPEB}}$ are centered around the true parameter value, the distribution is quite dispersed and substantially noisier than $\hat{\tau}_{0}^{\mathrm{oracle}}$. Given that the sample size is moderately large at 10,058, this demonstrates the slow convergence rate predicted by Theorem (ref). The simulation results for other specifications of $\tau_{0}$ and $\mu_{0}$ are qualitatively similar.
We revisit how patent examiners leniency (i.e., leniency in patent crafting) predicts the outcomes of the patents they decide to grant feng2020crafting. As background, for each patent application, an examiner from a specific art unit is randomly assigned to assess the application and determine its merit for patent granting. There is a significant interaction between patent applicants and examiners during the revision of claims until the patent is either granted or the application is withdrawn. All other factors being equal, a stricter examiner may raise numerous questions regarding prior art, appropriate citations, and required clarifications before granting the patent, while a lenient examiner may provide minimal feedback for similar patents. This crafting process could influence the quality and clarity of the patent, should it be granted, which in turn could affect its market value, litigation propensity, and citations.
In particular, we consider the following infeasible regression: \[ Y_{j}= \beta_0 \mu_{i(j)} + a_{ut(j)} + \epsilon_j \] where $j$ indexes the patent, $i$ the examiner, $u$ the art unit, and $t$ the filing year. The parameter $\beta_0$ represents the effect of examiner $i$'s leniency during the crafting process, denoted as $\mu_{i(j)}$, on the patent outcome $Y_j$. Following feng2020crafting, we use two measures of leniency. The first measure accounts for the average percentage change in the number of words per claim between the application and post-grant stages. The second measure records the percentage change in the total number of claims. These measures are constructed using the pre- and post-grant publication data\footnote{These data are publically available at \url{https://patentsview.org/download/pg_claims} and \url{https://patentsview.org/download/claims}.}.
An examiner who demands an increase in the length of the claims---typically to clarify distinctions from existing grants and enhance precision---is considered more careful and stringent. Similarly, an examiner who requests a reduction in the number of claims to limit the scope of the patent is also regarded as more stringent. As expected, these measures are, at best, noisy estimates of the true leniency $\mu_i$.
We focus on the citation outcomes of patents within three years of grant. Additionally, we report results on the probability of purchase and litigation by a Patent Assertion Entity (PAE). In this case, the outcome variable takes a value of 1 if the granted patent is purchased by a PAE and is also litigated in court for infringement. We obtain the set of patents litigated by PAE firms from the Stanford NPE Litigation Database, which categorizes assertors into several categories. We classify acquired patents, failed startups, individual-inventor-stated companies, and individuals as Patent Assertion Entities (PAEs). As documented in feng2020crafting, these entities acquire patents from third parties and generate revenue by asserting them against alleged infringers, commonly known as patent trolls. We also include additional cases from the Unified Patent and Lex Machina databases\footnote{We thank Dr. Tommaso Alba for providing these data.}. The citation data is publicly available from the USPTO and includes all citations received by a patent since its grant, updated annually.
The discussion below excludes the art unit and year fixed effects. In practice, we first purge these fixed effects from all other variables as discussed in Section (ref).\footnote{We consider only those examiners who serve in a single art unit and have granted at least 10 patents, leaving us with 4,615 examiners.} Let the measure of leniency for each patent $j$ reviewed by examiner $i$ be denoted by $E_{ij}$ and $N_i$ represents the number of patents granted by examiner $i$. Then $Y_i = \frac{1}{N_i} \sum_{j: i(j) = i}Y_{ij}$ is the examiner-level mean outcome\footnote{Here we index the patent level outcome by $Y_{ij}$ rather than $Y_j$ to clarify that the patent level variables are more granular than the examiner level.}, and $E_i= \frac{1}{N_i} \sum_{j: i(j) = i} E_{ij}$ is the examiner's leniency, for which $ E_i \mid \mu_i \sim N(\mu_i, \sigma_i^2). $ where $\sigma_i^2$ is proportional to the inverse of $N_i$.
Replacing $\mu_{i(j)}$ with $E_{i(j)}$ introduces attenuation bias. Accounting for the fact that we aggregate the patent level regression to the examiner level, the population least-square objective is \[ \mathbb{E} \Big[ \sum_{j: i(j) = i} (Y_{ij} - \delta_0 - \beta_0 \mu_i)^2 \Big ] = \mathbb{E}[N_i (Y_i - \delta_0 - \beta_0 \mu_i)^2] + \text{constant} \] Therefore, the population coefficient $\beta_0$ is the weighted least square coefficient at the examiner level, with weights proportional to $N_i$. The natural sample analogue, correcting for measurement error, leads to: \[ \hat \beta = \frac{\sum_i W_i Y_i E_i - (\sum_i W_i E_i)(\sum_i W_i Y_i)}{\sum_i W_i E_i^2 - (\sum_i W_i E_i)^2 - \sum_i W_i \sigma_i^2} \quad W_i = N_i/\sum_i N_i.\]
An additional concern may be that the left-hand side variable is also a measurement of a latent effect and contains measurement error. In this case, it is possible that the measurement error in both $Y_{ij}$ and $E_{ij}$ are correlated. To model this, we consider the following model \[
\sim N \left (
, \Sigma \right)\quad and \quad
\sim N \left (
, \Sigma_i \right) \] where the second display is the model for the examiner-specific sample averages $(Y_i, E_i)$ and $\Sigma_i = \Sigma/N_i$. The coefficient $\beta_0$ can be related to the population coefficient of a regression of $Y_i$ on $E_i$ by $ \beta = \frac{\cov(Y_i, E_i) - \Sigma_{i, 12}}{\var(E_i) - \Sigma_{i, 22}}. $ In the presence of heterogeneous $N_i$, the natural weighted least square estimator with the correction due to measurement error on both sides yields \[ \check{\beta} = \frac{\sum_i W_i Y_i E_i - (\sum_iW_i E_i) (\sum_i W_i Y_i) - \sum_i W_i \hat \Sigma_{i,12}}{\sum_i W_i E_i^2 - (\sum_i W_iE_i)^2 - \sum_i W_i \sigma_i^2} \] where $\hat \Sigma_{i,12}$ is estimated based on patent level data $(Y_{ij}, E_{ij})$ and normalized by $N_i$.
In contrast to the two proposed estimators above, the common regress-on-shrinkage approach constructs the posterior mean as: \[ \bar E_i = \hat \mu_0 + \frac{\hat \sigma_e^2}{\sigma_i^2 + \hat \sigma_e^2} (E_i - \hat \mu_0) \] where $\hat \mu_0$ and $\hat \sigma_e^2$ are often estimated via moments of $E_i$. The resulting weighted least square estimator is: \[ \tilde \beta = \frac{\sum_i W_i Y_i \bar E_i - (\sum_i W_i \bar E_i)(\sum_i W_i Y_i)}{\sum_i W_i \bar E_i^2 - (\sum_i W_i \bar E_i)^2}. \]
(ref) reports the estimates for $\beta_0$ using different methods. The column with label FE represents the OLS estimation without correcting for measurement error in the leniency measures. This estimator biases towards zero due to un-corrected measurement error. The next column reports the regress-on-shrinkage estimator. The last two columns (labeled as FE-Corrected and FE-Corrected-Twoside) report estimates based on the measurement error correction. FE-Corrected only considers measurement error in leniency measures while FE-Corrected-Twoside accounts for measurement error from both the left and right hand side variables. The standard errors in the bracket for our proposed method are obtained via bootstrap with 999 bootstrap repetitions.
For the PAE litigation outcome, there is little difference between the shrinkage method and the direct correction we proposed. However, for the citation outcome, the difference is noticeable. As reported in (ref), individual variances $\sigma_i$ appear to have a significant impact on the citation outcome, but not on the PAE litigation outcome. This suggests some evidence of a violation of the assumptions necessary to ensure the consistency of the regress-on-shrinkage estimator.
We revisit bau2020teacher, who compute teacher-value added in a development economics context. In their Table 8, they consider whether value-added predicts log salary of teachers, for teachers in the public sector and in the private sector. bau2020teacher find that value added does not predict salary for teachers in the public sector but do for those in the private sector.
For generating shrunken teacher-value added estimates, bau2020teacher do not directly compute from (ref) given a set of preliminary noisy estimates. Rather, both the signal variance $\hat\sigma_\mu^2$ and the idiosyncratic variances $\sigma_i^2$ are estimated jointly through a sample-splitting procedure. A somewhat unusual feature of this computation is that the estimated total variance of teacher estimates does not equal the sum of $\hat\sigma_\mu^2$ and $\sigma_i^2$. In what follows, we take $\E_n \sigma_i^2$ from this procedure as the sampling variance of the noisy estimates, and we compute $\hat\sigma_\mu^2 = \var_n(X_i) - \E_n\sigma_i^2$ as an analogue estimator. See (ref) for details.
We show alternative estimates in (ref). The differences in the estimates are broadly explained by different methods for estimating $\sigma_\mu^2$. In this empirical example, our proposed estimator yields results similar to those of the conventional method, reaffirming the authors' original findings under weaker conditions. This is consistent with the fact that the shrinkage factors do not appear to be strongly correlated with $\mu_i$.
This paper critically examines the widespread practice of using the empirical Bayes shrinkage estimator to correct for measurement errors in regression models incorporating individual latent effects. Our analysis reveals that this approach only provides reliable estimates for the regression coefficients under unnecessarily stringent conditions. We demonstrate that the classical correction, which we advocate, holds under weaker assumptions and cannot be asymptotically improved when the latent effect enters the regression model linearly. In cases where the latent effect enters non-linearly, empirical Bayes shrinkage leads to slower minimax estimation rates. These findings underscore the limitations of using the regress-on-shrinkage estimator as a method for correcting measurement error in regression models.