Repairing Locally Misspecified GMM: An Empirical Bayes Approach

Technical Summary: Repairing Locally Misspecified GMM: An Empirical Bayes Approach

Problem Statement
Generalized Method of Moments (GMM) is a standard technique for estimating parameters in overidentified models ($m > p$), where the number of moment restrictions exceeds the number of parameters. While foundational work by Sargan (1958) and Hansen (1982) established frameworks for testing overidentifying restrictions via the $J$-statistic, empirical applications frequently reject these restrictions. Standard econometric practice often treats such rejections with ambivalence, acknowledging that models are approximations, yet rarely quantifies the nature of the approximation error. When moment conditions are misspecified, standard GMM estimators can exhibit first-order bias and invalid inference. Existing approaches to local misspecification (e.g., Andrews, Gentzkow, and Shapiro 2017) typically treat specification errors as fixed constants. This paper addresses the gap in handling specification errors that are random and exchangeable across moment conditions, a scenario common in applied work with multiple instruments (e.g., Angrist and Krueger 1991) where the specific source of excludability violations is unknown.

Methodology
The paper develops a framework for GMM estimation under local misspecification where the population moment conditions are perturbed by a vector of specification errors $b \in \mathbb{R}^m$ that shrinks at the rate $n^{-1/2}$. The core innovation is treating the elements of $b$ as exchangeable random variables rather than fixed constants.

1. Exchangeable Misspecification Framework:
   The scaled sample moments are decomposed as $\sqrt{n}\hat{g}(\theta_0) = b + \varepsilon_n$, where $\varepsilon_n$ represents sampling noise and $b$ represents the specification error. The author assumes $b$ is exchangeable with bounded fourth moments, implying a common marginal mean $\bar{\mu}$ and variance $\bar{\sigma}^2$. The analysis conditions on the realized mean $\mu$ and variance $\sigma^2$ of the specific $b$ vector, treating them as hyperparameters.

2. Hyperparameter Estimation:
   The paper proposes estimators for the hyperparameters $\mu$ and $\sigma^2$ in an asymptotic regime where both the sample size $n$ and the number of moments $m$ grow, but $m^2/n \to 0$. This "moderately overidentified" regime avoids the first-order biases associated with "many-instruments" asymptotics.

   • Mean ($\mu$): Estimated via a regression of the sample residuals on the annihilator matrix applied to a vector of ones.
   • Variance ($\sigma^2$): Estimated by adjusting the residual sum of squares of the moments, subtracting the expected sampling noise, and normalizing by effective degrees of freedom. This estimator is closely related to the $J$-statistic but isolates the overdispersion attributable to misspecification.

3. Corrected Estimators:
   Two improved estimators for the structural parameters $\theta$ are proposed:

   • Bias-Corrected (BC) Estimator: A plug-in estimator that subtracts the estimated leading-order bias ($\hat{\mu}\hat{\Lambda}\mathbf{1}_m / \sqrt{n}$) from the standard GMM estimator. In linear IV models, this is shown to be numerically equivalent to a "Visual IV" approach (Angrist and Pischke 2009) that allows for an intercept in the regression of reduced forms on first stages.
   • Empirical Bayes (EB) Estimator: A shrinkage estimator that further adjusts the BC estimator by subtracting a Best Linear Predictor (BLP) of the remaining first-order estimation error. This step exploits the correlation between the masked moment conditions and the estimation error to reduce variance. The EB estimator is shown to be asymptotically equivalent to efficient GMM applied to the centered moment conditions.

4. Asymptotic Theory:
   Using a combinatorial central limit theorem, the paper establishes the asymptotic normality of both the BC and EB estimators. Crucially, the limiting distribution is derived conditional on the hyperparameters $(\mu, \sigma^2)$, allowing for misspecification-aware frequentist inference. The paper provides consistent variance estimators that account for both sampling uncertainty and the variability introduced by the specification errors.

Key Results

• Consistency: The hyperparameter estimators for $\mu$ and $\sigma^2$ are consistent under the joint asymptotic regime ($m, n \to \infty, m^2/n \to 0$).
• Risk Reduction: The EB estimator weakly improves upon the BC estimator in terms of leading-term risk (Mean Squared Error). The improvement is strict when the specification errors are non-zero and the model is not perfectly identified in a way that renders the shrinkage adjustment zero. This parallels the inadmissibility results of Brown (1990) regarding high-dimensional nuisance parameters.
• Inference: The proposed variance estimators enable confidence intervals with asymptotically valid unconditional coverage. Standard GMM intervals are shown to undercover significantly when misspecification is present, whereas the proposed intervals maintain correct size.
• Diagnostics: The paper proposes two specification tests: a Hausman-type test comparing the BC and EB estimators, and a test based on the covariance between specification errors and observed moment features (e.g., instrument group sizes) to detect departures from exchangeability.
• Simulation Evidence: Monte Carlo simulations in a linear IV setting with exchangeable exclusion violations demonstrate that the BC and EB estimators substantially reduce bias and Root Mean Squared Error (RMSE) compared to standard TSLS. The EB estimator further reduces RMSE relative to BC. The proposed variance estimators yield rejection rates close to nominal levels, whereas standard TSLS tests over-reject.
• Empirical Application: Revisiting Angrist and Krueger (1991) on returns to schooling using season-of-birth instruments, the author finds evidence of mean bias in the moment conditions. Applying the corrections shifts the TSLS estimates toward Ordinary Least Squares (OLS) estimates and reduces the sensitivity of the point estimates to the inclusion of different control variables.

Significance and Claims
The paper claims to offer a practical framework for "repairing" GMM estimates when overidentifying restrictions are rejected, moving beyond the binary decision of accepting or rejecting a model. By modeling specification errors as exchangeable, the author provides a method to quantify and correct for the bias and variance inflation caused by these errors.

The significance lies in:

1. Bridging Local Misspecification and Empirical Bayes: It adapts the local misspecification framework (Andrews, Gentzkow, and Shapiro 2017) to an exchangeable setting, allowing for the estimation of hyperparameters that characterize the distribution of errors.
2. Feasible Inference: It provides a complete inferential toolkit, including consistent standard errors and confidence intervals that remain valid even when the model is misspecified, addressing a common limitation in modern empirical work where $J$-statistics are often ignored.
3. Risk Improvement: It demonstrates that even in the presence of misspecification, one can improve upon standard GMM by shrinking estimates based on the estimated structure of the errors, effectively purging the "predictable" component of the bias.

The author notes that the methods are not entirely automatic; they require careful scaling of moment conditions to ensure the exchangeability assumption is plausible. The paper concludes that while the framework offers new opportunities for misspecification-aware estimation, it relies on the researcher's commitment to a specific encompassing model of misspecification.