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Optimal Invariant Similar Tests for Instrumental Variables Regression

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This paper considers tests of the parameter on endogenous variables in an instrumental variables regression model. The focus is on determining tests that have some optimal power properties. We start by considering a model with normally distributed errors and known error covariance matrix. We consider tests that are similar and satisfy a natural rotational invariance condition. We determine tests that maximize weighted average power (WAP) for arbitrary weight functions among invariant similar tests. Such tests include point optimal (PO) invariant similar tests. The results yield the power envelope for invariant similar tests. This allows one to assess and compare the power properties of existing tests, such as the Anderson-Rubin, Lagrange multiplier (LM), and conditional likelihood ratio (CLR) tests, and new optimal WAP and PO invariant similar tests. We find that the CLR test is quite close to being uniformly most powerful invariant among a class of two-sided tests. A new unconditional test, P*, also is found to have this property. For one-sided alternatives, no test achieves the invariant power envelope, but a new test -- the one-sided CLR test -- is found to be fairly close. The finite sample results of the paper are extended to the case of unknown error covariance matrix and possibly non-normal errors via weak instrument asymptotics. Strong instrument asymptotic results also are provided because we seek tests that perform well under both weak and strong instruments.

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  • Donald W.K. Andrews & Marcelo J. Moreira & James H. Stock, 2004. "Optimal Invariant Similar Tests for Instrumental Variables Regression," Cowles Foundation Discussion Papers 1476, Cowles Foundation for Research in Economics, Yale University.
  • Handle: RePEc:cwl:cwldpp:1476
    Note: CFP 1168.
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    1. Andrews, Donald W K, 1991. "Heteroskedasticity and Autocorrelation Consistent Covariance Matrix Estimation," Econometrica, Econometric Society, vol. 59(3), pages 817-858, May.
    2. Andrews, Donald W K & Monahan, J Christopher, 1992. "An Improved Heteroskedasticity and Autocorrelation Consistent Covariance Matrix Estimator," Econometrica, Econometric Society, vol. 60(4), pages 953-966, July.
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    1. Marcelo J. Moreira & Geert Ridder, 2020. "Efficiency Loss of Asymptotically Efficient Tests in an Instrumental Variables Regression," Papers 2008.13042, arXiv.org, revised Sep 2021.
    2. Dufour, Jean-Marie & Taamouti, Mohamed, 2007. "Further results on projection-based inference in IV regressions with weak, collinear or missing instruments," Journal of Econometrics, Elsevier, vol. 139(1), pages 133-153, July.
    3. Marcelo J. Moreira & Jack R. Porter & Gustavo A. Suarez, 2004. "Bootstrap and Higher-Order Expansion Validity When Instruments May Be Weak," NBER Technical Working Papers 0302, National Bureau of Economic Research, Inc.
    4. Russell Davidson & James G. MacKinnon, 2008. "Bootstrap inference in a linear equation estimated by instrumental variables," Econometrics Journal, Royal Economic Society, vol. 11(3), pages 443-477, November.
    5. Andrews, Donald W.K. & Marmer, Vadim, 2008. "Exactly distribution-free inference in instrumental variables regression with possibly weak instruments," Journal of Econometrics, Elsevier, vol. 142(1), pages 183-200, January.
    6. Richard Smith, 2005. "Weak instruments and empirical likelihood: a discussion of the papers by DWK Andrews and JH Stock and Y Kitamura," CeMMAP working papers CWP13/05, Centre for Microdata Methods and Practice, Institute for Fiscal Studies.
    7. Moreira, Humberto & Moreira, Marcelo J., 2019. "Optimal two-sided tests for instrumental variables regression with heteroskedastic and autocorrelated errors," Journal of Econometrics, Elsevier, vol. 213(2), pages 398-433.
    8. Justin McCrary & Heather Royer, 2011. "The Effect of Female Education on Fertility and Infant Health: Evidence from School Entry Policies Using Exact Date of Birth," American Economic Review, American Economic Association, vol. 101(1), pages 158-195, February.
    9. Donald W.K. Andrews & James H. Stock, 2005. "Inference with Weak Instruments," Cowles Foundation Discussion Papers 1530, Cowles Foundation for Research in Economics, Yale University.
    10. Andrews, Donald W.K. & Cheng, Xu & Guggenberger, Patrik, 2020. "Generic results for establishing the asymptotic size of confidence sets and tests," Journal of Econometrics, Elsevier, vol. 218(2), pages 496-531.
    11. Badinger, Harald, 2009. "Globalization, the output-inflation tradeoff and inflation," European Economic Review, Elsevier, vol. 53(8), pages 888-907, November.
    12. Russell Davidson & James G. MacKinnon, 2015. "Bootstrap Tests for Overidentification in Linear Regression Models," Econometrics, MDPI, Open Access Journal, vol. 3(4), pages 1-39, December.
    13. Mills, Benjamin & Moreira, Marcelo J. & Vilela, Lucas P., 2014. "Tests based on t-statistics for IV regression with weak instruments," Journal of Econometrics, Elsevier, vol. 182(2), pages 351-363.
    14. Andreea Balan-Cohen, 2008. "Healthy, Wealthy, and Wise? The Impact of the Old Age Assistance Program on Elderly Mortality in the United States," Discussion Papers Series, Department of Economics, Tufts University 0719, Department of Economics, Tufts University.
    15. Badinger, Harald, 2008. "Trade policy and productivity," European Economic Review, Elsevier, vol. 52(5), pages 867-891, July.
    16. Michael P. Murray, 2006. "Avoiding Invalid Instruments and Coping with Weak Instruments," Journal of Economic Perspectives, American Economic Association, vol. 20(4), pages 111-132, Fall.
    17. Anna Mikusheva & Brian P. Poi, 2006. "Tests and confidence sets with correct size when instruments are potentially weak," Stata Journal, StataCorp LP, vol. 6(3), pages 335-347, September.
    18. James G. MacKinnon, 2006. "Applications Of The Fast Double Bootstrap," Working Paper 1023, Economics Department, Queen's University.
    19. Harald Badinger, 2007. "Market size, trade, competition and productivity: evidence from OECD manufacturing industries," Applied Economics, Taylor & Francis Journals, vol. 39(17), pages 2143-2157.
    20. Adrian Pagan, 2007. "Weak Instruments: A Guide to the Literature," NCER Working Paper Series 13, National Centre for Econometric Research.

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    More about this item

    Keywords

    Instrumental variables regression; invariant tests; optimal tests; similar tests; weak instruments; weighted average power;
    All these keywords.

    JEL classification:

    • C12 - Mathematical and Quantitative Methods - - Econometric and Statistical Methods and Methodology: General - - - Hypothesis Testing: General
    • C30 - Mathematical and Quantitative Methods - - Multiple or Simultaneous Equation Models; Multiple Variables - - - General

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