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Bootstrap and Higher-Order Expansion Validity When Instruments May Be Weak

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  • Marcelo J. Moreira
  • Jack R. Porter
  • Gustavo A. Suarez

Abstract

It is well-known that size-adjustments based on Edgeworth expansions for the t-statistic perform poorly when instruments are weakly correlated with the endogenous explanatory variable. This paper shows, however, that the lack of Edgeworth expansions and bootstrap validity are not tied to the weak instrument framework, but instead depends on which test statistic is examined. In particular, Edgeworth expansions are valid for the score and conditional likelihood ratio approaches, even when the instruments are uncorrelated with the endogenous explanatory variable. Furthermore, there is a belief that the bootstrap method fails when instruments are weak, since it replaces parameters with inconsistent estimators. Contrary to this notion, we provide a theoretical proof that guarantees the validity of the bootstrap for the score test, as well as the validity of the conditional bootstrap for many conditional tests. Monte Carlo simulations show that the bootstrap actually decreases size distortions in both cases.

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Paper provided by Harvard - Institute of Economic Research in its series Harvard Institute of Economic Research Working Papers with number 2048.

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Date of creation: 2004
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Handle: RePEc:fth:harver:2048

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  1. Frank Kleibergen, 2002. "Pivotal Statistics for Testing Structural Parameters in Instrumental Variables Regression," Econometrica, Econometric Society, Econometric Society, vol. 70(5), pages 1781-1803, September.
  2. Donald W.K. Andrews & Marcelo Moreira & James H. Stock, 2004. "Optimal Invariant Similar Tests for Instrumental Variables Regression," NBER Technical Working Papers 0299, National Bureau of Economic Research, Inc.
  3. Douglas Staiger & James H. Stock, 1994. "Instrumental Variables Regression with Weak Instruments," NBER Technical Working Papers 0151, National Bureau of Economic Research, Inc.
  4. James H. Stock & Jonathan Wright, 2000. "GMM with Weak Identification," Econometrica, Econometric Society, Econometric Society, vol. 68(5), pages 1055-1096, September.
  5. Sargan, J D, 1976. "Econometric Estimators and the Edgeworth Approximation," Econometrica, Econometric Society, Econometric Society, vol. 44(3), pages 421-48, May.
  6. Nelson, C. & Startz, R., 1988. "Some Furthere Results On The Exact Small Sample Properties Of The Instrumental Variable Estimator," Discussion Papers in Economics at the University of Washington, Department of Economics at the University of Washington 88-06, Department of Economics at the University of Washington.
  7. Atsushi Inoue, 2006. "A bootstrap approach to moment selection," Econometrics Journal, Royal Economic Society, Royal Economic Society, vol. 9(1), pages 48-75, 03.
  8. Qumsiyeh, Maher B., 1990. "Edgeworth expansion in regression models," Journal of Multivariate Analysis, Elsevier, Elsevier, vol. 35(1), pages 86-101, October.
  9. Horowitz, Joel L., 2001. "The Bootstrap," Handbook of Econometrics, Elsevier, in: J.J. Heckman & E.E. Leamer (ed.), Handbook of Econometrics, edition 1, volume 5, chapter 52, pages 3159-3228 Elsevier.
  10. Donald W. K. Andrews, 2000. "Inconsistency of the Bootstrap when a Parameter Is on the Boundary of the Parameter Space," Econometrica, Econometric Society, Econometric Society, vol. 68(2), pages 399-406, March.
  11. Guggenberger, Patrik & Smith, Richard J., 2005. "Generalized Empirical Likelihood Estimators And Tests Under Partial, Weak, And Strong Identification," Econometric Theory, Cambridge University Press, Cambridge University Press, vol. 21(04), pages 667-709, August.
  12. Jean-Marie Dufour, 1997. "Some Impossibility Theorems in Econometrics with Applications to Structural and Dynamic Models," Econometrica, Econometric Society, Econometric Society, vol. 65(6), pages 1365-1388, November.
  13. Jinyong Hahn & Jerry Hausman & Guido Kuersteiner, 2004. "Estimation with weak instruments: Accuracy of higher-order bias and MSE approximations," Econometrics Journal, Royal Economic Society, Royal Economic Society, vol. 7(1), pages 272-306, 06.
  14. Frank Kleibergen, 2004. "Expansions of GMM statistics that indicate their properties under weak and/or many instruments and the bootstrap," Econometric Society 2004 North American Summer Meetings 408, Econometric Society.
  15. Phillips, Peter C B, 1977. "A General Theorem in the Theory of Asymptotic Expansions as Approximations to the Finite Sample Distributions of Econometric Estimators," Econometrica, Econometric Society, Econometric Society, vol. 45(6), pages 1517-34, September.
  16. Moreira, Marcelo J., 2009. "Tests with correct size when instruments can be arbitrarily weak," Journal of Econometrics, Elsevier, Elsevier, vol. 152(2), pages 131-140, October.
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Cited by:
  1. Russell Davidson & James G. MacKinnon, 2007. "Wild Bootstrap Tests For Iv Regression," Departmental Working Papers, McGill University, Department of Economics 2007-14, McGill University, Department of Economics.
  2. Russell Davidson & James MacKinnon, 2006. "Bootstrap Inference In A Linear Equation Estimated By Instrumental Variables," Departmental Working Papers, McGill University, Department of Economics 2006-21, McGill University, Department of Economics.
  3. Patrik Guggenberger & Richard Smith, 2005. "Generalized empirical likelihood tests in time series models with potential identification failure," CeMMAP working papers, Centre for Microdata Methods and Practice, Institute for Fiscal Studies CWP01/05, Centre for Microdata Methods and Practice, Institute for Fiscal Studies.
  4. Donald W.K. Andrews & James H. Stock, 2005. "Inference with Weak Instruments," Cowles Foundation Discussion Papers, Cowles Foundation for Research in Economics, Yale University 1530, Cowles Foundation for Research in Economics, Yale University.
  5. Russell Davidson & James G. MacKinnon, 2012. "Bootstrap Confidence Sets with Weak Instruments," Working Papers, Queen's University, Department of Economics 1278, Queen's University, Department of Economics.
  6. Angelica Gonzalez, 2007. "Empirical Likelihood Estimation in Dynamic Panel Models," ESE Discussion Papers, Edinburgh School of Economics, University of Edinburgh 168, Edinburgh School of Economics, University of Edinburgh.
  7. Moreira, Marcelo J. & Porter, Jack R. & Suarez, Gustavo A., 2009. "Bootstrap validity for the score test when instruments may be weak," Journal of Econometrics, Elsevier, Elsevier, vol. 149(1), pages 52-64, April.

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