Inference for Iterated GMM Under Misspecification and Clustering

Abstract

This paper develops a new distribution theory and inference methods for over-identified Generalized Method of Moments (GMM) estimation focusing on the iterated GMM estimator, allowing for moment misspecification, and for clustered dependence with heterogeneous and growing cluster sizes. This paper is the first to provide a rigorous theory for the iterated GMM estimator. We provide conditions for its existence by demonstrating that the iteration sequence is a contraction mapping. Our asymptotic theory allows the moments to be possibly misspecified, which is a general feature of approximate over-identified models. This form of moment misspecification causes bias in conventional standard error estimation. Our results show how to correct for this standard error bias. Our paper is also the first to provide a rigorous distribution theory for the GMM estimator under cluster dependence. Our distribution theory is asymptotic, and allows for heterogeneous and growing cluster sizes. Our results cover standard smooth moment condition models, including dynamic panels, which is a common application for GMM with cluster dependence. Our simulation results show that conventional heteroskedasticityrobust standard errors are highly biased under moment misspecification, severely understating estimation uncertainty, and resulting in severely over-sized hypothesis tests. In contrast, our misspecification-robust standard errors are approximately unbiased and properly sized under both correct specification and misspecification. We illustrate the method by extending the empirical work reported in Acemoglu, Johnson, Robinson, and Yared (2008, American Economic Review ) and Cervellati, Jung, Sunde, and Vischer (2014, American Economic Review ). Our results reveal an enormous effect of iterating the GMM estimator, demonstrating the arbitrariness of using one-step and two-step estimators. Our results also show a large effect of using misspecification robust standard errors instead of the Arellano-Bond standard errors. Our results support Acemoglu, Johnson, Robinson, and Yared's conclusion of an insignificant effect of income on democracy, but reveal that the heterogeneous effects documented by Cervellati, Jung, Sunde, and Vischer are less statistically significant than previously claimed.

A Doubly Corrected Robust Variance Estimator for Linear GMM

“…The expansion for the one-step GMM is (19)- (21) and those terms in (20) are O p (n −1/2 ). Thus, by considering the finite sample variation caused by these terms provide more accurate inference.…”

“…The doubly corrected variance estimator for the one-step GMM, V dc (θ 1 ), takes into account for the variations up to the order of O p (n −1/2 ) in the expansion (19)- (21). This correction is not considered in Windmeijer (2005).…”

“…whereθ is the iterated GMM. Assume that g n (θ 0 ) = O p (n −1/2 ) andθ − θ 0 = O p (n −1/2 ) whose sufficient conditions are provided in Hansen and Lee (2019). By applying the first-order Taylor expansion around θ 0 to g n (θ) and Ω n (θ) sequentially…”

Forecast Encompassing Tests for the Expected Shortfall

“…While such theory is available for smooth moment conditions (see e.g. Hall and Inoue (2003) and Hansen and Lee (2019)), its generalization to nonsmooth objective functions is not straight-forward and thus, we leave conditional ES encompassing tests based on misspecified GMM-estimation for future research.…”

Inference in Moment Inequality Models That Is Robust to Spurious Precision under Model Misspecification

“…The pseudo-true value for GMM minimizes a population quadratic form that depends on the weight matrix employed by the GMM estimator. References include White (1982), Gallant and White (1988), Hall and Inoue (2003), and Hansen and Lee (2019), among others. to misspeci…cation of the moment inequalities.…”