Sensitivity, Informativeness, and Misspecification in GMM Estimation

This paper develops misspecification-robust sensitivity and informativeness diagnostics for GMM estimators, evaluated at pseudo-true values. The sensitivity matrix nests that of Andrews, Gentzkow, and Shapiro (2017) under correct specification. The informativeness $\Delta$ measures the share of an estimator's asymptotic variance explained by sampling variation in the moments, a notion of structural efficiency that equals one under correct specification and can fall below one under misspecification, even when the Hansen $J$-test does not reject. We derive influence-function representations for one-step, two-step, iterated, and continuously updating GMM. We show that in minimum-distance estimation, estimating the optimal weight matrix adds estimator variance that the moments do not explain, lowering informativeness, while simpler weight matrices largely avoid it. The choice of weight matrix therefore involves a trade-off between classical efficiency and informativeness. In applications to the automobile demand model of Berry, Levinsohn, and Pakes (1995), the consumption insurance model of Blundell, Pistaferri, and Preston (2008), and the income-and-democracy regressions of Acemoglu, Johnson, Robinson, and Yared (2008), misspecification reorders sensitivity rankings, simpler weights preserve the informativeness that the optimal weight loses, and $\Delta$ detects structural-efficiency losses that the $J$-test does not.