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This paper introduces two new identiﬁcation- and singularity-robust conditional quasi-likelihood ratio (SR-CQLR) tests and a new identiﬁcation- and singularity-robust Anderson and Rubin (1949) (SR-AR) test for linear and nonlinear moment condition models. The paper shows that the tests have correct asymptotic size and are asymptotically similar (in a uniform sense) under very weak conditions. For two of the three tests, all that is required is that the moment functions and their derivatives have 2 + γ bounded moments for some γ > 0 in i.i.d. scenarios. In stationary strong mixing time series cases, the same condition suﬀices, but the magnitude of γ is related to the magnitude of the strong mixing numbers. For the third test, slightly stronger moment conditions and a (standard, though restrictive) multiplicative structure on the moment functions are imposed. For all three tests, 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 two SR-CQLR tests are 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 two SR-CQLR tests reduce asymptotically to Moreira’s CLR test when p = 1 in the homoskedastic linear IV model. The ﬁrst SR-CQLR test, which relies on the multiplicative structure on the moment functions, also does so for p ≥ 2.
Andrews, Donald W.K. and Guggenberger, Patrik, "Identification- and Singularity-Robust Inference for Moment Condition Models" (2011). Cowles Foundation Discussion Papers. 2393.