Testing identification strength

Abstract

We consider models defined by a set of moment restrictions that may be subject to weak identification. We propose a testing procedure to assess whether instruments are "too weak" for standard (Gaussian) asymptotic theory to be reliable. Since the validity of standard asymptotics for GMM rests upon a Taylor expansion of the first order conditions, we distinguish two cases: (i) models that are either linear or separable in the parameters of interest (ii) general models that are neither linear nor separable. Our testing procedure is similar in both cases, but our null hypothesis of weak identification for a nonlinear model is broader than the popular one. Our test is straightforward to apply and allows to test the null hypothesis of weak identification of specific subvectors without assuming identification of the components not under test. In the linear case, it can be seen as a generalization of the popular first-stage F-test but allows us to fix its shortcomings in case of heteroskedasticity. In simulations, our test is well behaved when compared to contenders, both in terms of size and power. In particular, the focus on subvectors allows us to have power to reject the null of weak identification on some components of interest. This observation may explain why, when applied to the estimation of the Elasticity of Intertemporal Substitution, our test is the only one to find matching results for every country under the two symmetric popular specifications: the intercept parameter is always found strongly identified, whereas the slope parameter is always found weakly identified.

Weak Identification in Discrete Choice Models

“…Using this characterization of varying identification strength, Antoine and Renault (2020) have devised a testing strategy that is capable of detecting (certain levels of) instrument strength in nonlinear models estimated by GMM. The proposed test, dubbed the distorted J-test (DJ test), is based on computing the GMM J-test statistic at a slightly perturbed value of the continuously updated GMM (CUGMM) estimator.…”

“…Interestingly, Antoine and Renault (2020) have demonstrated that their DJ test is akin to the standard rule-of-thumb when the model is linear and homoskedastic. In contrast, they stress (see also Windmeijer, 2019 for related work in the context of clustering) that this DJ test differs from standard "robustified" versions of the rule-of-thumb in case of a heteroskedastic linear model.…”

“…Herein, we adapt the general testing strategy of Antoine and Renault (2020) to the case of discrete choice models and construct a consistent test for the null hypothesis that the instruments are too weak to allow consistent point estimation. Following the nomenclature of Antoine and Renault (2020), we refer to this test as a distorted J-test (DJ test).…”

“…Herein, we adapt the general testing strategy of Antoine and Renault (2020) to the case of discrete choice models and construct a consistent test for the null hypothesis that the instruments are too weak to allow consistent point estimation. Following the nomenclature of Antoine and Renault (2020), we refer to this test as a distorted J-test (DJ test). Similar to Antoine and Renault (2020), we demonstrate that our DJ test can be interpreted as a natural "generalized rule-of-thumb" in the context of discrete choice models, in the sense that this test appropriately modifies the standard approach to account for both heteroskedasticity and non-linearity.…”

“…Following the nomenclature of Antoine and Renault (2020), we refer to this test as a distorted J-test (DJ test). Similar to Antoine and Renault (2020), we demonstrate that our DJ test can be interpreted as a natural "generalized rule-of-thumb" in the context of discrete choice models, in the sense that this test appropriately modifies the standard approach to account for both heteroskedasticity and non-linearity.…”