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This paper introduces a new identiﬁcation- and singularity-robust conditional quasi-likelihood ratio (SR-CQLR) test and a new identiﬁcation- and singularity-robust Anderson and Rubin (1949) (SR-AR) test for linear and nonlinear moment condition models. Both tests are very fast to compute. The paper shows that the tests have correct asymptotic size and are asymptotically similar (in a uniform sense) under very weak conditions. For example, in i.i.d. scenarios, all that is required is that the moment functions and their derivatives have 2 + γ bounded moments for some γ > 0: No conditions are placed on the expected Jacobian of the moment functions, on the eigenvalues of the variance matrix of the moment functions, or on the eigenvalues of the expected outer product of the (vectorized) orthogonalized sample Jacobian of the moment functions. The SR-CQLR test is shown to be asymptotically eﬀicient in a GMM sense under strong and semi-strong identiﬁcation (for all k ≥ p ; where k and p are the numbers of moment conditions and parameters, respectively). The SR-CQLR test reduces asymptotically to Moreira’s CLR test when p = 1 in the homoskedastic linear IV model. The same is true for p ≥ 2 in most, but not all, identiﬁcation scenarios. We also introduce versions of the SR-CQLR and SR-AR tests for subvector hypotheses and show that they have correct asymptotic size under the assumption that the parameters not under test are strongly identiﬁed. The subvector SR-CQLR test is shown to be asymptotically eﬀicient in a GMM sense under strong and semi-strong identiﬁcation.
Andrews, Donald W.K. and Guggenberger, Patrik, "Identification- and Singularity-Robust Inference for Moment Condition Models" (2015). Cowles Foundation Discussion Papers. 2394.