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Bootstrap Inference in a Linear Equation Estimated by Instrumental Variables

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  • Russell Davidson

    () (GREQAM and McGill University)

  • James G. MacKinnon

    () (Queen's University)

Abstract

We study several tests for the coefficient of the single right-hand-side endogenous variable in a linear equation estimated by instrumental variables. We show that all the test statistics--Student's t, Anderson-Rubin, Kleibergen's K, and likelihood ratio (LR)--can be written as functions of six random quantities. This leads to a number of interesting results about the properties of the tests under weak-instrument asymptotics. We then propose several new procedures for bootstrapping the three non-exact test statistics and a conditional version of the LR test. These use more efficient estimates of the parameters of the reduced-form equation than existing procedures. When the best of these new procedures is used, K and conditional LR have excellent performance under the null, and LR also performs very well. However, power considerations suggest that the conditional LR test, bootstrapped using this new procedure when the sample size is not large, is probably the method of choice.

Suggested Citation

  • Russell Davidson & James G. MacKinnon, 2006. "Bootstrap Inference in a Linear Equation Estimated by Instrumental Variables," Working Papers 1024, Queen's University, Department of Economics.
  • Handle: RePEc:qed:wpaper:1024
    as

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    References listed on IDEAS

    as
    1. D.S. Poskitt & C.L. Skeels, 2005. "Small Concentration Asymptotics and Instrumental Variables Inference," Department of Economics - Working Papers Series 948, The University of Melbourne.
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    3. Marcelo J. Moreira, 2003. "A Conditional Likelihood Ratio Test for Structural Models," Econometrica, Econometric Society, vol. 71(4), pages 1027-1048, July.
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    5. Stock, James H & Wright, Jonathan H & Yogo, Motohiro, 2002. "A Survey of Weak Instruments and Weak Identification in Generalized Method of Moments," Journal of Business & Economic Statistics, American Statistical Association, vol. 20(4), pages 518-529, October.
    6. 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.
    7. Davidson, Russell & MacKinnon, James G., 1999. "The Size Distortion Of Bootstrap Tests," Econometric Theory, Cambridge University Press, vol. 15(03), pages 361-376, June.
    8. 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.
    9. Marcelo J. Moreira & Jack R. Porter & Gustavo A. Suarez, 2004. "Bootstrap and Higher-Order Expansion Validity When Instruments May Be Weak," Harvard Institute of Economic Research Working Papers 2048, Harvard - Institute of Economic Research.
    10. Davidson, Russell & MacKinnon, James G., 2006. "The power of bootstrap and asymptotic tests," Journal of Econometrics, Elsevier, vol. 133(2), pages 421-441, August.
    11. Kleibergen, Frank, 2007. "Generalizing weak instrument robust IV statistics towards multiple parameters, unrestricted covariance matrices and identification statistics," Journal of Econometrics, Elsevier, vol. 139(1), pages 181-216, July.
    12. Jean-Marie Dufour, 1997. "Some Impossibility Theorems in Econometrics with Applications to Structural and Dynamic Models," Econometrica, Econometric Society, vol. 65(6), pages 1365-1388, November.
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    14. Davidson, Russell & MacKinnon, James G., 2010. "Wild Bootstrap Tests for IV Regression," Journal of Business & Economic Statistics, American Statistical Association, vol. 28(1), pages 128-144.
    15. Hillier, Grant, 2009. "Exact Properties Of The Conditional Likelihood Ratio Test In An Iv Regression Model," Econometric Theory, Cambridge University Press, vol. 25(04), pages 915-957, August.
    16. Frank Kleibergen, 2002. "Pivotal Statistics for Testing Structural Parameters in Instrumental Variables Regression," Econometrica, Econometric Society, vol. 70(5), pages 1781-1803, September.
    17. Phillips, P.C.B., 1983. "Exact small sample theory in the simultaneous equations model," Handbook of Econometrics,in: Z. Griliches† & M. D. Intriligator (ed.), Handbook of Econometrics, edition 1, volume 1, chapter 8, pages 449-516 Elsevier.
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    Citations

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    Cited by:

    1. Wenjie Wang, 2012. "Bootstrapping Anderson-Rubin Statistic and J Statistic in Linear IV Models with Many Instruments," KIER Working Papers 810, Kyoto University, Institute of Economic Research.
    2. Simon A. Broda & Raymond Kan, 2016. "On distributions of ratios," Biometrika, Biometrika Trust, vol. 103(1), pages 205-218.
    3. repec:eee:ecolet:v:157:y:2017:i:c:p:107-111 is not listed on IDEAS
    4. Russell Davidson & James G. MacKinnon, 2014. "Confidence sets based on inverting Anderson–Rubin tests," Econometrics Journal, Royal Economic Society, vol. 17(2), pages 39-58, June.
    5. Davidson, Russell & MacKinnon, James G., 2010. "Wild Bootstrap Tests for IV Regression," Journal of Business & Economic Statistics, American Statistical Association, vol. 28(1), pages 128-144.
    6. 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.
    7. Wenjie Wang & Qingfeng Liu, 2015. "Bootstrap-based Selection for Instrumental Variables Model," Economics Bulletin, AccessEcon, vol. 35(3), pages 1886-1896.
    8. Corinna Ghirelli, 2015. "Scars of early non-employment for low educated youth: evidence and policy lessons from Belgium," IZA Journal of European Labor Studies, Springer;Forschungsinstitut zur Zukunft der Arbeit GmbH (IZA), vol. 4(1), pages 1-34, December.
    9. Chau, Tak Wai, 2014. "On the equivalence of indirect inference and bootstrap bias correction for linear IV estimators," Economics Letters, Elsevier, vol. 123(3), pages 333-335.
    10. Corinna GHIRELLI, 2015. "Scars of early non-employment in a rigid labour market," Discussion Papers (IRES - Institut de Recherches Economiques et Sociales) 2015008, Université catholique de Louvain, Institut de Recherches Economiques et Sociales (IRES).
    11. Wang, Wenjie & Kaffo, Maximilien, 2016. "Bootstrap inference for instrumental variable models with many weak instruments," Journal of Econometrics, Elsevier, vol. 192(1), pages 231-268.
    12. James G. MacKinnon, 2007. "Bootstrap Hypothesis Testing," Working Papers 1127, Queen's University, Department of Economics.
    13. Noud P.A. van Giersbergen, 2011. "Bootstrapping Subset Test Statistics in IV Regression," UvA-Econometrics Working Papers 11-08, Universiteit van Amsterdam, Dept. of Econometrics.
    14. Corinna.Ghirelli, 2014. "The scarring effect of early non-employment," Working Papers of Faculty of Economics and Business Administration, Ghent University, Belgium 14/895, Ghent University, Faculty of Economics and Business Administration.
    15. Keith Finlay & Leandro M. Magnusson, 2014. "Bootstrap Methods for Inference with Cluster-Sample IV Models," Economics Discussion / Working Papers 14-12, The University of Western Australia, Department of Economics.

    More about this item

    Keywords

    bootstrap test; weak instruments; anderson-rubin test; conditional LR test; wald test; instrumental variables;

    JEL classification:

    • C12 - Mathematical and Quantitative Methods - - Econometric and Statistical Methods and Methodology: General - - - Hypothesis Testing: General
    • C15 - Mathematical and Quantitative Methods - - Econometric and Statistical Methods and Methodology: General - - - Statistical Simulation Methods: General
    • C30 - Mathematical and Quantitative Methods - - Multiple or Simultaneous Equation Models; Multiple Variables - - - General

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