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An influential paper by Kleibergen (2005, Econometrica 73, 1103–1123) introduces Lagrange multiplier (LM) and conditional likelihood ratio-like (CLR) tests for nonlinear moment condition models. These procedures aim to have good size performance even when the parameters are unidentified or poorly identified. However, the asymptotic size and similarity (in a uniform sense) of these procedures have not been determined in the literature. This paper does so. This paper shows that the LM test has correct asymptotic size and is asymptotically similar for a suitably chosen parameter space of null distributions. It shows that the CLR tests also have these properties when the dimension p of the unknown parameter θ equals 1. When p ≥ 2, however, the asymptotic size properties are found to depend on how the conditioning statistic, upon which the CLR tests depend, is weighted. Two weighting methods have been suggested in the literature. The paper shows that the CLR tests are guaranteed to have correct asymptotic size when p ≥ 2 when the weighting is based on an estimator of the variance of the sample moments, i.e., moment-variance weighting, combined with the Robin and Smith (2000, Econometric Theory 16, 151–175) rank statistic. The paper also determines a formula for the asymptotic size of the CLR test when the weighting is based on an estimator of the variance of the sample Jacobian. However, the results of the paper do not guarantee correct asymptotic size when p ≥ 2 with the Jacobian-variance weighting, combined with the Robin and Smith (2000, Econometric Theory 16, 151–175) rank statistic, because two key sample quantities are not necessarily asymptotically independent under some identification scenarios. Analogous results for confidence sets are provided. Even for the special case of a linear instrumental variable regression model with two or more right-hand side endogenous variables, the results of the paper are new to the literature.
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Inference for Iterated GMM Under Misspecification
This paper develops inference methods for the iterated overidentified Generalized Method of Moments (GMM) estimator. We provide conditions for the existence of the iterated estimator and an asymptotic distribution theory, which allows for mild misspecification. Moment misspecification causes bias in conventional GMM variance estimators, which can lead to severely oversized hypothesis tests. We show how to consistently estimate the correct asymptotic variance matrix. Our simulation results show that our methods are properly sized under both correct specification and mild to moderate misspecification. We illustrate the method with an application to the model of Acemoglu, Johnson, Robinson, and Yared (2008).
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- Award ID(s):
- 1656123
- PAR ID:
- 10293047
- Date Published:
- Journal Name:
- Econometrica
- Volume:
- 89
- Issue:
- 3
- ISSN:
- 0012-9682
- Page Range / eLocation ID:
- 1419 to 1447
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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- Free Publicly Accessible Full Text
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Accepted Manuscript1.0
- Journal Article:
- https://doi.org/10.3982/ECTA16274
