The purpose of an estimator is what it does: Misspecification, estimands, and over-identification
Isaiah Andrews, Jiafeng Chen, Otavio Tecchio
TL;DR
This paper addresses how misspecification in over-identified moment-condition models alters the estimand targeted by estimators, challenging the conventional emphasis on efficiency under a correctly specified model. It provides a structured framework distinguishing econometric and statistical misspecification, analyzes how estimator choice implicitly selects estimands under misspecification, and advocates misspecification-robust inference and broader reporting of Hansen's $J$-statistic. A key theoretical contribution shows that, under local misspecification, the range of estimators attainable at a fixed standard error is an interval around the efficient GMM estimate with width proportional to $\\sqrt{J}$, and that a researcher could weight-hack to achieve large $t$-statistics up to $\\sqrt{J}$. Practically, the work guides transparent reporting of misspecification diagnostics, clarifies the meaning of estimands in misspecified settings, and provides robust inference tools (including misspecification-robust standard errors and bootstrap) to improve empirical reliability in economics and related fields.
Abstract
In over-identified models, misspecification -- the norm rather than exception -- fundamentally changes what estimators estimate. Different estimators imply different estimands rather than different efficiency for the same target. A review of recent applications of generalized method of moments in the American Economic Review suggests widespread acceptance of this fact: There is little formal specification testing and widespread use of estimators that would be inefficient were the model correct, including the use of "hand-selected" moments and weighting matrices. Motivated by these observations, we review and synthesize recent results on estimation under model misspecification, providing guidelines for transparent and robust empirical research. We also provide a new theoretical result, showing that Hansen's J-statistic measures, asymptotically, the range of estimates achievable at a given standard error. Given the widespread use of inefficient estimators and the resulting researcher degrees of freedom, we thus particularly recommend the broader reporting of J-statistics.