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Identification- and Singularity-Robust Inference for Moment Condition

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Abstract

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 + gamma bounded moments for some gamma > 0 in i.i.d. scenarios. In stationary strong mixing time series cases, the same condition suffices, but the magnitude of gamma 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 greater than or equal to 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 greater than or equal to 2

Suggested Citation

  • Donald W. K. Andrews & Patrik Guggenberger, 2015. "Identification- and Singularity-Robust Inference for Moment Condition," Cowles Foundation Discussion Papers 1978, Cowles Foundation for Research in Economics, Yale University.
  • Handle: RePEc:cwl:cwldpp:1978
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    References listed on IDEAS

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    1. Motohiro Yogo, 2004. "Estimating the Elasticity of Intertemporal Substitution When Instruments Are Weak," The Review of Economics and Statistics, MIT Press, vol. 86(3), pages 797-810, August.
    2. Kleibergen, Frank & Paap, Richard, 2006. "Generalized reduced rank tests using the singular value decomposition," Journal of Econometrics, Elsevier, vol. 133(1), pages 97-126, July.
    3. Whitney K. Newey & Frank Windmeijer, 2009. "Generalized Method of Moments With Many Weak Moment Conditions," Econometrica, Econometric Society, vol. 77(3), pages 687-719, May.
    4. Marcelo J. Moreira, 2003. "A Conditional Likelihood Ratio Test for Structural Models," Econometrica, Econometric Society, vol. 71(4), pages 1027-1048, July.
    5. Donald W. K. Andrews & Marcelo J. Moreira & James H. Stock, 2006. "Optimal Two-Sided Invariant Similar Tests for Instrumental Variables Regression," Econometrica, Econometric Society, vol. 74(3), pages 715-752, May.
    6. Chernozhukov, Victor & Hansen, Christian & Jansson, Michael, 2009. "Admissible Invariant Similar Tests For Instrumental Variables Regression," Econometric Theory, Cambridge University Press, vol. 25(03), pages 806-818, June.
    7. Moreira, Humberto Ataíde & Moreira, Marcelo J., 2013. "Contributions to the Theory of Optimal Tests," FGV/EPGE Economics Working Papers (Ensaios Economicos da EPGE) 747, FGV/EPGE - Escola Brasileira de Economia e Finanças, Getulio Vargas Foundation (Brazil).
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    Keywords

    Asymptotics; Conditional likelihood ratio test; Confidence set; Identification; Inference; Moment conditions; Robust; Singular variance; Test; Weak identification; Weak instruments;

    JEL classification:

    • C10 - Mathematical and Quantitative Methods - - Econometric and Statistical Methods and Methodology: General - - - General
    • C12 - Mathematical and Quantitative Methods - - Econometric and Statistical Methods and Methodology: General - - - Hypothesis Testing: General

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