A doubly corrected robust variance estimator for linear GMM

Abstract

We propose a new finite sample corrected variance estimator for the linear generalized method of moments (GMM) including the one-step, two-step, and iterated estimators. Our formula additionally corrects for the over-identification bias in variance estimation on top of the commonly used finite sample correction of Windmeijer ( 2005) which corrects for the bias from estimating the efficient weight matrix, so is doubly corrected. Formal stochastic expansions are derived to show the proposed double correction estimates the variance of some higher-order terms in the expansion. In addition, the proposed double correction provides robustness to misspecification of the moment condition. In contrast, the conventional variance estimator and the Windmeijer correction are inconsistent under misspecification. That is, the proposed double correction formula provides a convenient way to obtain improved inference under correct specification and robustness against misspecification at the same time.

Inference for Iterated GMM Under Misspecification

“…The standard errors are moderately downward biased under strong identification but severely (60%) biased under weak identification. This is consistent with the finding of Hwang, Kang, and Lee (2021) that the Windmeijer formula only partially corrects the misspecification bias.…”

“…Surprisingly, this is not a coincidence. Hwang, Kang, and Lee (2021) show that the misspecification‐robust standard errors provide finite sample corrections up to the same order with the Windmeijer correction under correct specification for linear GMM assuming strong identification. Thus, it is strongly preferred to use our new robust standard error for linear models.…”

“…The Windmeijer formula corrects for the bias in the standard error of the linear two‐step and iterated GMM estimators by considering the extra variation arising from the efficient weight matrix being evaluated at an estimate rather than the true value. Hwang, Kang, and Lee (2021) showed that the misspecification‐robust standard errors provide the same order of finite sample correction with the Windmeijer standard error under correct specification. Since the Windmeijer formula is not robust to misspecification, these calculations show that our misspecification‐robust standard errors are more accurate than both conventional and Windmeijer standard errors.…”

Linear dynamic panel data models

“…This generates better coverage and is frequently used. Hwang et al (2021) extend this to provide valid standard errors even in the presence of misspecification.…”

Finite-sample corrected inference for two-step GMM in time series

“…Hwang and Sun (2017) and Martínez-Iriarte et al (2019) propose improved inferences for GMM methods using …xed smoothing asymptotics. A recent paper by Hwang et al (2019) points out a connection between the …nite-sample corrected and the misspeci…cation robust asymptotic variance formula in i.i.d. data.…”